User's Guide for MOSAICS Version 3.6∗
Contents
1. produces two plots with titles from title 2 and title 3 a Equivalent results using substitution are produced with the single title title Hair color Eye color Data Model amp MODEL 3 2 Global input variables The global variables below allow many of the details of the model fitting and mosaic display to be altered Since they all have default values it is only necessary to specify those you wish to change All character valued variables are case insensitive 3 2 1 Analysis options config is anumeric or character matrix specifying which marginal totals to fit when fitt ype USER is also specified config is ignored for all other fit types Each column specifies a high order marginal in the model either by the names of the variables or by their indices according to their order in vnames For example the log linear model AB AC BC for a three way table is specified by the 2 by 3 matrix config 1 1 2 3 3h or config A A B B C C The same model can be specified more easily row wise and then transposed config t 1 2 1 3 2 3 J Some SAS GRAPH fonts do not produce brackets and braces Use parentheses instead in model symbolic formulae 3 USING MOSAICS 7 devtype GF LR FT ADJ is a character string which specifies the type of deviations resid uals to be represented by shading devtype GF is the default GF calculates components of P2. 5 3 Example 3 Reordering variables This example shows the use of SAS IML itself to reorder the variables in a contingency table for the mosaic display It uses the same data as in the previous example The variables in a contingency table are reordered by the MARG function which calculates marginal totals when the model specified by the config parameter is the saturated model with the variables listed in the desired order For example for the four way table of the previous example the configuration 4 3 2 1 gives the same order of the variables created by the PROC SORT step mosaics sas includes an IML module transpos shown partly below which will reorder the variables in any table It also rearranges the values in the levels vnames and lnames variables in the same order The order parameter must be either a permutation of the integers 1 ncol dim ora permutation of the variable names in vnames start transpos dim table vnames lInames order X reorder the dimensions of an n way table if nrow order 1 then order order run marg loc newtab dim table order table newtab dim dim order vnames vnames order lnames lnames order finish The data table is defined listing the observations in the same order as in the DATA step marital shown in Example 2 Note that vnames and lnames conform to this order After the 20 21 22 23 24 25 26 27 5 EXAMPLES 23 cal
3. Multidimensional arrays in SAS IML In Proceedings of the SAS User s Group International Conference vol 25 pp 1420 1427 SAS Institute 27 Friendly M 2000 Visualizing Categorical Data Cary NC SAS Institute ISBN 1 58025 660 0 Friendly M 2002 A brief history of the mosaic display Journal of Computational and Graphical Statistics 11 1 89 107 1 Hartigan J A and Kleiner B 1981 Mosaics for contingency tables In W F Eddy ed Computer Science and Statistics Proceedings of the 13th Symposium on the Interface pp 268 273 New York NY Springer Verlag 1 27 10 Wang C M 1985 Applications and computing of mosaics Computational Statistics amp Data Analysis 3 89 97 27
4. The n way frequency table to be analyzed is described in SAS IML by four arrays called levels table dimensions table table frequencies vnames variable names and lnames variable labels shown in lines 6 11 below These arrays are specified in the run mosaic statement line 18 A great many options all of which have default values are specified by global variables in the PROC IML step e g lines 14 15 Hence the program is typically used as follows libname mosaic sasuser mosaics proc iml worksize 10000 symsize 10000 reset storage mosaic mosaic load module _all_ specify data table levels 2 2 2 variable levels table contingency table vnames Gender Admit Faculty variable names lnames Male Female Yes No Arts Science specify non default global inputs fittype USER config 1 3 1 2 run mosaic levels table vnames lnames plots title The n way contingency table to be analyzed is specified by the table parameter the names of the dimension factor variables and the names of the values that the dimension variables take on are specified in the vnames and 1names parameters respectively as described below In situations where the contingency table and factor variables are available in a SAS dataset the table levels and 1names matrices may be constructed with the readt ab module described in Section 3 5 Dataset Input Th
5. e User s Guide for MOSAICS i Version 3 6 m A Michael Friendly Psychology Department York University Contents 4 1 TheMOSAIC macro 13 4 2 The MOSMAT macro 15 1 Introduction 1 4 3 The TABLE macro 16 2 Installation Guide 2 5 Examples 16 2 1 How to obtain MOSAICS 3 5 1 Example 1 Direct use in IML 16 2 2 Installing MOSAICS 3 5 2 Input from SAS dataset 21 5 3 Example 3 Reordering variables 22 3 Using MOSAICS 4 5 4 Example 4 MOSMAT and TABLE 3 1 Input parameters 5 macroS 0 ee eee 23 3 2 Global input variables 6 5 5 UsingGENMOD 24 3 3 Graphicoptions 10 5 6 Sample datasets 25 3 4 Multiplecalls 11 3 5 SAS Dataset Input 11 6 Implementation 25 3 6 Fitting specialized models 12 6 1 Algorithm 27 6 2 Programstructure 28 4 Macro interface 13 63 MCHANEES s o o i alk ws Bs 28 1 Introduction The mosaic display proposed by Hartigan amp Kleiner 9 represents the counts in a contingency table directly by tiles whose area is proportional to the cell frequency This display generalizes readily to n way tables Friendly 1 2 3 4 5 extended the use of the mosaic display as a graphical tool for fitting log linear models The enhanced mosaic uses color and shading of the tiles to reflect the sign and magnitude of the residual from a specified log linear model Friendly also shows how the understandi
6. the rows for smaller factors must be padded with blank entries 3 USING MOSAICS 6 plots is a vector containing any of the integers 1 to n which specifies the list of marginal tables to be plotted If plots contains the value the marginal subtable for variables 1 to will be displayed For a 3 way table plots 1 2 3 displays each sequential plot showing the A AB and ABC marginal tables while plot s 3 displays only the final 3 way ABC mosaic title is a character string or vector of strings containing title s for the plots If title is a single character string it is used as the title for all plots Otherwise t it le may be a vector of up to max plots strings and title i is used as the tile for the plot produced by plots i If the number of strings is less than max plots the last string is used for all remaining plots Moreover if the title for a given plot contains the string MODEL upper case that string is replaced by the symbolic model description Similarly the string amp G2 or amp X2 is replaced by the LR Pearson chisquare value and df for the current model in the form G2 df value Enclose such titles in single quotes otherwise the SAS macro processor will complain about an Apparent symbolic reference For example the specifications plots 2 3 fittype JOINT title Hair color Eye color Data Model H E Hair color Eye color Data Model HE S
7. 1 95 Red 0 07 0 85 2428 1 73 Blond 5 2819 2023 0 61 7205 Factor 3 Sex Marginal totals MARGIN Male Female Black Brown 32 36 Black Hazel 10 5 Black Green 3 2 Black Blue 11 9 Brown Brown 38 81 Brown Hazel 25 29 Brown Green 15 14 Brown Blue 50 34 Red Brown 10 16 Red Hazel 7 7 Red Green g 7 Red Blue 10 7 Blond Brown 3 4 Blond Hazel 5 5 Blond Green 8 8 Blond Blue 30 64 MODEL DF CHISQ PROB Hair Eye Sex 15 G F 28 993 0 0161 L R 29 350 0 0145 Standardized Pearson deviations Male Female Black Brown 0 30 0 27 Black Hazel 1 28 A E S Black Green 0452 0 46 Black Blue 0 70 0 63 Brown Brown 2 07 1 86 Brown Hazel 0 19 0 17 Brown Green 0 57 0 52 Brown Blue 2 05 1 84 Red Brown 0 47 0 42 Red Hazel 0 30 O e27 Red Green 0 30 0 27 Red Blue 0 88 0 79 Blond Brown 0 07 0 06 Blond Hazel 0 26 0 23 21 22 23 24 25 26 27 28 29 30 31 32 33 5 EXAMPLES 21 Blond Green 0 32 05529 Blond Blue 1 84 1 65 l op fe 5 2 Example 2 PROC IML Input from SAS data set This example illustrates input of data from a SAS data set and the use of PROC SORT to rearrange the variables in a table to the order desired in the mosaic displays The data is a 2 table classified by Gender reported Pre marital sex Extra marital sex and Marital Status read in by the DATA step marital below Note that the variable marital varies most rapid
