I am seeking suggestions for suitable regression or statistical tests to measure the effect of a treatment in counts (or proportions), when a control group is not available. Let's say an event takes place at time $t$, e.g., a new fee to use the swimming pool inside a city park. We have count data for the prior and post periods ($t - k$ and $t+k$ respectively) for different sub-groups. These counts correspond to (i) visitations of the park, and (ii) use of the swimming pool. The sub-groups correspond to ethnicity, gender, age, etc.
The idea is that the counts for park use would not be affected by swimming pool fee, but the counts for swimming pool use would be affected by the fee. Given this scenario, is there a suitable regression or statistical test to quantify how the introduction of fee changes pool usage (immediately before and after the fee introduction) ? I am interested in measuring how this event changes the use of swimming pool, and how they relate to different sub-groups. For example, I want to know if the change in swimming pool usage is higher on elderly males than young females.
I was thinking of using proportions test to check whether the proportion of people who use the swimming pool to those who visit the park changes (in the pre and post periods) but was not sure if this is the best way to do this. Would love to hear your suggestions on this. Is there a regression method that is suitable for a problem like this?
