# Combine crude events and Hazard Ratio in Meta-analysis

I'm running a meta-analysis on several studies; most of them have crude event numbers that I can put in my software, but other ones have hazard ratios with 95% C.I. reported and no crude number reported. Question 1 is: is there a way to pool in a meta-analysis study that reports only crude events with study that reports only hazard ratios? If so, what software can I use to do that? Question 2 if the answer for Q1 is "no": is there a way to get from a hazard ratio the crude events number, knowing the number of patients?

• What are crude event numbers? – Michael R. Chernick Apr 20 '17 at 16:19
• For the trials where you only have the number of patients with an event, do you have something like patient-years of follow-up (ideally even patients with an event per patient-year of follow-up to first event or end of follow-up)? – Björn Apr 20 '17 at 16:26

Firstly, yes, this can be done. And it is probably best to try to get everything to the hazard ratio scale, because one would guess there's a good reason why time-to-event analyses were done in the first place (e.g. not all patients complete trial, event driven trials where not all patients were followed for the same duration etc.).

If you have a log-hazard ratio with standard error (easy to backcalculate from confidence intervals or p-values, if not given, or approximately from the total number of patients with an event), then everything is great and a standard approach would be to do an inverse variance (or corresponding random effects) meta-analysis of the estimate (log-hazard ratio) and its standard error.

Problems arise, if you do not have that information for all trials. It is the easiest to deal with, if you have a relatively constant hazard rate for events over time, in which case you can approximately assume an exponential time-to-event model. You can get an idea off whether that is appropriate, if some of the trials give you e.g. a Kaplan-Meier plot. Alternatively, you can think about whether it is plausible: e.g. if all patients are relatively stable hypertensive patients with a blood pressure of 160 to 170 before we start following them, then there is probably no reason why their hazard rate for death should differ too much or change much over time. In contrast, if we say start following patients the day after which they just survived a myocardial infarction, then there is probably much more of a risk that they died shortly thereafter (say within 30 days) and then the risk goes down over time.

If you have the number of patients with an event $y_{ij}$ in each treatment group $j=0,1$ of trial $i$ and the follow-up to first event or censoring $t_{ij}$ (i.e. summed up over all the patients), then you have a likelihood for that trial of $$(\exp(\text{log-HR}) \lambda_i)^{y_{ij}} \exp(e^{\text{log-HR} \times j} \lambda_i t_{ij}),$$ where $\text{log-HR}$ is the log-hazard ratio and $\lambda_i$ is a nuisance parameter for the control group hazard rate. In contrast, for the trials, for which you have the log-hazard ratio and its standard error, the likelihood is a normal pdf $\phi(\text{log-HR}), \text{SE}^2)$.

If you do not have this information, then this paper on the meta-analysis of aggregate data on medical event occurrence may be relevant and discusses a number of relevant options.

Getting from a hazard ratio and no further information back to the number of patients with an event is more or less impossible.

Finally, you always have the option of contacting the authors for extra information, checking clinical trial registries (and results databases), health authority briefing books, approved drug labels and so on.

• Thank you for your exhaustive reply. Some things I can't understand. Fact is, I have HRs, C.I., possibly a median time of FU and the number of the two whole treatment groups. The only thing I need and I don't have is the number of event; which, based on what you said, I would not be able to calculate in this setting. am I right? On the other hand, I have several studies with number of events and number of total group patients... but not Hazard Ratios. You can put that into your meta-analyzer, but how do you put event number for one study and Hazard Ratio for another study in your analysis? – pankevedmo Apr 20 '17 at 19:12
• a) Looking only at the number of patients with an event is problematic, if the distribution of drop-out times differs between groups. – Björn Apr 20 '17 at 20:06
• b) If you have the hazard ratio + CI (or standard error), then you can back-calculate the number of events in a 1:1 randomized trial the standard error of the log-hazard ratio is approximately $2/\sqrt{y_i}$, where $y_i$ is the total number of events in both treatment groups combined and the confidence interval is approximately $\log \text{HR} \pm 1.96 \text{SE}$. – Björn Apr 20 '17 at 20:11
• c) If you have different types of data, then you would normally not be able to use simple graphical user interface type meta-analysis programs (such as RevMan etc.) and would normally have to use some statistics package that lets you specify a different (log-)likelihood for each trial. Almost any serious statistics package can do that. Options include e.g. PROC NLMIXED or PROC MCMC in SAS, or Stan (easily accessed e.g. via the rstan R package), or stats4:::mle() in R, or many others. – Björn Apr 20 '17 at 20:17