Homework 3 Write Up vJMC
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This document describes analyses performed on data from the National Health and Aging Trends Study to explore dimension reduction techniques. Principal component analysis and factor analysis were used. PCA yielded the most interpretable results, identifying three components explaining 22% of variance: Balance/Mobility, Household/Family Size, and Basic Care. Factor analysis with different extraction methods also identified three factors, though interpretations varied slightly with cross-loadings. Orthogonal and oblique rotations were compared.
Homework 3 Write Up vJMC
- 1. Abstract Thispaperdescribesvariousdimensionreductiontechniquesonthe resultsof the National Healthand AgingTrendsStudy(NHATS),asurveythatis designedtounderstandthe life of agingadults. Principal ComponentAnalysis (PCA) andFactorAnalysis(FA) wereusedto exploredimensionreduction. Two Factor Analysismethodswere utilized:principal componentmethodandprincipal factormethod. Orthogonal andoblique rotationswereevaluated. Principal ComponentAnalysisissuperiortoFactor Analysisforthe dataavailable.
- 2. Data Preparation The NHATS datasetcontainsa response type variableusedtoidentify patientsansweringfor themselves. Only‘sampleperson’responses were analyzed.Table 1shows 4,510 records were analyzed. Table 1: Non-Proxy Respondents is3resptype Frequency Missing 179 Inapplicable 213 Sample Person 4510 Proxy 897 Table 1 indicatesmissingvaluesinthe dataset. Due totime constraints,onlycomplete variableswere analyzed;all variableswithmissing responses were removed. (Please seethe SAScode forproxyvariable identificationif desired). Table 2indicates 132 variablesare complete. Table 2: Variable Missing Response Count Category Freq Percent Cum Freq Cum Percent Zero 132 11.64 132 11.64% 0.0-0.05 273 24.07 405 35.71% 0.05-0.10 71 6.26 476 41.98% 0.10-0.15 18 1.59 494 43.56% 0.15-0.20 30 2.65 524 46.21% 0.20-0.25 9 0.79 533 47.00% 0.25-0.30 33 2.91 566 49.91% 0.30-0.40 10 0.88 576 50.79% 0.40-0.50 15 1.32 591 52.12% Greater0.5 543 47.88 1134 100.00% Table 3 identifies additional variables removed. Table 3: Additional Variable Removal Variable Reason Spid ID variable notappropriate is3resptype Constantnot appropriate cg3todaydat5 Date notappropriate cg3todaydat6 Date notappropriate w3anfinwgt0-w3anfinwgt56 Appearstobe resultsof an analysisonthe survey w3varstrat Appearstobe resultsof an analysisonthe survey w3varunit Appearstobe results of an analysisonthe survey
