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An Environment for Graphical Models

1. Print all models print all models Can an edge rejected by coherence now be rejected backward Did accept wrong edges in the start forward Set DecomposableMode off to compute asymptotic P values for non decomposable models set decomposable mode off 1 2 An Example Compute asymptotic P values backward Compute asymptotic P values forward Try to drop the significant interaction ABC drop interaction ABC Drop the edge AB and accept a non decomposable model drop edge AB Name the last accepted model current current Name current model base base Print all models print all models No edges can be removed from this model backward No edges should be added to the found model forward Print all models print all models Exit CoCo quit 29 It might be read as input to CoCo by coco lt log or the command SOURCE log in CoCo Part II User s Manual 30 Chapter 2 Commands Command names in this guide are written in upper case like in the command HELP Arguments to CoCo commands are capitalized if they are keywords like in the com mand SET DIARY On Off J or if they are numbers names sets models etc surrounded by and like in MENU NUMBER a Keywords and arguments surrounded by and are optional I
2. gc edges model gc a s list s list EndOfCommand models models a m list EndOfCommand table value name Observed 0 Probabilities 1 Expected 2 Unadjusted 3 F res 4 R 5 G res 6 R g 7 Adjusted 8 Standardized 9 2log q 10 Freeman Tukey 11 2 n m 12 Power 13 Index 14 Zero 15 Error 31 value name x value name y value name Current Base Complete Log Random table value name list of value names value name value name amp EndOfLine EndOfCommand B 1 Quick Reference Card 161 factor specification factor name unsigned integer unsigned integer factor name unsigned integer factor list factor specification factor specification factor names factor name factor name name list factor names unsigned integer list unsigned integer list factor names unsigned integer list specification Factors factor list Names name list list of cases marginal cell cutpoints unsigned integer list table 4 1 unsigned integer observations List list of cases Accumulated list list of cases Table table specification file specification observation file 4 Cutpoints factor name cutpoints Q table table Q list list of cases observations
3. SET LARGE On Off SET SORTED On Off Declare Subset of Variables to be Ordinal SET ORDINAL set B 1 18 Read Data Selection SELECT CASES set marginal cell OR SELECT CASES set marginal cell REJECT CASES set marginal cell OR REJECT CASES set marginal cell Redeclaring a Factor for Cutpoints after Entering of Specification REDEFINE FACTOR factor name of observed levels 4 of missing levels CUTPOINTS factor name cutpoints B 1 19 Skip Cases with Missing Values During Reading Ob servations SKIP MISSING B 1 20 Read Data Observations READ OBSERVATIONS observations READ TABLE table READ LIST list of cases Replace Observations with a Random Table with Margins as in the current Model SUBSTITUTE DATA WITH GENERATED TABLE 156 Appendix B Reference Manual section B 1 21 Read Data Incomplete table Structural Zeros READ Q TABLE set table READ Q LIST set list of cells CLEAN DATA B 1 22 Select use only cases with complete observations after reading observations EXCLUDE MISSING On Off J EXCLUDE MISSING In set B 1 23 Request the EM algorithm EM ON B 1 24 Data Description PRINT TABLE value name a b 1 DESCRIBE TABLE Uniform Normal Rankit value name a PLOT x value name y value name a LIST a CASE LIST a RETURN VECTOR value name a RETURN MATRIX list of value na
4. 1 80000 2 00000 2 20000 4 Ho Ho gt gt Ho 3 5000 2 2500 1 0000 0 2500 1 5000 2 7500 Adjusted K oO0o0D0OO0OORRRGBRBRNN TIME 0 104secs A histogram is omitted from the above output 12 gt Chapter 1 Introduction to CoCo gt Make a plot of the adjusted residuals against observed counts gt gt plot observed PLOT OF PLOT Unit horizontal Unit vertical Adjusted adjusted Adjusted BY Observed 2 5000 l 0 2500 O OOO a a a A NN TIME 5000 2500 0000 7500 5000 2500 0000 7500 5000 2500 0000 2500 5000 7500 0000 2500 5000 7500 0000 2500 kk Lxx 3 x 102 xxx l kk x kk I 2 2 2 Ix x ok 0 0000 25 0000 EXCLUDED 0 066secs 2 xk 2 1 ok 1 1 1 1 1 50 0000 75 0000 100 0000 125 0000 Observed 0 The likelihood ratio test statistic 2log Q the x test statistic and the power divergence 2nJ for one model under another model is computed by the command TEST The power A in computing the power divergence is by default set to 2 3 CoCo has to know two models to perform the test the current model and a base model The model first read is both base and current and is given the model number 1 The first model stays base until another m
5. LBD CD BE EF DF CE AF BF CF AB CABCF ABEF ADEF CAB AF BE BF CF DF EF 0519 R ABCEF ACEF BCEF R ABCEF ABCE ACEF R CACDEF ACDE ACEF R CABCEF ABCF ACEF LACEF ADEF 0519 R ABCEF ACEF BCEF ABCEF ABCE ACEF ABCE ABEF ABCEF LABCF ACEF CADEF ADE AEF CABCEF ABCF ABEF 17 18 Chapter 1 Introduction to CoCo AB ACF ADEF BCF BEF is not decomposable AF ABC ABE ADE BCF BEF DEF is not decomposable BF ABC ABE ACF ADEF is not decomposable EF ABCF ABE ADE ADF is not decomposable Sorted list BE 4 21 159 Exact 1000 CF 4 7 018 Exact 1000 DF 4 6 368 Exact 200 Edge selected Edge DF 2log Q P X2 P Models 0 0003 21 168 0 0003 R ABEF ABF AEF 0 0000 0 0000 0 1349 7 459 0 1135 R ABCF ABC ABF 0 1320 0 0930 0 1733 6 583 0 1596 R ADEF ADE AEF 0 1950 0 1750 DF CBE AD DE AE BC AC Rejected edges Accepted edges Model Edges eligible BF ABC ABE AF ABC ABE BD CD BE EF DF CE AF BF CF AB ABCF ABEF ADE CEF CF BF AF AB ACF ADE AEF is not decomposable DE BCF BEF is not decomposable gt AB ACF ADE AEF BCF BEF is n
6. SET ALGORITHM AI 1BICIDI B 1 12 Do only Graph Stuffs only for Debugging SET GRAPH MODE On Off B 1 13 Only Decomposable Models in Stepwise Model Selec tion and in the Global EH Search SET DECOMPOSABLE MODE On Off B 1 14 Computed Test Statistics and Choosing Tests and Sig nificance Level for Model Selection SET ADJUSTED DF On Off SET POWER LAMBDA Null lambda SET REUSE OF TESTS On Off SET TEST Lr Chi Power SET Aic Bic Ic Kappa kappa Ic On Ic Off SET ACCEPTANCE alpha SET REJECTION alpha SET COMPONENTS p value SET SEPARATORS p value B 1 15 Exact Tests in Tests and Model Selection SET EXACT TEST On All Deviance Off SET ASYMPTOTIC p value SET NUMBER OF TABLES Null number SET LIST OF NUMBERS OF TABLES list SET LIST OF NUMBERS OF TABLES integer list SET EXACT TEST FOR TOTAL TEST On Off SET EXACT TEST FOR PARTS OF TEST On Off SET EXACT TEST FOR UNPARTED TEST On Off SET SEED seed Random B 1 Quick Reference Card 155 SET EXACT EPSILON epsilon SET FAST On Off B 1 16 Read Data without selection etc READ DATA specification observations B 1 17 Read Data Specification READ SPECIFICATION specification 1 READ FACTORS factor list READ NAMES name list SET READ All Subset set SET DATASTRUCTURE All Necessary File Large
7. 160 generating class 65 170 global model selection See model selection Goodman and Kruskal s 7 79 Goodman and Kruskal s Gamma 40 73 Goodness of fit linear trend 79 Gopher 148 graphical 65 71 Graphical keyword 120 121 GraphMode mode 133 headlong See model selection Headlong keyword 110 HELP command 33 34 154 help file 33 HelpFile keyword 148 Hierarchical keyword 120 121 home directory 39 148 149 http www iesd auc dk pub packages CoCo iv 147 148 IC keyword 106 110 incomplete data or information See missing values incomplete tables See q tables incremental search See model selection independence graph 65 138 Index keyword 61 162 Information Criteria AIC Akaike s 100 BIC Bayesian 100 INIT PARSER command 35 154 INIT TAB file 35 initial models all n factor interactions 109 EH procedure 113 118 119 stepwise model selection 104 initial values for the IPS algorithm See q tables Input keyword 51 interaction terms 65 IPS algorithm 37 133 irreducible components 67 IS DECOMPOSABLE command 71 159 IS GRAPHICAL command 71 159 Index IS IN ONE CLIQUE command 71 159 IS SUBMODEL OF command 71 159 iterative proportional fitting See IPS algorithm iterative proportional scaling See IPS algorithm Jaundice 46 JOIN OF MODELS command 50 93 96 125 160 Kappa k 79 Kendall s Tau b 7 79 Large keyword 130 Last
8. ABCDEF TIME 0 007secs gt gt Exit CoCo gt gt quit TIME 0 002secs TOTAL TIME 15 017secs Chapter 1 Introduction to CoCo Models ABE ABC BE BC AE AC CAC AE AF BC BE BF CEAC AE BC BE BF AC ADE BC BDE CEAC ADE BC BE ACD ADE BC BE TAC ADE BC BE BCE ACE BE BC AE AC LBCF1 BC BF CDF BF BE BC ADE Ac BF BE BC ADE AC BEF BE BF CURRENT The model ADE AC BC BE BF found by coherent backward elimi nation with exact and local tests on the data from Reinis et al 1981 adds the 1 2 An Example 25 D Blood pressure A Smoking 6 of lipoproteins C Strenuous physical work B Strenuous mental work F Family anamnesi of coronary heart disease Figure 1 1 A final model from recursive coherent backward elimination with local and exact tests on data from Reinis et al 81 The TeX code of the figure is written by Xlisp CoCo edge lt BF gt to the first of the two models ADE AC BC BE F and ACE ADE BC F found in Edwards amp Havr nek 1985 Note also that we in the search observed that the interaction lt ABC gt is significant The model ACE ADE BC F from Edwards amp Havr nek 1985 will in the next part of the guide often be used as an example This model is selected mai
9. Below are described three strategies for selecting the dual to fit in each step of the EH search Which of the three strategies to use is set by the command SET Smallest Alternating Rough SEARCH or by a keyword argument of the EH command For the time being the strategy Smallest is default Which of the three strategies is the fastest depends on the situation Try them all three they need not give the same result since e g weakly rejected models may be accepted See the ReportFile for where CoCo uses computing time Find some models by e g BACKWARD and copy them to the EH procedure by the commands FIT ACCEPT and REJECT Before the automatic search is started you can do some directed search by the command FIT 10 9 1 Smallest Automatic With EH Smallest for each step classification of all not already classified models in a dual into the accepted and the rejected regions in the search both duals are found and the dual with the fewest not classified models is selected for the next step This will often minimize the number of fitted models but sometimes a lot of computing time is used for finding large duals that are not used 10 9 2 Rough Automatic With this strategy rough estimates of the dual sizes are found after each step and the dual with the smallest rough estimate of dual size is selected for the next step The rough estimate of dual size is the product of the number of edges or interaction terms in the model
10. DF CF CD BF BE BD AF AB LF BC ADE ACE TIME 0 074secs gt print all models Model no 4 LAB AF BD BE BF CD CF DF EF BC ADE ACE Model no 3 F BC AD DE AC AE CE Model no 2 ACE ADE BC F CURRENT Model no 1 EF DF DE CF CE CD BF BE BD BC AF AE AD AC AB BASE TIME 0 003secs 7 9 Miscellaneous Deprecated Commands Only DROP FACTOR factor Only REDUCE GENERATOR set Only REMOVE GENERATOR set Only REMOVE TOTAL INTERACTION set Drop Factor DROP FACTOR factor drops the factor factor in the current model The generated model is tested against base model The same effect can be achieved by DROP INTERACTIONS factor Reduce Generator With REDUCE GENERATOR set the generating class for the current model is searched for the set set If the set is found the set is substituted with all proper subsets of the set Subsets of other sets in the resulting set of sets are removed and the generated model is tested against the base model Remove Generator REMOVE GENERATOR set searches the generating class for the current model for the set set If the set is found it is removed from the generating class and the resulting model is tested against the base model Remove Total Interaction With REMOVE TOTAL INTERACTION set all interac tions between factors in set are removed in the curren
11. ExcludeMissing mode 50 51 100 103 exogenous 139 Expected keyword 162 explanatory 139 expression 66 EXTRACT command 120 161 extracting table values 64 F res keyword 162 factorization of a test 37 133 FACTORIZE command 73 FACTORIZE ONE EDGE command 50 76 78 83 87 125 133 159 FACTORIZE ONE INTERACTION command 50 83 90 125 133 159 Factors keyword 39 40 45 163 169 false 74 Fast mode 77 fep CoCo program 3 fep program 36 File keyword 51 130 FILL IN 94 FIND command 120 121 FIND DEVIANCE command 50 76 80 137 159 FIND DUAL command 120 121 161 FIND LOG L command 50 80 125 159 First keyword 51 Fisher s exact test 79 FIT command 118 122 161 FIX command 105 115 117 160 fix edges See model selection fixed zeros See q tables FixEdgeList mode 105 FixIn mode 105 114 116 117 FixOut mode 105 114 116 117 follow See model selection Follow keyword 108 112 FOREIGN FLAG DFOREIGNCALL 150 FOREIGN FLAG DFOREIGNCALL file 140 FORWARD command 13 37 50 83 97 99 101 109 112 125 133 160 forward selection See model selection Freeman Tukey residuals See residuals Freeman Tukey keyword 162 ftp 148 ftp iesd auc dk iv 147 148 G res keyword 162 Gamma 40 73 Gamma y 79 GENERATE DECOMPOSABLE MODEL command 50 93 125 160 GENERATE GRAPHICAL MODEL command 50 93 94 125
12. if the current model is graphical else BACKWARD Interactions 9 3 4 Model Control Separators Separators BACKWARD With this keyword in a Separators BACKWARD command then for each edge or interaction term all tests for conditional independence are performed For each edge to remove that is for each pair of vertices connected in the model the minimal sepa rators between the two vertices are found and the two vertices are tested conditionally independent given each separator If a limit separators limit is set by SET SEPARATORS separators limit then edges or interaction terms with a p value less than the limit separators limit for one separator are rejected in a Separators BACKWARD command Analogously if a limit components limit is set by the command SET COMPONENTS components limit and Partitioning is On then edges or interaction terms with a p value less than components limit for one part of the test are rejected in backward elimination 9 4 Forward Selection Only Reversed Sorted Separators Recursive Headlong Coherent Most FORWARD Edges Interactions All edges not included in the current model are in turn added to the current model and the current model is tested against the resulting model with the command FORWARD 9 4 1 Controlling the Output By default a test is printed for each edge or interaction term visited and a report of rejected eligible edges or
13. is entered as a list of sets enclosed in sharp parentheses for example the model with the two generators lt AB gt and lt BC gt is entered as LAB BC A shorter form can be used Comma ends sets whereas period ends both sets and generating classes so the set above set with the three variables A B and C can be also be entered as ABC and the above model can be entered as AB BC EndOfLine Return ends sets generating classes and sets of generating classes unless the EndOfLine is presided by the character amp The asterisk is an abbreviation for the set containing all declared factors and for the saturated model Period with no factors preceding is an abbreviation for the model with only main effects The uniform model model with no effects can be entered as or 2 6 Pausing and Front End Processor History Mechanism 35 2 5 1 Variable names of length longer than one character If variable names longer than one character are used then the first character of each name is and sets are entered as e g Sex Smoking Drinking or Sex Smoking Drinking and models can be entered as e g Sex Smoking Smoking Drinking or Sex Smoking Smoking Drinking 2 6 Pausing and Front End Processor History Mechanism No history mechanism is implemented in CoCo On Unix Use fep If not available on your system get your system administrator to install it or run CoCo as
14. model nil def graph 1 spin send graph 1 return dynamic coco spin plot list a b c send graph 1 spin location 700 500 A Headlong Backward Elimination at graph 1 can be performed by send graph 1 drop least significant edge p accepted send graph 1 graph p accepted p rejected send graph 1 graph p rejected random order send graph 1 graph random order coherent send graph 1 graph coherent headlong T recursive T x move 0 The Model Dynamic Spin Plot and Headlong Backward can also be made by selecting the Recursive backward and Dynamic Spin plot items from the graph menu Or the total example of the tutorial can be performed by the four lines load concatenate string xlisp cocohome Examples Testgraph setf manager return manager load concatenate string xlisp cocohome Examples TestDynamic load concatenate string xlisp cocohome Examples TestHeadlong Consider also load concatenate string xlisp cocohome Examples TestBlock load concatenate string xlisp cocohome Examples TestBootstrap load concatenate string xlisp cocohome Examples TestTeX load concatenate string xlisp cocohome Examples Krippendorf Part III Appendix 143 Appendix A Installing CoCo A 1 Availability The source code in C executable for Sun 4 Sparc and executable of the standalone version of CoCo for PC s
15. 0 set 7 number 0 base FALSE 137 Splus values 1 136 0 137 coco identifications 136 coco identifications a 136 current coco 136 FALSE 136 137 NULL 137 TRUE 136 137 standardized residuals See residuals Standardized keyword 162 statlib 150 templer stat cmu edu 150 176 STATUS command 36 40 114 120 124 126 129 130 151 154 STATUS command 36 125 stepwise model selection See model selection strategy See model selection structural zero fixed zeros See q tables Stuart s Tau c Te 79 SUBSTITUTE DATA WITH GENERATED TABLE command 42 157 Symmetric optimal prediction lambda A 79 Symmetric uncertainty coefficient U 79 T keyword 131 Table keyword 4 39 41 45 163 table value 34 TEST command 12 50 72 73 75 76 80 98 103 125 159 TEST ONE EDGE command 50 83 86 159 TestFormats mode 93 the observed count 43 Timer keyword 93 Total percents 58 Trace keyword 51 76 93 129 132 Trend Test 79 TRUE keyword 70 umnstat stat umn edu 150 Unadjusted keyword 162 Uncertainty coefficient Uasym 79 Uniform keyword 51 59 unobserved variables See missing values values 55 variables function 140 vertex 138 vertices 65 weak acceptance and rejection See coherence principle of Wermuth s method 99 WWW 148 Index XCOCO environment 150 XCOCOHOME environment 150 xlisp coco program 140 150 XLISP S
16. 119 exact tests 102 fix 105 115 follow 108 forward selection 109 graphical search EH procedure 117 stepwise model selection 108 111 headlong 107 110 hierarchical search EH procedure 118 stepwise model selection 109 111 incremental search 104 Information Criteria 100 initial models all n factor interactions 109 EH procedure 113 118 119 stepwise model selection 104 level of significance 99 101 171 model class 108 111 117 model control 102 109 112 ordinal variables 99 output 106 stepwise model selection decomposable search 98 graphical search 108 111 hierarchical search 109 111 initial models 104 strategy 107 110 117 122 alternating 122 headlong 107 110 rough 122 smallest 122 test statistic 99 Wermuth s method 99 Modified asymmetric optimal prediction lambda A 79 Monte Carlo approximation 76 more program 36 mv coco sparc coco program 149 mv scoco spare scoco o program 149 my alias file 34 N keyword 76 131 152 Names keyword 39 40 45 163 Necessary tables keyword 130 Necessary keyword 51 New S program 150 NewPage 36 125 NEXT MENU command 33 34 154 NO ACTION command 35 154 NORMAL TO DUAL command 65 66 96 158 Normal keyword 59 number of cells 67 number of cliques the edge is in 67 O keyword 131 observations 38 126 Observed keyword 58 60 162 Off keyword 124 On keyword 58 99 100 103 107 1
17. 2 7202 4580 7211 0598 17 2037 DF 1003 991 12 P 0 1417 Adjustment 755 755 0 757 12 P 0 1417 TIME 0 851secs gt keep log Keep Log set ON TIME 0 002secs gt quit TIME 0 001secs TOTAL TIME 1 947secs The statement SET LARGE On Off J is used to tell CoCo to dispose of marginal tables immediately after use that is to only store marginal tables for one term in the expression for the maximum likelihood estimate at time Note the time used for computing the adjusted degrees of freedom is larger than the time used to compute the test statistics Since the two models of the above example have common decompositions we should first have decomposed the two models For each possible common decomposition of the two models the set of vertices for the two parts of the graphs and the inter section between those two vertex sets are given in the output from PRINT COMMON DECOMPOSITIONS The number of vertices in each set are also printed 82 Chapter 6 Tests 6 5 Miscellaneous Commands for Model Control PARTITIONING TEST TEST ONE EDGE FACTORIZE ONE EDGE set FACTORIZE ONE INTERACTION set 6 5 1 Partitioning and Common Decompositions If B Bz is decomposition of B with respect to b C1 C2 is decomposition of C with respect to c and b c a Ugew d Uaec d Aj and Ugep d Udec d Az then B and C are said to have a common decomposition and if C is a sub generating class of B the te
18. 200 0 BE 8 11 364 0 Exact 200 0 CD 8 7 149 0 Exact 100 0 Edge selected CD Rejected edges Accepted edges Model Edges eligible EF AF Sorted list Edge DF 2log Q P AB 8 15 5740 Exact 1000 0 BF 8 15 744 0 Exact 200 0 CF 8 12 237 0 Exact 100 0 BE 8 11 364 0 Exact 200 0 DF 4 6 368 0 Exact 200 0 CE 8 12 735 0 Exact 200 0 Edge selected CE Rejected edges Accepted edges Model Edges eligible 0484 0530 0457 0850 1845 1750 1812 1850 5214 5600 ABCE ACDE BCEF CDEF ABCF ABEF ACDF ADEF1 ABCE ABEF ACDE ADEF1 ABCE ABCF ACDE ACDF 15 16 11 11 7 is not decomposable not decomposable not decomposable not decomposable is is is P Models 364 0 0 0560 733 0 0327 0 0450 478 0 1754 0 1550 268 0 1863 0 1850 067 0 5301 R ACDEF 0 5700 CCAD DE AE BC AC CBD CD BE EF DF CE AF BF CF AB ABCEF ADEF LEF DF CF CE BF BE AF AB 0484 0530 0457 0850 1403 1300 1812 1850 1733 1950 1207 2000 ABCE ABCF ADE ADF LABCE ADE BCEF DEF 15 16 11 11 12 is not decomposable is not decomposable P Models 364 0 0 0560 733 0 0327 R 0 0450 687 0 1651 R ABCEF 0 1300 268 0 1863 R 0 1850 583 0 1596 R 0 1750 666 0 1233 R 0 1950 CAD DE AE BC AC
19. 50 01 Max umol 1 3 Missing read observations print observed quit 46 Chapter 3 Reading Data These commands will result in Version 1 3 Friday March 17 12 00 00 MET 1995 Compiled with gcc a C compiler for Sun4 Compile time Mar 17 1995 13 00 00 Copyright c 1991 by Jens Henrik Badsberg Diary set ON gt set input data liver gt read specification gt redefine factor J Levels J gt 2 Missing J gt 1 gt cutpoints J Cutpoint J 1 gt 1 000 Cutpoint J 2 gt 4 000 gt 1 1 2 2 3 and 4 3 Missing 5 Missing gt redefine factor B Levels B gt 2 Missing B gt 1 gt cutpoints B Cutpoint B 1 gt 50 000 Cutpoint B 2 gt 900 000 gt 1 0 50 00 umol 1 2 50 01 Max umol 1 3 Missing gt read observations Ignoring cutpoint for factor B on datafile Finding all marginals 23 cases read gt print observed JB J 1 2 3 B 1 9 2 2 2 4 3 1 2 1 gt quit 3 8 Data Selection 47 3 8 Data Selection SELECT CASES set marginal cell OR SELECT CASES set marginal cell REJECT CASES set marginal cell OR REJECT CASES set marginal cell 3 8 1 Select or Reject Cases During the Reading of Data With the commands SELECT CASES set marginal cell and OR SELECT CASES set marginal cell between READ SPECIFICATION specification and READ OBSERVATIONS observations only cases in particular cells in one or more
20. AD 2 16 542 0 0003 16 427 0 0003 Exact 1000 0 0000 0 0000 AE ABC AD BE BF DE is not decomposable BC 2 682 296 0 0000 631 416 0 0000 Exact 1000 0 0000 0 0000 BE 2 20 929 0 0000 20 930 0 0000 Exact 1000 0 0000 0 0000 BF 1 4 732 0 0296 4 772 0 0289 Exact 1000 0 0300 0 0300 DE 2 18 319 0 0001 18 298 0 0001 Exact 1000 0 0000 0 0000 TIME 1 561secs gt gt Did accept wrong edges in the start gt gt forward 1 2 An Example 21 Edge DF 2log Q P X72 P Models AF 2 2 658 0 2647 2 648 0 2661 R ABF AB BF Exact 100 0 2300 0 2300 BD 4 2 504 0 6439 2 502 0 6443 ABDEJ ABE ADE Exact 20 0 7000 0 7000 CD ABC ABE ACD ADE BF is not decomposable CE 4 7 516 0 1110 7 602 0 1073 ABCE ABC ABE Exact 1000 0 1010 0 0940 CF 2 2 290 0 3182 2 426 0 2974 BCF1 BC BF Exact 100 0 3300 0 3100 DF ABC ABE ADE BF DF is not decomposable EF 2 2 405 0 3004 2 427 0 2972 BEF1 BE BF Exact 100 0 2700 0 2700 TIME 0 523secs gt gt Set DecomposableMode off to compute gt asymptotic P values for non decomposable models gt gt set decomposable mode off Decomposable mode set OFF TIME 0 001secs gt gt Compute asymptotic P values gt gt backward Edge DF 2log Q P X72 P Models AB 3 6 177 0 1033 6 162 0 1040 CABE ABC BE BC AE AC1 AC 2 23 8
21. Accepted Rejected B 1 39 EH Read and Fit Models or Force Models into Classes FIT Models models ACCEPT Models models REJECT Models models B 1 40 EH Copy Models between the Model List and Search Classes Models from List to Search Classes FIT Base Current Last Number lt a gt All models Interval of models lt a gt lt b gt ACCEPT Base Current Last Number lt a gt All models Interval of models lt a gt lt b gt REJECT Base Current Last Number lt a gt All models Interval of models lt a gt lt b gt Models from Search Classes to Model List EXTRACT Decomposable Graphical Hierarchical A dual R dual Accepted Rejected B 1 41 EH Find Duals FIND DUAL Decomposable Graphical Hierarchical A dual R dual Both duals B 1 42 EH Directed Search FIT Decomposable Graphical Hierarchical A dual R dual Both duals Smallest dual Largest dual B 1 43 EH Automatic Search EH Decomposable Graphical Hierarchical Smallest Alternating Rough B 1 44 EH Force a Dual into a Model Class ACCEPT Decomposable Graphical Hierarchical A dual R dual Both duals REJECT Decomposable Graphical Hierarchical A dual R dual Both duals 160 Appendix B Reference Manual section B 1 45 Arguments to commands Note the following symbols are meta symbols belonging to the Backus Na
22. Blot TestallSb fe so AL BR eee Be bat Ped Pek 157 B 1 32 Dispose of Tables 000000200045 158 B 1 33 Editing Models with Tests o o 158 B 1 34 Stepwise Edge or Interaction Selection and Elimination 158 B 1 35 Global Search The EH procedure 158 B 1 36 EH Fix Edges Interactions o o 158 B 1 37 EH Choose Model Class and Strategy 158 B 1 38 EH Dispose of Model Classes and Duals 159 B 1 39 EH Read and Fit Models or Force Models into Classes 159 B 1 40 EH Copy Models between the Model List and Search Classes 159 B 1 41 EH Find Dials a 2 24 2 4544 2 Pe ee ewe ee a 159 B 1 42 EH Directed Search o o 159 B 1 43 EH Automatic Search e o 159 B 1 44 EH Force a Dual into a Model Class 159 B 1 45 Arguments to commands o o 160 B 1 46 Codes of Table values for S Plus 161 Bibliography 165 Part I Tutorial Chapter 1 Introduction to CoCo 1 1 Starting and Finishing DOS On a PC Install CoCo as described in Appendix A make sure that your PATH is pointing to the directory with CoCo and just type COCO Macintosh On Macs If not installed Install CoCo as described in Appendix A Click the CoCo folder Unix On UNIX Workstations If not installed Install CoCo as described in Ap pendix A make sure that the directory with the shell script CoCo i
23. Describe the adjusted residuals in the marginal table BCDEF gt gt describe table rankit adjusted BCDEF DESCRIBE TABLE Adjusted NUMBER OF CELLS IN TABLE 32 NUMBER OF INVALID CELLS 0 NUMBER OF CELLS TO DESCRIBE 32 SORTED LIST 3 4465 2 2064 2 1418 0 9380 0 8524 0 7695 0 6893 0 4999 0 4912 0 4444 0 4315 0 2857 0 2795 0 2417 0 1614 0 1219 0 0463 0 1210 0 1490 0 2556 0 3147 0 7317 0 8584 0 9000 0 9719 1 1956 1 2852 1 3426 1 4399 1 7381 1 9406 4 1246 STATISTICS NUMBER OF VALUES 32 25 RANK 8 VALUE 0 4999 RANK 9 VALUE 0 4912 50 RANK 16 VALUE 0 1219 RANK 17 VALUE 0 0463 75 RANK 24 VALUE 0 9000 RANK 25 VALUE 0 9719 1 2 An Example MINIMUM MAX RUN SUM X SUM 1 X PROD X SUM X MEAN 2 SUM X MEAN 3 SUM X MEAN 74 RANKIT PLOT Unit horizontal Unit vertical 3 4465 MAXIMUM 1 MODE 3213 30 0000 58 12 506 8929 7837 1395 5294 0 1250 0 2000 4 1246 RANGE lt Eps MEAN HARMONIC MEAN GEOMETRIC MEAN VARIANCE SKEWNESS KURTOSIS 11 7 5711 0 1038 1 0358 0 6163 1 8370 0 1524 4 6907 20000 f 00000 80000 l 60000 40000 20000 00000 80000 l 60000 2 40000 20000 2 OE 00 0 20000 2 l 0 40000 k 0 60000 2 0 80000 1 00000 1 20000 H ox 1 40000 1 60000
24. cedures 8 1 Decomposable Mode ls 20 20 0020 0005 8 2 Computed Test Statistics and Choosing Tests and Significance Level 8 2 1 Computing Test Statistics 0 0 0 2 0 000 8 2 2 Selecting Test Statistic and Significance Level 8 3 Exact Tests in Tests and Model Selection o o 8 4 Partitioning merecia ed ly GR o dd ey a a Stepwise Model selection Backward and Forward 9 1 Selecting Test Statistic and Significance Level 9 2 Fixing of Edges and Interactions 00 0 9 3 Backward Elimination 0 02 00 0000 00 9 3 1 Controlling the Output o o 9 3 2 Recursive Search 0002 ee eee 9 3 3 Graphical or Non Graphical Models 9 34 Mod l Control aio Save eh a Bla ake oe ee 9 4 Forward Selection a eee eee eee 9 4 1 Controlling the Output 9 4 2 Recursive Search 0 2 200002 eee 9 4 3 Graphical or Non Graphical Models 94 4 Model Control y seri a eae oe er ees Ee ee ee 9 5 Asymmetry between Backward and Forward The EH procedure 10 1 Computed Tests and Selection Criteria 0 10 2 Fix Edges Interactions 2 0 0 00 000000000004 OZI SERIN A ADEA ea ee BAAS Se oY 10 22 PixOut lt lt con eee wee ee ORR ESS a 10 2 3 Read Base Model 0020000000 0s 10 2 4 Interaction between FixIn FixOut a
25. data file specification file observation file B 1 46 Codes of Table values for S Plus The codes used in S Plus for table values are noted in the parentheses in the above description of table value name Bibliography Agresti A 1990 Categorical data analysis Wiley Chichester Aickin M 1983 Linear Statistical Analysis of Discrete Data Wiley amp Sons Inc New York Asmussen S amp Edwards D 1983 Collapsibility and response variables in contin gency tables Biometrika 70 567 578 Badsberg J H 1986 Kontingenstabeller implementation og kompleksitet af algo ritmer for estimation og test i store kontingenstabeller Masters thesis Aalborg University Becker R A Chambers J M amp Wilks A R 1988 The New S Language A Programming Environment for Data Analysis and Graphics Wadsworth amp Brooks Cole Advanced Books amp Software Pacific Grove California Bishop Y M M Fienberg S E amp Holland P W 1975 Discrete Multivariate Analysis Theory and Practice MIT Press Cambridge Massachusetts Christensen R 1990 Log Linear Models Springer Verlag Darroch J N amp Ratcliff D 1972 Generalized iterative scaling for log linear models Annals of Mathematical Statistics 43 1470 1480 Darroch J N Lauritzen S L amp Speed T P 1980 Markov fields and log linear interaction models for contingency tables Annals of Statistics 8 522 539 Dawid
26. echo SCOCOHOME Functions sep in S Plus to access the directory Functions with the S function for commu nicating with CoCo Available S functions for interaction with CoCo are then listed by 1s pos 2 The function coco load will load CoCo into S Plus 12 1 S CoCo 135 coco start will then start CoCo in S Plus Ordinary CoCocommands are then expected CoCo is reading from standard input and writing to standard output The CoCo command END will bring you back to S Plus without disposing of the running CoCo session CoCo is not terminated but only left temporarily The running CoCo session can be resumed from S Plus with coco resume The running CoCo session within S Plus is removed with coco end If coco end is not used before leaving S Plus some temporary files will not be deleted The function coco init starts CoCo and returns immediately to S Plus without expecting CoCo commands coco start is a combined command of coco init and coco resume The returned value from coco start coco init and coco resume is TRUE if the command succeeds coco start and coco init may be given the optional arguments n 512 p 512 and q 512 for setting the initial size of the N array P array and Q array respectively The optional argument s 620000 to coco 1load is the expected size of the code added to S Plus and used in dyn load2 The size s of the code added to S Plus depends on the mach
27. marginal tables are read For example only cases with A 1 and B 2 or C 1 or D 2 are read by READ SPECIFICATION SELECT CASES AB 1 OR SELECT CASES C 1 OR SELECT CASES D 2 READ OBSERVATIONS 2 The command SELECT CASES set marginal cell disposes of previous select statements Use the command SELECT CASES to select all cases not rejected Cases in particular cells can be skipped with REJECT CASES set marginal cell and OR REJECT CASES set marginal cell For example READ SPECIFICATION SELECT CASES AB 1 2 OR SELECT CASES C 1 REJECT CASES CD 1 1 OR REJECT CASES D 2 READ OBSERVATIONS Cases with A 1 and B 2 or C 1 and not C 1 and D 1 or D 2 are read As with the command SELECT CASES set marginal cell the command REJECT CASES set marginal cell disposes of previous reject statements All cases selected are read after the command REJECT CASES Note that selection will result in zero cells in the read data if factors done selection on is read The selection is done on the result of REDEFINE FACTOR factor name 4 of observed levels 4 of missing levels and CUTPOINTS factor name cutpoints commands Cases with a particular level at a non binary factor can be excluded without making zero cells by reading a data list with the factor twice doing selection on one copy of the factor and grouping the other copy of the factor by the CUTPOINTS fa
28. npB JBUa Standardized e Grey 2log Q tata n jBua eL s n jgua n Ps jBua ee V jsuada ia V Taa Ta GeO n jBUa N PB jBua R F sais iJa la Vn jBua 1 Dp Bua T 1 G res Eonia vaa ZauBl PB jBua 1 PB jBua 56 Chapter 4 Description of Data and Fitted Values re A jBuaija ia yn paa Ea Freeman Tukey imuajenia LV Bua Vn jBua 1 4npB jBua 1 2 n m Dinvajazio 1 VG Bua VTPBlIBua Power 304D Wipuatjaci MiBua Yee 1 Index Index i Index i a 1 discat ts 1 Ny 25 ye0 Lv da e 1 if Fjatja ta A gA O otherwise Error Nia for q ia 0 The power is by default set to 2 3 It is changed by the statement SET POWER LAMBDA Null lambda 4 3 Printing Tables PRINT TABLE value name a b 1 By the command PRINT TABLE value name a the marginal table determined by the set a and with the values value name for the by default current model is printed The commands PRINT TABLE value name a DESCRIBE TABLE value name a RETURN VECTOR value name a RETURN MATRIX value names a PLOT x value name y value name a and LIST a can only be used if a current model exists See the next chapter use PRINT OBSERVED a or type READ MODEL The command PRINT OBSERVED a prints a table of marginal observed counts without expecting a current model PRINT TABLE Current Use value
29. or the dual representation of the model over all models in the class A or R Since it sometimes avoids finding large duals that are not used this can be effective By this strategy the complete search in Edwards amp Havr nek 1985 on a 6 dimensional table is done by CoCo in less than 700 milliseconds on a SPARCstation 10 9 3 Alternating Automatic The R dual and the A dual of the two model classes are in turn classified with the command EH Alternating In a few examples where alternating fitting the R dual and the A dual is optimal this is the fastest But often the strategy will start to select the largest dual and then this strategy will be the slowest 122 Chapter 10 The EH procedure The automatic search stops when either all models in D4 R A are accepted or all models in DR A R are rejected 10 10 Force a Dual into a Model Class ACCEPT Decomposable Graphical Hierarchical A dual R dual Both duals REJECT Decomposable Graphical Hierarchical A dual R dual Both duals The command ACCEPT R dual adds the R dual of the accepted models to the ac cepted models The command REJECT A dual adds the A dual of the rejected models to the rejected models If we add the A dual of the rejected models to the accepted models the search will be finished So this command does not exist The A dual of the accepted models can be added to the accepted models by ACCEPT A dual Similarly the R dual of t
30. results from the commands TEST FIND LOG L etc The arguments line length and page length in the command SET PAGE FORMATS line length page length control sizes of plots and histograms The sorted list gen erated by DESCRIBE TABLE is only printed if it can fit onto two pages determined by line length and page length Use STATUS Formats to see the current setting of formats 11 1 1 Pausing SET PAUSE On Off SET PAGING LENGTH number of lines With Pausing set on by SET PAUSE On Off J Default Off the output is paused for each written 22 lines after last NewPage if EndOfLine is found after the total command is read by CoCo E g if Pausing is on then CoCo will pause for each printed 22 lines after STATUS If Pausing is on then CoCo will not pause during the status command if STATUS or STATUS READ MODEL A B is entered The next 22 lines are printed by pressing Return If a is entered when pausing then the command will finish without pausing The number of lines to print between pausing are set by SET PAGING LENGTH number of lines Default 22 See also section 2 6 Use STATUS Formats to see the current setting of pausing 11 2 Input files 125 11 2 Input files 11 2 1 Input from keyboard or file SET KEYBOARD On Off With this switch one selects between reading from file or from standard input De fault From the data file After the command SET KEYBOARD On the arg
