SEM SR MODELS REX B KLINE PSYCHOLOGY, CONCORDIA UNIVERSITY E
Latents CFA: Exogenous only L M only SR: Exogenous or endogenous L M or M L E2
Fully latent SR EX EX2 EY EY2 EY3 EY4 X X2 Y Y2 Y3 Y4 A B C DB DC E3
Partially latent SR () EX EX2 EY3 EY4 X X2 Y3 Y4 A Y C DY DC E4
Partially latent SR (2) EY EY2 EY3 EY4 Y Y2 Y3 Y4 X B C D B D C E5
identify two parts measurement structural E6
2-step (sufficient) rule measure as cfa structure as path E7
(a) Original SR (b) Respecified as a CFA EX EX 2 EY EY 2 EY 3 EY 4 EX EX 2 EY EY 2 EY 3 EY 4 X X2 Y Y2 Y3 Y4 X X2 Y Y2 Y3 Y4 A B C A B C DB DC (c) Structural A B C DB DC E8
analyze -step original model E9
EX EX2 EY EY2 EY3 EY4 X X2 Y Y2 Y3 Y4 A B C DB DC E0
analyze 2-step respecify as cfa then structural E
(a) Original SR Model (b) Respecified as a CFA Model EX EX 2 EY EY 2 EY 3 EY 4 EX EX 2 EY EY 2 EY 3 EY 4 X X2 Y Y2 Y3 Y4 X X2 Y Y2 Y3 Y4 A B C A B C DB DC E2
analyze 4-step efa, cfa, structural (2) 4 indicators/l E3
R 2 size indicators factors E4
Single indicators () EX EX2 EY3 EY4 X X2 Y3 Y4 A Y C DY DC E5
Single indicators (2) EY EY2 EY3 EY4 Y Y2 Y3 Y4 X B C D B D C E6
Single indicators. Proportion error variance ( rxx) s 2 2. Fixed parameters E7
2.30s Y EX EX2 EY EY3 EY4 X X2 Y Y3 Y4 A B C D B D C E8
2.20s X EX EY EY2 EY3 EY4 X Y Y2 Y3 Y4 A B C D B D C E9
X DY Y X2 E20
( r ) 2 s ( r YY ) 2 s Y E EY X Y A C ( r 22 ) 2 s 2 E2 DY X2 B E2
Hayduk, L. A. & Littvay, L. (202). Should researchers use single indicators, best indicators, or multiple indicators in structural equation models? BMC Medical Research Methodology, 2(59). http://www.biomedcentral.com/47-2288/2/59 E22
EAS EGS EPL EInt EJo Acculturation Scale General Status Percent Life U.S. Interpersonal Job Acculturation Stress DSt SES Depression Scale DDS Education Income EEd EInc E23
EAS EGS EPL EInt EJo Acculturation Scale General Status Percent Life U.S. Interpersonal Job Acculturation Stress DSt SES Depression DDe Education Income EEd EInc Depression Scale EDS 2.30 s DS E24
v = 8; 8(9)/2 = 39 dfm = 39 20 = 9 E25
Model Direct effects Variances Covariances Total Acc GS Acc %L SES Inc Str Job Acc Str Str Dep SES Dep Acc, SES E (7) D (2) Acc GS SES %Li 20 E26
LISREL title: shen and takeuchi (200) error term for depression scale observed variables acculscl genstat perlife educ income interper job depscale latent variables: Accultur Ses Stress Depressi correlation matrix.00.44.00.69.54.00.37.08.24.00.23.05.26.29.00.2.08.08.08 -.03.00.09.06.04.0 -.02.38.00.03.02 -.02 -.07 -..37.46.00 E27
standard deviations 3.9 3.279 2.408 3.270 3.440 2.96 3.604 3.94 sample size is 983 relationships acculscl = *Accultur genstat perlife = Accultur educ = *Ses income = Ses interper = *Stress job = Stress depscale = *Depressi! depscale as single indicator Stress = Accultur Depressi = Ses Stress E28
set error variance of depscale to 3.06! fixes the error variance of the single indicator! rxx =.70, proportion of error variance =.30! sample variance is 0.200;.30 * 0.200 = 3.06 let the errors of genstat and perlife correlate path diagram LISREL output: ND = 3 SC RS end of program E29
