Variable selection for factor analysis and structural equation models

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Variable selection for factor analysis and
structural equation models
International Symposium on Structural Equation
Modeling, at Chicago, Dec. 13-15, 2000

• Yutaka Kano Akira Harada
• Osaka University

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SEM has come to Japan
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SEM in Japan

• 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

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SEM in Japan

• 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

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SEM in my class(graduate course)

• What does SEM can do?
• Path analysis, CFA, Multiple indicator analysis
• How to create a program file
• How to read an output file
• Fit index, standardization, decomposition of
  effects

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

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• 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
• Other topics
• EFA versus CFA
• Cautionary notes on causal analysis
• Improper solution
• Variable selection
• Software
• LISREL, EQS, AMOS, CALIS, SEPATH, etc

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Variable selection in factor analysis

• Exploratory analysis
• SEFA(Stepwise variable selection in EFA)
• http//koko15.hus.osaka-u.ac.jp/harada/sefa2001/s
  tepwise/
• Confirmatory analysis
• SCoFA(Stepwise Confirmatory FA)
• http//koko16.hus.osaka-u.ac.jp/harada/scofa/inpu
  t.html

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Input Data

• What SEFA or SCoFA needs are
• correlation matrix
• sample size
• the number of variables
• the number of factors
• and Internet!!

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Illustration

• Data
• 24 Psychological variables
• p24, n145, k4
• Joreskog(1978, Psychometrika)
• Analyzed it with EFA and CFA
• EFA.Chi-square227.14, P-value0.021
• CFA.Chi-square301.83, P-value0.001

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WebPage for input
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WebPage for input
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24 Psychological variablesExploratory analysis
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24 Psychological variablesConfirmatory analysis
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Specify factor loading matrix
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Original Model (p24)
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P-values for 24 models
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X3-deleted Model (p23)
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X3,X11-deleted Model (p22)
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Final results

• EFA
• Chi-square227.14(186), P-value0.021
• Delete X11
• Chi-square190.01(176), P-value0.107
• CFA
• Chi-square301.83(231), P-value0.001
• Delete X3, X11
• Chi-square220.17(189), P-value0.060

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Theory of SEFA and SCoFA

• 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

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Known quantities and goal
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Basic idea
We construct T02 as LM test
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Final formula for T2
Note This is Brownes (Browne 1982) statistic of
goodness-of-fit using general estimates
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Summary 1

• 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//koko15.hus.osaka-u.ac.jp/harada/sefa2001/s
  tepwise/
• SCoFA(Stepwise Confirmatory FA)
• http//koko16.hus.osaka-u.ac.jp/harada/scofa/inpu
  t.html

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Summary 2

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

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References for variable selection

• 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.