31. s will terminate the EM algorithm factorization of a test a BACKWARD command a FORWARD command or the computation of a dual in the EH procedure after SET INTERRUPT B On Default A Control C will return the action to the lisp listener when the interrupt is given within a CoCo function of CoCo loaded into XLISP STAT Chapter 12 CoCo in other systems CoCo can on UNIX systems be loaded dynamically into XLISP STAT Tierney 1991 and S Plus Becker Chambers amp Wilks 1988 and may be into New S if the function dyn load2 is available This feature is only tested on Sparc The purpose of loading CoCo dynamically into XLISP STAT is besides the system Xlisp CoCo A Tool for Graphical Models to be able to do further computations on fitted table values and statistics plot output from CoCo by high resolution graphics or e g do a model search by a strategy implemented in Lisp Data specification of table and the table of counts and models can be sent to the CoCo object Tables and test values can be returned When CoCo is loaded into XLISP STAT or S Plus the system demands more mem ory and is somewhat less stable than any of the 3 programs alone although it is a long time since the author unexpected has crashed the system If you are e g going to do a lot of computation in XLISP STAT or S Plus without using CoCo functions or you are doing model search in CoCo run XLISP STAT S Plus or CoCo respectively without lin
32. the first factor met in a is alternating at the fastest speed Example gt read model gt read model ACE ADE BC F gt test Test of F BC ADE ACE against ABCDEF St 2log Q Power X72 DF gt list AB A B Observed Ll 522 21 541 12 439 22 339 A B Expected 11 522 30 21 540 70 12 438 70 22 339 30 atistic 62 0779 59 7593 59 9956 Probabi 0 283706 0 293697 0 238292 0 184304 Adjusted 0 8228 1 3433 2 2441 1 6995 gr Asymptotic 0 0994 0 1396 0 1350 49 Residual F res 0 3036 0 3053 0 3036 0 4024 0 3036 0 9062 0 3036 0 4927 Standard 2log q 0 2969 0 3237 0 4825 1 7218 0 8091 2 7436 0 6149 1 1986 Adjusted 0 0994 0 1396 0 1350 49 Res F 0 2734 0 4458 0 7652 0 5854 Freeman 0 2798 0 4930 0 8250 0 6080 G res 1 7525 0 5794 15 2189 18 1416 2 n m 0 3138 0 4591 0 8778 0 5656 Res G 4 3730 5 6692 1 5189 8 5233 Power 0 5583 1 4953 2 4241 1 4852 In incomplete tables the observed count for structural zero cells is marked by the asterisk If EM is on then CASE LIST will for each cell with non zero count in the full table print the number n i of cases If the cell i is not marked as missing none of the levels for the cell are marked as missing i Z then the estimated probability P in the cell and an identification of the cell are printed else if the cell has le
33. will show the selected data structure After reading specification and SET READ Subset a you can change the choice of data structure with the commands SET DATASTRUCTURE A11 SET DATASTRUCTURE Necessary SET DATASTRUCTURE File SET LARGE On and SET LARGE Off If e g you have a 6 dimensional table with binary factors then read data with the three commands READ SPECIFICATION SET DATASTRUCTURE Necessary and READ OBSERVATIONS instead of READ DATA to avoid finding all marginal tables If there is space for reading the full table into an array but it is expected that the list of cases can be read faster than the full table it is expected that the table is sparsed the data structure File is selected and a message like If more than cases then use SET DATASTRUCTURE NECESSARY between READ SPECIFICATION and READ OBSERVATIONS is stated The time used to read tables and lists of cases depends on the computer When reading the data CoCo then uses time to find out which data structure is the fastest on your computer for the particular dataset In subsequent runs of CoCo on the dataset force CoCo to use the selected data structure with SET DATASTRUCTURE Necessary or SET DATASTRUCTURE File between the two commands READ SPECIFICATION and READ OBSERVATIONS If Large is off and there is sufficient workspace in the N array and in the P array when performing tests all necessary marginal tables are found and the IPS algorithm is run on al
34. 0 033578 2 2 8 0 009815 3 2 2 38 0 067956 0 111349 1 2 2 38 0 033578 0 111349 2 2 2 38 0 009815 0 111349 3 2 2 3 0 454953 0 522909 1 1 2 3 0 067956 0 522909 1 2 2 3 0 239503 0 273081 2 1 2 3 0 033578 0 273081 2 2 2 3 0 454953 0 823276 ix 1 2 3 0 239503 0 823276 2x i1 2 3 0 017471 0 823276 3 1 2 3 0 067956 0 823276 1 2 2 3 0 033578 0 823276 2 2 2 3 0 009815 0 823276 3 2 2 gt quit 4 8 Exporting Table Values 63 4 8 Exporting Table Values RETURN VECTOR value name a RETURN MATRIX list of value names a The command RETURN VECTOR value name a returns a vector with the values value name in the a marginal table Similar to PRINT TABLE value name a but without headings The print format is controlled by SET PRINT FORMATS format width decimals The cells in the printed vector are ordered according to the order of the factors in a The first factor met in a is alternating at the fastest speed If RETURN STANDARD VECTOR value name a is used then the values will be ordered according to the order of the factors in specification of the factors when declaring data If Dump is on See chapter 11 Options the vector is written to the DumpFile instead of the standard output Dump is by default off Dump is set on or off by the command SET DUMP On Off The default name on the DumpFile is tmp CoCo dump no pid number under Unix and C0C0 DMP in the current direc tor
35. 00 umol 1 5 200 01 1000 00 umol 1 6 gt 1000 00 umol 1 code 900 00 7 Missing Code or 999 99 List 4 123 12 1 27 36 3 49 59 1 15 39 3 53 01 2 22 23 1 49 99 1 50 01 1 50 00 1 8 00 1 12 00 1 73 00 5 7 00 8 00 1 9 00 1 6 00 4 108 00 1 0 00 2 900 00 1 999 99 3 999 99 4 Comments are inserted in the data file by as the first character at lines before the key words Names List Factors and Table The commands 3 7 Grouping of Factor Levels Cutpoints and Redefine Factors 45 set output diary diary l1 set input data liver read data print observed quit will then result in Version 1 3 Friday March 17 12 00 00 MET 1995 Compiled with gcc a C compiler for Sun4 Compile time Mar 17 1995 13 00 00 Copyright c 1991 by Jens Henrik Badsberg Diary set ON gt set input data liver gt read data Finding all marginals 23 cases read gt print observed JB J 1 2 3 4 5 B 1 1 0 0 0 0 2 8 1 1 0 2 3 2 0 1 0 0 4 0 0 0 2 0 5 0 0 0 0 0 6 0 1 0 0 0 7 1 0 1 1 1 gt quit The variables Jaundice and Bilirubin are both ordinal Without changing the file liver they can be redefined by the following commands set output diary diary 12 set input data liver read specification redefine factor J 2 1 cutpoints J 1 4 1 1 2 2 3 and 4 3 Missing 5 Missing redefine factor B 2 1 cutpoints B 50 00 900 00 1 0 50 00 umol 1 2
36. 119 R keyword 131 R f keyword 162 R g keyword 162 random edge order See model selection random table 42 random vector 56 Random keyword 51 56 162 rankit plot 59 Rankit keyword 59 READ BASE MODEL command 113 115 117 160 READ DATA command 4 5 38 39 44 45 130 157 READ FACTORS command 38 39 126 157 READ LIST command 38 39 126 157 READ LOCAL PARSER command 35 154 READ MODEL command 10 35 36 43 57 65 111 125 158 READ N INTERACTIONS command 65 109 158 READ NAMES command 38 39 126 157 READ OBSERVATIONS command 38 39 44 45 48 126 130 157 READ Q LIST command 42 44 158 READ Q TABLE command 42 43 158 READ SPECIFICATION command 38 39 48 126 130 157 READ TABLE command 38 39 126 157 READ ME file 148 Index reading data 40 specification 39 README DOS file 148 README Unix file 148 recursive 139 Recursive Separators command 112 Recursive keyword 106 107 110 REDEFINE FACTOR command 44 45 48 157 REDO FIX command 115 117 160 REDUCE GENERATOR command 50 97 125 133 160 Reinis 4 reinis81 dat 5 REJECT command 119 120 122 123 161 REJECT CASES command 48 157 rejected models 114 119 rejected region 119 REMOVE GENERATOR command 50 97 125 160 REMOVE TOTAL INTERACTION command 50 97 125 160 Report keyword 51 93 129 132 ReportFile keyword 122 128 129 132 ReportOn ke
37. 16 28 724 0 0257 27 465 0 0363 R ABCDEF ABCEF BCDEF Exact 1000 0 0420 0 0380 AB 16 22 652 0 1232 21 213 0 1702 R ABCDEF ACDEF BCDEF Exact 1000 0 1740 0 1770 CF 16 22 153 0 1381 20 746 0 1881 R ABCDEF ABCDE ABDEF Exact 100 0 1800 0 1400 BF 16 22 788 0 1193 23 124 0 1102 R ABCDEF ABCDE ACDEF Exact 1000 0 1890 0 1100 AF 16 21 305 0 1668 19 710 0 2331 R ABCDEF ABCDE BCDEF Exact 100 0 2700 0 2600 CE 16 18 629 0 2878 17 517 0 3526 R ABCDEF ABCDF ABDEF Exact 100 0 3700 0 3800 DF 16 18 345 0 3035 17 196 0 3728 R ABCDEF ABCDE ABCEF Exact 100 0 3700 0 3700 EF 16 18 316 0 3052 17 820 0 3341 R ABCDEF ABCDE ABCDF Exact 200 0 4000 0 3450 BE 16 17 226 0 3709 16 113 0 4454 R ABCDEF ABCDF ACDEF Exact 100 0 4600 0 4600 CD 16 14 808 0 5389 13 427 0 6422 R ABCDEF ABCEF ABDEF Exact 20 0 6000 0 6500 BD 16 12 226 0 7293 10 985 0 8109 R ABCDEF ABCEF ACDEF Exact 20 0 8500 0 9500 Edge selected BD Rejected edges CAD DE AE BC AC Accepted edges LBD CD BE EF DF CE AF BF CF AB Model ABCEF ACDEF Edges eligible CAB AF BE BF CD CE CF DF EF 1 2 An Example AF CE CF EF Sorted list Edge DF 2log Q P AB 8 15 574 0 Exact 1000 0 BF 8 15 744 0 Exact 200 0 DF 8 11 302 0 Exact
38. 3 33 36 39 64 125 162 163 39 86 162 39 162 33 39 162 0 35 13 34 39 162 0 75 39 162 BINDIR environment 149 COCOHOME environment 149 SCOCOHOME environment 149 XCOCOHOME environment 149 39 162 amp 33 35 162 _ 39 162 33 39 33 39 33 39 162 39 162 Functions file 135 Dynamic Spin plot 143 Recursive backward 143 Return 33 35 36 64 125 liver 45 46 probit 59 fep program 150 0 9 39 162 2 n m keyword 162 2 section graph 65 94 0 42 A Z 39 162 Index a z 39 162 A dual 114 119 ACCEPT command 119 120 122 123 161 accepted models 114 119 accepted region 119 Accumulated list keyword 41 59 163 ADD EDGES command 50 65 66 93 94 125 133 160 ADD FIX command 115 117 160 ADD INTERACTIONS command 50 65 93 95 125 160 adjacency matrix 66 adjacent 65 138 adjusted degrees of freedom See degrees of freedom adjusted residuals See residuals Adjusted keyword 162 AdjustedDF mode 99 137 AIC See Information Criteria Akaike s information criteria See Information Criteria All marginal tables keyword 130 All keyword 108 Alternating keyword 122 AND FIX command 105 160 arrow 139 association marginal 104 111 measures of 79 partial 109 association diagram 138 atoms 67 BACKWARD command 13 37 50 73 83 97
39. 32 Dispose of Tables DISPOSE OF TABLES DISPOSE OF Q TABLE set DISPOSE OF ALL Q TABLES DISPOSE OF PROBABILITIES B 1 33 Editing Models with Tests Only GENERATE DECOMPOSABLE MODEL Only GENERATE GRAPHICAL MODEL Only DROP FACTOR factor Only DROP EDGES edge list Only ADD EDGES edge list Only DROP INTERACTIONS gc Only ADD INTERACTIONS gc Only REDUCE GENERATOR set Only REMOVE GENERATOR set Only REMOVE TOTAL INTERACTION set Only MEET OF MODELS Only JOIN OF MODELS 7mm XM mr B 1 34 Stepwise Edge or Interaction Selection and Elimination FIX Edges edges gc AND FIX Edges edges gc Only Reversed Sorted Separators Recursive Headlong Coherent Follow All BACKWARD Edges Interactions Only Reversed Sorted Separators Recursive Headlong Coherent Most FORWARD Edges Interactions B 1 35 Global Search The EH procedure SET MAIN EFFECTS set READ BASE MODEL model B 1 36 EH Fix Edges Interactions FIX In Out edges gc ADD FIX In Out edges gc REDO FIX In Out B 1 37 EH Choose Model Class and Strategy SET Graphical Hierarchical SEARCH SET Smallest Alternating Rough SEARCH B 1 Quick Reference Card 159 B 1 38 EH Dispose of Model Classes and Duals DISPOSE OF EH A11 Duals A dual R Dual Classes
40. 5 Controlling Algorithms 129 After reading specification and the use of SET READ All Subset set CoCo chooses automatically between three internal data structures for marginal tables 1 All marginal tables If the dimension of the table is low less than or equal to 9 for binary factors in PC there is space for finding all marginal tables In e g a model search you will then save some time 2 Necessary tables If there is no space for all marginal tables but space for read ing the full table into an array and it is expected that the full table can be read faster than the list of cases the data structure Necessary tables is selected On UNIX the size of the N array is controlled by the option N size when invoking CoCo When performing a test necessary marginal tables are then found Necessary marginal tables are sufficient marginal tables intersections of sufficient marginal tables 3 File If the full table cannot fit into the memory dimension is higher than 15 on PC the data structure File is selected Observations are then placed on an internal file On the PC you are then able to compute log L for a model with 64 factors if the largest clique in the graph for the model has 14 vertices at the most and there are no more than 13 vertices in the largest non decomposable atom or more precisely if the state space of each irreducible component after adding a fill in can fit into the P array The command STATUS Limits
41. A P amp Skene A M 1979 Maximum likelihood estimation of observer error rates using the em algorithm Journal of the Royal Statistical Society Series C 28 1 20 28 Dempster A P 1972 Covariance selection Biometrika 28 157 175 Dixon W J 1983 BMDP Statistical Software University of California Press 1985 printing Edwards D 1989 A guide to MIM version 1 7 Research Report 12 Statistical Research Unit University of Copenhagen A Guide to MIM version 2 0 1991 162 Bibliography 163 Edwards D 1993 Some computational aspects of graphical model selection in J Antoch ed Computational Aspects of Model Choice Springer Verlag pp 187 210 Edwards D amp Havr nek T 1985 A fast procedure for model search in multidi mensional contingency tables Biometrika 72 339 351 Edwards D amp Havr nek T 1987 A fast model selection procedure for large families of models Journal of the American Statistical Association 82 205 211 Fienberg S E 1970 An iterative procedure for estimation in contingency tables Annals of Mathematical Statistics 41 907 917 Frydenberg M 1989 The chain graph Markov property Scandinavian Journal of Statistics 17 333 353 Frydenberg M amp Lauritzen S L 1989 Decomposition of maximum likelihood in mixed interaction models Biometrika 76 539 555 Fuchs C 1982 Maximum likelihood estimation and model selection in contin gency tables
42. A gt lt B gt lt C gt lt AB gt lt AC gt lt BC gt and lt ABC gt into all models 10 2 2 FixOut With the command FIX Out edges gc all terms containing a generator in the gen erating class are FixOut The FixOut is the dual representation of the maximal model considered see pp 340 Edwards amp Havr nek 1985 Note that only terms containing a FixOut generator are FixOut The command FIX Out ABC excludes the terms lt ABC gt lt ABCD gt lt ABCE gt lt ABCDE gt lt ABCF gt lt ABCDF gt lt ABCEF gt lt ABCDEF gt lt ABCG gt etc from models But the terms lt AB gt lt AC gt and lt BC gt are not excluded So to exclude edges in the graphical search only generators with cardinality 2 edges should be given to FixOut Using the commands FIX In edges gc and FIX Out edges gc and ADD FIX In edges gc ADD FIX Out edges gc REDO FIX In edges gc and REDO FIX Dut edges gc of this section will only set up fixing for future found models No interactions will be added to or removed from already found models Thus if the EH procedure already has been used the duals and model classes should be disposed of by the command DISPOSE OF EH Main effects SET MAIN EFFECTS set The factors to be included in the search are set with SET MAIN EFFECTS set default value that is all factors read are used in the EH procedure The EH procedure can also be restricted to a subset of the decl
43. Cliques ACE ADE BC F 2 Section Model is decomposable Test of F BC AD DE AC AE CE against ACE ADE BC F DF 2log Q P X72 P Models 1 0 0105 0 91851 0 0105 0 91851 ACEJ1 AC AE CE 1 2 8340 0 09229 2 8324 0 09238 ADEJ AE DE AD 2 2 2 8445 0 24117 2 8428 0 24137 ADE ACE CCAD DE CE AE AC TIME 0 044secs The command NORMAL TO DUAL will generate a model where the generating class for the generated model is the dual representation of the current model Together with MEET OF MODELS this can be used to do DROP INTERACTIONS gc on several models 7 8 Join of Models The join of the two models base and current is formed by the command JOIN OF MODELS The join is the smallest model containing both base and current model The resulting model will contain the maximal generators in both base and current The generation of the model is symmetric in base and current The current model is tested against the resulting model Continuing the example from the previous section gt join of models 4 AB AF BD BE BF CD CF DF EF BC ADE ACE Model is not graphical Cliques ABCDEF 2 Section Model is decomposable Test of ACE ADE BC F against AB AF BD BE BF CD CF DF EF BC ADE ACE 96 Chapter 7 Editing Models with Tests DF 2log Q P X72 P Models 9 17 5851 0 03991 18 2781 0 03176 ACE ADE BC EF
44. IPS ITERATIONS mazimum cycles Number of cycles difference between convergence criterion for last two cycles and computing time of the IPS algorithm for each non decomposable irreducible compo nent is reported in the ReportFile Use STATUS Ips to see the current values 11 5 2 Controlling the EM algorithm SET EM INITIAL Uniform First Last Mean Random Input SET EM EPSILON epsilon SET EM ITERATIONS maz The EM algorithm is selected by EM ON Initial values are chosen by the command SET EM INITIAL Uniform First Last Mean Random Input See section 3 9 4 for further details Cycles in the EM algorithm is repeated until the difference between values of the deviance for each cycle is less than the e set by SET EM EPSILON c SET EM ITERATIONS max controls the maximum cycles in the EM algorithm If Report is on the likelihood ratio test statistics are reported for each step in the EM algorithm If Trace is on the probability for each cell for each case is reported for each step in the EM algorithm Use STATUS Em to see the current values 11 5 3 Data structure Do not worry about the options of the section since CoCo has been become quite good at selecting the data structure But for those interested a short description of the data structures is given SET DATASTRUCTURE All Necessary File Large SET LARGE On Off SET HUGE On Off SET SORTED On Off 11
45. J Gori an K Hor kov D Stuch likov E Havr nek T amp Hrabovsk F 1981 Prognostick v znam rizikov ho profilu v prevenci ischemick choroby srdce Bratis lek Listy 76 137 150 Prog nostic significance of the risk profile in the prevention of coronary heart disease Rose D J Tarjan R E amp Lueker G S 1976 Algorithmic aspects of vertex elimination on graphs SIAM Journal on Computing 5 266 283 Sakamoto Y amp Akaike H 1978 Analysis of cross classified data by aic Ann Inst Statist Math 30 185 197 Sundberg R 1975 Some results about decomposable or markov type models for multidimensional contingency tables Distribution of marginals and partitioning of tests Scandinavian Journal of Statistics 2 771 779 Tarjan R E 1985 Decomposition by clique separators Discrete Mathematics 55 221 232 Tarjan R E Yannakakis M 1984 Simple linear time algorithms to test chordal ity of graphs test acyclicity of hypergraphs and selectively reduce acyclic hyper graphs SIAM Journal on Computing 13 566 579 Bibliography 165 Tierney L 1991 Generalized linear models in lisp stat Technical Report 557 School of Statistics University of Minnesota Wedelin D 1993 Efficient algorithms for probabilistic inference combinatorial optimization and the discovery of causal structure from data Ph d thesis De partment of Computer Sciences Chalmers University of
46. Partitioning is on and the decomposable models differ with more than one edge then the computation of exact test for each part of the test is controlled by SET EXACT TEST FOR PARTS OF TEST On Off J Default On and the computation of exact test for the total test is controlled by SET EXACT TEST 6 5 Miscellaneous Commands for Model Control FOR TOTAL TEST On Off Default On If Partitioning is off and the de composable models differ with more than one edge then exact test is only computed if SET EXACT TEST FOR UNPARTED TEST On Off is on Default On Example of the use of the commands PRINT COMMON DECOMPOSITIONS and DECOMPOSE MODELS a on the data from Edwards amp Havr nek 1985 see Intro duction gt read data 64 cells with 1841 cases read Finding all marginals TIME gt read model TIME 0 026secs 0 002secs gt read model ACDE ABCF TIME 0 001secs gt read model ADE ACE ABC ABF TIME gt make TIME base 2 0 001secs 0 001secs gt print common decompositions Decomposition of ADE ACE ABC ABF and ACDE ABCF Decompositions ABCF 4 AC C 2 ACDE 4 Selected ABCF 4 AC 2 ACDE 4 TIME 0 006secs gt decompose models AC Partitioning of ADE ACE ABC ABF and ACDE ABCF TIME gt print Model Model Model Model Model Model Model TIME 0 004secs all models no
47. a table of counts into CoCo in the normal set If enter table table is succeeded the value TRUE is returned else FALSE is returned and a warning message is printed If the command SET READ All Subset set has been used or data se lection has been done between calls to the functions enter names names levels missing NULL and enter table table the fully declared table still has to be entered The count 1 is entered for structural zero cells The function read model a in S Plus will add the model a to the list of models in CoCo The argument a to read model a has to be a single character string with the model written as models are written in CoCo 136 Chapter 12 CoCo in other systems The functions make current a and make base a in S Plus will make the model with number a the current or the base model respectively in CoCo base O makes the current model the base model current makes the last model the cur rent model These four commands return the value TRUE if the commands succeed else FALSE The commands return current number and return base number return respectively the model numbers for the current model and the base models if the command succeeds else NULL The commands return current return a text string with the generating class for the current model if the command succeeds else NULL Analogously with return base From S Plus the models read in CoCo are printed by print a
48. be added to the FixEdgeList by AND FIX Edges edges Following use of the FIX Edges edges will clear and replace the FixEdgeList 9 3 Backward Elimination Only Reversed Sorted Separators Recursive Headlong Coherent Follow All BACKWARD Edges Interactions The simplest form BACKWARD lOne could consider removing the command FIX Edges edges and then also set the fix for stepwise backward elimination and forward selection with the commands FIX In edges and FIX Out edges of the EH procedure This would also allow for different fixing for respectively backward elimination and forward selection But for comparability with former versions of CoCo separate fixes for the stepwise model selection and the global EH procedure respectively have been kept 9 3 Backward Elimination 105 of the backward elimination command will visit all edges in the current model if it is graphical and for each visited edge produce a test See section 9 3 3 about choos ing between backward elimination of edges and backward elimination of interaction terms With the command Recursive BACKWARD all edges in the current model are visited and the least significant edge is removed This step of the backward elimination is then repeated on the resulting model All edges are visited in the model and the least significant edge is removed This is repeated until all edges are significant in the resulting mode
49. begin ning of command names e g HELP read or HELP set and some documentation for that beginning might be printed HELP value name will give documentation for the table value value name HELP keyword is used to print documentation for special topics like limits overlays data etc See HELP help MENU NUMBER 0 will print a quick reference card With MENU NUMBER a NEXT MENU CURRENT MENU PREVIOUS MENU it is possible to step through the quick reference card If a command is removed from CoCo then HELP command name will not be able to reach help information on how the command is replaced See HELP help or HELP removed 2 2 Comments If is the first character of a command then characters from to or end of line are skipped 2 3 Make Delete and Print Alias PRINT ALIAS command name abbreviation MAKE ALIAS command name abbreviation DELETE ALIAS abbreviation Abbreviations for commands with long names can be seen with the com mand PRINT ALIAS command name The command HELP abbreviation will print the full command name with arguments for the command with abbreviation abbreviation New abbreviations can be declared with the command MAKE ALIAS command name abbreviation An abbreviation is deleted with DELETE ALIAS abbreviation The author uses the following abbreviations make alias quit end make alias end es make alias set inputfile data sid make alias set
50. command 101 102 109 112 156 SET SHORT command 124 125 155 SET SORTED command 39 129 157 SET TABLE FORMATS command 57 62 124 125 155 SET TEST command 99 102 156 SET TEST FORMATS command 124 125 155 SET TIMER command 128 155 SET TRACE command 132 155 setf function 140 Short keyword 107 110 ShortTestOutput mode 75 92 93 SHOW TESTS command 92 99 159 significance level 99 101 simple 80 SINK command 127 SKIP MISSING command 49 50 157 SLICE command 79 159 Smallest keyword 122 Somers D 79 Sorted keyword 106 110 SOURCE command 29 34 126 SourceFile keyword 126 127 152 Spearman rank correlation coefficient 79 specification 38 126 Splus functions attach paste getenv SCOCOHOME 175 Functions sep 150 attach paste unix echo SCOCOHOME Functions sep 135 base 137 coco end 136 coco init 136 coco load 135 136 coco resume 136 coco start 136 compute deviance 137 compute test 137 current 137 dyn load2 136 enter names names levels missing NULL 136 enter table table 136 Is pos 2 135 make base a 137 make current a 137 make current coco a 136 print all models print TRUE 137 read model a 136 return base 137 return base number 137 return current 137 return current number 137 return vector type
51. command 132 155 174 SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET DECOMPOSABLE MODE command 98 118 120 121 156 DIARY command 33 124 127 155 DUMP command 64 128 155 ECHO command 132 155 EM EPSILON command 51 52 129 156 EM INITIAL command 51 129 156 EM ITERATIONS command 51 52 129 156 EXACT EPSILON command 77 78 93 102 103 157 EXACT TEST command 13 76 78 93 100 102 103 156 EXACT TEST FOR PARTS command 103 EXACT TEST FOR PARTS OF TEST command 77 83 93 102 156 EXACT TEST FOR TOTAL TEST command 77 84 93 102 103 156 EXACT TEST FOR UNPARTED command 103 EXACT TEST FOR UNPARTED TEST command 77 84 93 102 156 FAST command 77 78 93 102 103 157 GRAPH MODE command 133 156 Graphical SEARCH command 118 Hierarchical SEARCH command 119 HUGE command 129 131 IC command 100 101 INPUTFILE COMMANDS command 126 155 INPUTFILE DATA command 5 38 126 155 INPUTFILE OBSERVATIONS command 38 126 155 INPUTFILE SPECIFICATION command 38 126 155 INTERRUPT command 133 155 SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET SET Index IPS command 129 156 IPS EPSILON command
52. edge 86 Chapter 6 Tests 6 5 3 Factorization of Test into Tests of One Edge Missing If C is a decomposable submodel of a decomposable model B and the two models differ by k edges then there exist sequences B Ap Aj Ax C of models such that for i 1 k Aj_1 and A differ with exactly one edge and A is decomposable The likelihood ratio statistics can then be computed by 2logQ 2 Xon nin 1og BE tAElA Petia Paria DA i A BAK i ia ES TON fags Pun PEO tAElA i PA ta 2 Y Y nar Basa I 1 k ia Ela PAX 5 2 log Q l 1 k The likelihood ratio for the test between the two decomposable models is by the command FACTORIZE ONE EDGE set 1 factorized into a product of likelihood ratios between decomposable models that differ with exactly one edge The sequence of models is not unique Choose between different orders of edges with SET ALGORITHM A B C before FACTORIZE ONE EDGE Algorithm A and B add edges to the current model by testing for decomposability by the algorithm TestForZeroFillIn of Tarjan amp Yannakakis 1984 Algorithm B tries the edges in the opposite order as algorithm A Algorithm C removes edges from the base model by looking for edges only in one generator If the optional argument set is given then the edges are tried according to the order of the factors in set The following example shows a factorization of the test of one of the final models from Edwards a
53. estimated by the IPS algorithm The list of tests is disposed of when reading Q tables see SET REUSE OF TESTS On Off in later sections section 6 6 and chapter 11 3 7 Grouping of Factor Levels Cutpoints and Re define Factors REDEFINE FACTOR factor name of observed levels 4 of missing levels CUTPOINTS factor name cutpoints If a factor has too many levels or is continuous the number of levels can be reduced with the command CUTPOINTS factor name cutpoints between reading specifications and observations Note that the tests in CoCo do not take into account an ordering or scaling on the levels Observations have to be entered as a list if cutpoints are to be used Before using CUTPOINTS factor name cutpoints the factor with name factor name has to be 44 Chapter 3 Reading Data declared with number of levels equal to number of groups that CUTPOINTS factor name cutpoints result in The number of cutpoints must be the number of result ing levels minus one The first group of the resulting factor will consist of values less than or equal to the first cutpoint values greater than the first cutpoint and less than or equal to the second cutpoint will be grouped in the second group etc Commands CUTPOINTS factor name cutpoints can be placed on the file with the list of observations and is read when the command READ DATA or READ OBSERVATIONS is used see the beginning of this chapter If two CU
54. example one of the final models from Edwards amp Havr nek 1985 and analyzed in the introduction is tested against the saturated model gt read model gt read model ACE ADE BC F gt test Test of F BC ADE ACE against ABCDEF Statistic Asymptotic 2log Q 62 0779 P 0 0994 Power 59 7593 P 0 1396 X 2 59 9956 P 0 1350 DF 49 gt The command TEST performs a test of current model against the base model The power A is by default set to 2 3 It is changed by the statement SET POWER LAMBDA A If 1 is entered the power divergence is not printed if the test output is in table format See the FACTORIZE and BACKWARD statements 6 1 2 Goodman and Kruskal s Gamma SET ORDINAL set If two factors are tested conditionally independent e g by the command TEST by the model editing commands described in chapter 7 or by procedures for model selection then Goodman and Kruskal s Gamma coefficient is computed besides the deviance Pearson s chi square y and the power divergence 2nJ gt Factors are declared to be ordinal by the command SET ORDINAL set 6 1 3 Dimension and Degrees of Freedom The degrees of freedom D F for the test of C under B is computed as the difference between the dimensions dim Pg and dim Pe for the two models D F dim Pg dim Pe The dimension for a model with generating class A is computed by 1 If A a is complete i e the generati
55. for your running version of CoCo are printed with STATUS Limits A 3 1 DOS PC The number of declared factors and factors used in the models is limited to 64 2 The maximum number of levels for each factor is set to 60 2 4 All marginal tables of observations are placed in an integer array N array with 32766 215 2 cells The estimated probabilities are placed in a real array P array with 10922 215 3 cells You are then able to compute log L for a model with 64 binary factors if the largest clique in the graph for the model has 14 vertices at the most and there are no more than 13 vertices in the largest non decomposable atom Except in the IPS algorithm all real computations are performed in double precision 8 bytes 15 to 16 significant digits Estimated probabilities are stored in normal real 6 bytes 11 to 12 significant digits If estimated probabilities instead are stored in double real the number of cells in the P array is reduced from 10922 215 3 to 8190 21 2 On the other hand if probabilities are stored in single real 4 bytes 7 to 8 significant digits the size of the P array can be increased to 16380 214 4 cells The maximal number of observations is set to 65532 216 4 It can be increased to 2147483644 2 4 at the cost of one dimension The size of the N array is then reduced from 32766 21 2 to 16383 21 1 If you are displeased with any of the limits please l
56. he Le 132 gt Contents 12 CoCo in other systems 133 12 A A RE 134 12 2 XLISP STAT CoCo o 137 12 2 1 Association Diagrams 000008 ee 137 12 2 2 Block Recursive Models 000 138 12 23 Miscellaneous cuide 4 ye ee ob DD REE ae a 138 12 2 4 A Tutorial Example o o 139 III Appendix 143 A Installing CoCo 145 Aad Availability rr Bote AAA te oe RAMS BO Shook oo PPE ke eA 145 AD Installation si beach aana e Beko eth RR IS A 146 Ads DOSYOn PRE o d aiun oe ee ee eb rs 2 ee 146 2 27 Macintosh oo a aise hbk ee A keke eS eee e e e 146 A 2 3 Unix On Workstations 02004 146 A 3 Limitations and Precision 2 0 e 149 ALS IN DOSER Ol niet Beh ket OA Seo oe tae See eee 149 A32 Macintoshi e cotas wa a A ae BA as 150 A 3 3 Unix Workstations o e e a 150 A 4 CoCo Datasets and Examples o e o 150 Acon Reporting Errors alitas O A os 150 NCGS IWAL da A dl de o gh OG 151 B Reference Manual section 152 B 1 Quick Reference Card e 152 B 1 1 End and Restart o 152 B 1 2 Help and Quick reference card 152 B L3 Abbreviations ier ir 24 4 aca we e RAR UY Ye oa A 152 Bila SAUS goad db ae OL Ge A a OR 152 B 1 5 Input from keyboard or file 152 B L 6 gt Input files 2 24 a eA ee a o oo A 153 BLT Onitput tiles ceci e a Goes D2 ESS ae 153 B 1 8 Diary Timer ete uu ee ee e
57. in the b marginal table times 100 are printed Then Row Column Total percents or percents of any other marginals table are printed E g print table observed AB print table observed AB B print table observed ABC A print table observed ABC BC To print Total percents let the b be the empty set print table observed AB Missing values If ExcludeMissing is On and the argument a of the command PRINT TABLE value name a is not a subset of the factors with complete information then an action similar to EXCLUDE MISSING In aU AgUAc is performed before printing the table where Ag is the factors of the base model and Ac is the factor of the current model Analogously for other commands of this chapter Consider using the command EXCLUDE MISSING In a before the command PRINT TABLE value name a 58 Chapter 4 Description of Data and Fitted Values 4 4 Printing Sparse Tables CASE LIST a With CASE LIST a the observed counts in the marginal table determined by the set a is printed For each cell with non zero count the cell count and an identification of the cell is printed The list printed by CASE LIST when the EM algorithm is not used can be read as an accumulated list by the keyword Accumulated list see section 3 4 3 See also section 4 7 4 5 Describing Tables DESCRIBE TABLE Uniform Normal Rankit value name a The DESCRIBE TABLE procedure gives a
58. keyword 51 latent variables 49 less program 36 level of significance 99 101 libscoco so file 150 likelihood ratio test statistic 2log Q 51 72 77 79 132 limits 34 151 linear trend 79 LIST command 10 39 55 57 61 158 list function 141 List keyword 39 41 45 49 163 loadcoco sp file 150 log 127 Log keyword 55 61 93 162 LogFile keyword 36 127 147 152 LogFileName keyword 36 main effects 65 111 MAKE ALIAS command 34 154 MAKE BASE command 13 66 158 MAKE CURRENT command 13 66 158 make coco function 140 make graph base model message 143 make graph current model message 143 Mantel Haenszel x 79 marginal association 111 McNemar s test of symmetry 79 Mean keyword 51 Index measures of association 79 MEET OF MODELS command 50 66 93 95 96 125 160 MENU NUMBER command 33 34 154 missing values 49 51 model class See model selection model control See model selection model dynamic spinning plot 139 model editing commands 93 model formulae 67 model manager 140 model number 12 66 model search See model selection model selection backward elimination 105 coherence 107 110 criteria 99 105 115 decomposable mode 98 118 decomposable search EH procedure 118 stepwise model selection 98 EH base model 116 EH procedure 113 decomposable search 118 graphical search 117 hierarchical search 118 initial models 113 118
59. levels for the factor 6 are marked as missing then let Zs 1 2 Zs denote the unmarked levels and let Tf 1 2 Zs denote the total set of lev els TIA Zs Z5 1 Z5 2 Zf is the set of marked levels The counts n t ier Z T Xscal is the total table The counts n i i J Ta XsecaZo is the table with only cases with complete information The cells Z Z have at least one level marked as missing Let p i be the maximum likelihood estimate on the table n i4 i ez for a model M4 where the union A of variables in the model is a subset of the declared variables A i is the maximum likelihood estimate for the same model but on the full table n 2 iez with no effect of variables in A A corresponding to the distribution on A A being uniform Then p i Blia Zac The a marginal estimated probability Dlia E jrjamia BU For any subset a of the declared variables A we have Pliava Blia Zanacl and P t P taua Z auayc Pliava Zacnac For a subset a of the declared variables A let B be the smallest set of variables onto which the model Ma is collapsible Asmussen amp Edwards 1983 and such that aNACBCA Recall that Ma is collapsible onto B if Pa tB Paltz 54 Chapter 4 Description of Data and Fitted Values MA is collapsible onto B if and only if in the 2 section graph for the model the boundary O c of every connected component c of the complement BY o
