Allowing correlations with residuals
Message/Author
 Jon Heron posted on Monday, May 12, 2014 - 1:51 am
Hi Bengt/Linda,

As you know, an oft-quoted assumption of growth models is that the growth factors are uncorrelated with the occasion level residuals.

We are currently considering the possibility, when modelling bodyweight, that heavier people at baseline may have weight that fluctuates to a greater extent about the population mean trajectory, whilst those lighter at baseline are more stable.

I am able to correlate the intercept with one occasion residual, but no more than this. Ideally I would have a single constrained correlation between I and all such residuals.

Any feel for whether this may be an issue with my dataset or a more fundamental problem with what I am attempting?

many thanks, Jon
 Bengt O. Muthen posted on Monday, May 12, 2014 - 4:11 pm
UG ex 3.9 uses a random coefficient to handle a heterogenous residual variance. I don't know if that idea can be used. There must be tricky identification issues given that the intercept factor is latent. Of course, one could try mixtures.
 Jon Heron posted on Tuesday, May 13, 2014 - 12:16 am
Thanks Bengt, looks promising
 Bengt O. Muthen posted on Tuesday, May 13, 2014 - 6:00 am
You may also find some ideas in our new paper that just got posted this morning as we are about to launch Mplus Version 7.2:

Asparouhov, T. & Muthén B. (2014). Structural Equation Models And Mixture Models With Continuous Non-Normal Skewed Distributions.
 Bengt O. Muthen posted on Tuesday, May 13, 2014 - 1:38 pm
I talked to Tihomir about the use of the new Mplus Version 7.2 features that I mentioned in my posting above. So here is an investigation to be made and a paper to be written:

Look at page 15 of this new paper which discusses the t-distribution (DISTRIBUTION = TDIST; in Mplus Version 7.2). Let's say that Y2 takes the role of the factor (growth factor or otherwise) and Y1 takes the role of the observed indicator of the factor. The conditional variance, that is the residual variance of Y1, is given in (34), where you see that the variance is a function of the df value "nu" as well the Y2 values, with larger residual variance values for Y2 values further away from its mean. That's what you wanted, right? The conditional mean is a function of Y2 as usual in normal theory so the mean growth is ok.

Your paper is due in a week.