Regarding modelling longitudinal variables using Two stage mixed effects modeling

Question

I have a question about the basic understating of key statistical methodology.

I came across the idea about two stage modelling to incorporate longitudinal predictors. Lets say there is a continuous longitudinal predictor $x_{it}$ and the dependent variable is a binary variable $y$.

First stage Model the continuous longitudinal predictor using Linear Mixed Effects model (LMM) So this is how my model looks like with respect to fixed effects($\beta$) and random effects($u$).

$x_{it}=\beta_0 + u_{0i} + (\beta_1+ u_{1i})t + \epsilon _{it}$

Second Stage So in the second stage I can use these random effects as predictors to model the response variable.

$logit(p(y_i=1))=\alpha_0 + \alpha_1\hat{u_{0i}} + \alpha_2\hat{u_{1i}}$

So My Question is what is the justification of using these random effects as predictors instead of longitudinal predictor ?

I got the point that if we use longitudinal predictor as a predictor to model the response, then we need to dependent predictors of same measurement $x_{i1},x_{i2},..,x_{it}$ .

Also I know that random effects are the estimated subjects deviations from the population average . So the random effects basically have the subject specific effects . Is this the real reason. Or is there theoretical justification ?

Thank you very much

Answer 1

I imagine that people could come up with a justification for the two step approach, but to me, it seems like a bit of a waste. Most critical from my perspective, if you run everything in a single model,

$logit(p(y_{it}=1))=\beta_0 + u_{0i} + (\beta_1 + u_{1i})x$

then the $u_{0i}$ and $u_{1i}$ remain latent variables, thus reducing measurement error that is induced when you predict them using an Empirical Bayes (EB) approach. Said differently, there is a fair degree of uncertainty about each individual's value of $u_{0i}$ and $u_{1i}$, and this uncertainty is preserved in the LMM above. In contrast, Empirical Bayes prediction assigns a single value to each person's $u_{0i}$ and $u_{1i}$. There is an associated standard error for the EB prediction, but you would need to go fully Bayesian to incorporate such uncertainty back into a model. Mark Lai has example code showing how to do this with lmer() and brms().

The only thing this single model does not presently have is a mean value for $x_{ij}$ in the prediction of $y_{ij}$. However, you could easily compute each person's mean x value ($\bar x_i$) and add it as a predictor to the model.