8. Alive 2 x Time 2 x Length 2 abortion Abortion opinion data Sex 2 x Status 2 x Support Abortion 2 berkeley Berkeley Admissions Data Admit 2 x Gender 2 x Dept 6 cancer Breast Cancer Patients Survival 2 x Grade 2 x Center 2 cesarean Risk factors for infection in cesarean births Infection 3 x Risk 2 x Antibiotics 2 x Planned 2 detergen Detergent preference data Temperature 2 x M User 2 x Preference 2 x Water softness 3 dyke Sources of knowledge of cancer Knowledge 2 x Reading 2 x Radio 2 x Lectures 2 x Newspaper 2 employ Employment Status Data EmployStatus 2 x Layoff 2 x LengthEmploy 6 gilby Clothing and intelligence rating of children Dullness 6 x Clothing 4 haireye Hair color Eye color data Hair 4 x Eye 4 x Sex 2 heckman Labour force participation of married women 1967 1971 1971 2 x 1970 2 x 1969 2 x 1968 2 x 1967 2 hoyt Minnesota High School Graduates Status 4 x Rank 3 x Occupation 7 x Sex 2 marital Pre Extramarital Sex and Marital Status Marital 2 x Extra 2 x Pre 2 x Gender 2 mobility Social Mobility data Son s Occupation 5 x Father s Occupation 5 suicide Suicide data Sex 2 x Age 5 x Method 6 titanic Survival on the Titanic Class 4 x Sex 2 x Age 2 x Survived 2 victims Repeat Victimization Data First Victimization 8 x Second Victimization 8 6 IMPLEMENTATION 27 6 1 Algorithm The process is a nat
9. 4 2 vnames Hair Eye Sex Variable names lnames Category names 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 5 EXAMPLES Model HairEye Sex EEEE a 1 1 1 i 1 1 1 1 1 1 1 1 71 1 1 m 1 1 gt 4 1 2 2 0 0 2 2 4 kuud Hazel GreerBlue q I E a lt 4 Brown Standardized residuals Male Female Black Brown Red Blond 18 Figure 2 Mosaic display for hair color eye color and sex The categories of sex are crossed with those of hair color but only the first occurrence is labeled Residuals from the model HE S are shown by shading Black Brown Red Blond hair color Brown Blue Hazel Green eye color Male Female r sex title Hair color Eye color data finish run haireye reset storage mosaic mosaic load module _all_ Fit models of joint independence fittype JOINT plots 2 3 split V H htext 1 6 run mosaic levels table vnames lnames plots title reorder eye colors brown hazel green blue table table 1 4 9 16 5 8 ilnames 2 lna
10. SAS dataset e Added devtype FT to calculate and display Freeman Tukey residuals e Character values of global input variables no longer need be entered in upper case Version 3 0 e Added ability to fit a sequence of Markov models fitt ype MARKOV for lag sequential data e Fit the equiprobability model for the display of the first variable Version 2 9 e Installation simplified by creating a separate file mosaicm sas to install IML modules e Filltypes changed to allow separate coding for positive and negative residuals and to provide grayscale shading levels e Added ability ce11f 1i11 to print a symbol in the cell symbolizing the value of the residual References 1 Friendly M 1991 SAS System for Statistical Graphics Cary NC SAS Institute 1st edn ISBN 1 55544 441 5 1 REFERENCES 30 2 Friendly M 1992 Mosaic displays for loglinear models In ASA Proceedings of the Statisti cal Graphics Section pp 61 68 Alexandria VA 1 3 Friendly M 1994 Mosaic displays for multi way contingency tables Journal of the American Statistical Association 89 190 200 1 17 4 Friendly M 1995 Conceptual and visual models for categorical data The American Statisti cian 49 153 160 1 Friendly M 1999 Extending mosaic displays Marginal conditional and partial views of categorical data Journal of Computational and Graphical Statistics 8 3 373 395 1 Friendly M 2000
11. baseline models of independence conditional or partial independence among the table variables see Table 1 A shorthand keyword may used to specify many models of interest Alternatively the user can specify and fit any log linear model which can be estimated by iterative proportional fitting IPF Specialized log linear models or poisson family GLMs which cannot be fit by IPF can be fit separately using SAS IML or PROC GENMOD These include models for square tables quasi independence symmetry etc models with linear effects for table variables linear x linear association and so forth Residuals for such models may be shown as mosaics using either the SAS IML module mosaicd or the resid parameter of the mac mosaic sas macro See Section 3 6 and Section 5 5 for examples The program can perform a correspondence analysis on marginal subtables to suggest a re ordering of the levels of each variable to make the patterns of association more coherent Models and tables with structural zeros are accommodated naturally A contingency table can be read from a SAS data set or entered in SAS IML as a table of frequencies together with variable name and factor level values A collection of sample contin gency tables in SAS IML format is suppplied in mosdata sas A SAS macro mac mosaic sas provides a more easily used interface to the SAS IML modules Another macro mac table sas makes it easy to construct and manipulate con tingen
12. readlab read level names reorder input transpos reorder the dimensions of an n way table Figure 5 Calling structure of the modules in MOSAICS The top level module mosaic simply validates the input parameters assigns default values for global variables and calls the module divide The steps in the algorithm described above are carried out by divide the calculation of the new tiles in step 5 is performed in dividel The other top level modules readt ab and t ranspos are included for convenience to input a frequency table into SAS IML from a SAS dataset and reorder table dimensions respectively 6 3 Changes Version 3 6 e Added an out stat global variable to generate an output data set with residuals fitted etc e Module reorder changed to transpos e Added a GOUT global variable for graphics catalog entry e Added WINDOW global variable to control window size Version 3 5 e Fixed conflict between the global variable DEVTYPE and the macro variable used for graphics device control e Changed circle blanking used for CELLF ILL to white black text depending on shading den sity REFERENCES 29 e Added control of threshold for CELLFILL You can now say CELLFILL DEV 1 0 and all absolute residuals gt 1 0 will have their values written in the tiles e Added calculation and display of adjusted residuals d Ja h e The default font now depends on device driver making it easier
13. vertically if s is odd or horizontally s even into ls tiles with the width s odd or height s even of each tile proportional to Mgh Mg This computational scheme has several desirable properties e At any stage the division of the tiles for the current variable is in proportion to the entries in each row of M divided by the row totals e We can draw the tiles representing the marginal frequencies at any stage not just the final stage as Hartigan and Kleiner 9 do e Fitting the model of joint independence of the current variable with all previous variables jointly is equivalent to testing independence of the rows and columns of the matrix M For example for a three way table the expected frequencies under the model A B C can be ex pressed in terms of the IJ x K matrix M as mij 4m4n m44 6 1 1 Spacing This procedure gives a mosaic of Ln l x l2 X X ln tiles with no spacing in which cells with small frequencies are difficult to see Following Hartigan and Kleiner 9 the tiles are separated with larger spacings at the earlier subdivisions to help preserve the visual impact of small counts For a four way table with vertical splitting on variables 1 and 3 the divisions of the first variable are spaced proportionally to 1 l 1 divisions between levels of the third variable are spaced proportionally to 1 hils a 1 This spacing of the tiles is accomplished by constructing an unspaced mosaic in a reduced area
