 # Calculate ratio and confidence interval for two hazard ratios

I’d like to investigate from published data if two different diagnostic evaluations have similar performance regarding survival. Usually the studies I’d like to pool report the binary outcome percentage and the hazard ratio of the two groups. As an example:
-diagnostic evaluation 1, group with high risk has an HR of 2.3 compared to group with low risk
-diagnostic evaluation 2, group with high risk has an HR of 2.0 compared to group with low risk
All the studies evaluate the same patients with both the two diagnostic methods.

I was thinking of calculating a ratio with 95%CI between the Hazard ratios and seeing if this would include or not 1. Would this be a formally correct approach?

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I think so, if high risk and low risk have amazingly tight definitions so that the estimates are estimating the same thing (they usually aren’t).

It is easy to compute the variance of the sum or difference of two log hazard ratios. You can get a standard error of the difference in log HRs and use that to get a fold-change margin of error and confidence limits for the ratio of ratios.

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Thank you very much for the reply. In this case the reported data should be very tight in definitions and refer to the same thing. Do you know if there Is a reference for this calculations or a similar worked example?

If I understand well:

• I can calculate the variance of both the HRs (as square(SE_HR) x sample_size)
• Then I can obtain the variance of the difference considering them independent and so without the need for the covariance Var_HR_1 - Var_HR_2, and i can back calculate the SE of the difference
• Then I calculate the HR ratio on the log scale
• I consider the SE of the difference and calculate the ratio with one of the HR on the log scale and use that to build the 95% CI (I call this SE_fold)? Ratio_HR ± 1.96 x SE_fold

I think you could use the general method described by Zou & Donner Stat Med 27(10):1693-1702 (see also the comment by Julia Singer: Stat Med 29(16):1757-1759)

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As an update to Zou & Donner, you may also use the MOVER-R approach by @RGNewcombe who provides a wonderful list of resources for CI calculation (including MOVER-D and MOVER-R) here.

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Thank you for your kind words! All this work is described more fully in my book, see https://www.routledge.com/Confidence-Intervals-for-Proportions-and-Related-Measures-of-Effect-Size/Newcombe/p/book/9781439812785 , which also includes a link to a zip file containing these and other related spreadsheets. Right now, they’re advertising SUPER SAVINGS – 20% OFF BOOKS & 35% OFF EBOOKS – SHOP NOW!

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