I have longitudinal data with three time points, following 13/14 years old adolescents 2 and 7 years later. I am interested in how the correlation between self-esteem and emotional problems are changing over time, and whether other variables (for example unstable self-concept) can predict this change (the correlation is actually increasing, whereas the self-concept is getting more and more stable).
For this, I thought it may be possible to do a random coefficient regression (as described in example 3.9 in your handbook) where self-esteem is the independent variable and depression the independent variable. As I have understood, the Hildreth-Houck regression gives random slopes representing a deviation from the coefficient mean for each individual. I further tried to construct a growth model with the random slopes as indicators:
s1 | dep1m ON sw1m; s1 WITH dep1m;
s2 | dep2m ON sw2m; s2 WITH dep2m;
s3 | dep3m ON sw3m; s3 WITH dep3m;
sw3m WITH sw2m; sw1m*;
dep2m WITH dep1m; dep3m WITH dep1m; dep3m WITH dep2m;
int slope | s1@0 s2@2 s3@7;
Unfortunately, I get the message that the psi-matrix is not positive definite. When I run the model (and I cannot check tech 4 with type=random).
My question is if it is possible to do such a growth curve model based on random slopes and whether I can interpret the slope of the growth curve as change in the relationship between depression and self-esteem over time. If yes, I would also be grateful to know if you have some suggestions to how I can find out why I get the error message.
I think this could be a difficult approach because it requires that deptm on swtm has variation across individuals for all time points t - perhaps the lack of such variation is that the non pos def Psi message is about. Actually, the random slopes refer to heteroscedasticity of the residual variances in Hildreth-Houck style. Note also that there is already a correlation between dep and sw by the fact that you regress dep on sw. I wonder if there isn't another way to do this. What about using a factor for each time point, influencing dep and sw (say with fixed factor variance so the covariance between dep and sw is picked up by one loading). The factor could then be regressed on a covariate. Or, a 2-level approach with a factor on level 1, where that loading is a random variable on level 2 somehow. Repeated measures being level 1 and people level 2. Just thinking out loud - haven't thought it through.
Thank you for your help! I have finally managed to get the model I have described before to run by fixing the variance of the intercept and the slope to 0, but I think you are right that this is quite a difficult approach.
I have tried to analyze the data as you suggested, by using a factor and fixing one of the factor loadings and the factor variance. But I am not quite sure how to interpret the regression of the factor on the covariate. Does the factor variance reflect the variation in the free factor loading (which again represents the covariation between dep and sw), and does the regression on the covariate represent how much the covariate influences the strength of the relationship between sw and dep? Another problem is that I cannot fix both one of the factor loadings and the factor variance to 1 - there is too little variance in dep and sw, such that I get a negative residual variance in one of the two observed variables. Therefore, I have to reduce the factor loading or the variance in the factor, which in turn, influences the regression coefficient.
I am not quite sure if your 2-level suggestion will actually help me to answer my research question. Somewhat unsure how I can get the covariate into the model, but I will try...
Maybe these are too complex approaches. Perhaps a simple start could be useful where you have your covariate self-concept predicting the two outcomes self-esteem and emotional problems at each of the 3 time points. With different coefficients for the 3 time points, you would get different amounts of correlation across the two outcomes explained by the covariate.