Efficiency gains from the addition of controls to matched sets in cohort studies

We evaluate relative efficiencies of added controls in matched cohort studies in terms of the asymptotic variance of the estimate of the log relative risk associated with a single binary exposure variable. We calculate expected contributions to the information based on the Cox proportional hazards model under the assumption that the failure times for exposed and unexposed individuals have independent exponential or Weibull distributions and that the censoring variable, when present, also has an exponential distribution or a Weibull distribution with a shape parameter common to that of the failure time distribution. With estimation of a relative risk of four or less, one gains little with an increase in the ratio of unexposed to exposed cohort members beyond five or so.