Solved – What are the pros and cons of using the logrank vs. the Mantel-Haenszel method for computing the Hazard Ratio in survival analysis

hazardsurvival

One way to summarize the comparison of two survival curves is to compute the hazard ratio (HR). There are (at least) two methods to compute this value.

  • Logrank method. As part of the Kaplan-Meier calculations, compute the number of observed events (deaths, usually) in each group ($Oa$, and $Ob$), and the number of expected events assuming a null hypothesis of no difference in survival ($Ea$ and $Eb$). The hazard ratio then is:
    $$
    HR= \frac{(Oa/Ea)}{(Ob/Eb)}
    $$
  • Mantel-Haenszel method. First compute V, which is the sum of the hypergeometric variances at each time point. Then compute the hazard ratio as:
    $$
    HR= \exp\left(\frac{(Oa-Ea)}{V}\right)
    $$
    I got both these equations from chapter 3 of Machin, Cheung and Parmar, Survival Analysis. That book states that the two methods usually give very similar methods, and indeed that is the case with the example in the book.

Someone sent me an example where the two methods differ by a factor of three. In this particular example, it is obvious that the logrank estimate is sensible, and the Mantel-Haenszel estimate is far off. My question is if anyone has any general advice for when it is best to choose the logrank estimate of the hazard ratio, and when it is best to choose the Mantel-Haenszel estimate? Does it have to do with sample size? Number of ties? Ratio of sample sizes?

Best Answer

I think I figured out the answer (to my own question). If the assumption of proportional hazards is true, the two methods give similar estimates of the hazard ratio. The discrepancy I found in one particular example, I now think, is due to the fact that that assumption is dubious.

If the assumption of proportional hazards is true, then a graph of log(time) vs. log(-log(St)) (where St is the proportional survival at time t) should show two parallel lines. Below is the graph created from the problem data set. It seems far from linear. If the assumption of proportional hazards is not valid, then the concept of a hazard ratio is meaningless, and so it doesn't matter which method is used to compute the hazard ratio.

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I wonder if the discrepancy between the logrank and Mantel-Haenszel estimates of the hazard ratio can be used as a method to test the assumption of proportional hazards?

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