# Calculate confidence interval for hazard ratio given maximum likelihood and standard error

I'm doing a meta-analysis and one of the studies presents the hazard ratio (HR) with the maximum likelihood estimate (MLE) with its standard error (SE). The rest of the studies give the HR and the confidence interval (CI) for the HR. Now, I know how to the the SE for the HR given the HR's CI. My question here is how can I get the HR's CI from the first study given only the MLE and its SE (the standard error of the MLE). This is my first meta-analysis and my first jump into this type of statistics so my question might be very bald for most of you.

I was looking here: http://www.mas.ncl.ac.uk/~nmf16/teaching/mas3311/week10.pdf but in the example they put they calculate the HR's CI for data when there is no more differences between two individuals. I guess the results from a survival analysis can't be interpreted that simple so I can't applied the same maths... Or maybe I'm just struggling with the interpretation of these concepts in general.

If anyone could give an example of how to do it correctly or give me a link to look at to understand better, I'd really appreciate it.

Thanks!

You just need to reverse the process you already know. I think you have been a bit confused by all the detail in the reference you cite which is being very precise about the meaning of the coefficients.

• Hi, are you referring to use the formula in the reference I cite? Or which process? Jan 22, 2016 at 13:40
• When you have the standard error and the estimate you just multiply the standard error by the multiplier for your chosen confidence coefficient (may be 1.96) and then use estimate plus and minus that as your confidence interval limits. You would want to do this on the log scale of course so the limits are symmetrical on that scale. Jan 25, 2016 at 13:25
• Hi, thanks for the comment. Maybe I didn't expressed in the clearest way possible on my question but the standard error given on the paper corresponds to the likelihood estimate not to the hazard ratio itself. That's my doubt. I know how to get the CIs given the SE of the hazard ratio but not when the SE given is the one of the MLE or maybe I'm interpreting this wrong? Jan 26, 2016 at 10:31
• Without seeing the paper I cannot be sure but I was assuming that the MLE to which they refer was the MLE of the hazard ratio. Do you have any reason to doubt that or does it seem that they were estimating something else? Jan 26, 2016 at 16:22
• Hmm, you assumed correctly, the MLE is of the hazard ratio but the SE is expressed next to the MLE not with the hazard ratio. This is the paper, I think it's open access: ncbi.nlm.nih.gov/pubmed/25353007 And the table I'm looking for the HR ratios is table 4. Really big help if you find any time to have a look but I can understand if you can't. Thanks a lot for still answering! :) Jan 27, 2016 at 11:30