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1130-CA-06X - about the IBM 1130 Computing System

1. F Application Program H20 0333 1 1130 Statistical System 1130 CA 06X Users Manual The 1130 Statistical System performs four major statistical functions regression analysis factor analysis analysis of variance and polynomial fitting This manual contains for each type of analysis performed a description of the computational algorithms used the form and content of the control cards operating instructions and sample problems Kristofer Sweger RR EE BILL a emn H LOMO mo m Second Edition H20 0333 1 is major revision obsoleting H20 0333 0 and incorporating TNL N20 1029 0 Specifications contained herein are subject to change from time to time Any such change will be reported in subsequent revisions or Technical Newsletters Copies of this and other IBM publications can be obtained through IBM branch offices Address comments concerning the contents of this publication to IBM Technical Publications Department 112 East Post Road White Plains N Y 10601 International Business Machines Corporation 1967 Y INTRODUCTION The 1130 Statistical System contains four major analysis programs 1 Stepwise Linear Regression 2 Factor Analysis 3 Analysis of Variance 4 Polynomial Fitting with Orthogonal Polynomials Each of these analysis programs is composed of a number of subroutines which are store
2. ORTH POLY NO SCALING JOB 1111 PAGE 6 IDENTIFICATION ya Yeys 1 1 0 10000E OsllOOOE 01 0O 14497E Ol 04 34972E 00 2 2 0 20000E 01 Oe71900E 0O 61166bE 06298334E 00 3 3 0 30000E 0O 11000E 02 O 10271E 02 0 72875E 09 4 4 O0 40000E 01 0O 12600E 02 0 13913E 02 0 13134E 01 5 5 0 50000 0O 14700E 02 0O 417043E 02 0423434E 01 6 6 Os60000E 01 Oel9SOOE 02 966 02 0 423899E 90 7 27 OsTOOCOE Oe25100E 02 21766 02 0233337E 8 8 0 80000E 01 0 23900E 02 0 23359 02 O6540AR4E 00 9 9 0 90000E Me231CNE n2 0e24439E 02 0413397E 91 10 10 0 10000E 02 0423600E 02 0 4 25007E 02 04 14079E 01 11 211 0 11000E 02 O426000E 02 0 25063bE 02 0293614E 00 12 12 0 12000E 02 O4624600E 02 0 24607E 02 0 74081E 02 JOA COMPLETED INPUT XEQ POLY 02 LOCALPOL24 POLSQ PCOEF PDER PFIT LOCALPOLY TRAN DATRD FMTRD PRNTB 020200 1111 ORTH POLY SCALING 0201010102010100 010000000000100 C12 I1 1X F2 0 F3 11 011 01011 020 02071 031 03110 040 04126 051 05147 060 06199 O71 07251 080 08239 091 09231 100 10236 111 11260 120 12246 1 PUNCHED OUTPUT 0 3636363E 00 0 2363636E 01 1 111124 1 1 0 3178914E 06 1575755 111124 1 2 0 1016575E 06 Maa d RR E PEE RE aaa 5 XEQ POLY 02 0 1772499E 02 0 1234159E 0 5789424E 111124 1 3 0 3039385t 06 0 1147578E 01 0 1937265E 01 OUTPUT LOCALPOL2 sPOLSQ sPCOEF PDERsPFIT LOCALP
3. K STEPWISE TEST ONE REGRESSION ANALYSIS DEPENDENT VARIABLE P6 RESIOUAL STANDARD DEVIATION 2349726 STANDARD ERROR OF THE MEAN 299071 MULTIPLE R 028456 MULTIPLE RSOR 0 7150 VARIABLE ENTERED P1 VARIABLE B COEF STD ERROR OF B PARTIAL R P1 0 4272 024813 0 1111 P3 1 8967 004145 024994 Ph 2 2333 025349 024654 PS 191167 093923 063375 CONSTANT 7122530 ANALYSIS OF VARIANCE TABLE SOURCE DaFo SUM OF SQUARES MEAN SQUARE MEAN i 0 21933 06 Oe21933E 06 REGRESSION 4 Oe90867E 05 0 22716E 05 ERROR 63 0 36205 C5 0e57468E 03 JOR COMPLETED 30 BETA COEF STD ERROR OF BETA 045228 0 1089 023184 0 0725 0 2967 021091 F 0 52617E 02 908 BETA COEF 020635 025063 093075 063200 0 39529E 02 2222 STD ERROR OF BETA 020715 021106 0 0736 0 1124 2 2 PRINCIPAL COMPONENTS AND FACTOR ANALYSIS The aim of factor analysis is to explain observed relationships among numerous variables in terms of simpler relations This simplification can take the form of producing a set of classificatory categories or creating a smaller set of hypothetical variables The usual procedure is to collect measurements on n variables over N persons or objects N should be appreciably larger than n To find out what goes with what among these n variables the n variables can be intercorrelated as they vary over the N objects This is done for all possible n n 1 2 pairings of the variables producing a square symmetrica
4. 10051 O1 0 8391e4E 00 0 48826E 00 0 31524E 00 0 54778E 01 0 371688 00 0 89269 00 04 91570E 00 0 10012 0 99368E 00 CORRELATION MATRIX INPUT MULTIPLE R 2 ON DIAGONAL XEQ FCTR L CALFCTR FMTRD DATRD PRNTB MXRAD TRAN LOCALFCTRIsTRIDI OR INVRS LOCALCDREL PRNT LOCALFCTR2 VECTR PRNT LOCALFCTR3 VARMX PROMX SCDRE RFOUT 020200 3333 Pl 3333 4 3333 4 3333 4 05 FACTOR ANALYSIS SAMPLE PROBLEM 040300000000000000000100920200020090 1010001910101009101010100 P2 1 P3 P4 l 0 64125T1E 00 2 0 7225732E 00 3 0 5798441E 1 1 3333 4 1 333323 1 333323 1 333323 1 333323 1 333321 1 1 d WW N mq 0 7999757E 0 112285 7E 0 1030857E 0 1016857E 0 9960000E 0 3500000E 00 03 03 03 02 02 0 7225T32E 0 8018028E 00 0 6567597E 0 8950214E 0 5141101E 0 2161068E 0 2601331 0 7545321F DUT PUT 00 0 5798441E 09 0 7999757E 00 00 0 656T7T59TE 00 0 8950214E 09 00 0 5689992E 00 0 7525222E 00 09 0 7525222E 00 0 8816457E 00 02 02 02 02 3333 PAGE 21 67 XEQ FCTR 05 LOCALFCTReFMTRD sDATRD os PRNTB oMXRAD STRAN LOCALFCTRI sTRIDI sQRs INVRS LOCALCOREL PRNT LOCALFCTR2 sVECTR sPRNT LOCALFCTR3 sVARMX sPROMX SCORF sRFOUT FACTOR ANALYSIS SAMPLE PROBLEM JOB 3333 PAGE 0 NUMBER OF VARIABLES INCUT TYPE SEQUENCE CHFCK VARIABLES ON CARD 1 VARTARLES ON CARD VARIABLES ON CARD 3 TRANSFORMATION SWITCH OuTPUT RAW CROSS PRODUCTS
5. 0 1005148E O1 01 0 8391422E 00 01 0 4882641E 00 0 3152475E OQ 00 0 4477804E O1 01 0 3768844E 00 01 0 8926919E 00 01 0 9157280E 00 01 0 10C1285E 01 0 9936804 00 QUTPUT 61 XEQ FCTR 05 LOCALFCTReFUTRD sDATRDePRNTB eMXRAD TRAN LOCALFCTRIsTRINDI sQReINVRS LOCALCORELsPRNT LOCALFCTR2 eVECTR ePRNT LOCALFCTR3 sVARMX ePROMX se SCORE sRFOUT 4 FACTOR ANALYSIS SAMPLE PROBLEM JOB 3333 PAGE 0 NUMBER OF VARIABLES INPUT TYPE SEQUENCE CHECK VARIABLES ON CARD 1 VARIABLES ON CARD 2 VARIARLES ON CARD 3 TRANSFORMATION SWITCH OUTPUT RAW CROSS PRODUCTS OUTPUT RESIDUAL CROSS PRODUCTS OUTPUT VARIANCE COVARIANCE OUTPUT CORRELATION FACTOR SCORES NUMRER OF FACTORS OPTION NUMAFR OF FACTORS OR PERCENT OF TRACE COMMUNALITY OPTION ROTATION OPTION NUMBER OF FACTORS TO ROTATE POOL ING OPTION LATENT VECTORS UNROTATED FACTOR MATRIX ORTHOGONAL TRANSFORMATION MATRIX ORTHOGONAL FACTOR MATRIX TRANSFORMATION MATRIX TO OBLIQUE REFERENCE VECTOR STRUCTURE OBLIQUE REFERENCE VECTOR STRUCTURE MATRIX CORRELATIONS AMONG OBLIQUE REFERENCE VECTORS OBLIQUE REFERENCE VECTOR PATTERN MATRIX CORRELATIONS BETWEEN REFERENCE VECTORS AND PRIMARY FACTORS OBLIQUE PRIMARY FACTOR STRUCTURE MATRIX CORRELATIONS AMONG JBLIQUE PRIMARY FACTORS OBLIQUE PRIMARY FACTOR PATTERN MATRIX FACTOR SCORE REGRESSION COEFFICIENTS 8 2 Os pe m ee VA M PO I pa OoOoooonr p 21291X 4F640 FACTOR
6. 0 1721 0 7213E 0 1900E 0 8116E 906 9541E 0e2230 0 1209 0 4183E 002326E 06 3517E 0e6528E OeS7O9E 045197E 0 1196 0 3081 0 1371 0 93631 06153578E 0 4082 01461 0 1680E 0e2616E 0 53981 0 51928 0 1110 0606166E 0e4678E 0 2106E 0 1612 026515E 066557E 0 208 7E 0 9257 0e5037E RESIDUAL 906 1317E 0461323E 0 7006 0 47151E 05 1500 8850 0049164 06 2925E 0s14T2E 0 1226 0 8966 0 5869E 051330E 021964E 0 1770 0e2346E Qe 4988E 0 1606E 023684E 006563E 90 53518E 0 1353 02 1006E 0 1115 0 57169 0 2809 70 6905E 0 8035E 08 2222 JOB PAGE 2222 10 PAGE 11 27 28 XEQ REGR LOCALREGR FMTRD PRNTB DATRD MXRAD TRAN LOCALREGR2 REGRE LOCALCOREL PRNT 020200 2222 NUMBER OF INPUT TYPE 03 STEPWISE TEST ONE STEPWISE TEST ONE VARIABLES SEQUENCE CHECK VARIABLES ON CARD 1 VARIABLES ON CARD 2 VARIABLES ON CARD 3 TRANSFORMA TION SWITCH CORRELATION MATRIX INPUT OUTPUT RAW CROSS PRODUCTS OUTPUT RESIDUAL CROSS PRODUCTS PRINT PREDICTED VALUES PRINT STEP POOLING OP DEPENDENT FeLEVEL TO REMOVE VARIABLES FeLEVEL TO ENTER VARIABLES TOLERANCE OUTPUT VARIANCE COVARIANCE 5 VALUE OUTPUT CORRELATION oO Ooo0o000000000 0 500 02300 0 00010 0 0 06030000 000000000000010006 500 300 00010 Pl P2 P3 P4 P5 P6
7. 0 21610E 02 0 926013 02 0 79453E 02 MATRIX OF CORRELATION COEFFICIENTS P3 0 72257E 00 045T7986F 0 72257E 00 0O 10 h00E 91 0 65675E 0 57984E 00 0C 65675E 09 O4 10CO9E 0 79997E 00 O89502E AN 0475252E Pe 00 0 479997E 09 00 0O0 89502E 00 01 49 75252E 09 on 01990900E 01 JOB 3333 JOB 3333 Jo8 33233 VARIANCE 0 26430E 04 0 46702E 03 0 67669E 03 0 56931E 04 JOB 3333 PAGE PAGE PAGE PAGE 2 3 5 63 64 FACTOR ANALYSIS SAMPLE PROBLEM 908 3333 PAGE 6 MATRIX OF CHARACTERISTIC VECTCRS P1 0 68 311E 00 04 54530E 09 P2 0 512656E O0 04 16668E 00 P3 0 66188E 09 0 81965E 00 P6 04 53892F 00 04 55082E 01 FACTOR ANALYSIS SAMPLE PROBLEM JOB 3333 PAGE 7 TRACE 4 0000 CHARACTERISTIC ROOTS CUMULs PERCENT OF TRACE 3 2134 80 3374 0 4301 914 0900 0 2753 0 0000 0 0810 0 0000 FACTOR ANALYSIS SAMPLE PROBLEM JOB 3333 PAGE 8 NORMALIZED UNROTATED FACTOR LOADINGS P1 86604E 00 0 35762 E 00 P2 Oe91901E 00 0 10931E 00 P3 0 8279RE O0 e0 53754E 00 P4 049660RE 00 0 36124E 01 COMMUNALITIES H 87793646E 00 0 8565298E 00 0 9745132E 00 029446225E 00 FACTOR ANALYSIS SAMPLE PROBLEM JOB 3333 PAGE NORMAL VARIMAX CRITERION NORMALIZED CYCLE CRITERION DIFFERFNCE EPSILON CRITERION 0 00116000 1 0202950188 0202850189 2 0418619206 0615760916 3 0418610206 0 s0000000n0 FACTOR ANALYSIS SAVPLE JOB 3333 PAGE 10 ORTHOGONAL TRANSFORMATION MATRIX VARIASLE 1 2 1
8. 0 68182 0 204 67 04 05 05 04 05 05 04 04 03 04 05 0 6 0 500 0 300 0200010 1 2 LOCALREGR oFMTRD oPRNTB sDATRD eMXRAD eo TRAN LOCALREGR2 REGRE LOCALCOREL PRNT MATRIX P3 0 49851E 0 17138 0 26448 0 27853E 0 25314E 0 66929 MATRIX P3 0 259893 0063273E 0 90575E Oe58802E 0 73100 0 2 667 04 05 05 04 05 05 P 0 21390E 0935929 0 27853E 0 30625 04 6487 0 17964 04 04 04 04 05 OF RAW CROSS PRODUCTS P5 06122022E 0 33215E 0 25314E 0946487E 0 54295E 0 12157E 05 05 05 05 06 OF RESIDUAL CROSS PRODUCTS 02 04 04 03 P4 0 66455E 0 37869E 0 58602 0 24096 04 09 70419E 05 0 59945F P5 03 70410592E 03 03 04 03 04 0 68782E 0 73100E 0 70419E 0410434E 9 23492 04 04 04 03 95 05 J08 JOB P6 0 28566 0 793643 0 66929 0 17964E 0 121297 02034641E 05 05 05 05 06 06 JOB 6 0215499E 0204 67 0 26667 00599465E 0 523492E 0 512707E Q 05 05 04 05 06 2222 2222 2222 PAGE PAGE PAGE i 23 24 STEPWISE TEST VARIABLE P1 P2 P3 P4 P5 P6 READY THE PUNCH WITM BLANK CARDS AND PRESS START ON THE VARIABLE P1 P2 P3 P4 P5 P6 P1 0 41909E 02 02010689E 02 038647 00 0 99187E 01 06 15809E 02 0 2313 4E 02 STEPWISE TEST ONE SUMMARY STATISTICS VARIABLE P
9. 65 01 0 8678567E 01 0 2561686E 00 OUTPUT XEQ POLY 02 LOCALPEOL 2 sPOLSO sOCOFF PDERSPFIT LOCALPOLYs TRAN sDATRD sFMTRDePRNTB ORTH POLY NO SCALING MAXIMUM DEGREE OF POLYNOMIAL 2 INPUT TYPE 1 POLYNOMIAL COEFFICIENTS 1 COMPUTE DERIVATIVES 1 ORDER OF DERIVATIVE 2 PREDICTED VALUFS l PUNCH SOLUTION VECTORS 1 SECONDARY INPUT TYPE 0 VARIANCE CRITERION 0 010000001 TRANSFORMATION SWITCH 0 SCALING 0 IGNORE POLYNOMIAL OUTPUT 0 12111 91 9 2 0 3 1 TRANSFORMATION MAX DEGREF OF POLYNOMIAL REACHEDe VARIANCE CRITERION NOT SATISFIED 103 l ORTH POLY NO SCALING JOB 1111 PAGE ORTHOGONAL POLYNOMIALS INENTIFICATION xX Y Y Y Ys 0 i 2 1 0 10000E 01 Q0 11000E Ol 0O 14497E 0 434972E 00 0O 10000E 0l 0 55000E 01 0 18333E 02 2 2 0 20000EF 01 0O0O 71900E Os61166E 0O0 98334E O0 0O 4 10000E 04 65000E 01 0683333E Ul 3 3 0 30000E Ul U l11U0U0UE 02 04 10271bE 02 Us72875E 00 UslVUOVE 04 35000E 01 0 33333 OU 4 4 9 40000E Ol Usl260VE 02 004 13913 02 04 13134E 0O4 10000E 0425000E 0 26 66 Ul 5 5 Q450000E 04 14700E 02 OslTOSG3E 02 0 2345434E O 10000 04 15000E 0596666E 6 6 O s560000E 01 0 19900E 02 0O 4 19661E 02 0 23899E OO 0O0 10000E Ol 0550000E OO 2 05411666 E 7 7 0 70000 O 25100E 02 0U 21766E 02 404 33337E 01 U 10000E OUsS50Q000UE QU 05
10. Control Cards Input Source Form Cross Products Form Subsidi ary and Cor relation Complete Regression Analysis Figure 17 Analysis of variance and stepwise multiple regression V 5 Control Cards input Initial Compute Orthogonal Polynomials YES Compute Polynomial Coefficients Figure 18 Curve fitting with orthogonal polynomials ed Value Pr Required Compute Predicted Values Deriv Required Compute Derivatives 115 Read and Print Control Cards Input Source 3 Orthogonal Correl Rotation Form Subsid iary and Correlation Matrices Perform Oblique Rotation Are Factors Re quested Compute and Output Factors Compute Factor Scores 3 y Figure 19 Factor analysis 116 gt CHAPTER 4 SAMPLE PROBLEM TIMING The table below gives times for each sample problem from the reading of the first monitor card to the end of the output listing The 1132 Printer was used as the output device Problem Regression analysis card input Regression analysis correlation matrix input Orthogonal polynomials card input no scaling Orthogonal polynomials cards scaling Orthogonal polynomials solution vector input no scaling Orthogonal polynomials solut
11. OUTPUT RESIDUAL CROSS PRODUCTS OUTPUT VARIANCE COVARIANCE OUTPUT CORRELATION FACTOR SCORES NUYBER OF FACTORS OPTION NUMBER OF FACTORS OR PERCENT OF TRACE COMMUNALITY OPTION ROTATION OPTION NUMRER OF FACTORS TO ROTATE BDNOLING OPTION LATENT VECTORS UNROTATED FACTOR MATRIX ORTHOGONAL TRANSFORMATION MATRIX ORTHOGONAL FACTOR MATRIX TRANSFORMATION MATRIX TO ORLIQUE REFERENCE VECTOR STRUCTURE OALIQUE REFERENCE VECTOR STRUCTURE MATRIX CORRELATIONS AMONG OBLIQUE REFERENCE VECTORS ORLIQUE REFERENCE VECTOR PATTFRN MATRIX CORRELATIONS BETWEEN REFERENCE VECTORS AND PRIMARY FACTORS ORLIOUF PRIMARY FACTOR STRUCTURE MATRIX FO v it e Orro ON ONN OF CO CQCQ0D OQ COW Ff CORRELATIONS AMONG OBLIQUE PRIMARY FACTORS OBLIQUE PRIMARY FACTOR PATTERN MATRIX FACTOR SCORE RFGRESSION COFFFICIENTS C e wx FACTOR ANALYSIS SAMPLE PROBLEM JOB 3333 PAGE 1 MATRIX OF CHARACTERISTIC VECTCRS P1 0 446728RE 00 7333 00 P2 04 52362E 00 31961 00 P3 0 43527E OO 0 81887E OO P4 04 56391E 00 0 4 56930E 01 FACTOR ANALYSIS SAYELE PRORLEM JOB 3333 PAGE 2 TRACE 209937 CHARACTERISTIC RGOTS CUMULe PERCENT OF TRACE 229564 102 1689 020298 103 1991 0 0061 020000 O 2 0864 0 0000 FACTOR ANALYSIS SAMPLE PRORLEM JOB 3333 PAGE 3 NORMALIZED UNROTATED FACTOR LOADINGS P1 0 80356E ON 0 81725E 01 D 0 90033E nn 0 55183E 01 p3 0 74R42E 00 04 15138 00 P 0 96961E 00 0 98294E 02 COMMUNALITIFS 50 6522407F 00 0 8136
12. factor loadings is computed by scaling each latent vector by the square root of its associated latent root that is f a V gt 1J 1j J where the m are factor loadings the 2 are elements of the latent vectors and the ur are latent roots Several methods are available to the user for estimating the rank m of the factor space for the purpose of retaining only m factors for output or subsequent rotation The value of m may be specified on a control card and arbitrarily accepted as the maximum number of factors The user may also request that only those factors whose latent roots are equal to or greater than 1 0 be retained An option is also available to retain only those factors which cumulatively account for an amount of variance equal to or less than a given percentage of the total variance This option could be used in a principal components analysis The matrix of factor loadings may be rotated to approximate simple structure in an orthogonal reference frame by the Normal Varimax method Kaiser 1958 for the case of uncorrelated factors k factors are rotated where k is equal to or less than m the rank of the factor space as determined above The Normal Varimax method develops a transformation matrix T over a cycle of rotations of each of the 1 pairs of orthogonal axes of the factor space taken in turn The angle of each rotation is chosen such that a function U of the factor matrix is maximized Complete cycles of r
13. when k 1 the predicted values for all step equations are printed When k 3 the predicted values for the equation containing three independent variables are printed The printout also contains the actual value of y and the difference between the predicted and the actual values If predicted values are requested when equations with fewer than p variables are not desired no predicted values are printed That is if column 23 24 contains a positive integer less than the integer in column 25 26 no predicted values are printed Print Steps of Regression cc 25 26 If this field contains a value of k all step equations containing k or more independent variables are printed For example if all steps are desired a value of one 1 forces the printout of the equations containing 1 2 m independent variables If this field is zero or blank no printout occurs Only a correlation matrix is calculated Pooling Option cc 27 28 When using the matrix input option cc 3 4 are 03 and when pooling sums of squares and cross products section 2 1 3 if the user desires that matrices be subtracted rather than added of aid in deletion of outlyers this field should be nonzero Number of the Dependent Variable cc 29 30 The regression analysis program uses the value punched in this field to rearrange the correlation matrix means standard deviation and variable names vectors such that the dependent variable a
14. 0 3099554E 0 599T12TE 0 2539TO6E 02 0 1247940E 0 56T9412E 02 0 4355013E 0 6800001E 02 PRP LPH PH HLH LS Em P oq pe de de m OS OS OS OS ER rm om mmo RA e Om PWN Oh WN OUTPUT OUTPUT 61 02 01 02 02 0 5134508bE 02 C 2554817E 0 8679906E 00 C 1007193E 0 1000COOE 01 0 1258658E 0 1258658E 00 0 1000000F 0 7519358E 00 0 1404377E 0 7860574E 00 0 3425673E 00 2 1956896E C9 29 879119T7E 00 0 7519358E 01 0 14043 TE 00 0 1300000F 00 O 6451673E c2 03 05 am z E 4 XEQ 21251X REGR 03 STEPWISE TEST ONE NUMBER OF VARIABLES INPUT TYPE SEQUENCE CHECK VARIABLES ON CARD 1 VARIABLES ON CARD 2 VARIABLES ON CARD 3 TRANSFORMATION SWITCH OUTPUT RAW CROSS PRODUCTS OUTPUT RESIDUAL CROSS PRODUCTS PRINT PREDICTED VALUES PRINT STEPS POOLING OPTION DEPENDENT VARIABLE FeLEVEL TO REMOVE VARIABLES FeLEVEL TO ENTER VARIABLES TOLERANCE VALUE OUTPUT VARIANCE COVARIANCE OUTPUT CORRELATION sF5a2 F 500 92F 502 92F 500 STEPWISE TEST VARIABLE P1 P2 P3 P PS P6 VARIABLE P1 P2 P3 P4 PS P6 P1 0 61357E 0 65381 0 949851 0 21390E 04 11022E 0 28566 STEPWISE TEST P1 0 28079E 90 71522E 0025893E 0 266455E 90410592E 0415499E ONE 04 04 04 05 05 ONE P2 0 653B1E 0 21681E 0 17138 0 35929 0 33515 0 79363E P2 04 0 71622t 03 02 03 04 04 0 58667 0 632734 0 378 65
15. 1 r 1 Experimental error par n 1 SSerror SSerror Par n 1 within cell Total npqr 1 SStotal where p number of levels in factor A q number of levels in factor B r number of levels in factor C n number of observations per cell To obtain other experimental designs from a complete factorial design the user should analyze the data as if it were a complete factorial design and then reconstruct his ANOVA table from the output Not all experimental designs can be handled by this technique notably Latin and Youden squares lattices and incomplete randomized blocks 73 74 Also this program does not handle repeated measurement designs that is replications must be considered as a factor For a detailed account of the various experimental designs see O Kempthorne Design and Analysis of Experiments John Wiley 1952 Single Classification Design A X B In this case the replications are considered as a factor B The error term is SSerror SS SSab giving the following reconstructed ANOVA table SS4 SSerror SStotal Two Way Classification with Cell Repetition A X B X C This differs from a random ized block design in that one is not interested in the recovery of interblock information Consequently the error term is SS SS oc 88 SS S error bc where factor C is the cell repetitions thus giving the following ANOVA table SS SSp SSab SSerror SStotal Randomized Block
16. 48556E 05 STEPWISE TEST ONE REGRESSION ANALYSIS DEPENDENT VARIABLE P6 RESIDUAL STANDARD DEVIATION 2520821 STANDARD ERROR OF THE MEAN 320416 MULTIPLE R 08235 MULTIPLE RSOR 0 6781 VARIABLE ENTERED P4 VARIABLE B COEF STD ERROR OF B P3 208275 0 2656 PG 1 7976 0 5150 CONSTANT 2167445 ANALYSIS OF VARIANCE TABLE SOURCE Defe SUM OF SQUARES MEAN 1 0 21933 06 REGRESSION 2 OeB6180E 05 ER amp OR 65 0e40892E 05 MEAN SQUARE 0 21933E 06 0 78516E 05 0 7357OE 03 PARTIAL R 0 7971 093972 MEAN SQUARE 00e21933E 06 Oe43090E 05 Oce62911E 03 JOB 2222 PAGE l BETA COEF STD ERROR OF BETA 0 7860 020760 F Oel0672E 03 JOB 2222 PAGE 2 BE TA COEF STD ERROR OF BETA 0 7548 0 0709 082475 0 0709 F 0e68493E 02 29 STEPWISE TEST ONE REGRESSION ANALYSIS DEPENDENT VARIABLE P6 RESTOUAL STANDARD DEVIATION 2329328 STANDARD ERROR OF THE MEAN 229022 MULTIPLE R 028435 MULTIPLE RSOR 067115 VARIABLE ENTERED P5 JOB 2222 VARIABLE B COEF STD ERROR OF B PARTIAL R P3 1 9583 0 4079 0 5145 PS 263123 025266 0 4811 P5 190355 0 3808 0 53217 CONSTANT 299105 ANALYSIS OF VARIANCE TABLC SOURCE DeFe SUM OF SQUARES MEAN SQUARE MEAN 1 Oe21933E 06 0 21933 06 REGRESSION 3 906415 05 0 30138 05 ERROR 64 0e36658E 05 Oe57276E 03 3 M
17. Option cc 37 62 In a complete factor analysis there are 13 additional matrices section 2 5 3 that the user has the option to output if desired The 13 remaining fields on the card are for this purpose Each of the two column fields may take four possible values described below Value Meaning 0 or blank No output 1 Print matrix 2 Print and punch matrix 3 Punch matrix The following gives the name and field column numbers for each matrix Column Matrix 37 38 Characteristic vectors A 39 40 Unrotated factors F 41 42 Orthogonal transformations T 43 44 Orthogonal factors G S 45 46 Transformation to oblique reference vector structure L 47 48 Oblique reference vector structure V 49 50 Correlations among oblique reference vectors V 51 52 Oblique reference vector pattern W 53 54 Correlations between reference vectors and primary factors D 55 56 Oblique primary factor structure S 57 58 Correlations among oblique primary factors 59 60 Oblique primary factor pattern P 61 62 Factor score regression coefficients Factor Analysis Option Card Summary Column Meaning 1 2 Number of variables 3 4 Input type and source 1 Raw data input from card reader 2 Raw data input from disk 3 Matrix input from card reader 0 6 Check sequence of raw data input 0 No 1 Yes 7 8 Number of variables on card 1 9 10 Number of variables on card 2 11 12 Number of variables on card 3 13 14 Transformation s
18. PRNT amp LOCALFCTR34 VARMX PROMX SCORE RFDUT 020200 3333 Pl FACTOR ANALYSIS SAMPLE PROBLEM 040100000000000101010202020200020000 1010101010101010101010101 P2 P3 P4 I21231X 4F6 0 i 0101 0201 0301 0401 0501 0601 0701 0801 0901 1001 1101 1201 1301 1401 1501 1601 1701 1801 1901 2001 2101 2201 2301 2401 2501 2601 2701 2801 2901 3001 3101 3201 3301 3401 3501 I 00900639900 75000159000041 0001010000920001 42000049 000119000098000131000068 00015 7000191000124000092 0001 78000104000119000097 0001470001 060C0L18000102 000128000108000116000109 000113000107000116000066 000094000107000115000044 000111000104000117000069 000139000110000104000117 0001570001070C0100000118 000169000111002075000157 000145000109000079000107 0000 79090095000096000069 000049000086000111000047 000 0480000770001 11000032 000041000069000106000022 000066000062000097000017 0001110000 7400 0092000045 0001640001040000886000097 0001700001170C0039000164 0002080001 35000053000246 0002370001480000580003 66 000169000152000061000230 0001140001370000730001 75 00 0106000130009077000178 0000970001230000860001 56 0000990001 10000092000125 000111000111000102000105 000068000108000108000081 000048000096000121000044 0000420000 78000123000020 0000340000730001 25000017 0000480000840001 25000014 PUNCHED CORRELATION MATRIX OUTPUT 3 33323 333321 1 2 3 4 l 2 3 4 l 0 1000000E 0 7225T32 E 0 5798441E 0 799975 TE
