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The General LISREL MODEL and Non-normality

This document outlines the General LISREL model, focusing on the concepts of non-normality in statistical data, particularly skewness and kurtosis. It examines the distinction between positive and negative skewness and compares various estimators, including Maximum Likelihood, Robust Maximum Likelihood, and Diagonally Weighted Least Squares, which are recommended based on the nature of the data. Understanding these concepts is crucial for accurate data analysis in statistics, especially for financial applications like loan satisfaction and customer loyalty.

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The General LISREL MODEL and Non-normality

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  1. The General LISREL MODELand Non-normality Ulf H. Olsson Professor of Statistics

  2. Branch Loan Satisfaction Loyalty Savings The General LISREL model Ulf H. Olsson

  3. 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

  4. Bivariate normal distribution Ulf H. Olsson

  5. 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

  6. 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

  7. Non-normality (Interval Scale continuous variables) • Skewness • Kurtosis Ulf H. Olsson

  8. Making Numbers S: sample covariance θ: parameter vector σ(θ): model implied covariance Ulf H. Olsson

  9. Making Numbers Ulf H. Olsson

  10. Making Numbers Ulf H. Olsson

  11. Making Numbers Ulf H. Olsson

  12. Making Numbers Ulf H. Olsson

  13. Making Numbers Generally Ulf H. Olsson

  14. 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

  15. 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

  16. Estimation • 1) No AC provided • ML, GLS or ULS • 2) AC provided • ML • WLS (ADF) • DWLS Ulf H. Olsson

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