Determining the value of additional surrogate exposure data for improving the estimate of an odds ratio

We consider the design of both cohort and case-control studies in which an initial ('stage 1') sample of complete data on an error-free disease indicator (D), a correct ('gold standard') dichotomous exposure measurement (X) and an error-prone exposure measurement (Z) are available. We calculate the amount of additional information on the odds ratio relating D to X that one can obtain from a second ('stage 2') sample of measurements only on D and Z. If one allows for differential measurement error in Z, there is often little advantage in having more than four times as much data in stage 2 data as in stage 1. With the assumption that a non-differential measurement error model is reasonable, larger amounts of stage 2 data can be useful. Simulations indicate that stage 1 samples of modest size (50 cases in case-control studies and 50 failures in cohort studies) yield sufficiently reliable estimates of needed parameters to assist in determining an appropriate size for the stage 2 sample. These ideas apply in settings either where the amount of stage 1 data is limited and fixed by external constraints or where one has gathered stage 1 data in advance to avoid collecting superfluous stage 2 data.