- 3. wa3dwlkadm Appearstobe resultsof an analysisonthe survey wc3dwaistadm Appearstobe resultsof an analysisonthe survey PCA and FA were performedon 67 variables. Part A: Most Interpretable Analysis, Principal ComponentAnalysis PCA yieldedthe mostinterpretable output.The primary advantage of PCA is utilizinganorthogonal rotationpreservingthe originalaxes’90degree intersection.PCA oblique rotationswere notconsidered. Proportionof variance explainedanda scree plotwere usedtodetermine the numberof principal components utilized. The the scree plotindicatesthreecomponents explaining22.08% of the variance. The typical 80% cutoff isnot achieveduntil the 34thcomponent. The marginal increasestovariance explainedafterthree components are notsignificantenoughtoincludeadditional variables. Figure 1: Scree Plot for PCA Table 4: PCA Eigenvalues of the Correlation Matrix Eigenvalue Difference Proportion Cumulative 1 8.10443241 4.26088803 0.121 0.121 2 3.84354438 0.99720043 0.0574 0.1783 3 2.84634395 0.36360199 0.0425 0.2208 4 2.48274197 0.11609628 0.0371 0.2579 5 2.36664569 0.1406731 0.0353 0.2932 6 2.22597259 0.1528232 0.0332 0.3264 7 2.07314939 0.2150905 0.0309 0.3574 8 1.85805889 0.09958468 0.0277 0.3851 9 1.75847422 0.0711056 0.0262 0.4113 10 1.68736862 0.18340598 0.0252 0.4365 11 1.50396263 0.07480332 0.0224 0.459 12 1.42915932 0.0588168 0.0213 0.4803
- 4. 13 1.37034251 0.17138215 0.0205 0.5007 14 1.19896036 0.05373733 0.0179 0.5186 15 1.14522303 0.0317402 0.0171 0.5357 16 1.11348283 0.01216687 0.0166 0.5524 17 1.10131596 0.0316017 0.0164 0.5688 18 1.06971427 0.01582539 0.016 0.5848 19 1.05388888 0.02873123 0.0157 0.6005 20 1.02515765 0.02984417 0.0153 0.6158 21 0.99531348 0.03519187 0.0149 0.6306 22 0.96012161 0.00329527 0.0143 0.645 23 0.95682633 0.02170117 0.0143 0.6593 24 0.93512516 0.02242507 0.014 0.6732 25 0.91270009 0.00799147 0.0136 0.6868 26 0.90470862 0.01428631 0.0135 0.7003 27 0.89042231 0.01045968 0.0133 0.7136 28 0.87996263 0.00503824 0.0131 0.7268 29 0.87492439 0.02732821 0.0131 0.7398 30 0.84759618 0.01264904 0.0127 0.7525 31 0.83494714 0.0121947 0.0125 0.7649 32 0.82275245 0.00774509 0.0123 0.7772 33 0.81500735 0.03093115 0.0122 0.7894 34 0.78407621 0.02142532 0.0117 0.8011 Table 5 shows the firstcomponent’sloadings.The colorcoding(Green=High;Yellow=Moderate; Red=Low) suggestsno cross-loadingswiththe othertwocomponents. Variablesloadedonthe first componentrelate tothe individual’sbalance andmovement.The firstcomponentwasnamed “Balance/Mobility”. Table 5: Loadings for Component 1 Original Variables Prin1 Prin2 Prin3 mo3dinsddevi R3 D MOVE INSIDEWITH DEVICES 0.243157 0.015635 0.030634 ba3dblftadm R3 D BALANCEFULL TANDEMADMIN 0.241131 0.02113 -0.1458 ba3dblstadm R3 D BALANCESEMI TANDEM ADMIN 0.23886 0.026577 -0.18704 sc3dbathhelp R3 D HAS HELP WHILE BATHING 0.23543 0.044858 0.000795 mo3doutdevi R3 D GO OUTSIDE USINGDEVICES 0.232603 0.025178 0.025245 ch3drchradm R3 D REPEAT CHAIRADMIN 0.232251 0.038432 -0.14685 ch3dschradm R3 D SINGLE CHAIRADMIN 0.214467 0.041418 -0.2161 ba3dblssadm R3 D BALANCESIDE BY SIDE ADMIN 0.213311 0.037197 -0.22803 mo3douthelp R3 D GO OUTSIDE USINGHELP 0.206145 0.057931 -0.00065