14. D AB BC CD DE MARKOV2 A B C ABC BCD ABC BCD CDE a Tn all cases the model A B is fit to a two way table or marginal table b The letters A B C refer to the table variables in the order of entry into the mosaic display fittype JOINT MUTUAL CONDIT PARTIAL MARKOV USER isa character string which specifies the type of sequential log linear models to fit fittype JOINT is the default For two way tables or two way margins of larger tables all fittypes fit the independence model The fittype values and the models they imply for sub tables of various size are summarized in Table 1 JOINT specifies sequential models of joint independence A B AB C ABC D These models specify that the last variable in a given plot is independent of all previous variables jointly Optionally the keyword JOINT may be followed by a digit k to specify which of the n ordered variables is independent of the rest jointly e g JOINT1 gives A BC MUTUAL specifies sequential models of mutual independence A B A B C A B C P CONDITK specifies sequential models of conditional independence which hypothesize that all previous variables are independent given the last i e A B AC BC AD BD CD For the 3 way model A and B are hypothesized to be conditionally independent given C for the 4 way model A B and C are conditionally independent given D 3 USING MOSAICS 8 Op
15. S12 313 89 19 353 207 17 8 120 205 202 391 138 279 131 244 53 138 94 299 5 EXAMPLES 22 351 24 317 24 The program lines below read this dataset and use formats to recode the category levels into more meaningful labels in a mosaic mosmat9 sas Sinclude catdata berkeley proc format value admit 1 Admit O0 Reject valu d pt T nan 2 R 3 oa 4 p 5 value sex M Male F Female stable data berkeley E 6 he r var Admit Gender Dep format admit admit weight freq char Y alt gender S sex order data out berkeley Ssmosmat data berkeley Admit vorder Admit Gender Dept Reject a dept dept sort no htext 3 Figure 4 Mosaic matrix for Berkeley admissions data The TABLE macro is used lines 4 6 translate the original variables into new variables which have the formatted equivalents as their values because SAS IML still cannot read formatted values It was desired to retain the values of Sex in the order Male Female so ORDER DATA was spec ified The sorted values Female Male produces a display where the labels are more crowded because there are fewer females The new factors in the data set are all character variables The MOSMAT macro line 10 produces Figure 4 SORT NO keeps the program from messing thing
16. V JOINT ROW COL Specifies whether and how to perform a cor respondence analysis to assist in reordering the levels of each factor variable as it is entered into the mosaic display Not performed if order NONE Otherwise order may be a character vector containing either DEV or JOINT to specify that the CA is performed on residuals from the model for the current subtable DEV or on residuals from the model of joint inde pendence for this subtable JOINT In addition order may contain either ROW or COL or both to specify which dimensions of the current subtable are considered for reordering The usual options for this reordering are order JOINT COL At present this analysis merely produces printed output which suggests an ordering but does not actually reorder the table or the mosaic display zeros is a matrix of the same size and shape as the input table containing entries of 0 or 1 where 0 indicates that the corresponding value in table is to be ignored or treated as missing or a structural zero Zero entries cause the corresponding cell frequency to be fitted exactly one degree of freedom is subtracted for each such zero The corresponding tile in the mosaic display is outlined in black If an entry in any marginal subtable in the order A AB ABC corresponds to an all zero margin that cell is treated similarly as a structural zero in the model for the corresponding subtable Note howev
17. ays see http math yorku ca SCS Online mosaics about htm1 but none except for the vcd package in R which provide the same degree of flexibility These programs have the following features mosaics sas produces graphical displays of an n way contingency table of any size Ex perience shows that tables of up to 5 or 6 dimensions can be usefully explored The main limitation is in the resolution of the display with large complex tables The order of variables in the mosaic is specified by the user Different orderings of the variables can show different aspects of the data For an unordered factor the order of its levels can be determinedcaveat to enhance understand ing of the pattern of association This ordering can be found from a correspondence analysis of the residuals from a model of independence The program can produce sequential displays of any or all of the marginal subtables A AB ABC and so forth up to the full n way table where A B C refer to the table variables in the order entered For each display the program fits a log linear model and depicts the residuals from the model by the color and shading of tiles in the mosaic The color and shading provide a visual rep resentation of the departures from the model or equivalently the associations among table variables which remain after the effects specified in the model have been accounted for The program can automatically construct and fit a wide set of
18. cy tables for use with mac mosaic sas macro Other SAS IML modules and macros extend the idea of mosaic displays to mosaic matrices mosmat sas both marginal and conditional and partial mosaic plots mospart sas Partial mosaics are included in the mac mosaic sas macro using the by parameter mosaic matrices have their own macro mac mosmat sas 2 Installation Guide Unsurprisingly you have to get the software and install it on your system before you can use it 2 INSTALLATION GUIDE 3 2 1 How to obtain MOSAICS The program mosaics sas and examples of its use are available from the host euclid psych yorku ca The directory http euclid psych yorku ca ftp sas mosaics contains two iden tical archives mosaics tar gz and mosaics zip as well as individual files 2 2 Installing MOSAICS mosaics sas consists of a collection of SAS IML modules which are designed to be called in a PROC IML step Because the program is large the modules are most conveniently stored in com piled form in a SAS IML storage catalog called MOSAIC MOSAIC The archive also includes sev eral macro programs notably mosaic sas and mosmat sas that provide the easiest way to use mosaic displays and do not require knowledge of or direct use of SAS IML You will probably want to add these macros to your SAS autocall library library name sasautos To install the programs in this way 1 Extract all the SAS and other files e g mosaics sas and mosa
19. determined by the space parameter then expanding to include the necessary spacing The boxes array is initialized as length 100 100 x y length of box area boxes 0 0 lowerleft x y I length space length x y Then the x y coordinates are multiplied by a diagonal matrix of scale factors sf calculated from the maximum zx y values of the upper right corners ur of the tiles ur boxes 1 3 boxes 2 4 max max ur 1l max ur 2 sf 100 100 max 6 IMPLEMENTATION 28 6 2 Program structure mosaics sas consists of 22 SAS IML modules subroutines and functions The calling structure of the modules is shown in Figure 5 mosaic check inputs assign default values divide fit models and draw the mosaic display reduc find reduced model for factors 1 f mfit fits a specified model chisq calculate chisquares df calculate degrees of freedom terms find all terms in a loglinear model vars_in x find variables in a term modname expand config into string for model label dividel divide the mosaic for the next variable spac space the tiles in the current display labels calculate label placements gboxes X draw the current display fillbox x custom shading glegend draw legend readtab read input frequencies level names
20. e parameters for the run mosaic statement are Parameter Description levels is a numeric vector which specifies the number of variables and the dimensions of the contingency table If levels is n x 1 then the table has n dimensions and the number of levels of variable is leve1s i The order of the variables in levels is the order they are entered into the mosaic display table is a matrix or vector giving the frequency fij of observations in each cell of the table The table variables are arranged in accordance with the conventions of the SAS IML IPF and MARG functions so the first variable varies most rapidly across the columns of table and the last variable varies most slowly down the rows The table must be complete If you use PROC FREQ to sum a larger data set use the SPARSE option on the TABLES statement so that all combinations are created In addition table must conform to levels as follows If table is J rows by J columns the product of all entries in Levels must be IJ Moreover J must equal the product of the first k entries of Levels for some k That is the columns must correspond to the combina tions of one or more of the first k factors vnames isa1xncharacter vector of variable factor names in an order corresponding to levels lnames is a character matrix of labels for the variable levels one row for each variable The number of columns is the maximum value in levels When the number of levels are unequal