19. Principal Components and Factor 35 Beove The ee De 3 OE 42 2 2 3 Summary of Output Statistics 45 2 2 4 Job ExecullOl e cdd c we ous eS v You Xue OR d 46 2 20 Dalla MDOE uos oos ne aee EO Se Se dE qc ON 56 2 2 6 Matrix Input Output 57 2 2 7 Operating Instructions osoa soea 58 A 2 2 8 Sample Problem i ek Gee 2 2 9 References ur 71 2 9 Analysis of Variance deret Se i ue Eod e e ies 72 2 3 1 Tests of Significance dde ud 76 2 3 2 Job Execution doe pork NI 78 2 3 3 Analysis of Variance Table Generation 81 2 3 4 Data PUT s e CORO eis ee RUP aha AS SR 83 2 3 5 Operating Instructions s s o e a a Soa 026622 ce ee eee 84 2 9 6 Sample Problem axo wm Schr Sw xU X bodas e 86 2 4 Least Squares Curve Fitting by Orthogonal Polynomials 89 2 4 1 Summary of Output gt ess sec cw ee Se 92 2 4 2 Job Execution nM EE pot 93 2 4 4 Data Input 5 e cs we 99 5 2 4 4 Operating Instructions s s s ee e e e e m e eses 100 2 4 5 Sample Problem bs Mr dic ee se po 102 2 5 General No
20. a disk containing the 1130 Disk Monitor System as described in section 1 1 The following decks should be preceded by a cold start card placed in the card reader hopper and the buttons IMMEDIATE STOP console RESET console START card reader and PROGRAM LOAD console should be pressed A blank card should be placed after the last deck in the card reader hopper DECKS LABELS ANOVA NOVA STORE STOR GET GETO ANOV2 NOV2 SDOP SDOP MNSQ MNSQ REPRT RPRT FMAT FMAT FMTRD FMRD DATRD DTRD GMPYX GMPY GDIVX GDIV PRNTB PRNB TRAN TRAN Used in all four analysis types B Execution from Disk Once the component subroutines and main calling programs are on the disk the execution of a job requires the monitor control cards program control cards and data cards to be placed in the card reader The deck should be preceded by a cold start card To initiate processing the buttons IMMEDIATE STOP and RESET console START card reader and PROGRAM LOAD console should be pressed The order in which the cards are placed in the card reader for either matrix or raw data input is shown in Figures 12 and 13 Table Generation Deck lt including the end of deck indicator End of Dota E Data Deck Variable ormat Input Output Z Units Monitor Control Final table generator card Figure 12 ANOVA card reader input Ed 85 86 Table Generation Deck including the end of de
21. add operation is terminated when a card with a blank or negative job number field is encountered unless the pooling option ce 27 28 of the option card is nonzero In this case the read add operation terminates at the first detection of a blank or negative job number field and the second succeeding matrix is subtracted from the previous matrix This operation is terminated when the second blank or negative job number is encountered If the subtraction option is used the second set of matrices must also include its associated raw sums and sums of squares vectors for proper analysis completion Predicted values are not available with this option Also high and low values are not calculated for the observations on variables If outlyers are detected the user has two options available if he wishes to reanalyze ignoring these outlyers 1 He can eliminate the data cards containing the outlyers and rerun the entire analysis 2 He can prepare cards according to the format given above under Format Description either by hand or by using the program To use the program he must run the analysis using only data cards associated with outlyers The option card must request raw cross products matrix output and may note that the dependent variable is zero So that the analysis will terminate after the correlation matrix is calculated If the user allows an entire regression analysis to be computed using only the outlyer cards a terminatio
22. but must be included with every job processed The first program operated on by this system Should be preceded by a cold start card PROGRAM CONTROL CARDS The program control cards communicate the data specific parameters and output options to the program There are five possible card types necessary for execution Input output units card Job title card 1 2 3 Option card described below 4 Variable name card described below Variable format card Four of the control cards are required in every job The variable format card is necessary only if source data is to be processed See General Operating Instructions section 1 2 t OPTION CARD Number of Variables cc 1 2 This field must be punched with a nonzero integer n which is less than or equal to 30 The value contained in n is the total number of variables tobe processed Input Type and Source cc 3 4 This field allows the user to specify the input device 1442 card reader or disk and indirectly the type of input analysis to be undertaken in the input program The three possible values that may be punched in this field are described below Value Meaning 1 Raw data will be read from the 1442 card reader and transferred to the disk It will be retained there for use by this or other programs until destroyed by input from one of the four programs in this system Raw sums and raw sums of cross products will be accumulated Dat
23. cards within a case are together and that the order of cards is consistent If the option is chosen cc 5 6 on option card this field must be punched with an integer that is in ascending sequence for all cards in the case If sequence checking is not desired the field may be blank and may consist of one blank column 3 4 n Floating Observation on variable xj Any number may be etc point F punched in this field Decimal points are not required The remaining fields on the card are reserved for variable observations If there are more variables than can fit on the first card a second and a third card may be used The particular card columns for each field are arbitrary Following the data deck the user must include a card containing a negative integer in the identification field This card signals the end of data 2 1 4 Matrix Input Output It is possible to obtain punched card output of a number of matrices see section 2 5 3 and vectors with the regression program This program is designed to also input some of these matrices at a later time for further analysis or processing In addition matrices from another program or source if punched in the proper format may also be used as input This section is devoted to a description of various possible forms of analysis with the output options available in each program Format Description Matrices are punched rowwise five elements to a card in the FORTRAN E
24. found and the calculation of F ratios for this case presents some difficulties For a two factor model A X B the F ratios are A MS MS p B MS MS p AB MS ib MS igor For the three factor case A X B X C we have the following F ratios A MS MS be f MS MSp MS MS jo Sa C MS MS bo MS AB MS b MS bec AC MS MS be BC M5 MS he ABO MPG MS8 ror In the case of the random effects model the interactions should be tested for significance first because if they are found to be significant there is little point in testing the main effects for significance 3 Model III Mixed Model The mixed model is probably the most common form used in analysis of variance Here some factors are fixed and others are random The calculation of the F ratios in this case depends on which factors are fixed and which are random An example is given with two fixed factors and one random factor A X B X C Design with Factor A a random effect A MS MS a error B MS MS b ct C MS MS C ac MS hb MS ror AC MS MS ac error BC MS MS be ABC MS E abc MP ene 2 3 2 Job Execution b To perform an analysis of variance the user must supply four sets of cards to the program 1 Monitor control cards 2 Program control cards 3 Data cards 4 Table output specification cards Random effects Monitor Control Car
25. initial computation of the best fitting polynomial See General Operating Instructions section 1 2 93 94 Maximum Degree of Polynomial cc 1 2 This field should be punched with an integer n which is less than or equal to ten The program attempts to fit a polynomial to the data points until the variance criterion punched in columns 17 26 is satisfied by successive degree polynomials If the variance criterion is not satisfied when the degree of the polynomial reaches n the program prints a message to this effect and continues using the solution to the nth degree polynomial The value of n should be less than m 1 where m is the number of the data points read Input Source cc 3 4 This field must be punched with an integer n which assumes a value of either one 1 or two 2 If n is equal to 1 the data points followed by a negative identification card are read from the 1442 card reader If n is equal to 2 the data points are read from the disk having previously been transferred there by a program using input mode 1 The data on the disk is destroyed by any program using input mode 1 As noted below Secondary Input Sources input type 3 also destroys disk data Coefficients of Fitted Polynomial cc 5 6 In the determination of the best fitting polynomial the computation involves only the orthogonal polynomials and the three associated vectors called the polynomial solution vectors The orthogonal polynomials
26. may be desirable to calculate for each observation a set of factor scores in place of the scores on the original variables These scores are a weighted summation of the original scores on each of the variables for a given observation These scores can then give a profile score for each observation over the factors extracted 2 2 1 Mathematics of Principal Components and Factor Analysis Principal Components Analysis Let us consider n points in a space of p dimensions when the x s are expressed in standard score form that is mean 0 variance 1 The line with current coordinates X is The sum of squares of the distances from the n points on to this line is n where p 2 emp s x om i 1 m M I p If this is a stationary value the partial derivatives with respect to the m s vanish and 7 PENNE 0 LL 2 Gp and since gt Xi 0 m g k a constant j Thus the origin lies on the line 1 and we may take all the m s to be zero thus n Yx n and we have e Im e 1 2 3 4 9 35 36 Then we can find stationary values of S for variations in g subject to equation 2 above If we consider an undetermined multiplier this gives us n MnY x 8 70 k 12 p 6 jl d which gives us the set of p equations g 1 A Bol io TU Ep 1p 0 Bol So po 4 EA 0 7 Eliminating the g s we get
27. no additional vectors or matrices punched with the punched output The matrix is computed even if the no output option is chosen Output Correlation Matrix cc 21 22 This field s used to indicate whether the correlation matrix defined by C r L i j 1 2 n ij 5 8 1 where is an element of the variance covariance matrix and S 5 are the standard deviations of the th and je variables respectively is to be printed punched printed and punched or not presented The four 4 possible values contained in this field are given above under Output Raw Sums of Cross Products The punched output of the correlation matrix includes the number of cases and cards containing the vectors of means and standard deviations The matrix is generated even if the no output option is chosen Compute and Print Predicted Values and Residuals cc 23 24 This field is used to indicate the regression step to begin computing and printing the predicted values defined as Y bo DX bk where si b are the coefficients of the regression equation for the Ke step and Xi are the source data elements If this field is blank or zero the predicted values are not computed If the field is negative predicted values are printed only on the final step If it contains a nonzero value k the predicted values are computed for each regression step equation containing k or more variables For example
28. or floating point format Each card is identified as to its job number matrix number row number and column number of the first element on the card The specific card columns occupied by the identification fields and matrix elements are shown below Column Meaning 1 4 Job number 5 6 Matrix identification number section 2 5 3 7 8 Column number of first element on card 9 10 Row number 11 24 Matrix element 0 XXXXXXXEXNN 25 38 Matrix element 39 52 Matrix element 53 66 Matrix element 67 80 Matrix element Note that all matrices are punched and read under a fixed format Hence a variable format card is not allowed when using punched card matrices as input Most matrices have a unique identification number However there are a few cases where two or three vectors have the same identification and are always punched together For these cases see section 2 5 3 17 18 Regression with Correlation Matrix Input The punched output option of the correlation matrix includes the punchout of the number of cases matrix 21 and means and standard deviation vectors matrix 23 This complete output can be used as input to initiate another analysis without the necessity of reprocessing the raw data used to generate the matrices To use the correlation matrix set as input the user places the punched output behind the variable names card followed by a card that contains a negative number in the job number field The program r
29. should take place before data entry In orthogonal polynomials if the order of the polynomial is high and or the range of x is large the elements of the polynomials will change rapidly in magnitude as the order of the polynomial increases so as to even exceed the range of the floating point number resulting in underflow or overflow If data entered into this program is such that this happens as evidenced for example by the residuals the user can elect to transform or scale the dependent and or independent variables A useful option in this program automatically scales the independent variable into a range such that the magnitudes of the successive polynomial elements are approximately uniform In summary the programs in this system are data dependent as is the case for many computer programs In some programs definite accuracy characteristics can be stated In data dependent routines these statements are difficult to make 113 114 CHAPTER 3 GENERAL FLOWCHARTS The following charts Figures 17 18 19 describe generally the programs in this package More detailed flowcharts and listings are available in the Systems Manual for this package The Systems Manual is not distributed with this program unless specifically requested Analysis of Variance System Flow Control Cards Compute Deviates and Mean Squares Generate ANOVA Table Stepwise Multiple Regression System Flow Read and Print
30. sum of four of them into one column of the observation matrix the sum should be placed in the column of one of the original variables and the program would have access to the resulting ten variables for its analysis In this specific instance the program would exit because of a singularity 2 5 2 Notes on Correlation and Eigen Analysis The regression and factor analysis programs contain options that in proper combination cause program termination when the correlation matrix and the latent roots and vectors have been calculated For example the no print option cc 25 26 of the option card used with the option for printing the correlation matrix gives this facility in the regression program With the matrix input option to factor analysis and using option 2 for the number of factors rotation option 0 and communality option 0 eigenvalues of matrices can be obtained However the number of eigenvectors is limited to ten the maximum number of rotatable factors 111 2 5 3 Punched Matrix Output Matrix Number Dimension Name 1 Raw cross products matrix 2 NxN Adjusted cross products matrix 3 NxN Variance covariance matrix 2 4 NxN Correlation coefficients matrix 5 NxN Characteristic vectors 6 NxK Principal axis factor matrix T KxK Orthogonal transformation matrix 8 NxK Orthogonal factor matrix 9 KxK Transformation to oblique reference structure matrix 10 NxK Oblique reference vector structure matrix 11 KxK Correlation
31. the je variable Inthis case the variance of the squared correlations of the common parts of the variables with a factor are now being maximized The oblique rotational scheme is developed from the above normalized Varimax solution The Promax rotation simply takes the Varimax rotated factor loading matrix and generates a pattern matrix from it by powering allthe elements in the original matrix We can define a matrix P P such that k 1 a 1 ET 5 with k gt 1 Each element of this matrix is except for the sign that remains unchanged the kth power of the corresponding element in the row column normalized orthogonal matrix We then find the least squares fit of the orthogonal matrix of factor loadings to the pattern matrix generated by equation 5 L G G ap 6 where L the unnormalized transformation matrix of the reference vector structure G the orthogonal rotated matrix and P the matrix derived by equation 5 above Finally the columns of L are normalized so that their sums of squares are equal to unity Computation of Factor Scores Direct Estimation Factor scores can be computed for each observation on the factors extracted from the original correlation matrix See Harman 1960 Chapter 16 in list of references at end of this chapter These are computed using the following equations Let F matrix of factor loadings V the orthonormal matrix of eigenvectors A diagonal matrix of l
32. trace of the communality adjusted correlation matrix Constant for Number of Factors cc 27 28 This field is used in conjunction with cc 25 26 A two 2 in the previous field implies that this field will contain an integer m which is equal to the number of factors to compute If 25 26 contains a three 3 this field should contain an integer P which is the percentage of factor variance Communality Estimation Options cc 29 30 This field is used to offer the user a choice of three methods of estimating the communality or common variance of the reduced factor space Before the characteristic roots and vectors are computed the program places the communality estimate on the principal diagonal of the matrix to be factored Three possible values may be punched in this field and they are described below Iteration on communalities is not performed automatically and is discussed in section 2 2 6 Value Meaning 0 No change to the matrix 1 The absolute value of the largest off diagonal element in a row will be used as the communality estimate for that variable 2 The square of the multiple correlation coefficient between variable i and all other variables in the matrix will be used as the communality estimate for the ith variable This is done for all i Rotation Switch cc 31 32 This field is used to indicate the type of rotation to simple structure that will be used on the principal axis factor matrix The user has the ch
33. 0 1122857TE 0 1030857 0 1016857E 0 9960000 C 3500000E Ol 0 7225732E 00 0 10000COE 30 0 6567597TE 00 0 8950214E 03 0 5141101E 03 0 2161068E 03 0 2601331E 02 0 7545321E 02 00 0 5798441E O0 0 7999757E 00 01 0 6567T597E 00 0 8950C214 4E 00 00 0 1000000E 0 0 7525222 00 00 0 7525222E 00 0 1000000E 01 02 02 02 02 A PUNCHED FACTOR SCORES OUTPUT 125 0 195204 225 0 296384 TE 325 0 4612540E 425 0 2389709E 25 0 3506941E 62 5 0 1578879E 725 0 3429788E 825 0 6738971E 925 0 8490992E 1025 0 399951 8E 1125 0 1011253E 1225 0 2248286E 1325 0 2765937E 1425 0 1946555E 1525 0 1136240E 1625 0 249322TE 1725 0 1984453E 1825 0 1837948E 1925 0 1365285E 2025 0 1992324E 2125 0 3228456E 2225 0 3100469E 2325 0 3355663E 2425 0 304406 2E 2525 0 3732885E 2625 0 1751099E 2125 0 2006 701 2825 0 2129229E 2925 0 1194178E 3025 0 4921422E 3125 0 2693805E 3225 0 3135445E 3325 0 2451350E 3425 0 2694848E 3525 0 2337661E 01 0 2331642E 00 0 1594353E 00 0 1131953E 01 01 0 7728281 OC 01 0 5253528E 00 01 0 5796633E 00 00 0 5622065E 00 02 0 5635983E 00 00 0 5561192E 00 01 0 6032414E 00 01 0 5582308E 01 01 0 1632854E O0 0 1167346E 01 0 9881128E 00 01 0 1907818E 00 01 0 4555233E 00 01 0 4183342E 00 01 0 2014065E 90 GO 0 2584076E 00 01 0 5247329E 00 01 0 6974968E 00 0 2611022E O1 01 9 2021247E 1 Ol 0 175577T5E 00 0 1551542 E Ol
34. 0 7966 6044 026044 027966 FACTOR ANALYSIS SAMPLE PROBLEM J08 3333 PAGE 11 ORTHOGONAL FACTOR MATRIX VARIMAX VARIABLE 1 2 P1 0 9061 0 2385 P2 7982 024684 P3 0 3346 0 9287 PA 027914 0 45551 FACTOR ANALYSIS SAMPLE PROBLEM JOB 3333 PAGE 12 TRANSFORMATION TO OBLIQUE REFERENCE VECTOR STRCTRe VARIABLE 1 2 1 029322 043853 2 0 361 7 0 9227 FACTOR ANALYSIS SAMPLE OROBLE JOB 3333 PAGE 13 CORRELATIONS AMONG OBLIQUE REFERENCE VECTORS VARTASLE 1 2 1 1 0000 0 6931 2 0 6931 41 0000 FACTOR ANALYSIS SAMPLE PROBLEM J08 3333 PAGE 14 OBLIQUE REFERENCE VECTOR STRUCTURE MATRIX VARIABLE 1 2 Pt 0 758 6 0 1290 p2 0 05746 041246 P3 020239 0 7279 PA 05370 0 2072 FACTOR ANALYSIS SAMPLE PROBLEM 408 3333 PAGE 15 OBLIQUE REFERENCE VECTOR PATTERN MATRIX VARIABLE 1 2 P1 2162874 0 7632 P2 21 2722 1 0063 P3 049249 16 3690 P4 916 3099 191151 65 66 FACTOR ANALYSIS SAMPLE PROBLEM VARIABLE i 2 FACTOR ANALYSIS SAMPLE PROBLEM VARTABLE 1 2 FACTOR ANALYSIS SAMPLE PROBLEM VARIABLE P1 p2 P3 PG FACTOR ANALYSIS SAMPLE PROBLEM VARTABLE P 1 P2 P3 P4 FACTOR ANALYSIS SAMPLE PROBLEM VARIABLE P1 P2 Ps P4 1 0 7208 0 0000 1 1 0000 0 6931 1 049280 9 049170 0 6667 06 9442 1 190521 0 7972 0 0332 0 7450 1 22699865 029614 024487 0 8373 2 04 0000 0 7208 2 70 6931 1 0000 2 0 5502 0 7254 04 9868 0 8038 2 204 17
35. 00 06 14043 0 34256E P5 00 704 195658E 00 0 87911 00 0675193E 01 1 042 00 0 10000 00 0O 64516E J08 P6 0 82055E 01 0 74961E 00 0 78605E 00 0634256E 00 Qe64516E 00 Oe1G000E 01 2222 PAGE 4 wo STEPWISE TEST REGRESSION ANALYSIS DEPENDENT VARIABLE RESIDUAL STANDARD DEVIATION STANDARD ERROR OF THE MEAN MULTIPLE R MULTIPLE RSOR VARIABLE ENTERED VARIABLE B OEF P3 209442 CONSTANT 2660998 SOURCE MEAN REGRESSION ERROR STEPWISE TEST ONE REGRESSION ANALYSIS P6 271 1238 3 2892 0 7860 0 618 P3 STD ERROR OF B PARTIAL R 002850 0 7860 NALYSIS OF VARIANCE TABLE DeFe SUM OF SQUARES MEAN SQUARE 1 21933 06 0 21933E 06 l 0 7851S5E 05 0 78516E 05 66 0 48556E 05 Oe73570E 03 DEPENDENT VARIABLE P6 RESIDUAL STANDARD DEVIATION 25 0821 STANDARD ERROR OF THE MEAN 320416 MULTIPLE R 008235 MULTIPLE RSQR 0 678 VARIABLE ENTERED P4 VARIABLE 8 P3 228275 P4 1 97976 CONSTANT 2107445 STD ERROR OF B 0 2656 0 65150 ANALYSIS OF VARIANCE TABLE PARTIAL R 0 7971 Qo 3972 MEAN SQUARE SOURCE Defoe SUM OF SQUARES MEAN 0 21933E 06 REGRESSION 2 0986180 05 ERROR 65 Oe40892E 05 0 21933E 06 0943090 05 Oe62911E 03 JOB 2222 PAGF 6 BE TA COEF STO ERROR OF BETA 0 7860 0 0760 F 0 10672E 03 BETA COEF STO ERROR OF BETA 0 7548 000709 0 2475 000709 F 0a68493E 02 25 STEPWISE TEST ONE REG
36. 0082 0401 001750000900130001800001500023 0501 003000002302300002000003300064 0601 002000001000060003300001300016 0701 005500000700140003400001600012 0801 006000000600080005000001100027 0901 0013000008002 70001 500001900048 1001 005000001800360001800002 700050 1101 005000000300100001400001400012 1201 0030000008002 70001000002500013 9 1301 002000000600300001500002100020 1401 002000000800100002500001800023 1501 001000002202200001100004600118 1601 004000001301300002800001700050 1701 000500002600120000730004800063 ij 1801 000250002302300000100003600150 1901 014000000300100003500000500072 2001 002500001500250000280003300054 2101 003500002801400000010004600109 2201 003500000600060005000001000010 2301 002500003503500005700003800125 2401 000500001100200003400001600044 2501 002000001101100000500002000048 2601 007000003203200006600003800105 2701 004000000800100004500001200009 2801 015000002302300000150004900130 2901 001000003803800002200004300160 3001 003500001500500001500003300048 22 3101 3201 3301 3401 3501 3601 3701 3801 3901 4001 4101 4201 4301 4401 4501 4601 4101 4801 4901 5001 5101 5201 5301 5401 5501 5601 5701 5801 5901 6001 6101 6201 6301 6401 6501 6601 6101 6801 1 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 2222 222223 222223 222223 222223 222223 222223 222221 013000000600120003 700000900036 002000002502500001000003500150 012000000500170000300002100078 004000000