- 5. Table 6 showsthe secondcomponent’s loadings.Nocross-loadingappearwiththe othertwo components. Variablesloadedonthe second componentrelatetothe individual’shouseholdandfamily size.The secondcomponentwasnamed“Household/FamilySize”. Table 6: Loadings for Component 2 Original Variables Prin1 Prin2 Prin3 hh3dlvngarrg R3 D LIVINGARRANGEMENT 0.008616 0.324399 -0.08289 hh3dhshldnum R3 D TOTAL NUMBER IN HOUSEHOLD -0.0119 0.30869 -0.09222 hh3dhshldchd R3 D TOTAL CHILDREN IN HOUSEHOLD 0.029299 0.263683 -0.06513 cs3dnumchild R3 D NUMBER OF CHILDREN 0.004973 0.191259 -0.0091 cs3dnumdaugh R3 D NUMBER OF DAUGHTERS 0.014425 0.148269 -0.01082 cs3dnumson R3 D NUMBER OF SONS -0.00682 0.14089 -0.00298 Table 7 showsthe thirdcomponent’s loadings. Nocross-loadingappearwiththe othertwo components. Variablesloadedonthe third componentrelate tothe individual’s daily ability totake care of themselves.The thirdprincipal wasnamed“BasicCare”. Table 7: Loadings for Component 3 Original Variables Prin1 Prin2 Prin3 sc3ddressfdf R3 D DIFF LEVEL DRESSINGSELF 0.086347 0.0818 0.356976 sc3ddresdevi R3 D USES DEVICES WHILE DRESSING 0.089676 0.061617 0.290535 sc3dbathsfdf R3 D DIFF BATHINGSELF NO HELP 0.011438 0.05149 0.28953 sc3dtoilsfdf R3 D DIFF LEVEL TOILETING SELF 0.095177 0.08114 0.265979 mo3dbedsfdf R3 D GET OUT OFBED 0.08032 0.072452 0.259304 sc3deatsfdf R3 D DIFF EATINGBY SELF WO HELP 0.108138 0.071426 0.238821 In summary,the PCA generatesorthogonal components.Three componentswereselectedtorepresent the 67 variables.These three componentsexplainabout22% of the variance inthe 67 variables.Since addingadditional components marginallyincreasedthe variance proportion explained, three components were utilized, supportedbythescree plotabove.No cross-loadingsoccurredacross components.Withnocross-loadings,the constructsallowedforsimplecomponentinterpretation. .
- 6. Part B: Compare and Contrast the Various Techniques As statedinthe abstract, several dimensionreduction techniques were explored.Thissectionseeksto explainsimilaritiesanddifferencesof the varioustechniques. Figure 2 throughFigure 4 comparesthe scree plotsforthe PCA andthe FA withtwodifferentextraction methods. Figures2and 3 are identical..Figure3differs somewhat withlessdistinctionbetween factors three andfour. . Consequently,the PCA andthe FA (principal componentmethod)agree,extracting three components/factors.The FA (principalfactormethod) extractedtwocomponents.Note thatthe issue of proportionof variance explainedissimilarto the PCA for both FAs (see appendixproportion of variance explainedtablesifdesired). Figure 2: Scree Plot for PCA
- 7. Figure 3: Scree Plot for FA (Principle Component Method)
- 8. Figure 4: Scree Plot for FA (Principle Factor Method) Each technique identified similarconstructs.However,the constructs varieddependingonanalysistype (PCA or FA) and the numberof componentsselected(the FA with3factorshad slightlydifferent constructsthan the FA with2 factors). The FA (principal components method) withanorthogonal rotationresultedinthree factorsextracted. Factors one and twohad three variablesthatwere cross-loaded(R3DMOVE INSIDEWITH DEVICES,R3
