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A variant of the case-cohort design is proposed for the situation in which a correlate of the exposure (or prognostic factor) of interest is available for all cohort members, and exposure information is to be collected for a case-cohort sample. The cohort is stratified according to the correlate, and the subcohort is selected by stratified random sampling. A number of possible methods for the analysis of such exposure stratified case-cohort samples are presented, some of their statistical properties developed, and approximate relative efficiency and optimal allocation to the strata discussed. The methods are compared to each other, and to randomly sampled case-cohort studies, in a limited computer simulation study. We found that all of the proposed analysis methods performed well and were more efficient than a randomly sampled case-cohort study.