37. 1 P2 P3 P P5 P6 APUN STEPWISE TEST ONE P1 0210000E 01 0 17646E 00 02051345E 02 0 25548E 00 04 19568E 00 0 82055E 701 ONE p2 10689 098756 3 06094437E 0056521E 0 10266 0 30548E VARIANCE COVARIANCE MATRIX P3 0 38647E 00 0094437E 02 0013518EF 03 Oe87764E 0 10910E 03 0 39802E 03 PA 0499187E 0956521 0987754E 0035965E 10510 0589470 5 01 04 15809E 01 0210266E OL 10910 02 70410510 02 15573 02 205435063E NO OF CASES LOW 0425000E 00 0 20000 01 0e40000E 00 0 10000E 01 05009000E 01 0 70000E HIGH 0e30000E 0 38000E 0 38000E 0Os48000E 0 58000E 0e20800E 68 02 02 02 02 02 03 AVERAGE 0eS9955E 062529390E 0 10425E 003099S5E 0 25397E 0056 794E 02 02 02 01 02 92 P2 0 17646E O 10000E 0e86799E O 1L0071E 0 87911E 0 74961E oQ 01 00 00 00 00 PUNCH AND CONSOLE JOB 56 0e23136E 02 Q 30548E 03 0939802 03 Oe8947CE 02 0e35063E 03 0 18966E 04 JOB STDe DEVe Os64737E 0 93575E 06411627E 0 59971E 0 12479 0e43550E 2222 2222 PAGE PAGE VARI ANCE 0241909E 00e87563E 0 13318 0939965 0 915573 0 18966E 02 02 03 02 03 04 TURN CONSOLE SWITCH 15 ONe MATRIX OF CORRELATION COEFFICIENTS P3 00 1345E 02 Oe86799E 00 0910000 01 06122586E 00 0975193 00 0 78605E 00 P4 0 29548E 0 10071 0 12586 0 100
38. 11666E Ve 8 OS B80UOOEF UI Ue239J9uE 02 Q 23359E 02 Us5408 4E OO Us lLUVOVE O sl5UUuE Ul 96 vi 9 9 O0 amp S0900F OC4 23100E 02 0 4 24439E 02 04 13397E O04 10000E 01 Us2500UE 05 56066E Ul 10 I2 0 12000E 02 2 236900E C2 0 25007E 02 0 14079E O 10000 O43500C0E 0O 33333E 00 11 11 0 11000E 02 0 2600UE 02 004 25063 E 02 06693614E 00 VelVOOVE O445000E Ul 0083333E ul 12 i Q 120U0E U2 Us246UVE 02 U 246U07E 02 O 74081E 02 UelVUDVE Ul UseS5U0UVE O 0O 515333t v2 ALPHA Ge 6500uU0E Ol UseG5UUUE Ul VeGSUUVE Ul BETA Uesll9S1GE 40493333E Ul Gesb7ud5E vai C 0 17726E 02 Oe2lOVSZ2E OL 60525016E UU ANALYSIS OF VARIANCE VARIATION SOURCE DeFa SUM OF SGUARES MEAN SQUARE NEGREE 1 COMPONENT 1 0 463378E 03 O0e63378E 03 RESIDUALS INEGREE 1 REGRe 10 Oell254E 03 OellZ254E O2 SEGRE 2 COMPONENT 1 0 87583E O2 0 487583EF O02 RESIDUALS DEGREE 2 REGRe 9 Oe24957E 02 Oe27730E ORTH POLY NO SCALING JOB 1111 PAGE 2 COEFFICIENTS OF FITTED POLYNOMIAL 0 0 3729547E 01 1 005435437E 01 2 0 42561686 00 ORTH POLY NO SCALING JOB 1111 PAGE 3 IDENTIFICATION x ys DERIVe ORDER 1200000 1244972 1 2 3 3200000 10627124 1 2 5 5200000 17204362 l 2 7 7200000 212676624 1 2 9 9 00000 24043972 1 2 11 11200000 25206385 1 2 104 DERIVe VALUE 4 492309 20 451233 3089842 0051233 2087375 0251233 1284907 0251233 0282440 0051233 0220027 0251233 z
39. 2222 4 1 0 1000000E 01 0 1764650E 00 0 5134508E 2222 4 1 2 0 1764650E 00 0 1000000E 01 0 8679906E 2222 4 1 3 0 5134508E 02 0 8679906E 00 0 1000000E 2222 4 1 4 0 2554817E 00 0 1007193E 00 0 1258658E 2222 4 1 5 0 1956896E 00 0 8791197E 00 0 7519358E 2222 4 1 6 0 8205512E 01 0 7496167E 00 0 7860574E 2222 4 6 1 0 8205512E 01 2222 4 6 2 0 149616TE 00 2222 4 6 3 0 7860574E 00 2222 4 6 4 0 3425673E 00 2222 4 6 5 0 6451673E 00 2222 4 6 6 0 1000000E 01 222223 1 1 0 6995588E 01 0 6473750E 222223 1 2 0 1525000E 02 0 9357534E 222223 1 3 0 1042514E 02 0 1162702E 02 222223 1 4 0 3099554E 01 0 5997127E 222223 1 5 0 2539706E 02 0 1247940E 02 222223 1 6 0 5679412E 02 0 4355013E 02 222221 1 1 0 6800001E 02 1 OUTPUT XEQ REGR 03 LOCALREGR FMTRD PRNTBSDATRDsMXRAD TRAN LOCALREGR2 REGRE LOCALCOREL sPRNT 02 0 25548 7E 00 0 1007193E 01 0 1258658E 00 0 1000000E 00 0 14043TTE 00 0 3425673E JOB 00 0 1956896E 00 0 8T791197TE CO 0 7519358E C1 0 1404377E 00 0 1000000E 00 0 6451673t 2222 PAGE z STEPWISE TEST ONE REGRESSION ANALYSIS DEPENDENT VARIABLE P6 RFSIDUAL STANDARD DEVIATION 27 1238 STANDARD ERROR OF THE MFAN 302892 MULTIPLE R 0 7860 MULTIPLE RSQR 0 6178 VARIABLE ENTERED VARTAALE B COEF STD ERROR OF B PARTIAL R P3 209442 0 2850 027660 CONSTANT 2620996 ANALYSIS OF VARIANCE TABLE SOURCE Defo SUM OF SQUARES MEAN 2 0 21933 06 REGRESSION 1 0 78516 05 ERROR 66 0
40. 25 Identifier 7 20 Score for variable 1 0 XXXXXXXE XX 21 34 Score for variable 2 Ht 63 76 Score for variable 5 H If more than five factors have been rotated card I 1 has a format as follows Columns Elements 1 6 Blank 7 20 Score for variable 6 The factor scores are computed from the Varimax solution if an oblique rotation has not been requested otherwise they are computed from the oblique solution Number of Factors to Compute cc 25 26 The number punched in this field is used as a switch setting in the program to determine the number of factors to compute from the characteristic roots and vectors There are four 4 possible values that may be punched and they are described below Value Meaning 0 or blank No factors will be computed 1 Only those factors will be computed whose characteristic vectors have associated characteristic roots greater than or equal to one 1 2 Compute a fixed number of factors m The value of m will appear in cc 27 28 m must not be greater than ten 3 Compute factors whose variance accounts jointly for no more than P percent of the total variance The value of P appears in cc 27 28 The variance of a factor is the characteristic root associated with a particular characteristic vector The Value Meaning 3 cont total variance is defined as the trace of the matrix The percentage is computed by adding characteristic roots and forming the ratio of this sum to the
41. 292025 1442266 2 721033 9896 91 25 395 87 576044674 35 2 4 LEAST SQUARES CURVE FITTING BY ORTHOGONAL POLYNOMIALS Given m points Xj Yi i 1 m the X set not necessarily being equally spaced this program will determine a polynomial of specified degree n or less 2 n a a xtax tax y 1 2 n which best approximates these points in the least squares sense n should not be specified greater than ten m of course must be greater than n and for practical purposes should be considerably greater than n The program allows the number of points m to be as high as 149 It is difficult to envisage a requirement such that n 7 will not suffice however the program has been successfully tested on polynomials of order 16 with 134 data points witu accurate results To maintain maximum accuracy the program uses orthogonal polynomials as described by G E Forsythe The process of finding the polynomial is accomplished by beginning with a first and a second degree polynomial and evaluating a variance criterion to determine whether the second degree will offer a better fit than the first If the variance criterion is satisfied within a specified tolerance the program accepts the second degree polynomial computed to be the best fitting polynomial If not the next order polynomial is computed and compared to the second degree The process con tinues until the variance criterion is satisfied or a specified maximum
42. 3333 PAGE 9 CORR BET REFERENCE VECTORS AND PRIMARY FACTORS JOB 3333 PAGE 10 CORR AMONG OBLIQUE PRIMARY FACTORS JOB 3333 PAGE 11 OBLIQUE PRIMARY FACTOR STRUCTURE MATRIX JOB 3333 PAGE 12 OBLIQUE PRIMARY FACTOR LOADINGS 9 2 9 References Businger P A Eigenvalues of a real symmetric matrix by the QR method Algorithm 253 Communications of the Association for Computing Machinery April 1965 vol 8 no 4 Guttman L Some Necessary Conditions for Common Factor Analysis Psychometrika 1954 19 149 162 Harman H Modern Factor Analysis University of Chicago Press 1960 pp 289 308 pp 261 288 pp 349 356 Hendrickson and White Promax A quick method for rotation to oblique simple structure British Journal of Statistical Psychology 1964 XVII 65 70 Horst P Factor Analysis of Data Matrices Holt Rinehart and Winston Inc New York Chicago San Francisco Toronto London 1965 p 214 HOW FORTRAN Subroutine using non iterative methods of Householder Ortega and Wilkinson solves for eigenvalues and corresponding eigenvectors of a real symmetric matrix Program Writeup F2 BC HOW Berkeley Division University of California 1962 Kaiser The Varimax criterion for analytical rotation in factor analysis Psychometrika 1958 23 187 200 Wilkinson J H The calculation of the eigenvectors of codiagonal matrices The Computer Journal 1958 vol 1 p 90 Wilkinson J H Ho
43. 515E 00 065801334 00 09407521E 00 FACTOR ANALYSIS SAMPLE PROBLEM JOB 3333 PAGE 4 NORMAL VARIMAX CRITERION NORMALIZED CYCLE CRITERION DIFFERENCE EPSILON CRITERION 0 00116000 1 6200035874 0200035874 2 0202373140 0202337265 3 0 023731 40 0 00000020 FACTOR ANALYSIS SAMPLE PROBLEM JOB 3333 PAGE 5 asd ORTHOGONAL FACTOR MATRIX VARIMAX i VARIARLE 1 2 P1 0 6504 004786 p2 027044 0 5633 P3 024599 096071 P4 007121 0 6580 FACTOR ANALYSIS SAMPLE PROBLEM JOB 3333 PAGE 6 TRANSFORMATION TO OBLIQUE REFERENCE VECTOR STRCTRe VARTABLE 1 2 1 0 8730 04 3683 2 024876 029296 FACTOR ANALYSIS SAMPLE PROBLEM JOB 3333 PAGE 7 CORRELATIONS AMONG OBLIQUE REFERENCE VECTORS VARTASLE 1 2 1 120000 0 7749 2 067749 1 0000 69 TO FACTOR VARIABLE P1 03 FACTOR ANALYSIS SAMPLE PRORLEM VAR TABLE 1 2 FACTOR ANALYSIS SAMPLE PRORLEM VARIARLE 1 2 FACTOR ANALYSIS SAMPLE PROBLEM VARIARLE P1 P2 P3 P4 FACTOR ANALYSIS SAMPLE PROBLEM VARTARLE P1 P P3 Ph JOR COMPLETED ANALYSIS SAMPLE PROBLEM 1 0 53344 20 634053 04 1054 0 23008 0 6320 9 090 1 1 40000 204 7749 1 0 7810 0 8624 046512 029045 1 206 5291 045384 0 1668 0 4760 2 0 2054 0 2642 0 3950 0 3494 2 0 000N 0 62320 2 1 0000 2 0 7351 048353 20 7543 0 9218 2 0 53250 0 64181 0 6250 0 5529 JOB 3333 PAGE 8 OBLIQUE REFERENCE VECTOR STRUCTURE MATRIX JOB
44. 6 Transformation Switch 0 No transformation Transformation 4 8 Number of levels for factor 1 9 10 Number of levels for factor 2 11 12 Number of levels for factor 3 13 14 Number of levels for factor 4 Data previously entered under mode 1 is available for mode 2 until destroyed by input mode 1 from this or one other of the system s four main programs g 2 3 3 Analysis of Variance Table Generation The operation of the program is designed to handle a general four factorial design The program will read the data form the deviates and accumulate the sums of squares as if all four factors were always present As a result of this operation certain accumu lation and storage areas in general have cells that are not used unless all four factors are present In forming the analysis of variance table for a particular design the user has the option of pooling component sums of squares to form the error sums of squares specific to the design The sums of squares are located in storage and can be accessed by the table generator cards The table components and index are shown below Subscript Component 1 A 2 B 3 C 4 D 5 AB 6 AC T AD 8 BC 9 BD 10 CD 11 ABC 12 ABD 13 ACD 14 BCD 15 ABCD For a two factor experiment the sums of squares are located in cells with subscripts 1 2 5 For a three factor experiment the sums of squares are located in cells with subscripts 1 2 3 5 6 8 11 For a four factor experi
45. 6F Ol 0698334E 00 3 3 0 12727E OellQOOE 02 0610271E 02 0 72875E 00 4 4 90909 00 12600 C2 513913 02 0 513134E 5 04 55545E 00 0 14700E 02 0 17043E 02 0 23434E 01 6 6 518181 00 9900 02 02419560 02 0823902E 00 1 7 0 18181 E 00 0 25100 02 0 421766 02 0 2323337E 8 0 54565E 00 0623900E 02 0 23359E 02 0254086E 00 9 9 0 90909E 00 0 23100E 02 0 24439E 02 0 13397E 10 10 0 12127 0 23600 02 0 25007E 02 04 14079 11 11 0 16363 0 26000E 02 0 25063E 02 0 93614E 00 12 12 Oe20000E 01 0624600E 02 0 4246507E 02 0 74310E 02 JOR COMPLETED INPUT XEQ POLY 02 LOCALPOL2 POLSQs PCOEF PDER PFIT LOCALPOLY4 TRAN DATRD FMTRD PRNTB 020200 1111 ORTH POLY NO SCALING 020301010201000100000000000000000 CI25 I1 1X F2 1 F3 0 SOLUTION VECTOR INPUT 0 3636363E 00 0 1636363E O 111124 1 1 0 6500000E 01 0 1191666E 02 0 1772499E 02 111124 1 2 0 6500000E 0 9333328E 0 2105245E 01 111124 1 3 0 6500000E 0 8678567E 01 0 2561686E 00 010 05000 020 55000 031 10000 1 These numbers were produced in the first sample problem and as explained there are not used However the card is necessary OUTPUT 107 108 XEQ POLY 02 LOCALPOL2 POLSQ sPCOEF PDERs PFIT LOCALPOLY TRAN sDATRD sFMTRDsPRNTB ORTH POLY NO SCALING SOLUTION VECTOR INPUT MAXIMUM DEGREE OF POLYNOMIAL 2 INPUT TYPE 3 POLYNOMI
46. 90 0 1728 140098 0 2875 2 0 1404 0 0652 1 0582 0 06641 CORR BET REFERENCE VECTORS AND PRIMARY FACTORS CORR AMONG OBLIQUE PRIMARY FACTORS OBLIQUE PRIMARY FACTOR STRUCTURE MATRIX OBLIQUE PRIMARY FACTOR LOADINGS FACTOR SCORE REGRESSION COEFFICIENTS 8 FACTOR ANALYSIS SAMPLE PROBLEM IDENTIFICATION 1 1 1 0 19520 01 2 2 0 29638 00 3 3 0 46125E 00 4 4 0423897E 02 5 5 0 35069 6 6 0 15788E 7 7 0034297E 00 8 8 0 67389E 02 9 9 Os R4909E 00 10 10 0239995E 01 11 11 0 10112 02 12 12 04 22482bE 13 13 0 27659E 14 14 02e19465E Ol 15 15 Oell362E 01 16 16 0 24932 0 17 17 0419844 Ol 18 18 0 18379 O1 19 19 0 13652E 00 20 20 0419923E 01 21 21 70 432284E 01 22 22 31004 01 23 23 20 33556E 01 24 24 20 30440E 01 25 25 0037328E 00 26 26 0 17510E 27 27 0 20067 28 25 0221292E 01 29 29 0 11941E 01 30 30 0e49214E 00 31 31 0 26938 32 32 0931354E 33 33 00245213E 34 34 0 26948E 01 35 35 0 23376 01 JOA COMPLETED JOB FACTOR SCORES 2 0 23316 0 15943E 0 11319 0 77282 E 00 0 52535E 00 06 57966E 00 0e56220E 00 0 565359E 00 06 55611 00 20 60324E 00 06 56823E 01 0216328E 00 Oell67BE 01 0 98811E 00 19078 00 0245552E 00 02e41833E 00 0 20140 00 0e25840E 00 0e52473E 00 0 69749E 00 0 26110E Ol 04 20212 0 17557E Ol OelS515E
47. 900075001900001700023 003000000700350002600001200042 008000002002000002200003000072 009090000600086002500001500020 006000001200400001200002000036 008000002600160001100003500056 00150000 1500300001600002900036 007000001000090010000001200026 008000002802800004200004000108 002000003403400000900004200106 006000000400080003600001100016 015000003203200001 800004400104 01 7000001101100002300001400047 016000000200050001800001100027 003000001800160001100003200012 006000000300040001 300001500007 0140000008001 10002000001 700018 006000001 400090000700002900028 0018000012002 40001500002100025 0150000003001 50000800001300011 018000000600550005 700000900020 005000001200200004100001600014 03000000 1101100002000002200038 029000000800800001000002200103 001800002402400001100003800106 01300000 2602600001 700003800063 019000002902900048000002900208 011000001701700001600002500032 010000001500500003500001900028 006000001000500001000002600032 005000002202200001200003900100 001000001500500000800002900050 017000000900300013000001000080 005000003003500000900005800065 001300001000130009000001000025 PUNCHED CORRELATION MATRIX 1 0 1000000E C1 0 1764650E 2 0 1764650E 00 0 1000000E 3 0 5134508E 02 0 8679906E 4 0 255481 7E 00 0 1007193E 5 0 1956896E 00 0 8791197E 0 8205512E 01 0 7496167E 0 8205512E Ol 0 7496167E 00 0 7860574E 00 0 3425673E 00 0 6451673E 00 0 1000000E 01 0 6995588E 0 6473T50E 0 1225000 02 0 9357534E 0 1042514E 02 O 1162TO02E
48. AL COEFFICIENTS 1 COMPUTE DERIVATIVES 1 ORDER OF DERIVATIVE 2 PREDICTED VALUES 1 PUNCH SOLUTION VECTORS 0 SECONDARY INPUT TYPE 1 VARIANCE CRITERION 0 000000000 TRANSFORMATION SWITCH 0 SCALING 0 IGNORE POLYNOMIAL OUTPUT 0 12511 461 26 300 X X NO TRANSFORMATION ORTH POLY NO SCALING SOLUTION VECTOR INPUT COEFFICIENTS OF FITTED POLYNOMIAL 0 Ne3729551E l 065435436E 01 2 20 2561686E 00 JOB 908 1111 1111 PAGE PAGE 0 l ORTH POLY NO SCALING SOLUTION VECTOR INPUT 908 1111 PAGE 2 IDENTIFICATION x Ys DERIV ORDER DERIVe VALUE 3 1200000 1044971 1 4 92309 2 0451233 ORTH POLY NO SCALING SOLUTION VECTOR INPUT JoB 1111 PAGE 3 IDENTIFICATION x Y ya Y y 1 1 0 50000 00 0O 400C00E 00 0 10758E O1 0 10758E 01 2 2 0 55000E 01 0400000E 00 0 18415 02 0 18416E 02 3 3 0 s10000E QeOONNCE 00 04 14497E O414497E JOR COMPLETED 3 4 v INPUT XEQ POLY 02 OCALPOL2 POLSQ PCOEF PDER PFIT LOCALPOLY TRAN DATRD FMTRD PRNTB 020200 1111 ORTH POLY SCALING SOLUTION VECTOR INPUT 020301010201000100000000000000000 CI2 I11 1X F2 1 F 3 0 0 3636363E 00 0 2363636E 1 111124 1 1 0 3178914E 06 0 1575755E O 1772499E 02 111124 1 2 0 1016575E 06 0 1234159E 0 5789424E 111124 1 3 0 3039385E 06 0 1147578E 01 0 1937265E Ol O10 05000 020 55000 031 10000 1 OUTPUT XEQ POLY 02 LOCALPOL 2 sPOLSG sP
49. ANALYSIS SAM LE PROBLEM J08 3333 PAGE 1 2 MATRIX OF RAW CROSS PRODUCTS 62 VARIABLE Pi P2 P3 Pe P1 0 53114E 06 0 43242E 06 0637325E 06 0O0449693E 06 P2 0 43242E 06 0 38781E 06 0235432E 06 0 40897E 06 P3 0 37325E O6 0635432E 06 0638490E 06 0230425E 06 Pe 90 9693 06 O0 40897E 06 0 30425E 06 0454077E 06 FACTOR ANALYSIS SAMPLE PROBLEM VARTAALE Pl 02 P3 P4 0 89865F 05 0 21295 05 061586878E 05 04 12553E 0 26365 05 0 12553E 05 0410550E 06 204 49520E 95 04 50219F 01 P 0 27295E 05 04 26365E MATRIX OF RESIDUAL CROSS PRODUCTS P3 05 05 PS 961090550E 06 OeS9620E 05 0423007E 05 04 50219 05 FACTOR ANALYSIS SAMPLE PROBLEM VARIABLE pl 02 D3 P4 0 26430E 04 0eR0279E 03 04 77566E 03 0 36920E 03 0 31032 04 14594F O4 04 14770E O456931E Pi P2 FACTOR ANALYSIS SAMPLE PROBLEM SUMMARY STATISTICS VARTABLE 1 P1 2 P2 3 P3 4 P4 VARIARLF Pl 52 P3 Py 05 0 19356E 06 VARIANCE COVARIANCE MATRIX P3 0 80279E 93 0 77546E 3 P4 0 31032F 04 0 46702E 03 0 36920E 03 1459 0 67669E 03 04 14770E 04 NO OF CASES 35 LOW 0 34000E 02 0 62000E 02 0939000 06 0 14000E 02 HIGH 0 23700E 03 0 15200 03 0 15900 03 0 s36600E 3 FACTOR ANALYSIS SAMPLE PRORLE e10000E 01 Pl P2 AVERAGE 0 11228E 03 0 10308E 03 0 10168 03 0 996n0E 02 STD DEVe 0 51411 02
50. COEF sPDER PFIT LOCALPOLY sTRAN sDATRD sFMTRDsPRNTB ORTH POLY SCALING SOLUTION VECTOR INPUT JOB 1111 PAGE MAXIMUM DEGREE OF POLYNOMIAL INPUT TYPE POLYNOMIAL COEFFICIENTS COMPUTE DERIVATIVES M vu N ORDER OF DERIVATIVE PREDICTED VALUES PUNCH SOLUTION VECTORS SECONDARY INPUT TYPE Fr Oe FN VARIANCE CRITERION 000000000 TRANSFORMATION SWITCH 0 0 0 SCALING IGNORE POLYNOMIAL OUTPUT T2ellelXeF2eleF320 X VALUES HAVF BEEN TRANSFORMED TO X 0 3636363E 00 X 0 23263636 Olle ORTH POLY SCALINGs SOLUTION VECTOR INPUT JOB 1111 PAGE 1 COEFFICIENTS OF FITTED POLYNOMIAL 0 0 2077764E 02 i Oe5789424E 01 2 0 193726858E 01 109 ORTH POLY SCALINGs SOLUTION VECTOR INPUT 08 1111 PAGE 2 IDENTIFICATION x DERIV ORDER DERIVe VALUE 3 71 499999 le44974 l 13453848 2 3 87453 l A ON Qn 9 eeM dictct XXms rtQ M MOD QM C 9 ORTH POLY SCALINGs SOLUTION VECTOR INPUT 08 1111 PAGE 3 IDENTIFICATION x Y Y Y Y 1 1 04 21818E 00 10758 O1 0010758E 01 2 2 705036363E 00 00 0618416E 02 OclB416E 02 3 3 0 19999E 01 00 00 0O 14497E Ol 0e14497E JOB COMPLETED US w 110 2 5 GENERAL NOTES ON THE PROGRAMS 2 5 1 Transformations This feature was added to aid users who are familiar with prog
51. E 0 1600 0 1200E 0 2700E 0s4800E 0 5000 0 1200 0 1300 0 2000 092300 0 1180 0 5000E 0 6300 0 1500 0 7200 0 5400 0 1090E 0 1000 0 1290 0 4600 0e4800E 0 1050 0 9000E 001300E 0 1600 0e4800E 0 3600E 0s1500 067800 0 2300 0e4200E 0 7200 0e2000 0 3600 05 5600 02e3600E STEPWISE TEST ONE CASE JOB COMPLETED ACTUAL 0e2600E 0 1086 Osl060E 0 1600 0 1040 06 4 TOOE 0 2706 Oel200E Oe 7000E Osl800E 0e2800E 062500E 0 1100 0 2000E Oe1400E Qe3800E 0 1030 0ei060E 026300E 0 2080 0 3200 0 2800E 063200 0 1000E 0e5000E 0e8000E 096500 092500 PREDICTED VALUES PREDICTED 0 8855E 0 8627 E 008871 042273E 0s 8497E 0 2262E 0 2921E 0e2627E 0 2899E Os4188E 002154E 03535330E 0 3209E 0 2718E 049473E 045035E 0 5647E 0e 8290E 0 2002E 024203E 0 7818E 0 2371E 0 1213 0 2821E 0e4391E 001196E 0 2580 0 1038E 0 s1241E 0 4992E 0 2489E 04 8833 0 93121E 0 2510E 0 2587E 067851 0 2655E 0 3391E 0 467A4E 0 4103 PREDICTED VALUES PREDICTED 0 3917E 051093 0 1130 0 2315 06121190E 0e4764E 042283E 004125E 0 2172 E 043026E 023696E 0e3087E 0 2430E 0e3964E Os31L TOE 0 6146 0 5311E 048993 E 0 9984E Oe2014E 0e6718E 054153E Oe4206E 0 8884 0 4283E 0 5190E 0 1340E 0 3303E RESIDUAL 062455E 00e2127 0 6717 0 2 682 0e 2097E 0a6627E
52. OLY TRAN eDATRD sFMTRDeoPRNTB 105 J08 1111 PAGE ORTH POLY SCALING MAXIMUM DEGREE OF POLYNOMIAL 2 INPUT TYPE 1 POLYNOMIAL COEFFICIENTS i COMPUTE DERIVATIVES 1 ORDER OF OFRIVATIVE 2 PREDICTED VALUES 1 PUNCH SOLUTION VECTORS 1 SECONDARY INPUT TYPE 0 VARTANCE CRITERION 0e010000001 TRANSFORMATION SWITCH 0 SCALING IGNORE POLYNOMIAL OUTPUT o t 1291191 9 2 09 3 1 X VALUES HAVE BEEN TRANSFORMED TO 5 043636363E OQ x 062363636 01 MAX DEGREE OF POLYNOMIAL REACHEDe VARIANCE CRITERION NOT SATISFIEO ORTH DALY SCALING JOB 1111 PAGE l ORTHOSONAL POLYNOMIALS IDENTIFICATION x Y Ye YeYa 0 H e 1 O 19999E 0 11000E 0O 414497E 0 34974E 00 12000000 2 000000 20424242 2 2 0 16363F 0 71000E OeG61166E 01 0 4 98334E 00 1 000000 21 636363 1910192834 i 3 3 0 12727E 0O 11000E 02 0 10271E 02 0 72875E 00 1 000000 21 272721 02044076 r 4 04 90909E CO 0 12600E 02 0 13913E 02 04 1313 4E 1 000000 02909091 0 749301 5 5 0 54545E 00 0 14700E 02 0 17043E 02 0 234354E 1 000000 00545454 21 278235 6 6 0 18181E 00 0O0 19900E 02 0O 19660E 02 0 23902E 00 1 000000 72065181818 21 542697 7 7 0 18181E 00 0 25100E 02 0 21766E 02 0 33337E 1 000000 0 181817 214542691 A 8 0054545E 00 23900 02 23359 02 54086 00 12000000 00545454 21 278235 9 0 90909 00 0 23100E 02 0 4 24439E 02 0 13397E Ol 1 000000 00909090 90 749310 10 10 Cel2727E
53. Ol 0 23600E 02 O425007E 02 0 14079E Ol 1 000000 1 272726 02044077 11 11 0 16363E 0 26000E 02 Ce250H3E 02 0 4936154E 00 1 000000 19636363 19101928 12 12 0 20000E Ve24QUUE 02 Dse24607TE 02 764310 02 12000000 le 9999 20424242 ALPHA 0 31789E 06 0 10165E 06 0 30393F U6 BETA 04 15757E 0O0 412391E 0O0511475E Ul C 0 17724E 02 0 57896t Ol 0 19372 READY PUNCH WITH BLANK CARDS AND PRESS START ON THE PUNCH AND CONSOLE TURN CONSOLE SWITCH 15 ON ANALYSIS OF VARIANCE VARIATION SOURCE DeFe SUM OF SQUARES MEAN SQUARE x DEGREE 1 COMPONENT 1 0 63378 03 0 63378E 03 RESIDUALS DEGREE 1 REGRe 10 0 11254E 03 0 11254E 02 DEGREE 2 COMPONENT 0 87584E 02 0 87584E 02 i RESIDUALSIDEGREE 2 REGRe 9 0024956E 02 O 27729E PAGE 2 JOB 1111 ORTH POLY SCALING COEFFICIENTS OF FITTED POLYNOMIAL 0 062077764E 02 1 065789424E 01 2 01937265E 01 106 ORTH POLY SCALING JOB IDENTIFICATION x Ys DERIV ORDER DERIVe VALUE 1 71699999 1044974 1 13053848 2 3 87453 3 1 27272 10927124 1 10972066 2 3 87453 5 0 54545 17 04340 1 7 90280 2 3 87453 7 0418181 21 76622 1 5 08496 2 3 87453 9 0290909 24043971 1 2026712 2 3 87453 11 1963636 25406386 1 0955071 2 3 87453 1111 PAGE 3 ORTH POLY SCALING JOB 1111 PAGE amp IDENTIFICATION xt Yoys 1 0019999E 01 OsllOOOE 014497E Ol 70534974E 00 2 2 516363 Ol 0 71000 066116
54. RESSION ANALYSIS DEPENDENT VARIABLE RESIDUAL STANDARD DEVIATION STANDARD ERROR OF THE MEAN MULT IPLE R MULTIPLE RSQR VARIABLE ENTERED P6 2399328 209022 0208435 0 7115 P5 JOB 2222 PAGE VARIABLE B COEF STO ERROR OF B PART IAL R BETA COEF STD ERROR OF BETA P3 1 9583 064079 0 5145 065228 001089 P4 263123 0 5266 004811 003184 000725 P5 110355 0 3808 0 3217 0 2967 001091 CONSTANT 209105 ANALYSIS OF VARIANCE TABLE SOURCE DeFe SUM OF SQUARES MEAN SQUARE F MEAN l 0e21933E 06 062121933E 06 REGRESSION 3 0 90415E 05 00e30138E 05 06532617E 02 ERROR 64 0036658E 05 0 57278E 03 ba H 2 Y Y SS SSS SS SS vn SAG E MGR ERR n s n E 2222 PAGE 9 STEPWISE TEST ONE J08 REGRESSION ANALYSIS DEPENDENT VARIABLE P6 RESIDUAL STANDARD DEVIATION 2369726 STANDARD ERROR OF THE MEAN 299071 MULTIPLE R 028456 MULTIPLE RSOR 0 7150 VARIABLE ENTERED Pl 26 VARIABLE B COEF STD ERROR OF 8 PARTIAL R BETA COEF STD ERROR OF BETA 3 Pl 0 4272 004813 061111 020635 020715 P3 108967 004145 024994 025063 001106 P4 202333 005349 024654 003075 0 0736 P5 191167 093923 0 3375 003200 0 1124 CONSTANT 122530 ANALYSIS OF VARIANCE TABLE SOURCE DeFe SUM OF SQUARES MEAN SQUARE F MEAN 1 0 21933 06 09621933E 06 REGRESSION amp 0 90867 05 0422716 05 Oe39529E 02 ERROR 63 0e36205E 05 OeS7T468E 03 STEPWISE TEST ONE CASE VQ O 0 UU 4 UN ACTUAL 0 6400 0 6500 0 8200 0 2300 0 6400