- 9. D GO OUTSIDE USING DEVICES,andR3 D HASHELP WHILE BATHING) makingfactor namingsomewhat difficult. Table 8 shows factorloadings. Respectively,factornamesare “Balance/Mobility”,“BasicCare”, and “LivingSituation”. The FA (principal components method) withanoblique rotationresultedinthree factorsextracted. Factors one and three hadthree variablesthatwere cross-loaded(R3DMOVE INSIDEWITH DEVICES,R3 D GO OUTSIDE USING DEVICES,andR3 D HASHELP WHILE BATHING) makingfactor namingsomewhat difficult. giventhe constructsappearinginbothfactors. Table 9 shows factor loadings. Respectively, factor namesare “Balance/Mobility”,“LivingSituation”,and“BasicCare”. Orthogonal andoblique rotationsyieldedcross-loadingswithsimilarunderlyingfactorconstructs, althoughthe orderchanges (see the factornamesforfactors 2 and 3 on each rotation). The FA underlyingconstructs are similartoPCA’sexceptfor“Household/FamilySize”(PCA) and“Living Situation”(FA). Table 8: Factor Analysis (Principal Component Method) Varimax Rotation Loadings Principal Component Method Varimax: Rotated Factor Pattern Variables Factor1 Factor2 Factor3 ba3dblstadm R3 D BALANCESEMI TANDEM ADMIN 0.73954 0.10285 0.08479 ba3dblssadm R3 D BALANCESIDE BY SIDE ADMIN 0.72131 0.01224 0.04049
- 10. ch3dschradm R3 D SINGLE CHAIRADMIN 0.71414 0.03236 0.03521 ba3dblftadm R3 D BALANCEFULL TANDEMADMIN 0.70476 0.16156 0.10325 ch3drchradm R3 D REPEAT CHAIRADMIN 0.69017 0.15415 0.0642 sc3dbathhelp R3 D HAS HELP WHILE BATHING 0.56182 0.36775 0.07781 ba3dblopadm R3 D BALANCE1 LEG OPEYE ADMIN 0.54851 0.15885 0.11217 mo3dinsddevi R3 D MOVE INSIDEWITH DEVICES 0.54333 0.40873 0.14323 mo3doutdevi R3 D GO OUTSIDE USINGDEVICES 0.52657 0.38956 0.11702 mo3dbedhelp R3 D HELP TO GET OUT OF BED 0.51139 0.26718 -0.01122 mo3douthelp R3 D GO OUTSIDE USINGHELP 0.49874 0.32787 0.03261 sc3dtoilhelp R3 D HAS HELP WHILE TOILETING 0.49039 0.264 0.02888 mo3dinsdhelp R3 D MOVE INSIDEWITH HELP 0.46837 0.34108 0.02566 mo3dbeddevi R3 D DEVICE USE 2 GET OUT OF BED 0.44116 0.33094 0.0561 Variables Factor1 Factor2 Factor3 sc3ddressfdf R3 D DIFF LEVEL DRESSINGSELF -0.10711 0.66025 -0.03818 sc3ddreshelp R3 D HAS HELP WHILE DRESSING 0.23071 0.61092 -0.0569 sc3ddresdevi R3 D USES DEVICES WHILE DRESSING -0.04339 0.56398 -0.00829 sc3dtoilsfdf R3 D DIFF LEVEL TOILETING SELF -0.00185 0.54591 -0.04537 sc3deatsfdf R3 D DIFF EATINGBY SELF WO HELP 0.05082 0.52311 -0.02237 mo3dbedsfdf R3 D GET OUT OFBED -0.03287 0.51102 -0.04025 mo3dinsdsfdf R3 D MOVE INSIDESELF 0.10391 0.51095 0.00383 sc3dbathsfdf R3 D DIFF BATHINGSELF NO HELP -0.22798 0.44272 -0.04332 mo3dinsddevi R3 D MOVE INSIDEWITH DEVICES 0.54333 0.40873 0.14323 mo3doutdevi R3 D GO OUTSIDE USINGDEVICES 0.52657 0.38956 0.11702 sc3dbathhelp R3 D HAS HELP WHILE BATHING 0.56182 0.36775 0.07781 Principal Component Method Varimax: Rotated Factor Pattern Variables Factor1 Factor2 Factor3 r3dresid R3 D RESIDENTIALCARE STATUS 0.30321 0.07174 0.72094 r2dresid R2 D RESIDENTIALCARE STATUS 0.17627 0.05828 0.67861 fl3hotype R3 F RE HT TYPE OF