I am analyzing the effect of two randomized interventions on 1) first-time and, 2) second-time event attendance. All variables are binary (0 vs. 1). Second attendance was conditional on first attendance. A large percentage (~50%) of people did not show up for first attendance; an even smaller percentage of first-time attendees (~50%*~20% = 10%) showed up the second time.
Intervention one was designed to increase both attendances; intervention two, just second attendance. Neither increased the probability of first or second attendance, but the interaction between second attendance and intervention was significant for both. The probability of second attendance was greater for both interventions relative to first.
The issue is that we are trying to test the effect of both interventions on second attendance *given* first attendance. We are also controlling for household using a random intercept.
I was advised an instrumental variable (IV) analysis would adjust for first attendance (Sussman BJ, Hayward, RA, BMJ, 2010). I regressed first attendance on intervention controlling for household, and then regressed second attendance on the predicted probability of first attendance (the instrument variable). I did this separately for both interventions, and the instrument variable was significant for both. My questions are:
- Is IV appropriate given both interventions have a non-significant association to first attendance? I've read the instrument variable (randomized intervention), should be related to the treatment/biomarker (first attendance).
- If appropriate, how do I interpret the instrument variable coefficients?
- Are there better analyses which measure second attendance for first-time attendees only?