55. a cards Descriptions of the form and content of each card set follow MONITOR CONTROL CARDS The monitor control cards are necessary to initiate program loading from the disk and to establish the necessary communication with the monitor A general description of cards may be found in IBM 1130 Disk Monitor System Reference Manual C26 3750 A regression analysis requires the following cards CC 1 4 8 16 17 XEQ REGR 03 LOCALREGR FMTRD PRNTB DATRD MXRAD TRAN LOCALCOREL PRNT LOCALREGR2 REGRE The monitor control cards do not change from job to job within one analysis type but must be included with every job processed The first program operated on by this system should be preceded by a cold start card PROGRAM CONTROL CARDS The program control cards communicate the data specific parameters and output options to the program There are five possible card types necessary for execution Input output units card Job title card Option card described below Variable name card described below ao A N HB Variable format card Four of the control cards are required in every job The variable format card which specifies data format is not necessary if matrix data is to be processed OPTION CARD Number of Variables cc 1 2 This field must be punched with a nonzero integer n which is less than or equal to 30 The value of integer n gives the total number of independent and dependent variabl
56. a will be read until a card with a negative number in the identification field is encountered section 2 2 5 2 Raw data will be read from the disk Raw sums and raw sums of cross products will be accumulated Data will be read until a negative integer in the identification field is encountered 3 A previously computed matrix or matrices will be read from the 1442 card reader Matrix cards will be read until a negative job number field is encountered see Pooling section 2 2 6 Sequence Checking ce 5 6 This field is used to indicate that raw data input from the card reader cc 3 4 contains a 1 is to be sequence checked A value of zero or a blank field implies that no sequence check will be made A value of one 1 implies that the cards will be sequence checked The sequence checking process consists of an equal comparison check of the case identification field for all cards in a case and an ascending sequence check of the card number field If an error in either of these conditions is encountered the program prints a message and the job is terminated Number of Variables on Card 1 cc 7 8 When a data vector contains more variables than will fit on one card the user must indicate to the program the number of variables punched on each card This field must be punched with the number of variables on the first card If there is only one card per case this field must be blank or zero Number of Variables on Card 2 cc 9 10 S
57. alues for externally supplied data set kth_order derivatives of the polynomial at user specified points where k is the order of the polynomial Scaling equation used if required Analysis of Variance Table 2 2 4 2 Job Execution To perform a polynomial regression analysis the user must supply three sets of cards to the program 1 Monitor control cards 2 Program control cards 3 Data cards Monitor Control Cards The monitor control cards are necessary to initiate program loading from the disk and to establish the necessary communication with the monitor A general description of the cards may be found in IBM 1130 Disk Monitor System Reference Manual C26 3750 The orthogonal polynomial program requires the following cards CC 1 4 8 16 i V v XEQ POLY 02 LOCALPOLY TRAN DATRD FMTRD PRNTB LOCALPOL2 POLSQ PCOEF PDER PFIT Monitor control cards do not change from job to job but must be included with every job processed The first program operated on by this system should be preceded by a cold start card Program Control Cards The program control cards communicate the data specific parameters and output options to the program The four possible card types are described below 1 Input output units card 2 Job title card 3 Option cards described below 4 Variable format card TYPE I OPTION CARD This type of option card is to be used when data is being entered into the program for the
58. ame as cc 7 8 except that this field indicates the number of variables on the second card of the data 41 48 Number of Variables on Card 3 cc 11 12 Same as cc 7 8 except that this field indicates the number of variables on the third card of the data Transformation Switch cc 13 14 If the value in this field is nonzero a user written transformation subroutine is called after each data record is read and before any computation takes place If the value in this field is zero or blank the transformation subroutine is not called The use of transformations is discussed in section 2 5 1 Output Raw Sums of Cross Products cc 15 16 This field is used to indicate whether the raw sums and sums of raw cross products matrix are to be printed punched printed and punched or not presented The four 4 possible values of this field are described below The computation to generate the matrix is performed even if the no output option is chosen Value Meaning 0 or blank No output 1 Matrix will be printed 2 Matrix will be printed and punched 3 Matrix will be punched Punched output of the raw sums of cross products matrix includes the number of observations and the vector of raw sums and sums of squares This entire output must be entered on the pooling option section 2 2 6 Output Residual Cross Products cc 17 18 This field is used to indicate whether the residual cross products matrix defined as u C
59. are generated from the solution vectors whenever itis necessary Hence the actual coefficients of the fitted polynomial are not required for evaluation of derivatives of the computation of estimated values However if desired they may be printed by punching one 1 in the field If this field is left blank or contains a zero the coefficients are not printed Evaluate Derivative Switch cc 7 8 This field is used to indicate whether derivatives are to be evaluated at a selected list of points If this field contains a zero or is blank the derivatives are not computed If this field contains a one 1 each data point is examined to determine whether the derivative computation indicator for that point is nonzero When a nonzero indicator is located for a particular value of x the program evaluates the kth order derivatives where k is an integer punched in 9 10 The printout includes the 1 2 kth order derivatives and the estimated value of y for that particular value of x Maximum Order Derivatives cc 9 10 If 7 8 contains a one 1 this field must be punched with an integer k which is less than or equal to the order of the polynomial It indicates the maximum order derivative to be computed when a nonzero derivative computation indicator is located The printout includes all lower order derivatives as well as the maximum Predicted Values cc 11 12 The value of y as estimated is printed for each data point
60. ariable is to enter the equation A variable is entered into the equation if it significantly reduces the sample residual variance The computed variance is compared to this criterion value to determine whether it does decrease this variance The typical values and the form of the punched data are exactly as described above for the remove variable criterion 13 A number twice the size of the removal factor can be tried if the user is not sure of the correct number to be used in this field Tolerance For Ill Condition cc 39 44 A poorly conditioned matrix occurs when an independent variable is approximately a linear combination of other independent variables This number is associated in the program with the size of the pivot element If the pivot element is less than the tolerance level the associated variable is not entered into the model on the iteration in which the condition occurs A tolerance level of zero is not to be advised unless the user is sure that his matrix is not ill conditioned Regression Analysis Option Card Summary Column Meaning 1 2 Number of variables 3 4 Input type and source 1 Raw data input from card reader 2 Raw data input from disk 3 Matrix input from card reader 9 6 Check sequence of raw data input 0 No l Yes 1 8 Number of variables on card 1 9 10 Number of variables on card 2 must be blank if there is only one card per observation 11 12 Number of variables on card 3 13 14 Transforma
61. at each row of the factor loading matrix shall have at least one near zero factor loading and that at least one of the others shall be large positive In general the fewer the number of large loadings and the larger the number of near zero loadings in a row the more simple the structure of that particular variable The more variables that can be expressed in the simplest form in terms of a relatively large number of near zero loadings the more simple is the structure of the factor loading matrix However the knowledge that there are a relatively large number of zeros in each row and column of the matrix of factor loadings is insufficient the relative positions of the zeros and high loadings in the matrix must also be taken into account For every pair of factors the factor loadings should be arranged so that not many variables have high loadings on both If variables have high loadings on the same factor this implies that they tend to measure the same factor Also for any pair of factors a number of variables should have high loadings on one factor and near zero loadings on the other According to these criteria if the loadings of one factor were to be plotted against another we should find that some variables would cluster about zero some would be 33 34 high on one axis and low on the other and vice versa for other variables Thus the procedure for finding the best simple structure matrix for any given set of variables would give a
62. atent roots eigenvalues Then 1 2 F VA 1 g F is an orthogonal matrix but not orthonormal Since V is orthonormal v2evt 2 From the principal components model we know Z BS US eS 3 where Z vector of scores for each observation B matrix of common factor loadings U diagonal matrix of uniqueness 17h e diagonal matrix of errors Therefore s rz 4 since zb ob o2 yA V7 5 SO P sa e 6 S Alrz 7 Therefore the factor score for an observation on factor 1 would be 8 7 A fig o d pie 8 TERMS USED IN FACTOR ANALYSIS Communality Sum of squares of factor loadings for any given variable that is the total variance due to factors which this variable shares with other variables in the matrix Covariance Mean product of deviations of variable x and variable y from their means 1 n Z x X y y 41 Diagonal matrix A square matrix having zeros in all positions except those on the diagonal from upper left to lower right Direction cosine One of a set of cosines of angles defined for a point each angle being measured between one of the reference axes and the vector connecting the point with the origin Factor loading Correlation of any particular variable with the factor being extracted 9 Factor matrix Matrix whose entries are the factor loadings obtained from a factor analysis it generally is arranged so that it has as many columns as factors
63. ation 2 are calculated and presented using x Since the y s are not transformed the y s estimated y s and residuals calculated at x are the same as those which would have been calculated at x if and Pj had been obtained without transforming 91 2 5 The coefficients a from i y ax i 0 are actually a from i y 2 aix 14 i 0 If in addition to transforming x the user elects to punch a 8 and c for later use in obtaining y s from a new set of x s that is for later use in prediction the transformation is retained and the new x s will be transformed as were the original x s y s for these new x s will be the same as the estimated y s would have been if transformations had not been performed Derivatives are calculated using equation 14 dy dx dx dx dx The elements of P x will now as j increases be of approximately the same size for allj The elements of Pj x are related to those of Pi by the factor j I 2 4 1 Summary of Output 10 92 x y for all cases x y if scaling is elected Predicted value of y for 3rd qth kth order polynomials Residuals for 3 7 order polynomials that is if k 13 Y Ypred is given for k 3 7 11 and 13 Orthogonal polynomials of all degrees to k Polynomial solution vectors 8 and c used to generate orthogonal polynomials Cocfficients of fitted polynomial Predicted v
64. ber of observations summed to obtain A AB Sum of all observations at level ab n Number of observations summed to obtain AB ABC Sum of all observations at level abc A ijk ijk n Number of observations summed to obtain ABC abc ijk Thus a general formula for the main effect due to factor A is Ralston A and Wilf H S Mathematical Methods for Digital Computers New York John Wiley and Sons Inc 1960 pp 221 230 where G the grand total of all observations and n is the number of observations summed to obtain G The main effect due to factor B has the form ZB 2 The general computational formula for the variation due to the AB interaction is z AB E BH esee ab nb a b The general computational formula for the variation due to the ABC interaction is 2 Z ABC g2 o lA SS SS SS _ SS SS SS 8SS abc D be Ds a b e ab ac bc The following table shows the layout of the complete table for the analysis of variance of a complete factorial design ANOVA TABLE FOR A COMPLETE FACTORIAL DESIGN Sum of Source of Variation D F Squares Mean Square A Main effect p 1 SSa SS p 1 B Main effect q 1 SSp SSp q 1 C Main effect r 1 SSe 58 1 1 AB Interaction p 1 q 1 SSab SSab j p 1 q 1 AC Interaction p 1 r 1 SSac SSac p 1 r 1 BC Interaction q 1 r 1 SSpe SSbe q 1 r 1 ABC Interaction p 1 q 1 r 1 SSabe SSabc P 1 q
65. bles in the matrix may be assigned a four character name to aid in the identification of the output The card is punched in four column fields and each field corresponds to the variable to be identified for example field 3 columns 9 12 will be the name of row and column 3 on all matrix output At most 20 names can appear on one card If there are more than 20 variables in the analysis a second card having the same format as the first must be included in the control card deck 95 96 Column Meaning 1 4 Name of variable 1 5 8 Name of variable 2 4N 3 4N Name of variable N 2 2 5 Data Input Raw data input to the program consists of a set of observations made on several different variables The variables for each observation are punched on one two or three cards according to the following general format Field Type Meaning 1 Integer I Identification field Any numeric information that serves to identify the particular observation is punched in this field It must be greater than zero and should be different for each observation 2 Integer I Card number within observation If it is not possible for one card to contain all the variables they may be continued on a second and a third card as necessary The user has the option of sequence checking the cards to ensure that all cards within a case are together and that the order of cards is consistent If the option is chosen cc 5 6 on option card this field m
66. ck De indicator Input Output B Units Monitor Final table generator card Figure 13 ANOVA disk input 2 3 6 Sample Problem The data for this sample problem was taken with permission from page 276 Statistical Theory in Research by R L Anderson and T Bancroft McGraw Hill Book Company Inc New York 1952 INPUT XEQ ANOVA 02 LOCALANOVA FMTRD PRNTB DATRD STORE OCALANOV2 SDOP MNSQ REPRT 020200 3333 TEST ANOVA I 0301010602302 411 F4 0 111 161 112 192 121 145 122 232 131 172 132 227 211 166 212 253 221 231 222 231 231 204 232 214 311 113 312 208 321 131 322 190 331 104 332 144 411 103 412 171 421 158 422 171 431 135 432 146 511 132 512 196 521 176 522 242 531 178 532 186 611 180 612 198 621 216 622 238 631 175 632 230 BLOCKS 0101 FERTILIZER 02 VARIETY 03 F XV 08 ERROR 1050611 OUTPUT XEQ ANOVA 02 LOCALANOVA FMTRDsPRNTB DATRD STORE LOCALANOV2 SDOP 3 MNSQ REPRT ST ANOVA T NUMBER OF FACTORS INPUT MODE TRANSFORMATION SWITCH NUMBER OF LEVELS FACTOR 1 NUMBER OF LEVELS FACTOR 2 NUMBER OF LEVELS FACTOR 3 NUMBER OF LEVELS FACTOR 4 411 F44 0 On uU 87 88 ANALYSIS OF VARIANCE TABLE FOR 6 X 3 X 2 X O EXPERIMENT COMPONENT BLOCKS FERTILIZER VARIETY F XV ERROR TOTAL JOB COMPLETED SUM OF DEGREES OF MEAN SQUARES FREEDOM SQUARE 24938691 5 4987278 4034200 2 2017200 1729225 1 17
67. control card possibly signaling a new analysis will be operated on If this is not desired the following should be done It is possible that the format specification card is incorrect Ifthis is so the entire deck to be analyzed must be rerun However if a specific data card is in error the reader hopper and the stacker should be emptied Pressing the nonprocess runout button will clear the card read punch and the second card in the stacker will be the card containing the error After correcting this card the user should place it and the third stacker card at the front of the deck that was withdrawn from the hopper place this entire deck in the reader hopper and press START on the card reader and console to continue processing 2 Other error conditions are signaled by a printed message and or the program exits to the monitor The monitor will read cards until a monitor control card is met that is the next job to be done or will stop when the reader hopper is empty For a list of the error messages see chapter 5 3 When an analysis is terminated successfully an end of job message is printed and control is relinquished to the monitor 4 When the user calis for output on cards a message is written reminding the operator to enter blank cards if console entry switch 15 is not on The computer then pauses to allow input of blank cards 1f console entry switch 15 is on no reminder is given It is possible in this case to destro
68. ctors estimate the communalities and run a proper factor analysis Rotation of the Factor Loadings In using the Varimax criterion for rotation to orthogonal simple structure we have already defined verbally what we mean by simple structure Mathematically the simplicity of the factorial composition of the jth variable can be defined as the variance of the squared loadings for the test 2 2 du c2 q rZ a Za Yr 1 J 15 B JS where j 1 2 n variables and s 1 2 r factors and ajs is the factor loading of the jth variable on the stb factor To obtain the total criterion for the entire factor matrix we can sum over the variables thus q Z eray 2 j 8 JS B JS This criterion can be modified if we define the simplicity of a factor as the variance of its squared loadings 2 2 2 o s ex n 3 and for the criterion for all the factors define the maximum simplicity of a factor matrix as the maximization of ye v gt nS as 3 S S S j which is the variance of squared loadings by columns rather than by rows This is the row Varimax rotation However this will exhibit a systematic bias because of the divergent weights that implicitly are attached to the variables by their 39 40 communalities Therefore the normalized Varimax rotation weights each variable by its communality thus B 2122 2 2 2 2 vaE OE Gyu A 4 where hi is the communality of
69. d by input mode 1 from one of the four system programs See General Operating Instructions Chapter 1 T9 Transformation Switch cc 5 6 If the value in this field is equal to zero the transformation program is not used If the value is 1 the transformation program is called after each data item has been read and before the item is stored in the appropriate storage cell The transformation program itself is a user written FORTRAN program which is discussed in section 2 5 1 Number of Levels for Factor 1 cc 7 8 This field must be punched with an integer n which indicates the number of levels in the first factor For example n would be equal to three for a 3X4X5 factorial design n should be less than ten Number of Levels for Factor 2 cc 9 10 Same as cc 7 8 This field is for factor 2 Number of Levels for Factor 3 cc 11 12 Same as cc 7 8 This field is for factor 3 Number of Levels for Factor 4 cc 13 14 Same as cc 7 8 This field is for factor 4 Columns 11 12 and 13 14 may be left blank for a two factorial experiment However the program does not operate for fewer than two factors All four factors whether used or not must be accounted for on the variable format card The product of the levels of the factors is limited to 2000 Analysis of Variance Option Card Summary Column Meaning 1 2 Number of factors 3 4 Input Mode 1 Source data from card reader 2 Source data from disk 5 5
70. d on the disk and are called into core storage when required for the execution of a particular job The logic flow of the programs and the type of analysis to be performed is controlled by a main program which reads the user supplied parameter cards and calls in the appropriate link at the proper time Although the programs imply different techniques a common approach can be used in executing a job for any analysis Chapter 2 of this manual is divided into four parts that describe completely the necessary parameters and monitor control cards for execution Special features available with this package provide added user flexibility In all four major programs data card formats can be specified by the user e Stepwise Linear Regression Matrix input and output are allowed with a pooling option which provides for the combining of raw cross products matrices either by addition or subtraction This allows the combining of input from different sources or of input that is available at different times without requiring recalculation of these matrices The subtraction feature gives flexibility for the handling of outlyers Residuals are available on option e Factor Analysis The pooling options and matrix input output are available as described above for stepwise regression Factor scores are calculated on option and punched on option Several options for the handling of communalities are available Oblique and orthogonal rotations are allo
71. de on the disk together section 2 1 5 names additional routines to be placed on the disk B Execution from disk Once the component subroutines and main calling programs are on the disk the execution of a job requires the monitor control cards program control cards and data cards to be placed in the card reader The deck should be preoeded by a cold start card To initiate processing the buttons IMMEDIATE STOP and RESET console START card reader and PROGRAM LOAD console should be pressed The order in which the cards are placed in the card reader for either matrix or raw data input is shown in Figures 8 9 and 10 Used in all four analysis types Used in regression analysis Optional Blank Output Negative Identification Data Deck Variable Format Variable Names Input Output 1 Units Monitor Control Figure 8 Factor analysis card order card reader input Optiona Blank Output Variable Names Input Output B Units Monitor Control Figure 9 Factor analysis card order disk input Optional Blan Output Negative Identification to be added to matrix 1 Variable Names M Monitor Control Figure 10 Factor analysis card order matrix input 59 60 2 2 8 Sample Problem INPUT XEQ FCTR 05 LOCALFCTR FMTRD OATROs PRNTB MXRAD TRAN LOCALFCTRI TRIOI QR INVRS LOCALCOREL PRNT LOCALFCTR2 VECTR
