Empirical comparison of methods for estimating latent variable interactions

114 Pages·2000·3.7 MB·English

by  Moulder, Bradley C

ANEMPIRICALCOMPARISONOFMETHODSFORESTIMATINGLATENT
VARIABLEINTERACTIONS
By
BRADLEYC.MOULDER
ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLINPARTIAL
FULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOF
DOCTOROFPHILOSOPHY
UNIVERSITYOFFLORIDA
2000
ACKNOWLEDGMENTS
Iwouldliketothanksomeofthepeoplewhohaveprovidedsupportformy
educationwhichhavemadethispaperpossible. Iwouldfirstliketothankmywife,Kim,
whohasmadesomanysacrificesoverthelastseveralyears. 1haveappreciatedeachand
everyone. Iwouldalsoliketothankmyparentswhosewordswerealwayssupportive
andwhoseexamplesattesttoabeliefinthevalueofeducationthatwasneverfarfrom
mymind.
Iwouldliketothankallthemembersofmycommittee. Eachofthemis
exceptionalandIhavetakensomethingfromeachoftheirexamples. Iwouldliketo
thankJamesAlgina. Hisloveofresearchisinfectiousandhisexampleasbotha
researcherandinstructorsetatrulyhighmark. 1wouldliketothankMargaretBradley.
TheexperiencesIhadworkingwithherhavebeenavaluableresourceprovidinga
groundinginbothpracticalanalysisandexperimentaldesign. Iwouldliketothank
LindaCrocker. IntheyearsIworkedwithher,Iverymuchappreciatedthatshealways
actedasthoughIwasastudentfirst. Ialsoappreciatethemanyopportunitiesshemade
availabletome. IwouldalsoliketothankDavidMiller. Hisveryapproachablemanner
hasmadeavaluableresourceofthebreadthofknowledgeandexperiencehepossesses.
ii
TABLEOFCONTENTS
page
ACKNOWLEDGMENTS ii
LISTOFTABLES v
LISTOFFIGURES vi
INTRODUCTION 1
MeasurementError ^
MultipleRegressionandStructuralEquationModeling 2
LatentVariableInteraction 4
IndicantProductApproaches 4
Two-StepApproaches 13
Two-StageLeastSquaresApproach 15
StatementoftheProblem 18
METHOD 19
DataSimulation 19
EffectSize 19
SquaredMultipleCorrelation 19
CorrelationAmongVariables 20
ReliabilityofObservedVariables 20
SampleSize 20
SimulationProper 21
ParameterRecovery 26
ComparisonofMethods 28
RESULTS '. '. 30
Bias 32
StandardErrorRatios 33
Gamma4 33
Psi 39
Gamma1andGamma2 41
Gamma3 47
iii
MeanSquaredError ^
Gamma4 51
Psi 53
Gamma1,2,and3 53
Type1ErrorRateandPower 54
Gamma4 54
Gamma 62
1
Gamma2 63
Gamma3 65
FitStatistics 65
Chi-squaredtestofExactFit 65
ComparativeFitIndex 68
Non-NormedFitIndex 69
StandardizedRootMeanSquaredResidual 70
DISCUSSION 73
Fit 75
HypothesisTesting 75
ConfidenceIntervals 76
Conclusion 77
Limitations 77
APPENDIX SUPPLEMENTARYTABLES 79
REFERENCES 98
:
BIOGRAPHICALSKETCH 101
iv
LISTOFTABLES
Table page
1. KennyandJuddParametersandEstimates 6
2. PerformanceoftheKenny-JuddModelintheJaccardandWanStudy 7
3. Yang-JonssonParametersandEstimates 10
4. ComparisonofParametersandEstimatesforJoreskog-YangApproaches 13
5. Ping'sEstimatesoftheKenny-JuddParameters 14
6. BoUen'sEstimatesoftheKenny-JuddParameters 18
7. SimulationParameterValues 25
8. ProportionsofMeanSquareandVarianceComponentSumfor74
StandardErrorRatios 34
9. ProportionsofMeanSquareandVarianceComponentSumfor\j/
StandardErrorRatios 39
10. ProportionsofMeanSquareandVarianceComponentSumfor73
StandardErrorRatios 48
11. ProportionsofMeanSquareandVarianceComponentSumfor73Power 55
12. ProportionsofMeanSquareandVarianceComponentSumfor74Power 56
13. ProportionsofMeanSquareandVarianceComponentSumfor71
Type1ErrorRate 62
14. ProportionsofMeanSquareandVarianceComponentSumfor72
Type1ErrorRate 64
15. ProportionsofMeanSquareandVarianceComponentSumfor73
Power 66
16. ProportionsofMeanSquareandVarianceComponentSumfor 67
17. ProportionsofMeanSquareandVarianceComponentSumforCFI 68
18. ProportionsofMeanSquareandVarianceComponentSumforNNFI 69
19. ProportionsofMeanSquareandVarianceComponentSumforStandardized
RootMeanSquaredResidual 71
20. OverviewofEffects 74
21. BiasinEstimatesofGamma4 79
