Estimation Methods

The TCALIS procedure provides five methods of estimation specified by the METHOD= option:

Each estimation method is based on finding parameter estimates that minimize a badness-of-fit or discrepancy function, which measures the difference between the observed sample covariance matrix and the fitted (or predicted) covariance matrix, given the model and the parameter estimates. See the section Estimation Criteria in Chapter 88, The TCALIS Procedure, for formulas, or refer to Loehlin (1987, pp. 54–62) and Bollen (1989, pp. 104–123) for further discussion.

The default is METHOD=ML, which is the most popular method for applications. The option METHOD=GLS usually produces very similar results to those produced by METHOD=ML. Asymptotically, ML and GLS are the same. Both methods assume a multivariate normal distribution in the population. The WLS method with default weight matrix, which is equivalent to the asymptotically distribution free (ADF) method, yields asymptotically normal estimates regardless of the distribution in the population. When the multivariate normal assumption is in doubt, especially if the variables have high kurtosis, you should seriously consider the WLS method. When a correlation matrix is analyzed, only the WLS can produce correct standard error estimates. However, in order to use the WLS method with the expected statistical properties, sample size must be large. Several thousands might be a minimum requirement.

The ULS and DWLS methods yield reasonable estimates under less restrictive assumptions. You can apply these methods to normal or nonnormal situations, or to covariance or correlation matrices. The drawback is that the statistical qualities of the estimates seem to be unknown. For this reason, PROC TCALIS does not provide standard errors or test statistics with these two methods.

You cannot use METHOD=ML or METHOD=GLS if the observed covariance matrix is singular. You could either remove variables involved in the linear dependencies or use less restrictive estimation methods like ULS. Specifying METHOD=ML assumes that the predicted covariance matrix is nonsingular. If ML fails because of a singular predicted covariance matrix, you need to examine whether the model specification leads to the singularity. If so, modify the model specification to eliminate the problem. If not, you probably need to use other estimation methods.

You should remove outliers and try to transform variables that are skewed or heavy-tailed. This applies to all estimation methods, since all the estimation methods depend on the sample covariance matrix, and the sample covariance matrix is a poor estimator for distributions with high kurtosis (Bollen 1989, pp. 415–418; Huber 1981; Hampel et al. 1986). PROC TCALIS displays estimates of univariate and multivariate kurtosis (Bollen 1989, pp. 418–425) if you specify the KURTOSIS option in the PROC TCALIS statement.