72. degree reached The degree of the last polynomial is assumed to be the best fitting polynomial for the data The essential characteristics of the method are as follows n Lety cP x 1 where each P x is a polynomial of degree j By minimizing m n M 2 i 1 j 0 and letting m Pik E x P x 3 i 1 m S 2 y P x 4 Forsythe G E Generation and Use of Orthogonal Polynomials for Data Fitting with a Digital Computer J Soc Indust Appl Math 5 1957 74 78 89 a set of normal equations are obtained n 5 2 ek ue j20 1 2 n Equation 5 consists of a set of n 1 simultaneous equations with n 1 unknowns However as the polynomials P x are orthogonal then 0 forj k jk 2 2 P x forj k Under this condition the coefficients c can now be evaluated by S C y 0 1 2 n 1 JJ The polynomials P x are defined recursively by P Q0 0 P ox 1 x a PX B P 18 P a P9 8 P G9 j j 1 where m 2 X p X ee 150 j 1 2 i 2 Pj x i 1 m 2 P x _ tl j i ro B H 8 E i 1 2 n 1 8 0 2 p x 5 6 7 8 9 10 The polynomial solution vectors ei and 8 are then used to compute aj the coefficients of the equation n 2 y a taxta x a X 11 for the degree n In principle the coefficients a could be used in equation 11 to co
73. ds The monitor control cards are necessary to initiate program loading from the disk and to establish the necessary communication with the monitor A general description of the cards may be found in IBM 1130 Disk Monitor System Reference Manual C 26 3750 An analysis of variance requires the following cards CC 1 4 8 16 17 XEQ ANOVA 02 LOCALANOVA FMTRD PRNTB DATRD STORE LOCALANOV2 SDOP MNSQ REPRT The monitor control cards do not change from job to job but must be included with every job processed Program Control Cards The program control cards communicate the data specific parameters and output options to the program The five card types are described below In addition control cards are necessary for defining the format and content of the ANOVA table section 2 3 3 1 Input output units card 2 Job title card 3 Option card described below 4 Variable format card 5 Table generation card section 2 3 3 Option Card Number of Factors cc 1 2 This field is punched with an integer n less than or equal to 4 n is the total number of factors in the experiment Input Mode cc 3 4 This field must be punched with an integer n which may take the values 1 or 2 If n is equal to 1 the program reads the raw data from the 1442 card reader If n 2 the raw data is read from the disk where it has previously been transferred by a program using input mode number one The data is retained until destroye
74. e punched output of the correlation matrix includes the number of cases and cards containing the vectors of means and standard deviations The matrix is generated even if the no output option is chosen Factor Scores cc 23 24 This field is used to indicate whether factor scores are to be computed Ifa value of zero 0 is punched or the field is left blank the factor score computation is suppressed When this field contains a one 1 factor scores and factor score regression coefficients are computed A two 2 in this field causes scores to be punched Upon entry to the program SCORE which computes the scores the program assumes that the necessary matrices and data have been set up for the analysis Hence either an orthogonal or an oblique rotation must be performed before entry to the SCORE routine The number of scores to compute is equivalent to the number of rotated factors Hence it is not possible to compute factor scores from the unrotated principal axis factor matrix In addition to the factor matrix and an auxiliary matrix computed by the rotation output program the scores program requires the data file to be on the disk and the means and standard deviations vectors to be located in common storage If the entire factor analysis is being done from the raw data these operations are performed automatically by the program 49 50 Factor Score Punched Output Format Card I Columns Elements 1 4 Factor score number I 5 6
75. eads the number of cases means standard deviations and correlation matrix storing each in its appropriate location and then initiates the analysis as specified on the option card No matrix output is possible in this case Also observations are not read hence predicted values and certain summary statistics are not available Pooling Sums of Squares and Cross Products cc 27 28 In the regression and factor analysis programs a considerable amount of processing time is devoted to accumulating raw sums and raw cross products as each data vector is read If there is a large amount of source data or if there is some logical division in the data set it is frequently desirable to obtain partial punched output of the raw sums of squares and cross products These partial outputs can then be added together to complete the total analysis in another job Both programs allow this type of analysis By choosing the punchout option for this matrix the program includes in the punchout the number of cases and raw sums and sums of squares vectors in addition to the raw cross products matrix Any number of these matrices may be punched and used later to complete the analysis The user simply stacks each output set one after the other following the variable names card The program reads the matrices examining the matrix identification and row and column numbers to determine the location or group of locations to which the matrix is to be added The read
76. er where a specially constructed correlation matrix is given rather than carried over from previous computation For example the user may desire to place specific communality values on the diagonal of this matrix see section 2 2 6 Options are also available to consider each i communality as the maximum absolute off diagonal element in the 15 vector of the correlation matrix or as the squared multiple correlation for the itb variable 2 All n of the latent roots of the correlation matrix with diagonal elements chosen as indicated above are computed by a Householder HOW 1962 tridiagonalization followed by the use of the QR algorithm The k latent vectors are computed by Wilkinson s HOW 1962 method These latent roots eigenvalues are solutions to the matrix equation R A A 0 where R is the correlation matrix is a diagonal matrix of the latent roots and A is a matrix of the latent vectors eigenvectors The total variance accounted for by the principal components of the correlation matrix is evaluated by its trace the sum of its diagonal elements The percentage of this total variance accounted for successively by each latent root is computed and presented cumulatively for the first through the nii root This percentage is presented whether or not a principal components analysis ri 1 is being performed If ri 1 the output should be ignored The latent vectors ai are normalized and a matrix F of
77. es to be processed Input Type and Source cc 3 4 This field allows the user to specify the input device 1442 card reader or disk and indirectly the type of input analysis to be undertaken in the input program The three possible values that may be punched in this field are described below Value Meaning 1 Raw data will be read from the 1442 card reader and transferred to the disk where it will be retained until destroyed by input from See General Operating Instructions section 1 2 10 Value Meaning 1 one of the four programs in this system Until destroyed option cont 2 below can be used to read this data from the disk 2 Raw data will be read from the disk Raw sums and raw sums of cross products will be accumulated Data will be read until a negative number in the identification field is encountered section 2 1 3 3 A previously computed matrix or matrices will be read from the 1442 card reader Matrix cards will be read until a negative job number field is encountered see Pooling section 2 1 4 Sequence Checking Within Observations cc 5 6 This field is used to indicate that raw data input from the card reader cc 3 4 contains a 1 is to be sequence checked A value of zero or a blank field implies that no sequence check will be made A value of one 1 implies that the cards will be sequence checked The sequence checking process consists of an equal comparison check of the case identi
78. extracted and as many rows as variables a Hyperplane Space of N 1 dimensions defined by a reference vector perpendicular to it For example in two dimensions either coordinate axis is the hyperplane of the other in three dimensions the plane defined by any two coordinate axes is the hyperplane of the third Normalize To divide each of a set of numbers by the square root of the sum of squares of all numbers in the set so that the sum of squares of the new set is 1 00 2 2 2 The Program Given a set of observations numbering 499 or fewer containing measures on n 30 variables xi X a square symmetric n x n correlation matrix R and vectors of means and standard deviations are computed From this or a given correlation matrix a factor matrix is extracted containg n or fewer vectors of factor coefficients This factor matrix may be rotated to approximate simple structure by an analytical criterion in either an orthogonal or an oblique reference frame Given the original data X together with its means and standard deviations this rotated factor matrix G may be used to compute factor measurements factor scores by a regression model Communalities are treated as the diagonal elements of the correlation matrix These elements are computed equal to 1 0 and should be retained as such for the computation of the principal components factor matrix They may be specified by some predetermined values howev
79. f punched card output This is discussed briefly in sections 1 3 4 and 1 4 Inthe following a detailed explanation of the mechanics of this operation is given Consider first that stacking is not being done only one job is being run 1f the user places one blank card behind his input deck it is unnecessary to press the card reader and console start buttons to complete the reading of the input data If the user has asked for punched output in his analysis definition option card an adequate number of blank cards should be in the hopper following the data section 1 4 If this is not the case the computer halts waiting for the entry of blank cards After these are placed in the reader hopper the start button should be pressed on the card reader and console to continue processing If jobs have been stacked section 1 4 and if following the card with a negative identification field signifying end of data there is another analysis deck the next job it is possible to destroy this next deck if punched output is being requested in the current job In this case if console switch 15 is down off a message is written reminding the user to place blank cards in the hopper then the computer pauses If this occurs the card reader hopper which contains the next job to be run should be emptied the non process runout button on the card reader should be pressed and the last two cards in the stacker XEQ and a LOCAL card should be placed at
80. fication field for all cards in a case and an ascending sequence check of the card number field If an error in either of these conditions is encountered the program prints a message and the job is terminated Number of Variables on Card 1 cc 7 8 When a data vector contains more variables than will fit on one card the user must indicate to the program the number of variables punched on each card This field must be punched with the number of variables on the first card If there is only one card per case this field must be blank or zero Number of Variables on Card 2 cc 9 10 Same as cc 7 8 except that this field indicates the number of variables on the second card of the data Number of Variables on Card 3 cc 11 12 Same as ce 7 8 except that this field indicates the number of variables on the third card of the data Transformation Switch cc 13 14 If the value in this field is nonzero a user written transformation subroutine is called after each data record is read and before any computation takes place If the value in this field is zero or blank the transformation subroutine is not called The use of transformations is discussed in section 2 5 1 a Output Raw Sums of Cross Products cc 15 16 This field is used to indicate whether the raw sums and sums of raw cross products matrix are to be printed punched printed and punched or not presented The four 4 possible values of this fie
81. g the following reconstructed ANOVA table A B D C blocks A XB AXD BXD AXBXD Error Total 2 3 1 Tests of Significance SS a SS SS d SS S8 b Pd 55 4 SS ibd SS SS SS SS 88 SS SS abc abcd acd bed SS otal The output of this program consists of the sums of squares and mean squares for all the main effects and interactions together with the error mean square In general these main effects and interactions are tested for significance by dividing the mean square for the particular effect or interaction by the appropriate error term The difficulty arises in the choice of the appropriate error term A brief account of how to choose the correct error term is given below This account is by no means com prehensive and if the user is in any doubt as to the error term to use in his own case he should consult H Scheffe The Analysis of Variance John Wiley 1959 The basis for the choice of error term in ANOVA F tests is the type of structural model used for the analysis of variance Three models are discussed below these should cover the majority of cases 1 Model I Fixed Effects The fixed effects model is applicable when the factors used in the experiment include all possible levels for each factor and when inferences are not made about any levels not included Examples of this are such factors as sex where there are only two levels possible or training methods fo
82. gh 1130 FORTRAN does not allow the use of an object time format definition a specially written format processing program is employed to enable the user to specify the format of his data by means of a FORTRAN like statement The format statement may contain almost all the specifications included in a normal FORTRAN format statement as described in 1130 FORTRAN Language C28 5933 pp 11 15 with the following exceptions 1 Only I E F or X data specifications are allowed 2 Continuation cards are not allowed 3 The use of a slash is not permitted 4 Internal parentheses in the format specification are allowed The format card is punched with parentheses surrounding the specifications in columns 1 80 as shown in Figure 3 CC P F1 0 F5 2 F7 5 3F2 1 Figure 3 Format card example Note For each data card within an observation set in case more than one card is required per observation there must be a variable format card preceding the data deck These format cards must be in the same order as are the cards in the observation sets At most the user will supply three format cards 1 3 PROGRAM PAUSES AND MESSAGES 1 Pause 10 An illegal character in a numeric field has been encountered in reading data The program will print the card and the approximate column where the error was detected Pressing START on the card reader and console will cause the remaining data cards to be read and ignored The next monitor
83. hange which should take place before loading the statistical system is listed below Program Name IOCS Card Indentification cc 73 80 COREL CORL 10 POLY POLY 20 POL2 POL2 20 REGR REGR 20 REGR2 RGR2 10 ANOVA NOVA 20 ANOV2 NOV2 20 FCTR FCTR 10 FCTRI FCT1 20 FCTR2 FCT2 20 FCTR3 FCT3 20 In the routine PRNTB a Card PRNB 150 should be changed to read LIBF TYPEZ b Cards PRNB 70 PRNB 130 should be omitted In the decks distributed with this system identifying labels are given in cc 73 76 These four characters do not allow labels to agree perfectly with names of programs When referencing programs keep this distinction in mind 1 2 CONTROL CARDS Each of the four programs included in the 1130 Statistical System requires monitor and program control cards These cards are described in the job execution section of the specific program being considered However certain program control cards are standard for all programs Their descriptions follow For any control card numbers specified as integers I this includes all numbers used for program control should be specified as follows 1 All numbers should be shifted to the right of their fields right justified unless left justification is specifically called for 2 Blanks and zeros are synonymous STANDARD PROGRAM CONTROL CARDS Input Output Units Card a The function of the input output units card Figure 1 is to assign logical unit numbers to each of the I O device
84. he number should include the decimal point which may be placed anywhere in the field No blank columns are allowed The variance criterion is used to determine when the best fitting polynomial has been computed The process of fitting the points involves the computation of successively higher degree polynomials As each degree computation is completed a variance criterion is developed When the difference between any two successive variances is less than the variance criterion punched in this field the best fitting polynomial is assumed to have the degree of the last polynomial computed If however this condition is not met before the maximum degree polynomial as defined in cc 1 2 is satisfied the maximum degree is the degree used If the user has no feeling for the magnitude of this number 01 may be used Transformation Switch cc 27 28 If this field is nonzero a user written transformation routine is calied see section 2 5 1 TRAN is called before any scaling that the user might elect to have the program perform 95 Scaling Switch cc 29 30 if this field is nonzero x is scaled by a linear transformation to x such that the elements of x are in the range 2 2 All calculations then deal with the data set x y see section 2 4 Polynomial and Residual Output cc 31 32 If this field is nonzero only the coefficients of the polynomial y ag a1X t t ay xl are presented the residuals y y y and P x are n
85. ij ij n where c are the elements of sums of raw cross products matrix and 8 s are the raw sums of the jth and 8 variables respectively and n is the number of cases is to be printed punched printed and punched or not presented The four 4 possible values are described above under Output Raw Sums of Cross Produets The matrix is computed even if the no output option is chosen Output Variance Covariance Matrix cc 19 20 This field is used to indicate whether the variance covariance matrix defined as u i j 1 2 n J n 1 where Wij is an element of the residual cross products matrix and n is the number of cases is to be printed printed and punched punched or not presented The four 4 possible values that may occur in this field are as is given for the above matrices There are no additional vectors or matrices punched with the punched output The matrix is computed even if the no output option is chosen Output Correlation Matrix cc 21 22 This field is used to indicate whether the correlation matrix defined by EUM i j 1 2 n ij 5 5 l J where c is an element of the variance covariance matrix and 5 P are the standard deviations of the fh and jib variables respectively is to be printed punched printed and punched or not presented The four 4 possible values contained in this field are as is given for other matrices above Th
86. ion Promax including Varimax Constant for number of factors to rotate 0 Rotate the number of factors determined by the option chosen in cc 25 26 above k Rotate a number of factors equal to the minimum of k ten and or the number of factors determined by the option above in cc 25 26 Pooling option see Sections 2 5 3 and 2 2 4 00 Add matrices with ID 1 Nonzero Subtract matrices with ID 1 Column Meaning Note In columns 37 60 the matrix output options are as follows 0 No output 1 Print only 2 Print and punch 3 Punch 37 38 Output the latent vectors A 39 40 Output the unrotated factor matrix F 41 42 Output the orthogonal transformation matrix T 43 44 Output the orthogonal factor matrix G 45 46 Output the transformation matrix to oblique vector structure L 47 48 Output the oblique reference vector structure matrix V 49 50 Output the correlations among oblique reference vectors 51 52 Output the oblique reference vector pattern matrix W 53 54 Output the correlations between reference vectors and primary factors D 05 56 Output the oblique primary factor structure matrix S 57 58 Output the correlations among oblique primary factors d 59 60 Output the oblique primary factor pattern matrix P 61 62 Output the factor score regression coefficients VARIABLE NAME CARD In the factor analysis program there are a number of matrix printouts that the user may request The varia
87. ion vector input scaling Analysis of variance Principal components analysis card input Factor analysis correlation matrix input Time min sec 6 02 3 04 2 35 2 35 2 00 2 00 2 14 5 48 3 45 117 CHAPTER 5 ERROR MESSAGES Following is a list of error messages presented to the user on the optional printer or the typewriter Common to All Programs 1 AN ILLEGAL CHARACTER HAS BEEN ENCOUNTERED IN COLUMN N OF THE ABOVE FORMAT CARD CHANGE CARD AND RERUN JOB Action Correct the format card and rerun the job See section 1 2 Variable Format Card 2 AN ILLEGAL CHARACTER HAS BEEN ENCOUNTERED IN APPROXIMATELY COLUMN N OF THE ABOVE DATA CARD CHANGE CARD AND RERUN JOB Action The format card and or the data card is in error Correct the card s and rerun the job See sections 1 3 1 and 1 2 Variable Format Card and the data input section pertaining to the particular analysis being run 3 INVALID INPUT OPTION JOB TERMINATED Action Data input mode is not 01 02 or 03 See the section discussing the option card input mode for the particular analysis being run Common to Regression and Factor Analysis 4 CARD ID IS OUT OF SEQUENCE RERUN JOB Action Check sequence number of card revise and rerun job See section 2 1 3 or 2 2 5 Regression Messages 5 MEAN SQUARE NONPOSITIVE JOB TERMINATED Action A format specification error could have caused data to be converted incorrectly or an il
88. istic values Trace Cumulative percentage of trace of each characteristic value Unrotated factor matrix Orthogonal transformation matrix Orthogonal factor matrix Transformation matrix to oblique reference vector structure Oblique reference vector structure matrix Correlations among oblique reference vectors Oblique reference vector pattern matrix Oblique primary factor structure matrix Correlations among oblique primary factors Oblique primary factor pattern matrix Factor score regression coefficients Factor scores Communalities 45 46 2 2 4 Job Execution To perform a factor or principal components analysis the user must supply three sets of cards to the program 1 Monitor control cards 2 Program control cards 3 Data cards Descriptions of the form and content of each card set follow MONITOR CONTROL CARDS The monitor control cards are necessary to initiate program loading from the disk and to establish the necessary communication with the monitor A general description of the cards may be found in IBM1130 Disk Monitor Reference Manual C26 3750 A factor analysis requires the following monitor cards Cc 1 4 8 16 17 j XEQ FCTR 04 LOCALFCTR FMTRD DATRD PRNTB MXRAD TRAN LOCALFCTRI TRIDI QR INVRS LOCALFCTR2 VECTR PRNT LOCALFCTR3 VARMX PROMX SCORE RFOUT LOCALCOREL PRNT The monitor control cards do not change from job to job within one analysis