Covariance Matrix interper job depscale acculscl genstat perlife -------- -------- -------- -------- -------- -------- interper 8.768 job 4.055 2.989 depscale 3.499 5.295 0.202 acculscl.08.02 0.299 9.728 genstat 0.777 0.709 0.209 4.500 0.752 perlife 0.570 0.347-0.54 5.82 4.264 5.798 educ 0.775 0.8-0.73 3.774 0.858.890 income -0.306-0.248 -.209 2.468 0.564 2.54 Covariance Matrix educ income -------- -------- educ 0.693 income 3.262.834 Total Variance = 80.763 Generalized Variance = 47092.055 Largest Eigenvalue = 22.83 Smallest Eigenvalue =.972 Condition Number = 3.354 E30
Parameter Specifications LAMBDA-Y Stress Depressi -------- -------- interper 0 0 job 0 depscale 0 0 LAMBDA-X Accultur Ses -------- -------- acculscl 0 0 genstat 2 0 perlife 3 0 educ 0 0 income 0 4 Loadings, indicators Endogenous factors Loadings, indicators Exogenous factors E3
BETA Stress Depressi -------- -------- Stress 0 0 Depressi 5 0 Endogenous factor Exogenous factor GAMMA Accultur Ses -------- -------- Stress 6 0 Depressi 0 7 Exogenous factor Endogenous factor E32
PHI Accultur Ses -------- -------- Accultur 8 Ses 9 0 PSI Stress Depressi -------- -------- 2 Variances, covariances Exogenous factors Disturbance variances, Endogenous factors E33
THETA-EPS interper job depscale -------- -------- -------- 3 4 0 THETA-DELTA Errors, indicators Endogenous factors Errors, indicators Exogenous factors acculscl genstat perlife educ income -------- -------- -------- -------- -------- acculscl 5 genstat 0 6 perlife 0 7 8 educ 0 0 0 9 income 0 0 0 0 20 E34
Number of Iterations = 9 LISREL Estimates (Maximum Likelihood) LAMBDA-Y Stress Depressi -------- -------- interper.000 - - job.454 - - (0.23).863 depscale - -.000 E35
LAMBDA-X Accultur Ses -------- -------- acculscl.000 - - genstat 0.469 - - (0.046) 0.239 perlife 0.54 - - (0.043) 2.49 educ - -.000 income - - 0.68 (0.089) 7.622 E36
BETA Stress Depressi -------- -------- Stress - - - - Depressi.32 - - (0.4).634 GAMMA Accultur Ses -------- -------- Stress 0.092 - - (0.023) 3.947 Depressi - - -0.257 (0.060) -4.280 E37
Covariance Matrix of ETA and KSI Stress Depressi Accultur Ses -------- -------- -------- -------- Stress 2.775 Depressi 3.577 7.3 Accultur 0.884 0.2 9.575 Ses 0.344-0.790 3.724 4.844 E38
PHI Accultur Ses -------- -------- Accultur 9.575 (0.825).608 Ses 3.724 4.844 (0.345) (0.732) 0.80 6.63 E39
PSI Note: This matrix is diagonal. Stress Depressi -------- -------- 2.694 2.87 (0.354) (0.465) 7.607 4.705 Squared Multiple Correlations for Structural Equations Stress Depressi -------- -------- 0.029 0.693 E40
THETA-EPS interper job depscale -------- -------- -------- 5.992 7.2 3.060 (0.349) (0.564) 7.74 2.626 Squared Multiple Correlations for Y - Variables interper job depscale -------- -------- -------- 0.37 0.452 0.699 E4
THETA-DELTA acculscl genstat perlife educ income -------- -------- -------- -------- -------- acculscl 0.53 (0.699) 0.29 genstat - - 8.643 (0.420) 20.579 perlife - -.83 2.993 (0.248) (0.245) 7.388 2.204 educ - - - - - - 5.849 (0.666) 8.786 income - - - - - - - - 9.588 (0.524) 8.33 E42