60. of the forward selection step and such a model is inserted in the model list for each step of the forward selection Models with a p value or minus the IC value greater than or equal to the value set by the command SET ACCEPTANCE alpha are accepted and models containing accepted models can be accepted by Coherence in forward selection When the commands SET Aic SET Bic SET IC Kappa integer or SET IC On are used then both the acceptance and the rejection limit are set to 0 If the acceptance limit is set to a value lower than the rejection limit then the rejection limit is also set to the entered value Analogously if the rejection limit is set to a 8 3 Exact Tests in Tests and Model Selection 101 value greater than the acceptance limit then the acceptance limit is set to the entered value Model control Some further model checking is possible If the two models to be tested against each other have common decompositions then the test can be partitioned The command SET PARTITIONING On of the last section in this chapter will control this partitioning of tests The deviance for the total test is then the sum of the deviances for each part of the test If a limit components limit is set by SET COMPONENTS components limit default 1 0 then tests with a p value lower than components limit for any part of the test are rejected in the backward elimination and forward selection If two variables are accepted conditiona
61. of the power divergence is set by the command SET POWER LAMBDA Null lambda If the command SET POWER LAMBDA Null is used or the argument A is set to 0 or 1 then the power divergence 2n1 is not computed and not printed for computed tests The adjusted number of degrees of freedom is only computed if AdjustedDF is set On by SET ADJUSTED DF On Off On by default If reuse of tests is on set by the command SET REUSE OF TESTS On Off then changing options by the commands SET ADJUSTED DF On Off J and SET POWER LAMBDA Null lambda will not result in changes in the used test statistics for already computed tests See 6 6 and use SHOW TESTS or DISPOSE OF TESTS 8 2 2 Selecting Test Statistic and Significance Level SET ORDINAL set SET TEST Lr Chi Power SET Aic Bic Ic Kappa kappa Ic On Ic Off In the model selection models are accepted or rejected according to the limits set by the commands SET ACCEPTANCE alpha and SET REJECTION alpha The test statistic to compute the p value or IC value from is selected according to the option set by the command SET TEST Lr Chi Power unless two ordi nal variables are tested conditionally independent Goodman and Kruskal s Gamma coefficient is used when two ordinal variables are tested conditionally independent also when an information criterion is selected Ordinal variables are declared by the command SET ORDINAL set see als
62. outputfile diary sod make alias set outputfile report sor make alias read data rd make alias read specification rs make alias read observations rovi make alias read model rm make alias print formula pf make alias print all models pam These abbreviations can be placed on the file e g my alias and read by the command SOURCE file name SOURCE my alias See also the next section about 34 Chapter 2 Commands the local parser The file cocorc COCO SRC under DOS is read when CoCo is started See also the chapter 11 about source files 2 4 Reading a Local Parser READ LOCAL PARSER If you are running a public CoCo on workstations and declaring new abbreviations a local version of the parser table is generated in the current directory Next time you start CoCo up in that directory the local version of the parser table with your abbreviations can be read in with READ LOCAL PARSER The following two commands are of no interest to the normal user INIT PARSER NO ACTION INIT PARSER is used by the author to create a new parser table from the file INIT TAB With the command NO ACTION CoCo enters a mode where commands are read but no action is performed Exit this mode by END 2 5 Sets and Models A set is entered as a list of factor names enclosed in sharp parentheses e g the set with the three variables A B and C is entered as ABC A model generating class e g in the command READ MODEL gc
63. send graph 1 send graph 1 send graph 1 send graph 1 vertex position vertex position vertex position vertex position vertex position vertex position wal gu mAN pl E pun list list list list list list 0 20 o 20 35 0 30 40 40 0 30 40 Change the color on the vertex labels for all graphs send graph 1 item color vertex label red Move the vertex labels for vertex C and E redraw redraw redraw redraw redraw redraw send graph 1 vertex label position E list 3 4 send graph 1 vertex label position C list 11 2 0 redraw T Add control buttons to graph 1 send graph 1 add controls Let graph 3 have same vertex positions as graph 1 send graph 3 slot value positions send graph 1 slot value positions A Model Manager is created by nil nil nil nil nil T 142 Chapter 12 CoCo in other systems setf manager return manager The following will create a Model Dynamic Spin Plot for graph 1 The val ues in the plot are re evaluated and the plot is redrawn when the graph for the spin plot receives one of the two messages make graph current model and make graph base model let a send this graph return vector adjusted model current b send this graph return vector adjusted model base c send this graph return vector observed
64. several calls to the procedure might give different results and a sample of models with all edges significant to a selected level of significance can be generated Add only the most significant edge interaction Most FORWARD By default if a headlong forward selection is not performed all significant edges or interaction terms are added in each step of the forward selection With the command Most FORWARD only the most significant edge or interaction term is added in each step of the forward selection 9 4 3 Graphical or Non Graphical Models This section is analogous to the section with the same name for the command BACKWARD Edges FORWARD Edges With the command FORWARD Edges the 2 section graph for the current model is created In turn each edge not present in this 2 section graph is then added to the 2 section graph and the current model is then tested against the model with the generating class described by the cliques in the resulting graph By FORWARD Edges all pairs of factors without an edge in the graph for the cur rent model are tested conditionally independent given the rest factors of the model With FORWARD Edges and a current model with only main effects can be read by the command READ MODEL all first order marginal associations are tested Interactions FORWARD Interactions FORWARD Interactions will for each minimal interaction term generator in the dual representation of the current model not pr
65. started the files cocorc COCO SRC under DOS of the home directory of CoCo and cocorc COCO SRC under DOS of the current directory are read by CoCo as a source file 11 3 Output files Besides on the standard output file CoCo can write on a diary a dump file a log file and a report file Use STATUS Files to see the current or default file names 126 Options for Controlling Input Output and Algorithms 11 3 1 Standard Output Sink SET OUTPUTFILE RESULTS file name SET OUTPUTFILE COMMANDS file name On DOS the output from CoCo seen on the screen is divided into two parts 1 results from procedures and 2 output from the parser The output from procedures can then be redirected to a printer without loosing the prompt etc on the screen with SET OUTPUTFILE RESULTS LPR If you want to see what is going on on the screen Dump the reply from the parser somewhere SET OUTPUTFILE COMMANDS garbage and get the diary on the screen SET DIARY On SET OUTPUTFILE DIARY CON On Unix the two commands SET OUTPUTFILE RESULTS file name and SET OUTPUTFILE COMMANDS file name are the same and will redirect the standard out put to the file file name The abbreviation SINK file name for SET OUTPUTFILE RESULTS file name is available 11 3 2 Diary SET OUTPUTFILE DIARY file name SET KEEP DIARY On Off SET DIARY On Off J A diary of everything shown on the screen can be kept with the command SET DIARY O
66. step are eligible in the next step of the recursive forward selection together with the edges visited in the current step with a p value found between the two limits If not the keyword Coherent is used then of course all edges not in the resulting model are eligible in the next step If the p value or minus the IC value for a model in this stepwise edge or inter action selection is greater than the value set by SET ACCEPTANCE alpha then the model is accepted i e the edge or interaction term is not added By coherence these accepted edges are then not considered in sub sequential steps of the forward selection Consider the edges with a p value less than or equal to the value set by SET REJECTION alpha in a step of the forward selection Then in each step of the forward selection these edges are added to the model of the current step and the 110 Chapter 9 Stepwise Model selection Backward and Forward resulting model is used in the next step The resulting model of each step is inserted in the model list The recursive forward elimination continues only as long as there are found edges or interaction terms with a p value greater than this rejection limit Only if the p value for the edge to be added is smaller than this rejection limit a new model is inserted in the model list Not to favour the removal of edges with e g a low lexicographical order the edges in this headlong forward selection are visited in a random order Hence
67. stored test or if the current number of tables is equal to 0 You may force CoCo to compute new exact p values for the same test without disposing of the list of tests by changing the number of generated tables e g from 1000 to 999 Stored values of 2Log Q may depend on factorization and partitioning Note that if a test is computed by factorization or partitioning then the sum of x statistics is only approximately the x statistic for the full test Analogously with power divergence If a test is reused it is annotated with Reuse or an r in ShortTestOutput Tests in a sorted list of tests are always marked by r Chapter 7 Editing Models with Tests Only GENERATE DECOMPOSABLE MODEL Only GENERATE GRAPHICAL MODEL Only DROP EDGES edge list Only ADD EDGES edge list Only DROP INTERACTIONS gc Only ADD INTERACTIONS gc Only MEET OF MODELS Only JOIN OF MODELS mmama ee ee ee The commands in this chapter edit the current model If not the keyword Only is used a test is computed The computed test statistics will depend on the use of SET ORDINAL set and SET ADJUSTED DF On Off and on the value set by SET POWER LAMBDA Null lambda See the following chapter Exact tests might be selected by SET EXACT TEST On All Deviance Off If ExactTest is selected then the commands SET ASYMPTOTIC p value SET NUMBER OF TABLES Null number SET LIST OF NUMBER
68. the adjusted residuals in the marginal table BCDEF Print a rankit plot describe table rankit adjusted BCDEF Make a plot of the adjusted residuals against observed counts plot observed adjusted Read the saturated model read model Name this model current and then the current model base current base Choose exact test and compute also exact test P value for Pearson s chi spuare set exact test all Let CoCo decide on the number of tables to generate set number of tables null Ignore non decomposable models set decomposable mode on Do not compute the adjusted number of degrees of freedom set adjusted df off 28 Chapter 1 Introduction to CoCo Do not compute power divergence set power lambda null Change the size of a page to fit into this guide set page formats 72 256 Change the test formats set test formats 8 3 6 4 Do a recursive backward elimination where tests are done against the previously accepted model follow and edge exclusions once rejected remain rejected coherent Print only the sorted list of tests only sorted only sorted recursive coherent follow backward Print all models the sequence of accepted models print all models Name the last accepted model current current Name current model base base
69. the current model is graphical else False IS GRAPHICAL gc with the argument gc will return True if the model gc is graphical else False Analogously with IS DECOMPOSABLE and IS DECOMPOSABLE gc True is returned if the model is decomposable IS SUBMODEL OF without any arguments will return True if the current model is a submodel of the base model IS SUBMODEL OF gc with the argument gc will return True if the model gc is a submodel of the current model IS SUBMODEL OF gc1 gc2 with the arguments gci and gc will return True if the model gc1 is a submodel of the model gcz The command IS IN ONE CLIQUE factor factor with the two arguments factor and factor will return True if the edge factor factorg is an edge of the current model IS IN ONE CLIQUE factor factorg gc with the argu ments factori factor2 and gc will return True if the edge factor factor2 is an edge of the model with generating class gc 5 6 Common Decompositions PRINT COMMON DECOMPOSITIONS DECOMPOSE MODELS set The command PRINT COMMON DECOMPOSITIONS prints common decompositions for the current and base models If the current and the base models are both decomposable with respect to a they are decomposed with the command DECOMPOSE MODELS a This will result in 4 new models which are added to the model list See also section 6 5 1 Partitioning a
70. the entered specification 3 3 Ordinal Variables SET ORDINAL set With this command the factors of the subset set of the factors are declared to be ordinal Then when two factors are tested conditionally independent Goodman and Kruskal s Gamma coefficient is also computed The argument set has to be a set with the names of the ordinal factors 3 4 Observations Observations can either be entered as a table or as a list The table is read cell by cell in a standard cell order A list is a list of cases each case consisting of the levels of the factors in the same order as the factors are defined or some values to be recoded by the command CUTPOINTS factor name cutpoints Characters including and when no cutpoints for the variable reading is declared not a digit are interpreted as separators between integers The characters and are also abbreviations for the lowest level marked as missing The integers must be greater than or equal to 1 and less than 30 if not CUTPOINTS factor name cutpoints is used see section 3 7 The 6 cases in the 2 x 2 x 3 table 3 1 factors declared with one of the first two specifications shown in the previous section can be entered as 40 Chapter 3 Reading Data C 1 1 0 0 C 2 0 1 C 3 2 0 0 Table 3 1 A simple data set 3 4 1 Table Table 1000 0011 1200 or as 3 4 2 List List 111 222 113 21 21 x 122 When readin
71. 0 0003 Exact 1000 0 0000 0000 AE 2 25 887 0 0000 25 828 0 0000 o BC 1 688 782 0 0000 638 896 0 0000 BE 1 20 740 0 0000 20 767 0 0000 BF 1 4 732 0 0296 4 772 0 0289 Exact 1000 0 0300 0 0300 DE 2 18 319 0 0001 18 298 0 0001 Exact 1000 0 0000 0 0000 TIME 0 135secs 23 CURRENT Models BE BC AE AC CEBE BC AE R CADE CAE DEJ AC ADE BC BE CEAC AD BC BE DE BE BC AE AC CEBE AE AC BE BC AE AC CEBC AE ACT BF B FJ R ADEJ AD AE 24 gt gt No edges should be added to the found model gt gt forward Edge DF 2log Q P X72 P AB 3 6 177 0 1033 6 162 0 1040 AF 1 1 388 0 2388 1 389 0 2385 BD 2 1 397 0 4974 1 396 0 4976 CD 2 1 082 0 5823 1 082 0 5823 CE 3 7 365 0 0611 7 456 0 0587 CF 2 2 290 0 3182 2 426 0 2974 Exact 100 0 3300 0 3100 DF 1 1 057 0 3038 1 062 0 3027 EF 2 2 405 0 3004 2 427 0 2972 Exact 100 0 2700 0 2700 TIME 0 214secs gt gt Print all models gt gt print all models Model no 10 AC ADE BC BE BF BASE Model no 9 BF BC AC ABE ADE Model no 8 ABC ABE ADE BF Model no 7 ABC ABE ABF ADE Model no 6 ABCF ABE ADE Model no 5 ABCF ABEF ADE Model no 4 ABCF ABEF ADEF Model no 3 ABCEF ADEF Model no 2 ABCEF ACDEF Model no 1
72. 05 0 0000 23 929 0 0000 R ABC AB BC Exact 1000 0 0000 0 0000 AD 2 16 542 0 0003 16 427 0 0003 R ADE AE DE Exact 1000 0 0000 0 0000 AE 3 26 077 0 0000 25 991 0 0000 CADE ABE DE BE AD AB BC 2 682 296 0 0000 631 416 0 0000 R ABC AB AC Exact 1000 0 0000 0 0000 BE 2 20 929 0 0000 20 930 0 0000 R ABE AB AE Exact 1000 0 0000 0 0000 BF 1 4 732 0 0296 4 772 0 0289 R BF B F Exact 1000 0 0300 0 0300 DE 2 18 319 0 0001 18 298 0 0001 R ADE AD AE Exact 1000 0 0000 0 0000 TIME 0 076secs gt forward 22 Chapter 1 Introduction to CoCo Edge DF 2log Q P X 2 P Models AF 2 2 658 0 2647 2 648 0 2661 R ABF AB BF Exact 100 0 2300 0 2300 BD 4 2 504 0 6439 2 502 0 6443 R ABDE ABE ADE Exact 20 0 7000 0 7000 cD 2 1 084 0 5817 1 084 0 5817 CABC ABE ACD ADE ABC ABE ADE CE 4 7 516 0 1110 7 602 0 1073 R ABCE ABC ABE Exact 1000 0 1010 0 0940 CF 2 2 290 0 3182 2 426 0 2974 R BCF BC BF Exact 100 0 3300 0 3100 DF 1 1 054 0 3045 1 059 0 3035 ABE ADE BF DF CABE ADE BF EF 2 2 405 0 3004 2 427 0 2972 R BEF1 BE BF Exact 100 gt 0 2700 0 2700 TIME 0 072secs gt gt Try to drop the significant interaction ABC gt gt drop interaction ABC 9 BF BC AC ABE ADE Model is not graphical C
73. 0598 Dimension 991 TIME 0 037secs gt print common decompositions Decomposition of ABI ACEHILP CEHILOP EILPQ CEHIW HLMO DHMO LPQR EHVW EHJ LNO UVW GIK and ABI ACEHILP CEHILOP EILPQ CEHIW EHJW HLMO DHMO LPQR EHVW LNO UVW GIK Decompositions ABCEGHI JKLMNOPQRUVW 19 HMO 3 DHMO 4 ABCDEGHI JKLMNOPQUVW 19 LPQ 3 LPQR 4 ABCDEGHIKLMNOPQRW 17 EHW 3 EHJUVW 6 ABCDEGHI JKLMOPQRUVW 19 LO 2 LNO 3 ABCDEGHI JKLMNOPQRVW 19 VW 2 UVW 3 ABCDEHI JLMNOPQRUVW 18 I 1 GIK 3 Selected ABCDEGHIKLMNOPQRW 17 EHwW 3 EHJUVW 6 TIME 0 040secs gt set adjusted off Adjusted df set OFF TIME 0 002secs 6 4 Computation of log L in Large Models 81 gt find 2log q Test of ABI ACEHILP CEHILOP EILPQ CEHIW HLMO DHMO LPQR EHVW EHJ LNO UVW GIK against ABI ACEHILP CEHILOP EILPQ CEHIW EHJW HLMO DHMO LPQR CEHVW LNO UVW GIK 2log Q 2 7202 4580 7211 0598 17 2037 DF 1003 991 12 P 0 1417 TIME 0 006secs gt set adjusted on Adjusted df set ON TIME 0 001secs gt find 2log q Test of ABI ACEHILP CEHILOP EILPQ CEHIW HLMO DHMO LPQR EHVW EHJ LNO UVW GIK against ABI ACEHILP CEHILOP EILPQ CEHIW EHJW HLMO DHMO LPQR CEHVW LNO UVW GIK 2log Q
74. 09 112 124 Only keyword 65 93 94 97 106 110 Optimal prediction lambda Aasym 79 OR REJECT CASES command 48 157 172 OR SELECT CASES command 48 157 ordering 44 overlays 34 overview graph 140 P keyword 76 131 152 PageFormats mode 93 pager program 36 partial association 109 PARTITIONING command 133 PARTITIONING TEST command 50 80 83 86 125 159 Partitioning mode 77 83 84 109 112 Patefield s method for exact tests 76 102 PATH file 3 Pausing mode 36 125 Pearson product moment correlation coefficient 79 Pearson residuals See standardized residuals Pearson s chi square x 72 Pearson s chi square x 79 Percents Row Column and Total 58 Phi and maximum of Phi 79 PLOT command 10 55 57 60 61 158 power divergence 2n1 72 Power keyword 162 PREVIOUS MENU command 33 34 154 PRINT command 67 PRINT ALIAS command 34 154 PRINT COMMON DECOMPOSITIONS command 71 80 82 84 86 159 PRINT FORMULA command 67 133 159 PRINT MODEL command 66 80 83 159 PRINT OBSERVED command 5 57 PRINT STANDARD TABLE command 58 158 PRINT TABLE command 10 43 55 57 58 64 125 158 Index PRINT VERTEX ORDERING command 67 70 133 159 Probabilities keyword 162 Q list command 44 Q list keyword 163 Q table keyword 163 q tables 42 quasi independence conditional 42 quit 3 r 92 R dual 114
75. 1 3 Compiled with gcc a C compiler for Sun4 Friday March 17 12 00 00 MET 1995 Compile time Mar 17 1995 13 00 00 Copyright c 1991 by Jens Henrik Badsberg Licensed to Diary set ON gt read data 64 cells with Finding all marginals 0 083secs TIME 1841 cases read 1 2 An Example gt Change the size of a page to fit into this guide gt gt gt set page formats 72 52 TIME 0 000secs gt gt Describe the observed counts in the saturated table gt print also a uniform plot gt gt describe table uniform observed DESCRIBE TABLE Observed NUMBER OF CELLS IN TABLE 64 NUMBER OF INVALID CELLS 0 NUMBER OF CELLS TO DESCRIBE 64 SORTED LIST 0 0000 1 0000 2 0000 2 0000 3 0000 3 0000 3 0000 4 0000 4 0000 4 0000 4 0000 4 0000 5 0000 5 0000 5 0000 7 0000 7 0000 7 0000 7 0000 7 0000 8 0000 9 0000 9 0000 9 0000 9 0000 11 0000 11 0000 12 0000 12 0000 13 0000 13 0000 14 0000 14 0000 14 0000 14 0000 16 0000 16 0000 17 0000 17 0000 21 0000 23 0000 23 0000 24 0000 25 0000 32 0000 33 0000 35 0000 40 0000 44 0000 50 0000 51 0000 57 0000 63 0000 66 0000 67 0000 67 0000 70 0000 73 0000 80 0000 80 0000 109 0000 112 0000 129 0000 145 0000 STATISTICS NUMBER OF VALUES 64 25 RANK 16 VALUE 7 0000 RANK 17 VALUE 7 0000 50 RANK 32 VALUE 14 0000 RANK 33 VALUE 14 0000 75 RANK 48 VALUE 40 0000 RANK 49 VALUE 44 0000 MINIMUM 0 0000 MAXIMUM 145 0000 RANGE 145 0000 MAX R
76. 12 80 33 109 67 7 9 23 32 70 66 50 80 7 13 24 25 73 57 51 63 7 16 5 7 21 9 9 17 1 4 4 35 11 8 14 17 5 2 7 3 14 14 9 16 2 3 4 O 13 11 5 14 4 4 enter table n read model read model ACDE ABCF read model ACE ADE BC F compute test compute deviance 12 2 XLISP STAT CoCo 137 make base 2 print all models x110 plot return vector 0 return vector 8 xlab Observed counts return vector 0 ylab Adjusted residuals return vector 8 coco resume print all models end coco end q0 12 2 XLISP STAT CoCo The system Xlisp CoCo A Tool for Graphical Models is on UNIX systems obtained by loading CoCo dynamically into XLISP STAT This feature is only tested on Sparc See Part III Part III of this thesis is a guide to Xlisp CoCo See this guide for further docu mentation Some of the highlights of the system are 12 2 1 Association Diagrams The association diagram is a graph window with the independence graph of an as sociation model The model can be causal or not and the variables can be discrete ordinal continuous etc The independence graph association or interaction graph is a simple undirected graph with as many vertices as the table has dimension A vertex for each variable Two vertices are adjacent in the independence graph for the model unless the two variables are conditionally ind
77. 129 156 IPS ITERATIONS command 129 156 KEEP DIARY command 127 155 KEEP DUMP command 64 128 155 KEEP LOG command 36 127 155 KEEP REPORT command 128 155 KEYBOARD command 38 126 154 KEYBOARD On command 4 LARGE command 39 82 129 130 157 LIST OF NUMBER OF TABLES command 102 LIST OF NUMBERS OF TABLES command 77 93 102 156 LOG command 127 155 MAIN EFFECTS command 115 116 160 NOTE command 132 155 NUMBER OF TABLES command 76 77 92 93 102 156 OPTION command 132 155 ORDINAL command 40 73 93 99 157 OUTPUTFILE COMMANDS command 127 155 OUTPUTFILE DIARY command 127 155 OUTPUTFILE DIARY CON command 127 OUTPUTFILE DUMP command 64 128 155 OUTPUTFILE LOG command 127 155 OUTPUTFILE REPORT command 128 155 OUTPUTFILE RESULTS command 127 155 PAGE FORMATS command 6 57 59 61 125 155 PAGING LENGTH command 36 125 155 PARTITIONING command 83 93 102 103 156 PAUSE command 36 125 155 Index SET POWER LAMBDA command 57 73 93 99 156 SET PRINT FORMATS command 59 64 125 155 SET READ command 49 50 115 130 136 157 SET READ All command 49 SET READ Subset command 49 SET REJECTION command 99 101 108 110 156 SET REPORT command 132 155 SET REUSE OF TESTS command 44 92 93 99 156 SET Rough SEARCH command 114 SET SEED command 51 77 78 93 102 103 108 156 SET SEPARATORS
78. 139 Model Manager The model search by association diagrams may generate numerous windows with graphs and to handle these windows the model manager has been made The model manager is an overview graph where each point in the graph corresponds to a model Tests can be performed by dragging edges between points in the model manager Model Selection with Resampling To show the advantages of including CoCo in XLISP STAT a method for resampling from the case list and do a backward elimination on each sample has been imple mented The edges of the association diagram can be drawn with a width growing with the reciprocal of the p value or growing with BIC the Bayesian information cri terion But we can also let the width of the edges be proportional to the number of times the edges are found in the final models of headlong backward eliminations on random subsets of the cases This method is implemented in a few lines of Lisp code and the full implementation is shown in part III Xlisp CoCo A Tool for Graphical Models of the thesis as an example of what can be made by the end user in XLISP STAT extended with CoCo 12 2 4 A Tutorial Example This tutorial example is also given in Part III of this thesis The data used is that of table 1 1 from Reinis et al 1981 also used in the introduction Start XLISP STAT with the script xlisp coco to set up some environment vari ables needed by CoCo and to load the Lisp functions for inte
79. 140 0000 gt 150 0000 gt 160 0000 COUNTS Cell Number of Cumm of total count cells with number of number of count cells cells loaa gt E o 1 1 0 0156 1 1 2 0 0156 2 2 4 0 0312 31 31 7 0 0469 4 5 12 0 0781 5 3 15 0 0469 7 5 20 0 0781 8 1 21 0 0156 9 4 25 0 0625 11 2 27 0 0312 al 4 o 67 2 56 0 0312 70 1 57 0 0156 73 1 58 0 0156 80 2 60 0 0312 109 1 61 0 0156 112 1 62 0 0156 129 1 63 0 0156 145 1 64 0 0156 do Oo0o0DO0OO0O0O0OoOooOoOo PROOOOOOO 10 Chapter 1 Introduction to CoCo TIME 0 126secs The hierarchical log linear interaction model described by the generating class gc is read by READ MODEL 9c for example gt read model AB BC Observed counts estimated counts probabilities and residuals absolute adjusted standard Freeman Tukey etc can be printed in tables or described with the com mands e g PRINT TABLE Probabilities set and DESCRIBE TABLE Adjusted Residuals set or the values can be listed or plotted against each other with LIST set and PLOT 2 value name y value name set See chapter 4 for details gt gt Read model ACE ADE BC F gt gt read model ACE ADE BC F TIME 0 017secs Change the page size gt set page formats 72 36 TIME 0 000secs gt gt
80. 2secs gt print formular P ABE ADE BF BC AC I ABCDEF NCI B 1 N I BF yo a N I AE 1 N I AB o 1 N I ADE de Stee N I ABE E E P AB BC ACI I ABC 1 0000000 TIME 0 005secs gt drop edge AB 4 L AC ADE BC BE BF Model is graphical Graph is not decomposable Generating class for Fill In ACE ADE BCE BF Test of AC ADE BC BE BF against ABCDEF DF 21log Q P x 2 P Models 49 58 2814 0 17094 57 5445 0 18832 ABCDEF BF BE BC ADE AC TIME 0 029secs 5 4 Formulas 69 gt current TIME 0 001secs gt print formular P AC ADE BC BE BF I ABCDEF N I B 1 N I BF FNT 1 N I AE 7 1 N I ADE 7 1 P BE BC AE AC I ABCE 1 0000000 TIME 0 004secs gt quit TIME 0 000secs TOTAL TIME 0 187secs The model ADE ABE ABC BF is the resulting decomposable model from the coherent backward elimination with exact tests on data from Reinis et al 1981 The closed form expression for the maximum likelihood estimate for the cell probabilities is n iagc N taBE n iape Nise nliap tar n ip n Plia PRINT VERTEX ORDERING shows that a perfect vertex elimination ordering is F D E C B and A If the vertices are removed from the graph in that order then the boundary C V around the removed vertex in the rest of th
81. 3 Exclude Missing in Subset Use only Cases with Com plete Information for a Specific Subset of Factors EXCLUDE MISSING In set 50 Chapter 3 Reading Data Cases with complete information in a particular subset of the factors may be con sidered with the command EXCLUDE MISSING In set One can then print describe etc marginal tables of the set marginal table The command finds the set marginal table of observed counts with cases with complete information for the factors set The status of EXCLUDE MISSING On Off is unchanged ExcludeMissing has to be on before use of EXCLUDE MISSING In set The considered models have to have generating classes with union of factors a subset of the set set used in EXCLUDE MISSING In set Commands resulting in a test will cancel the EXCLUDE MISSING In set command 3 9 4 EM algorithm EM ON SET EM INITIAL Uniform First Last Mean Random Input SET EM EPSILON epsilon SET EM ITERATIONS maz SET DATASTRUCTURE All Necessary File Large Given the data and a model the probability of each level for a missing factor at a case can be estimated by the EM algorithm CoCo is told to use the EM algorithm by EM ON If Report is on the likelihood ratio test statistics are reported for each step in the EM algorithm If Trace is on the probability for each cell for each case is reported for each step in the EM algorithm The data structure Necessa
82. 54 0 0790 0 0804 9 97146 30102 02336 24443 01471 26905 78057 47972 59560 07528 Chapter 6 Tests Models CEBC AC AB CEAC BC AF F A CAB AF BF CAF AB LCD ACE ADE CLABC ABF ADE ACE LAB AC BC LAB BF AF ACD ACE ADE ADE ACE CD CDE ACD ACE ADE ADE ACDE ACE ACD ADE ACE ACD CDE ACDE ABF ABC CEACE ADE F BC Note that the edge lt BF gt and interaction lt ABC gt are significant Most is the deviance is due to those two interactions 6 6 Reuse of Tests 91 6 6 Reuse of Tests SET REUSE OF TESTS On Off SHOW TESTS DISPOSE OF TESTS All performed tests are stored and re used i e each performed test is pushed into a structure and when a test is to be performed this structure is searched for a test between the two models of the test This feature is turned on and off with SET REUSE OF TESTS On Off The total list of performed tests is printed or disposed of with SHOW TESTS and DISPOSE OF TESTS respectively The list of tests is disposed of when the data is changed by reading observations or Q tables Also the exact p values are re used Exact tests are reused if the current number of tables SET NUMBER OF TABLES Null number is equal to the number of tables generated to compute the
83. 6 1 3 Dimension and Degrees of Freedom 621 42 Adjusted DEl 202 2 4 2 be gee ee a A Aa A A A 0 23 ERAGE PSUS isis Ah dias a atar corte ier AAA Sy sas tatty 6 3 Measures of Associations 6 4 Computation of log L in Large Models o 6 5 Miscellaneous Commands for Model Control 6 6 6 5 1 Partitioning and Common Decompositions 6 5 2 Testing One Edge 2 2 2 2 0 02 ee ee eee 6 5 3 Factorization of Test into Tests of One Edge Missing 6 5 4 Factorization of Test into Tests of One Missing Interaction Reuses0f Tests ae eee kA a oo Sa Re ee ie a 7 Editing Models with Tests 7 1 7 2 7 3 Generate Graphical Model o e e Add Fill In Generate Decomposable Model Drop Cd Sess e to a o A amp cata le eo vii 48 48 49 49 49 50 52 52 54 56 58 58 59 60 64 64 65 65 66 70 70 71 71 71 72 72 73 75 78 79 82 82 85 86 89 91 viii 10 Contents TE Add Edg s geam ra A G ee e Oe ae G8 Drop Intetaction lt 2 2 ead aa Re a A a 7 6 Add Interaction 0002000 eee ee eee Tat Meetior Models 2 ee eee eae oe Se et ee te ae a 28 JO Of Models oc sre nite Sapna te bday Rok aa eA re 7 9 Miscellaneous Deprecated Commands 7 10 Edit the Models without Test Only o Controlling the Computed Test Statistics in the Model Selection Pro
84. 99 101 105 112 122 125 133 160 backward elimination See model selection BASE command 12 66 94 158 Base keyword 56 61 162 Bayesian information criteria See Information Criteria BIC See Information Criteria Bilirubin 45 46 BINDIR environment 149 150 Block recursive models 139 167 button 97 CASE LIST command 41 59 61 62 158 cell counts 56 cell values 55 Chain graph models 139 CLEAN DATA command 42 43 158 cliques 65 closed form expression for maximum likelihood estimates 67 cmdtool program 36 Cochran Armitage Trend Test 79 coco lt log file 29 coco N65536 P65535 Q1024 command 131 coco 096000 R32000 T command 131 CoCo file 3 133 coco file 149 COCO program 3 148 CoCo program 3 COCO DAT file 39 148 149 COCO EXE file 148 COCO HLP file 148 149 COCO OVR file 148 coco sparc file 149 COCO SRC file 35 126 coco sun3 file 149 COCO TAB file 148 149 COCO DIA file 127 COCO DMP file 64 128 COCO LOG file 127 COCO RPT file 128 cocoapi S file 134 149 cocoapix I sp file 150 cocograph lsp file 150 COCOHOME environment 149 150 cocometh lsp file 134 150 coherence 101 107 108 principle of 113 coherent See model selection Coherent keyword 107 110 COLLAPSE MODEL command 65 66 158 collapsible 54 66 83 86 Column percents 58 168 complete 65 Complete ke
85. AALBORG UNIVERSITY An Environment for Graphical Models Part 11 A Guide to CoCo Jens Henrik Badsberg Jens Henrik Badsberg 2001 A Guide to CoCo Journal of Statistical Software Volume 6 issue 4 http www jstatsoft org v06 Fredrik Bajers Vej 7 DK 9220 Aalborg Denmark INSTITUTE OF ELECTRONIC SYSTEMS DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Phone 45 98 15 85 22 Telefax 45 98 15 81 29 An Environment for Graphical Models Part Il A Guide to CoCo Jens Henrik Badsberg March 1995 Second edition Jens Henrik Badsberg 2001 A Guide to CoCo Journal of Statistical Software Volume 6 issue 4 http www jstatsoft org v06 Part II of a thesis submitted to the Faculty of Technology and Science at Aalborg University for the degree of Doctor of Philosophy Fredrik Bajers Vej 7 DK 9220 Aalborg st Denmark INSTITUTE FOR ELECTRONIC SYSTEMS DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE Ka Tel 45 98 15 85 22 TELEX 69 790 aub dk This book has been produced with IATFX by the author and has been typeset in Times Roman typeface The book has been prepared on Sun Sparcstations running SunOS UNIX and OpenWindows and it has been printed in PostScript on a Hewlett Packard LaserJet 4m Xlisp Stat is developed by Luke Tierney Xlisp is developed by David Betz TEX is a registered trademark of American Mathematical Society IATFX is develo
86. AB AF ABCDE ABCE ABDE CCABCF ABC ABF ABCDF ABCD ABCF ABCDEF ABCDF ABCDE ABCDEF CLADE ACE F Bc 6 5 Miscellaneous Commands for Model Control 89 6 5 4 Factorization of Test into Tests of One Missing Interac tion If C is a submodel of B then sequences B Ap Aj Ay C of models exist which for i 1 k Aj_1 and A differ with exactly one interaction term The likelihood ratio statistics can then be computed by 2logQ 2 Y n ia log PB ta Be ia tAElA i i 2 Y ni a log Pelia Ba ia DA alla Pan ia PA ia P PA2 ia Papa ia Pe ia tAElA 2 Y YE ns 6 I 1 k iA Ela Pa ia 5 2log Q l 1 k Interactions not in the current model but in base are added to the current model one at a time and for each interaction the generated model is tested against the previous model with the command FACTORIZE ONE INTERACTION set If not the optional argument set is given then the interactions are added according to cardinality If set is given then first the first factor in set is added then the second factor in set is added then the interaction among the first two factors in set is tried then the interactions among the first three factors in set are tried according to cardinality etc If all factors are binary the test of current against base is factorized into tests with one degree of freedom Since most
87. ACKWARD and FORWARD Comments are inserted in the source file with the character The following is a copy of the diary edited a little some blank lines removed echo of comments removed page breaks removed and inserted Version 1 3 Friday March 17 12 00 00 MET 1995 Compiled with gcc a C compiler for Sun4 Compile time Mar 17 1995 13 00 00 Copyright c 1991 by Jens Henrik Badsberg Licensed to 14 Chapter 1 Introduction to CoCo gt gt Set diary on gt gt set diary on Diary set ON gt gt Keep the log gt gt set keep log Keep Log set ON gt gt Set timer on gt gt set timer on Timer set ON TIME 0 002secs gt gt Set the input file gt gt set input data reinis81 dat TIME 0 003secs gt gt Read data gt gt read data 64 cells with 1841 cases read Finding all marginals TIME 0 025secs gt gt Read the saturated model gt gt read model TIME 0 001secs gt gt Name this model current and then the current model base gt gt current TIME 0 001secs gt base TIME 0 001secs 1 2 An Example 15 gt gt Choose exact test and gt compute also exact test P value for Pearson s chi spuare gt gt set exact test all Exact test set ON Only Exact Deviance set OFF TIME 0 001secs gt gt Let CoCo decide on the number of tables to generate gt gt set number of
88. ARD Edges the 2 section graph for the current model is created In turn each edge in the 2 section graph is then dropped and the model with the generating class described by the cliques in the resulting graph is tested against the base model By BACKWARD Edges all pairs of factors with an edge in the graph for the current model are tested conditionally independent given the remaining factors of the model Interactions BACKWARD Interactions 108 Chapter 9 Stepwise Model selection Backward and Forward BACKWARD Interactions will for each maximal interaction term in the current model remove the interaction term and then test the resulting model against the base model This is computed by in turn adding the interaction terms for the cur rent model to the dual representation for the current model With BACKWARD Interactions and a current model with only n th order effects READ N INTERACTIONS n or READ N INTERACTIONS n set all n th order partial associations are tested For sparse contingency tables Whittaker 1990 advocates the use of first order log linear models i e hierarchical log linear models with at most two factor interac tions The model with all two factor interactions can be read by the command READ N INTERACTIONS 1 and then the backward elimination from this model can be performed by the command Recursive BACKWARD Interactions By default the command BACKWARD is the same as the command BACKWARD Edges
89. D Missing values can in CoCo be handled in the following three ways e Skip all cases with missing values This is selected by the command SKIP MISSING Cases with missing values among the subset of variables to be read set by SET READ Subset set are then skipped during the reading of cases e Use all cases without missing values for relevant factors After the command EXCLUDE MISSING On all cases without missing values for relevant factors are used in each test This is turned off again by the com mand EXCLUDE MISSING Off and all cases are used The command EXCLUDE 3 9 Missing Values Incomplete Observations 49 MISSING In set will exclude the cases with variables having levels marked as missing for any of the variables in the set set of factors The command is used after entering the data e Use the EM algorithm to estimate the values of unobserved factors This is selected by the command EM ON after entering the data The current implementation is tested on Fuchs 1982 for situations where factors at random are unobserved and on Dawid amp Skene 1979 with a latent variable i e a variable unobserved for all cases 3 9 1 Skip Missing Read only Cases with Complete Informa tion SKIP MISSING If the command SKIP MISSING is used between reading specification and data only cases with complete information in the subset to be read are read Cases with variables with values marked as missing among the subset va