22. StandardErrorRatiosforGamma4 82
23. StandardErrorRatiosforPsi 85
?4.,^ , RootMeanSquaredErrorforvj/ 88
25. Type1ErrorRateforGamma4 90
Type1ErrorRateforGamma1 92
27. Type1ErrorRateforGamma2 95
vi
LISTOFFIGURES
Figure ^ ms.
1. BiasinEstimatesofGamma4 33
2. StandardErrorRatiosbyReliabilityforGamma4:RobustStandardErrors...35
3. StandardErrorRatiosbyReliabilityforGamma4:OrdinaryStandardErrors
35
4. StandardErrorRatiosbyMultipleCorrelationforGamma4:RobustStandard
Errors 36
5. StandardErrorRatiosbyMultipleCorrelationforGamma4:OrdinaryStandard
Errors 37
6. StandardErrorRatiosbySampleSizeforGamma4:RobustStandardErrors
38
7. StandardErrorRatiosbySampleSizeforGamma4:OrdinaryStandardErrors
38
8. StandardErrorRatiosbyReliabilityforPsi:RobustStandardErrors 40
9. StandardErrorRatiosbyReliabilityforPsi:OrdinaryStandardErrors 41
10. StandardErrorRatiosbyReliabilityforGamma1~ExpandedView:Robust
StandardErrors 42
11. StandardErrorRatiosforGamma2~ExpandedView:RobustStandardErrors
43
12. StandardErrorRatiosbyReliabilityforGamma1: RobustStandardErrors
43
13. StandardErrorRatiosforGamma2:RobustStandardErrors 44
vii
14. StandardErrorRatiosbyReliabilityforGamma1:OrdinaryStandardErrors
45
15.
16. StandardErrorRatiosforGamma2:OrdinaryStandardErrors 45
17. StandardErrorRatiosbyp]„,.,. forGamma1 46
18. StandardErrorRatiosbyyO,i . forGamma2 46
19. StandardErrorRatiosbyReliabilityforGamma3:RobustStandardErrors..48
20. StandardErrorRatiosbyReliabilityforGamma3:RobustStandardErrors..49
21. StandardErrorRatiosbyReliabilityforGamma3:OrdinaryStandardErrors49
21. StandardErrorRatiosbyReliabilityforGamma3:OrdinaryStandardErrors50
22. StandardErrorRatiosbyp]„^ forGamma3 51
23. MeanSquaredErrorsforGamma4bySampleSize 52
24. MeanSquaredErrorsforGamma4byProcedure 52
25. MeanSquaredErrorsforGamma1andGamma2 53
26. MeanSquaredErrorsforGamma3 54
27. Type1ErrorRateforGamma4byMATRIX 55
28. Powerby ,andNforGamma4:RobustStandardErrors 57
29. Powerbyp,,,andNforGamma4:OrdinaryStandardErrors 59
30. PowerbyaL.^.^jandNforGamma4:RobustStandardErrors 60
31. PowerbyaL.^.cj^"'^^fo'"Gamma4:OrdinaryStandardErrors 61
32. Type1ErrorRateUsingRobustandOrdinaryStandardErrorsforGamma1
63
33. Type1ErrorRateUsingRobustandOrdinaryStandardErrorsforGamma2
64
viii
34. Gamma3PowerfortheJoreskog-YangModelandRobustStandardErrors.66
35. Chi-SquaredbyProcedure,N,and/?,, 68
36. CFIbyPROCEDUREandp,'„, 69
37. NNFIbyp,^,^,^^ andPROCEDURE 70
38. StandardizedRootMeanSquaredResidualbyNandp„ 71
39. StandardizedRootMeanSquaredResidualbyPROCEDUREandp„ 72
40. StandardizedRootMeanSquaredResidualbyMATRIX,PROCEDURE,and
Samplesize 72
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AbstractofDissertationPresentedtotheGraduateSchoolinPartialFulfillmentofthe
RequirementsfortheDegreeofDoctorofPhilosophy
ANEMPIRICALCOMPARISONOFMETHODSFORESTIMATINGLATENT
VARIABLEINTERACTIONS
By
BradleyC.Moulder
December2000
Chair:JamesJ.Algina
MajorDepartment:EducationalPsychology
Researchersinthesocialsciencesandelsewhereusemultipleregressionto
evaluaterelationshipsamongobservedvariables. Insimplelinearregression,itiswell
knownthatthepresenceofmeasurementerrortendstodecreasepowerintestsoftheP
coefficient. Theinfluenceofmeasurementerrorbecomesmoretroublesomewith
multipleregressionaspcoefficientsareunpredictablychanged,sometimes
overestimatingandothertimesunderestimatingpopulationPs. Onesolutionforthis
problemistoevaluaterelationshipsamonglatentvariablesusingstructuralequation
modeling.
Whenresearchershypothesizethepresenceofinteractionamongvariables,this
solutionhasuntilrecentlynotbeenavailable. Thisisironicgiventheparticular
sensitivityinteractiontermshaveshowntomeasurementerror. Anumberofapproaches
tointeractionusingstructuralequationmodelinghaverecentlybeenproposed. However,
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