Squared Multiple Correlations for X - Variables acculscl genstat perlife educ income -------- -------- -------- -------- -------- 0.984 0.96 0.484 0.453 0.90 E43
Goodness of Fit Statistics Degrees of Freedom for (C)-(C2) 6 Maximum Likelihood Ratio Chi-Square (C) 59.775 (P = 0.0000) Browne's (984) ADF Chi-Square (C2_NT) 60.075 (P = 0.0000) Root Mean Square Error of Approximation (RMSEA) 0.0528 90 Percent Confidence Interval for RMSEA (0.0389 ; 0.0673) P-Value for Test of Close Fit (RMSEA < 0.05) 0.350 Chi-Square for Independence Model (28 df) 896.404 Comparative Fit Index (CFI) 0.977 Root Mean Square Residual (RMR) 0.323 Standardized RMR 0.032 E44
Fitted Covariance Matrix interper job depscale acculscl genstat perlife -------- -------- -------- -------- -------- -------- interper 8.768 job 4.036 2.989 depscale 3.577 5.20 0.73 acculscl 0.884.285 0.2 9.728 genstat 0.45 0.603 0.099 4.494 0.752 perlife 0.479 0.696 0.4 5.83 4.264 5.798 educ 0.344 0.500-0.790 3.724.748 2.06 income 0.234 0.340-0.538 2.535.90.372 Fitted Covariance Matrix educ income -------- -------- educ 0.693 income 3.298.834 E45
Fitted Residuals interper job depscale acculscl genstat perlife -------- -------- -------- -------- -------- -------- interper 0.000 job 0.020 0.000 depscale -0.078 0.095 0.029 acculscl 0.224-0.274 0.088 0.000 genstat 0.362 0.06 0. 0.006 0.000 perlife 0.092-0.349-0.268-0.00 0.000 0.000 educ 0.43-0.382 0.059 0.050-0.890-0.26 income -0.540-0.588-0.670-0.068-0.626 0.78 Fitted Residuals educ income -------- -------- educ 0.000 income -0.036 0.000 E46
Summary Statistics for Fitted Residuals Smallest Fitted Residual = -0.890 Median Fitted Residual = 0.000 Largest Fitted Residual = 0.78 Stemleaf Plot - 8 9-6 73-4 94-2 8577-0 3874000000000 0 2356999 2 26 4 3 6 8 E47
Standardized Residuals interper job depscale acculscl genstat perlife -------- -------- -------- -------- -------- -------- interper 0.000 job 0.69 0.000 depscale -.4.83 0.798 acculscl.20 -.58 0.733 0.000 genstat.249 0.309 0.364 0.536 0.000 perlife 0.48 -.638 -.438-0.088 0.000 0.000 educ.462 -.086 0.286.535-3.480 -.484 income -.699 -.535-2.237-0.729-2.058 4.797 Standardized Residuals educ income -------- -------- educ 0.000 income -0.649 0.000 E48
Summary Statistics for Standardized Residuals Smallest Standardized Residual = -3.480 Median Standardized Residual = 0.000 Largest Standardized Residual = 4.797 Stemleaf Plot - 3 5-2 2-7655544 - 0 7600000000 0 33455678 2558 2 3 4 8 E49
E50
Standardized Solution LAMBDA-Y Stress Depressi -------- -------- interper.666 - - job 2.422 - - depscale - - 2.667 LAMBDA-X Accultur Ses -------- -------- acculscl 3.094 - - genstat.452 - - perlife.675 - - educ - - 2.20 income - -.499... E5
Completely Standardized Solution LAMBDA-Y Stress Depressi -------- -------- interper 0.563 - - job 0.672 - - depscale - - 0.836 LAMBDA-X Accultur Ses -------- -------- acculscl 0.992 - - genstat 0.443 - - perlife 0.696 - - educ - - 0.673 income - - 0.436 E52
