# The estimated logarithm of the hazard ratio is approximately normally distributed

According to this document: The estimated logarithm of the hazard ratio is approximately normally distributed with variance (1/d1) + (1/d2), where d1 and d2 are the numbers of events in the two treatment groups.

Do you have a reference for this statement? Or at least can you say me which estimator is used?

The fact that this is approximately normally distributed relies on the central limit theorem (CLT), so will be a better approximation in large samples. The CLT works better for the log of any ratio (risk ratio, odds ratio, hazard ratio..) than for the ratio itself.

In suitably large samples, I think this is a good approximation to the variance in two situations:

1. The hazard in each group is constant over time (regardless of the hazard ratio)
2. The proportional hazards assumption holds and the hazard ratio is close to 1

I think it may become a fairly crude assumption in situations far from these, i.e. if the hazards vary considerably over time and the hazard ratio is far from 1. Whether you can do better depends on what information is available. If you have access to the full data you can fit a proportional hazards model and get the variance of the log hazard ratio from that. If you only have the info in a published paper, various other approximations have been developed by meta-analysts. These two references are taken from the Cochrane Handbook:

1. M. K. B. Parmar, V. Torri, and L. Stewart (1998). "Extracting summary statistics to perform meta-analyses of the published literature for survival endpoints." Statistics in Medicine 17 (24):2815-2834.
2. Paula R. Williamson, Catrin Tudur Smith, Jane L. Hutton, and Anthony G. Marson. "Aggregate data meta-analysis with time-to-event outcomes". Statistics in Medicine 21 (22):3337-3351, 2002.

In Parmar et al, the expression you give would follow from using observed numbers in place of expected in their equation (5), or combining equations (6) and (12). Equations (5) and (6) are based on logrank methods. They reference Kalbfleisch & Prentice for equation (12) but I don't have that to hand, so maybe someone who does might like to check it and add to this.