90. DATASTRUCTURE All Necessary File Large SET LARGE On Off J SET SORTED On Off These commands control the selected data structure see the chapter 11 Options 3 2 Specification The specification declares the number of factors order of factors and name and number of levels non missing and missing levels for each factor Factor names in CoCo consist either all of a single character or names of any length can be used If factor names consist of only a single character then the name can be any character except slash E ga z A Z 0 9 and A 54 55 5 and Some of the special characters may be reserved in later versions of CoCo Note that CoCo distinguishes between lower case and upper case letters when reading factor names The characters and parentheses lt gt and can be used as factor names but should not since it would cause trouble when reading sets and models If names consisting of more than one character are to be used then all names must be declared with as the first character The specification starts either with the key word Factors or with the key word Names 3 2 1 Factor by Factor Factors If Factors is found a list of items separated by and ended by is expected The number of items defines the number of factors and the order of the items defines the order of factors used when reading data Each item consists of a n
91. EF ABCDE ABCDF ABCDF ABCD ABCF ABCDE ABCE ABDE CABCF ABC ABF CABDE ABE ADE CABCE ABC ACE ABF1 LAB AF ABC1 CAC Bc CCAF LA FJ ABCDEF1 F ADE ACE BC 88 gt set number of tables null TIME 0 001secs gt set factorize a TIME 0 002secs gt dispose tests TIME 0 001secs gt factorize one edge Test of ACE ADE BC F against ABCDEF Edge DF 21log Q P AB 2 5 988 0 0501 Exact 1000 0 0430 AF 1 1 069 0 3012 Exact 100 0 3500 BE 4 6 771 0 1485 Exact 200 0 1650 BD 4 2 504 0 6439 Exact 20 0 6000 BF 2 6 321 0 0424 Exact 1000 0 0430 cD 8 4 893 0 7705 Exact 20 0 6500 CF 4 7 018 0 1349 Exact 100 0 2400 DF 8 9 199 0 3252 Exact 100 0 3100 EF 16 18 316 0 3052 Exact 100 0 3600 49 62 078 0 0994 Exact 200 0 1750 TIME 5 645secs gt quit TIME 0 000secs TOTAL TIME 5 581secs 17 61 953 070 944 502 349 800 459 086 820 983 o0o00O0O0O0O0O0O0O0000000000n0S gt P 0510 0460 3010 3500 1389 1550 6443 6000 0418 0400 7801 6500 1135 1900 3346 3300 3341 3500 1008 1200 Note that the edge lt BF gt is significant Chapter 6 Tests Models aBc11 AC Bc AF11 CCA F CCABCE ACE ABC CCABDE ADE ABE ABF11
92. IC value is minimized The command SET IC Kappa kappa changes the constant used in computa tion of the information criterion and sets the computation of this criterion on The constant is to 2 if the command SET Aic is used and is set to log total number of cases if the command SET Bic is used When ExcludeMissing is On the number of used cases may differ for the different parts of a partitioned test and the value BIC can not be computed Note that the commands SET Aic SET Bic SET IC Kappa integer or SET IC On besides choosing an information criterion also sets both the acceptance and the rejection limit to 0 Asymptotic or Exact p values SET EXACT TEST On All Deviance Off Exact p values are used according to whether exact test are chosen by the command SET EXACT TEST See the next section about exact tests Acceptance and Rejection of Models SET ACCEPTANCE alpha SET REJECTION alpha SET COMPONENTS p value 100 Controlling the Computed Tests Rejected models seeped models S l model rejected by Undetermined edge Edge removed h in BACKWARD Considered in the next Model added to the SO EREE T de step of BACKWARD model list in BACKWARD Edges added in FORWARD and FORWARD Super model accepted by is de coherence in FORWARD E g 5 E g 20 p value st E g 0 0 E g 3 0 IC Rejection limit Acceptance limit Figure 8 1 Significance levels for stepwise model selection and th
93. Incomplete Tables If the cells B C 2 1 B C 1 2 and B C 2 3 in the previously used ex ample are structurally zero this can be specified by using the character for structural zero cells when entering the observations as a table Table 14 gt 2 24 1 L2 A Or after reading the observations without specifying structural zero cells the struc tural zero cells can be specified by the READ Q TABLE set table command gt READ Q TABLE ABC 1100 0011 1100 O or is entered for structural zero cells 1 for non structural zero cells Since a cell in this table is structural zero if and only if the corresponding cell in the BC marginal table is zero this table can be specified by gt READ Q TABLE BC 10 o 1 10 If more than one marginal table for structural zero cells is specified then a cell in the full table is set to structural zero by CoCo if and only if the corresponding cell in one of the marginal tables is zero Thus 1At the time of writing this guide the table entered by READ Q TABLE set table should be a table of integers This should not be much of a restriction since the table can just be multiplied with a factor 42 Chapter 3 Reading Data gt READ Q TABLE AB gt READ Q TABLE BC 10 11 10 o 1 10 is equivalent to gt READ Q TABLE ABC 1000 0011 1000 or since all the counts in the non zero cells in the example are 1 Table 1 saa e 8 A If a Q table can b
94. MBER OF TABLES See chapter 6 for details Note that if the power divergence 2n14 or Pearson s chi square x is selected by the command SET TEST Lr Chi Power and exact p values are only computed for the likelihood ratio test statistics because of the use of the command SET EXACT 102 Controlling the Computed Tests TEST Deviance then the asymptotic p value will be used in the model selection procedures Please disregard the following options before you is familiar with exact tests in CoCo The options SET EXACT TEST FOR TOTAL TEST SET EXACT TEST FOR PARTS and SET EXACT TEST FOR UNPARTED will control computing of exact tests for respectively the total of a test that can be partitioned the parts of a partitioned test and for tests not partitioned The command SET PARTITIONING of the following section will control partitioning of tests that is partitioning of the two models by common decompositions and thus collapsibility of tests The command SET SEED seed Random will set the seed for the random number generator used when generating the exact test The command SET SEED Random will set this seed to some random value depending on the process number of the CoCo session and the total used computing time The value set by SET EXACT EPSILON e is to do with round of errors and one of two algorithms to compute the exact p value by is selected by SET FAST See chapter 6 for further details 8 4 Partitioning SET PARTITI
95. ONING On Off SET ALGORITHM A1BICID The command SET PARTITIONING of the following section will control partitioning of tests that is partitioning of the two models by common decompositions The tests by the command TEST the model editing commands and in model selection are then collapsed to the smallest tables on which the tests can be performed When ExcludeMissing is On the number of used cases may differ for the different parts of a partitioned test The command SET ALGORITHM will set an option to choose between different algorithms for factorization computation of the Gamma coefficient exact p value fill in etc See chapter 6 and the online help information of CoCo for further details Chapter 9 Stepwise Model selection Backward and Forward This chapter is on stepwise model selection in CoCo The incremental search in CoCo is performed by forward inclusion Dempster 1972 and backward elimination Wermuth 1976 of edges or interaction terms In the backward elimination of edges in graphical models edges are sequentially eliminated from the independence graph In the forward selection the edges are sequentially added to the independence graph Wermuth 1976 considers stepwise edge elimination on decomposable models Two resent books Christensen 1990 and Whittaker 1990 each contain a chapter on model selection In Whittaker 1990 with the title Graphical Models the model selection is of course treat
96. ORMATS format width decimals SET PAGE FORMATS line length page length Print formats for tables of cell values The format of the numbers printed in the table produced by PRINT TABLE is controlled by the command SET TABLE FORMATS width decimals prob decimals expt decimals residual The values set by the arguments width and decimal residual of this command are also used in the procedure DESCRIBE TABLE Print formats for test statistics in tables The print width and number of decimals of the statistics and probabilities in tables of tests are set with the command SET TEST FORMATS width statistic decimals statistic width probabil ity decimals probability With the command SET SHORT Off you get detailed output from the com mands GENERATE DECOMPOSABLE MODEL GENERATE GRAPHICAL MODEL DROP EDGES edge list ADD EDGES edge list DROP INTERACTIONS gc ADD INTERACTIONS gc MEET OF MODELS JOIN OF MODELS DROP FACTOR factor REMOVE TOTAL INTERACTION set REMOVE GENERATOR set REDUCE GENERATOR set FORWARD EDGES INTERACTIONS BACKWARD EDGES INTERACTIONS FACTORIZE ONE EDGE FACTORIZE ONE INTERACTION and PARTITIONING TEST e i not in ta ble format Use SET SHORT On to get the results of these commands in a table Print formats for all other commands The command SET PRINT FORMATS format width decimals sets the width and number of decimals used when printing
97. OTIC p value exact tests are computed Default 1 i e exact tests for all tests if ExactTest is on SET NUMBER OF TABLES Null number sets the number of tables to gen erate in each exact test Default 1000 If the number of tables is set to 0 by SET NUMBER OF TABLES Null or SET NUMBER OF TABLES 0 then first 20 tables are gen erated Let m be the number of these generated tables with deviance greater than or equal to the observed deviance If m is equal 0 1 or 2 then the exact test will be based on 1000 additional random tables if m is equal 3 4 or 5 then the exact test will be based on 500 additional random tables if m is equal 6 7 or 8 then the exact test will be based on 100 additional random tables and if m is greater than 8 then the exact test will be based on 20 additional random tables This variable number of tables to generate can be changed by SET LIST OF NUMBERS OF TABLES initial number list of pairs The first argument is the ini tial number of tables to generate The second argument is a list of pairs where the first item of each pair is the upper limit of m and the second item is the number of tables to generate The list is terminated by a pair where the first item is greater than or equal to the initial number of generated tables E g the default list is entered by SET LIST OF NUMBERS OF TABLES 20 2 1000 5 500 8 100 20 20 Please disregard the following options before you is familiar with ex
98. Off The following sections will describe how the specification and observations have to be formatted When reading from file the file with data specification and observations is named and reset with the command SET INPUTFILE DATA file name If specification and observations are not in the same file the two files are named and reset with SET INPUTFILE SPECIFICATION file name and SET INPUTFILE OBSERVATIONS file name After naming files the specification and observations are read with READ DATA without arguments If data are read from key board the specification and observations are expected as arguments to READ DATA specification observations 37 38 Chapter 3 Reading Data If data selection is to be done it has to be declared between reading specifica tion and observations so READ DATA specification observations 1 has to be divided in to the two actions READ SPECIFICATION specification 1 and READ OBSERVATIONS observations See section 3 8 on data selection The command READ LIST list of cases 1 is the same command as READ OBSERVATIONS observations except that the key word LIST is not expected Analogous to READ NAMES name list READ FACTORS factor list and READ TABLE table On the key words Names List Factors and Table see the following sections Specifications and observations are defaults read from the file COCO DAT in the home directory for CoCo SET
99. S OF TABLES list SET SEED seed Random SET EXACT EPSILON e and SET FAST On Off should be noted See the section 6 2 about exact tests in chapter 6 Tests Partitioning of tests is controlled by SET PARTITIONING On Off The commands SET EXACT TEST FOR TOTAL TEST On Off SET EXACT TEST FOR PARTS OF TEST On Off and SET EXACT TEST FOR UNPARTED TEST On Off will control for which parts of tests to compute exact tests Reuse of test is controlled by SET REUSE OF TESTS On Off and SET NUMBER OF TABLES Null number The output might depend on the op tions TestFormats ShortTestOutput PageFormats Report Trace Timer Diary Log etc Finally the convergence criterion for the IPS and EM algorithm choice of how to handle missing values and data structure will have effect on the computing time 92 7 1 Generate Graphical Model 93 7 1 Generate Graphical Model With GENERATE GRAPHICAL MODEL the 2 section graph for the current model is made The generated model is the model with the generating class equal to the cliques in the 2 section graph for the current model the maximal interaction terms are the cliques in the 2 section graph for the current model The current model stays current The generated model is added to the model list It can be disposed of with DISPOSE OF MODEL Last or it can be named current with the command CURRENT If base and current are the same models then
100. TAT program 149 150 XLISPSTAT environment 150 Yates corrected x 79 Yule s Q 79 Yule s Y 79 zero partial association 65 76 Zero keyword 56 162 Zeros by structure fixed See q tables
101. TPOINTS factor name cutpoints commands for the same factor name are found on the observation file the first of them is used and the others ignored If the specification and the list of observations are written on files it is annoying to have to edit the file with specification to experiment with different numbers of cutpoints for a specific factor The specification has to be edited since the number of cutpoints must be equal to the number of declared levels minus one But the declared number of levels can be changed by the command REDEFINE FACTOR factor name of observed levels 4 of missing levels after reading the specification but before using the CUTPOINTS factor name cutpoints command and reading data The number of non missing levels is set to of observed levels and the number of levels marked as missing is set to 4 of missing levels for factor factor name The total number of levels is set to 4 of observed levels of missing levels Consider the following artificial data file liver where cutpoints for the continuous variable Bilirubin are placed on the data file Jaundice No 1 Mild 2 Moderate 3 Severe 4 Not coded 5 Bilirubin in umol 1 0 lt 1 umol 1 900 00 gt 1000 umol 1 999 99 No measurement Missing on form Factors J41 B61 Cutpoints B 0 50 00 100 00 200 00 899 99 900 00 di lt T 2 1 50 00 umol 1 3 50 01 100 00 umol 1 4 100 01 200
102. TS OF ALL NECESSARY SERVICING REPAIR OR CORREC TION IN NO EVENT I AND OR ANY OTHER PARTY WHO MAY MODIFY AND REDISTRIBUTE CoCo MAY BE LIABLE TO YOU FOR DAMAGES INCLUD ING ANY LOST PROFITS LOST MONIES OR OTHER SPECIAL INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BE ING RENDERED INACCURATE OR LOSSES SUSTAINED BY THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH PROGRAMS NOT DISTRIBUTED BY ME THE PROGRAM EVEN IF YOU HAVE BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES OR FOR ANY CLAIM BY ANY OTHER PARTY Appendix B Reference Manual section B 1 Quick Reference Card B 1 1 End and Restart END RESTART B 1 2 Help and Quick reference card HELP command name abbreviation MENU NUMBER a NEXT MENU CURRENT MENU PREVIOUS MENU B 1 3 Abbreviations PRINT ALIAS command name abbreviation MAKE ALIAS command name abbreviation DELETE ALIAS abbreviation READ LOCAL PARSER INIT PARSER NO ACTION B 1 4 Status STATUS Specification Observations Files Formats Fix Tests Exact Ips Em EH Limits Other B 1 5 Input from keyboard or file SET KEYBOARD On Off 152 B 1 Quick Reference Card B 1 6 Input files SET SET SET SET INPUTFILE SPECIFICATION file name INPUTFILE OBSERVATIONS file name INPUTFILE DATA file name INPUTFILE COMMANDS file n
103. Technology University of G teborg G teborg Sweden Wermuth N 1976 Model search among multiplicative models Biometrics 32 253 264 Wermuth N amp Lauritzen S L 1983 Graphical and recursive models for contin gency tables Biometrika 70 537 552 Wermuth N amp Lauritzen S L 1990 On substantive research hypotheses condi tional independence graphs and graphical chain models with discussion Journal of the Royal Statistical Society Series B 52 21 72 Whittaker J 1990 Graphical Models in Applied Multivariate Statistics Wiley Chichester Index 39 162 lt 39 gt 3 39 y Gamma 79 k Kappa 79 A Symmetric optimal prediction lambda 79 A Modified asymmetric optimal prediction lambda 79 Aasym Optimal prediction lambda 79 39 162 39 162 Tp Kendall s Tau b 79 Te Stuart s Tau c 79 A 39 162 n th order effects 109 n th order partial associations 109 33 39 162 39 162 Y 141 39 162 39 162 5 35 39 41 49 59 62 115 162 39 86 162 35 39 40 162 39 42 43 55 162 1 163 2log q keyword 162 35 39 41 49 162 EXE file 148 cocorc file 35 126 4 39 162 163 39 40 163 tmp CoCo diary no pid number file 127 tmp CoCo dump no pid number file 64 128 166 tmp CoCo log no pid number file 127 tmp CoCo report no pid number file 128 36 39 162
104. UN 5 MODE 4 0000 lt Eps 1 SUM X 1841 0000 MEAN 28 7656 SUM 1 X HARMONIC MEAN PROD X 0 0000 GEOMETRIC MEAN 0 0000 SUM X MEAN 2 7 096E 04 VARIANCE 1108 6794 SUM X MEAN 3 3 956E 06 SKEWNESS 1 6744 SUM X MEAN 4 4 113E 08 KURTOSIS 5 2282 UNIFORM PLOT Unit horizontal 2 5000 o0o00O0O0O0O0O000O0O000000000000000000000000000000o 00000 97500 95000 92500 90000 87500 85000 82500 80000 77500 75000 72500 70000 67500 65000 62500 60000 57500 55000 52500 50000 47500 45000 42500 40000 37500 35000 32500 30000 27500 25000 22500 20000 17500 15000 12500 10000 07500 05000 02500 OE 00 NO ND N XxXN RN N NX XNN 2 lx 12 1 Chapter 1 Introduction to CoCo 4 4 100 0000 0 0000 25 0000 50 0000 75 0000 125 0000 Observed 1 2 An Example HISTOGRAM Unit horizontal 1 Unit vertical 10 0000 0 0000 gt gt kA k kkk 10 0000 gt Fae a a a 2 a kk k kkk kkk kk 20 0000 gt akak ak ke ak k k k 30 0000 gt 40 0000 gt x 50 0000 gt xx 60 0000 gt xx 70 0000 gt ok 80 0000 gt xx 90 0000 gt 100 0000 gt 110 0000 gt ox 120 0000 gt 130 0000 gt
105. User s Manual 2 Commands 2 1 2 2 2 3 2 4 2 5 2 6 2 1 2 8 Online HEGE igre grete on a aa a a Comments Tegosa ots ek a e he Be Make Delete and Print Alias o e Reading a Local Parser o 00500 Sets and Models tesi ais de da a A e e Ae e 2 5 1 Variable names of length longer than one character Pausing and Front End Processor History Mechanism Print formats Tests Diary Source Sink Log etc Interrupts ci a een Sa a ea AA A A Se ey 3 Reading Data 3 1 3 2 3 3 3 4 Keyboard and Files veda a aS REG a a Specification cc en Ad dd doe A o da Se 3 2 1 Factor by Factor Factors o o 3 2 2 Attributes one by one Names o a OrdinalVariables stes tias e See tow E ER ee oak OBSEEVALIONS is A A A ht et E RRR BE SD EA SAGA Table sarera Be be dt dde bec be oy ob Als ee eo te BPS Ss IS a Re ee a a e 3 4 3 Accumulated List o e 00002000 Replacing the Observations with a Random Data Set Initial values to the PS algorithm o o 3 6 1 Structural Zeros Incomplete Tables 2 Grouping of Factor Levels Cutpoints and Redefine Factors Data S lection 22 54 2 Gee Ace ee See A A BE tee i 3 8 1 Select or Reject Cases During the Reading of Data vi 30 32 32 33 33 34 34 35 35 36 36 Contents 3 9 3 8 2 Read only a Subs
106. a sub process in emacs On DOS use the F3 key SET PAUSE On Off J SET PAGING LENGTH number of lines At some commands and especially on some fast machines it has become a problem to reach to read output from CoCo before it scrolls out of the top of the screen With Pausing set on by SET PAUSE On Off J Default Off the output is paused for each written 22 lines after last NewPage if EndOfLine is found after the total command is read by CoCo E g if Pausing is on then CoCo will pause for each printed 22 lines after STATUS If Pausing is on then CoCo will not pause during the status command if STATUS or STATUS READ MODEL A B is entered The next 22 lines are printed by pressing Return If a is entered when pausing then the command will finish without pausing The number of lines to print between pausing are set by SET PAGING LENGTH number of lines Default 22 Note that not only printing but also computation in CoCo are paused It is not recommended to pipe the output from CoCo into pagers like more or less since you will not see entered characters to CoCo before you have typed return Use cmdtool run CoCo as a sub process in emacs use Control S and Control Q or set the diary on and use some pager on the diary A log of the CoCo session is kept Unless you use the command SET KEEP LOG On or change the LogFileName the LogFile is removed on Unix when CoCo termi
107. ach other have no common decomposition If the deviance is wanted for the test of a large current against base then print first common decompositions with the command PRINT COMMON DECOMPOSITIONS see the next section If the two models have a common decomposition with respect to the set a they are decomposed into 4 new models B Bz C and Cy with DECOMPOSE MODELS a The 4 models can be seen with PRINT MODEL A11 The test statistic can then be computed as the sum of the statistic for the test of C against 6 and of the statistic for the test of C2 against Bo The test procedures of CoCo performs possible common decompositions as the com mand PARTITIONING TEST before computing the test statistics except when using the command TEST Example of computation of log L for test between two 23 dimensional models gt set input data user jhb Examples CoCo ex7a dat TIME 0 002secs 80 Chapter 6 Tests gt read specification TIME 0 004secs gt skip missing TIME 0 001secs gt read observations 710 cases read TIME 0 721secs gt set large Large set ON TIME 0 002secs gt read model ABI ACEHILP CEHILOP EILPQ CEHIW EHJW HLMO amp gt DHMO LPQR EHVW LNO UVW GIK TIME 0 004secs gt find log 1 Log L 7202 4580 Dimension 1003 TIME 0 206secs gt read model ABI ACEHILP CEHILOP EILPQ CEHIW HLMO amp gt DHMO LPQR EHVW EHJ LNO UVW GIK TIME 0 003secs gt find log 1 Log L 7211
108. act tests in CoCo If ExactTest and Partitioning are on then exact tests are computed for de composable models that differ with one edge If Partitioning is on and the de composable models differ with more than one edge then the computation of exact tests for each part of the test is controlled by SET EXACT TEST FOR PARTS OF TEST On Off J Default On and the computation of exact tests for the total test is controlled by SET EXACT TEST FOR TOTAL TEST On Off J Default On If Partitioning is off and the decomposable models differ with more than one edge then the exact test is only computed if SET EXACT TEST FOR UNPARTED TEST On Off ison Default On If Fast is on SET FAST On Off Default Off then for each sufficient marginal table of counts generated each edge in the list of edges given by FACTORIZE 6 2 Exact Tests 77 ONE EDGE a term to the deviance is computed and for each generation of all suffi cient marginals for the base model the terms for the deviances are added and com pared with the observed deviance else for each generation of all sufficient marginals for the base model the deviance between the base model and the current model is computed This is the fastest for decomposable models which only differ with few edges but if the models differ with many edges the computing time is the smallest if fast is off In later versions of CoCo the author might have decided what to do and the co
109. actors for coronary heart disease Start CoCo and give the command SET KEYBOARD On to indicate that you want to read data from the keyboard Then type READ DATA CoCo will respond with the prompt DATA gt indicating that CoCo is expecting data Declare the table with Factors A2 B2 C2 D2 E2 F2 and give the keyword Table to tell CoCo that the data is to be entered in table form Then type the 64 cell counts and end with gt keyboard Keyboard set ON gt read data DATA gt Factors A2 B2 C2 D2 E2 F2 DATA gt Table DATA gt 44 40 112 67 129 145 1223 3512 80 33 109 67 7 9 DATA gt 23 32 70 66 50 80 713 2425 73 57 51 63 7 16 DATA gt 5 7 21 9 9 17 1 4 4 3 11 8 14 17 5 2 DATA gt 7 3 14 14 9 16 2 3 4 0 13 11 5 14 4 4 64 cells with 1841 cases read Finding all marginals gt The table can also be entered as a list of observations See chapter 3 for details on 1 2 An Example 5 this and for details on the declaration of factors The declaration of factors and the table can be placed on a file e g reinis81 dat Factors A2 B2 C2 D2 E2 F2 Table 44 40 112 67 129 145 12 23 35 12 80 33 109 67 7 9 23 32 70 66 50 80 7 13 24 25 73 57 51 63 7 16 5 7 21 9 9 17 1 4 4 3 11 8 14 17 5 2 7 3 14 14 9 16 2 3 4 0 13 11 5 14 4 4 This file is then declared to CoCo by SET INPUTFILE DATA reinis81 dat and then read by READ DATA without arguments CoCo A program for estimation test and model search in very large C
110. akly accepted and weakly rejected models and duals The command STATUS Search will show the accepted models the rejected mod els the A dual the R dual the SearchBase the base model used in the EH procedure the FixIn and the FixOut The command DISPOSE OF EH will dispose of the accepted models the rejected models the A dual and the R dual A simple model search among first decomposable models then among all graphical models and finally among hierarchical models can be performed by EH Decomposable EH Graphical EH Hierarchical The smallest dual in each step is fitted by these three commands If instead rough estimates of the dual sizes should be found in each step and the dual with the smallest rough estimate is to be fitted in each step then the commands SET Rough SEARCH EH Decomposable EH Graphical EH Hierarchical can be used 114 Chapter 10 The EH procedure 10 1 Computed Tests and Selection Criteria Chapter 8 will describe how to set the options for the computed test statistics select the test statistic to classify the models according to and how to set the level of significance 10 2 Fix Edges Interactions FIX In Out edges gc ADD FIX In Out edges gc REDO FIX In Out 10 2 1 FixIn With the command FIX In edges gc edges generators in the generating class are forced into all models in the EH procedure The command FIX In ABC would force the terms lt gt lt
111. ame B 1 7 Output files SET SET SET SET SET SET SET SET SET SET OUTPUTFILE RESULTS file name OUTPUTFILE COMMANDS file name OUTPUTFILE DIARY file name OUTPUTFILE LOG file name OUTPUTFILE DUMP file name OUTPUTFILE REPORT file name KEEP DIARY On Off J KEEP LOG On Off KEEP DUMP On Off KEEP REPORT On Off B 1 8 Diary Timer etc SET SET SET SET SET SET SET SET SET SET SET SET DIARY On Off LOG On Off 1 DATA LOG On Off DUMP On Off REPORT On Off TIMER On Off ECHO On Off NOTE On Off TRACE On Off DEBUG On Off J OPTION On Off number value 1 INTERRUPT A B On Off y B 1 9 Print Formats SET SET SET SET SET SET SET PRINT FORMATS format width decimals TABLE FORMATS width dec prob dec expt dec diff TEST FORMATS 2 width x dec prob width prob dec PAGE FORMATS line length page length PAUSE On Off PAGING LENGTH number of lines SHORT On Off J 153 154 Appendix B Reference Manual section B 1 10 Controlling the IPS and EM algorithm SET IPS Cell Sum SET IPS EPSILON epsilon SET IPS ITERATIONS mas SET EM INITIAL Uniform First Last Mean Random Input SET EM EPSILON epsilon SET EM ITERATIONS maz B 1 11 Partitioning and Factorizes SET PARTITIONING On Off
112. ame a character number of non missing levels an integer and an optional number of levels to be in terpreted as missing values see section 3 9 Missing Values Incomplete observations for example Factors A2 B2 C21 or Factors Sex 2 Smoking 2 Drinking 2 1 3 3 Ordinal Variables 39 3 2 2 Attributes one by one Names These specification could also be entered as Names ABC 222 001 and Names Sex Smoking Drinking 222 00 1 respectively After the key word Names a list of characters giving names and order of factors is expected Then a list of integers giving number of non missing levels for each factor is expected An optional list of number of levels for each factor to be marked as missing can be entered If this list is not entered end the list of numbers of non missing levels with for example Names ABC 22 2 Factor levels are integers from 1 up to the total number of levels number of non missing levels plus the number of levels marked as missing New users are often confused by the above two ways of declaring the variables The form Factors is useful to specify the factors one by one Names is useful if the data is entered as a list The names number of levels and number of levels to marked as missing can then be written as headings to the columns of levels for each factor The form Names is also used when defining a table from S Plus or XLISP STAT see chapter 12 Use STATUS Specification to see
113. and Macintosh for this version of CoCo is available free of charge for non commercial use The source code may only be read and edited for the purpose of porting CoCo to other machines No new features may be added to CoCo and no parts of the program may be included in other systems or new interface procedures to New S S Plus XLISP STAT or any other extendable system may be made without the written permission from the author The source code in C and executable for Sun 4 Sparc both with Lisp code for the graphics and executable of the standalone version of CoCo for PC s and Macintosh can be obtained by anonymous ftp over internet from ftp iesd auc dk or by WWW from http www iesd auc dk pub packages CoCo Or CoCo is available on disks 3 5 or 5 25 High density You should however by prepared to bear the costs of copying e g by supplying a disk or tape and a stamped mailing envelope This guide and the guide to CoCo is also available in TeX and postscript code from the ftp site or printed from Aalborg University If you have problems with CoCo or find bugs please contact the author If possible send on disk or by Email the LogFile for the session which caused the problem I will do my best to solve your problems and provide updates if necessary If you have comments on CoCo suggestions for improvements or new facilities that you feel would make the software more useful to you please feel free to contact me Jens Henrik Bad
114. ared factors by the command READ BASE MODEL model or by giving sets of cardinality one to the FIX Out command But CoCo works more efficiently if the command SET READ Subset 10 2 Fix Edges Interactions 115 subset is used when reading data instead of using FIX Out SET MAIN EFFECTS or READ BASE MODEL commands in the EH procedure 10 2 3 Read Base Model READ BASE MODEL model With the command READ BASE MODEL model a base model is read for the EH procedure Tests in the EH procedure are then performed against the read model It is not the same model as the base model used outside the EH procedure since READ BASE MODEL model effects FixIn and Fix0ut Terms not in the base model are FixOut i e FixOut is set to the dual representation of SearchBase Since there is no distinction between FixOut due to the FixOut command and FixOut due to READ BASE MODEL model and since a large SearchBase would give less FixOut due to SearchBase previous FixOut is cancelled by the use of the command READ BASE MODEL model Use REDO FIX Out to redo previous FixOut 10 2 4 Interaction between FixIn FixOut and SearchBase Effect of FixIn on FixOut FixIn terms are removed from FixOut i e FixIn is added to the normal representation of FixOut the maximal terms in the models Effect of FixIn on SearchBase FixIn has no effect on SearchBase but only edges in SearchBase are Fixed In FixIn is reduced It could have been chosen that t
115. ation area then the size of the probation area is set to one third of the overlay buffer size The option T will cause trace of swapping 11 5 5 Unix Size of N P and Q arrays coco N65536 P65535 Q1024 Options for setting the initial size of N P and Q arrays at run time at Unix If the table spanned by the factors of a test can not fit onto the N array then Pearson s chi square and the power divergence can not be computed and if all suffi cient marginal tables of the two models of a test can not fit into the P array then the adjusted degrees of freedom can not be computed The state space of each irreducible component after adding a fill in has to fit into the P array for CoCo to be able to compute the deviance On Unix size of the N P and Q arrays are increased automatic by CoCo when needed and as long as the size do not get larger than 1048576 27 This upper limit is set to prevent CoCo from allocating too big arrays These big arrays will enable CoCo to compute e g the adjusted degrees of freedom in large models and much computing time will thus be required The deviance can often be found without these big arrays faster If bigger arrays are wanted for computing Pearson s chi square the power diver gence the adjusted degrees of freedom and exact tests select these with the command line options N and P Note that the option N size controls the selected data structure 11 6 Miscellaneous Options for Tracin
116. available in TeX and postscript code from the ftp site or printed from Aalborg University Disclaimer CoCo is an experimental program It has not been extensively tested The author of CoCo the University of Aalborg and any other part assisting the author of CoCo in distributing CoCo take no responsibility for losses or damages resulting directly or indirectly from the use of this program CoCo is an evolving system Over the time new features will be introduced and existing features that do not work may be changed Every effort will be made to keep CoCo consistent with the information in this guide but if this is not possible the help information in CoCo and XLISP STAT should give accurate information about the current use of a command Acknowledgments Many thanks go to Professor Steffen Lilholt Lauritzen who inspired the creation of CoCo Also thanks to David Edwards and Svend Kreiner for useful comments and sug gestions to new features in CoCo Thanks to Flemming Skjoth Soren H jsgaard Bo Thiesson and J rgen Greve especially Flemming Skjgth who in their study patiently used and tested CoCo and found numerous errors Thanks to my sister Annette Badsberg for correcting the English language of ear lier versions of this guide Aalborg Danmark March 15th 1995 Jens Henrik Badsberg Contents I Tutorial 1 Introduction to CoCo 1 1 1 2 Starting and Finishing esa sga a a de An Example ico ee 4 RR REA aed ES we II
117. ble CAB AB AC ADE BC BE BF is not decomposable TIME 12 096secs gt gt Print all models the sequence of accepted models gt gt print all models Model no 8 ABC ABE ADE BF Model no 7 ABC ABE ABF ADE Model no 6 ABCF ABE ADE Model no 5 ABCF ABEF ADE Model no 4 ABCF ABEF ADEF Model no 3 ABCEF ADEF Model no 2 ABCEF ACDEF Model no 1 ABCDEF BASE CURRENT TIME 0 005secs gt gt Name the last accepted model current gt gt current TIME 0 001secs 20 gt gt Name current model base gt gt base TIME 0 000secs gt gt Print all models gt gt print all models Chapter 1 Introduction to CoCo CURRENT LABC1 LAB BC LADEJ CAE DE LABC1 LAB ACI LABE1 LAB AE BF B FJ LADEJ CCAD AE Model no 8 ABC ABE ADE BF BASE Model no 7 ABC ABE ABF ADE Model no 6 ABCF ABE ADE Model no 5 ABCF ABEF ADE Model no 4 ABCF ABEF ADEF Model no 3 ABCEF ADEF Model no 2 ABCEF ACDEF Model no 1 ABCDEF TIME 0 005secs gt gt Can an edge rejected by coherence now be rejected gt gt backward Edge DF 2log Q P X72 P Models AB AC ADE BC BE BF is not decomposable AC 2 23 805 0 0000 23 929 0 0000 Exact 1000 0 0000 0 0000
118. clares individual cells in the set marginal table to be structural zero i e adds structurally zero cells to the set marginal Q table Thus gt READ Q TABLE AB gt READ Q TABLE BC 10 11 10 o 1 10 is equivalent to 3 7 Grouping of Factor Levels Cutpoints and Redefine Factors 43 gt READ Q TABLE BC 10 o 1 10 gt READ Q LIST AB 21 or gt READ Q LIST ABC Data gt 2 1 1 Data gt 1 2 1 Data gt 2 2 1 Data gt 1 1 2 Data gt 2 1 2 Data gt 2 1 3 Data gt 1 2 3 Data gt 2 2 3 Data gt If the a marginal Q table exists when READ Q LIST a list of cells is used then the list of cells is added to the existing Q table If the specified table contains or is a marginal table in an existing Q table a warning message is printed Commands Q list a table and Q list a list of cells can be placed on the file with the list of observations and is read when the command READ DATA or READ OBSERVATIONS is used see the beginning of this chapter DISPOSE OF Q TABLE set DISPOSE OF ALL Q TABLES These two commands dispose of the specified set marginal Q table and all spec ifications of structural zeros respectively Since the estimated probabilities depend on the read Q tables the estimated prob abilities and tests are disposed of when reading Q tables Models are not disposed of but when the estimated probabilities are used after the reading of the Q table the probabilities are re
