Population-Average vs Subject-Specific Interpretation in 3-Level Mixed Effects Model with Random Intercepts

Question

Most of the population-average versus subject-specific interpretations I've come across refer to 2-level mixed-effects models, but in practice, we may encounter situations where we have to provide such interpretations for a 3-level mixed-effects model.

Assume a situation where several patients are measured repeatedly over time and the patients are nested in hospitals such that the outcome measured for each patient is a count outcome.

Also, assume that we are measuring two time-varying covariates x1 and x2 for each patient, such that x1 is continuous and x2 is binary. For simplicity, assume x1 = time, where time is coded as 0, 1, 2, 3, etc. for regularly spaced occasions. For x2, the only possible patterns of values are 0, 0, 0, ..., 0 (i.e., all zeroes) or 0, 1, 1, ..., 1 (i.e., a single zero followed by ones).

One possible model formulated for the ensuing data is a Poisson mixed effects model that would look like this:

$$\log(E(y_{ijk} \mid time_{ijk}, x2_{ijk})) = \beta_{0} + \beta_{1}*time_{ijk} + \beta_{2}*x2_{ijk} + v_{i} + w_{ij}$$

where $v_{i}$ is a random intercept associated with the i-th hospital and $w_{ij}$ is a random intercept associated with the j-th patient in the i-th hospital. (The index $k$ is reserved for the temporal occasion.)

My questions are:

Question 1

Will $\beta_{1}$ and $\beta_{2}$ have a population-average interpretation given that the model includes only random intercept terms?

Question 2

If a population-average interpretation for $\beta_{2}$ is suitable, will it look like this:

For any particular time occasion (e.g., time = 2), the average value of the count outcome $y$ for patients for whom $x2 = 1$ differs from the average value of the count outcome for patients for whom $x2 = 0$ by a multiplicative factor of $\exp(\beta_{2})$, regardless of the hospital the patients come from?

Question 3

In contrast, will a subject-specific interpretation for $\beta_{2}$ look like this:

For any particular time occasion (e.g., time = 2), increasing the value of $x2$ from 0 to 1 for a typical patient in a typical hospital will be associated with an increase (if $\beta_{2} > 0$) or decrease (if $\beta_{2} < 0$) in the average value of the count outcome $y$ given by a multiplicative factor of $exp(\beta_{2})$?

For this last interpretation, can we replace "for a typical patient in a typical hospital" with "for any patient in any hospital" given that the model includes only random intercepts?

Question 4

For an added twist, assume that the model is now expanded to include an interaction between the two time-varying predictor variables:

$$\log(E(y_{ijk} \mid time_{ijk}, x2_{ijk})) = \beta_{0} + \beta_{1}*time_{ijk} + \beta_{2}*x2_{ijk} + \beta_{3}*time_{ijk}*x2_{ijk} + v_{i} + w_{ij}$$

How will we interpret $\beta_{2}$ and $\beta_{3}$ in this model?

Answer 1

Indeed, because the model only includes random intercepts terms, the marginal mean of your Poisson outcome will be

$$E(Y_{ijk}) = \exp \bigl (\beta_0^* + \beta_1 \texttt{time}_{ijk} + \beta_2 \texttt{x2}_{ijk} + v_i + w_{ij}\bigr ),$$

where

$$\beta_0^* = \beta_0 + \frac{\sigma_v^2}{2} + \frac{\sigma_w^2}{2},$$

with $\sigma_v^2$ and $\sigma_w^2$ the variances of the two random intercepts terms.

Hence, the coefficients $\beta_1$ and $\beta_2$ will have the classical marginal interpretation. That is, you could interpret them in exactly the same way as if you have fitted a simple Poisson regression in your data without any random effects. The same will also hold for the coefficient for $\beta_3$.