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

2. Ulf H. Olsson Bivariate normal distribution

3. Ulf H. Olsson Positive vs. Negative Skewness Exhibit 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.

4. Ulf H. Olsson Low vs. High Kurtosis Exhibit 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.

5. Ulf H. Olsson Non-normality Skewness Kurtosis Ordinal Scale Interval Scale

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

7. Ulf H. Olsson Making Numbers

8. Ulf H. Olsson Making Numbers

9. Ulf H. Olsson Making Numbers

10. Ulf H. Olsson Making Numbers

11. Ulf H. Olsson Making Numbers Generally

12. Ulf H. Olsson ESTIMATORS Maximum Likelihood (ML) NWLS Log Likelihood RML Generalized Least Squares (GLS) Asymptotic Distribution Free (ADF) Robust ML (Satorra-Bentler correction) Diagonally Weighted Least Squares(DWLS) Unweighted Least Squares(ULS)

13. 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.

14. Ulf H. Olsson Estimation 1) No AC provided ML or GLS 2) AC provided ML WLS (ADF) Robust ML 3) Continuous or Ordinal

15. Ulf H. Olsson Ordinal Variables In practice, observed or measured variables are often ordinal However, ordinality is often ignored and numbers such as 1,2,3, etc. representing ordered categories, are treated as continuous variables. But, this is incorrect!