I need to estimate a proportion based on independent measurements of the numerator and the denominator. Both the numerator and denominator are measured as counts.
I know from theory that the proportion must be less than one, but in practice the numerator can be higher than the denominator because both are estimates.
To make it concrete, these are cell counts in pre-treated vs post-treated biological samples. I can measure cells in independent before-treatment and after-treatment samples, but imperfectly, and I know the treatment leads to fewer cells. I want to estimate what proportion of cells survive the treatment (with an appropriate interval to reflect uncertainty).
How could I go about this? My initial thought is a Bayesian negative binomial model with a truncated prior on the treatment effect but it feels like there should be a simpler way!
Edit: An example as requested - usually the post-treatment counts are less than the pre-sample counts, eg 4 cells in 1ml of sample post-treated compare to 10 cells in 1ml pre-treated. In one or two cases I see, say 12 or 15 cells in 1ml treated sample compared to 10 in 1ml of pre-treated sample. (Cells do cluster so I'm not concerned that the treated numbers look high given the theory). I am comparing around 20 treatments to each control if that's useful information!