Calculating group size from hazard ratio

Question

I am undertaking a systematic review and as part of the data extraction process I would like to know how many patients were analysed in the treatment and control arm (in a particular subgroup). In one included study, the only information given (for this particular subgroup) is the total number of patients analysed, the number of events in each arm and the hazard ratio with the CI. Based only on this information, is it possible to calculate the number of patients analysed in each arm (treatment arm and control arm)? Thank you very much in advance for your help.

Answer 1

If this was a designed trial, do you have any reason to suppose that there were not approximately equal numbers in each arm within this subgroup?

The simplest way to get the information would be to ask the authors of the study; they should be prepared to provide supporting information to those who inquire about published work.

Failing that, I see no way to calculate the numbers, but you might be able to make a reasonable estimate. The hazard ratio represents the relative risk of events between the 2 arms. You might try an estimate assuming that the relative numbers of events would have been proportional to the hazard ratio if there were equal numbers in each arm, and solving for the numbers in each arm that would be consistent with the observed numbers of events and total number in the subgroup. The accuracy of this estimate will depend on the ways that the analysis was performed, the timings and distributions between arms of censored cases, etc., and might not work if almost all cases in an arm either had events or did not have events. But for your purposes that might be better than nothing, provided you specify in your review how you made your estimate and how errors in your estimate might influence the conclusions of your review.