21. earson goodness of fit chisquare dj fij Miz Mi where fij is the estimated expected frequency under the model LR calculates components of the likelihood ratio deviance chisquare dj sign fi iig 2 Fig log fig titag Fig Mig FT calculates Freeman Tukey residuals di fij y fij 1 47 1 ADJ Appending ADJ to one of the above options causes adjusted residuals d v1 h where h is the diagonal element of the hat matrix to be calculated Because 0 lt h lt 1 the adjusted residuals are always larger in magnitude than the unadjusted values however adjusted residuals have the property that their standard errors are equal so their values are more comparable over cells in the contingency table Adjusted residuals require additional computation it becomes necessary to construct the design matrix X and then calculate X TWX JH however experience shows that they provide better visual display of the patterns of association than do ordinary Pearson or LR residuals Table 1 Log linear models corresponding to the various fittype values recognized by MOSAICS fittype 3 way 4 way 5 way MUTUAL ATIC Aec D AJ B C D1 TE JOINT AB C ABC D ABCE E JOINT A BC A BCD A BCDE CONDIT AC BC AD BD CD AE BE CE DE CONDIT1 AB AC AB AC AD AB AC AD AE PARTIAL AC BC ACD BCD ADE BDE CDE MARKOV1 AB BC AB BC C
22. el separated by example to fit the model of no three way association specify config 1 2 1 3 2 3 or using variable names config A B A C B C Note that the numbers refer to the variables after they have been reordered either sorting the data set or by the vorder parameter For vorder Specifies either the names of the variables or their indices in the desired order in the mosaic Note that using the VORDER parameter keeps the factor levels in their order in the data whereas the SORT parameter arranges factor levels in sorted order lorder Specifies a reordering of the levels of one or more variables of the form A a2 al a3 B b2 b3 b4 b1 where separates different variables and separates the name of a variable from the desired order of the levels sort Specifies whether and how the input data set is to be sorted to produce the desired order of variables in the mosaic SORT YES sorts the data in the reverse order that they are listed in the VAR paraemter so that the variables are entered in the order given in the VAR parameter Otherwise SORT lists the variable names possibly with the DESENDING or NOTSORTED options in the reverse of the desired order e g SORT C DESCENDING B DESCENDING A resid Specifies that externally calculated residuals are contained in the variable named by the resid parameter Here is an example title Alcohol Cigarette and Marijuana Use by High School Se
23. er that tables with zero margins may not always have estimable models If the table contains zero frequencies which should be treated as structural zeros assign the zeros matrix like this zeros table gt 0 For a square table to fit a model of quasi independence ignoring the diagonal entries assign the zeros matrix like this assuming a 4 x 4 table zeros J 4 4 I 4 3 USING MOSAICS 9 3 2 2 Display options abbrev If abbrev gt 0 variable names are abbreviated to that many letters in the model formula and in the plot title if title MODEL cellfill NONE SIGN SIZE DEV FREQ min Provides the ability to display a symbol in the cell representing the coded value of large residuals This is particularly useful for black and white output where it is difficult to portray both sign and magnitude distinctly NONE Nothing default SIGN Draws or symbols in the cell whose number corresponds to the shading density SIZE Draws or symbols in the cell whose size corresponds to the shading density DEV Writes the value of the standardized residual in the cell using format 6 1 FREQ Writes the value of the cell frequency in the cell using format 6 0 If a numeric value min is also specified e g celL1f il1 DEV 27 then only cells whose residual exceeds that value in magnitude are so identified colors isa character vector of one or two elements specifying the colors used for positive and
24. fit simply using the matrix operations and functions available in SAS IML For example the model of symmetry for a square table has expected frequencies rij fij fii 2 The fitted frequencies and residuals can be calculated in SAS IML as fit 2 dev f fit sqrt fit where f is a square table of observed frequencies 4 MACRO INTERFACE 13 MOSAICS includes an additional program mosaicd sas designed for situations such as this where the fitted values and residuals are calculated externally e g with IML programming state ments or with PROC CATMOD or PROC GENMOD The mosaicdis then called instead of mosaic The residuals are supplied as a dev parameter which replaces the plots parameter of mosaic The following example uses mosaicd to fit a model of symmetry to a 4 x 4 table of women classified by visual acuity ratings of their left and right eyes proc iml worksize 10000 symsize 10000 dim 4 4 Unaided distant vision data Bishop etal p 284 px Left eye grade 1520 266 124 66 234 1512 432 TB 117 362 1772 205 36 82 T79 492 title Unaided distant vision Symmetry vnames Right Eye Left Eye lnames rHigh 727 3 Low THigh T27 13 TLOW J reset storage mosaic mosaic load module _all_ include sasuser mosaics mosaicd sas fit f 2 dev f fit sqrt fit run mosaicd dim f vnames lnames dev title The sample
25. for BBfix goptions device PSLEPSFC gaccess gsasfile gend 0A x gepilog showpage OA x only for 6 07 gsflen 80 gsfmode replace Smend 3 4 Multiple calls The mosaic module may be called repeatedly in one PROC IML step However global variables which are set in one call remain in force To restore these values to their default setting use the SAS IML free statement For example to revert to the default fit type of joint independence use the statement free fittype before the next run mosaic statement 3 5 SAS Dataset Input A contingency table and its index factor variables may be read into SAS IML in the format required for MOSAICS using the readt ab module as shown in the following example The factors in the 2x 3 x 2 table are gender occup and heart The dataset heart has 12 observations one observation per cell 20 21 22 23 24 25 26 3 USING MOSAICS 12 Sex Occupation and heart disease Karger 1980 data heart input gender occup 4 heart Disease input freq output heart No Dis input freq output cards Male WhiteCol 158 3155 Female WhiteCol 52 3082 Male BlueCol 87 2829 Female BlueCol 16 416 Male Unemp1l 254 759 Female Unempl 431 10283 i proc sort data heart by heart occup gender proc iml worksize 10000 symsize 10000 title Sex Occupation and Heart Disease reset storage mosa
26. ic mosaic load module _all_ vnames Gender Occup Heart run readtab heart freq vnames table levels plots 2 ncol levels run mosaic levels table vnames lnames plots tit The readtab routine reads the index factor variables from the input dataset heart and deter mines the order of the factor variables according to which variable is actually varying most rapidly in the input dataset The variable names vector vnames can be given in any order it is reordered to correspond to the order of observations in the input dataset Note that if you sort the dataset as in the example above character valued index variables are ar ranged in alphabetical order For example the levels of occup are arranged in the order BlueCol Unempl WhiteCol which may or may not be what you want The PROC SORT step can be omitted in which case the levels are ordered according to their order in the input dataset You can also use the DESCENDING option in the PROC SORT step to reverse the order of the levels of a given factor For example to reverse the levels of the gender variable use proc sort data heart by heart occup descending gender 3 6 Fitting specialized models For square tables or tables with ordered factors a wide variety of specialized models are available which cannot be specified as any IPF configuration for a hierarchical loglinear model However many of these models can be
