In a cohort study, a sample of children is studied at two time points ($t_1$ and $t_2$) that are several years apart. At each time point, the BMI (body mass index), blood pressure ($\mathrm{BP}$), sex, age and physical activity (a positive continuous variable) are measured for each child.

The research question we would like to answer is: Is the change in physical activity between $t_1$ and $t_2$ associated with the blood pressure at $t_2$?

I am unsure about my model choice. My idea was to fit the following model:

$$ \mathrm{BP}_{t_2, i}=\beta_0 + \beta_1\cdot\mathrm{BP}_{t_1,i} + \beta_2\cdot \Delta\mathrm{Physical Activity}_{i} + \beta_3\cdot \mathrm{age}_i + \beta_4\cdot \mathrm{sex}_i + \beta_5\cdot \mathrm{BMI}_{t_1,i} + \beta_6\cdot\mathrm{BMI}_{t_2,i} + \epsilon_i $$

Where $\Delta \mathrm{Physical Activity}_{i}$ is the change in physical activity between $t_1$ and $t_2$ of the $i$th child.

This is in the vein of an ANCOVA as described by Vickers et al. (2001). It adjusts for baseline blood pressure ($\mathrm{BP}_{t_1,i}$) as well as for age, sex and BMI. That the BMI is included twice worries me, however. Maybe it would be better to just adjust for BMI at $t_2$.

My questions are:

  1. Is this a sensible model for answering the question or are there more suitable models (for example a lag-model or a model using only differences)?
  2. Is it okay to include the BMI of both time points or is there a better way to include time-varying covariates? One of the obvious problems I anticipate is collinearity.
  3. Would you recommend including the physical activity in the form of a difference or include both time points (i.e. $\mathrm{Physical Activity}_{t_1,i}$ and $\mathrm{Physical Activity}_{t_2,i}$)?

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