Signature
SE_beta = (log(HR_U) - log(HR_L)) / (2 * z)
| Inputs | Definition | Unit |
|---|---|---|
HR_U | Upper limit of the published interval, above HR_L | ratio, no unit |
HR_L | Lower limit of the published interval, above zero | ratio, no unit |
z | Standard normal value for the coverage of the published interval: 1.96 for 95%, 1.64 for 90% and 2.58 for 99% | none |
SE_beta | Standard error of the natural logarithm of the reported hazard ratio | log hazard ratio, no unit |
|---|
Function
Relative hazard function
Maps the hazards of an event in a treatment group and a comparator group at the same time t to their ratio. Under proportional hazards the ratio is constant over time, and it can then be applied to a baseline survival curve or a baseline transition probability to obtain absolute outcomes for the treatment group.
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Implementations
Excel
Log hazard ratio standard error from named interval limits
With the published limits in cells named LowerHR and UpperHR and the standard normal value in ZValue, the formula returns the standard error, held in a cell named SELogHR.
=(LN(UpperHR)-LN(LowerHR))/(2*ZValue)
Assumptions
Reported interval symmetric on the log scale
The published interval was computed on the log scale from a large-sample standard error, as Cox and logrank analyses usually report it, and its limits are given to enough significant figures; rounding of the limits passes straight into the standard error.
Coverage of the reported interval known
The coverage of the interval is stated, so that z can be matched to it. Most trial reports give 95% intervals.
Worked examples
Bladder cancer trial interval of 0.71 to 1.02
Tierney and colleagues' bladder cancer example reports a hazard ratio of 0.85 with a 95% interval of 0.71 to 1.02, giving a standard error of about 0.0924 for the log hazard ratio and a variance of about 0.0085, as in the paper.
HR_U = 1.02; HR_L = 0.71; z = 1.96; SE_beta = 0.0924
Round trip from the 0.5158 to 0.9501 interval
Applied to the interval from HE-EX-HR-009, the formula returns about 0.1558, the standard error that produced it (computed here for illustration).
HR_U = 0.9501; HR_L = 0.5158; z = 1.96; SE_beta = 0.1558
Common errors
Taking the interval width on the ratio scale
Dividing the untransformed width, 1.02 minus 0.71, by 3.92 gives about 0.0791 instead of 0.0924, understating the standard error of the log hazard ratio by about 14% in the bladder cancer example (computed here for illustration).
Using 1.96 for an interval with other coverage
If the same limits were a 99% interval, z is 2.58 and the standard error about 0.0702; using 1.96 would overstate it by a factor of about 1.32 (computed here for illustration).
Sources
Variance of the log hazard ratio from a confidence interval
Tierney JF, Stewart LA, Ghersi D, Burdett S, Sydes MR. Practical methods for incorporating summary time-to-event data into meta-analysis. Trials. 2007;8:16. Section 3, equation 10: the variance of the log hazard ratio is the square of the difference between the logs of the upper and lower limits divided by 2 times the z value for the upper limit; the bladder cancer example (hazard ratio 0.85, 95% interval 0.71 to 1.02) gives 0.0085.
Standard error of a ratio measure from its confidence limits
Higgins JPT, Li T, Deeks JJ (editors). Chapter 6: Choosing effect measures and computing estimates of effect (last updated August 2023). In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 6.3.2 with 6.3.1: for a ratio measure such as the hazard ratio, the natural logs of the confidence limits are taken and the standard error of the log ratio is their difference divided by 3.92 for a 95% interval, 3.29 for 90% and 5.15 for 99%.
Canonical Identity
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