LCGA continuous vs count
Message/Author
 Chris Greenwood posted on Monday, March 04, 2019 - 7:41 pm
Hello,

I am working with non-normal data.

I have run a LCGA two ways: (1) specifying the indicators as continuous, and (2) specifying the indicators as counts (negative binomial).

I understand that the assumption of within class normality does not apply to LCGA, as the variance of the latent growth parameters are set to 0.

However, I feel I am struggling to understand what changing the specification of the indicators has on the inner workings of the model. My main concerns are in regard to normality.

Is it that in the continuous model - the LCGA is explaining the var/covar of a normal distribution, but in the count model - the LCGA is explaining the var/covar of a negative binomial distribution?

 Bengt O. Muthen posted on Tuesday, March 05, 2019 - 2:41 pm
Within-class normality refers to the observed variables, not the latent ones.

In the non-mixture case for continuous-normal outcomes, variances and covariances (and means) are sufficient statistics and are therefore the only quantities that the model needs to fit. In the non-mixture case for count outcomes, there are no such sufficient statistics so the model needs to be fitted to raw data.

In the mixture case, there are never any such sufficient statistics but the model needs to be fitted to raw data.
 Chris Greenwood posted on Tuesday, March 05, 2019 - 3:32 pm
Thanks very much for the response.

As an extension to your comment on within-class normality, does this all apply to GMMs as well.

Specifically, I was under the impression for GMMs that the assumptions of normality were related to the distribution of the random intercept, slope, etc.

See Post 1 and 2: http://www.statmodel.com/discussion/messages/13/3364.html?1215532906

and this paper:
Kreuter, F., Muthén, B., 2008. Analyzing criminal trajectory profiles: Bridging multilevel and group-based approaches using growth mixture modeling, Journal of Quantitative Criminology.

 Bengt O. Muthen posted on Tuesday, March 05, 2019 - 5:15 pm
Yes, GMMs also assume normality within class for the outcomes. It's true that this happens when both the growth factors and the residuals are normal within class - so there isn't much of a difference here. With numerical integration with respect to continuous latent variables such as growth factors, e.g. with categorical or count outcomes, the assumption of normal latent variables is explicitly made.

Mplus also allows within-class non-normality that Mplus achieves for skew-normal and skew-t.
 Chris Greenwood posted on Tuesday, March 05, 2019 - 5:44 pm
Thanks again.

Last question (promise):

If I understand correctly when the observed indicators are specified as counts (nb; negative binomial) in the GMM, the latent growth parameters are still explicitly assumed as normal.

How does changing the specification of indicators from continuous to counts (nb) change the estimation of growth parameter variation?
Is it just that growth parameters are now estimated with an appropriate link?

Sorry if this is a broad question.

Cheers.
 Bengt O. Muthen posted on Wednesday, March 06, 2019 - 1:36 pm
First question: yes.

Last question: Yes.