Planning domain sizes in cluster sampling

Multi-stage cluster sampling is a common sampling design of social surveys because populations of interest are often structured by, or partitioned into, disjoint organizational and administrative units. The need to use cluster sampling can conflict with survey planners' goal to select a sample that contains a specific number of elements from certain domains of interest. This can be a complex problem if sampling units, i.e. clusters, cut across the domains of interest, as it is often the case. For example, an analysis require sufficient observations from certain age and gender categories. But the population is clustered within schools, hospitals, establishments, or municipalities and hence age-gender categories cannot be used for stratification. We propose a quadratic optimization approach to define inclusion probabilities that can be used for drawing balanced cluster samples that comply with predefined sample sizes from domains of interest. Henceforth the clusters may cut across domains. We also provide an application of the proposed solution to the domain size problem for an existing social survey on migration and emigration in Germany.