89. l conditioned matrix for example one with high correlations between independent variables could have caused inaccuracy in the inversion of the correlation matrix or in the calculation of mean squares 6 NO MORE DEGREES OF FREEDOM JOB TERMINATED Action The number of parameters being estimated is larger than the number of observations Increase the number of observations or accept a model with fewer parameters 7 NO MORE VARIABLES SATISFY THE VARIANCE CRITERION JOB TERMINATED Action Modify the variance criterion section 2 1 2 or accept one of the models produced 118 HZU U333 1 IBM Int rnational Business Machines Corporation Data Processing Division 112 East Post Road White Plains N Y 10601 USA Only IBM World Trade Corporation 821 United Nations Plaza New York New York 10017 International wht V S f ut payaitg TI 0 0cH MINE De r gn
90. l correlation matrix R The process of factor analysis or principal component analysis is designed to resolve this correlation matrix into an n x k factor matrix in which the number of factors k is usually considerably smaller than n the number of variables These factors may be considered as underlying influences which in further measurement can be substituted for the more numerous original variables and which largely account for the correla tions among the latter In analyzing the structure of a correlation matrix two approaches can be taken Formally they resemble one another to a certain extent but they have in fact rather different aims One method is principal component analysis the other is factor analysis The former method is a relatively simple technique of breaking down a correlation matrix into a set of orthogonal uncorrelated components equal in number to the original variables These correspond to the latent roots eigenvalues and accompanying latent vectors eigenvectors of the matrix The method has the property that the roots are extracted in descending order of magnitude this is important if only a few of the components are to be used for summarizing the data These vectors are mutually orthogonal and the components derived from them are uncorrelated Although a few components may extract a large proportion of the total variance of the original variables all components are required to reproduce the correlati
91. ld are described below The computation to generate the matrix is performed even if the no output option is chosen Value Meaning 0 or blank No output 1 Matrix will be printed 2 Matrix will be printed and punched 3 Matrix will be punched Punched output of the raw sums of cross products matrix includes the number of observations and the vector of raw sums and sums of squares This entire output must be entered on the pooling option section 2 1 3 Output Residual Cross Products cc 17 18 This field is used to indicate whether the residual cross products matrix defined as where c are the elements of sums of raw cross products matrix 8 are the raw J sums of the u and je variables respectively and n is the number of cases is to be printed punched printed and punched or not presented The four 4 possible values are described above under Output Raw Sums of Cross Produets The matrix is computed even if the no output option is chosen Output Variance Covariance Matrix cc 19 20 This field is used to indicate whether the variance covariance matrix defined as u sce i j 1 2 n n 1 where uj is an element of the residual cross products matrix and n is the number of cases or sum of weights is to be printed printed and punched punched or not presented The four 4 possible values that may occur in this field are as given for the above matrices 11 12 There are
92. le correlation of a variable with all of the other variables constitutes a lower bound on the communality The true communality lies somewhere between the squared multiple correlation and one To date no method has been found of arriving at the true communality The problem is further complicated by the fact that the communality estimates and the number of factors extracted are mutually interdependent the communality estimates chosen and the number of factors chosen determine when the residual correlation matrix drops to zero Since both of these values must be solved for simultaneously and this is impossible one has to start with one fixed and allow the other to be decided on by the program Probably the best method to use is to fix the number of factors and by iteration find communalities that exactly fit the off diagonals to give that number of factors In other words decide on number of factors insert the squared multiple correlations as initial estimates of the communalities and factor the correlation matrix from which a new set of communalities is obtained using these new values refactor the matrix again This process is continued until the change in communalities between successive factorings becomes trivial A final problem is deciding the number of factors to extract In the absence of prior knowledge of the number of common factors in the correlation matrix the safest course to adopt is Guttmann s lower bound theore
93. le structure rotations of the factor loading matrix are performed The criteria for oblique simple structure are identical to those for orthogonal simple structure with the exception that the orthogonality restriction is relaxed that is the factors need not be uncorrelated with one another Basically rotation to oblique simple structure is achieved by first performing a Varimax rotation and then using the Varimax factor loading matrix to rotate obliquely The effects of oblique rotation are usually to maximize the high loadings on each factor and minimize the near zero loadings The clusters of variables on each factor as derived from the Varimax rotation will not be altered the effect simply is to give a cleaner solution When an oblique rotation is used the situation is rather more complex than in the orthogonal case There are in fact three matrices that must be understood in relating factors and variables 1 The factor structure matrix which gives the correlations between factors and variables 2 The factor pattern matrix which gives the loadings of factors on variables 3 The factor estimate matrix which gives the beta weights for estimation of factors from variables The final feature is the estimation of factor scores for both the orthogonal and the oblique case When a given number of factors have been extracted from a large number of variables and they have been identified by rotation of the factor loading matrix it
94. lways follows the independent variables The user must therefore indicate to the program the number of the dependent variable in this field The value punched must be greater than zero and less than or equal to 30 A value of zero implies a regression analysis is not desired and the program will exit after the correlation analysis is complete Variance Criterion to Remove Variables cc 31 34 This field is used to determine whether an independent variable when removed from the equation significantly increases the sample residual variance The significance of the increase is determined by comparing it to the variance criterion as punched in this field If the computed variance measure is greater than the criterion the variable is removed from the equation The form of the number to be punehed is xxxx A decimal point may replace any x If there is no decimal point the number is taken to be xxxx The size of this number depends on the information available to the particular analysis If the user has no idea about the size to be used a number between 005 and 05 may be acceptable It is possible for the user to set criterion levels for entry and removal of variables that cause cycling of variables into and out of the model The program does not check this cycling possibility Variance Criterion to Enter Variables cc 35 38 This field has a similar function to the previous field except that the value punched is to determine whether a v
95. m 1954 which demonstrates that eigenvalues with roots less than 1 0 are statistically insignificant so that one could use this to set an upper bound on the number of factors to extract Anoth r possibility which can be used in conjunction with the method of principal components is to extract components until a prespecified amount of the total variance in the correlation matrix has been extracted This can be done most successfully when there is some idea as to the amount of error variance in the correlation matrix that is the reliability of the measurements is known Thus a slightly smaller proportion of the variance can be extracted than is known to be common variance since to take out more factors is to include error variance In addition to finding out the number of factors or components required to account adequately for an observed set of variables we may also be interested in finding out or defining what these variables are When a predetermined number of factors are extracted from a correlation matrix we have a matrix of factor loadings with k columns for the k factors and n rows for the n original variables These factor loadings for variables vo on factors 1 2 k are the correlations of the newly discovered factors with the original variables The concept of simple structure applies to the factor loading matrix As has been pointed out above the factor loading matrix consists of numbe
96. m would be A AB The remaining two digit fields have the same effect as cc 23 24 but are used to add additional components before printing the line For example if cc 21 22 contained a 2 cc 23 24 contained a 6 and 25 26 contained a 9 the sums of squares mean square and degrees of freedom would be printed as the cumulative summary of B AC BD Table Generator Card Summary Meaning Alphameric heading for analysis of variance component END of table indicator 0 More cards to follow 1 No more cards to follow l Skip to a new page before line is printed Table component to be printed Table component to be pooled N Add the component with subscript N corres table to the first component defined in cc 21 22 Same as cc 23 24 Same as cc 23 24 2 Column Meaning 29 30 Same as cc 23 24 31 32 Same as cc 23 24 33 34 Same as cc 23 24 35 36 Same as cc 23 24 37 38 Same as cc 23 24 39 40 Same as cc 23 24 41 42 Same as cc 23 24 43 44 Same as cc 23 24 45 46 Same as cc 23 24 47 48 Same as cc 23 24 49 50 Same as cc 23 24 2 3 4 Data Input To set up the data for the analysis of variance the user must identify each item of data as to its factor and level and punch this information on a card along with the data item Hence each data card will have five fields as follows Field Type Meaning 1 Integer 1 Number of level factor 1 2 Integer 1 Number of level factor 2 3 Integer I Nu
97. mber of level factor 3 4 Integer 1 Number of level factor 4 5 Floating point F Observation 4 Disk working storage allows input of 499 observations The particular columns occupied by each field are arbitrary The user describes the format of the card by means of a variable format card which is entered into the program behind the option card On the format card provision must be made for all four level fields even though all four fields are not necessary in the particular analysis Figure 11 shows a sample data card from a two factor design 83 84 CC 123 45 6 7 11 12 Data card 2 4 32 CC Format card 212 211 0 Figure 11 Sample cards from two factorial design In normal usage the data items are punched one to a card with the appropriate identification Following the data deck there must be an end of deck indicator card which is a card containing a negative number in the first field The order of cards is arbitrary as the cards are rearranged in proper order before the analysis takes place 2 3 5 Operating Instructions A Using the analysis of variance program when the total 1130 Statistical System has not been stored on the disk If the user wishes to load only the set of programs that allow analyses of variance the following programs must be compiled or assembled and stored on the disk Each deck begins with a card punched as FOR and ends with an STORE card The user should use
98. ment the sums of squares are located in cells with subscripts 1 15 Table Generator Card Format To print the proper component and compute the appropriate error term for a particular design a set of table cards indicating the appropriate terms must be punched A description of this card is given below Column Meaning 1 16 Row heading for component This field may contain any 16 or fewer characters that serve to identify the row of the table 81 82 Column 17 20 21 22 23 24 25 26 27 28 49 50 Column 21 22 23 24 25 26 21 28 Meaning Print and end control 0 or blank Normal card 1 Skip to a new page and print column headings and title information before printing line for this component i Last card No more component cards will follow The analysis will be complete after this card is processed The residual and total line will be printed Subscript of the correspondence table component to be printed or used in accumulation of the sums of squares For example if this field contains a 5 the component AB will be used for either printing the remainder of the card is blank or accumulation Subscript of the cell in the correspondence table to be added to the cell used in cc 21 22 This procedure is used for adding components to form the sums of squares For example if cc 21 22 contained a 1 and cc 23 24 contained a 4 the printed sums of souares mean Square and degrees of freedo
99. milarly variables are not entered or removed from the equation on the basis of a second variance criterion which indicates that the variable does not offer any significant improvement in the goodness of fit The general method of solution to determine the coefficients by b b is to compute the matrix of correlation coefficients from the source data This matrix will contain the correlations between all the independent variables and the dependent variable By applying a Gaussian elimination inversion process a stepwise inverse of the correlation matrix is computed Multiplying this inverse by a vector containing the dependent variable correlated with each independent variable forms the normalized regression coefficients The inversion process is carried out for one variable at a time As each variable is processed it is compared to the variance criterion to determine its significance If the variable is to be entered the coefficients for the equation containing a subset of the total number of variables in the analysis are computed and made available for printout and use in the next step of the analysis Because of the nature of the computational process the elements of several subsidiary statistics are also available If the user elects to print each regression step as it is computed these statistics will be printed with the regression coefficients The following book can be used as a reference Ralston A and Wilf H S Mathematical Meth
100. mpute the fitted values for any given argument array However aj may change rapidly as n changes therefore it would be necessary to compute a with great precision To avoid this difficulty fitted values use Cj and B to compute P x from equation 8 and simultaneously compute the fitted valued from equation 1 Similarly the derivative computation uses Cj a j and B to compute the rth derivative of P 159 from the recurrence relation dP dp aq dP F r p J 12 dx Y dx dx J ax j70 1 2 n 1 for any given set of arguments If n and or the range of x is large the elements of the orthogonal vectors generated change rapidly in size this imposes severe restrictions on accuracy This becomes evident to the user by viewing the changes in the elements of successive vectors P jU9 as j increases or by viewing residuals which in this case may tend to increase rather than decrease as j increases To aid in circumventing this problem the user is allowed to elect on option to have the program transform X to X such that X is in the range 2 2 This transformation will cause elements of P 19 to remain approximately uniform in size as j increases The transformation is 57 0 3 om h order statistic from the set x where x is the i i When this scaling is used the following points must be considered 1 The values of cj and Pj x equ
101. n general as a increases the dependence of the rotated factors or their obliquity increases A transformation matrix is computed that rotates the orthogonal factor matrix into an oblique reference vector structure matrix V which is a least squares fit to the pattern matrix P described above This transformation matrix L is derived in unnormalized form from the following matrix equation Ls c ay 1a p After the transformation matrix L has been column normalized the reference vector structure matrix V is obtained from V GL The correlation matrix of the reference vectors y is computed by v L L and the reference vector pattern matrix W is then developed by W VV l To derive the primary factor structure from the reference vector solution described above the diagonal matrix D of the correlations among the reference vectors and the primary factors is computed by taking 1 C V 11 where the di are diagonal elements of D and eu are diagonal elements of the matrix ii y 1 The primary factor structure matrix S is then determined by S WD and the primary factor pattern matrix is derived by P VD 1 The matrix of correlations between the primary factors P is computed by Dy lp Factor measurements factor scores are computed by the short regression method Harman 1960 The diagonal matrix of uniquenesses U is obtained by taking 2 us E l h where the h are the communalities computed from G the o
102. n indication of which variables were the best measures for which factors The factors would then be defined in terms of the variables that have relatively high loadings on them In other words simple structure is the application of Occam s Razor to the factor loading matrix since it aims to explain the configuration of variables in such a way that each factor is represented by only a few variables that is it loads or correlates with the smallest possible number of variables Practically simple structure is realized by several computational techniques The best known is the Varimax rotation Kaiser 1958 which aims to maximize the fourth power of the factor loadings this amounts essentially to maximizing the scatter among the loadings Since a few highs means several lows this leads to finding a position in which there are many low loadings Thus Varimax aims to prevent a variable being simultaneously highly loaded on two factors The Varimax rotation retains the property of orthogonality among the factors that is the factors remain uncorrelated In other words the clusters of variables that load highly on each of the factors are essentially uncorrelated with one another In actual practice this seems rather unlikely It would seem more plausible that the clusters of variables or factors which are obtained from a factor analysis are probably correlated positively or negatively with one another It is for this reason that oblique simp
103. n with some error condition may result for example mean j square nonpositive In any case matrices 1 21 and 22 will be punched see section 2 5 3 These matrices should be used as the second set of matrices for input using the subtraction option 2 1 5 Operating Instructions A Using the regression analysis program when the total 1130 Statistical System has not been stored on the disk If the user wishes to load only the set of programs that allow regression analyses the following programs must be compiled or assembled and stored on the disk Each deck begins with a card punched as FOR and ends with an ani STORE card The user should use a disk containing the 1130 Disk Monitor System as described in section 1 1 The following decks should be preceded by a cold start card placed in the card reader hopper and the buttons IMMEDIATE STOP console RESET console START card reader and PROGRAM LOAD console should be pressed A blank card should be placed after the last deck in the card reader hopper DECKS LABELS REGR REGR COREL CORL PRNT PRNT FMTRD FMRD DATRD DTRD PRNTB PRNB GMPYX GMPY GDIVX GDIV MXRAD MXRD REGR2 RGR2 REGRE RGRE FMAT FMAT TRAN TRAN In addition regression and factor analysis programs must reside on the disk together e section 2 2 7 names additional routines to be placed on the disk B Execution from disk Once the component subroutines and main calling programs are
104. ods for Digital Computers New York John Wiley and Sons Inc 1960 2 1 1 Summary of Output Statistics High and low value of each variable Means of each variable 1 2 3 Standard deviation of each variable 4 Sample variance for each variable 5 Matrix of raw cross products 6 Matrix of residual cross products 7 Variance covariance matrix 8 Matrix of correlation coefficients 9 Residual Standard Deviation 10 Standard error of the mean of the predicted dependent variable 11 Multiple correlation coefficient sum of squares due to regression 12 Square of the multiple correlation coefficient adjusted total sum of squares 13 Degrees of freedom 14 Regression coefficients B 15 Standard error of regression coefficients 16 Partial correlation coefficients ajn 2 where n denotes the dependent variable and a is an element of the stepwise inverse of the correlation matrix 17 Normalized regression coefficients B Bj B S Sy where Sj is the standard deviation of the ith independent variable 18 Standard error of normalized regression coefficients 19 For each data case the predicted value and difference between the predicted value and the actual value 20 Analysis of variance table 2 1 2 Job Execution To perform a regression analysis the user must supply three sets of cards to the program 1 Monitor control cards 2 Program control cards 3 Dat
105. of the output The card is punched in four column fields and each field corresponds to the variable to be identified for example field 3 columns 9 12 will be the name of row and column 3 on all matrix output At most 20 names can appear on one card If there are more than 20 variables in the analysis a second card having the same format as the first must be included in the control card deck 5 Column Meani 1 4 Name of variable 1 5 8 Name of variable 2 4N 3 4N Name of variable N Not pertinent when matrix input is used When correlation matrix input is used matrices are not available for output 15 16 CC 1 4 5 8 9 12 13 16 GRP 1 ANL XX Y Figure 4 Variable name card example 2 1 3 Data Input Raw data input to the regression program consists of a set of observations made on several different variables The variables for each observation are punched on one two or three cards according to the following general form Field Type Meaning 1 Integer I Identification field Any numeric information that serves to identify the particular observation is punched in this field It must be greater than zero and should be different for each observation 2 Integer I Card number within observation If it is not possible for one card to contain all the variables they may be continued on a second and a third card as necessary The user has the option of sequence checking the cards to ensure that all
