Maximizing the Overlap of Sample Units for Two Designs with Simultaneous Selection

Lawrence R. Ernst


It is demonstrated, using transportation theory, that controlled selection can be used to solve the following sampling problem. Sample units are to be selected with probability proportional to size for two designs, both one unit per stratum, denoted as D1 and D2, with generally different stratifications. The goal of the problem is to simultaneously select the sample units for the two designs in a manner which maximizes the expected number of units that are in both samples. The procedure differs from previous overlap procedures in that it yields a better overlap, but is only applicable when the two samples can be selected simultaneously. An important special case occurs when the probability of selection for each unit in D1 sample units are a subset of the D2 sample units. A proposed, but since cancelled, expansion of the Current Population Survey, which is discussed, would have been a potential application of this special case. Variance formulas for estimators of total under the controlled selection procedure are also presented. In addition, it is demonstrated that hte procedure can eeasily be modifies to minimize expected overlap instead of maximizing it.