Multivariate growth model or something else?

[martinmodrak]

@jd_c

Along those same lines, @martinmodrak what do you think of one of my original ideas where I use this model:
brm(stress ~ 0 + (1 + time|id)
to find slopes and intercepts for each person for stress over time, and then use those slopes and intercepts (and their sd’s from the posterior samples) in a measurement error model to predict the outcome or predict the change in outcome computed either as differences in values from timepoints or as slopes taken from this model:
brm(outcome ~ 0 + (1 + time|id)
??
I think that might have a similar concept as what you suggest but maybe use all of the data in a single predictor for the slope, instead of several predictors for the differences at the different timepoints. Any thoughts on that?

In broad terms, I would advise against this. If I understand you correctly, what you describe is a heuristic approach to implement something like a bigger joint model. I.e.:

1. There is some true, unobserved level of stress (stress_true)
2. The observed stress is a noisy observation of stress_true
3. stress_true is predicted by a linear model
4. stress_true and some form of “trend in stress_true at given time” is then used as a predictor for outcome

If that would describe your assumed model, I would encourage you to use Stan to implement this model (it can’t AFAIK really be implemented in brms without a lot of hacks, which might in the end be more difficult than just using Stan). If I understand you correctly, what you describe is hacky way to implement this in two steps. I would expect it to face similar challenges as e.g. at Using posteriors as new priors The hacky way might give similar results to the full model, but it also might not and there is no easy way to tell - the two step procedure throws a lot of the nice guarantees Stan gives you out of the window.

Additionally, I would note that it is not clear which of the two models (using observed differences as predictors vs. the joint model described above) uses “all of the data” or uses the data “better” in some sense. The joint model IMHO has the potential to be a bit more efficient, but it is also making stronger assumptions about the data by enforcing a linear model for stress_true whereas just using the differences avoids those assumptions (but it makes the additional assumption that stress is interval scaled and thus difference is a meaningful quantity). So if the linear assumption for stress_true is correct, the joint model will “use all the data” but if it is incorrect, the joint model could artificially smooth-out some features of the observed stess levels while using differences as predictors will “use all the data” because it avoids this smoothing.

Finally, if you are going to use differences in stress as predictor, you may also want to use outcome at previous time as predictor (but you usally don’t want to use difference in outcome as response as discussed e.g. Statistical Thinking - Statistical Errors in the Medical Literature)

Does that make sense?