Regressions Among Random Coefficients
Message/Author
 Rich Jones posted on Thursday, February 17, 2000 - 8:38 am
I have a question about alternative parameterizations for latent growth modeling.

Two equivalent parameterizations of a LGM are to (1) freely estimate the correlation between initial status and change factors, or (2) regress the slope factor on the initital status factor (and fixing the correlation to zero). These two parameterizations will not, however, produce equivalent parameter estimates for the mean and variance of the slope factor (although naturally the posterior estimates of slopes and intercepts are equivalent in the two parameterizations).

However, when interest extends to how exogenous variables relate to initial status and growth factors, the two parameterizations can lead to very different inferences.

I'm afraid there is something I am missing regarding choosing models or interpreting output from the two parameterizations. Does anyone have any suggestions or references that might help?
 Dieter Urban posted on Friday, February 18, 2000 - 8:13 am
Rich,
you can find some information on a related problem in:
Rovine, M.J./Molenaar, P.C.M., 1998, The covariance between level and shape in the latent growth curve model with estimated basis vector coefficients. Methods of Psychological Research Online (3) (www.pabst-publishers.de/mpr/)
However, concerning your special problem: a bivariate regression should show the same results as a bivariate correlation. How did you get the difference between both?
 Bengt O. Muthen posted on Friday, February 18, 2000 - 9:43 am
With an intercept and a slope factor and no covariates, the model that covaries them and the model that regresses the slope on the intercept gives the same fit and estimates of the growth factor means, variances, and covariance. With covariates, the model interpretation differs for the two alternatives. Regressing the slope on the intercept, the covariate has both a direct and indirect effect on the slope so the coefficient for the slope regressed on the covariate is different because it is a partial regression coefficient. In my view, the choice between the two models should be substantively-driven. The regression approach may be motivated if the intercept is defined as the initial status and the slope the change thereafter. This was the case in Muthen-Curran's 1997 Psych Methods article where initial status was pre-intervention status which affected how much the intervention caused change.
 Apsalam posted on Tuesday, January 31, 2006 - 8:54 pm
Hi Bengt and Linda,

I have a cross sectional structural model where A causes B. I then collected data on A and B at three time points, and I’m running parallel-process multiple indicator growth models.

For my parallel process model to be consistent with my ‘A causes B’ hypothesis, I’m letting the intercept factor for A predict the intercept factor for B (i.e. regressing B on A), but because I have no theories about rates of change I am freely estimating the covariance between the two slope factors and the intercept for B, i.e. in my model, the intercept for B is an endogenous variable, predicted by the intercept for A, while all other growth factors are exogenous variables with freely estimated covariances.

Is the growth model I describe consistent with my A causes B hypothesis?
 Linda K. Muthen posted on Wednesday, February 01, 2006 - 8:51 am
The model you have says where people start on process A predicts where they start on process B. You may want to add that where they start on process A predicts where how they grow on process B.
 apsalam posted on Wednesday, February 01, 2006 - 11:45 am
Thank you that is very helpful.
 Chuck He posted on Tuesday, October 02, 2007 - 5:16 am
I have a question about Random Effects.
My model is F1-F2-F3, while all of them are latent variables. F4, another latent variable, is included in this model. I would like to find how F4 has effect on the relationship between F1 and F2 (F1-F2). The following is my scripts. However, whenever I run this programme, it tells me the dimensions of integration and total number of integration points and then stops.

TITLE: Hierarchical regression
DATA: FILE IS 111.TXT;
Variable: NAMES ARE m1-m7 o1-o8 s1-s4 gs1-gs6;
ANALYSIS: TYPE = RANDOM;
ALGORITHM=INTEGRATION;
MODEL: f1 BY m1-m7;
f2 BY s1-s4;
f3 BY o1-o8;
f4 BY gs1-gs6;
f2 on f1;
f3 on f2;
s | f2 on f1;
s with f2;
s on f4;

Does anyone have solutions on this problem?

Thanks, Chuck
 Linda K. Muthen posted on Tuesday, October 02, 2007 - 8:40 am
It sounds like you want the interaction between f1 and f2. You would use the XWITH option for that. If this does not solve your problem, please send your input, data, output, and license number to support@statmodel.com.
 Chuck He posted on Tuesday, October 02, 2007 - 9:05 am
Hi, Linda,
Thanks for your response. However, it is not what I want.
Anyway, I will send all information together with the license number to you.
Thanks and best regards,
Chuck
 Chuck He posted on Thursday, October 04, 2007 - 1:21 am
Hi, Linda:
I have solved this problem. I misput one parameter in my model.
Thanks, Chuck
 Karoline Brobakke posted on Monday, March 18, 2013 - 2:58 am
Hi,
I have run a parallel process model (depression and stress) with regressions among the random effects (Your example 6.13 in the manual).

The slope for depression shows an overall stable mean and sig. variance. While the slope for stress shows a slight decline and sig. variance. The regression coefficients from the intercept of one process to the slope factor of the other are both negative.

Because some individuals decline and others increase in both processes, I am uncertain about the correct interpretation of the regression coefficients.

Is it correct to interpret this so that individuals with lower initial status on either process either increase faster or decline more slowly on the other process (depending on whether they have positive or negative slopes)?

Karoline
 Bengt O. Muthen posted on Monday, March 18, 2013 - 9:07 am
Yes.
 Tammy Kochel posted on Tuesday, June 11, 2013 - 2:26 pm
If two parallel processes are included in a latent growth model, but the hypothesis predicts only the growth factors of one of those processes, does including the growth for the second process "control" for that change over time, even though nothing is regressed upon it? Thanks. Tammy
 Bengt O. Muthen posted on Tuesday, June 11, 2013 - 2:36 pm
No.