# Group-level random effect correlated with within-group independent variable

Dear StackExchange community,

# Question:

According to, e.g., Wikipedia, random effects models assume "that the individual specific effects are uncorrelated with the independent variables."

I understand that to mean that, in the context of a shared frailty model (see below), a control variable that varies on the group level must not be correlated with the group level frailty. But:

Question: are variables that vary within groups allowed to be correlated with the group-level frailty?

Example: imagine that

• one is modeling the time for siblings to reach their terminal educational degree,
• the control variables include dummy variables for each kind of terminal degree (BSc, MSc, etc.),
• the shared frailty at the sibling level represents the sibling-level speed, i.e., tendency to finish degrees faster

Imagine also, that people who run through their education at a faster pace also tend to continue into higher levels of education. Faster families will be overrepresented among higher degrees. Therefore, the researcher wants to correct for the sibling-level speed. But at the same time, the degree dummies will be correlated with the frailty.

Would the shared frailty model be valid in such a case? Would the model correctly estimate the associations between types of terminal degrees and the the time to completion?

# Background:

I would like to estimate a shared frailty model (see these lecture notes; the following draws on this document). It is a a random effects survival analysis model where unobserved heterogeneity between groups are modelled as a so-called shared random effect common to each group. It takes the form:

$h(t_{ij})=h_0(t)\exp(\beta' \mathbf x_{ij} + \varphi'w_i)$,

where $h(t_{ij})$ is the hazard of individual $i$ in group $j$ at time $t$ given the baseline hazard, $h_0$, the control variables for each individual, $\mathbf x_{ij}$, and the shared frailty for group $i$, $w_i$.

The $w_i$ are assumed to be drawn from an independent sample from a distribution with mean 0 and variance 1.

I hope my question makes sense and I look forward to hear your answers.