I would like to parameterize a LCM with binary repeated measures in the following way: for each repeated binary measure I would like to assume an underlying continuous normal variable (i.e. y*) with the threshold fixed to 0 and the unconditional variance fixed to 1. This should allow the means of the y* to be estimated. With these constraints it should be possible to estimate means variances and covariances for the latent curve factors. Also, in this parameterization there should be no need for scale factors.
So I know how to constrain threholds, but I have been unable to constrain the unconditional variances of the y*'s, instead I can only constrain the residual variances of the y's. Also, I don't know how to prevent the scale factors from being estimated (other than moving to a logit link).
Do you have any advice or code for this specification?
A clarification: after reading M+ Web Notes #4 (Muthen and Asperouhov 2002) it seems to me that the scale factors are the inverted standard deviation of y*'s. If this is correct, I should be able to achieve the specification I describe above by fixing the scale factors to 1, the threshholds to 0, and estimating the growth factor means in the Delta parameterization. Right? If so, how do I constrain the scale factors?
There are several ways in Mplus to do categorical growth modeling. I am actually writing a little note on the topic right now. In a regular growth model, not only the y* mean but also the y* variance changes over time because it is influenced by the random slope which has different impact (different time scores) at different time points. So holding y* variances constant over time would not be doing growth modeling. But perhaps you are interested in another model. Scale factors enable y* variances to change over time. That is the default "Delta" parameterization in the WLSMV estimator. There is also a Theta parameterization. And there is the ML approach.
But to answer your question about how to hold y* variances equal across time. This can be accomplished using Model Constraint, where you express the y* variances in terms of the model parameters and constrain the variances.
Ok, I've read the section Model Constraint and still have been unable to constrain the y* variances. To illustrate my approach I'll describe what I did for the first repeat measure using the Theta parameterization: For the first time point I labeled both the latent intercept factor variance (p1) and the residual variance for the first time point (p2) and then specified under MODEL CONSTRAINT: p1 + p2 =1; (this should be the definition of the y* variance for the first time point as the other latent growth factor variances and covariances equal 0 and the factor loading of the latent intercept factor equals one) . M+ didn't like this and gave me an error message.
Still problems. Unfortunately, the data is confidential. I'll send the input and output though. Thanks.
Dave Flora posted on Thursday, December 13, 2007 - 2:01 pm
Hi, Related to this post, I am fitting a growth model to binary variables with MLR estimation and the probit link (although I realize logit is the default). I prefer to identify the model by constraining the thresholds to be equal and the mean of the intercept factor to zero. If I am understanding everything correctly, this specification forces the variance of u* to change over time. Typically information about the variance of u* would come from the scale factors, correct? But since I'm using algorithm = integration, they are not provided. I would like to calculate the estimated probabilities at each time point, which is why I am interested in the variance of u*. Do I have to either free the thresholds, use the logit link, or use WLS(MV) to be able to calculate the estimated probabilities?
The estimated probabilities at each timepoint are computed also with MLR and in the case of numerical integration - they are provided both in Tech7 (or in Residuals) and in the graphics part (see Estimated probabilites).
Dave Flora posted on Monday, December 17, 2007 - 4:29 pm
Thanks; that helps but now I see I only get the estimated probabilities when there are no covariates. I actually have two dichotomous covariates; is there a straightforward way to get the estimated probabilities for the four groups at each timepoint?
I didn't realize you needed numerical integration for your model. This is because you have binary outcomes for your growth model. In this case, you cannot compute the probabilities by hand. I would suggest using the plot without covariates.
Dave Flora posted on Wednesday, December 19, 2007 - 2:02 pm
ok, well I would really like to know the predicted probabilities for each group at each time point. After all, the actual data are binary not continuous, and it would be good to see how the model matches up with the observed data. Sounds like something for version 5.1? But I do appreciate very much all that is possible with Mplus.