On the covariance of two correlated log-odds ratios

In many applications two correlated estimates of an effect size need to be considered simultaneously to be combined or compared. Apparently, there is a need for calculating their covariance, which however requires access to the individual data that may not be available to a researcher performing the analysis. We present a simple and efficient method for calculating the covariance of two correlated log-odds ratios. The method is very simple, is based on the well-known large sample approximations, can be applied using only data that are available in the published reports and more importantly, is very general, because it is shown to encompass several previously derived estimates (multiple outcomes, multiple treatments, dose-response models, mutually exclusive outcomes, genetic association studies) as special cases. By encompassing the previous approaches in a unified framework, the method allows easily deriving estimates for the covariance concerning problems that were not easy to be obtained otherwise. We show that the method can be used to derive the covariance of log-odds ratios from matched and unmatched case-control studies that use the same cases, a situation that has been addressed in the past only using individual data. Future applications of the method are discussed.