Normality and GMM
Message/Author
 Jon Hill posted on Monday, July 07, 2008 - 12:39 pm
In both GMM and standard growth curve modeling the distribution of the random intercept and slope terms should be normally distributed, correct? Could a possible impetus for running a GMM be that a standard growth curve model does not meet the normality assumption but a properly specified GMM of the same data has roughly normal within class variation? This seems to make sense to me if the underlying data truly are a function of multiple subpopulations.

If my thinking here is correct, are there any tests that take into account improvement in normality for a GMM with k and opposed k-1 classes?
 Bengt O. Muthen posted on Monday, July 07, 2008 - 3:39 pm

For your second question - I am not sure I understand what you mean. Do you mean better fitting of non-normality using a GMM with k as opposed to k-1 classes? If so, the Mplus Tech13 is aimed at this, but may not be a sensitive enough test.
 Jon Hill posted on Tuesday, July 08, 2008 - 5:45 am
Thank you for your response. I guess I am asking is there a statistical test that shows that a GMM with k classes is significantly more normal (regarding the distribution of intercept and slope terms within classes) than a k-1 class model?

If a standard growth curve model (1 class) is misspecified, then the trajectories will cluster together around multiple distinct trajectories and probably not be normally distributed around some grand trajectory, right? I would think part of finding a GMM with the correct number of classes would involve increasing the classes until the within class variation is roughly normal. So I guess I am wondering if there is a test statistic that tests this?
 Bengt O. Muthen posted on Tuesday, July 08, 2008 - 9:01 am
I don't understand the phrase "test that shows that a GMM with k classes is significantly more normal...than a k-1 class model". A distribution (here you consider the distribution for growth factors) gets less normal the more classes you add, not more normal.

In any case, regarding your first question, no there is not a direct, neatly contained test of the distribution for growth factors related to mixtures. You can use a non-parametric representation of the growth factors using mixtures, as explained in my 2004 Kaplan chapter on GMM.

Regarding your last question, I now see what you mean by "more normal" within class - that's a heuristic way of describing the observed data. No test I know of is formulated that way. Instead, the way to settle on the number of classes (with within-class normality specified) is to use BIC and the Mplus BLRT.