27. icm sas etc to a direc tory sasuser mosaics or c sasuser mosaics say perserving the folder names mac and doc in the archive 2 In the files mosaicm sas and mosdata sas edit the libname and filename state ments to correspond to this directory On a Unix system these might be Change the path in the following filename statement to point to the installed location of mosaics sas filename mosaics sasuser mosaics Change the path in the libname to point to where the compiled modules will be stored ordinarily the same directory libname mosaic sasuser mosaics On Windows you should use something like filename mosaics c sasuser mosaics libname mosaic ce sasuser mosaics 3 You may wish to change some of the program default values in the module globals in mosaics sas particularly the font value As of V3 5 this is set to font SWISS unless the current graphics device amp SYSDEVIC is one of the Postscript drivers e g PS COLOR PSMONO PSLEPS in which case the program uses the hardware Helvetica font font hwps1009 because the resulting output graphic files are much smaller and can be potentially edited 4 To store the modules in compiled form run the mosaicm sas program with the command sas mosaicm 5 Optionally install the sample data sets see Section 5 6 Sample data sets by running sas mosdata These steps need only be do
28. itting process using the mosaicd module described in Section 3 6 5 6 Sample data sets A variety of contingency tables are supplied with the MOSAICS distribution in the filemosdata sas These are listed in Table 2 with the variable names and dimensions given in their order as in vnames Each data set is stored as a SAS IML module containing definitions for the variables title dim vnames lnames and table used in the run mosaics statement Note that the variable dim corresponds to levels in the arguments to mosaic See the module haireye in Example 1 The program mosdata sas is set up so that running it will create a SAS IML storage catalog MOSDATA in the MOSAIC library Once this has been done any dataset may be obtained by loading the module from MOSAIC MOSDATA and running it For example the previous example could be done using the module marital as shown below proc iml reset storage mosaic mosdata load module marital run marital reset storage mosaic mosaic load module _all_ ord 4 3 2 1 run reorder dim table vnames lnames ord split V H plots 2 4 run mosaic dim table vnames Ilnames plots title quit 6 Implementation This section describes the algorithm for the construction of mosaic displays and provides some notes on the structure of the program 6 IMPLEMENTATION 26 Table 2 Mosaics data sets Module Title name Variable names dimensions bartlett Bartlett data
29. l to transpos the variables table levels vnames and lnames have been rearranged so that Gender is the first variable in the mosaic and Marital status is last proc iml worksize 10000 symsize 10000 x define the data variables table 17 4 Women Yes Yes 54 25 Women Yes XY 36 4 Women No Ky 214 322 Women No x 28 11 Men Yes 60 42 Men Yes 17 4 Men No ay 68 130 Men No 222 2 Marital Extra Pre Gender Divorced Married Extra Sex Yes No Pre Sex Yes No Women Men title Pre Extramarital Sex and Marital Status reset storage mosaic mosaic load module _all_ ord 4 3 2 1 run transpos levels table vnames lnames ord split V H plots 2 4 run mosaic levels table vnames lnames plots title quit Note that the order of variables could also be specified using their names case doesn t matter in line 23 as ord gender pr xtra marital 5 4 Example 4 MOSMAT and TABLE macros The data step below creates the datasetberkeley a2 x 2 x 6 table classifying 4526 applicants to graduate school at U C Berkeley in 1971 by Admission Gender and Department title Berkeley Admissions data data berkeley do dept 1 to 6 do gender M FEF do admit 1 0 input freq output end end end Admit Rej Admit Rej cards
30. lgorithm in SAS TML Readers who wish to avoid SAS IML should start with the example in Section 4 1 and Section 5 5 5 1 Example 1 Direct use in IML The program below shows the use of MOSAICS to produce a set of different mosaic displays for a 4 x 4 x 2 table of 592 people classified by hair color eye color and sex The module hai reye creates the variables table levels vnames lnames andtitle Since the variables are to be entered into the mosaic in the order hair color eye color and sex the table variable is created as a 2 x 16 matrix with hair color varying most rapidly across the columns and sex varying down the two rows Note that the Lnames variable is a 3 x 4 matrix and the last row contains two blank values The statement run haireye creates these variables in the SAS IML workspace 5 EXAMPLES 17 The first run mosaics statement produces two plots whose tiles show the Hair Eye marginal table and the full three way table Since fittype is not specified the model HairEye Sex in which Sex is independent of hair color and eye color jointly is fit to the three way table split V H specifies that the first division of the mosaic is in the vertical direction The printed output pro duced from this run is shown below gt 4 o 3 7 8 a i g A p fo o B I 1 A OM on S Y V Brown Standardized residuals Black Brown Red Bl
31. ly and the variable gender varies most slowly in the observations in the data set The desired order of the variables in the mosaic is Gender Pre Extra and Marital In the table variable in SAS IML the first variable Gender must vary most rapidly This is accomplished by sorting the observations with the variables listed in the reverse order on the by statement in the PROC SORT step data marital input gender pre extra marital Divorced input freq output marital Married input freq output cards Women Yes Yes 17 4 Vomen Yes No 54 25 Women No Yes 36 4 Vomen No No 214 322 Men Yes Yes 28 11 Men Yes No 60 42 Men No Yes 17 4 Men No No 68 130 i proc sort data marital by marital extra pre gender Inthe PROC IML step the statement use marital accesses the data set The variable freq from the data set is read into the IML table variable a 16 x 1 matrix Note that the levels of the character variables gender pre and extra are sorted alphabetically so the category labels in lnames must appear in this order proc iml worksize 10000 symsize 10000 use marital read all var freq into table levels 222 2 vnames Gender Pre Extra Marital lnames Men 7 Women oes Pre Sex No Yes Extra Sex No Yes Divorced Married title Pre Extramarital Sex and Marital Status reset s
32. m in this way simply access the program witha include statement filename mosaics path to mosaics sas proc iml Sinclude mosaics On some platforms you may need to add a path specification to the include statement or use a filename statement to specify the location of the mosaics sas file in the operating system file structure 2 2 1 Public Use On most platforms SAS IML requires by default that the user have Read Write access to the library accessed by the Load command Therefore if the MOSAICS modules are stored in compiled form and are to be accessed publicly on a network users must specify access readon1y on the libname statement libname mosaic sasuser mosaics access readonly You can place this statement in the system wide autoexec sas file Alternatively copy the mosaics sas file to any public readable directory and instruct users to load them using the include statement as described above 3 Using MOSAICS You can use MOSAICS either through a SAS IML step or through the mosaic macro Section 4 1 The macro is easier to use but IML is somewhat more flexible If you are using IML the contingency table can either be defined directly with IML statements or input from a SAS dataset Section 3 5 Section 5 2 Unless you are quite comfortable with SAS IML you should probably start with the macro inter face so skip to Section 4 and read this section later 3 USING MOSAICS 5 3 1 Input parameters
