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There is a lot to address here, and while I appreciate you providing so much detail, I will only focus on some of the most important issues I see. So, the first thing I notice from looking at your model structure is that your model is perhaps misspecified. For one, the beta distribution is defined only for positive $\alpha, \beta$. However, your ...


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One option would be to do survival analysis and use a Cox model to predict the probability that each individual already contacted will "survive" 30 days without responding, then subtract those predicted probabilities from 1 (with 0s for all the people who've already replied) and sum the results to get an estimate of the eventual response rate. In this model, ...


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