A simple method for combining binomial counts or proportions with hazard ratios for evidence synthesis of time-to-event data

In studies with time-to-event data, outcomes may be reported as hazard ratios (HR) or binomial counts/proportions at a specific time point. If the intent is to synthesise evidence by performing a meta-analysis or network meta-analysis (NMA) using the HR as the measure of treatment effect, studies that only report binomial data cannot be included in the network. Methods for converting binomial data to HRs were investigated, so that studies reporting binomial data only could be included in a network of HR data. Estimating the log HR is relatively straightforward under the assumptions of proportional hazards and minimal censoring at the binomial data time point. Estimating the standard error of the log HR is harder, but a simple method based on using a Taylor series expansion to approximate the variance is proposed. Thus, we have 2 easy-to-calculate equations for the log HR and variance. The performance of the method was assessed using simulations and data from a NMA of multiple sclerosis treatments. In the simulation, our binomial method produced very similar HRs to those from survival analysis when censoring rates were low, and also when censoring rates were high but the event rate was low. In all situations, it outperformed using relative risk to approximate the HR. In the NMA, results were consistent between reported HRs and HRs derived from binomial data for studies that reported both types of data. This method may be useful for easily incorporating trials reporting binomial data into an evidence synthesis of HR data, under certain assumptions.