The General (LISREL) SEM model

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The General (LISREL) SEM model

• Ulf H. Olsson
• Professor of statistics

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Making Numbers
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CFA and SEM
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CFA and SEM
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CFA and SEM

• No differences in estimation and testing
• Many estimators
• ML
• GLS
• ULS
• WLS
• DWLS

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Notation and Background

• Satorra Bentler, 1988, equation 4.1

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C1 for ML and GLS
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The four different chi-squares

• C1 is N-1 times the minimum value of a
  fit-function
• C2 is N-1 times the minimum value of a weighted
  (involving a weight matrix) fit function under
  multivariate normality
• C3 is the Satorra-Bentler Scaled chi-square
• C4 is N-1 times the minimum value of a weighted
  (involving a weight matrix) fit function under
  multivariate non-normality

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Asymptotic covariance matrix not provided
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Asymptotic covariance matrix provided

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ESTIMATORS

• If the data are continuous and approximately
  follow a multivariate Normal distribution, then
  the Method of Maximum Likelihood is recommended.
• If the data are continuous and approximately do
  not follow a multivariate Normal distribution and
  the sample size is not large, then the Robust
  Maximum Likelihood Method is recommended. This
  method will require an estimate of the asymptotic
  covariance matrix of the sample variances and
  covariances.
• If the data are ordinal, categorical or mixed,
  then the Diagonally Weighted Least Squares (DWLS)
  method for Polychoric correlation matrices is
  recommended. This method will require an estimate
  of the asymptotic covariance matrix of the sample
  correlations.

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Problems with the chi-square test

• The chi-square tends to be large in large samples
  if the model does not hold
• It is based on the assumption that the model
  holds in the population
• It is assumed that the observed variables comes
  from a multivariate normal distribution
• gt The chi-square test might be to strict, since
  it is based on unreasonable assumptions?!

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Alternative test- Testing Close fit
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How to Use RMSEA

• Use the 90 Confidence interval for EA
• Use The P-value for EA
• RMSEA as a descriptive Measure
• RMSEAlt 0.05 Good Fit
• 0.05 lt RMSEA lt 0.08 Acceptable Fit
• RMSEA gt 0.10 Not Acceptable Fit

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Other Fit Indices

• CN
• RMR
• GFI 1-(Fm/Fn)
• AGFI 1 (k(k1)/(2df)) (1-GFI)
• Evaluation of Reliability
• MI Modification Indices

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Nested Models and parsimony

• Modification Indices
• ?chi-sq is chi-sq with df ?df
• Nested Models
• Re-specification (Modification indices)

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RMSEA
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RMSEA
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LISREL SYNTAX