33. mes 2 1 3 4 2 plots 2 3 run mosaic levels table vnames lnames plots title plots 3 fittype MUTUAL run mosaic levels table vnames lnames plots title quit Generalized Mosaic Display Version 2 9 5 EXAMPLES 19 Model Hair Eye Sex 5 q i 1 i 1 i 1 J 1 f I niu 5 I 1 gt 4 Hazel GreerBlue 1 2 2 0 0 2 2 4 a lt 4 Brown Standardized residuals Maldemale Black Brown Red Blond Figure 3 Mosaic display for hair color eye color and sex showing residuals from the model of complete independence H E S TITLE Hair color Eye color data VNAMES LEVELS LNAMES Hair 4 Black Brown Red Blond Eye 4 Brown Hazel Green Blue Sex 2 Male Female Global options Gl FITTYPE DEVTYPE FILLTYPE SPLIT SHAD JOINT GF M45 V H 2 4 Factor 1 Hair Marginal totals MARGIN Black Brown Red Blond 108 286 71 127 Factor 2 Eye Marginal totals MARGIN Brown Hazel Green Blue Black 68 15 5 20 Brown 119 54 29 84 Red 26 14 14 17 Blond J 10 16 94 5 EXAMPLES MODEL DF CHISO PROB Hair Eye 9 G F 138 290 0 0000 Tie RS 146 444 0 0000 Standardized Pearson deviations Brown Hazel Green Blue Black 4 40 0 48 SIIS 3 07 Brown 1 23 1635 0 435
34. ne once 6 To cause SAS to search automatically for the macros mosaic and mosmat If you already have a SAS autocall library set up you can simply copy all the files in the mac directory to your local SASAUTOS directory Otherwise add a line like one of the following to your autoexec sas file options sasautos c sasuser mosaics mac sasautos or 3 USING MOSAICS 4 options sasautos sasuser mosaics mac SASROOT sasautos For Unix systems the distribution archives include a rudimentary Makefile which carries out the steps above but you must first edit the 1ilbname and filename statements in step 2 then type make install or make n install to see what it s going to do In applications the modules are loaded into the SAS IML workspace with either the load or Sinclude statement as follows libname mosaic sasuser mosaics proc iml reset storage mosaic mosaic load module _all_ On most platforms a 1ilbname statement is needed to specify the location of the MOSAICS library in the operating system file structure Note This requires that you have Read Write access to the MOSAICS library even if the MOSAICS modules are only loaded See Public Use below for a solution Alternatively it is possible to store and use the program in source form This avoids the need to maintain and access the SAS IML catalog but means that the program is compiled each time it is run To use the progra
35. neg ative residuals The default is BLUE RED Fora monochrome display specify colors BLACK and use two distinct fill patterns for the fill type suchas filltype MO M45 orfilltype GRAY M45 filltype M45 LR M0 GRAY HLS is a character vector of one or two elements which specifies the type of fill pattern to use for shading filltype 1 is used for positive residu als filltype 2 if present is used for negative residuals If only one value is specified a complementary value for negative residuals is generated internally filltype HLS HLS is the default which usually looks best for color output M45 uses SAS GRAPH patterns MdN135 and Md45 with hatching at 45 and 135 d is the density value determined from the residual and the shade parameter LR uses SAS GRAPH patterns Ld and Rd MO uses SAS GRAPH patterns MANO and MdN90 with hatching at 0 and 90 GRAYstep uses solid greyscale fill using the patterns GRAYnn starting from GRAYF0O for density 1 and increasing darkness by step for each successive density level The default for step is 16 so GRAY gives GRAYF 0 GRAYEO GRAYDO and so forth HLS uses solid color varying fill based on the HLS color scheme The colors are selected at tempting to vary the lightness in approximately equal steps For this option the colors values must be selected from the following hue names RED GREEN BLUE MAGENTA CYAN YELLOW fuzz is a numeric value which specifies
36. ng of patterns of association can be enhanced by reordering the rows and columns to make the pattern more coherent Mosaic displays actually have a long history 8 This document is not intended as a tutorial on mosaic displays per se or on their use in data analysis and visualization Refer to Friendly 2 3 5 for details of the method and examples of its use in fitting log linear models The most complete discussion with many computational examples is contained in Visualizing Categorical Data 7 There is also e An online web application with several sets of sample data ht tp math yorku ca SCS Online mosaics You can submit your own data through a form or uploaded file This weblet always runs the current production version of MOSAICS but not all options are available in the web interface e A brief tutorial introduction http math yorku ca SCS Online mosaics about htm1 to mosaic displays This document is an updated version of User s Guide to MOSAICS A SAS IML program for Mosaic Displays York University Dept of Psychology Report 206 1992 This work is supported by Grant 8150 from the National Sciences and Engineering Research Council of Canada This version created November 23 2005 2 INSTALLATION GUIDE 2 This report describes the use and implementation of MOSAICS a collection of SAS IML pro grams and macros for producing mosaic displays There are now a variety of other implementations of mosaic displ
37. nior Source Agresti 1996 p 152 data druguse input alcohol cigaret marijuan Mar input freq output marijuan Mar input freq output Cig 911 538 Cig 44 456 Cig 3 43 Cig 2 279 goptions hsize 7in vsize 7in smosaic var alcohol cigaret marijuan count freq plots 2 3 fittype condit title str Alcohol Cigarette and Marijuana Use 4 2 The MOSMAT macro The MOSMAT macro uses the MOSAICS and MOSMAT SAS IML programs to create a scatterplot matrix of mosaic displays for all pairs of categorical variables Each pairwise plot shows the marginal frequencies to the order specified by the PLOT S parame ter When PLOTS 2 these are the bivariate margins and the residuals from marginal independence are shown by shading When PLOTS gt 2 the observed frequencies in a higher order marginal table are displayed and the model fit to that marginal table is determined by the FITTYPE parameter The keyword parameters and their default values are listed below Either the VAR or the VORDER parameter is required Smacro mosmat data _last_ Name of input dataset var Names of factor variables y 5 EXAMPLES 16 count count Name of the frequency variable Ej fittype joint Type of models to fit config User model for fittype USER EJ devtype gf Residual type F shade shading levels for residuals ard plots 2 which plots to p
38. ond Figure 1 Two way mosaic for hair color and eye color Positive deviations from independence have solid outlines and are shaded blue Negative deviations have dashed outlines and are shaded red The two levels of shading density correspond to standardized deviations greater than 2 and 4 in absolute value The second run mosaics statement line 33 fits the same models but reorders the eye colors in the table to better display the pattern of association between hair color and eye color in the two way table It is also necessary to rearrange the eye color labels in row 2 of 1names This reordering is based on a correspondence analysis of residuals in the two way table as described in 3 carried out separately See the order global variable in Section 3 2 Note that the global variables split and htext specified in the first mosaic continue to be used here The plots produced from this call are shown in Figure and Figure 2 The third run mosaics statement line 37 plots only the three way display showing residuals from the model in which hair color eye color and sex are mutually independent This plot is shown in Figure 3 goptions vsize 7in hsize 7in R square plot environment proc iml worksize 10000 symsize 10000 start haireye x Hair color eye color data table JE brown blu hazel green 32 53 10 3 11 50 10 30 10 25 7 5 3 15 7 8 M 36 66 16 4 9 34 7 64 5 29 7 5 2 14 7 8 E levels 4
39. ou wish Or change the default values in the globals module The program cannot access global fonts assigned with the GOPTIONS FTEXT and HTEXT options Instead you may specify a desired font with the IML global font and ht ext variables For some output devices e g PostScript specifying a hardware font e g font hwps1009 for Helvetica can yield an enormous reduction in the size of the generated graphic output files By default the program uses the Helvetica hardware font when it detects a PostScript device and uses the SWISS font otherwise 3 3 1 EPS Output Some output devices such as Encapsulated Postscript and GIF require that each figure be written to a separate output file Mosaics contains a gskip module which handles this automatically for EPS output It uses three global SAS macro variables DEVTYP Device type Use 3let devtyp eps for EPS output Ordinarily let devtyp screen for Display Manager DISPLAY Display option Use let display ON for ordinary use Setting DISPLAY OFF suppresses graphic output for all devices FIG Figure number Initialize to 1 let fig 1 Listed below is a macro EPS which I use to initialize graphics options for EPS output sglobal fig gsasfile devtyp Smacro eps let devtyp EPS let fig 1 let gsasfile grfout eps Sput gsasfile is amp gsasfile filename gsasfile amp gsasfile A A o goptions horigin 5in vorigin 5in override