119. ctor name cutpoints commands 48 Chapter 3 Reading Data 3 8 2 Read only a Subset of Variables SET READ All Subset set SET READ Subset a causes only a subset of the declared factors to be read The command must be used after reading specification and before data selection and before reading observations Data selection can still be performed at all declared factors SET READ A11 will cancel the effect of SET READ Subset a 3 9 Missing Values Incomplete Observations One may want to exclude cases with particular levels at some factors in tests If e g a factor A on the form has the 3 levels 1 Yes 2 No and 3 Not able to decide Yes or No and a fourth level is added during transformation of data from forms to data in computer 4 No information about factor A Then one may want to mark level 3 and 4 as missing values The levels to be marked have to be the levels with the highest numerical value The factor A is declared together with a factor C with a total of 4 levels one of which is to be marked as missing and two binary factors B and D as Factors A22 B2 C31 D2 or Names ABCD 2232 20 When declaring factors the characters and are abbreviations of 1 When reading data in the form List the characters are abbreviations of the lowest level marked as missing For example at the factor A declared in this section is equal to 3 and at the factor C equal to 4 in the following list List a C
120. cussion Scan dinavian Journal of Statistics 16 273 306 Lauritzen S L 1991 The EM algorithm for graphical association models with miss ing data Technical Report R 91 05 Department of Mathematics and Computer Science Aalborg University Denmark Lauritzen S L 1993 Graphical association models DRAFT Technical Report IR 93 2001 Department of Mathematics and Computer Science Aalborg University Denmark Lauritzen S L Dawid A P Larsen B N amp Leimer H G 1990 Independence properties of directed Markov fields Networks 20 491 505 Lauritzen S L amp Wermuth N 1989 Graphical models for associations between variables some of which are qualitative and some quantitative Annals of Statis tics 17 31 57 Lauritzen S L Thiesson B amp Spiegelhalter D J 1992 Diagnostic systems cre ated by model selection methods a case study Technical report The university of Aalborg Institute for Electronic Systems Department of Mathematics and Computer Science Leimer H G 1993 Optimal decomposition by clique separators Discrete Mathe matics 113 90 123 Patefield W M 1981 An efficient method of generating random R C tables with given row and column totals Algorithm AS 159 Appl Statist 30 91 97 Read T R C amp Cressie N A C 1988 Goodness of fit Statistics for Discrete Multivariate Data Springer Verlag New York Reinis Z Pokorny J Basika V Tiserov
121. detailed univariate description of val ues in marginal tables It gives statistics a histogram uniform and quantile quantile plot and if short a list of sorted values and for observed counts a table with extreme cell counts DESCRIBE TABLE Normal Print Normal probit plot Rankit Print Rankit plot Uniform Print Uniform plot Current Use values from current model Base Use values from base model value name Name of value to be described a Describe values in the a marginal table With DESCRIBE TABLE value name a the marginal table determined by the set a and with values computed as in the previous section is described If a sorted list of the values can fit onto two pages it is printed The size of a page is controlled by page length and line length in the statement SET PAGE FORMATS line length page length The mean arithmetic harmonic and geometric variance skewness kurtosis me dian mode minimum maximum and range are computed The command SET PRINT FORMATS format width decimals can be used to change the formats for the printed statistics A histogram and a Uniform plot are printed With the key words Uniform Rankit and Normal immediately after TABLE in the DESCRIBE TABLE statement you can choose between different plots The plots are made to fit a page Let 20 14 q be the sorted list of values to be described In the uniform plot zx is then plotted a
122. dges with a p value or minus the IC value less than or equal to the value set by the command SET REJECTION alpha are rejected Sub models can be rejected by Coherence in the backward elimination Not to favour the removal of edges with e g a low lexicographical order the edges in this headlong backward elimination are visited in a random order Hence several calls to the procedure might give different results and a sample of models with all edges significant to a selected level of significance can be generated The command SET SEED seed Random will set the seed for the used random number generator Follow Recursive Follow BACKWARD Tests are if this keyword is given performed against the previously accepted model In the first step of the Recursive Follow BACKWARD command the tests will be performed against the current model In the next step the tests are performed against the model accepted in the first step etc If the keyword Follow not is given then all the tests will be performed against the base model Remove all non significant edges or interactions in one step All BACKWARD By default only the least significant edge or interaction term is removed All non significant edges are removed by one step of the backward elimination by the keyword A11 i e all non significant edges in the current model is removed from the model 9 3 3 Graphical or Non Graphical Models Edges BACKWARD Edges With the command BACKW
123. e EH procedure SET SEPARATORS p value A hypothesis is accepted if the p value is greater than a given limit Thus to be able to classify models by the same expressions regardless of whether p values or an information criterion is used minus the change in the information criterion is used to classify models If the p value or minus the IC value for a model in the stepwise edge or interaction selection or in the EH procedure is greater than the value set by the command SET ACCEPTANCE alpha then the model is accepted else the model is rejected in the EH procedure In backward elimination edges or interaction terms with a p value greater than this limit are eligible for removal The recursive backward elimination continues only as long as there are edges or interaction terms with a p value greater than this acceptance limit Only if the p value for the edge to remove is greater than this acceptance limit a new model is inserted in the model list Models with a p value or minus the IC value lower than or equal to the value set by the command SET REJECTION alpha are rejected and sub models can be rejected by Coherence in backward elimination Consider in each step of a forward selection all visited edges or interaction terms not in the current model and with a p value lower than or equal to the value set by SET REJECTION alpha and consider the model with these edges added to the current model This model will be the result
124. e Se S 153 B 1 9 Print Formats a ne da ateen y e a a a 153 B 1 10 Controlling the IPS and EM algorithm 154 B 1 11 Partitioning and Factorizes o oo 154 B 1 12 Do only Graph Stuffs only for Debugging 154 B 1 13 Only Decomposable Models in Stepwise Model Selection and in the Global EH Search aoaaa o 154 B 1 14 Computed Test Statistics and Choosing Tests and Significance Level for Model Selection ooa 154 B 1 15 Exact Tests in Tests and Model Selection 154 B 1 16 Read Data without selection etc 155 B 1 17 Read Data Specification e e 155 B 1 18 Read Data Selection 155 Contents xi B 1 19 Skip Cases with Missing Values During Reading Observations 155 B 1 20 Read Data Observations 004 155 B 1 21 Read Data Incomplete table Structural Zeros 156 B 1 22 Select use only cases with complete observations after reading Observations a A UP a A eit dd te 156 B 1 23 Request the EM algorithm 0 0 156 B 1 24 Data Description cn ee es 156 B 1 25 Read Model o 0200002 cee eee 156 B 1 26 Edit Model s tors de ote ee ee Be Se a ew a 156 B 1 27 Moving pointers in the Model list 156 B 1 28 Describe Print and Dispose of Models 157 B 1 29 Common decompositions of models 157 B130 Tests it halk eee ae BS Be eek 157
125. e factorized into a product of Q tables it should be since it saves space and computing time Tf the a marginal Q table exists when READ Q TABLE a table is used then the existing Q table is overwritten If the specified table contains or is a marginal table in an existing Q table a warning message is printed The used Q table will still be the product of all defined Q tables The cell A B C 2 1 3 is structural zero but the observed count in this cell was 2 Such illegal observations are removed with the command CLEAN DATA If illegal observations are not removed computed statistics probabilities expected counts residuals etc will be incorrect A simple test is that READ MODEL PRINT TABLE Probabilities should return the value 1 The PRINT TABLE value name a and DESCRIBE TABLE value name a commands are described in later chapters In tables with structural zeros the fol lowing is useful PRINT TABLE Complete Observed a will print a table with the observed count in the cell if the cell is not structurally zero else PRINT TABLE Zero a will print a table with 0 in the cell if the cell is structurally zero else 1 PRINT TABLE Errors a will print a table with the observed counts in the cell if the cell is structurally zero else DESCRIBE TABLE Errors a will give a description of the observed counts in cells with structural zeroes The command READ Q LIST set list of cells de
126. e graph will be complete Complete V is TRUE if the boundary is complete Removal of the edges DE BF BE BC AD and AC will result in a decomposable model since they are only in one clique Removal of AE or AB will result in a non decomposable model See the output from DESCRIBE CURRENT The model ADE ABE AC BC BF generated by removing the interaction lt ABC gt is not graphical but the 2 section graph is decomposable The model is not graphical since the clique lt ABC gt in the 2 section graph is not in the generating class for the model See the result from DROP INTERACTIONS ABC The expression for the maximum likelihood estimate for the cell probabilities is _ PeaBy ec acy liaec tape N iapde iB Pia n iaB N iag nliB The model ADE AC BC BE BF generated by removing the edge lt AB gt is graphical but not decomposable The graph contains the 4 cycle lt AC gt lt CB gt lt BE gt lt EA gt See the output from DROP EDGES AB The expression for the maximum likelihood estimate for the cell probabilities is _ PuBE BC AE AC tABCE M taDE Mier Plia n iag n ip 70 Chapter 5 Models 5 5 Is Graphical Is Decomposable Is Submodel and Is In One Clique IS GRAPHICAL gc IS DECOMPOSABLE gc IS SUBMODEL OF ge 1 gee 1 IS IN ONE CLIQUE factori factora gc l IS GRAPHICAL without any arguments will return True if
127. e than 14 binary vertices 24 on workstations One model can be tested against another by using partitioning and collapsibility of tests or by computing the deviance for each model by summing over only non zero cells in the sufficient marginal tables as described in chapter 2 of Part I These tests can be performed between any two large or very large models but if the parts of the test involve tests on large tables then only the deviance can be computed and the adjusted degrees of freedom cannot be computed if the parts involve large models The procedures for backward elimination and forward selection of edges in graphical models are still useful on large tables Vari ous runs of model selection by respectively forward selection and backward elimination has been performed on the 121 dimensional table with 10 000 cases of Wedelin 1993 In large tables the observations are placed as a case list on an internal file This crude classification of tables follows the classification of datasets by Huber 1994 Tiny datasets are suitable for black boards a small set can be printed on a few pages a medium set fits on a floppy disk a large dataset requires a tape and a huge dataset requires many tapes Other programs Other programs for analysis of log linear interaction models for contingency tables are DIGRAM Kreiner 1989 and MIM Edwards 1989 CoCo finds closed form expres sion for estimates in decomposable models handles i
128. ed decomposable models Lauritzen 1993 The sequence of edges given by FACTORIZE ONE EDGE is used to generate the sufficient marginal tables for the base model from the observed sufficient marginal tables for the current model An exact test between a decomposable base and decomposable current model is done by the command EXACT TEST Exact tests between models generated by the commands EXACT TEST FACTORIZE ONE EDGE the model editing commands commands for stepwise model selection and the EH procedure are controlled by the following commands SET EXACT TEST On All Deviance Off SET ASYMPTOTIC p value SET NUMBER OF TABLES Null number 76 Chapter 6 Tests SET LIST OF NUMBERS OF TABLES list SET EXACT TEST FOR TOTAL TEST On Off SET EXACT TEST FOR PARTS OF TEST On Off SET EXACT TEST FOR UNPARTED TEST On Off SET EXACT EPSILON epsilon SET SEED seed Random SET FAST On Off The command SET EXACT TEST On A11 Deviance Off with no ar guments sets the computation of ExactTest on or off Default Off With SET EXACT TEST Deviance it is possible to compute exact tests for only the deviance i e the likelihood ratio test statistics 2log Q This does not save much computing time but exact p values for the power divergence 2n1 and Pearson s chi square x are typically less interesting Only if the asymptotic p value is less than the p value set by SET ASYMPT
129. ed in relation to graphical models but also in Christensen 1990 an excellent discussion of model search on graphical model is given Edwards 1993 considers some computational aspects of both the stepwise edge selection and the EH procedure see the next chapter In CoCo the backward elimination and the forward selection can be started from any initial model The backward elimination procedure is not restricted to work on edges but can also perform stepwise elimination of interaction terms Analogously with the forward selection Steps of the backward elimination and the forward selec tion can be mixed together in any order and between each step any set of edges or interaction terms can be added or eliminated The backward elimination from the saturated model has the advantage that the tests are not performed under models later to be found rejected In the forward selection from the uniform model the initial tests will be performed under models likely to be rejected But in the forward selection the initial tests often will be collapsible to small tables and much larger tables can be handled The marginal associations tested in the first step of a forward selection from the uniform model might also be of some interest The resulting models from each cycle of the backward elimination and the resulting model from the forward selection are added to the linked list of models in CoCo 103 104 Chapter 9 Stepwise Model selection Backward and Fo
130. ent model New models can be generated from the current model with the commands DROP EDGES edge list ADD EDGES edge list DROP INTERACTIONS gc ADD INTERACTIONS gc etc see chapter 7 Editing Models with Tests With Only before these model editing commands no test is produced when generating the new model With READ MODEL gc models are entered by their generating classes and CoCo writes models by their generating classes the maximal interaction terms in the log linear interaction models The 2 section graph for the model with generating class C is the graph with vertices e a for some a C and edges e1 e2 C a for some a C The 2 section graph is a simple undirected graph where the edges only contains two vertices The model is graphical if and only if the cliques in the 2 section graph for the model are the generating class C for the model The cliques are the maximal complete subsets of the graph A subset of the graph is complete if all pairs of vertices in the subset are mutual adjacent Two vertices are adjacent if there is an edge between them 64 5 2 The Model List Shifting Base and Current 65 The command DROP EDGES e e2 will add the statement that e1 and e2 are conditionally independent given the rest variables in the current graphical model See Lauritzen 1982 or Whittaker 1990 The command COLLAPSE MODEL set will collapse the current model onto the smallest set containing
131. ependent given the other variables The association diagram is a very appealing working tool for model selection The layout of the graph the association diagram can be edited by dragging vertices points for variables with the mouse and the edges will then follow the vertices New association diagrams can be created from a diagram by adding or dropping edges by the mouse Tests of two variables conditionally independent can be performed by clicking on the edge between the two variables by the mouse Edges are then drawn with a width proportional to the significance of the edges Edges can also be labeled with a value computed from a test statistic for removing the edge e g a p value or an C value Causal models can be handled in the association diagrams Methods for backward elimination and forward selection of edges in association diagrams are implemented In the backward elimination of edges the edges are redrawn with a width proportional to the significance of the edges as they are visited The edge will be drawn with a color depending on whether the edge is fitted or not If the test of an edge is not computed then the edge will be drawn with a color depending on whether the edge is fixed the edge is rejected by coherence removing the edge will result in a non decomposable model etc 138 Chapter 12 CoCo in other systems 12 2 2 Block Recursive Models Block recursive models Chain graph models are models with both symmetric and asym
132. er 59 9956 P 0 1350 0 1350 X 2 59 9956 P 0 1350 0 1350 DF 49 49 TIME 0 912secs gt quit TIME 0 000secs TOTAL TIME 4 889secs 6 3 Measures of Associations The following command will compute lots of statistics for two factors independent etc given other factors SLICE factor name a factor name b set 1 For each slice 2 dimensional table of factor name a by factor name b given configurations of the factors set the following measures of associations statistics is computed and printed For each configurations of the factors in the set set the 2 dimensional table of factor name a by factor name b is printed with margins and on each slice 2 dimensional table the following statistics is computed and printed Fisher s exact test Pearson x test G likelihood ratio test Continuity adjusted x7 Yates corrected x McNemar s test of symmetry Kappa Cramer s V Phi and maximum of Phi Contingency Coefficient C and maximum of Contingency Co efficient C Cross product ratio alpha and logarithm of Cross product ratio alpha Mantel Haenszel x Yule s Q Yule s Y Cochran Armitage Trend Test with Good ness of fit linear trend Pearson product moment correlation coefficient Spearman rank correlation coefficient m Kendall s Tau b Te Stuart s Tau c Somers D Good man and Kruskal s 7 y Gamma Aasym Optimal prediction lambda A Symmetric optimal prediction
133. es of the likelihood ratio test statistics 2log Q the deviance for each cycle is less than the e 3 9 Missing Values Incomplete Observations 51 set by SET EM EPSILON e The command SET EM ITERATIONS maz controls the maximum number of cycles in the EM algorithm The computing time is proportional to the sum over different cases of the product of number of levels at the unobserved factors at the case times some polynomium in the dimension Not too many cases should have more than a single missing value If a latent variable is used there should not be many other missing values and if cases have more than a single missing value around 5 it should be at a limited number of cases The computing time depends on the product over missing factors of observable level for the case with the largest product of number of observable levels for factors marked as missing ReportOn is recommended to get the likelihood ratio test statistic reported for each step in the EM algorithm The commands for semi automatic and automatic search are over night commands when the EM algorithm is used The EM algorithm in CoCo is tested on data from Fuchs 1982 for an example with sporadic missing observations and on Dawid amp Skene 1979 for an example with latent class variables Chapter 4 Description of Data and Fitted Values The two first sections of this chapter will describe how table values are computed These values can be printed in table
134. esent in the current model add the interaction term to the model i e generate a model with the interaction added and then test the current model against the resulting model By default the message FORWARD is the same as the message FORWARD Edges if the current model is graphical else it is the command FORWARD Interactions 9 5 Asymmetry between Backward and Forward 111 9 4 4 Model Control Separators This is analogous to the same keyword at the command BACKWARD Separators FORWARD With this keyword in a Recursive Separators FORWARD command for each edge or interaction term all tests for conditional independence are performed For each edge to add that is for each pair of vertices not connected in the model the min imal separators between the two vertices are found and the two vertices are tested conditionally independent given each separator If a limit separators limit is set by SET SEPARATORS separators limit then edges or interaction terms with a p value less than the limit separators limit for one separator are rejected in a Separators FORWARD command Thus these edges are to be added Analogously if a limit components limit is set by the command SET COMPONENTS components limit and Partitioning is On then edges or interaction terms with a p value less than components limit for one part of the test are rejected i e the edges are added in forward selection 9 5 Asymmetry between Backward and F
135. et me know See section A 1 Availability of this appendix 150 Appendix A Installing CoCo A 3 2 Macintosh The maximum number of declared factors and factors used in models is set to 128 27 The maximum number of observations is 2147483644 2 4 The initial size of both the N and P array is set to 4094 212 The cells in the N array are longs and the cells in the P array are floats All real computations are performed in double A 3 3 Unix Workstations The maximum number of declared factors and factors used in models is set to 256 2 This can be increased by changing constants and recompiling A version with constants set to handle 4096 214 factors has been compiled and run on a SPARCstation with 600 Mbytes swap space The maximum number of observations is 2147483644 23 4 In the version compiled for workstations the initial size of both the N and P array is set to 65536 21 These arrays are increased automatic by CoCo when needed up to a limit of 1048576 220 If bigger arrays are needed these must be given by the N and P command line options The cells in the N array are longs and the cells in the P array are floats Each on 4 bytes 218 cells per Mbyte All real computations are performed in long real 8 bytes 15 to 16 significant digits A running CoCo with space for a 20 dimensional binary table with 1048576 cells in the N array and a 20 dimensional binary table in the P array will then need a
136. et of Variables Missing Values Incomplete Observations 3 9 1 Skip Missing Read only Cases with Complete Information 3 9 2 Exclude Missing On Off Use Maximal Number of Cases with Complete Information in each Test 3 9 3 Exclude Missing in Subset Use only Cases with Complete Information for a Specific Subset of Factors 3 9 4 gt EM algorithm saa a la hc eK A Se e 4 Description of Data and Fitted Values 4 1 Notation and Computation of Marginal Values 432 Values seu bh be he Ca a dod ba bo eA ed daa Avot UPrinting Tables Pe oe tte ee OE eh th th ae ae ar Ben Bead bab te Rete 4 4 Printing Sparse Tables o 4 5 Describing Tables o eee ee eee ANG IR O A O NS AT ists eek foe tee ah aw a ee ee AAS ak Eee BG 4 8 Exporting Table Values o o 5 Models 5 1 Reading Models o aa ai a ee eee eee 5 2 The Model List Shifting Base and Current 5 3 Printing Describing and Disposing of Models 5 4 5 5 Formulas see te RR RE OE SER eee ee a a Is Graphical Is Decomposable Is Submodel and Is In One Clique 5 6 Common Decompositions 0 00000 eee eee 6 Tests 6 1 The Main Test Procedure 0 000000 eee eee 6 1 1 2log Q Pearson s x and Power Divergence 6 1 2 Goodman and Kruskal s Gamma
137. f B in A is contained in a generator of Max Asmussen Edwards 1983 See the chapter 5 Models about 2 section graphs Then the estimated probabilities in the a marginal are computed as Palia 2 Pali 5 5 gt O pal JBUa ja ta kAua kBUa jBua l lAUa kAua gt O PaGe lTanac JBuUa Ja ta So Palau JBUa Ja ta II II The factor c i used in computing the adjusted residual Haberman 1978 for a given decomposable model is found by di pati pi Y 9 aCA n ia where v a a C A is the index or adjusted replication number for subsets of sets in the generating class for the model The index v a is the power to which the marginal counts n ia are raised in the closed form expression for the maximum likelihood estimate of the probability in a decomposable model See decomposition of models in chapter 5 4 2 Values The values for the PRINT TABLE DESCRIBE TABLE PLOT RETURN VECTOR RETURN MATRIX and LIST statements are then computed as follows If a is a subset of B the values are computed by summing over cells in the B marginal table If the set a contains factors not in B the values are computed by summing over cells in the B U a marginal table the table determined by the union of the set a and B where no effect of the extra factors is assumed when estimating probabilities If the optional key word Complete is given the value is only computed for non struc
138. f keywords are surrounded by and and separated by either one of the keywords can be given If keywords or arguments are surrounded by and and separated by one of the keywords or arguments must be given When reading commands CoCo does not distinguish between lower and upper case letters But note that CoCo distinguishes between lower and upper case letters when reading factor names Several commands separated by can be entered on the same line A command consisting of more than one word can be divided between two lines with amp The current version of CoCo only uses the first 4 characters of each word of a command name to recognize the command Commands do not have to be ended by the character EndOfLine Return works as well 2 1 Online HELP HELP command name abbreviation MENU NUMBER a NEXT MENU CURRENT MENU PREVIOUS MENU An introduction to CoCo is given by the command HELP without any arguments The command HELP can be given a command name as an argument The command HELP command name will print the full command name with arguments and a description of the command with name command name is printed when this de scription is entered in the help file HELP help will give a description of how to use help 32 2 2 Comments 33 If only the beginning of a command name is given to the help command HELP start of command name then CoCo will show expected continuation of the
139. ferent numbers of cutpoints 45 direct estimates 67 DISPOSE OF command 67 DISPOSE OF ALL Q TABLES command 44 132 160 DISPOSE OF EH command 114 115 119 161 DISPOSE OF FORMULA command 67 159 DISPOSE OF MODEL command 67 94 132 159 DISPOSE OF PROBABILITIES command 132 160 DISPOSE OF Q TABLE command 44 132 160 DISPOSE OF TABLES command 132 160 DISPOSE OF TESTS command 92 99 159 DROP command 75 DROP EDGES command 50 65 66 70 75 93 94 125 133 160 DROP FACTOR command 50 97 125 160 DROP INTERACTIONS command 50 65 66 70 93 97 125 160 dual representation 66 DUAL TO NORMAL command 65 66 158 Dump keyword 64 107 128 DumpFile keyword 64 107 128 dyn load2 program 134 e 162 Index edges 65 EH command 114 117 119 121 122 161 EH procedure See model selection 113 EM algorithm 49 EM ON command 50 51 124 129 158 EM keyword 62 EM algorithm 37 51 133 emacs program 36 END command 35 136 154 EndOfFile 126 EndOfLine 33 35 36 64 125 endogenous 139 enter names message 140 141 Error keyword 162 estimated expected cell counts 56 EXACT TEST command 13 50 76 98 159 exact test 37 133 conditional 76 102 ExactTest See model selection ExactTest mode 51 77 83 93 118 137 Examples 152 EXAMPLES file 148 EXCLUDE MISSING command 49 51 58 83 124 158 ExcludeMissing 58
140. ff SET DUMP On Off Output from the two commands RETURN VECTOR value name set and RETURN MATRIX list of value names set is copied to the DumpFile if Dump is on Default Off The default name of the DumpFile is COCO DMP in the current directory un der DOS and tmp CoCo dump no pid number under Unix Another DumpFile can be named by the command SET OUTPUTFILE DUMP file name Note that this com mand rewrites the file file name and that under DOS the file COCO DMP is rewrit ten every time CoCo is invoked If the DumpFile is not renamed the DumpFile is re moved when CoCo terminates normally unless the command SET KEEP DUMP On Off is used 11 3 5 Report file SET OUTPUTFILE REPORT file name SET KEEP REPORT On Off Computing time used in the IPS algorithm number of cycles in the IPS algorithm computing time used in the EM algorithm number of cycles in the EM algorithm time used to find duals in the EH procedure etc are reported in the ReportFile This reporting cannot be turned off The default name of the ReportFile is COCO RPT in the current directory under DOS and tmp CoCo report no pid number under Unix Another ReportFile can be named by SET OUTPUTFILE REPORT file name Note that this command rewrites the file file name and that under DOS the file COCO RPT is rewritten every time CoCo is invoked If the ReportFile is not renamed the ReportFile is removed when CoCo te
141. for this example and the example of section 6 4 originated in an epidemi ological panel study in County of Copenhagen the so called Glostrup Study In this example the statement DROP EDGES was used to generate a model See the next chapter about DROP statements From the saturated model a model without the edge SW is generated and a test of the generated model against the sat urated model is performed The first test output is given in the ShortTestQutput format After adjustment the degree of freedom is equal to 40 The adjustment labeled by 0 is 88 In the normal test output generated by the TEST statement both the p values for the unadjusted and adjusted degrees of freedom is printed second and third column 6 2 Exact Tests 75 If the FIND DEVIANCE statement is used for the test instead of the TEST statement then the number of zeros Z B c and Z C n are printed In this case 230 and 142 respectively Also the observed number of zeros Z B n in the sufficient marginal tables minus number of zeros in intersections of sufficient marginal tables plus for the base model is printed In this case 258 If Trace is on the cumulated number of zeros for each sufficient marginal table is printed when computing Z B rnc and Z C n Since all cells in the sufficient marginal tables have to be visited in order to compute the adjustment the computing time used to compute the adjustment is comparable to the time used t
142. found but with the interaction lt ABC gt 5 4 Formulas gt read model ADE ABE ABC BF 0 002secs TIME gt set page formats 70 65 TIME 0 001secs gt print vertex ordering V Order mH o awe F TIME v cw Complete V 6 0 TRUE 5 Al TRUE 4 AB TRUE 2 AE TRUE 3 AB TRUE 1 B TRUE 0 007secs gt describe current 2 ADE ABE ABC BF Model is graphical Graph is decomposable CAF CF BD CD DF EF CE Dual rep Adjacency matrix ABCD ok ok H mu o awe XA Cliques Cells 0 00 A A BNE Log L Dimension TIME Edges DE BF BE BC AD AC AE Ar aS NS ANS a BCDE ACEF AB AE ABD B AB Expression RZZZZZZZZ OMAN CI RAHRAHAHA I B BF AE AB ADE ABE ABC 0000000 17 0 016secs MU MU NA NA ON ON NON KA XA XA Xk XX XX XX 67 68 Chapter 5 Models gt set power lambda null Power Divergence not printed TIME 0 001secs gt set adjusted df off Adjusted df set OFF TIME 0 000secs gt drop interaction ABC 3 ABE ADE BF BC AC Model is not graphical Cliques ABC ABE ADE BF 2 Section Graph is decomposable Test of ABE ADE BF BC AC against ABCDEF DF 2log Q P X72 P Models 47 58 0908 0 12885 57 0962 0 14863 CCABCDEF AC BC BF ADE ABE TIME 0 033secs gt current TIME 0 00
143. found in the directory DATASETS CoCo runs on IBM XT ATs and compatible with or without coprocessor 8087 80287 80387 and with or without expanded memory EMS If the numeric coprocessor is not present it is emulated If EMS is present the overlay file is loaded into the expanded memory when sufficient space is available See the sections DOS Overlays in the chapter Miscellaneous Options for in A Guide to CoCo about how to control the size of the overlay buffer See the README DOS file for Run Time Errors A 2 2 Macintosh Read the disk with CoCo for Macintosh or uncompress the sit Hqx file Click the folder CoCo to start CoCo A 2 3 Unix On Workstations Read the latest READ ME and README Unix file Copy the contents of the disk or tape into a directory the build directory of CoCo Or obtain relevant files by anonymous ftp over internet from ftp iesd auc dk or by WWW from http www iesd auc dk pub packages CoCo and uncompress Change to the directory A 2 Installation 147 1 Set the variables BINDIR COCOHOME SCOCOHOME XCOCOHOME to appropriate values in the Makefile BINDIR e g home local bin is where the script for starting CoCo goes The value of COCOHOME e g home local lib coco is the home directory of CoCo SCOCOHOME e g home local lib coco or home local lib Splus local is where the functions to use CoCo in S Plus go The XLISP STAT interface functions are copied t
144. g and Debugging 131 11 5 6 The workspace in the heap DISPOSE OF TABLES DISPOSE OF PROBABILITIES DISPOSE OF ALL Q TABLES DISPOSE OF Q TABLE set The two commands DISPOSE OF TABLES and DISPOSE OF PROBABILITIES will dispose of sufficient marginal tables and tables of estimated cell probabilities These tables can safely be disposed of when necessary to give CoCo more workspace since these are found again when needed Disposing of these tables will not release much space in the heap On a PC remember to dispose of unused models with the command DISPOSE OF MODEL before using all space in the heap The commands DISPOSE OF ALL Q TABLES and DISPOSE OF Q TABLE set disposes of tables of initial values for the IPS algorithm Disposing of these tables will not release much space in the heap The command RESTART clears all tables model specifications etc 11 6 Miscellaneous Options for Tracing and Debug ging 11 6 1 Echo and Note SET ECHO On Off J SET NOTE On Off J The echo from the command parser can be turned on and off with the switch SET ECHO On Off J Default Off The SET NOTE On Off controls a note when some commands have finished Default Off 11 6 2 Report Trace and Debug SET REPORT On Off SET TRACE On Off SET DEBUG On Off J SET OPTION On Off number value 1 Report On will cause reporting of likelihood ratio test statistics 2log Q e i the deviance f
145. g from key board the list or table has to be ended with a slash The characters and are abbreviations for the lowest level marked as missing Thus at the factor C 3 and are all codes for the level marked as missing If List is replaced by Accumulated list then an accumulated list is expected Each identification of a cell has to be preceded by the number of cases in that cell 3 4 3 Accumulated List Accumulated list 1 PNR PR RNRANE NRPRPNPR N XxX WD 1 The accumulated list for data entered by List or Table can be printed by CASE LIST see section 4 4 3 5 Replacing the Observations with a Random Data Set 41 3 5 Replacing the Observations with a Random Data Set SUBSTITUTE DATA WITH GENERATED TABLE This command will replace the table of observed counts with a random table of counts where the sufficient marginals for the current model are unchanged The current model has to be decomposable The table is generated as when computing an exact p value 3 6 Initial values to the IPS algorithm READ Q TABLE set table READ Q LIST set list of cells CLEAN DATA Initial values integers to the IPS algorithm are entered by READ Q TABLE set table Zero 0 is entered for cells to be zero by structure Cells to be zero by structure can also be entered by the command READ Q LIST set list of cells By cells to be zero by structure incomplete tables can be handled 3 6 1 Structural Zeros
146. gainst i n where i is the rank of x Normal and Rankit are normal probability plots quantile quantile plots The empirical quantiles are plotted against the quantiles of a standard normal distribution In the rankit plot x is plotted against 6 1 3i 1 3n 1 and in the normal plot 2 is plotted against d7 i 1 2 n B7 is the inverse of the standard normal distribution function See Haberman 1978 pp 20 for rankit plots 4 6 Plots 59 If value name is Observed and no log transformation is used a table showing the number of cells with extreme cell counts is printed With DESCRIBE OBSERVED a the observed counts in the marginal table deter mined by the set a are described without expecting a current model Examples of DESCRIBE TABLE statements describe observed describe table expected describe table probabilities ABCDE describe table unadjusted describe table g res describe table random current observed ABEF describe table random current observed describe table random base expected ABEF describe table log observed describe table log 2 n m describe table log freeman tukey describe table rankit 2 n m describe table uniform normal standard describe table uniform rankit adjusted describe table rankit normal adjusted describe table rankit normal uniform adjusted describe table complete log observed describe table errors See cha
