Optimal design and efficiency of two-phase case-control studies with error-prone and error-free exposure measures

This paper addresses optimal design and efficiency of two-phase (2P) case-control studies in which the first phase uses an error-prone exposure measure, Z, while the second phase measures true, dichotomous exposure, X, in a subset of subjects. Optimal design of a separate second phase, to be added to a preexisting study, is also investigated. Differential misclassification is assumed throughout. Results are also applicable to 2P cohort studies with error-prone and error-free measures of disease status but error-free exposure measures. While software based on the mean score method of Reilly and Pepe (1995, Biometrika 82, 299--314) can find optimal designs given pilot data, the lack of simple formulae makes it difficult to generalize about efficiency compared to one-phase (1P) studies based on X alone. Here, formulae for the optimal ratios of cases to controls and first- to second-phase sizes, and the optimal second-phase stratified sampling fractions, given a fixed budget, are given. The maximum efficiency of 2P designs compared to a 1P design is deduced and is shown to be bounded from above by a function of the sensitivities and specificities of Z. The efficiency of 'balanced' separate second-phase designs (Breslow and Cain, 1988, Biometrika 75, 11--20)-in which equal numbers of subjects are chosen from each first-phase strata-compared to optimal design is deduced, enabling situations where balanced designs are nearly optimal to be identified.