40. plot split is acharacter vector consisting of the letters V and H which specifies the directions in which the variables divide the unit square of the mosaic display If solit H V the default the mosaic alternates between horizontal and vertical splitting If the number of elements in split is less than the maximum number in plots the elements in split are reused cyclically vlabels is an integer from 0 to the number of variables in the table It specifies that variable names in addition to level names are to be used to label the first vlabels variables The default is vlabels 2 meaning variable names are used in plots of the first two variables only 3 2 3 Other options gout is a character string which specifies the name of the graphics catalog The default is GSEG the default graphics catalog name is a character string up to 7 characters which specifies the prefix for the names of the graphs in the graphics catalog The default is MOSAIC outstat is a character string containing the name of an optional output data set containing the fol lowing variables RESIDUAL FITTED and FREQ The variable FACTORS gives the number of factors in a given mosaic display and LABELS gives the cell labels for each cell in the given table verbose NONE FIT BOX is a character vector of one or more words which controls verbose or detailed output If verbose contains FIT additional details of the fitting process fitted frequencies ma
41. program moseye sas included in the distribution archives illustrates how models of quasi independence and quasi symmetry can also be fit with MOSAICS 4 Macro interface 4 1 The MOSAIC macro The MOSAIC macro provides an easily used macro interface to the MOSAICS and MOSAICD SAS IML programs Using the SAS IML programs directly means that you must compose a proc iml step and invoke the mosaic module as described in Section 3 1 The MOSAIC macro may be used with any SAS dataset in frequency form e g the output from PROC FREQ The macro simply creates the proc iml step reads the input dataset see Sec tion 3 5 and runs the mosaic module If your data is in case form or you wish to collapse over some table variables you must use PROC FREQ first to construct the contingency table to be analyzed The TABLE macro may be used for this purpose It has the advantage of allowing formatted values of the table factors to be used by the mosaics program Ordinarily the program fits a model specified by the itt ype parameter and displays resid uals from this model in the mosaic for each marginal subtable specified by the PLOTS parameter However if you have already fit a model and calculated residuals some other way e g using PROC CATMOD or PROC GENMOD specify a RES ID variable in the macro call The macro will then call the mosaicd module as described in Section 3 6 The MOSAIC macro is easier to use but is not as flexible as direc
42. rginal proportions are printed If verbose contains BOX additional details of the drawing process tile dimensions label placement are printed window is a numeric vector of 4 elements containing the world coordinates of the lower left and upper right coordinates of the graphics window used for the mosaic display The actual mosaic fills the region 0 0 100 100 The default window is set to 16 16 108 108 to allow for text labels and a title 3 2 4 Caveats There is one caveat imposed by this use of global variables The mosaic module should not be called from an IML module with its own arguments since this would cause all variables defined within that module to inaccessible as global variables The mosaic module may be called either in immediate mode as in the example in secrefsec ex direct or from an IML module defined without arguments 3 3 Graphic options MOSAICS assumes that the vertical and horizontal dimensions of the plot are equal so you should include a goptions statement specifying equal values for hsize and vsize if the default values for your device are unequal For example goptions hsize 7 in vsize 7 in 3 USING MOSAICS 11 By default the program uses shades of the colors blue and red to draw the tiles corresponding to positive and negative residuals It cannot respect the global colors options on the goptions statement You can specify the IML global colors variable to change these assignments if y
43. roduce mf colors blue red colors for and residuals fill HLS HLS fill type for and residuals ara split V H split directions vorder order of variables in mosaic htext height of text labels xf font font for text labels ay title title for plot s space room for spacing the tiles aA fuzz smallest abs resid treated as zero abbrev abbreviate variable names in model sort YES Sort variables first 4 3 The TABLE macro The TABLE macro constructs a grouped frequency table suitable for input to the MOSAIC macro or the MOSMAT macro The input data may be individual observations or a contingency table which may be collapsed to fewer variables Factor variables may be converted to character using user supplied formats See Section 5 4 for an example Smacro table data _last_ Name of input dataset var Names of all factor variables char Force factor variables to character weight Name of a frequency variable order Specifies the order of the variable levels format List of var format pairs out table Name of output dataset 5 Examples The examples below were written sequentially as the MOSAICS package developed so the initial examples Section 5 l Section 5 3 illustrates its use within SAS IML The macro interface was developed later and PROC GENMOD now allows a wider class of models to be fit than could be handled by the IPF a
44. rt variables resid Name of residual variable fuzz Fuzz value for residuals near 0 order Do CA on marginal tables lorder Reorder levels of one or more variables legend Legend for shading levels H V or NONE outstat Name of an output data set of fit statistics Zeros 0 1 variable where 0 indicates structural 0 name mosaic base name of graphic catalog entries gout name of graphic catalog The parameters for the mosaic macro are like those of the SAS IML program see Section 3 1 except data Specifies the name of the input dataset Should contain one observation per cell the variables listed in VAR and COUNT and possibly RESID and BY var Specifies the names of the factor variables for the contingency table Abbreviated variable lists are not allowed The levels of the factor variables may be character or numeric but are used as is in the input data That is a numeric variable with an attached user defined format appears as numeric You may omit the VAR variables if variable names are used in the VORDER parameter by Specifies the names of one or more By variables Partial mosaic plots are produced for each combination of the levels of the BY variables The BY variable s must be listed among the VAR variables count Specifies the names of the frequency variable in the dataset 4 MACRO INTERFACE 15 fy config For a user specified model config gives the terms in the mod
45. s up by sorting the data 5 5 Example 5 Using PROC GENMOD and the MOSAIC macro It was mentioned earlier that MOSAICS and the MOSAIC macro can be used to display the results of models fit using PROC GENMOD or PROC CATMOD Indeed this is often the easiest way to use MOSAICS and to visualize the results of a fitted model It also allows you to fit more complex models than can be handled by the IPF algorithm used internally in MOSAICS 6 IMPLEMENTATION 25 We illustrate the process using the marital dataset shown in Section 5 2 fitting the model GPE PM EM with PROC GENMOD mosaic5g sas include catdata marital proc genmod data marital class Gender Pre Extra Marital model count Gender Pre Extra Pre Marital Extra Marital dist poisson obstats residuals ods output obstats obstats Smosaic data obstats var Gender Pre Extra Marital vorder Marital Extra Pre Gender resid streschi The essential idea is to fit this as a Poisson regression model for the count variable lines 5 6 and obtain a dataset containing residuals using the ODS OUTPUT statement line 7 The obstats dataset contains the original variables plus various residuals calculated by PROC GENMOD one of which is the standardized adjusted Pearson residual called st reschi Feed ing the obstats dataset to the mosaic macro line 9 and specifying resid streschi in the macro call causes the program to bypass its built in IPF f
