Factor mixture analysis in Mplus

I'm using factor mixture analysis (FMA) for the estimation of latent classes (latent categories) and latent traits (latent dimensions) in the same model. I think it's great that this model provides an opportunity for combining factors and classes. Muthen even goes so far as to say that performing a Latent Class Analysis (LCA) or Factor Analysis (FA) is actually performing a restricted Factor Mixture Analysis (FMA), because in these models, the dimensional, or categorical, nature of the latent variables are not taken into account.

I was using the paper "MODELS AND STRATEGIES FOR FMA" on the statmodel website, to see how one would go about fitting such a model. It provides much more practical details about how to do a factor mixture analysis than most of the Muthen papers on FMA, which is great. At the same time, they leave some issues unresolved, and some issues just made me wonder. What follows are the questions / problems I came across, and my opinion on these issues.




Latent (co)variances
While their fmm-2 to fmm-5 models allow for the estimation of factor covariance matrices, only in fmm-3 to fmm-5, covariances between latent traits are modelled. I have been wondering for some time, why only class specific factor variances are modelled in fmm-2, while factor covariances are fixed to zero. The answer is simple and straightforward: by definition, latent classes do not permit covariances between the variables they explain, within classes. Covariances between the latent factors are supposed to be explained by the classes, so no covariations is allowed. However, variation on the latent traits within classes is allowed, by estimating variances for the factors, within class.


Prepare to wait and wait and wait

Estimating (c0)variances for the latent traits requires numerical integration. This is invoked by adding the line "ALGORITHM = INTEGRATION" in the ANALYSIS command. Be prepared to wait a long time: it may take very long, depending on you sample size, number of observed variables, and number of latent variables. If you want a faster, but less precise solution, add the line "INTEGRATION = STANDARD(7)" (the default is 15). This should speed up things.

Obviously, more factors and more classes take more time. My model, consisting of two factors and 28 binary indicators, running on an HP probook 6550b in Mplus 6, took 24 hours to complete the estimation of a four class model (with the default precision, using 100 random starts and 10 final stage iterations). And then some authors advice to obtain likelihood ratio tests for the k vs. k-1 model comparison. I simply gave up, this would take me a week.


Comparing FMA results to FA or LCA results

I you want to compare the outputs of an exploratory (EFA) or confirmatory factor analysis (consisting of only latent traits, not latent classes) to an FMA model, you may want to obtain values for AIC, BIC and SSABIC to be able to compare an exclusively dimensional model to a factor mixture model. In order to obtain these, use ML estimation (or a robust variation of it: MLM, MLMV or MLR), by typing the line "ESTIMATOR = ML;" in the ANALYSIS command.

How many factors, and how many classes should we fit to the data? The authors don't seem to have an answer, either. In the introduction, they seem to argue that the number of factors and classes of the best fitting models, when only LCA of only FA is applied to the data should be taken as an upper limit for the number of factors and classes in an FMA. Later on however, they note "Because a large number of factors computationally intensive to fit with the already complicated FMM's, the fitting of models stopped with two factors." Yes, and then we didn't even try to obtain the Lo-Mendell-Rubin Likelihood ratio test (TECH 11), or the bootstrapped Likelihood ratio test (TECH 14). I'd suggest you just inspect AIC, BIC and SSABIC values, and especially: check if the model makes sense! A class solution which includes near zero proportions is not a useful solution, so try a solution with less classes.


Different loadings and tresholds (or intercepts) for each class?

Why would one want to estimate different loadings and treshold (or intercepts) in every class? It may result in better model fit, but how are we to interpret the resulting factorial structure? It seems in such a case, we are measuring a different latent variable in every class. Personally, I feel comparing models with non-invariant loadings and tresholds (or intercepts) across classes, turns the factor mixture model into an analysis of measurement invariance across latent classes, which seems to be an altogether different subject.