Testing mixture SEM models

Because mixture LST models are special variants of mixture structural equation models, estimation and testing methods for this general approach can also be used to estimate and test mixture LST models (Arminger et al., 1999; Lubke & Muth´en, 2005; Muth´en, 2001; Muth´en & Shedden, 1999). In order to choose the number of latent classes, models with several latent classes can be estimated and compared. Information criteria such as the Akaike In-formation Criterion (AIC), Bayes InIn-formation Criterion (BIC), sample-size adjusted BIC (aBIC), consistent AIC (CAIC), and the Integrated Classifi-cation Likelihood BIC (ICL-BIC) can be applied to select the number of appropriate classes by choosing the solution with the lowest value of an in-formation coefficient (Lubke & Muth´en, 2005; McLachlan & Peel, 2000). The ICL-BIC is equal to the BIC plus a function of the posterior probabilities for each class. The ICL-BIC increases when the posterior probabilities are low.

In the case of mixture models, McLachlan and Peel (2000) showed that the AIC is too liberal and inconsistent when assessing the number of classes but that the BIC, consistent AIC and ICL-BIC are more appropriate. Therefore, we will restrict the goodness of fit information criteria we use to the BIC, the adjusted BIC, the CAIC and the ICL-BIC. However, the ranking of different class solutions with respect to their information criteria is only descriptive and there is no statistical test based on these coefficients. To circumvent this problem, a likelihood ratio test could be applied. However, the regularity conditions of the likelihood ratio test are not fulfilled, since, under the null hypothesis, the mixing proportions are on the boundary of the parameter space. Lo, Mendell, and Rubin (2001) proposed an adjusted likelihood ratio test (aLRT) to compare the fit of a model with c and c−1 classes. The null hypothesis of this test is that a model with c−1 classes fits the data.

Therefore, if the p value is smaller than .05, a model with c classes fits the data better than a model with c−1 classes (Lubke & Muth´en, 2005). Ano-ther solution to obtain a correct estimate of the likelihood ratio test is to use a bootstrap procedure (boot LRT). The parametric bootstrap likelihood ratio test (McLachlan & Peel, 2000) uses bootstrap samples to estimate the distribution of the log-likelihood difference test statistic. According to a si-mulation study of Nylund, Asparouhov, and Muth´en (2006) the boot LRT proved to be a valid test for deciding about the number of classes in mixture

modeling.

All these indices are useful to compare the number of classes. It is im-portant, however, to note that information coefficients as well as the aLRT and the bootstrap LRT are relative indices of fit and thus cannot determine the absolute best model. There are other approaches for evaluating mixture models that are described by McLachlan and Peel (2000).

Chapitre 4

Application of mixture LST models : Well-being

Mixture latent state-trait (LST) models apply one LST model per class.

These models may differ in their parameter estimations to fit theC subpopu-lations better. In this chapter, to illustrate the usefulness of the mixture LST model, an application to empirical data will now be presented. This data will first be analyzed with a traditional LST model and then by a mixture LST model. It will be shown that the mixture LST model yields additional infor-mations on the data. In a third step, the mixture LST model will again be extended to a mixture LST model that also allows for covariates of change.

The covariates of change are observed or latent variables that may corre-late with the occasion-specific variables. Furthermore, the results of three Monte-Carlo simulation studies will be reported that have been conducted to scrutinize whether the parameters of the model with and without covariates of change can be appropriately estimated and whether the fit criteria are reliably calculated in the applications presented. These Monte-Carlo studies were extended to several sample sizes and several occasions of measurement to investigate the influence of the sample size and the model complexity on the parameter estimate biases and the behavior of the fit coefficients. Some guidelines for determining the minimum sample size necessary for obtaining appropriate results depending on the number of occasions will be delineated.

4.1 Data description

Design and sample. The study is a reanalysis of a data set ¹ collected in Germany utilized by Eid et al. (1994). Subjects were 292 females and 211

1This data set is available on the zpid website : www.zpid.de

males between 17 and 78 years of age (mean age : 31.2 years). The scales (described below) were administered four times with a time lag of three weeks between each occasion of measurement. Complete data were available for 501 participants.

Measures. The short version of a pleasantness-unpleasantness adjective checklist (Steyer, Schwenkmezger, Notz, & Eid, 1994) was used on each oc-casion of measurement. This short checklist consists of 8 adjectives measuring the happy-sad dimension of momentary mood with a 5-point intensity scale ranging from 1 (not at all) to 5 (very much). Item are gl¨ucklich (happy), ungl¨ucklich (unhappy), wohl (well), unwohl (unwell), gut (good), schlecht (bad), zufrieden (satisfied), and unzufrieden (dissatisfied). The negative items were recoded. This scale was divided into two test halves containing four ad-jectives each and the two forms were used as observed variables (indicators) in the LST model. Each test half represents the sum of the responses to four items and runs thus from 4 to 20. Higher values indicate a more posi-tive mood. The observed variables are denoted P U_ik with i indicating the indicator and k the occasion of measurement.