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

Ulf H. Olsson 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 one on the left is positively skewed. The one 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

Ulf H. Olsson Making Numbers

Ulf H. Olsson Making Numbers

Ulf H. Olsson Making Numbers Generally

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

Ulf H. Olsson Ordinal Variables In practice, observed or measured variables are ofte 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 Estimation 1) No AC provided ML 2) AC provided and Polychoric Correlation WLS (ADF)

Ulf H. Olsson Examples Efficacy data From raw data to Model SPSS-file PSF-file