HOME 0.22451 0.05105 0.65303 r1dresid R1 D RESIDENTIALCARE STATUS 0.09795 0.02918 0.61536 fl3structure R3 F RE STRUCTURE OF SP DWELLING 0.17103 0.09872 0.6075 re3dresistrct R3 D RESIDNCEPHYSICAL STRCTUR 0.14559 0.0999 0.41032 Table 9: Factor Analysis (Principal Component Method) HK rotation Loadings Variables Factor1 Factor2 Factor3 ba3dblstadm R3 D BALANCESEMI TANDEM ADMIN 0.74059 0.17125 0.15287 ba3dblftadm R3 D BALANCEFULL TANDEMADMIN 0.72626 0.19045 0.20903
- 11. ch3drchradm R3 D REPEAT CHAIRADMIN 0.70745 0.1496 0.20072 ba3dblssadm R3 D BALANCESIDE BY SIDE ADMIN 0.69314 0.11826 0.06132 ch3dschradm R3 D SINGLE CHAIRADMIN 0.69195 0.11384 0.08092 sc3dbathhelp R3 D HAS HELP WHILE BATHING 0.6499 0.16619 0.40505 mo3dinsddevi R3 D MOVE INSIDEWITH DEVICES 0.64906 0.23228 0.44452 mo3doutdevi R3 D GO OUTSIDE USINGDEVICES 0.62557 0.203 0.42431 Variables Factor1 Factor2 Factor3 r3dresid R3 D RESIDENTIALCARE STATUS 0.36014 0.75263 0.09064 r2dresid R2 D RESIDENTIALCARE STATUS 0.23234 0.69604 0.06864 fl3hotype R3 F RE HT TYPE OF HOME 0.27433 0.67528 0.06478 fl3structure R3 F RE STRUCTURE OF SP DWELLING 0.23447 0.62819 0.10879 r1dresid R1 D RESIDENTIALCARE STATUS 0.14471 0.62268 0.03441 re3dresistrct R3 D RESIDNCEPHYSICAL STRCTUR 0.1969 0.43012 0.10868 Variables Factor1 Factor2 Factor3 sc3ddressfdf R3 D DIFF LEVEL DRESSINGSELF 0.09224 0.0027 0.65149 sc3ddreshelp R3 D HAS HELP WHILE DRESSING 0.39782 0.0164 0.62536 sc3ddresdevi R3 D USES DEVICES WHILE DRESSING 0.12627 0.03156 0.55973 sc3dtoilsfdf R3 D DIFF LEVEL TOILETING SELF 0.15785 -0.00216 0.54462 sc3deatsfdf R3 D DIFF EATINGBY SELF WO HELP 0.20279 0.02447 0.5254 mo3dinsdsfdf R3 D MOVE INSIDESELF 0.25152 0.05516 0.51684 mo3dbedsfdf R3 D GET OUT OFBED 0.11827 -0.00316 0.50768 mo3dinsddevi R3 D MOVE INSIDEWITH DEVICES 0.64906 0.23228 0.44452 sc3dbathsfdf R3 D DIFF BATHINGSELF NO HELP -0.08805 -0.03245 0.42623 mo3doutdevi R3 D GO OUTSIDE USINGDEVICES 0.62557 0.203 0.42431 sc3dbathhelp R3 D HAS HELP WHILE BATHING 0.6499 0.16619 0.40505 The FA(principal factormethod) withanorthogonal rotationresultedintwofactorsextracted.Nocross- loadingsoccurbetweenthesetwofactors butare suspectedif more factorswere kept. Table 10 shows factor loadings. Respectively, factornames are “Balance/Mobility/BasicCare”and“LivingSituation”.
- 12. The FA (principal factormethod) withanoblique rotationresultedintwofactorsextracted.Nocross- loadings appearacross the factors,but are suspectedif more factorswere kept. Table 11 shows factor loadings. Respectively,factornamesare “Balance/Mobility/BasicCare”and“LivingSituation”. Giventhe FA (principal factormethod) results,nocrossloadingsorfactornames changes (underlying constructs) occur fromrotationto rotation.Generally,the same three underlyingconstructsfrom FA (principal components method).The firstcomponentforFactorAnalysis (principal factormethod) appearsto combine the “Balance/Mobility”and“BasicCare” constructs. Table 10: Factor Analysis (Principal Factor