A Latent growth curve model with a time-invariant covariant can be specified with (at least) two ways, by using the multigroup model approach or the "regression approach".
By regression approach I mean regressing intercept and slope on the time-invarying covariate, and by multigroup model I mean running two models, one multigroup model with no restrictions and one with slopes restricted to equality across the groups and then using model comparison tests to see whether the slopes are different.
I recently learned that the multigroup approach has certain benefits over the regression approach, as the latter assumes that the outcome variable (LGC indicator variable) has similar properties at different levels of the covariate (e.g. homogeneous variances), whereas the multigroup approach does not.
This is very clear to me when it comes to categorical covariates such as gender. We can use a multigroup model with gender as the group variable, or we can check whether men and women have homogeneous variances on the LGC indicator variable at each time point and if they do, we can also use the regression approach.
However, what if we have continuous time-invariant covariate? Can we use the regression approach? If so, should we first check whether the LGC indicator variable has similar variance (and other properties) at different levels of the covariate? If so, which levels should we check and how?
Or, should we divide the continuous covariate into categorical levels and run a multigroup model using those levels as groups? This at least feels incorrect, but I'm not sure.