The General LISREL MODEL and Nonnormality Ulf H

Bivariate normal distribution Ulf H. Olsson

Positive vs. Negative Skewness Exhibit 1 These graphs illustrate the notion of skewness. Both

Low vs. High Kurtosis Exhibit 1 These graphs illustrate the notion of kurtosis. The

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

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

ESTIMATORS • Maximum Likelihood (ML) • NWLS • Log Likelihood • RML • •

ESTIMATORS • If the data are continuous and approximately follow a multivariate Normal distribution,

Estimation • 1) No AC provided • ML or GLS • 2) AC provided

Ordinal Variables • In practice, observed or measured variables are often ordinal • However,

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

Bivariate normal distribution 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 on the left is positively skewed. The on the right is negatively skewed. 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. Ulf H. Olsson

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

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

Making Numbers Ulf H. Olsson

Making Numbers Generally 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) 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 or GLS • 2) AC provided • ML • WLS (ADF) • Robust ML • 3) Continuous or Ordinal 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! Ulf H. Olsson