147. ght be expected in computing the tests and fitting the models by the IPS algorithm the usefulness of the decomposable search is doubtful unless exact tests are to be used If ExactTest is on then DecomposableMode also should be on If ExactTest is on and DecomposableMode is off then asymptotic p values will be used for the non decomposable models Note that the commands EH Decomposable EH Graphical and EH Hierarchical set DecomposableMode to off If no models are entered or found by EH procedure the search is started with the saturated model accepted and the model with only main effects rejected 10 3 3 Hierarchical Search In the hierarchical search we for each step either fit all not classified models in the hierarchical A dual or all not classified models in the hierarchical R dual Da R The hierarchical A dual D4 R of the rejected models is the minimal models in the class of models that adds at least one interaction term to each rejected model Dr A The hierarchical R dual DR A of the accepted models is the maximal models in the class of models that sets at least one interaction term in all accepted models to zero 118 Chapter 10 The EH procedure The command SET Hierarchical SEARCH will set the search module up to a hierarchical search and EH Hierarchical will perform a model selection among hierar chical models i e the hierarchical A dual and R dual are used If the hierarchical search follows a graphica
148. he Search Classes The following commands are used to copy models between the model list and the classes of models in the EH procedure 10 6 1 Models from List to Search Classes FIT Base Current Last Number lt a gt All models Interval of models lt a gt lt b gt ACCEPT Base Current Last Number lt a gt All models Interval of models lt a gt lt b gt REJECT Base Current Last Number lt a gt All models Interval of models lt a gt lt b gt Models from the model list are fitted and put into a model class in the EH procedure according to whether the models are accepted or rejected or models from the model list are forced into a model class of the EH procedure regardless of whether the model is accepted or rejected 10 6 2 Models from Search Classes to Model List EXTRACT Decomposable Graphical Hierarchical A dual R dual Accepted Rejected This command is used to copy a model class or a dual from the EH procedure to the model list 10 7 Find Duals FIND DUAL Decomposable Graphical Hierarchical A dual R dual Both duals the commands FIND A dual FIND R dual and FIND Both duals can be used if you want to see the dual by STATUS Search before fitting models If the keyword Decomposable Graphical or Hierarchical is given then the class of models given by the keyword is used in the FIND DUAL command and the requested class is also selected in s
149. he rejected models can be added to the rejected models by REJECT R dual Chapter 11 Miscellaneous Options for Controlling Input Output and Algorithms The SET commands of this chapter and of chapter 8 perform simple changes of a few variables Since EXCLUDE MISSING In set EXCLUDE MISSING On Off and EM ON change the whole data set they are not SET commands Commands with the two optional key words On and Off like SET DIARY On Off are switches SET DIARY On turns the diary on and SET DIARY Off turns the diary off SET DIARY with no argument turns the diary on if it is off off if it is on The command STATUS gives information about factors declared names on files data selection declared data structure selected print formats IPS convergence crite rion diary time echo workspace etc STATUS Specification Observations Files Formats Fix Tests Exact Ips Em EH Limits Other E g STATUS EH prints accepted and rejected models duals and FixIn FixOut and BaseModel if any Use STATUS Files to see the current or default file names RESTART The command RESTART will restart CoCo e i clear all settings in CoCo 11 1 Print formats SET TABLE FORMATS width dec prob dec expt dec diff SET TEST FORMATS 2 width a dec prob width prob dec SET SHORT On Off 123 124 Options for Controlling Input Output and Algorithms SET PRINT F
150. he FIT R dual command extra FIT A dual commands will only produce output without using much computing time when there are no more models to fit in the A dual 10 8 5 Fit Both Duals FIT Both duals fits all not classified models in both duals as they are Note that this need not give the same as FIND A dual followed by FIND R dual 10 9 Automatic Search EH Decomposable Graphical Hierarchical Smallest Alternating Rough This command is used to do a recursive search by the EH procedure If the keyword Decomposable Graphical or Hierarchical is given then the class of models given by the keyword is used in the EH command and the requested class is also selected in subsequential EH commands needing a model class else the class of models to use is selected by the options set by the commands SET Graphical Hierarchical SEARCH and SET DECOMPOSABLE MODE On Off or other commands setting model class 10 9 Automatic Search 121 If the keyword Smallest Alternating or Rough is given then the strategy given by the keyword is used and the strategy will also be used in subsequential EH commands else the strategy set by the command SET Smallest Alternating Rough SEARCH or previous use of the EH command is used The three commands EH Decomposable EH Graphical EH Hierarchical will do a graphical search ignoring non decomposable models a graphical search and a hierarchical search respectively
151. he SearchBase with the FixIn command is extended with edges in FixIn but not in SearchBase The extended SearchBase is printed by the use of the command FIX In edges gc but not used Effect of FixOut on FixIn FixOut terms in FixIn are removed from FixIn i e FixIn is restricted to normal representation of FixOut Effect of FixOut on SearchBase FixOut has no effect on SearchBase but terms not in the SearchBase are added to FixOut Effect of SearchBase on FixIn Only terms in the SearchBase can be FixIn FixIn thus is reduced by SearchBase 116 Chapter 10 The EH procedure Effect of SearchBase on FixOut Terms not included in the base model are FixOut i e FixOut is set to the dual representation of SearchBase Since there is no distinction between FixOut due to the FIX Out command and due to READ BASE MODEL and since a large SearchBase would give less FixOut due to SearchBase previous FixOut is cancelled Use REDO FIX Out to redo previous FixOut 10 2 5 Add FixIn and FixOut A new FIX Out command will clear the FixOut The command ADD FIX Out edges gc will add edges gc to the previous FixOut The FixIn is effected as described in a previous section by the argument edges gc in the ADD FIX Out command Similarly a new FIX In edges gc command will clear the FixIn The command ADD FIX In will add edges gc to the previous FixIn The FixOut is affected as described in the previous section by the argument edge
152. he class strictly contains the decomposable models e g Haberman 1974 See Darroch Lauritzen amp Speed 1980 for how graphical models are defined by the close connection between the theory of Markov fields and that of log linear interaction models for contingency tables Besides incomplete tables and tables with incomplete observations can be handled and exact tests between any two nested decomposable models computed CoCo is a program designed to perform estimation and tests in large contingency tables By using graph theoretical results Rose Tarjan amp Lueker 1976 Tarjan amp Yannakakis 1984 Tarjan 1985 Leimer 1993 the hierarchical log linear interaction models are decomposed See also chapter 3 of Part I The IPS algorithm is not used on the full table but only on the non decomposable irreducible components chapter 2 of Part I Furthermore the optimized version of the IPS algorithm of Jirousek 1991 is used on these non decomposable atoms If one model is tested against another and the two models have common decom positions then the test can be partitioned in tests on smaller tables Goodman 1971 Collapsibility Asmussen amp Edwards 1983 of models and tests is used If two large sparse models many factors but few interactions have no common decompositions they can be tested against each other by computing the deviance for each model not by summing over all cells in the full table but by summing over only cells in s
153. he current model the base model the last model model number a all models and finally models with numbers between a and b The generating classes for the models are printed If a list of models is printed the current and base model are marked A more detailed description of the models is obtained by replacing PRINT with DESCRIBE DESCRIBE MODEL Base Current Last Number a All models Interval of models a b The command DESCRIBE CURRENT gives a very detailed data independent de scription of the current model with adjacency matrix an expression for the maximum 66 Chapter 5 Models likelihood estimate of cell probabilities the number of cells in sufficient marginal ta bles and intersections of sufficient marginal tables and for each edge the number of cliques the edge is in If a model is decomposable and an edge is only in one clique the model resulting from removing the edge is decomposable See the next section for examples Models are disposed of with the commands DISPOSE OF As with PRINT and DESCRIBE six DISPOSE OF commands are available DISPOSE OF MODEL Base Current Last Number a A11 models Interval of models a b 1 5 4 Formulas PRINT FORMULA PRINT VERTEX ORDERING DISPOSE OF FORMULA Two subgraphs of a graph are a decomposition of a graph with respect to a subset a of the vertices if the graph is the union of two subgraphs and the intersection a be
154. ict both the stepwise forward selection the backward elimination and the global EH search procedure to decomposable models I e the search proce dures will not try to visit non decomposable models but edges or interactions can still by removed by other commands so that a non decomposable model is generated It is attractive to restrict model selection to the class of decomposable models for a variety of reasons Firstly computation efficiency the ability to use explicit formulae for the maximum likelihood estimates can reduce computational time substantially Secondly exact conditional tests for nested decomposable models can be calculated 97 98 Controlling the Computed Tests A backward elimination by the command BACKWARD with DecomposableMode set On will give the model selection proposed in Wermuth 1976 8 2 Computed Test Statistics and Choosing Tests and Significance Level This section will describe options for how to compute test statistics select test statis tics and set limits for when to accept or to reject a model in the model selection by the commands BACKWARD and FORWARD and in the EH procedure Also a short description of the Information Criterion is given 8 2 1 Computing Test Statistics SET POWER LAMBDA Null lambda SET ADJUSTED DF On Off J SET REUSE OF TESTS On Off The computed test statistics in all the test and model search procedures depend on values set by these commands The power A
155. in Frydenberg 1989 Model selection methods for creating a diagnostic system by causal models on discrete vari ables are discussed in Lauritzen Thiesson amp Spiegelhalter 1992 Also the text books Lauritzen 1993 Whittaker 1990 and Christensen 1990 contain chapters on block recursive models The tool for association diagrams implemented in the Lisp extension of CoCo are able to handle block recursive models In these diagrams incremental model search on block recursive models can be performed by backward elimination and forward selection 12 2 3 Miscellaneous Besides introducing association diagrams with the ability to handle block recursive models and dumping TeX picture code for the diagrams this tool also introduces the model dynamic spinning plot and the model manager Also a method for doing a model selection with resampling is implemented Model Dynamic Spinning Plots The model dynamic spinning plot is a spinning plot where the values in the plot are updated when models are modified Spinning plots are rotatable three dimensional scatterplots In the model dynamic spinning plot each of the variables are linked to a model and when the model is modified the plot is redrawn with the new values of the model These spinning plots can of course be linked to other spinning plots histograms etc and points of one plot highlighted when the corresponding points are brushed in other plots or histograms 12 2 XLISP STAT CoCo
156. ine you are using the compiler used options to the compiler and the version of CoCo If you get an error message from coco start coco init or coco load then use the size the message suggests Some versions of the object file loaded into S Plus allow for handling more than one CoCo object in the same S Plus session At the same time in S Plus CoCo can then be used on different datasets A list coco identifications of integers in frame 0 is used to identify the individual CoCo objects The values of the integers have no meaning for the user current coco is the identification of the default CoCo object used in the following functions The functions make current coco a in S Plus will make the a the started or initiated CoCo object the current CoCo object assign coco identifications a to current coco The function enter names names levels missing NULL enters a spec ification of a table to CoCo names is a single text string with the factor names a character for each factor levels is a vector of integers giving the total number of levels for each factor missing is an optional vector of integers giving the number of missing levels If enter names names levels missing NULL is succeeded the value TRUE is returned else a warning message is printed and FALSE is returned A table of counts is entered by enter table table table is a vector of integers giving the cell counts ordered as when reading
157. interaction found to be non significant is eliminated In each step of this backward elimination edges with p value less than a given limit e g 1 or 5 are rejected and when the principle of weak rejection coherence is applied they are not considered in sub sequential steps The first edge found in each step with p value greater than a second limit e g 20 is removed All eligible edges not visited in the current step are eligible in the next step together with the edges visited in the current step with a found p value between the two limits If not the keyword Coherent is used then of course all edges of the model are eligible in the next step This gives a faster elimination than a backward elimination where all eligible edges in each step have to be visited in order to find the least significant edge If the p value or minus the IC value for a model in this stepwise edge or inter action selection is greater than the value set by SET ACCEPTANCE alpha then the model is accepted I e in the backward elimination edges or interaction terms are 9 3 Backward Elimination 107 eligible for removal if the p values of the edges are greater than this limit The recursive backward elimination continues only as long as there are edges or interaction terms with a p value greater than this acceptance limit Only if the p value for the edge to be removed is greater than this acceptance limit a new model is inserted in the model list E
158. interaction terms the model resulting of each step of the 9 4 Forward Selection 109 forward selection etc is given between steps of the forward selection The keywords Only Reversed Sorted and Short is as at the command BACKWARD except that the p values by default are ordered in decreasing order IC increasing in the sorted list 9 4 2 Recursive Search Also the keywords Recursive Coherent and Headlong are as at the command BACKWARD A model where rejected edges or interaction terms are added to the current model is added to the model list for each step of the Recursive FORWARD command Recursive Recursive FORWARD Forward selection is done recursively until no more edges have to be added according to the selected level of significance Coherent Recursive Coherent FORWARD Once an edge or interaction term is accepted in the forward elimination process it is no more tested for adding Headlong Headlong Recursive FORWARD or Headlong Recursive Coherent FORWARD In each step of this forward selection edges or interaction terms with p value greater than a given limit e g 10 or 20 are not added i e it is accepted that the edge can be omitted and is not considered in sub sequential steps when the principle of weak acceptance coherence is applied The first edge found in each step with a p value less than a second limit e g 1 is added to the model All not visited eligible edges in the current
159. is tested against base model 7 7 Meet of Models The meet of the two models base and current is formed by the command MEET OF MODELS The meet is the largest model contained in both base and current model The resulting model will contain the maximal generators that are a subset of at least one generator in both base and current For each generator in base and for each generator in current the intersection of the two generators is formed and the resulting model is the generating class with the maximal of these intersections The generation of the model is symmetric in base and current The resulting model is tested against the current model In the following example all interactions in a final model from Edwards amp Havr nek 1985 see the Introduction are reduced to first order interactions gt set page formats 78 256 TIME 0 001secs gt set power lambda null Power Divergence not printed TIME 0 001secs gt set adjusted df off Adjusted df set OFF TIME 0 001secs gt read data 64 cells with 1841 cases read Finding all marginals TIME 0 024secs 7 8 Join of Models 95 gt read n interactions 1 ABCDEF 1 EF DF DE CF CE CD BF BE BD BC AF AE AD AC AB Model is not graphical Cliques ABCDEF 2 Section Model is decomposable TIME 0 011secs gt read model ACE ADE BC F TIME 0 001secs gt meet of models 3 F BC AD DE AC AE CE Model is not graphical
160. king the two programs together A CoCo object file is linked together with S Plus or XLISP STAT Data specifi cation of table and the table of counts and models can be sent to the CoCo object Table and test values can be returned Interface functions for all the CoCo commands are implemented only for XLISP STAT Only the most necessary interface functions are implemented for S Plus The remaining CoCo commands are for the time being used in S Plus by handling the control over to CoCo and reading commands in normal CoCo syntax Note that this section of CoCo might develop faster than other parts Returned values especially error messages and values when an error occurs arguments to func tions and methods and even names on functions and methods may change List the files cocoapi S and cocometh 1sp for an up to date documentation of the version you are running 133 134 Chapter 12 CoCo in other systems Figure 12 1 Plot from New S by call of functions in CoCo 121 S CoCo A version of CoCo an object file without main program but with some interface procedures is linked together with S Plus see Becker et al 1988 Start S Plus with the script S coco to set up some environment variables to CoCo The script S coco is available on your system if you or your system administrator has done step 4 and 5 of the installation of CoCo see appendix A S Plus of course has to be available on your system Use attach paste unix
161. l When the keyword Recursive is given the resulting accepted model from each step is added to the model list in CoCo 9 3 1 Controlling the Output By default a test is printed for each edge or interaction term visited and a report of rejected eligible edges the model resulting of the step of the recursive backward elimination etc is given between steps of the backward elimination Sorted Sorted Recursive BACKWARD Also a sorted list of the tests is printed for each step of the backward elimination The list is sorted according to the selected p value statistics if IC is selected The p values are ordered in increasing order IC decreasing The least significant edge or interaction term is then the last edge in the sorted list Reversed Reversed Sorted Recursive BACKWARD If this keyword is used the sorted list of tests is reversed Only Only Sorted BACKWARD Only the sorted list of tests is printed This has the advantage of saving paper if only the sorted list is wanted But when running interactively especially if exact p values are computed and on large tables there will be a lot of computation without any results shown since the whole list of tests has to be computed for each step of the backward elimination before any output is printed The keyword can of course be combined with the keyword Reversed Only BACKWARD If the keyword Only is given without the keyword Sorted being given then all printing from
162. l non decomposable atoms before computing log L If Large is on marginal tables of observations and probabilities are found and used in computing log L one at a time e i you force CoCo to dispose of sufficient marginal tables as soon as the are used 130 Options for Controlling Input Output and Algorithms With SET HUGE Off you prevent CoCo from finding the deviance without storing the sufficient marginal tables when the tables are to big to store 11 5 4 DOS Overlays coco D96000 R32000 T By default the size of the overlay buffer is set to 96 000 Bytes The probation area is one third of the overlay buffer size The size of the overlay buffer and the probation area can be controlled by the options 0 and R when invoking CoCo If you increase the size of the overlay buffer there will be used less computing time on swapping but the size off the heap for models tests etc will decrease If the disk is continuously accessed during your commands not just a few times for each command you should increase the size of the overlay buffer If the computation of the adjusted number of degrees of freedom is set of by SET ADJUSTED DF Off then the space needed for overlays is smaller Or if you run out of space in the heap for models tests classes of models and duals in the EH procedure etc you could restart with a smaller overlay buffer size minimum 31 000 Bytes If the overlay buffer size is changed but not the size of the prob
163. l search the regions found in the graph ical search are used as a starting point in the hierarchical search else the search is started with the saturated model accepted and the model with only main effects re jected or any models can be entered into the two classes 10 4 Dispose of the Model Classes and Duals DISPOSE OF EH All Duals A dual R Dual Classes Accepted Rejected The command DISPOSE OF EH will dispose of the accepted models the rejected mod els the A dual and the R dual 10 5 Read and Fit Models or Force Models into Classes 10 5 1 Fit Some Models FIT Models models With FIT Models models you can enter a list of models that is classified into accepted and rejected models and start the model search from there The argument models is a set of models a set of generating classes as a generating class is a set of sets e g FIT Models A B C LAB AC Bc or the shorter form FIT Models A B C AB AC BC can be used 10 5 2 Reading Accepted and Rejected Models ACCEPT Models models REJECT Models models With the commands ACCEPT Models models and REJECT Models models the entered models are forced into the accepted region and the rejected region re spectively The argument models is a set of models as at the command FIT Models models 10 6 Copy Models Between the Models List and the Search Classes 119 10 6 Copy Models Between the Models List and t
164. l set up the EH procedure to a graph ical search and EH Graphical will perform a model selection among graphical models i e the graphical A dual and R dual are used 10 3 2 Decomposable Mode By a naive approach the selection can be restricted to decomposable models non decomposable models in the duals to fit are simply ignored Non decomposable models in the search are ignored with the command SET DECOMPOSABLE MODE On before commands EH and FIT The search is then performed among only decomposable models This can also be performed by EH Decomposable After a search among only decomposable models a search among graphical models can be performed with the result of the decomposable search as a starting point DecomposableMode is turned off again by typing SET DECOMPOSABLE MODE Off Since non decomposable models in the graphical dual are just ignored in the de composable search there might still be graphical and decomposable models to fit when both the set D4 R A of not classified models in the graphical A dual and the set D A R of not classified models in the graphical R dual contain no more decom posable models So the decomposable search should be followed by a graphical search One might consider adding edges to or removing edges from the non decomposable models in the graphical duals in the models selection among decomposable models Since a lot of computing time in the search module is used to find duals and not as it mi
165. lambda A Modified asymmetric optimal prediction lambda Un certainty coefficient Uasym Symmetric uncertainty coefficient U Agresti 1990 and Appendix A 5 of the manual for the BMDP statistical computer package Dixon 1983 are good sources for references and for formulas for computing the above measures of associations with standard errors 6 4 Computation of log L in Large Models 79 6 4 Computation of log L in Large Models FIND LOG L FIND DEVIANCE Since the maximum likelihood estimate can be expressed as Pali c II nig a II PB in k 1 u l 1 v the log L can be computed as logL Y log pa i 0 El S n i logfe II n ia Pg ib tel k 1 u l 1 v nlog c 5 v ap 5 N ia log n ia k 1 u jap Clay Y Y n i log bp s l 1 v ty EIo Hence we only need to sum over cells in sufficient marginal tables FIND LOG L computes log likelihood by the method i e by only summing up over cells in necessary marginal tables This is effective in simple models of large dimensions Differences between log likelihood values for base and current the deviance are printed with FIND DEVIANCE but be careful with errors caused by rounding off This method is also used to compute the deviance by any test procedure of CoCo if the saturated table is to big to fit into the memory after performing common decompositions This method should only be used if the two models to be tested against e
166. lass 0 0040 122 11 Miscellaneous Options for Controlling Input Output and Algo rithms 123 111 Pri formats sic a Big eee Are ee ne Bw E eta 123 IVA PAM cris het te Se gts SS Se Ald aan tiie ee PSS 124 12 gt npt files s ee git take Acca aye a ere oo ted Ec Ee ge ete A 125 11 2 1 Input from keyboard or file o 125 11 22 Data fil s 2 0 24 al a id a es 125 11 2 3 Standard Input Source o 125 1133 Output flesse ri in a tada 125 11 3 1 Standard Output Sink o e 126 LA DICEN Ax AA EDS 126 AR A A tk ie te ete 126 11 34 Di mp tlet ree rr adds BD ee ea oes 127 1133 5 Rep ortetile s csc ae ERA AI E a 127 TLA Timer bh es da Be a et A A a A 127 11 5 Controlling AlgorithmMs o e e 127 11 5 1 Controlling the IPS algorithM 128 11 5 2 Controlling the EM algorithM 128 11 05 37 Data structure tt tr ce A a ie de 128 11 54 DOS Overlays ac oda oe og DD Dee ae a oe a 130 11 5 5 Unix Size of N P and Q arrays 130 11 5 6 The workspace in the heap 00 131 11 6 Miscellaneous Options for Tracing and Debugging 131 1126 1 Echo and Note suslu Bee eee ea ww RE ok ede et 131 11 6 2 Report Trace and Debug 0 131 11 6 8 Graph mode 0 00002 eee eee 132 Eer NNT OPUS cave Pear n Cece ae al tel Bo aa toate Diy Bi Beha Bp Aa ke oie a ee
167. liques ABC ABE ADE BF 2 Section Graph is decomposable Test of BF BC AC ABE ADE against ABC ABE ADE BF DF 2log Q P X72 P Models 1 5 986 0 0144 5 951 0 0147 CCABC BC AC AB TIME 0 026secs gt gt Drop the edge AB and accept a non decomposable model gt gt drop edge AB 10 AC ADE BC BE BF Model is graphical Graph is not decomposable Generating class for Fill In ACE ADE BCE BF Test of AC ADE BC BE BF against ABC ABE ADE BF DF 2log Q P X 2 P Models 3 6 177 0 1033 6 162 0 1040 ABE ABC BE BC AE AC1 TIME 0 032secs 1 2 An Example gt gt Name the last accepted model current gt gt current TIME 0 001secs gt gt Name current model base gt gt base TIME 0 001secs gt gt Print all models gt gt print all models Model no 10 AC ADE BC BE BF BASE Model no 9 BF BC AC ABE ADE Model no 8 ABC ABE ADE BF Model no 7 ABC ABE ABF ADE Model no 6 ABCF ABE ADE Model no 5 ABCF ABEF ADE Model no 4 ABCF ABEF ADEF Model no 3 ABCEF ADEF Model no 2 ABCEF ACDEF Model no 1 ABCDEF TIME 0 006secs gt gt No edges can be removed from this model gt gt backward Edge DF 2log Q P xO P AC 1 30 292 0 0000 30 211 0 0000 AD 2 16 542 0 0003 16 427
168. little more than 8 Mbytes of memory CoCo has been started with space for a 26 dimensional binary table with 67108864 cells in both the N array and the P array on a SPARCstation with 600 Mbytes swap space A 4 CoCo Datasets and Examples Datasets for CoCo and examples of SourceFiles for running these datasets can be found in the directory Datasets in the home directory for CoCo The examples are to illustrate the use of CoCo They are not meant to be good practice of data analysis The directory Examples contains tests runs These examples are not even instruc tive A 5 Reporting Errors Report errors by sending the LogFile DOS COCO LOG Macintosh CoCo log no and Unix tmp CoCo log 004 X pid number and data files to the author Please note whether the error appears again if you run CoCo with the LogFile as input On DOS rename the LogFile before starting CoCo again Note machine type operating system version of operating system and version of CoCo A 6 Warranty 151 A 6 Warranty NO WARRANTY I PROVIDE ABSOLUTELY NO WARRANTY EXCEPT WHEN OTHERWISE STATED IN WRITING I AND OR OTHER PARTIES PROVIDE CoCo AS IS WITHOUT WARRANTY OF ANY KIND EITHER EXPRESSED OR IMPLIED INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MER CHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE THE EN TIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU SHOULD THE CoCo PROGRAM PROVE DEFECTIVE YOU AS SUME THE COS
169. ll models print TRUE If print is FALSE then descriptions of the models are printed No values are returned The function compute test in S Plus will return the unadjusted number of degrees of freedom the adjustment of DF if AdjustedDF is on in CoCo the likelihood ratio test statistic 2Log Q Pearsons chi square x and the power di vergence 2nJ and if ExactTest is on in CoCo the exact P values The function compute deviance in S Plus will return values as the command FIND DEVIANCE in CoCo And what it is all about The function return vector type 0 set number 0 base FALSE in S Plus will return a vector with the table values type from the set marginal table computed under the current model See sec tion B 1 46 in the Quick reference card for how CoCo value names are coded into the integers type If base is TRUE then the values from the base model are re turned If number is different from 0 then the values from model with that number are returned The model ACE ADE BC F is tested against ABCDEF on the data from Reinis et al 1981 and the adjusted residuals in ACE ADE BC F are plotted against the standardized residuals in ACDE ABCF in S Plus by S coco attach paste getenv SCOCOHOME Functions sep coco load coco init enter names ABCDEF c 2 2 2 2 2 2 c 0 0 0 0 0 0 n lt c 44 40 112 67 129 145 12 23 35
170. lly independent in a graph then the two variables should by conditionally independent given any separator of the two variables in the graph If a limit separators limit is set by SET SEPARATORS separators limit default 1 0 then models with a p value lower than separators limit for one separator are rejected in the stepwise model selection when this model control is requested by the keyword Separators of the command for backward elimination of forward selection 8 3 Exact Tests in Tests and Model Selection SET EXACT TEST On All Deviance Off SET ASYMPTOTIC p value SET NUMBER OF TABLES Null number SET LIST OF NUMBERS OF TABLES integer list SET EXACT TEST FOR TOTAL TEST On Off SET EXACT TEST FOR PARTS OF TEST On Off SET EXACT TEST FOR UNPARTED TEST On Off SET SEED seed Random SET EXACT EPSILON e SET FAST On Off If exact p values are computed in the model selection then these p values are used Exact test is set on by the command SET EXACT TEST On Exact tests are only computed if the asymptotic p value is smaller than the value set by SET ASYMPTOTIC p value 1 by default The number of tables to generate in the Monte Carlo simula tion of the exact p value is set by the command SET NUMBER OF TABLES A variating number of tables is selected by the command SET NUMBER OF TABLES Variating and the number of tables to be generated is then set by SET LIST OF NU
171. lude an accepted model and weakly rejected models are models that are included in a rejected model In turn a boundary below the weakly accepted models and a boundary above the weakly rejected models the R dual Dr A and the A dual D4 R respectively are fitted In CoCo the global search can be started with the accepted class containing the saturated model and the class of rejected models containing the model with only main effects or any models can be entered into the two classes The search stops when either all models in D4 R A are accepted or all models in DR A R are rejected The search module is only usable for small tables The dimension of the table should not exceed 10 too much There should not be too many choices between acceptable models If there is no clear association between some factors given others i e there are many marginal associations but no clear conditional associations then all models in Dr A will be accepted for several fits of DR A and the duals will explode in size i e if one can choose between too many acceptable models When the temptation to use this algorithm is the greatest it should not be used If one of the search commands described in this chapter is used then the search module is initiated i e initial classes of models and duals are found for the EH procedure By default the saturated model is accepted and the model with only main effects is rejected 112 113 Figure 10 1 We
172. m CoCo is not only useful for working statisticians and other scientists analyzing discrete data by contingency tables but also in courses teaching analysis of discrete data and graphical models The guide is probably useful in such courses but the guide to CoCo is not a text book on contingency table and should only be used in courses assisted by text books as e g Lauritzen 1982 Availability The source code in C executable for Sun 4 Sparc and executable of the standalone version of CoCo for PC s and Macintosh for this version of CoCo is available free of charge for non commercial use The source code may only be read and edited for the purpose of porting CoCo to other machines No new features may be added to CoCo and no parts of the program may be included in other systems or new interface procedures to New S S Plus XLISP STAT or any other extendable system may be made without the written permission from the author The source code in C and executable for Sun 4 Sparc both with Lisp code for the graphics and executable of the standalone version of CoCo for PC s and Macintosh can be obtained by anonymous ftp over internet from ftp iesd auc dk or by WWW from http www iesd auc dk pub packages CoCo Or CoCo is available on disks 3 5 or 5 25 High density You should however by prepared to bear the costs of copying e g by supplying a disk or tape and a stamped mailing envelope This guide and the guide to CoCo is also
173. mes a PRINT STANDARD TABLE value name a RETURN STANDARD VECTOR value name a B 1 25 Read Model READ MODEL gc READ N INTERACTIONS order set B 1 26 Edit Model COLLAPSE MODEL set NORMAL TO DUAL DUAL TO NORMAL B 1 27 Moving pointers in the Model list BASE CURRENT MAKE BASE a MAKE CURRENT a B 1 Quick Reference Card 157 B 1 28 Describe Print and Dispose of Models PRINT FORMULA PRINT VERTEX ORDERING DISPOSE OF FORMULA PRINT MODEL Base Current Last Number a A11 models Interval of models a b DESCRIBE MODEL Base Current Last Number a All models Interval of models a b DISPOSE OF MODEL Base Current Last Number a All models Interval of models a b Return Characteristics of Model IS GRAPHICAL gc IS DECOMPOSABLE gc IS SUBMODEL OF gci gc2 J IS IN ONE CLIQUE factor factor2 gc 1 B 1 29 Common decompositions of models PRINT COMMON DECOMPOSITIONS DECOMPOSE MODELS set B 1 30 Tests TEST FIND LOG L FIND DEVIANCE EXACT TEST PARTITIONING TEST TEST ONE EDGE FACTORIZE ONE EDGE set FACTORIZE ONE INTERACTION set Compute Lots of Statistics for a and b Independent etc Given the Factors set SLICE factor name a factor name b set 1 B 1 31 Test list SHOW TESTS DISPOSE OF TESTS 158 Appendix B Reference Manual section B 1
174. metric associations between variables The symmetric associations have been dealt with so far in this thesis At the asymmetric association directed association between two variables the second variable might depend on the first variable but the first variable is independent of the second variable The first variable is then explanatory to the second variable the response variable In the graph we then draw an arrow from the first variable to the second variable The explanatory variables are also called the exogenous variables and the response variables the endogenous variables The endogenous variables have an undirected graph associated with them Each endogenous factor is the end point for one or more direct edges The directed edges can originate at either exogenous or endogenous variables In recursive models response variables of some explanatory variables cannot be explanatory variables to the same explanatory variables i e in the graph there can not be any oriented cycles cycles where one goes along undirected edges and in the direction of arrows Conditional independence statements for recursive causal models are examined in Wermuth amp Lauritzen 1983 The so called block recursive graphical models or graphical chain models are further treated in Lauritzen amp Wermuth 1989 Lauritzen 1989 Wermuth amp Lauritzen 1990 Lauritzen Dawid Larsen amp Leimer 1990 The chain graph Markov property is investigated in detail
