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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?

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Yes, this is correct, that is, one can easily reconstruct the (log) odds ratio for other group comparisons, but unfortunately, it is also correct that one cannot reconstruct the standard error of this derived log odds ratio, unless one has information on the number of events in group A. The problem is that log(odds(B)/odds(A)) and log(odds(C)/odds(A)) are not independent due to reuse of information from group A and the degree of dependence depends on log(odds(A)).

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    $\begingroup$ The covariances between the coefficient estimates need to be taken into account for these other ratios, following the formula for the variance of a sum of correlated variables. Those covariances can involve considerations beyond just the numbers of events. They are calculated as part of the fitting process but aren't reported in typical model summaries. That said, weighting the studies by the numbers of cases (or by the numbers of events, as you suggest) will be better than nothing. $\endgroup$
    – EdM
    Commented Apr 5, 2023 at 15:46
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    $\begingroup$ 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. $\endgroup$
    – Wolfgang
    Commented Apr 5, 2023 at 18:22

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