I'm currently working on a nested data set consisting of 100 subjects which answered several questions at home on five consecutive days (ecological momentary assessment). Among them, they were asked if they adhered to the study protocoll on each given day (no/ yes), leading to 0 (no) or 1 (yes) in the data for each of the five days. The continous outcome variable was also assessed on each day.

The reserach question is whether beeing adherent on a given day has an effect on the outcome. Now, to explore the effect of the adherence on the outcome variable, I built a nested multilevel model (days within subjects) predicting the outcome by some covariates, a random intercept (high ICC) and the variable coding for adherence (0 or 1).

Normally, I would proceed and disentangle within-subject variations (person mean centered) from between-subject variations (grand mean centering of the person means) for the adherence variable. However, it seems rather odd to me to center a dichotomous variable. On the other hand, I know that it is necessary in order to get a clear picture of the within-subject effect. When not centering, I could enter the 0/1 adherence variable as a factor. However, in this case, it would confound within- and between-subject variations (because subjects do not only differ as compared to themselves but also in their total amount of adherence in comparision to the group, the grand mean).

Do you have any advice on whether I should center the 0/1 variable? If yes, how would you center? If no, how would you proceed?