175. mmands SET EXACT TEST All Deviance and SET FAST On Off are removed SET SEED seed sets the initial value seed for the random number generator SET SEED Random sets a random initial value computed from CPU time process id and or real time The epsilon set by SET EXACT EPSILON epsilon is added to test statistics for the generated table when deciding whether the generate statistics is greater than or equal to the observed statistics In the following example a exact test for A being conditionally independent of D given B C E and F and an exact test for one of the final models against the saturated is computed on the data from Edwards amp Havranek 1985 gt read data 64 cells with 1841 cases read Finding all marginals TIME 0 025secs gt set exact on Exact test set ON TIME 0 002secs gt set power lambda null Power Divergence not printed TIME 0 002secs gt read model TIME 0 001secs gt drop edge AD 2 ABCEF BCDEF Model is graphical Model is decomposable Test of ABCEF BCDEF against ABCDEF DF 0 21log Q P X 2 P Models 16 O 28 7241 0 02566 27 4652 0 03630 ABCDEF BCDEF ABCEF Exact 1000 0 04000 TIME 0 904secs 78 Chapter 6 Tests gt read model ACE ADE BC F TIME 0 002secs gt exact test Test of F BC ADE ACE against ABCDEF Statistic Asymptotic Adjusted Exact 2log Q 62 0779 P 0 0994 0 0994 0 1500 Pow
176. mp Havr nek 1985 against the saturated model gt read data 64 cells with 1841 cases read Finding all marginals TIME 0 024secs gt read model TIME 0 001secs gt read model ACE ADE BC F TIME 0 001secs gt print all models Model no 2 ACE ADE BC F CURRENT Model no 1 ABCDEF BASE TIME 0 002secs 6 5 Miscellaneous Commands for Model Control gt test Test of F BC ADE ACE against ABCDEF Statistic Asymptotic 2log Q 62 0779 P 0 0994 Power 59 7593 P 0 1396 X72 59 9956 P 0 1350 DF 49 TIME 0 019secs gt set power lambda null Power Divergence not printed TIME 0 002secs gt set adjusted df off Adjusted df set OFF TIME 0 002secs gt set test formats 8 3 6 4 TIME 0 002secs gt set factorize c TIME 0 001secs gt factorize one edge FEDCBA Test of LACE ADE BC F against ABCDEF Edge DF 2log Q P X72 EF 16 18 316 0 3052 17 820 0 3341 DF 8 9 199 0 3252 9 086 0 3346 CD 8 4 893 0 7705 4 800 0 7801 CF 4 7 018 0 1349 7 459 0 1135 BD 4 2 504 0 6439 2 502 0 6443 BE 4 6 771 0 1485 6 944 0 1389 BF 2 6 321 0 0424 6 349 0 0418 AB 2 5 988 0 0501 5 953 0 0510 AF 1 1 069 0 3012 1 070 0 3010 49 62 078 0 0994 61 983 0 1008 TIME 0 040secs gt set exact test all Exact test set ON Only Exact Deviance set OFF 0 001secs TIME 87 Adjusted 0 0994 0 1396 0 1350 49 Models ABCD
177. n Off J Default Off The default name of the DiaryFile is tmp CoCo diary no pid number under Unix and COCO DIA in the current di rectory under DOS Another DiaryFile can be named by SET OUTPUTFILE DIARY file name Note that this command rewrites the file file name and that under DOS the file COCO DIA is rewritten every time CoCo is invoked If the DiaryFile is not renamed the DiaryFile is removed when CoCo terminates normally unless the command SET KEEP DIARY On Off is used 11 3 3 Log file SET OUTPUTFILE LOG file name SET KEEP LOG On Off SET LOG On Off J SET DATA LOG On Off A log of input to CoCo is written to the LogFile unless the commands SET LOG On Off and SET DATA LOG On Off are used Default Both On The LogFile can be read from of CoCo as SourceFile Input read from DataFiles is copied to the LogFile as comments The default name of the LogFile is COCO LOG in the current directory under DOS and tmp CoCo log no pid number under Unix Another LogFile can be named by SET OUTPUTFILE LOG file name Note that this command rewrites the file file name and that under DOS the file COCO LOG is rewritten every time CoCo is invoked If the LogFile is not renamed the LogFile is removed when CoCo terminates normally unless the command SET KEEP LOG On Off is used 11 4 Timer 127 11 3 4 Dump file SET OUTPUTFILE DUMP file name SET KEEP DUMP On O
178. n which no exact test is available the adjustment is not correct The adjusted D F computed by the above algorithm can be negative The following example is used to explain how output with adjusted degrees of freedom is to be read gt set input data user jhb Examples CoCo ksl d TIME 0 003secs gt read data 1082 cases read TIME 0 981secs gt set power lambda null Power Divergence not printed TIME 0 002secs gt read model TIME 0 002secs gt drop edge SW 2 FKHBSAGY FKHBAWGY Model is graphical Model is decomposable Test of FKHBSAGY FKHBAWGY against FKHBSAWGY 74 Chapter 6 Tests DF 0 21log Q P X72 P Models 40 88 47 1253 0 20390 38 6293 0 53186 FKHBSAWGY FKHBAWGY FKHBSAGY TIME 0 048secs gt print all models Model no 2 FKHBSAGY FKHBAWGY Model no 1 FKHBSAWGY BASE CURRENT TIME 0 001secs gt current TIME 0 001secs gt test Test of FKHBAWGY FKHBSAGY against FKHBSAWGY Re use Statistic Asymptotic Adjusted 2log Q 47 1253 P 1 0000 0 2039 Power 38 6293 P 1 0000 0 5319 X72 38 6293 P 1 0000 0 5319 DF 128 40 TIME 0 009secs gt find 2log Q Test of FKHBSAGY FKHBAWGY against FKHBSAWGY 2log Q 2 5548 1011 5571 6637 47 1253 DF 511 383 128 P 1 0000 Adjustment 230 142 88 258 40 P 0 2039 TIME 0 020secs gt quit TIME 0 000secs TOTAL TIME 1 144secs The data
179. nates normally See chapter 11 about the LogFile and use the command STATUS Files to see default filenames 36 Chapter 2 Commands 2 7 Print formats Tests Diary Source Sink Log etc See chapter 11 Options 2 8 Interrupts On Unix a Control C will terminate the IPS algorithm and the computation of an exact test Control or two close Control C s will terminate the EM algorithm fac torization of a test a BACKWARD command a FORWARD command or the compu tation of a dual in the EH procedure See section 11 7 Interrupts in chapter 11 Miscellaneous Options for for how to set interrupts off Chapter 3 Reading Data The command READ DATA is used to read observations and specifications of factors from input or from a file 3 1 Keyboard and Files SET KEYBOARD On Off SET INPUTFILE DATA file name SET INPUTFILE SPECIFICATION file name SET INPUTFILE OBSERVATIONS file name READ DATA specification observations 1 READ SPECIFICATION specification 1 READ FACTORS factor list READ NAMES name list READ OBSERVATIONS observations READ TABLE table READ LIST list of cases The data consists of two parts specification and observations The specification is the definition of variables and table The observations are the cell counts in the table Data can either be read from keyboard or file This is controlled by the command SET KEYBOARD On
180. ncomplete tables and tables with incomplete observations computes exact tests between any two nested decomposable models handles larger tables and has commands for semi automatic and automatic search DIGRAM handles recursive graphical models on contingency tables MIM also handles continuous variables in mixed interaction models CG distributions Both DIGRAM and MIM runs only on Personal Computers and are on these machines able to present graphical models as graphs Using this Tutorial This guide is the second volume of the thesis A Environment for Graphical Models Volume 3 Xlisp4 CoCo A Tool for Graphical Models of the thesis is a guide to CoCo within XLISP STAT Chapter 1 of this guide is a Tutorial Introduction to CoCo A semi automatic model search with exact tests is performed on the data from Reinis Pokorny Basik Tiserov Gorican Hor kov Stuchlikov Havr nek amp Hrabovsky 1981 and some tables are printed plotted and described This part of the guide is most useful for users who are familiar with analysis of discrete data and who want a quick introduc tion to CoCo Chapters 3 to 12 make up the User Manual Commands are described in detail arranged according to the step in the analysis to which they belong Some examples are given Some of the used methods algorithms are described The appendix contains parts of a Reference Manual a Quick Reference Card and an Installation Guide The progra
181. nd Common Decompositions Chapter 6 Tests 6 1 The Main Test Procedure This section will describe how test statistics and degrees of freedom are computed by the command TEST by the model editing commands described in chapter 7 and by procedures for model selection in CoCo The likelihood ratio test statistic 2log Q i e the deviance Pearson s chi square and the power divergence are by these com mands computed as follows 6 1 1 2log Q Pearson s y and Power Divergence TEST If the generating class C for the current model is a sub generating class of the gener ating class B for the base model then the deviance Pearson s chi square x and the power divergence 2n1 for the hypothesis that p Pe under p Pg is computed as 2logQ 2 5 n ia log Polia Pelia tAElA 2 m npo ia npelia x 7 pa npclia E pa ie by the command TEST where pe and f are the maximum likelihood estimates of p under Pe and Pg respectively and A is the union of variables in C and B For each cell in the table spanned by factors in both models a term for the statistics is computed and the terms are summarized If the table spanned by the factors of the test is to big to fit into the memory af ter performing common decompositions then Pearson s chi square x and the power 71 72 Chapter 6 Tests divergence 2n14 is not be computed and the deviance is computed by the method described in section 6 4 In this
182. nd SearchBase 10 2 5 Add FixIn and FixOut 2 0 0 0 0 000000008 10 2 6 Redo FixIn and FixOut 0 2 202 004 10 3 Selecting Model Class and Search Strategy o 10 3 1 Graphical Search 2 o 10 3 2 Decomposable Mode 00004 10 3 3 Hierarchical Search 2 2 ee 10 4 Dispose of the Model Classes and Duals 10 5 Read and Fit Models or Force Models into Classes 97 97 98 98 98 101 102 103 104 104 104 105 106 107 108 108 108 109 110 111 111 Contents ix 10 5 1 Fit Some Models 0000000000084 118 10 5 2 Reading Accepted and Rejected Models 118 10 6 Copy Models Between the Models List and the Search Classes 119 10 6 1 Models from List to Search Classes 119 10 6 2 Models from Search Classes to Model List 2 119 LOSE Emad Wales oe srs cd op as a os Se A ehh ey Be 119 10 8 Directed Searchin cs tcc a See aye eB AALS hoe Boe a 119 10 8 1 Fit Smallest Dual o 120 10 8 2 Fit Largest Dual e arm dog ob Ve a Se ie G 120 10 83 Fit R D l dara da ee A E 120 10 84 Fit A Dual hdd As ee eee amp OR RARER es 120 10 8 5 2F it Both Duals ria aa a eS 120 10 9 Automatic Search 120 10 9 1 Smallest Automatic 121 10 9 2 Rough Automatic 2 o 121 10 9 3 Alternating Automatic e 0 000008 121 10 10Force a Dual into a Model C
183. ng class contains only one generator dim Pa J sea Zo 1 2 If A is the union of two generating classes A and Az dim P4 dim Pa dim Pa dim Pa ra2 where A A Az is the meet a1 Naz a1 A1 a2 Az between the two generating classes Lauritzen 1982 6 1 The Main Test Procedure 73 6 1 4 Adjusted D F For A a complete denote the number of cells with zero count in the a marginal table m ta i er by Z A m Z a m of elements in m ia i er M ta 0 For A a union of two generating classes A and Ag set Z A m a Z A m Z A2 m re Z Az A Az m For a decomposable model the number Z A n is the number of zeros in sufficient marginal plus the index v a times the number of zeros in intersecting a marginal tables Then the adjustment of the degrees of freedom is computed as Z B tne Z C n and adjusted degrees of freedom D Fagj dim Pg dim Pe 2 B me Z C n where me is the expected counts in the current model C Z C n is approximately the number of parameters not estimable in the current model C Z B c is the number of parameters used in the base model to describe the not estimable effects in the current model So Z B mc Z C n is a false reduction in the number of parameters The adjusted D F is probably only correct for decomposable models on connected tables But in decomposable models exact tests are available In the non decomposable models i
184. ngth Examples of PLOT statements plot observed expected plot observed unadjusted plot observed adjusted plot observed standard plot observed 2log Q plot observed freeman tukey plot observed 2 n m plot observed power plot f res r f plot f res g res plot log observed log expected plot index observed plot index adjusted ABCD plot base adjusted current adjusted plot base log expected current adjusted plot index complete observed plot index error See chapter 1 for an example of output See chapter 12 CoCo in other systems for how to return values from CoCo for high resolution graphics in S Plus and XLISP STAT 4 7 Lists LIST a CASE LIST LIST a lists observed counts expected counts estimated probabilities and residuals unadjusted adjusted standardized etc in the marginal table determined by the set a with values computed by summing over cells in the table determined by the union of the set a and the set B containing the intersection between a and the model set and on which the model is collapsible 4 7 Lists 61 The list can be used for e g further computation tabulation high resolution graphics etc The used print format in the list is controlled by the command SET TABLE FORMATS width dec prob dec expt dec diff The cells in the printed list are ordered according to the order of the factors in a
185. nly because it is decomposable The log file after the run of CoCo to make output in this chapter looks like Name the diary file set output diary diary 2 Set diary on set diary on Keep the log set keep log 26 Chapter 1 Introduction to CoCo Set timer on set timer on Set the input file set input data reinis81 dat Read data read data Data Factors A2 B2 C2 D2 E2 F2 Data Data Table Data 44 40 112 67 Data 129 145 12 23 Data 35 12 80 33 Data 109 67 7 9 Data 23 32 70 66 Data 50 80 7 13 Data 24 25 73 57 Data 51 63 7 16 Data 5 7 21 9 Data 9 17 1 4 Data 4 3 11 8 Data 14 17 5 2 Data 7 3 14 14 Data 9 16 2 3 Data 4 0 13 11 Data 5 14 4 4 Change the print width used when printing tables set table formats 5 2 2 2 Print counts in the full table print observed ABCDEF Print counts in the marginal table AB print observed AB Print the total count print observed Change the size of a page to fit into this guide set page formats 72 52 1 2 An Example 27 gt gt Describe the observed counts in the saturated table gt print also a uniform plot gt gt describe table uniform observed Read model ACE ADE BC F read model ACE ADE BC F Change the page size set page formats 72 36 Describe
186. no no no no no no 7 PNW B HOD ACDE ABCF ACE ADE ABF ABC ADE ACE ABC ABF CURRENT ACDE ABCF BASE ABCDEF 0 005secs 84 Chapter 6 Tests gt read model ACE ADE BC F TIME 0 002secs gt print common decompositions Decomposition of ACE ADE BC F and ACDE ABCF Decompositions ABCF 4 Ac 2 ACDE 4 Selected ABCF 4 AC 2 ACDE 4 TIME 0 006secs gt set power lambda null Power Divergence not printed TIME 0 002secs gt partitioning Test of ACE ADE BC F against ACDE ABCF DF 0 21log Q P X72 P Models 9 O 20 3954 0 01559 20 0744 0 01738 ABCF AC BC F 4 0 2 2147 0 69633 2 2138 0 69651 ACDE ACE ADE 13 O 22 6101 0 04625 22 2881 0 05069 ACDE ABCF CADE ACE F Bc TIME 0 021secs gt make base 3 TIME 0 001secs gt print common decompositions Decomposition of ACE ADE BC F and LADE ACE ABC ABF Decompositions ABCF 4 Ac 2 ACDE 4 ABCEF 5 AE C 2 ADE 3 Selected ABCF 4 AC 2 ACDE 4 TIME 0 007secs gt set short off Short test output set OFF TIME 0 002secs 6 5 Miscellaneous Commands for Model Control 85 gt partitioning Test of ACE ADE BC F against ADE ACE ABC ABF Partition of ABCDEF in ABCF ACDE by AC Test on ABCF Tes
187. nv SCOCOHOME Functions sep is then unnecessary to get access to the CoCo S functions To install Xlisp CoCo do 6a Set the environment variable XCOCO to scoco o if you are using XLISP STAT version 2 1 Release 2 or to libscoco so if you are using XLISP STAT 2 1 Re lease 3 39 or later Edit also the environment variable XLISPSTAT if necessary 6b Type make installxcoco This will edit the script xlisp coco insert the two environment variables XLISPSTAT XCOCOHOME and COCOHOME and place the result in BINDIR write the file loadcoco 1sp for loading necessary interface files copy scoco o or libscoco so to COCOHOME and copy the files cocoapix 1sp cocometh 1sp cocograph 1sp etc to XCOCOHOME Do not do step manually since other actions might have been added since the writing of this XLISP STAT must be compiled with FOREIGN_FLAG DFOREIGNCALL to make dy namic loading working See the Makefile for XLISP STAT If you are running Solaris then you should use version 2 1 Release 3 39 Beta or later of XLISP STAT After successfully having installed Xlisp CoCo you should compile the Lisp file for Xlisp CoCo Change to the directory XCOCOHOME and type make This will compile the Lisp files for Xlisp CoCo and create a saved workspace for Xlisp CoCo By the saved workspace the upstart of Xlisp CoCo will be much faster and by the byte compilation the program will run faster XLISP STAT by L
188. o XCOCOHOME e g home local lib coco Isp 2a Move the executable coco sparc or coco sun3 to coco by e g the command mv coco sparc coco unless you have uncompressed a tar file with only one file coco or 2b Compile CoCo by make coco Installation of CoCo is then done by 3 Type make installcoco This will copy necessary files coco COCO TAB COCO HLP and COCO DAT to the home directory COCOHOME for CoCo edit the script CoCo insert the environment variable COCOHOME and place the result in BINDIR You only need to do step 4 and 5 or 6 if you want to use CoCo in S Plus or XLISP STAT 4a Move the object files scoco sparc or scoco sun3 to scoco o by e g the com mand mv scoco sparc scoco o unless you have uncompressed a tar file with only one executable file scoco o or use 4b make scoco o to make the object file scoco o To install S CoCo do 5a Edit also the environment variable S if necessary 5b Type make installscoco This will edit the script S coco insert the en vironment variables S SCOCOHOME and COCOHOME and then place the result in BINDIR copy scoco o to COCOHOME make the interface functions to S Plus by running S Plus on cocoapi S and copy the resulting S functions to SCOCOHOME Functions 148 Appendix A Installing CoCo The CoCo S functions SCOCOHOME Functions could be moved to some directory in the default S Plus search list The command attach paste gete
189. o mplete and InCo mplete Co ntingency tables Version 1 3 Friday March 17 12 00 00 MET 1995 Compiled with gcc a C compiler for Sun4 Copyright c 1991 by Jens Henrik Badsberg gt set inputfile data reinis81 dat gt read data 64 cells with 1841 cases read Finding all marginals gt The declaration of factors and the table or list of cases can be placed on separate files See chapter 3 Chapter 3 also describes how to read incomplete tables group levels make data selection and handle missing values in lists of observations After reading the data any marginal table of observations can be printed with the command PRINT OBSERVED set where set is a subset of the declared factors The asterisk is an abbreviation of the set containing all factors so in the following PRINT OBSERVED ABCDEF is equivalent to PRINT OBSERVED gt print observed ABCDEF ABCDEF w 44 35 23 24 40 12 32 25 gt print observed AB AB 1 522 541 2 439 339 112 80 70 73 21 11 14 13 67 33 66 57 14 11 129 145 109 50 51 14 67 80 63 17 17 16 14 Chapter 1 Introduction to CoCo N 23 13 16 DESCRIBE OBSERVED set where set is a subset of the declared factors will give a univariate description of the values in the marginal table The SET PAGE FORMATS line length page length is used to make the plot fit into this guide Version
190. o compute e g the deviance So if you are not using the ad justed degrees of freedom anyway the computing of it may be turned off with the SET ADJUSTED DF On Off statement The current implementation of the algorithm for finding the adjustment of the degrees of freedom requires space for large models If CoCo is unable to compute the adjustment for a test and the program is running under Unix increase the value given by the N option when starting CoCo and the P option if the models are non decomposable see section 11 5 5 6 2 Exact Tests EXACT TEST The distribution of the likelihood ratio statistic is poorly approximated by the Chi Square distribution But exact p values can be computed to any accuracy by Monte Carlo approximation Random tables with margins equal to observed suffi cient marginal tables for the hypotheses are generated and it is counted how large a proportion of the generated tables have a test statistic larger than the observed The exact test for the removal of one edge in the saturated model is computed by the method in Kreiner 1987 If two decomposable models differ with one edge the test can be collapsed to a test of removal of one edge in a saturated model The algorithm by Patefield 1981 is used to generate the individual tables The Monte Carlo algorithm of Patefield 1981 can also be extended to give approx imation to any degree of accuracy of exact p values for tests between any two nest
191. o section 3 3 If the Gamma coefficient or the information criterion is computed then this value is used to determine whether to accept or reject models Thus these two values cannot be computed and printed without using the value in the model selection 8 2 Computed Test Statistics and Choosing Tests and Significance Level 99 If both the Gamma coefficient and a information criterion are computed for a model in a model selection then the Gamma coefficient is used to decide whether to accept or to reject the model Selection by Information Criteria The Information Criteria 1C Sakamoto amp Akaike 1978 for model A is computed as IC Date dim A AICA ai DA 2 dim A and BIC D a log n dim A where D4 is the deviance of A relative to the saturated model the likelihood ratio test statistic for A tested under the saturated model The change of Information Criteria is then in backward elimination and forward selection computed by A IC 4 1Cg 10 A D ag s D F 18 A AIC ag A D ag 2 D F 4g and A BIC ag A D asg log n D F 48 where A D 18 Dg D4 is the difference between the deviances for the two models considered and D F 18 is the adjusted or unadjusted number of degrees of freedom for the test of B under A An adjusted number of degrees of freedom is used if it is set on by the SET ADJUSTED DF command If B is a submodel of A then A D 18 and D F 1g is positive In a model search the
192. odel is declared as base Additional read models are given an increasing number The last read model is the current model The current model can be declared as base with the command BASE Model number 1 2 An Example 13 a is named base or current with the commands MAKE BASE a and MAKE CURRENT a respectively In the example the model ACE ADE BC F can be tested against ABCDEF by gt read model gt base gt read model ACE ADE BC F gt test Test of F BC ADE ACE against ABCDEF Statistic Asymptotic Adjusted 2log Q 62 0779 P 0 0994 0 0994 Power 59 7593 P 0 1396 0 1396 X72 59 9956 P 0 1350 0 1350 DF 49 49 gt The first column of figures is the test statistic the second is the asymptotic p values based on an unadjusted number of degrees of freedom and the third is the p values based on an adjusted number of degrees of freedom With SET EXACT TEST On exact test is turned on The exact test for one de composable model against another decomposable model containing the model is then computed by EXACT TEST gt set exact test on EXACT TEST SET ON gt exact test Test of F BC ADE ACE against ABCDEF Statistic Asymptotic Adjusted Exact 2log Q 62 0779 P 0 0994 0 0994 0 1550 Power 59 7593 P 0 1396 0 1396 0 1630 X72 59 9956 P 0 1350 0 1350 0 1570 DF 49 49 gt The following is an example of a model search with the commands B
193. of the models are not decomposable the factorization takes time The following example shows a factorization of the test of one of the final models from Edwards amp Havr nek 1985 against some model gt read data 64 cells with 1841 cases read Finding all marginals TIME 0 026secs gt read model ABF ABC ACDE TIME 0 003secs gt read model ACE ADE BC F TIME 0 002secs gt print all models Model no 2 ACE ADE BC F CURRENT Model no 1 ABF ABC ACDE BASE TIME 0 003secs 90 gt test Test of F BC ADE ACE against ACDE ABC ABF 2log Q Power X72 DF TIME Statistic 15 5920 P 15 4412 P 15 3837 P 0 028secs gt set power lambda null Power Divergence not printed TIME gt set adjusted df off 0 001secs Adjusted df set OFF TIME gt factorize one interaction Test of ACE ADE BC F 0 001secs against ABF ABC ACDE Edge AB AF BF CD ABC ABF ACD CDE ACDE TIME gt quit TIME TOTAL TIME DF 210g Q 1 31E 03 1 5 0687 0955 3559 5 9864 m 2255 0 0777 15 0 148secs 0 001secs 0 255secs 4997 2816 5922 Asymptotic 0 0754 0 0790 0 0804 9 P X 2 97113 1 28E 03 30125 1 0697 02399 5 1413 24425 1 3549 01442 5 9506 26829 1 2216 78050 0 0776 47961 0 4995 59566 0 2817 07541 15 5980 Adjusted 0 07
194. or each cycle in the EM algorithm and cumulated number of zeros in computing the adjusted number of degrees of freedom Default Off Report Off will not suppress printing on the ReportFile of time used in the IPS algorithm and time used to find duals Trace will give a more detailed reporting Default Off The two commands SET DEBUG On Off and SET OPTION On Off number value are used by the author for debugging Default Off 132 Options for Controlling Input Output and Algorithms 11 6 3 Graph mode SET GRAPH MODE On Off After the command SET GRAPH MODE On estimation and tests are skipped De fault Off You are then able to investigate graph theoretical results of using the com mands PRINT FORMULA PRINT VERTEX ORDERING PARTITIONING FACTORIZE ONE EDGE FACTORIZE ONE INTERACTION DROP EDGES ADD EDGES REDUCE GENERATOR set etc in very large graphs GraphMode is most safely turned off again by ending CoCo and restarting since the IPS algorithm is suppressed when reading models with GraphMode on 11 7 Interrupts SET INTERRUPT A B On Off On unix the command SET INTERRUPT A Off will re install the normal Control C interrupt handler After use of SET INTERRUPT A On Default a Control C will terminate the computation of an exact test and the IPS algorithm SET INTERRUPT B Off will re install the normal Control interrupt handler Control or two close Control C
195. orward The backward elimination and forward selection commands of course differ by remov ing and adding edges or interaction terms respectively In the backward elimination tests are performed against the previously accepted model if the keyword Follow is given else if the keyword Follow is not used then all the tests will be performed against the base model In the forward selection the model of the current step of the selection is tested against the model resulting from adding an edge or interaction term In the backward elimination only the least significant edge is removed in each step by default whereas in the forward selection all significant edges are added in each step by default Chapter 10 The EH procedure By the principles of weak acceptance and weak rejection coherence the search space of all hierarchical models on the contingency table is divided into two sets of models the class of minimal acceptable models and the class of maximal rejected models Hence after using the Fast Procedure for Models Search the EH procedure by Edwards amp Havr nek 1985 and Edwards amp Havr nek 1987 any model can be labeled as accepted or rejected By the EH procedure the models or sub models of a specific base model the model SearchBase read by the command READ BASE MODEL model are classified into two regions A and R weakly accepted models and weakly rejected models Weakly accepted models are models that inc
196. ot decomposable Sorted list Edge DF 2log Q P X72 P Models CF 4 7 018 0 1349 7 459 0 1135 R ABCF ABC ABF Exact 1000 Exact 100 0 1320 0 0930 EF 4 3 951 0 4127 4 075 0 3959 R ABEF ABE ABF 0 3900 0 3800 EF Edge selected Rejected edges Accepted edges Model Edges eligible AB LACF ADE Sorted list CBE AD DE AE BC AC CBD CD BE EF DF CE AF BF CF AB ABCF ABE ADE CAB AF BF CF BCF BE is not decomposable Edge DF 2log Q P X72 P Models BF 4 11 151 0 0249 13 302 0 0099 R ABCF ABC ACF Exact 1000 0 0270 0 0060 AF 4 7 386 0 1169 7 336 0 1192 R ABCF ABC BCF Exact 200 0 1200 0 1200 1 2 An Example 19 CF 4 7 018 0 1349 7 459 0 1135 R ABCF ABC ABF Exact 1000 0 1320 0 0930 Edge selected CF Rejected edges IBF BE AD DE AE BC AC Accepted edges LBD CD BE EF DF CE AF BF CF AB Model CABC ABE ABF ADE Edges eligible CAF AB AB AC ADE AF BC BE BF is not decomposable Sorted list Edge DF 2log Q P X72 P Models AF 2 2 658 0 2647 2 648 0 2661 R ABF AB BF Exact 100 0 2300 0 2300 Edge selected AF Rejected edges BF BE AD DE AE BC AC Accepted edges CBD CD BE EF DF CE AF BF CF AB Model ABC ABE ADE BF Edges eligi
197. ped by Leslie Lamport PostScript is a registered trademark of Adobe Systems Inc S PLUS is a registered trademarks of Statistical Sciences Inc New S is a trademark of AT amp T Bell Laboratories UNIX is a registered trademark of AT amp T Bell Laboratories X Window System is a trademark of the Massachusetts Institute of Technology Sun SunOS Solaris and OpenWindows are registered trademarks of Sun Microsystems Inc Macintosh is a trademark of Macintosh Laboratory Inc licensed to Apple Computer Inc LaserJet is a trademark of Hewlett Packard Co Preface CoCo is a program for estimation test and model search among hierarchical interac tion models for large complete contingency tables The name CoCo is derivated of Co mplete Co ntingency tables since the initial program could only handle com plete tables but the program has been enhanced to handle incomplete tables CoCo works especially efficiently on graphical models and some of the commands are designed to handle graphical models Graphical models are log linear interaction models for contingency tables that can be represented by a simple undirected graph with as many vertices as the table has dimension Further all these models can be given an interpretation in terms of conditional independence and the interpretation can be read directly off the graph in the form of a Markov property The class of graphical model is a proper subclass of the hierarchical models but t
198. pter 1 for examples of output 4 6 Plots PLOT x value name y value name a A scatter plot of one value against another value in the same marginal table is made by the PLOT statement PLOT Current Use values from current model Base Use values from base model value name Name of x value Current Use values from current model Base Use values from base model value name Name of y value a Plot the values in the a marginal table The command PLOT x value name y value name a plots the values z value name in the a marginal table against the values y value name in the same 60 Chapter 4 Description of Data and Fitted Values table The values are by default computed in the current model Values for the current model can be plotted against values for the base model with the key words Current and Base With the keyword Complete only values for non structural zero cells is plotted and with Log value name the value is log transformed A value can be plotted against the number of the cell in the a marginal table in a standard order by letting 2 value name or y value name be Index This may be useful to find cells with large residuals patterns in the residuals or changes in residuals between models by making the same plot for more models The plots are made to fit a page The size of a page is controlled by page length and line length in the statement SET PAGE FORMATS line length page le
199. rfacing CoCo To be able to edit and redo entered lines in XLISP STAT you may want to use fep fep xlisptcoco To create a CoCo object use the function make coco def reinis coco object make coco By using the function def instead of setf the created objects variables etc can be listed by the function variables The specification of the table is then sent to the CoCo object by the method enter names send reinis coco object enter names 2 QU gr WGN pl E nen 2 2 222 2 00000 0 lMake sure that you or your system administrator has done step 4 and 5 of the installation of CoCo see appendix A and that XLISP STAT is available on your system XLISP STAT must be compiled with FOREIGN_FLAG DFOREIGNCALL to make the dynamic loading working See the Makefile for XLISP STAT 140 Chapter 12 CoCo in other systems The method enter names has three arguments A list with the names text string of the variables a list of integers with the number of unmarked levels for each factor and an optional list of integers with the number of levels to be marked as missing The variable n to hold the table of observed counts in a list is defined by def n 44 40 112 67 129 145 12 23 35 12 80 33 109 67 T 9 23 32 70 66 50 80 7 13 24 25 73 57 51 63 7 16 5 7 21 9 9 17 1 4 4 3 11 8 14 17 5 2 7 3 14 14 9 16 2 3 4 0 13 11 5 14 4 4 Instead of the form list could have been used But if
200. riables to be read are skipped The subset of factors to read is controlled by the command SET READ All Subset set 3 9 2 Exclude Missing On Off Use Maximal Number of Cases with Complete Information in each Test EXCLUDE MISSING On Off EXCLUDE MISSING On Off and EXCLUDE MISSING In set is used after reading the data without using the command SKIP MISSING All cases are read into the program if not some cases are selected or rejected After the command EXCLUDE MISSING On only cases with complete information about all used factors are considered ExcludeMissing may be turned off again with EXCLUDE MISSING Off Consider the tables onto which the tests in the commands TEST FIND LOG L FIND DEVIANCE EXACT TEST TEST ONE EDGE PARTITIONING TEST GENERATE DECOMPOSABLE MODEL GENERATE GRAPHICAL MODEL DROP EDGES edge list ADD EDGES edge list DROP INTERACTIONS gc ADD INTERACTIONS gc DROP FACTOR factor MEET OF MODELS JOIN OF MODELS REDUCE GENERATOR set REMOVE GENERATOR set REMOVE TOTAL INTERACTION set BACKWARD Edges Interactions FORWARD Edges Interactions FACTORIZE ONE EDGE FACTORIZE ONE INTERACTION and the tests in the EH procedure are col lapsible Then if ExcludeMissing is on then all cases with complete information in these tables are included in the tests Hence the number of observations entering in to tests may be varying from test to test 3 9
201. rminates normally unless the command SET KEEP REPORT On Off is used 11 4 Timer SET TIMER On Off Printing of computer time used on each command can be turned on and off with SET TIMER SET TIMER On and SET TIMER Off Default Off 11 5 Controlling Algorithms SET ALGORITHM A1BICID The command SET ALGORITHM will set an option to choose between different algo rithms for factorization computation of the Gamma coefficient exact p value fill in etc See chapter 6 and the online help information of CoCo for further details 128 Options for Controlling Input Output and Algorithms 11 5 1 Controlling the IPS algorithm SET IPS Cell Sun SET IPS EPSILON epsilon SET IPS ITERATIONS maz The convergence criterion for the IPS algorithm can either be convergence of cell probabilities or convergence of log L If cell probabilities are chosen the IPS algorithm is repeated until the difference between estimated cell probabilities from one cycle to the next for all cells is less than epsilon or until the maximum number of cycles is obtained The command SET IPS Cell sets convergence criterion to cell probabilities SET IPS Sum sets convergence criterion to log L The IPS algorithm is repeated until the difference between estimated log L values from one cycle to the next is less than epsilon or until maximum cycles Epsilon and maximum cy cles are set with SET IPS EPSILON epsilon and SET
202. rward 9 1 Selecting Test Statistic and Significance Level The previous chapter will describe how to set the options for the computed test statistics select the test statistic and how to set the level of significance 9 2 Fixing of Edges and Interactions FIX Edges edges gc y AND FIX Edges edges gc y The command FIX Edges gc will fix the edges and interaction terms given by the argument of the command in the models considered in the stepwise model selection commands In a backward elimination of interactions the interaction terms a subset of a gen erator in the generating class set by FIX Edges gc are not eligible for removal Also in a forward selection adding interaction terms containing a fixed generator to the model may not be considered In backward elimination of edges on graphical models the edges i e first order interactions given by FIX Edges edges are not tested for removal and are never added in forward selection of edges Thus in a forward selection of edges on graphical models an edge can be fixed without being containing a fixed generator Edges and interactions in the FixEdgeList can still be added or removed by the model editing commands i e by the Drop Reduce Remove and Add commands The FixEdgeList set by FIX Edges edges and AND FIX Edges edges is in dependent of the FixIn and Fix0ut used in the EH procedure described in the fol lowing chapter Edges interactions can
203. ry should be selected just after reading the specifi cations The data structure File can be used but since the propagation methods described in Lauritzen 1991 are not implemented in CoCo it will be very slow If the data structure File is selected missing values in tables that are too large to be stored in the memory can be handled if the sufficient marginals for the model can be stored in the memory But it will take time Initial values for the EM algorithm are set by SET EM INITIAL Uniform First Last Mean Random Input Uniform means a uniform distribu tion First Last and Mean sets initially the variables with values marked as missing to lowest observable level highest observable level and rounded mean value of observed factors for the case respectively Mean is useful to control the value of latent variates With Input the initial values can be given as input to CoCo By Input the initial value of a factor with a value marked as missing is set to a level equal to the coded level for this factor of the case minus the number of non missing un marked levels for this factor Random sets missing levels to a random observable level The same random generator as used in ExactTest is used SET SEED seed sets the initial value seed for the random number generator SET SEED Random sets a random initial value computed from CPU time and or real time Cycles in the EM algorithm is repeated until the difference between valu