Method) Varimax Rotation Loadings Variables Factor1 Factor2 ba3dblftadm R3 D BALANCEFULL TANDEMADMIN 0.67257 0.1144 ba3dblstadm R3 D BALANCESEMI TANDEM ADMIN 0.67013 0.10253 mo3dinsddevi R3 D MOVE INSIDEWITH DEVICES 0.66544 0.12642 mo3doutdevi R3 D GO OUTSIDE USINGDEVICES 0.66313 0.10111 ch3drchradm R3 D REPEAT CHAIRADMIN 0.64799 0.07684 sc3dbathhelp R3 D HAS HELP WHILE BATHING 0.64241 0.07238 ch3dschradm R3 D SINGLE CHAIRADMIN 0.61861 0.05753 ba3dblssadm R3 D BALANCESIDE BY SIDE ADMIN 0.61376 0.06468 mo3douthelp R3 D GO OUTSIDE USINGHELP 0.60484 0.02143 mo3dinsdhelp R3 D MOVE INSIDEWITH HELP 0.56214 0.01913 mo3dbedhelp R3 D HELP TO GET OUT OF BED 0.55256 -0.00587 sc3dtoilhelp R3 D HAS HELP WHILE TOILETING 0.53497 0.03001 ba3dblopadm R3 D BALANCE1 LEG OPEYE ADMIN 0.53227 0.11461 mo3dbeddevi R3 D DEVICE USE 2 GET OUT OF BED 0.5307 0.04794 sc3ddreshelp R3 D HAS HELP WHILE DRESSING 0.52141 -0.09173 Variables Factor1 Factor2 r3dresid R3 D RESIDENTIALCARE STATUS 0.29959 0.70681 r2dresid R2 D RESIDENTIALCARE STATUS 0.18654 0.6611 fl3hotype R3 F RE HT TYPE OF HOME 0.2217 0.6414 r1dresid R1 D RESIDENTIALCARE STATUS 0.1048 0.59284 fl3structure R3 F RE STRUCTURE OF SP DWELLING 0.20215 0.55675 re3dresistrct R3 D RESIDNCEPHYSICAL STRCTUR 0.17897 0.37216 Table 11: Factor Analysis (Principal Factor Method) HK rotation Loadings Variable Factor1 Factor2 ba3dblftadm R3 D BALANCEFULL TANDEMADMIN 0.66481 0.0715 ba3dblstadm R3 D BALANCESEMI TANDEM ADMIN 0.6639 0.05958 mo3doutdevi R3 D GO OUTSIDE USINGDEVICES 0.65701 0.05861 mo3dinsddevi R3 D MOVE INSIDEWITH DEVICES 0.65602 0.0842 ch3drchradm R3 D REPEAT CHAIRADMIN 0.64489 0.03493
- 13. sc3dbathhelp R3 D HAS HELP WHILE BATHING 0.63982 0.03077 ch3dschradm R3 D SINGLE CHAIRADMIN 0.61771 0.01726 ba3dblssadm R3 D BALANCESIDE BY SIDE ADMIN 0.61188 0.02485 mo3douthelp R3 D GO OUTSIDE USINGHELP 0.60853 -0.01854 mo3dinsdhelp R3 D MOVE INSIDEWITH HELP 0.56567 -0.01803 mo3dbedhelp R3 D HELP TO GET OUT OF BED 0.55926 -0.04281 sc3ddreshelp R3 D HAS HELP WHILE DRESSING 0.53902 -0.12806 sc3dtoilhelp R3 D HAS HELP WHILE TOILETING 0.53678 -0.00516 mo3dbeddevi R3 D DEVICE USE 2 GET OUT OF BED 0.53011 0.01337 ba3dblopadm R3 D BALANCE1 LEG OPEYE ADMIN 0.52297 0.08106 Variable Factor1 Factor2 r3dresid R3 D RESIDENTIALCARE STATUS 0.21023 0.69885 r2dresid R2 D RESIDENTIALCARE STATUS 0.10195 0.65991 fl3hotype R3 F RE HT TYPE OF HOME 0.14007 0.63752 r1dresid R1 D RESIDENTIALCARE STATUS 0.02827 0.59593 fl3structure R3 F RE STRUCTURE OF SP DWELLING 0.13139 0.55274 re3dresistrct R3 D RESIDNCEPHYSICAL STRCTUR 0.13214 0.36656 In summary,the extractionmethodaffectedthe numberof factors selected.Consequently,the extractionmethodhad affectedthe underlyingconstructs.Rotationsproducedmarginal adjustmentsto results,hadminimal impactonfactorcoefficientestimates,andfailedtoaddresscross-loadingissues. In thiscase,the extractionmethodhadthe greatestimpactonthe results. Appendix Table 1A: Eigenvalues for Factor Analysis (Principal Component Method) Eigenvaluesof the Correlation Matrix: Total = 67 Average = 1 Eigenvalue Difference Proportion Cumulative 1 8.10443241 4.26088803 0.121 0.121 2 3.84354438 0.99720043 0.0574 0.1783 3 2.84634395 0.36360199 0.0425 0.2208
