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 James L. Lewis posted on Sunday, November 04, 2007 - 9:01 pm
I have a "3-level" Growth Model (2 level in MPLUS I believe) with time points nested in individuals and individuals nested in organizations. The DVs are manifest, and I am running separate models for each DV. To achieve convergence and a positive definite solution, I have to constrain the random variances associated with intercept and slope (on the individual and/or organization level) to zero in different case. I also have time varying covariates and have to sometimes set their random effects to zero in different cases on different levels to converge and/or get a positve definite solution. In a perfect situation, of course, I would like to estimate the random effect for everything (intercept, slope, and time varying covariates) in every case on every level.

My question - is it permissible to "explain variance" using a covariate on a random effect that has been constrained to zero? It seems counter-intuitive that it would be, but I have at least one stat-book type reference where it seems that this is done, and I have until recently assumed it was appropriate. For example if my random slope variance has been constrained to zero, can I still do a Slope x Covariate interaction to test for slope differences? Any help would be appreciated. Thanks.
 Linda K. Muthen posted on Monday, November 05, 2007 - 7:17 am
In a conditional model, it is the residual variance of a random slope or growth factor that is fixed to zero not the variance. It is fine to regress such a random effect on a covariate. Also, when covariates are added to the model, it can make random effects have non-zero variances whereas without covariates they have zero variances.
 peter lekkas posted on Thursday, July 05, 2018 - 1:13 am
Hi,
There is discussion that the general way of specifying the effects of time-invariant covariates might be too restrictive i.e. this ”mediated model” specification assumes that all of the effects of the time-invariant variables are captured by their impact on the growth parameters. Whereas it might be the case that these covariates directly affect the outcome variables.
Moreover, the literature states that a model that includes both the direct and mediated effects of a time-constant variable is not identified. But Stoel et al. 2004 (SEM) have shown that the mediated model is nested within the direct model such that a LR chi-square test can be used to determine whether the direct model is warranted.
This said, I am a bit uncertain as to the syntax for this “direct effect” model i.e. which restrictions to apply. Would it be like this?

I S | Y1@0 Y2@1 Y3@2;
I on A1 A2 A3 @0;
S on A1 A2 A3 @0;
Y1 on A1;
Y2 on A2;
Y3 on A3;

And, then the nested “mediated” (or standard spec’ed) model like this:

I S | Y1@0 Y2@1 Y3@2;
I on A1 A2 A3;
S on A1 A2 A3;
Y1 on A1@0;
Y2 on A2@0;
Y3 on A3@0;

Kind thanks in advance
 Bengt O. Muthen posted on Thursday, July 05, 2018 - 5:41 pm
A model with both the indirect and all direct effects is not identified. One approach is to include the direct effects for one outcome at a time (regressed on all covariates) and see which are significant.
 peter lekkas posted on Thursday, July 05, 2018 - 9:58 pm
Tx Bengt - yes I understand that a model where both the indirect and direct paths are unconstrained won't identify however when I constrain either the direct or indirect paths and run these models separately (as per the annotation below) identification is respectively possible for each of these models. I just want to double check I have the syntax/specification right such that I can go with the nested LL_diff test as per Stoel et al 2004 (SEM). Any further thoughts on this would be greatly appreciated?

I S | Y1@0 Y2@1 Y3@2;
I on A1 A2 A3 @0;! time constant mediated effects constrained to zero
S on A1 A2 A3 @0;
Y1 on A1;
Y2 on A2;
Y3 on A3;

And, then the nested “mediated” (or standard spec’ed) model like this:

I S | Y1@0 Y2@1 Y3@2;
I on A1 A2 A3;
S on A1 A2 A3;
Y1 on A1@0; ! direct effects constrained to zeros for time constant effects
Y2 on A2@0;
Y3 on A3@0;
 Bengt O. Muthen posted on Friday, July 06, 2018 - 5:52 pm
I don't know about the Stoel article. For your first model, perhaps you want instead the 9 parameters

y1 on a1-a3;
y2 on a1-a3;
y3 on a1-a3;

Otherwise, the second model with its 6 ON parameters isn't nested within the first model.
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