# Estimating the standard error or confidence interval of a hazard (or odds) ratio from other reported contrasts

I am performing a meta-analysis and have a dataset of hazard ratios and odds ratios collected from primary studies. Suppose for a study we have a recorded hazard ratio (HR) (or odds ratios (OR)) (both point estimates and confidence limits) for control (A) vs. several treatments (B,C,D) e.g. A vs. B, A vs. C, A vs. D. The reference group is always A. Is it possible to estimate point estimates and standard errors or confidence intervals for a contrast that doesn't involve A using the information from the study e.g. B vs. C? It looks to me that we can indeed estimate the point estimate e.g. on the log scale: B-C = (B-A) - (C-A). But am I correct that we cannot reliably estimate the standard error or confidence limits of the corresponding HR or OR's? There is sample size information for each group A through D, I'm not sure if that will be enough to estimate the standard error or confidence limits?

• The covariance between the two log odds ratios is $1/x_A + 1/(n_A-x_A)$ where $x_A$ denotes the number of events in group A and $n_A$ the number of people in group A. So in this case, the covariance can be easily calculated when the number of events in group A (and the group size) is known. Apr 5, 2023 at 18:22