Structural Equation Models and Mixture Models With Continuous Nonnormal Skewed Distributions

Abstract

In this article we describe a structural equation modeling (SEM) framework that allows nonnormal skewed distributions for the continuous observed and latent variables. This framework is based on the multivariate restricted skew t distribution. We demonstrate the advantages of skewed SEM over standard SEM modeling and challenge the notion that structural equation models should be based only on sample means and covariances. The skewed continuous distributions are also very useful in finite mixture modeling as they prevent the formation of spurious classes formed purely to compensate for deviations in the distributions from the standard bell curve distribution. This framework is implemented in Mplus Version 7.2.

Keywords:

• skew-T distributions
• mixture models
• non-normal distributions
• skewed latent variables

Notes

¹ In Mplus language the δ parameter for a variable Y is referred to as and the degrees of freedom parameter is referred to as .

² This joint test can be done in Mplus with the Model Test command.

³ In the Mplus language, these parameters are referred to as {Y} and {η}.

⁴ Complex survey features of stratification, weights, and clustering are also handled in Mplus.

⁵ In Mplus the final estimated λ is reported at the end of the technical 8 output section and it should be monitored. In most cases a value above 0.001 is evidence that the parameter estimates are away from the boundary condition. If, however, the value becomes less than 0.001, Mplus will suggest that multiple random starting values are used to verify that the most optimal solution is reached. Even if the most optimal value is not reached, however, the model can still be interpreted and used. It might be difficult to run a huge number of starting values to search for the best solution when λ = 0 and the log-likelihood has many local maxima. The optimal estimation, understanding and handling of the case λ = 0, might still be out of reach with the current algorithm implemented in Mplus Version 7.2. What makes things even more complicated is that this case appears only for real data sets and not for simulated data; that is, it is difficult to demonstrate the λ = 0 case with a simulation study.

⁶ The test of model fit within the .same family of distributions can be obtained automatically in Mplus with the H1MODEL option of the OUTPUT command. The test of fit is not computed by default, as with standard SEM because the estimation of the H1 model might be more difficult than the estimation of the H0 model and might take longer to estimate, especially if multiple random starting values are used. Thus the HI model will be estimated only if it is requested.

⁷ In the Mplus language to obtain the sequential unrestricted model estimation one has to use the H1MODEL(sequential) option of the OUTPUT command.

⁸ We thank James Nonnemaker for providng the data to us.