# Introduction to Structural Equation Modeling with Latent Variables

### Estimation Methods

The CALIS procedure provides six methods of estimation specified by the METHOD= option:

 DWLS diagonally weighted least squares FIML full-information maximum likelihood GLS normal theory generalized least squares ML maximum likelihood for multivariate normal distributions MLSB maximum likelihood with Satorra-Bentler adjustments ULS unweighted least squares WLS weighted least squares for arbitrary distributions

Each estimation method is based on finding parameter estimates that minimize a discrepancy (badness-of-fit) function, which measures the difference between the observed sample covariance matrix and the fitted (predicted) covariance matrix, given the model and the parameter estimates. The difference between the observed sample mean vector and the fitted (predicted) mean vector is also taken into account when the mean structures are modeled. See the section Estimation Criteria in Chapter 29: The CALIS Procedure, for formulas, or see Loehlin (1987, pp. 54–62) and Bollen (1989, pp. 104–123) for further discussion.

The default estimation 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. If your data contain random missing values and it is important to use the information from those incomplete observations, you might want to use the FIML method, which provides a sound treatment of missing values in data. METHOD=ML and METHOD=FIML are essentially the same method when you do not have missing values (see Example 29.16 of Chapter 29: The CALIS Procedure). Asymptotically, ML and GLS are the same. Both methods assume a multivariate normal distribution in the population. The WLS method with the default weight matrix is equivalent to the asymptotically distribution-free (ADF) method, which 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 WLS can produce correct standard error estimates. However, in order to use the WLS method with the expected statistical properties, the sample size must be large. Several thousand observations might be a minimum requirement.

If you do not have a large enough sample to use the ADF method, you can consider two viable ML alternatives to deal with nonnormal distributions. Both require raw data input. The first alternative is to use a robust ML estimation by specifying the ROBUST option. This method downweights outlying observations that depart from the multivariate normal distribution during estimation. Hence, the estimation is robust to outliers. The second alternative is to use ML estimation with the Satorra-Bentler adjustments (METHOD=MLSB) for nonnormality (Satorra and Bentler 1994). This method takes the nonnormality into account in computing chi-square model fit statistics and standard errors. As a result, you can still have accurate statistical inferences about model fit and the parameter estimates even if data are nonnormal.

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 CALIS does not provide standard errors or test statistics with these two methods.

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

Except when you use robust ML, METHOD=MLSB, or METHOD=WLS, you should remove outliers and try to transform variables that are skewed or heavy-tailed. This is because all other 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 CALIS displays estimates of univariate and multivariate kurtosis (Bollen 1989, pp. 418–425) if you specify the KURTOSIS option in the PROC CALIS statement.

If you analyze raw data, you can use the RESIDUAL option to detect outliers and leverage observations. You can also use the PLOTS=CASERESID option to display various high-quality graphical plots for case-level residuals, outliers, and leverage observations. If outliers are present, you can either remove them from the analysis or consider using robust estimation that downweights the outliers during estimation.

See the section Estimation Criteria in Chapter 29: The CALIS Procedure, for details about the estimation criteria. See the section Case-Level Residuals, Outliers, Leverage Observations, and Residual Diagnostics of the same Chapter for details about case-level residual diagnostics.