BETA Stress Depressi -------- -------- Stress - - - - Depressi 0.825 - - GAMMA Accultur Ses -------- -------- Stress 0.7 - - Depressi - - -0.22 E53
Correlation Matrix of ETA and KSI Stress Depressi Accultur Ses -------- -------- -------- -------- Stress.000 Depressi 0.805.000 Accultur 0.7 0.026.000 Ses 0.094-0.35 0.547.000 PSI Note: This matrix is diagonal. Stress Depressi -------- -------- 0.97 0.307 E54
THETA-EPS interper job depscale -------- -------- -------- 0.683 0.548 0.30 THETA-DELTA acculscl genstat perlife educ income -------- -------- -------- -------- -------- acculscl 0.06 genstat - - 0.804 perlife - - 0.232 0.56 educ - - - - - - 0.547 income - - - - - - - - 0.80 E55
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Correlation Residuals (EQS) STANDARDIZED RESIDUAL MATRIX: ACCULSCL GENSTAT PERLIFE EDUC INCOME V V2 V3 V4 V5 ACCULSCL V.000 GENSTAT V2.00.000 PERLIFE V3.000.000.000 EDUC V4.005 -.083 -.06.000 INCOME V5 -.006 -.056.094 -.003.000 INTERPER V6.024.037.03.044 -.053 JOB V7 -.024.009 -.040 -.032 -.047 DEPSCL V8.009.0 -.035.006 -.06 INTERPER JOB DEPSCL V6 V7 V8 INTERPER V6.000 JOB V7.002.000 DEPSCL V8 -.008.008.003 E57
L M M L E58
reflect all cfa models classical theory assumes... E59
reflect exchangeable M high, positive rij unidimensional L E60
Reflective? Income SES Occupation Education Residence E6
M L form heterogeneous L no error, composite E62
exchangeable M form high, positive rij any pattern (, 0, +) E63
(a) L M (effect indicators) (b) M L (causal indicators) E E2 E3 E4 X X2 X3 X4 X X2 X3 X4 Latent variable Latent variable DLV E64
(c) M C (composite indicators) X X2 X3 X4 Composite E65
whole model is sr form identified? pls path modeling E66
Requirements Formative L emits 2 Formative L cannot receive Indirect effects... E67
EAS EGS EPL EInt EJo Acculturation Scale General Status Percent Life U.S. Interpersonal Job Acculturation Stress DSt SES Depression DDe Education Income EEd EInc Depression Scale EDS 2.30 s DS E68
construct validity form exogenous M unmodeled, error E69
( r) s 2 ( r22) s 2 2 ( r33) s 2 3 EX EY EY 3 X X2 X3 X4 X X2 X3 Latent variable A B C DLV Latent variable DLV E70
Bollen, K. A., & Bauldry, S. (20). Three Cs in measurement models: Causal indicators, composite Indicators, and covariates. Psychological Methods, 6, 265 284. Diamantopoulos, A. (Ed.). (2008). Formative indicators [Special issue]. Journal of Business Research, 6(2). Grace, J. B., & Bollen, K. A. (2008). Representing general theoretical concepts in structural equation models: The role of composite variables. Environmental and Ecological Statistics, 5, 9 23. E7
regression & pca pls composites prediction E72
combine predictors? pls no theory no identification E73
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Henseler, J., & Wang, H. (200) (Eds.) Handbook of partial least squares: Concepts, methods and applications. Berlin: Springer-Verlag. E75
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