Estimating the relative hazard by the ratio of logarithms of event-free proportions

Clinical trials typically examine associations between an intervention and the occurrence of a clinical event. The association is often reported as a relative risk, more rarely as an odds ratio. Unfortunately, when the scientific interest lies with the ratio of incidence rates, both these statistics are inaccurate: the odds ratio is too extreme, and the relative risk too conservative. These biases are particularly strong when the outcomes are common. This paper describes an alternative statistic, the ratio of logarithms of event-free proportions (or relative log survival), which is simple to compute yet unbiased vis-à-vis the relative hazard. A formula to compute the sampling error of this statistic is also provided. Multivariate analysis can be conducted using complementary log-log regression. Precise knowledge of event occurrence times is not required for these analyses. Relative log survival may be particularly useful for meta-analyses of trials in which the proportion of events varies between studies.