1 International Symposium on Structural Equation Modeling at

1 International Symposium on Structural Equation Modeling, at Chicago, Dec. 13 -15, 2000 Variable selection for factor analysis and structural equation models Yutaka Kano & Akira Harada Osaka University

2 SEM has come to Japan

3 SEM in Japan s Japanese Books • Toyoda (1992). CSA with SAS • Toyoda, et al. (1992). Exploring Causality: An Introduction to CSA • Kano (1997). CSA with Amos, Eqs and Lisrel • Toyoda (1998). SEM: Introductory Course • Toyoda (editor, 1998). SEM: Case Studies • Yamamoto and Onodera (editor, 1999). CSA with Amos • Toyoda (2000). SEM: Advanced Course

4 SEM in Japan s Tutorial Seminar (organized by academic society) • Behaviormetric Society of Japan • 1995, 1998, 2000 • Japan Statistical Society • 1999 • Japan Psychological Association • 1998 • Japanese Association of Educational Psychology • 1999

5 SEM in my class (graduate course) 1. What does SEM can do? • Path analysis, CFA, Multiple indicator analysis 2. How to create a program file 3. How to read an output file • Fit index, standardization, decomposition of effects

6 CFA and model modification 4. • • Hypotheses on loadings Analysis of MTMM matrix LM and Wald tests MIMIC model Extended models 5. • • Mean structure model Multi-sample analysis with mean structure Model with binary independent variables

7 6. Other useful models • Analysis of experimental data with SEM • • Longitudinal data and 3 -mode data analysis • • 7. Latent curve model Additive model, direct-product model, PARAFAC Other topics • • 8. Anove, Ancova, Manova, Latent mean analysis EFA versus CFA Cautionary notes on causal analysis Improper solution Variable selection Software • LISREL, EQS, AMOS, CALIS, SEPATH, etc

8 Variable selection in factor analysis s Exploratory analysis • SEFA(Stepwise variable selection in EFA) • http: //koko 15. hus. osakau. ac. jp/~harada/sefa 2001/stepwise/ s Confirmatory analysis • SCo. FA(Stepwise Confirmatory FA) • http: //koko 16. hus. osakau. ac. jp/~harada/scofa/input. html

9 Input Data s What SEFA or SCo. FA needs are • • correlation matrix sample size the number of variables the number of factors • and Internet!!

10 Illustration s Data • 24 Psychological variables • p=24, n=145, k=4 • Joreskog(1978, Psychometrika) • Analyzed it with EFA and CFA • EFA…. Chi-square=227. 14, P-value=0. 021 • CFA…. Chi-square=301. 83, P-value=0. 001

11 Web. Page for input

12 Web. Page for input

13 24 Psychological variables: Exploratory analysis

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17 24 Psychological variables: Confirmatory analysis

18 Specify factor loading matrix

Original Model (p=24) 19

P-values for 24 models 20

X 3 -deleted Model (p=23) 21

X 3, X 11 -deleted Model (p=22) 22

23 Final results s EFA • Chi-square=227. 14(186), P-value=0. 021 • Delete X 11 • Chi-square=190. 01(176), P-value=0. 107 s CFA • Chi-square=301. 83(231), P-value=0. 001 • Delete X 3, X 11 • Chi-square=220. 17(189), P-value=0. 060

24 Theory of SEFA and SCo. FA s s s Obtain estimates for a current model Construct predicted chi-square for each one-variable-deleted model using the estimates, without tedious iterations We will take a sort of LM approach

25 Known quantities and goal

26 Basic idea We construct T 02’ as LM test

27 Final formula for T 2 Note: This is Browne’s (Browne 1982) statistic of goodness-of-fit using general estimates

28 Summary 1 s s We introduced goodness-of-fit as a criteria for variable selection in factor analysis You can easily access the programs on the internet • SEFA(Stepwise variable selection in EFA) • http: //koko 15. hus. osakau. ac. jp/~harada/sefa 2001/stepwise/ • SCo. FA(Stepwise Confirmatory FA) • http: //koko 16. hus. osakau. ac. jp/~harada/scofa/input. html

29 Summary 2 s They print predicted values of fit indices for each one-variable-deleted model [one-variable-added models] • Chi-square, GFI, AGFI, CFI, IFI, RMSEA s s They will be useful for many situations including scale construction High communality variables can be inconsistent

References for variable selection s s s Kano, Y. (in press). Variable selection for structural models. Journal of Statistical Inference and Planning. Kano, Y. and Harada, A. (2000). Stepwise variable selection in factor analysis. Psychometrika, 65, 7 -22. Kano, Y. and Ihara, M. (1994). Identification of inconsistent variates in factor analysis. Psychometrika, Vol. 59, 5 -20. 30