- 14. 4 2.48274197 0.11609628 0.0371 0.2579 5 2.36664569 0.1406731 0.0353 0.2932 6 2.22597259 0.1528232 0.0332 0.3264 7 2.07314939 0.2150905 0.0309 0.3574 8 1.85805889 0.09958468 0.0277 0.3851 9 1.75847422 0.0711056 0.0262 0.4113 10 1.68736862 0.18340598 0.0252 0.4365 11 1.50396263 0.07480332 0.0224 0.459 12 1.42915932 0.0588168 0.0213 0.4803 13 1.37034251 0.17138215 0.0205 0.5007 14 1.19896036 0.05373733 0.0179 0.5186 15 1.14522303 0.0317402 0.0171 0.5357 16 1.11348283 0.01216687 0.0166 0.5524 17 1.10131596 0.0316017 0.0164 0.5688 18 1.06971427 0.01582539 0.016 0.5848 19 1.05388888 0.02873123 0.0157 0.6005 20 1.02515765 0.02984417 0.0153 0.6158 21 0.99531348 0.03519187 0.0149 0.6306 22 0.96012161 0.00329527 0.0143 0.645 23 0.95682633 0.02170117 0.0143 0.6593 24 0.93512516 0.02242507 0.014 0.6732 25 0.91270009 0.00799147 0.0136 0.6868 26 0.90470862 0.01428631 0.0135 0.7003 27 0.89042231 0.01045968 0.0133 0.7136 28 0.87996263 0.00503824 0.0131 0.7268 29 0.87492439 0.02732821 0.0131 0.7398 30 0.84759618 0.01264904 0.0127 0.7525 31 0.83494714 0.0121947 0.0125 0.7649 32 0.82275245 0.00774509 0.0123 0.7772 33 0.81500735 0.03093115 0.0122 0.7894 34 0.78407621 0.02142532 0.0117 0.8011 Table 2A: Eigenvalues for Factor Analysis (Principal Factor Method) Eigenvalues of the Reduced Correlation Matrix: Total = 32.3955993 Average = 0.48351641 Eigenvalue Difference Proportion Cumulative 1 7.73330704 4.23896275 0.2387 0.2387 2 3.49434429 1.08954725 0.1079 0.3466 3 2.40479704 0.24482524 0.0742 0.4208
- 15. 4 2.1599718 0.10379065 0.0667 0.4875 5 2.05618115 0.18565635 0.0635 0.551 6 1.87052481 0.19085254 0.0577 0.6087 7 1.67967227 0.2414357 0.0518 0.6605 8 1.43823657 0.1215954 0.0444 0.7049 9 1.31664117 0.13323556 0.0406 0.7455 10 1.18340561 0.15368915 0.0365 0.782 11 1.02971646 0.11167301 0.0318 0.8138 12 0.91804345 0.07614242 0.0283 0.8421 13 0.84190103 0.15110322 0.026 0.8681 14 0.69079781 0.06639623 0.0213 0.8894 15 0.62440158 0.09343712 0.0193 0.9087 16 0.53096447 0.05971991 0.0164 0.9251 17 0.47124456 0.04644624 0.0145 0.9396 18 0.42479832 0.00671234 0.0131 0.9527 19 0.41808598 0.01849774 0.0129 0.9656 20 0.39958823 0.04419685 0.0123 0.9779 21 0.35539139 0.01500231 0.011 0.9889 22 0.34038908 0.02386483 0.0105 0.9994 23 0.31652425 0.03101839 0.0098 1.0092 24 0.28550586 0.01444408 0.0088 1.018 25 0.27106178 0.04748383 0.0084 1.0264 26 0.22357795 0.02711126 0.0069 1.0333 27 0.19646669 0.01686769 0.0061 1.0394 28 0.17959899 0.01702985 0.0055 1.0449 29 0.16256914 0.02428301 0.005 1.0499 30 0.13828613 0.0087325 0.0043 1.0542 31 0.12955363 0.01023608 0.004 1.0582 32 0.11931755 0.01735586 0.0037 1.0619 33 0.10196169 0.01138343 0.0031 1.065 34 0.09057826 0.00477873 0.0028 1.0678 SAS Code