46. t use of the SAS IML programs e Factor levels are labelled using the values of the factor variables in the input dataset You cannot simply attach a SAS format to a factor to convert numeric values to character labels but you can use a DATA step to create character equivalents of numeric variables using the put function or use the TABLE macro 4 MACRO INTERFACE 14 e You cannot reorder the factors or the levels of a factor as flexibly as you can in SAS IML If you use the SORT parameter take care that an ordered factor Low Medium High is not sorted alphabetically Usage The mosaic macro is called with the keyword parameters below Either the VAR or the VORDER parameter is required Smosaic data _last_ Name of input dataset var Names of all factor variable count count Name of the frequency variable Name s of BY variables fittype joint Type of models to fit config User model for fittype USER devtype gf Residual type shade 2 4 shading levels for residuals plots which plots to produce colors blue red colors for and residuals fill HLS HLS fill type for and residuals split V H split directions vorder order of variables in mosaic htext 1 5 height of text labels font font for text labels title title for plot s space room for spacing the tiles cellfill write residual in the cell vlabels Number of variable names used as plot labels sort Pre so
47. the smallest absolute residual to be considered equal to zero Cells with d lt fuzz are outlined in black The default is fuzz 0 20 htext is anumeric value which specifies the height of text labels in character cells The default is htext 1 3 The program attempts to avoid overlap of category labels but this cannot always be achieved Adjust ht ext or make the labels shorter if they collide legend H V NONE Orientation of legend for shading of residual values in mosaic tiles V specifies a vertical legend at the right of the display H specifies a horizontal legend beneath the display Default NONE shade is a vector of up to 5 values of d which specify the boundaries between shading levels If shade 2 4 the default then the shading density number d is d residuals 0 0 lt d x2 1 2 lt di lt 4 2 48 dis 3 USING MOSAICS 10 Standardized deviations are often referred to a standard Gaussian distribution under the as sumption that the model fits these values roughly correspond to two tailed probabilities p lt 05 and p lt 0001 that a given value of d exceeds 2 or 4 respectively Use shade a big number to suppress all shading space is a vector of two values which specify the x y percent of the plotting area reserved for spacing between the tiles of the mosaic The default value is 10 times the number of variables allocated to each of the vertical and horizontal directions in the
48. tionally the keyword CONDIT may be followed by a digit k to specify which of the n ordered variables is conditioned upon PARTIAL specifies sequential models of partial independence of the first pair of variables conditioning on all remaining variables one at a time A B AC BC ACD BC D For the 3 way model A and B are hypothesized to be conditionally independent given C for the 4 way model A and B are conditionally independent given C and D MARKOVk specifies a sequential series of Markov chain models fit to the table whose dimen sions are assumed to represent discrete ordered time points such as lags in a sequen tial analysis The keyword MARKOV can be optionally followed by a digit to specify the order of the Markov chains e g fittype MARKOV2 specifies a second order Markov chain First order is assumed if not specified Such models assume that the table dimensions are ordered in time e g Lag0 Lag1 Lag2 MARKOV or MARKOV1 fits the models A B AB BC AB BC CD where the categories at each lag are associated only with those at the previous lag MARKOV2 fits the models A B A B C ABC BCD ABC BCD CDE USER If fittype USER specify the hypothesized model in the global matrix config The models for plots of marginal tables are based on reducing the hypothesized configu ration eliminating all variables not participating in the current plot order NONE DE
49. to get PS EPS output in Win doze Added NAME global variable for graph names in the graphics catalog Fixed a bug in the calculation of adjusted residuals Added CELLFILL FREQ to display cell frequency in the tiles Added ABBREV global to abbreviate variable names in models and titles Version 3 4 e Added vlabels global variable to control the number of variables for which variable names are used in the display fuzz now sets line style solid e Global variables are now set in a separate module to make changing defaults easier e In transpos module you can now specify the variable names in the new order rather than indices The config configuration may also be specified using variable names e Added code for models of joint independence and conditional independence in which any vari able may be specified as the jointly independent or conditioning one Version 3 3 e Added a GSKIP module for EPS or GIF output to separately named graphics files Requires a global macro variable amp DEVTYP EPS Version 3 2 e Added zeros global input matrix to handle structural zeros e Added ability to display chisquare value in the mosaic title for each plot by using amp G2 or amp X2 in the title string e Changed default values to filltype HLS HLS colors BLUE RED since this is what I always use now except for monochrome output Version 3 1 e Added readt ab routine for easier input from a
50. torage mosaic mosaic load module _all_ split V H htext 1 6 plots 2 4 run mosaic levels table vnames lnames plots title 34 35 36 37 38 39 40 41 5 EXAMPLES 22 plots 4 fittype USER title Model GPE PM EM config 1 2 3 2 4 4 3 0 0 run mosaic levels table vnames lnames plots title The first run mosaic statement produces plots of the 2 way to 4 way tables fitting models of joint independence The second run mosaic statement produces a plot of the 4 way table fitting the model GPE PM EM specified by the config variable and fittype USER This model treats G P and E as explanatory and M as a response This is equivalent to the logit model with main effects of premarital sex and extramarital sex on marital status Using the readt ab routine this example can be simplified as follows The routine constructs the table levels and lnames variables But note that the values of the Pre and Extra variables are both simply Yes or No proc iml worksize 10000 symsize 10000 vnames Gender Pre Extra Marital run readtab marital freq vnames table levels lnames title Pre Extramarital Sex and Marital Status reset storage mosaic mosaic load module _all_ split V H htext 1 6 plots 2 4 run mosaic levels table vnames lnames plots title
51. urally recursive one which can be implemented easily in a language which sup ports recursion and multi dimensional arrays such as APL or S R Wang 10 describes a FORTRAN implementation of mosaic displays which simulates multi dimensional arrays by subscripting a vec tor The following algorithm which uses two dimensional arrays is much simpler A general scheme for handling multi dimensional arrays in SAS IML is described in 6 1 Denote the number of levels of the n variables by l1 ln and let Ls be their cumulative products IT _ At step s 0 start with one tile a square of size 100 x 100 and let Lo 1 2 The tiles in the mosaic are represented by an array B of four columns called boxes in the program Columns and 2 give the x y location of the lower left corner of the tile columns 3 and 4 give the horizontal and vertical lengths of the tile At step0 B 0 0 100 100 There is one row for each tile The following steps are repeated for each variable s 1 n 3 For variable s find the marginal frequencies of variables s 1 n a vector of length Ls with the levels of variable s varying most rapidly 4 Reshape this vector row wise to a matrix M Myn of Ls rows and l columns The array M is called margin in the program See the arrays labeled Marginal totals the printed output The rows of M correspond to the tiles of the previous variables at step s 1 5 Each old tile is then divided
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