The General LISREL MODELand Non-normality

Ulf H. Olsson

Professor of Statistics

Ulf H. Olsson

The General LISREL model

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Ulf H. Olsson

Syntax• DA NI=? NO=??? MA=CM• CM FI=?????.cov• MO NX=? NY=?? NK=? NE=? BE=FU,FI• PA LX• 1 0 0• Etc…

• PA LY• 1 0• 1 0• Etc…• FR……• FI …..• pd• ou

Ulf H. Olsson

Bivariate normal distribution

Ulf H. Olsson

Positive vs. Negative SkewnessExhibit 1

These graphs illustrate the notion of skewness. Both PDFs have the same expectation and variance. The one on the left is positively skewed. The one on the right is negatively skewed.

Ulf H. Olsson

Low vs. High KurtosisExhibit 1

                                                                                              

           

These graphs illustrate the notion of kurtosis. The PDF on the right has higher kurtosis than the PDF on the left. It is more peaked at the center, and it has fatter tails.

Ulf H. Olsson

Non-normality (Interval Scale continuous variables)• Skewness• Kurtosis

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Ulf H. Olsson

Making Numbers

))(())'(( 1 sUs

S: sample covariance

θ: parameter vector

σ(θ): model implied covariance

Ulf H. Olsson

Making Numbers

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Ulf H. Olsson

Making Numbers

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Ulf H. Olsson

Making Numbers

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Ulf H. Olsson

Making Numbers

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Ulf H. Olsson

Making Numbers

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Generally

Ulf H. Olsson

ESTIMATORS

• Maximum Likelihood (ML)• NWLS• RML

• Generalized Least Squares (GLS)• Asymptotic Distribution Free (ADF)• Diagonally Weighted Least Squares(DWLS)• Unweighted Least Squares(ULS)

Ulf H. Olsson

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.

Ulf H. Olsson

Estimation

• 1) No AC provided• ML, GLS or ULS

• 2) AC provided• ML• WLS (ADF)• DWLS