Sample allocation for overlapping domains in a physician survey with a limited population

We consider an analytic survey in which each survey unit can respond with respect to one or more domains to which it belongs. In this situation, the optimal sample allocation is constrained by the number of available units in each stratum defined by a set of domains. We present optimal sample allocation rules for this situation, and other rules that apply when there are additional constraints on sample sizes or variances by domain. We apply these rules to our motivating example, the design of a survey of physicians on their experiences with health plans, in which each physician can only be asked about her experience with one plan regardless of her number of affiliations.