106. oice of choosing an orthogonal rotation normal Varimax and or an oblique rotation Promax The process by which an oblique rotation is computed requires that an orthogonal rotation be performed first Hence in addition to the oblique rotation matrices the user has the option of obtaining the output of the orthogonal rotation Three possible values may be punched in this field and they are described below Value Meaning 0 or blank No rotation will be performed 1 Orthogonal rotation 2 Oblique rotation includes an orthogonal rotation ol 02 Number of Factors to Rotate cc 33 34 This field is used in conjunction with the rotation switch described in the previous field The value punched in this field determines the number of factors to rotate Two possible conditions can arise If the user does not know the number of factors to rotate it is suggested that the field be left blank or zero The number of factors to rotate is then chosen on the basis of one of the options in cc 25 26 However ifa value of k appears k factors are rotated if this number is less than or equal to the number of factors computed In any case k must be less than or equal to ten Pooling Option cc 35 36 When using the matrix input output option 03 in cc 3 4 and when pooling sums of squares and cross products section 2 1 4 if the user desires that matrices be subtracted rather than added this field should be nonzero Factor Matrix Output
107. on the disk the execution of a job requires the monitor control cards program control cards and data cards to be Used in all four analysis types Used in factor analysis 19 placed in the card reader The deck should be preceded by a cold start card To initiate processing the buttons IMMEDIATE STOP and RESET console START card reader and printer and PROGRAM LOAD console should be pressed The order in which the cards are placed in the card reader for either matrix or raw data input is shown in Figures 5 6 and 7 Optional Blank eti DL Negative dent Data Deck ariable Forma Variable Names Option Card Job Title Card Monitor Control Cards Figure 5 Regression card order card reader input Optional Blank Output Variable Names Input Output E Units Monitor Control Figure 6 Regression card order disk input 20 Optional Blank Output Negative Identification to be added to matrix 1 Variable Names I Monitor Control Figure 7 Regression card order matrix input 2 1 6 Sample Problem INPUT XEQ REGR 03 LOCALREGR FMTRD PRNTB DATRD MXRAD4 TRAN LOCALREGR2 REGRE LOCALCOREL PRNT 020200 2222 STEPMISE TEST ONE 06010000 0001010102 1010006 500 300 00010 Pl P2 P3 P4 P5 P6 212 1X 4F5 25 F5 0 2F5 2 2F5 0 0101 002500002502500001500003400064 0201 013000002102100000870003600065 0301 00350000220220000043000410
108. onents and factor analysis In the former we consider all the variance common variance and extract its orthogonal components In factor analysis on the other hand we take into account that some of the variance is going to be due to error and some to variance that is quite specific to a certain variable In this sense the factor analytic model is more realistic and the principal components method in spite of its mathematical simplicity is misleading Therefore let us assume that we have n observations on p variables X Using the method of principal components we have S ies X Ph o e x 2 Thus the principal components equations are ape Xy ig ik 37 38 However in the factor analysis model if Fu 21k cov x xj aa Zf fj 4 k m j and if we substitute its expected value for ES dan D ifk 0 ifk m we have cov Xx x 8 Ay 5 k If the model is Xi aufi 4 eii 6 we have cov x x 2 8 ik ik var e d 7 Now we are required to estimate the coefficient aij We can operate on the estimated matrix 5 8 8 d var e This is the same as the former matrix except for the principal diagonals where each term is increased by var ei Thus we would like to have in the main diagonal not 2 2 Za var e but only a In other words if we are not to bias the estimates of the ik k a s we must remove var e from the diagonal terms This is equivalent to s
109. ons between the variables exactly Note that when the principal components method is employed no hypothesis need be made about the original variables They need not even be random variables although in practice their values are usually regarded as a sample from some population Factor analysis on the other hand seeks to account for or explain the matrix of correlations by a minimum or at least a small number of hypothetical variables or factors Factor analysis asks the question Does a random variable F exist such that the partial correlations between pairs of variables are zero after the effect of F4 has been removed If the correlation matrix is still unexplained the question is asked whether two random variables F and Fo exist so that the partial correlations between pairs of variables are zero after the effects of both of these variables have been removed etc Thus it may be said that principal components analysis is variance oriented whereas factor analysis is covariance oriented As has been noted above the number of factors needed to explain the correlations is fixed by the data itself in the sense that when the factor extraction process leaves a 31 32 residual correlation matrix of approximately zero all of the covariance present has been accounted for However this brings up the biggest problem in factor analysis When a set of variables is intercorrelated we have a set of n n 1 2 correlations Thi
110. ot listed A This option is helpful if solution vectors are punched for later evaluation of y at new points x when at x is not desired Curve Fitting Type I Option Card Summary Column Meaning 1 2 Maximum degree of polynomial to be fitted 3 4 Input source 1 Raw data input from card reader 2 Raw data input from disk 5 6 Coefficients of fitted polynomial switch 0 Do not print 1 Print 7 8 Evaluate derivatives switch 0 No derivatives 1 Derivatives 9 10 Maximum order derivative 11 12 Predicted values 0 Do not print 1 Print 13 14 Polynomial solution vectors punch switch 0 Do not punch 1 Punch lt 15 16 Must be zero or blank 17 26 Variance criterion x 27 28 Transformation switch to user written program 0 Do not call TRAN 1 Call TRAN 29 30 Nonzero Scale x into 2 2 31 32 Nonzero Do not print polynomials predicted values or residuals 5 TYPE II OPTION CARD This type of option card is to be used whenever data is entered into the program with a previously computed set of polynomial solution vectors Degree of Polynomial cc 1 2 This field is used to transmit the degree of the previously computed polynomial solution vectors that are to be read by the program Input Type cc 3 4 This field must be punched with a three 3 This program uses this number to read in the polynomial solution vectors that were punched from a previous analysis In addition to these solu
111. otations are performed until U is not significantly increased by an additional cycle U is computed by an evaluation of the following expression 4 2 n k n Bi i AX m i 1j 1 i 1 1 i 1 h j where k is the number of factors n the number of variables and an element of the h 2 factor matrix under rotation for the ie variable on the m factor h represents the th communality of the i variable computed using only the k factors under rotation The final rotated factor matrix G is derived by the matrix multiplication G FT where T is the complete Varimax transformation matrix and F is the unrotated factor matrix This matrix of k factors may also be rotated to approximate simple structure in an oblique reference frame by the Promax method Hendrickson and White A pattern 43 44 matrix P describing a factor matrix rotated to approximate simple structure in oblique axes may be accurately estimated by a matrix whose elements are functions of the elements of the orthogonal matrix rotated by the Varimax method This matrix can be derived by the following operation 811 LU 1 where Ps is an element of the pattern matrix P and gi is an element of the orthogonally rotated factor matrix G The value of a is four 4 According to Hendrickson and White 1964 four is the optimal value in the majority of cases The user can however easily change this number in the program PROMX I
112. plete output can be used as input to initiate another analysis without the necessity of reprocessing the source data that was used to generate the matrices To use the correlation matrix set as input the user places the punched output behind the variable names card followed by a blank card or one that contains a negative number in the job number field The program reads the number of cases means and standard deviations and correlation matrix stores each in its appropriate location and then initiates the analysis as specified on the option card Pooling Sums of Squares and Cross Products cc 35 36 The topic is discussed fully in section 2 1 4 When raw sums of squares matrices have been previously punched by this program or by hand in accordance with the above format description they can be stacked and will be combined by using this option When the option is given the number 0 they will be added If a nonzero field is used all matrices will be added until the first negative left justified job number field cc 1 4 is encountered Subsequent matrices will be subtracted until the second negative problem number card is encountered Iterating on Communalities This program does not iterate on communalities which is a desirable feature mentioned in section 2 2 However by electing to punch the correlation matrix one can insert the estimated communalities on the diagonal of this matrix and iterate using matrix input The factor
113. r factors In general the two main criteria for deciding on the number of components to retain are a when the value of the A s falls below 1 0 or b when the first n components account for a fixed percentage of the total variance Factor Analysis Model Basically the mathematics and computational steps in factor analysis are similar to those required by the method of principal components The difference lies in the fact that in component analysis we begin with a set of observations and look for components in the hope that we shall be able to effect a reduction in the dimensions of variation and that we can give some physical meaning to the components thus extracted Using the factor analytic model we begin with a theoretical model and try to find out whether it agrees with the data and if it does to estimate its parameters Let us begin as in the method of principal components with a matrix of observations Xij and consider whether they can arise from a situation with the following structure p x id t b S C where i 1 p 1 In this equation 1 the f are factors that can appear in more than one x s isa factor specific to the variable and ej is an error term At this level the model is undetermined and in fact by using the method of principal components we can always express the x s in terms of f s without invoking specific or error terms at all This is the basic difference between principal comp
114. r one 1 or two 2 If a one 1 is entered the data points followed by a negative identification card are read from the 1442 card reader and onto the disk destroying previously stored data If a two 2 is entered the data points are read from the disk If no data points are entered the format card is still required and the first card succeeding the solution vector deck is read as the new data card If this card is blank ay at x zero is evaluated If the user written program TRAN was called for initial analysis of the data the user should be sure that it is called to operate on the new data Curve Fitting Type II Option Card Summary Column Meaning 1 2 Degree of polynomial 3 4 Input type 3 Polynomial solution vectors from card reader 5 6 Coefficients of fitted polynomial 0 No l Yes 1 8 Evaluate derivatives 0 No 1 Yes 9 10 Maximum order derivatives Column 11 12 13 14 15 16 17 26 27 28 29 32 2 4 3 Data Input Meaning Compute estimated values 0 No 1 Yes Not used Secondary input source 1 Raw data from card reader 2 Raw data from disk Not used Transformations 0 Do not call TRAN 1 Call TRAN Not used however x is scaled if the x s were scaled for the previous analysis Raw data input to the program consists of a set of points x y punched on cards one point to a card with associated identification The general form for this input can be described in
115. r teaching a specific skill or in a drug experiment the treatments factor where one is inter ested in the drugs used and would not obviously want to make inferences to other drugs not included in the experiment In the case of a fixed factor the investigator is interested only in the levels of the variable studied in the experiment and not in any others In this case the computation of the F ratio is relatively simple The F ratios are calculated using the error term as a divisor for example MS 2 MS for the A main effect MS p MS AB interaction etc error error 2 Model II Random Effects The random effects model applies when the experiment involves only a random sample of the set of treatments about which the experimenter wants to make inferences For example to study the effects of a certain drug say alcohol on driving skill one would have several different levels doses of alcohol within the drug factor However all possible levels of alcohol could not be used so one takes what is considered to be a random sample of the levels within the factor and then makes inferences about other levels Another example would be the following To study the effects of level of illumination on productivity in a factory the luminance factor would be a random effects factor since all possible levels of luminance would not be used in the experiment but only a sample of them An analysis of variance with all random effects is rarely
116. ramming see the manuals 1130 FORTRAN Language C26 5933 and 1130 Disk Monitor System C26 3750 in adding transformation capability to the system Currently a subroutine TRAN is included in the package and is called on option by each main program subsequent to the reading of each observation The current routine returns to its calling program immediately In implementing such a subroutine the following points should be considered 1 In the regression and factor analysis programs the observation row X is in COMMON storage and can be reached by use of the COMMON statement in the user written program The row X contains one observation on Xj Xy and TRAN could be written using the row X as an argument 2 In the orthogonal polynomial program TRAN is called after each reading of xij yi which are elements of vectors X Y in COMMON TRAN could have arguments x and or y or could use the COMMON statement 3 For the analysis of variance program TRAN should include the argument DATA containing the observation 4 If a large transformation program is prepared by the user storage requirements may call for the use of LOCAL monitor facilities 5 Transformations that modify the number of variables in the observation require modification to the program supplying and analyzing the data Such modifications require programming knowledge of the package For example if one originally entered ten variables and wished to transform the
117. rinted Evaluate Derivative Switch cc 7 8 This field is used to indicate whether derivatives are to be evaluated at a selected list of points If this field contains a zero or a blank the derivatives are not computed If this field contains a one 1 each data point is examined to determine whether the derivative computation indicator for that point is nonzero When a nonzero indicator is located for a particular value of x the program evaluates the kth order derivatives where k is an integer punched in cc 9 10 The printout includes the 1 2 kth_order derivatives and the estimated values of y for that particular value of x 97 98 Maximum Order Derivatives cc 9 10 If cc 7 8 contains a one 1 this field must be punched with an integer k which is less than or equal to the order of the polynomial It indicates the maximum order derivative to be computed when a nonzero derivative computation indicator is located The printout includes all lower order derivatives as well as the maximum If the order is given as zero no derivatives are calculated Estimated Value Switch cc 11 12 This field is used to indicate whether estimated values are to be computed for the values of x read in by the programs If this field contains a zero or is left blank the estimated values are not computed If a one 1 is punched the estimated values are computed Secondary Input Sources cc 15 16 This field must be punched with an intege
118. rol Figure 14 Orthogonal polynomial card order card reader input Optional Blank Output Job Title Input Output Units Monitor Control Figure 15 Orthogonal polynomial card order disk input End of Data Polynomial Solution Vectors Variable Format Input Output Units including scaling constants card Monitor Control Figure 16 Orthogonal polynomial card order solution vector input 101 2 4 5 Sample Problem The data for this sample problem was taken with permission from page 213 Statistical Theory in Research by R L Anderson and T A Bancroft McGraw Hill Book Company Inc New York 1952 INPUT XEQ POLY 02 LOCALPOL2 POLSQ PCOEF PDER PFIT LOCALPOLY TRANs DATRD FMTRD PRNTB 020200 1111 ORTH POLY NO SCALING 0201010102010100 010000000000000 I2 I151X F2 0 F3 1 011 01011 020 02071 031 03110 040 04126 O51 05147 The numbers on this first card 060 06199 071 07251 are valid only when the user 080 08239 elects to scale see section 091 092231 2 4 2 When scaling is not 100 10236 performed they reflect prior 111 11260 core status and should be 120 12246 ignored that is they can take on any value PUNCHED OUTPUT 0 3636363E 90 0 1636363E 111124 1 1 65 O1 O 11 1666E 02 O 1772499E 02 111124 1 2 0 6500000EF 0 9333328E 0 2105245E 111124 1 3
119. rs such that each number corresponds to a given factor and a given variable A particular element in the matrix indicates the extent to which that factor is represented in a given variable However the particular configuration of numbers obtained in an unrotated factor loading matrix is largely a function of the particular method used in extracting the latent roots and vectors of the correlation matrix and may have no empirical meaning The concept of simple structure was developed as a number of criteria for the orthogonal rotation of the original factor loading matrix into such a position that the factors extracted are readily identifiable in terms of the original variables Simple structure is a nonmathematical concept that sets up several criteria for rotation The first of these is the existence of a positive manifold This means that all other things being equal the factor loading matrix should have a minimum number of negative values But for many factor loading matrices corresponding to particular correlation matrices we may still have a very large number of factor loading matrices which equally well account for the intercorrelations and which all have mostly positive values To further restrict the selection of the particular matrix by which we shall define the primary variables we require that each column of the matrix shall have a small number of high factor loadings and a large number of near zero loadings We also require th
120. rthogonal factor matrix or P the oblique factor matrix depending on which scores are requested For the case of uncorrelated or orthogonal factors the matrix Q is developed by Q I G U la where G is the orthogonal factor matrix Varimax solution Factor scores are formed by the operation f where f is a factor score matrix Bisa matrix of factor score regression coefficients and Z the vector of standardized data B is obtained by the operation Q GU The elements of Z are computed by g j where ej is an observation in a data matrix for the oo sample case on 3 the 2 variable and x is the mean and P the standard deviation of the variable For the case of oblique or correlated factors the procedure is much the same except for the definition Q p l 1 P U P where P is the primary factor pattern matrix and the matrix of intercorrelations of the primary factors Also here Pv 2 2 3 Summary of Output Statistics N N Ne me LH oz Oo C N e 0 Qi 4 BW KF c wa n C ao WO N Ha High and low value of each variable Means of each variable Standard deviation of each variable Sample variance for each variable Matrix of raw cross products Matrix of residual cross products Variance covariance matrix Matrix of correlation coefficients Matrix of characteristic vectors Character
121. s among oblique reference vectors 12 NxK Oblique reference vector pattern matrix 13 KxK Correlations between reference vectors and primary factors 14 NxK Oblique primary factor structure matrix dins 15 KxK Correlations among oblique primary factors 16 NxK Oblique primary factor pattern matrix 17 NxK Factor score regression coefficients 21 1x1 Number of cases The following matrices include two or three vectors which are punched as column vectors where the column dimension indicates the number of elements in the vector No of Elements Matrix Number Dimension on Each Card Meaning 22 Nx2 2 Raw sums raw cums of squares 29 Nx2 2 Means standard deviations 3 24 Nx3 3 Alpha Beta C orthogonal polynomial solution vectors In the above N number of variables K number of rotated factors N order of the polynomial 112 2 5 4 Scaling The programs in the 1130 Statistical System allow large data sets and a general input format Thus there is 2 possibility that scaling will be necessary In regression and factor analysis a pooling option is allowed that uses the raw cross products matrix If the number of observations is large or if some observed variable readings are quite large some inaccuracies may become evident in this matrix Sometimes scaling by use of the Format statement can aid in the solution of this problem In other situations a transformation of the variable may help It is also possible that scaling
122. s leaves unanswered the question of what to put in the diagonal of the matrix since we need a complete matrix for the process of factor analysis Two solutions to this problem exist 1 put ones in the diagonal the method of principal components on the grounds that except for errors in measurement a variable should correlate perfectly with itself 2 insert values into the diagonal known as communalities the term communality means the amount of variance of the variable accounted for by all the common factors together This will obviously be less than the total variance since some of the variance in any correlation matrix will be error variance and some variance specific to that variable This second solution is called factor analysis Using the method of principal components it is possible to account for many variables by a few factors since the first few principal components usually account for most of the variance However unlike the factor analysis model the variance accounted for will include both specific factor and error variance Factor analysis in putting communality estimates instead of ones in the diagonal attempts to partition the common factor variance from specific factor and error variance Unfortunately use of the factor analysis model leaves the problem of deciding what values to use for communalities The communality of a variable is the most that it has in common with other variables thus the squared multip
