# How to pool (average) hazard ratios?

Let's say the task did not allow me to use either the MICE package or RMS / HMISC to perform survival analysis on multiply imputed data. I had to impute 5 different data sets and calculate separate hazard ratios in each of these.

I need to pool the hazard ratio from each set to one "average" / "pooled" hazard ratio but how is that done? I've searched the web but can't find no formula or package that would facilitate this.

Now, there are plenty of information on pooling hazard ratios for meta-analyses but recall that my estimates originate from the same data which was used to impute 5 complete data sets.

## 1 Answer

Averaging the coefficients just means that: compute the hazard ratios in each of your dataset, add them and divide by 5 (= the number of imputed datasets).

However, Rubin's rules assume that the sampling distribtution (actually, the posterior distribution) of the estimates is normal. This is more likely to be true for log(hazard ratios) than hazard ratios. Remember that a normal distribution covers all negative and positive numbers, while a hazard ratio cannot be negative. So more often the log-hazard ratios are combined using Rubin's rules and than the combined log-hazard ratio is transformed to hazard ratios.

• Thanks @Maarten Buis. Can you recall any reference with the equation to perform the calculation? – Adam Robinsson Sep 25 '15 at 8:09
• Any book on missing data includes the Rubin's rules, e.g.: Paul D. Allison (2002) Missing Data. Thousand Oaks: Sage. Joseph L. Schafer (1997) Analysis of Incomplete Multivariate Data. Boca Raton: Chapman & Hall/CRC. Roderick J.A. Little and Donald B. Rubin (2002) Statistical Analysis with Missing Data. second edition. Hoboken: Wiley Interscience. – Maarten Buis Sep 25 '15 at 9:11
• Indeed I think it would hold better for the log(hazard ratios), i.e. the regression coefficient from the Cox model. This way, the maximum likelihood estimate follows an asymptotic normal distribution. – Theodor Sep 25 '15 at 14:17
• I really appreciate the answers, however the source I found dealt with this issue in terms of pure mathematical equations, which makes it difficulr for me to apply. Did I understand it correctly that the hazard ratio, lower and upper bound is obtained by averaging the log of each one of these? Sounds a bit too simple... No standarderror to consider? 😎 – Adam Robinsson Sep 25 '15 at 15:05
• biomedcentral.com/1471-2288/9/57 was a bit equatious... – Adam Robinsson Sep 25 '15 at 15:07