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

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

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

8 Variable selection in factor analysis s Exploratory analysis SEFA(Stepwise variable selection in EFA) u.ac.jp/~harada/sefa2001/stepwise/ u.ac.jp/~harada/sefa2001/stepwise/ s Confirmatory analysis SCoFA(Stepwise Confirmatory FA) u.ac.jp/~harada/scofa/input.htmlhttp://koko16.hus.osaka- u.ac.jp/~harada/scofa/input.html

9 Input Data s What SEFA or SCoFA 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 WebPage for input

12 WebPage for input

13 24 Psychological variables: Exploratory analysis

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17 24 Psychological variables: Confirmatory analysis

18 Specify factor loading matrix

19 Original Model (p=24)

20 P-values for 24 models

21 X3-deleted Model (p=23)

22 X3,X11-deleted Model (p=22)

23 Final results s EFA Chi-square=227.14(186), P-value=0.021 Delete X11 Chi-square=190.01(176), P-value=0.107 s CFA Chi-square=301.83(231), P-value=0.001 Delete X3, X11 Chi-square=220.17(189), P-value=0.060

24 Theory of SEFA and SCoFA s Obtain estimates for a current model s Construct predicted chi-square for each one-variable-deleted model using the estimates, without tedious iterations s 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 T2 Note: This is Browne’s (Browne 1982) statistic of goodness-of-fit using general estimates

28 Summary 1 s We introduced goodness-of-fit as a criteria for variable selection in factor analysis s You can easily access the programs on the internet SEFA(Stepwise variable selection in EFA) u.ac.jp/~harada/sefa2001/stepwise/ u.ac.jp/~harada/sefa2001/stepwise/ SCoFA(Stepwise Confirmatory FA) u.ac.jp/~harada/scofa/input.htmlhttp://koko16.hus.osaka- u.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 They will be useful for many situations including scale construction s High communality variables can be inconsistent

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