123. s used throughout the program Each subroutine that requires the use of an I O device has been programmed with symbolic unit designations This card fixes a number to a specific I O device Column Meaning 1 2 The unit for input of all control cards and source data Normally it is set equal to the logical number of the 1442 card reader which is 02 3 4 The unit used for card output of computed matrices Normally it is set equal to the logical number of the 1442 card punch which is 02 5 6 Output switch 0 1132 Printer output 1 Typewriter output CC 12 34 56 02 02 00 Figure 1 1 O units card printer Job Title Card r a The job title card Figure 2 allows the user to assign a job number and title information for the job to be processed This information is used only for labeling and is not used for processing in any program The job number and title contained on the card are printed as the heading line on each page of output produced The job number appears in the first four columns of any punched card output produced id CC 4 9 21 Multiple regression for class 3 data 10 7 64 Figure 2 Typical job title card Column Meaning 1 4 Job number 5 8 This field is not used 9 80 Title information These columns may contain any legitimate key punchable characters that serve to identify the job Variable Format Card Each program was designed to allow some flexibility in the input of data Althou
124. scores computation reads the original data matrix X from the disk If X is not on the disk which is the case if any card reader input with other data has been read subsequent to the reading of X then X must be reread under input mode 1 before this matrix input option is used 57 58 2 2 7 Operating Instructions A Using the factor analysis program when the total 1130 Statistical System has not been stored on the disk If the user wishes to load only the set of programs that allow this type of analysis the following programs must be compiled or assembled and stored on the disk Each deck begins with a card punched as FOR and ends with an STORE card The user should use a disk containing the 1130 Disk Monitor System as described in section 1 1 The following decks should be preceded by a cold start card placed in the card reader hopper and the buttons IMMEDIATE STOP console RESET console START card reader and PROGRAM LOAD console should be pressed A blank card should be placed after the last deck in the card reader hopper DECKS LABELS FCTR FCTR FCTR1 FCTI1 FCTR2 FCT2 FCTR3 FCT3 FMTRD FMRD DATRD DTRD GMPYX GMPY GDIVX GDIV PRNTB PRNB COREL CORL PRNT PRNT MX RAD MXRD INVRS INVS XMAX XMAX TRIDI TRID QR QR VECTR VCTR COVEC CVEC RFOUT ROUT PROMX PRMX VARMX VRMX RPRNT RPNT MATIN MATN SCORE SCOR FMAT FMAT TRAN TRAN In addition regression and factor analysis programs must resi
125. t be compiled or assembled and stored on the disk Each deck begins with card punched as FOR and ends with an STORE card The user should use a disk containing the 1130 Disk Monitor System as described in section 1 1 The following decks should be preceded by a cold start card placed in the card reader hopper and the buttons IMMEDIATE STOP console RESET console START card reader and PROGRAM LOAD console should be pressed A blank card should be placed after the last deck in the card reader hopper DECKS LABELS POLY POLY POL2 POL2 POLSQ PLSQ PCOEF PCOF PFIT PFIT PDER PDER FMAT FMAT FMTRD FMRD DATRD DTRD PRNTB PRNB GMPYX GMPY GDIVX GDIV TRAN TRAN B Execution from Disk Once the component subroutines and main calling programs are on the disk the execution of a job requires the monitor control cards program control cards and data cards to be placed in the card reader The deck should be preceded by a cold E start card To initiate processing the buttons IMMEDIATE STOP and RESET console START card reader and PROGRAM LOAD console should be pressed The order in which the cards are placed in the card reader for solution vector disk or raw data input is shown in Figures 14 15 and 16 Used in all four analysis types 100 Optional Blank Output Negative Identification Data Deck Variable Format Input Output Units Monitor Cont
126. terms of individual fields for each item on the card Field 1 Type Meaning Integer D Card identification Any numeric information that serves to identify the point x y The number punched in this field must be greater than zero Integer Derivative computation indicator The program computes Floating point F Floating point F derivatives at any point specified in the data set If the user wishes to have a derivative of the polynomial evaluated at the point punched on this card this field should contain a one 1 The order of derivative to be computed is specified on the option card If this field contains a zero 0 the derivative is not evaluated at this point The value of x Any floating point number is allowed The values of x need not be equally spaced Scaling of data may be required The value of y Any floating point number is allowed 99 The particular card columns for each field are arbitrary as long as all four fields are present on the card Following the data deck the user must include a card containing a negative integer in the identification field This card signals the program that no more data points are to be processed 2 4 4 Operating Instructions A Using the polynomial regression program when the total 1130 Statistical System has not been stored on the disk If the user wishes to load only the set of programs that allow this type of analysis the following programs mus
127. tes on the Programs PX EE 111 vy 2 9 1 Transformations 2 oi Sad E MONDE d Seo eov dede o 111 2 5 2 Notes on Correlation and Eigen Analysis seso 111 2 5 3 Punched Matrix Output lr 112 J SCAVNE METTE 113 CHAPTER 3 GENERAL FLOWCHARTS 52 22 52 22 52 5 5e5e 114 CHAPTER 4 SAMPLE PROBLEM TIMING c e e rr n or on ln 117 CHAPTER 5 ERROR MESSAGES PD el ee Pee ae we Oe 118 a CHAPTER 1 GENERAL OPERATING INSTRUCTIONS 1 1 SYSTEM GENERATION After preparing a disk with the monitor system 1130 Disk Monitor System 1180 05 001 described in manual C26 3750 the distributed source cards preceded by a cold start card can be loaded into the card reader hopper and compiled and stored on the disk It is not however necessary to store the total 1130 Statistical System on the disk before using any one of its four major programs Within the discussion given for each specific program is information pertaining to loading the particular set of routines necessary for that analysis type The distributed decks consider that the operating system will use the 1132 Printer as output If the console typewriter is to be used as the output device monitor generation must consider this and the IOCS card in each main program must be replaced by an IOCS card stating IOCS CARD TYPEWRITER DISK Identifying information for this exc
128. the characteristic equation of the correlation matrix r AlI 0 8 For a known this gives p roots in To each root corresponds a set of g s for which S has a stationary value Also using equation 6 we find from equation 5 S p 9 It follows from equation 9 that the root which gives the minimum S is the one with the largest If we choose the largest root of equation 8 we have the line required The sum of squares of distances of the points from it is a minimum and the variate measured along it has the maximum variance The variate is given by p Vi 10 This indicates that the set of g s relates to 1 If we multiply equation 6 by xy and sum over k we can see that the variance of V4 is pL If we now look for the direction perpendicular to the first line for which the sum of squares of perpendiculars is at a minimum we find the line corresponding to A 9 the second largest root etc Thus we have transformed to a new set of variates V that are uncorrelated and have variances Ais gree _ in decreasing order Note that p 2A since the sum of the roots is the sum of p units in the main diagonal Also if the original variables are normally distributed we can regard the V s as splitting off independent components of variance E 9 Ap from the total variable p Thus we can select from this set of p variates V the first n components and consider these as ou
129. the front of the next job to be run Blank cards should then be placed in the reader hopper followed by the next job to be run and START pressed on the card reader and console to continue processing 1 6 MACHINE AND SYSTEMS CONFIGURATION The 1130 Statistical System is designed to operate on an 8K 1130 Computing System with disk storage 1131 Model IT and 1442 Card Read Punch the 1132 Printer is optional It is written to operate under the 1130 Disk Monitor System 1130 05 001 1 7 PROGRAMMING LANGUAGE IBM 1130 FORTRAN and the IBM 1130 Assembler Language 1 8 REFERENCE MATERIAL IBM 1130 Disk Monitor System Reference Manual C26 3750 IBM 1130 Assembler Language 0626 5927 IBM 1130 FORTRAN Language C26 5933 CHAPTER 2 PROGRAMS 2 1 STEPWISE LINEAR REGRESSION From sets of observations numbering 499 or fewer containing measures on a dependent variable y and n independent variables x1 x2 where the total number of variables is less than or equal to 30 the stepwise linear regression analysis will determine the coefficients of a linear equation of the form y by bx b x bX 1 2 nn which best approximates the observations in the least squares sense The independent variables xj X are entered into the equation on the basis of a variance criterion supplied by the user which enables the program to determine which variable makes the greatest improvement in goodness of fit Si
130. tion switch 0 No transformation 1 Transformation 15 16 Output raw cross products matrix 0 No 1 Print 2 Print and punch 3 Punch 17 18 Output adjusted cross products matrix 0 No 1 Print 2 Print and punch 3 Punch 19 20 Output variance covariance matrix 0 No 1 Print 2 Print and punch 3 Punch Not pertinent when matrix input is used When correlation matrix input is used matrices are not available for output Column Meaning 21 22 Output correlation matrix 0 No 1 Print 2 Print and punch 3 Punch 23 24 Output predicted values 1 Print predicted values for last step only 0 Do not print predicted values k Print predicted values for models containing k or more independent variables 25 26 Output steps of regression 0 Print no regression steps Exit after correlation analysis k Print all steps for models containing k or more independent variables 27 28 Pooling option see sections 2 5 3 and 2 1 4 Zero Add matrices with ID 1 Nonzero Subtract matrices with ID 1 29 30 Number of dependent variable 31 34 Variance criterion to remove variables 35 38 Variance criterion to enter variables 39 44 Tolerance for colinearity VARIABLE NAME CARD Figure 4 In the multiple regression program there are a number of matrix printouts that the user may request The variables in the matrix may be assigned a four character name to aid in the identification
131. tion vectors necessary scaling constants from the previous analysis are also read It is necessary to keep all punched output from analyses in the order in which it was punched for later input In effect the solution vectors represent the coefficients of th fitted polynomial Hence this option is to be used when it is desired to use the fitted polynomial to compute additional estimated values and or to compute additional derivatives for points other than those used in the initial analysis The data points are read under the secondary input type indicated in cc 15 16 If no data points are read for evaluation of y only coefficients are calculated However in this case these have already been calculated for the previous analysis If the secondary input type is the card reader previously read input which was placed on the disk is destroyed Coefficients of Fitted Polynomial cc 5 6 In the determination of the best fitting po ynomial the computation involves only the orthogonal polynomial and the three associated vectors called the polynomial solution vectors The orthogonal polynomials are generated from the solution vectors whenever it is necessary Hence the actual coefficients of the fitted polynomial are not required for evaluation of derivatives or the computation of estimated values However if desired they may be printed by punching a one 1 in this field If this field is left blank or contains a zero the coefficients are not p
132. ubstituting communalities for unity in the diagonals of the standardized matrix Possibly the best way of estimating the communalities is to start with the squared multiple correlations in the diagonal since this is a lower bound on the communalities We then perform an analysis of the data and arrive at certain factors deciding on the number of factors by taking those with latent roots gt 1 0 those factors accounting for a certain percentage of the variance or some other method We can then use the coefficients occurring in those factors to estimate new communalities iterate and proceed until the communalities converge Basically what this process amounts to is that we assume m factors and assume that they account for as much as possible of the e variance this determines the communalities and consequently the error variances But this does not mean that we have estimated the actual error variances that occur in practice We have estimated only what they would be if the number of factors is what we think it is and the error variances are minimal In other words this would suggest that in practice care should be exercised in computing a factor analysis If one has no idea of the number of factors to extract possibly the best solution is to compute a principal components solution of various numbers of factors rotate and decide which set of factors gives the best empirical meaning Then using this number of fa
133. useholder s method for the solution of the algebraic eigen problem The Computer Journal 1960 vol 3 p 23 71 2 3 ANALYSIS OF VARIANCE From the experimental observations on a variable x this program will compute an analysis of variance for a complete factorial design for a maximum of four 4 factors The method used in the program is essentially that described by H O Hartley This method is particularly useful since it can be extended to accommodate a great many experimental designs The extension to other experimental designs is accomplished by a very simple procedure The program performs a factorial analysis and then allows the user to pool certain components of the analysis of variance table in accordance with the summary instructions that specifically apply to the particular design desired For example a two or three factor design can result in the following analysis of variance tables e Single classification e Two way classification with cell repetition e Randomized block with two factor treatments e Split plot e Split split plot e Three factor randomized blocks By utilizing a special report generator the user has flexibility in choosing the appropriate components to pool in forming the error term or terms to accommodate the above designs or any other similar designs Once the data is contained in storage the sum of squares is computed as follows Let A Sum of all the observations at level a n Num
134. ust be punched with an integer that is in ascending sequence for all cards in the case If sequence checking is not desired the field may be blank and may consist of one blank column H Floating Variable xq Any number may be punched in this etc point F field Decimal points are not required The remaining fields on the card are reserved for variables If there are more variables than can fit on the first card a second and a third card may be used Following the data deck the user must include a card containing a negative integer in the identification field This card signals the end of data 2 2 6 Matrix Input Output It is possible to obtain punched card output of a number of matrices see section 2 5 3 and vectors with this program This program is designed to also input some of these matrices at a later time for further analysis or processing In addition matrices from another program or source if punched in the program format may also be used as input This section is devoted to a description of various possible forms of analysis with the output options available in each program Format Description See the matrix format given under Format Description in section 2 1 4 Factor Analysis with Correlation Matrix Input The punched output option of the correlation matrix includes the punchout of the number of cases matrix 21 and means and standard deviation vectors matrix 23 This com
135. wed e Analysis of Variance A table generation feature allows output from the factorial design analysis in standard format e Orthogonal Polynomials These can be calculated for both equally spaced and unequally spaced intervals Derivatives are calculated on option Scaling of input is allowed on option CONTENTS CHAPTER 1 GENERAL OPERATING 1 CN 1 1 System Generation lae 1 2 1 2 Control Cards Sur pos uode 1 1 3 Program Pauses and Messages 4 1 4 Stacking Sequential Program Operation 4 1 5 Typewriter and Punched Card Output 5 1 6 Machine and Systems Configuration ert 9 1 7 Programming Language 5 1 8 Reference Material 1 456666 Vo V V S T E s 6 y CHAPTER 4 PROGRAMS 4 66 2s bee UE Ss cw BeOS SKS 7 2 1 Stepwise Linear 7 j 2 1 1 Summary of Output Statistics 7 2 1 2 Job Execution CEEE 8 2 1 3 Data Input 5 45275 Sn a a Sr ah 2c dite Sa ue od qoa UN 16 2 1 4 Matrix Input Output 17 2 1 5 Operating Instructions 19 2 1 6 Sample Problem 5 294 3 59 99 4o OS 21 2 2 Principal Components and Factor Analysis 31 2 2 1 Mathematics of
136. witch 0 No transformation l Transformation 15 16 Output raw cross products matrix 0 No 1 Print 2 Print and punch 3 Punch 17 18 Output adjusted cross products matrix 0 No l Print 2 Print and punch 3 Punch 19 20 Output variance covariance matrix 0 No 1 Print 2 Print and punch 3 Punch 21 22 Output correlation matrix 0 No 1 Print 2 Print and punch 3 Punch Not available when correlation matrix is used as input 53 Column 23 24 25 26 27 28 29 30 31 32 33 34 35 36 54 Meaning Factor score options 0 Do not compute factor scores 1 Compute and print factor scores 2 Compute print and punch factor scores Number of factors options 0 Do not compute factors 1 Compute factors for latent roots 21 0 only 2 Compute m factors where m is given in cc 27 28 3 Compute factors accounting jointly for no more than p percent of the total variance where p is given in ce 27 28 Constant for number of factors option if appropriate Communality options 0 Use diagonal values of correlation matrix normally unity unless otherwise specified in a given matrix 1 Use maximum absolute off diagonal element in each vector of the correlation matrix 2 Use the squared multiple correlation coefficient for each variable Rotation options 0 Do not perform any rotations 1 Perform an orthogonal rotation Varimax only 2 Perform an oblique rotat
137. with Two Treatment Factors A XB X C Here the third factor C is blocks In this case one is interested in finding out whether there are significant differences between blocks so the error term is computed from SS SS SS 58 error ac be abe thus giving an ANOVA table as follows 55 SS SS blocks SSerror 3S otal Split Plot Design A X B X C In this case let factor A main treatments B sub treatments and C blocks Then appropriate error terms are calculated as follows 3 SS SSc 3 b SB error j Pe j nT The ANOVA table becomes Main treatment A SS Blocks C SS Error a SS Subtreatment B SS Interaction AXB SB b Error 98 t 95 be Total SS o tal Split Split Plot Design A XB XC X D A Main treatment P Subtreatment C Sub subtreatment D Blocks Consequently there will be three separate error terms a SS SS error ac b error ae sar be c SS error D SS c SS dcb This gives the following reconstructed ANOVA table dca A C SS a SS Error a SS ac B 58 The factors in this case are Seah 16 AXB Error b D AXD BXD AXBXD Error Total SS jc SS cb SS SS dca i dcab Three Factor Randomized Blocks A X B XC X D Let factor C blocks The error term becomes SS SS error ac SS SS SS TuS more abc acd bed b TAS shed Thus givin
138. x y is also estimated and printed with the orthogonal polynomials unless the user does not elect to have them printed cc 31 32 Punch Solution Vectors Switch cc 13 14 If this field contains a one 1 the polynomial solution vectors are punched in the standard matrix format section 2 1 4 If the field contains a zero or is left blank the solution vectors are not punched To maintain maximum numerical accuracy in computing the coefficients of the fitted polynomial derivatives and estimated values the orthogonal polynomials are used for the computation However to conserve storage space the orthogonal polynomials are recomputed each time they are used The polynomial solution vectors as functions of the data points are used as parametric vectors in this computation to avoid making more than one pass through the original data In effect the solution vectors are used throughout the program to represent the coefficients of the fitted polynomials Hence if the user expects to use the polynomial to compute additional values or derivatives the solution vectors should be punched out In addition to the solution vectors the first output card includes the scaling constants required for evaluating y and derivatives at x If scaling is not performed this card is still punched and read but not used under input mode 3 Variance Criterion cc 17 26 This field must be punched with a positive floating point number of the form OXXXXXXXXX T
139. y the next analysis deck See sections 1 4 and 1 5 1 4 STACKING SEQUENTIAL PROGRAM OPERATION Stacking of jobs is permitted Each job must be a complete deck as defined in the job execution section of each program However when a program option card calls for output on the IBM 1442 Card Punch the negative identification card following the input data must be succeeded by blank cards For each matrix requested in factor analysis or regression analysis it is wise to place at least n 5 2n 2 blank cards behind the data deck where n is the number of variables processed For orthogonal polynomials n 1 blank cards should be included where n is the order of the polynomial requested When factor scores are to be punched 2n blank cards should be included where n is the number of observations processed It is advisable to place extra blank cards in the hopper because an insufficient number could result in the destruction of a part of the next analysis deck After an analysis is completed cards are read until the next monitor control card is met 1 5 TYPEWRITER AND PUNCHED CARD OUTPUT For typewriter output the 1130 Statistical System uses the same format statements as are used for the printer A user electing to use this device heavily may desire to modify output using Assembler Language routines calling on the typewriter tabulation feature All system programs excepting analysis of variance allow user selection o

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