Optimal sample allocation for design-consistent regression in a cancer services survey when design variables are known for aggregates.

We consider optimal sampling rates in element-sampling designs when the anticipated analysis is survey-weighted linear regression and the estimands of interest are linear combinations of regression coefficients from one or more models. Methods are first developed assuming that exact design information is available in the sampling frame and then generalized to situations in which some design variables are available only as aggregates for groups of potential subjects, or from inaccurate or old data. We also consider design for estimation of combinations of coefficients from more than one model. A further generalization allows for flexible combinations of coefficients chosen to improve estimation of one effect while controlling for another. Potential applications include estimation of means for several sets of overlapping domains, or improving estimates for subpopulations such as minority races by disproportionate sampling of geographic areas. In the motivating problem of designing a survey on care received by cancer patients (the CanCORS study), potential design information included block-level census data on race/ethnicity and poverty as well as individual-level data. In one study site, an unequal-probability sampling design using the subjectss residential addresses and census data would have reduced the variance of the estimator of an income effect by 25%, or by 38% if the subjects' races were also known. With flexible weighting of the income contrasts by race, the variance of the estimator would be reduced by 26% using residential addresses alone and by 52% using addresses and races. Our methods would be useful in studies in which geographic oversampling by race-ethnicity or socioeconomic characteristics is considered, or in any study in which characteristics available in sampling frames are measured with error.