Setup: I'm testing if the association between pairs of individuals for a trait (BMI) changes over time. I have repeated measures, where each individual in a pair gives BMI data at 7 points in time. Below is a simplified data frame in long format with Pair ID (the identifier given to each pair of individuals), BMI measurements for both individuals at each point in time (BMI_1 and BMI_2), and a time variable with seven intervals, coded as continuous.
| Pair_ID | BMI_1 | BMI_2 | Time |
|---|---|---|---|
| 1 | 25 | 22 | 1 |
| 1 | 23 | 24 | 2 |
| 1 | 22 | 31 | 3 |
| 1 | 20 | 27 | 4 |
| 1 | 30 | 26 | 5 |
| 1 | 31 | 21 | 6 |
| 1 | 19 | 18 | 7 |
| 2 | 21 | 17 | 1 |
| 2 | 22 | 27 | 2 |
| 2 | 24 | 22 | 3 |
| 2 | 25 | 20 | 4 |
First, I'm mainly interested in testing the within-pair association (the regression coefficient of BMI_2, below) and whether it changes over time (the interaction between BMI_2 and Time). I'd like to exclude any between-pair effects, so that I'm only testing associated over time within pairs.
I was planning on fitting a linear mixed model of the form:
lmer(BMI_1 ~ BMI_2 * Time + (BMI_2 | Pair_ID), Data)
I understand the parameters of the model (e.g., random slopes/intercepts), and that the BMI_2 * Time interaction tests whether the relationship between BMI_1 and BMI_2 is moderated by time.
However, I'm unsure how to identify the (mean) within-pair regression coefficients, and whether my approach is even suitable for this.
Second, I'm interested in understanding whether there is variation between pairs in the BMI_2 * Time interaction (i.e., the variance in slopes among pairs) - for example, does the associated between BMI_1 and BMI_2 increase over time in some pairs but not others?
For this, I was considering fitting a model like this:
lmer(BMI_1 ~ BMI_2 * Time + (BMI_2 : Time | Pair_ID), Data)
and then looking at the variance in the BMI_2 : Time random effect. As I understand it, large variance would imply that this interaction effect varied a lot between pairs.
Any help on these questions (especially the first question) would be greatly appreciated.
P.s., sorry if the question is poorly formatted. It's my first attempt.
Answering for completeness. @benimwolfspelz's comment is spot on. This is known as "contextual effects" in some areas of applied work. The idea is to split the variable into between and within components by mean-centering each group and fitting the mean-centred variable (which will estimate the within component) and the group means (which will estimate the between component).