The CALIS Procedure

Structural equation modeling is an important statistical tool in social and behavioral sciences. Structural equations express relationships among a system of variables that can be either observed variables (manifest variables) or unobserved hypothetical variables (latent variables). For an introduction to latent variable models, see Loehlin (2004); Bollen (1989b); Everitt (1984), or Long (1983); and for manifest variables with measurement errors, see Fuller (1987).

In structural models, as opposed to functional models, all variables are taken to be random rather than having fixed levels. For maximum likelihood (ML, the default) and generalized least squares (GLS) estimation in PROC CALIS, the random variables are assumed to have an approximately multivariate normal distribution. Nonnormality, especially high kurtosis, can produce poor estimates and grossly incorrect standard errors and hypothesis tests, even in large samples. Consequently, the assumption of normality is much more important than in models with nonstochastic exogenous variables. You should remove outliers and consider transformations of nonnormal variables before using PROC CALIS with maximum likelihood (default) or generalized least squares estimation.

Alternatively, several approaches are available to deal with the nonnormality issue. If the number of observations is sufficiently large, you can use Browne’s asymptotically distribution-free (ADF; Browne 1982) estimation method. However, there is no definite guideline for how large a sample needs to be in order to use ADF estimation. Simulation studies usually show that several thousand observations might be required. If you use maximum likelihood estimation, the Satorra-Bentler scaled chi-square test statistics and the associated sandwich-type standard error estimates (Satorra and Bentler 1994) might be a viable solution even if your data are not normal. In PROC CALIS, you can apply the Satorra-Bentler method by using the METHOD= MLSB option. In the psychometric literature, the Satorra-Bentler method is sometimes referred to as robust ML. However, PROC CALIS reserves the term "robust" for another estimation procedure. When you use METHOD=ML (default) with the ROBUST option, the iterative estimation downweights model outliers so that they have less impact on the estimation. Henceforth, this is the robust ML method that the CALIS procedure refers to.

When your data contain missing values, you might consider the full information maximum likelihood (FIML) method. Assuming that the data are missing at random (MAR; see Rubin 1976), the FIML method uses all available data to perform estimation within the maximum likelihood framework. In contrast, all other estimation methods perform listwise deletion of incomplete observations before estimation. Hence, when the data are scarce and you need to use as much information as possible, FIML estimation might provide a viable solution.

You can use the CALIS procedure to estimate parameters and test hypotheses for constrained and unconstrained problems in various situations, including but not limited to the following:

To specify models in PROC CALIS, you can use a variety of modeling languages:

Various modeling languages are provided to suit a wide range of researchers’ background and modeling philosophy. However, statistical situations might arise where one modeling language is more convenient than the others. This will be discussed in the section Which Modeling Language?.

In addition to basic model specification, you can set various parameter constraints in PROC CALIS. Equality constraints on parameters can be achieved by simply giving the same parameter names in different parts of the model. Boundary , linear , and nonlinear constraints are supported as well. If parameters in the model are dependent on additional parameters, you can define the dependence by using the PARAMETERS and the SAS programming statements .

Before the data are analyzed, researchers might be interested in studying some statistical properties of the data. PROC CALIS can provide the following statistical summary of the data:

After a model is fitted and accepted by the researcher, PROC CALIS can provide the following supplementary statistical analysis:

When fitting a model, you need to choose an estimation method. The following estimation methods are supported in the CALIS procedure:

Estimation methods implemented in PROC CALIS do not exhaust all alternatives in the field. For example, the partial least squares (PLS) method is not implemented. See the section Estimation Criteria for details about estimation criteria used in PROC CALIS. Note that there is a SAS/STAT procedure called PROC PLS, which employs the partial least squares technique but for a different class of models than those of PROC CALIS. For general path analysis with latent variables, consider using PROC CALIS.

All estimation methods need some starting values for the parameter estimates. You can provide starting values for any parameters. If there is any estimate without a starting value provided, PROC CALIS determines the starting value by using one or any combination of the following methods:

Although no methods for initial estimates are completely foolproof, the initial estimation methods provided by PROC CALIS behave reasonably well in most common applications.

With initial estimates, PROC CALIS will iterate the solutions so as to achieve the optimum solution as defined by the estimation criterion. This is a process known as optimization. Because numerical problems can occur in any optimization process, the CALIS procedure offers several optimization algorithms so that you can choose alternative algorithms when the one being used fails. The following optimization algorithms are supported in PROC CALIS:

In addition to the ability to save output tables as data sets by using the ODS OUTPUT statement, PROC CALIS supports the following types of output data sets so that you can save your analysis results for later use:

The OUTEST= , OUTMODEL= , and OUTWGT= data sets can be used as input data sets for subsequent analyses. That is, in addition to the input data provided by the DATA= option, PROC CALIS supports the following input data sets for various purposes in the analysis:

The CALIS procedure uses ODS Graphics to create high-quality graphs as part of its output. You can produce the following graphical output by specifying the PLOTS= option or the PATHDIAGRAM statement:

See Chapter 21: Statistical Graphics Using ODS, for general information about ODS Graphics. See the section ODS Graphics and the PLOTS= option for specific information about the statistical graphics available with the CALIS procedure. For more information about producing customized path diagrams, see the options of the PATHDIAGRAM statement.