  To generalize inferences from a randomized trial to the target population of
all trial-eligible individuals, investigators can use nested trial designs,
where the randomized individuals are nested within a cohort of trial-eligible
individuals, including those who are not offered or refuse randomization. In
these designs, data on baseline covariates are collected from the entire
cohort, and treatment and outcome data need only be collected from randomized
individuals. In this paper, we describe nested trial designs that improve
research economy by collecting additional baseline covariate data after
sub-sampling non-randomized individuals (i.e., a two-stage design), using
sampling probabilities that may depend on the initial set of baseline
covariates available from all individuals in the cohort. We propose an
estimator for the potential outcome mean in the target population of all
trial-eligible individuals and show that our estimator is doubly robust, in the
sense that it is consistent when either the model for the conditional outcome
mean among randomized individuals or the model for the probability of trial
participation is correctly specified. We assess the impact of sub-sampling on
the asymptotic variance of our estimator and examine the estimator's
finite-sample performance in a simulation study. We illustrate the methods
using data from the Coronary Artery Surgery Study (CASS).
