Two-phase designs with failure time processes subject to nonsusceptibility

Epidemiological studies based on 2-phase designs help ensure efficient use of limited resources in situations where certain covariates are prohibitively expensive to measure for a full cohort. Typically, these designs involve 2 steps: In phase I, data on an outcome and inexpensive covariates are acquired, and in phase II, a subsample is chosen in which the costly variable of interest is measured. For right-censored data, 2-phase designs have been primarily based on the Cox model. We develop efficient 2-phase design strategies for settings involving a fraction of long-term survivors due to nonsusceptibility. Using mixture models accommodating a nonsusceptible fraction, we consider 3 regression frameworks, including (a) a logistic "cure" model, (b) a proportional hazards model for those who are susceptible, and (c) regression models for susceptibility and failure time in those susceptible. Importantly, we introduce a novel class of bivariate residual-dependent designs to address the unique challenges presented in scenario (c), which involves 2 parameters of interest. Extensive simulation studies demonstrate the superiority of our approach over various phase II subsampling schemes. We illustrate the method through applications to the Prostate, Lung, Colorectal, and Ovarian Cancer Screening Trial.