Minmax designs for planning the second phase in a two-phase case-control study

Two-phase designs, in which a subsample with validation or complete information data is sampled stratified both on outcome and covariate from a first-phase study with incomplete data, have been proposed over 10 years ago and have been proven to result in efficient estimates with respect to standard designs. The efficiency depends, however, on the sampling fractions within each stratum. Our aim is to present a strategy for obtaining an optimized design, i.e. sampling fractions, that makes use of available phase-one data of an existing case-control study considered as the first-phase sample when the focus is on estimating a parameter vector. No global optimal design exists and local optimal designs depend on scenarios comprising the true disease model and the association between the phase-one and phase-two information. We develop an admissibility test that rejects scenarios inconsistent with the phase-one data and, for the selected scenarios, determine a minmax D- or A-optimal design that protects against worst-case scenarios. This work is applied on two examples.