204. s described plotted listed or exported to files by the commands of the following sections The values can be marginal counts probabil ities expected values and residuals as expected minus observed counts standardized adjusted etc 4 1 Notation and Computation of Marginal Values Let A denote declared variables For each variable Y A the set Zs 1 2 Zs is the set of unmarked levels 5 for the variable A cell in the table is then an element i is sea in the product I of level sets i ia is E A T Ta X5eaTs The number of cells in the table is Za se Zs The counts n i Jiez are the contingency table The number A of variables is the dimension of the table For a subset a of the declared variables A the a marginal table is obtained by only classifying the observations according to variables in a The marginal table on a has then cells la a is a Ta X seals and marginal counts nia ee n j The number of cells in the a marginal table is Za 5 Zs If the table is incomplete then a table q iez with q i 0 for cells that are structurally zero is defined The Q table might be defined as a product of a marginal tables q ta i ez a A 52 4 1 Notation and Computation of Marginal Values 53 Figure 4 1 Collaps of Mx onto B containing an A 0 if3JaE A qa ia 0 a i J aalia 4 1 if Va A da ia 1 art Tuc alia otherwise If some
205. s from current model Base Use values from base model value name Name of value to print in table a Print values in the a marginal table The format of the numbers printed in the table is controlled by the command SET TABLE FORMATS width dec prob dec expt dec diff The layout of the table is controlled by the command SET PAGE FORMATS line length page length See chapter 11 Examples of PRINT TABLE statements print observed print table expected 4 3 Printing Tables 57 print table probabilities ABCDE print table random observed print table random base expected ABCD print table unadjusted ABCD print table f res print table adjusted ABCDE print table log 2log Q print table log power print table log standard print table complete observed ABCDEF print table error ABCD print table zero AB For examples of output see chapter 1 The cells in the printed table are ordered according to the order of the factors in a The first factor met in a is alternating at the fastest speed With PRINT STANDARD TABLE value name a the cells are ordered according to the order of the factors in the specification of the factors when declaring the table Row Column Total percents or percents of any other marginal When value name is Observed and the optional set b is given then the counts in the a marginal table divided by the counts in the corresponding cell
206. s gc in the ADD FIX In command 10 2 6 Redo FixIn and FixOut The command REDO FIX Out will redo the combination of the last FIX Out command and following ADD FIX Out commands FixIn is then edited by the total specified FixOut Similarly will the command REDO FIX In redo the combination of the last FIX In command and following ADD FIX In commands FixOut is then edited by the totally specified FixIn 10 3 Selecting Model Class and Search Strategy SET Graphical Hierarchical SEARCH SET Smallest Alternating Rough SEARCH The option set by these command can also be given as an argument to the command EH But by setting the option with this command e g the calls of the directed search is shorter to type Describing the command gives the possibility of describing the sub classes to which the EH procedure is restricted 10 3 1 Graphical Search In the graphical search for each step either all not classified models in the graphical A dual are fitted or all not classified models in the graphical R dual are fitted DI R The graphical A dual D R of the rejected models is the minimal models in the class of models that adds at least one edge to each rejected model D A The graphical R dual DA A of the accepted models is the maximal models in the class of models that omits at least one edge from each accepted model 10 3 Selecting Model Class and Search Strategy 117 The command SET Graphical SEARCH wil
207. s in your PATH Type CoCo or fep CoCo CoCo will then start with something like CoCo z A program for estimation test and model search in very large Co mplete and InCo mplete Co ntingency tables Version 1 3 Friday March 17 12 00 00 MET 1995 Compiled with gcc a C compiler for Sun4 Copyright c 1991 by Jens Henrik Badsberg gt is the normal prompt from CoCo If CoCo is expecting integer or real numbers filenames factor names sets generating classes data etc the prompt is changed Give the expected data to CoCo or try some numbers names or CoCo is ended with quit 1 2 An Example The data of table 1 1 from Reinis et al 1981 is also used in Edwards amp Havr nek 1985 4 Chapter 1 Introduction to CoCo B No Yes A No Yes No Yes F E D C Negative lt 3 lt 140 No 44 40 112 67 Yes 129 145 12 23 gt 140 No 35 12 80 33 Yes 109 67 7 9 gt 3 lt 140 No 23 32 70 66 Yes 50 80 7 13 gt 140 No 24 25 73 57 Yes 51 63 7 16 Positive lt 3 lt 140 No 5 7 21 Yes 9 17 1 gt 140 No 4 3 11 Yes 14 17 5 2 gt 3 lt 140 No 7 3 14 14 Yes 9 16 2 3 gt 140 No 4 0 13 11 Yes 5 14 4 4 A smoking B strenuous mental work C strenuous physical work D systolic blood pressure E ratio of a to 8 lipoproteins F family anamnesis of coronary heart disease Table 1 1 Risk f
208. s useful and will terminate after an acceptable computing time Also exact tests by Monte Carlo simulation and the EM algorithm are useful on these tables Before any computation all the marginal tables are found to reduce the computing time unless some cases have unobserved variables iii Medium tables tables with up to 20 variables Any test between two hierarchical models can be performed The procedures for backward elimination and forward selection of edges in graphical models are useful In these tables only sufficient tables of observed counts and tables of estimated probabilities for few tables can be stored internally in the computer Large tables tables with more than 20 variables We do not say that all models on large tables are large e g a model with only main effects is called a small model If all the tables of the sufficient marginals of a model cannot fit in the com puter memory at one time together with the state spaces of all the non decomposable components of the model then the model is large A model is very large if any of the sufficient marginal tables cannot fit in the computer memory We cannot handle very large models if we cannot at least store the state spaces of the non decomposable components of the model one by one Such models are called huge models Thus in very large models the largest clique of the fill in graphs of non decomposable atoms of the graph of the very large model cannot have mor
209. sberg Department of Mathematics and Computer Science Institute for Electronic Systems Aalborg University Fredrik Bajers Vej 7 DK 9220 Aalborg DENMARK Fax 45 98 15 81 29 Phone 45 98 15 85 22 ext 5074 Email coco iesd auc dk 145 146 Appendix A Installing CoCo A 2 Installation The following section is the installation guide for both the standalone version of CoCo and of the interface functions for the graphics The graphics will only run on Unix systems with XLISP STAT XLISP STAT must on your system be compiled so it is able to do dynamic loading A 2 1 DOS On PC Read the latest READ ME and README DOS file Copy the contents of the distribution disk into a directory the home directory for CoCo Add the home directory of CoCo to your path The files COCO EXE COCO OVR and COCO TAB must be in the home directory for CoCo The two files COCO HLP and COCO DAT should also be there If the HelpFile COCO HLP not is in the directory there will be no online help information available in CoCo The default DataFile COCO DAT can be replaced by another data file By copy b coco exe coco ovr the overlay file is attached to the end of the EXE file COCO EXE You can then remove COCO OVR Type COCO to start CoCo Test examples are found in the directory EXAMPLES Run the test examples with runall or runsome runall will take approximately 24 hours and use three Mbytes of diskspace to diaries Datasets and CoCo source files are
210. set and onto which the model is collapsible The command DUAL TO NORMAL will generate a model where the generating class for the current model is the dual representation of the generated model The dual representation is the minimal interaction terms set to zero Together with MEET OF MODELS this can be used to do DROP INTERACTIONS gc on several models see chapter 7 Editing Models with Tests Analogously with NORMAL TO DUAL See chapter 10 Search for classes of models 5 2 The Model List Shifting Base and Current BASE CURRENT MAKE BASE a MAKE CURRENT a In tests the model called current is tested against base The model first read is both base and current and is given the model number 1 The first model stays base until another model is declared as base Additional read models are given an increasing number The model last read is the current model The current model can be declared as base with the command BASE Model number a is named base or current with the commands MAKE BASE a and MAKE CURRENT a respectively The last model generated as a result of the procedures DROP EDGES edge list ADD EDGES edge list DROP INTERACTIONS gc etc is the last model and can be named current with the command CURRENT 5 3 Printing Describing and Disposing of Models PRINT MODEL Base Current Last Number a All models Interval of models a b These six commands respectively prints t
211. st statistic can be partitioned as 2logQ 2 Xon n ia log mia ia EIn o PB ia PB in n ia n e NOES Pe ta Pez ia M ia n 2 Y nin og Dy 4 Y nlin flo A TA C2 Ca ihz ia Ela ina Elaz 2logQ 2log Qs If B2 Cz then 2 log Q2 0 and the test is collapsed onto A This is used in TEST ONE EDGE see the next section If base and current have common decompositions then they are printed with the command PRINT COMMON DECOMPOSITIONS If the two models base 6 and current C are decomposable with respect to the set a they are decomposed into 4 new models B B2 C and C2 with DECOMPOSE MODELS a The 4 models can be seen with PRINT MODEL A11 The command PARTITIONING TEST performs recursive and automatic common decompositions on base and current Tests are performed If tests are collapsible it is stated If cases with incomplete information Missing Values are read and EXCLUDE MISSING On has been used each test performed as a result of PARTITIONING TEST is performed with all the cases with complete information in the marginal table on which each test is collapsible With the command SET PARTITIONING On before BACKWARD FORWARD FACTORIZE ONE EDGE set etc each test caused by the command is parti tioned This is turned off again with SET PARTITIONING Off If ExactTest and Partitioning is on then exact test is computed for decomposable models that dif fer with one edge If
212. t model The resulting model is tested against the base model 7 10 Edit the Models without Test Only With Only before one of the Drop Reduce Remove or Add commands then the test is skipped Only is a button It is effective until after the next model editing command or until after the next BACKWARD or FORWARD command Chapter 8 Controlling the Computed Test Statistics in the Model Selection Procedures The commands in this chapter will control the criterion for judging the suitability of models in model selection in the CoCo i e the commands will set options for how to compute test statistics select test statistics and set limits for when to accept or reject a model in the model selection by the commands BACKWARD and FORWARD and in the EH procedure Also a command for restricting the model class of the selection procedures to the sub class of decomposable models is described in this chapter Besides by the options set by commands in this chapter the model selection may depend on choice of how to handle missing values options for controlling the IPS algorithm options for controlling output as options for print formats timer and diary etc Options controlling the computed test statistics and options controlling exact tests will also effect the values computed by the commands TEST EXACT TEST etc and the model editing commands 8 1 Decomposable Mode ls SET DECOMPOSABLE MODE On Off This option will restr
213. t of AC BC F against ABC ABF Statistic Asymptotic Adjusted 2log Q 13 3773 P 0 0199 0 0199 Power 13 1500 P 0 0218 0 0218 X72 13 1500 P 0 0218 0 0218 DF 5 5 Collaps Models ADE ACE and ACE ADE on ACDE identical TIME 0 021secs gt quit TIME 0 001secs TOTAL TIME 0 174secs In the output from PRINT COMMON DECOMPOSITIONS all common decompositions are printed For each decomposition a line with the 3 sets A1 a and Ag is printed The cardinality of the sets is noted The decomposition where the largest of the two cardinalities of Ay and Az respectively is minimum is selected Note in the short test output from PARTITIONING TEST The characters and are used to express that the parts of the test adds up to the total test 6 5 2 Testing One Edge If two decomposable models base and current differ with exactly one edge a 6 then both models are collapsible onto the clique a containing the edge a 3 in the base model The a marginal table of maximum likelihood estimates is determined by Peta N ia n and M E N ia a M ta B Pelia O aye E g the deviance can be found as 9 log Q 2 5 n ia log NINE ta la by only summing over cells in the a marginal table The command TEST ONE EDGE finds the test statistics by summing only over cells in the table on which the test is collapsible if the two models are decomposable and differ with exactly one
214. tables Observed counts estimated counts and probabilities and residual absolute ad justed standard Freeman Tukey etc can be listed plotted pairwise printed in tables and given an univariate description with mean variance median range etc Finally to make the use of CoCo somewhat easier and more flexible some data selection is possible and cases with missing values incomplete observations can be excluded at various levels in CoCo or the EM algorithm applied CoCo can exclude all observations with any variable marked as unobserved when reading the observa tions exclude observations with variables unobserved in a given set after reading the observations or exclude observations with relevant variables for a given test marked as unobserved when performing the test CoCo can be loaded into New S S Plus and XLISP STAT Functions for returning statistics and residuals from CoCo e g high resolution plotting are provided The program is originally designed to test methods described in the dissertation Badsberg 1986 Size of Tables The range of the sizes of the tables CoCo can handle can be classified as follows Tiny tables 2 and 3 dimensional tables In these tables a wide range of measures of associations on the 2 dimensional tables given other variables can be computed Small tables tables with up to between 7 and 10 variables On small tables the global model search procedure of Edwards amp Havr nek 1985 i
215. tables null TIME 0 001secs gt gt Ignore non decomposable models gt gt set decomposable mode on Decomposable mode set ON TIME 0 001secs gt gt Do not compute the adjusted number of degrees of freedom gt gt set adjusted df off Adjusted df set OFF TIME 0 001secs gt gt Do not compute power divergence gt gt set power lambda null Power Divergence not printed TIME 0 002secs gt gt Change the size of a page to fit into this guide gt gt set page formats 72 256 TIME 0 002secs gt gt Change the test formats gt gt set test formats 8 3 6 4 TIME 0 000secs 16 Chapter 1 Introduction to CoCo gt gt Do a recursive backward elimination gt where tests are done against the previously accepted model follow gt and edge exclusions once rejected remain rejected coherent gt Print only the sorted list of tests only sorted gt gt only sorted recursive coherent follow backward Sorted list Edge DF 21log Q P x 2 P Models AC 16 42 804 0 0005 41 617 0 0007 R ABCDEF ABDEF BCDEF Exact 1000 0 0000 0 0000 BC 16 684 989 0 0000 631 304 0 0000 R ABCDEF ABDEF ACDEF Exact 1000 0 0000 0 0000 AE 16 40 024 0 0010 38 762 0 0015 R ABCDEF ABCDF BCDEF Exact 1000 0 0030 0 0020 DE 16 31 059 0 0132 30 222 0 0168 R ABCDEF ABCDF ABCEF Exact 1000 0 0310 0 0230 AD
216. th DISPOSE OF MODEL Last or it can be named current with the command CURRENT The command BASE names the current model base The following commands in this chapter treat base current and last model in the same way 7 4 Add Edges The command ADD EDGES edge list adds the edges edge list to the 2 section graph for the current model and tests the resulting graphical model against the base model If base and current are the same models then the current model is tested against the generated model else the generated model is tested against base model if not the key word Only is used 94 Chapter 7 Editing Models with Tests 7 5 Drop Interaction With the command DROP INTERACTIONS gc all interaction terms which contain a set in gc are set equal to 0 In CoCo this is computed as follows The command DROP INTERACTIONS gc adds the sets gc to the dual representation for the current model Supersets of other sets in the resulting set of sets are removed and normal representation for that dual representation is found The generated model is tested against the base model 7 6 Add Interaction The command ADD INTERACTIONS gc adds the sets gc to the generating class for the current model Subsets of other sets in the resulting set of sets are removed and the generated model is tested against the base model If base and current are the same models then the current model is tested against the generated model else the generated model
217. the backward elimination will be eliminated The only result is then the models being inserted into the model list 106 Chapter 9 Stepwise Model selection Backward and Forward Short Short BACKWARD By this keyword only a short report of the elimination is given For each edge or interaction term removed the corresponding test is printed If Dump is set On and the keyword Short is given then for each completed step of the backward elimination i e each removal of an edge or interaction the DumpFile is rewound and a report of rejected accepted eligible edges and the model resulting of the step is printed on the DumpFile In the following step of the backward elimination each tested edge and the selected test statistic for the test of the edge is printed on the DumpFile This may be useful for monitoring a model selection on a large table and for recovering a terminated model selection 9 3 2 Recursive Search Recursive If the keyword Recursive is given then the backward elimination is done recursively until no more edges or interaction terms can be removed according to the selected significance level Coherent Recursive Coherent BACKWARD Once an edge or interaction term in the backward elimination process is rejected it is no more tested for removal Headlong Headlong Recursive BACKWARD or Headlong Recursive Coherent BACKWARD This keyword gives a recursive backward elimination strategy where the first edge or
218. the current model is tested against the generated model else the generated model is tested against base model if not the key word Only is used The command BASE names the current model base 7 2 Add Fill In Generate Decomposable Model A FILL IN is made adding extra edges which is minimal in the sense that no subset of the fill in will make the graph decomposable The minimal FILL IN is added to the 2 section graph for the current model The generated model is the model with the generating class equal to the cliques in that decomposable 2 section graph Note The added FILL IN is not unique e g there are two ways to add an edge to a graph with only four vertices and consisting of a four cycle in order that the graph becomes decomposable and is not minimum there may be another FILL IN not a subset of the selected with fewer edges If base is equal to current then the current model is tested against the generated model else the generated model is tested against base model if not the key word Only is used 7 3 Drop Edges With DROP EDGES edge list the edges edge list in the 2 section graph for the current model are removed and the generated model is tested against the base model Note that the generated model is graphical Use DROP INTERACTIONS gc if a first order interaction is to be removed from a hierarchical model The current model stays current The generated model is added to the model list It can be disposed of wi
219. the list is long you will get a stack error The list n containing the table of counts is then sent to the CoCo object by send reinis coco object enter table n The following three messages will read the 3 models into the CoCo objects and create model objects for these def model 1 send reinis coco object make model def model 2 send reinis coco object make model ABCF ACDE def model 3 send reinis coco object make model ACE ADE BC F and graphs for the 1st and 3rd model is created by def graph 1 send model 1 make graph title Reinis CoCo Graph def graph 3 send Model 3 make graph location list 700 50 title Reinis ACE ADE BC F You can now move vertices set variable labels drop and add edges etc with the mouse and perform e g a model search by selecting items from the menu See Part II of this thesis To set labels on the vertices in the 1st graph send graph 1 vertex label A A Smoking send graph 1 vertex label B B Mental send graph 1 vertex label C C Physical send graph 1 vertex label D D Blood pressure send graph 1 vertex label E E Ratio of lipoproteins send graph 1 vertex label F F Family anamnesi 12 2 XLISP STAT CoCo 141 Figure 12 2 The result of a Headlong Backward on Graph 1 with added p values The bit mat dump is created by Xlisp CoCo and move the vertices in the graph send graph 1 send graph 1
220. tural zero cells in the a marginal table Values for the structural zero cells are excluded from plots and the univariate descriptions and in tables and lists the char acter is printed for structural zero cells If the optional key word Log is given the values are log transformed before returned printed described plotted or listed but not before summing over cells in the BUa marginal table 4 2 Values 55 If the optional key word Random is given then the values are computed in a random table with sufficient marginals as in the current model or in the base model if keyword Base is given See Aickin 1983 pp 152 for F res R F G res and R G Bishop Fienberg amp Holland 1975 pp 136 for Standardized 2log Q and Freeman Tukey Haberman 1974 pp 138 for Adjusted Standardized 2log Q Freeman Tukey and 2 n m Haberman 1978 pp 272 for Adjusted and Standardized and Read amp Cressie 1988 for Power value name Base Compute the values under the base model Current Compute the values under the current model default Complete Compute only values with q ia 4 0 See definition of value Zero Log Natural logarithm of the values Random Compute the values in a random table Observed Nia Y j j 1 MI Probabilities 2400 Y jija ia Pal gt invajezio PB JauB Expected mia npa ta Unadjusted Mia Nia Adjusted Ejs are njeue nPB Bua VaiJa la Jeli n jBUa NPB jBUa
221. tween the two subgraphs is complete The graph is decomposed in to the two subgraphs The subgraphs might be decomposed in to further subgraphs If the graph and subgraphs can be decomposed recursively until the subgraphs are complete the graph is decomposable The maximum likelihood estimate for the cell probabilities can be expressed as Pali e nla k 1 u if the model is decomposable The constant c is equal to 1 Z4c where A is the union of variables in the model See Lauritzen 1982 and chapter 2 of Part I If the model can be decomposed but the model not is decomposable then the maximum likelihood estimate for the cell probabilities is found by fali c II n ia 00 II PB i k 1 u l 1 v where B is the generating classes for irreducible components atoms that cannot be decomposed further The irreducible components az and B are vertex sets for the subgraphs into which the graph is decomposed The maximum likelihood estimates Pg for the cell probabilities on the non decomposable irreducible components B are found by the IPS algorithm the iterative proportional fitting iterative proportional scaling or Deming Staphan algorithm see e g Darroch amp Ratcliff 1972 Fienberg 1970 Haberman 1972 Jirousek 1991 and chapter 2 of Part I The formula and vertex ordering is printed with PRINT FORMULA and PRINT VERTEX ORDERING respectively The following continuous the Introduction the final model
222. ubsequential EH commands needing a model class else the class of models to use is selected by the options set by the commands SET Graphical Hierarchical SEARCH and SET DECOMPOSABLE MODE On Off or other commands setting model class 10 8 Directed Search FIT Decomposable Graphical Hierarchical A dual R dual Both duals Smallest dual Largest dual This command is used to do a single step of the EH procedure 120 Chapter 10 The EH procedure The keywords Decomposable Graphical and Hierarchical is as for the FIND DUAL command 10 8 1 Fit Smallest Dual With the command FIT Smallest dual both duals are found if not already found and all not classified models in the dual with the fewest not classified models are fitted and classified into accepted and rejected models 10 8 2 Fit Largest Dual With the command FIT Largest dual both duals are found if not already found and all not classified models in the largest dual are fitted and classified into accepted and rejected models 10 8 3 Fit R Dual The command FIT R dual finds if not found and fits the R dual Repeated use of FIT R dual gives a backward elimination When there are no more models to fit then the command FIT R dual will use little computing time so extra FIT R dual commands will only produce output without using much computing time 10 8 4 Fit A Dual The command FIT A dual finds if not found and fits the A dual As at t
223. ufficient marginal tables chapter 2 of Part I Tests between models with an unlimited number of factors can then be performed on a PC if the largest clique in the fill in graphs of non decomposable atoms of graphs for the models has no more than 14 binary vertices 24 on workstations il The likelihood ratio test statistic Pearson s x test statistic and the power diver gence statistics can be computed In sparse tables an adjusted number of degrees of freedom is computed Exact tests between any two nested decomposable models can be performed Tables with structural zeros Incomplete tables can be declared and a modified version of the IPS algorithm is used on these Commands for interactive model search among graphical models are implemented A semi automatic model search is possible by backward elimination and forward selec tion The model search from Edwards amp Havr nek 1985 is included in the program Besides functions for tests and model search CoCo also has some procedures for model control The test of one decomposable model against another decomposable model can be factorized into a sequence of tests with one edge Sundberg 1975 Fry denberg amp Lauritzen 1989 and the test between any two submodels can be factorized in tests with one interaction Functions for producing these factorizations are avail able in CoCo CoCo also has a function for computing a wide range of measures of associations on 2 dimensional
224. uke Tierney can be obtained by anonymous ftp over internet from umnstat stat umn edu 128 101 51 1 or mail a message containing the line send index from xlispstat to statlib templer stat cmu edu for how to find out to obtain XLISP STAT from the statlib archive XLISP STAT can be obtained free of charge fep is a very useful tool when running CoCo and XLISP STAT In the manual page to fep the general purpose front end processor by K Utashiro Software Research Associates Inc Japan the Email address utashiro sra junet is found fep can be obtained free of charge S Plus by Statistical Sciences Inc P O Box 85625 Seattle WA 98145 1625 U S A is a commercial product It adds features to New S by Becker Chambers and Wilks see Becker et al 1988 for New S Electronic mail Europe sales statsci co uk mktg statsci co uk support statsci co uk USA sales statsci com AS Limitations and Precision 149 Dos Macintosh Unix Maximum number of factors 64 128 256 Maximum number of levels 60 124 252 Maximal cell count 65536 2147483644 2147483644 N array initial 32766 4096 65536 maximum do 27 storage mode int long long P array initial 10922 4096 65536 maximum do Dot storage mode real float float Q array initial 1024 1024 1024 maximum do ea storage mode int long long Table A 1 Limits and Storage Mode A 3 Limitations and Precision Limitations
225. uments to the commands READ SPECIFICATION specification READ FACTORS factor list READ NAMES name list READ OBSERVATIONS observations READ LIST list of cases and READ TABLE table are read from where the command SET KEYBOARD On is read normally keyboard else from the source file where the command is read After SET KEYBOARD Off the arguments to the six read commands are read from the default data file or the data file named by the commands SET INPUTFILE SPECIFICATION file name SET INPUTFILE OBSERVATIONS file name and SET INPUTFILE DATA file name 11 2 2 Data files SET INPUTFILE SPECIFICATION file name SET INPUTFILE OBSERVATIONS file name SET INPUTFILE DATA file name These commands set the names on DataFiles from which the specification and the observations are read The Specification and the observations can be read from separate files Note that the files are rewinded when the commands are used See section 3 1 for further details Use STATUS Files to see the setting of inputfiles 11 2 3 Standard Input Source SET INPUTFILE COMMANDS file name The abbreviation SOURCE file name is available After this command input is read from the SourceFile file name When EndOfFile is found on the file file name input is expected from the file or standard input where the command SET INPUTFILE COMMANDS file name SOURCE file name was given When CoCo is
226. ur formal ism and not symbols of CoCo The curly brackets denote possible repetition of the enclosed symbols zero or more times name a z A Z O 9 C Z name filename a z A Z O 9 filename command name name name abbreviation command name name digit 0 9 sign unsigned integer digit digit y integer sign unsigned integer unsigned integer real integer integer unsigned integer integer e integer integer unsigned integer e integer unsigned integer list unsigned integer unsigned integer format width decimals width dec prob dec expt dec diff x width a dec prob width prob dec line length page length number of lines a b value order of observed levels of missing levels max seed number unsigned integer lambda kappa p value epsilon real EndOfCommand EndOfLine factor a z A Z O0 9 4 A 1 1 lt gt name factor name factor factor name factor EndOfLine set a a factor name y 1 gc a set a set a models a gc a 4 gc a y s list 4 Taone factor name m list s list s list set set a factor name factor name EndOfCommand
227. vels marked as missing i T A Z then for each cell j T jp ip in the A b marginal table determined by the factors b with unmarked levels i Zs for 6 b in the cell i the estimated probability j of the cell j the sum p t of these probabilities and an identification of the cell 7 is printed The levels js for factors Y A b with marked levels in the cell to describe are in the identification marked by the asterisk The following example is on data from the Glostrup Study see section 6 1 3 62 Chapter 4 Description of Data and Fitted Values Version 1 3 Friday March 17 12 00 00 MET 1995 Compiled with gcc a C compiler for Sun4 Compile time Mar 17 1995 13 00 00 Copyright c 1991 by Jens Henrik Badsberg Licensed to gt set input data ex7a dat gt read specification gt set read subset ABC gt set datastructure necessary Datastructure selected NECESSARY gt read observations 1160 cases read gt read model gt set em on gt case list Count P j P ifb A B C 129 0 148546 1 1 1 8 0 009212 2 1 1 46 0 148546 0 157759 1 1 1 46 0 009212 0 157759 2 1 1 46 5 35230E 17 0 157759 3 1 1 11 0 017385 1 2 1 1 0 001580 2 1 10 0 017385 0 018966 1 2 1 10 0 001580 0 018966 2 2 1 10 1 73891E 11 0 018966 3 2 1 388 0 454953 1 1 2 203 0 239503 1 2 15 0 017471 3 1 2 212 0 454953 0 711927 1 1 2 212 0 239503 0 711927 2 1 2 212 0 017471 0 711927 3 1 2 55 0 067956 1 2 2 27
228. with missing data Journal of the American Statistical Association 77 378 270 278 Goodman L A 1971 Partitioning of chi square analysis of marginal contingency tables and estimation of expected frequencies in multidimensional contingency tables Journal of the American Statistical Association 66 339 344 Haberman S J 1972 Log linear fit for contingency tables Algorithm AS 51 Appl Statist 21 218 227 Haberman S J 1974 Log linear Models For Frequency Data Univ of Chicago Press Chicago Illinois Haberman S J 1978 Analysis of Qualitative Data Academic Press Inc London Huber P J 1994 Huge data sets in R Dutter amp W Grossmann eds Com putational Statistics Physica Verlag Heidelberg pp 3 13 COMPSTAT 1994 Vienna Jirousek R 1991 Solution of the marginal problem and decomposable distributions Kybernetike 27 5 403 412 Kreiner S 1987 Analysis of multidimensional contingency tables by exact condi tional tests Techniques and strategies Scandinavian Journal of Statistics 14 97 112 Kreiner S 1989 User s guide to DIGRAM a program for discrete graphical mod elling Technical Report 89 10 Statistical Research Unit University of Copen hagen Lauritzen S L 1982 Lectures on Contingency Tables 3rd edition 1989 Aalborg University of Aalborg Press Denmark 164 Bibliography Lauritzen S L 1989 Mixed graphical association models with dis
229. y under DOS Another DumpFile can be named by SET OUTPUTFILE DUMP file name Note that this command rewrites the file file name and that under DOS the file COCO DMP is rewritten every time CoCo is invoked If the DumpFile is not renamed the DumpFile is removed when CoCo terminates normally unless the command SET KEEP DUMP On Off is used The command RETURN MATRIX value names a writes row by row to a matrix where the columns contain the values value names for the a marginal table If Dump is on the matrix is written to the DumpFile instead of the standard output Note that the list of value names is terminated by or an EndOfLine Return The cells in the printed matrix are ordered according to the order of the factors in a The first factor met in a is alternating at the fastest speed Chapter 5 Models 5 1 Reading Models READ MODEL gc READ N INTERACTIONS order set COLLAPSE MODEL set NORMAL TO DUAL DUAL TO NORMAL A model is read with the command READ MODEL gc where gc is the generating class for the model for example READ MODEL AB BC The model with only main effects is read by READ MODEL and the saturated model by READ MODEL The command READ N INTERACTIONS a set will read a model with only and all a th order interactions among factors in set A gener ator clique with n factors is a n 1 th order interaction The model last read is the curr
230. yword 52 residuals adjusted 56 crude standardized 56 Freeman Tukey 56 index plots 56 Pearson 56 standardized 56 response 139 RESTART command 124 132 154 restricted incremental search 105 RETURN MATRIX command 55 57 64 128 158 RETURN STANDARD VECTOR command 64 158 RETURN VECTOR command 55 57 64 128 158 Reuse keyword 92 93 Reversed keyword 106 110 173 Rough keyword 122 Row percents 58 Run Time Errors 148 runall program 148 runsome program 148 S coco program 135 149 S environment 149 S functions 149 S Plus program 149 150 saturated model 65 113 scaling 44 scoco o file 149 150 scoco sparc file 149 scoco sun3 file 149 SCOCOHOME environment 149 SCOCOHOME Functions file 149 150 SELECT CASES command 48 157 send index from xlispstat program 150 separable tables 74 Separators command 112 separators 109 Separators keyword 102 SET command 99 101 124 156 SET Graphical Hierarchical SEARCH command 117 120 121 160 SET Smallest Alternating Rough SEARCH command 117 122 160 SET ACCEPTANCE command 99 101 107 110 156 SET ADJUSTED DF command 76 93 99 100 131 156 SET ALGORITHM command 87 103 128 156 SET ASYMPTOTIC command 76 77 93 102 156 SET COMPONENTS command 100 102 109 112 156 SET DATA LOG command 127 155 SET DATASTRUCTURE command 39 51 129 130 157 SET DEBUG
231. yword 55 61 162 conditional independence 108 111 112 quasi independence 42 conditional independence graph 65 conditional quasi independence 42 conditionally independent 66 connected tables 74 Contingency Coefficient C 79 Continuity adjusted x 79 continuous variables See cutpoints Control 37 133 Control C 37 133 Control Q 36 Control S 36 copy b coco exe coco ovr program 148 Cramer s V 79 criteria See model selection Cross product ratio alpha 79 crude standardized residuals See standardized residuals CURRENT command 66 94 158 CURRENT MENU command 33 34 154 Current keyword 61 162 CUTPOINTS command 40 44 45 48 157 cutpoints 45 Cutpoints keyword 163 data 4 34 DATA gt file 4 DataFile keyword 126 127 148 DATASETS file 148 Datasets keyword 152 decomposable 67 71 Decomposable keyword 120 121 DecomposableMode mode 99 118 DECOMPOSE MODELS command 71 80 83 84 159 decomposed 67 decomposition 67 def function 140 default file names 36 degrees of freedom adjusted 73 Index DELETE ALIAS command 34 154 Deming Staphan algorithm See IPS algorithm DESCRIBE command 67 DESCRIBE CURRENT command 66 70 DESCRIBE MODEL command 66 159 DESCRIBE OBSERVED command 6 60 DESCRIBE TABLE command 10 43 55 57 59 60 125 158 deviance 51 72 77 129 132 diary 127 Diary keyword 93 DiaryFile keyword 127 dif

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