Optimum sample size determination in stratified case‐control studies with cost considerations

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Abstract
We investigate sample size determination for Cochran's test for stratified case‐control studies when samples of cases and controls are allocated to maximize the asymptotic efficiency of Cochran's test subject to fixed total cost with cost per control varying by strata. We consider two situations typical of strata‐matched case‐control studies: when one samples both cases and controls and when cases are given and one samples controls. In each situation we develop and study an asymptotic method for finding the sample size required for a specific power under the optimum allocation proposed by Nam and Fears. Also, for the second situation, we investigate an asymptotic method for determining the common ratio, k, in one‐to‐k stratamatched case‐control studies without cost consideration for a given power. When cases are given, neither the optimum nor the standard control sample sizes appear in a closed form; we present numerical methods for calculating these sample sizes and illustrate them with examples. We find the reduction in total cost obtained under the optimum allocation compared to standard allocation more pronounced as the differences in stratum‐specific costs of sampling controls increase.