Hazard Ratio Calculator
A hazard ratio calculator recovers the statistics you need from a published survival result. Enter a hazard ratio with either its confidence interval or its standard error, and read back the log hazard ratio, the standard error, a p-value, and a recomputed interval ready for pooling.
Recover the log hazard ratio, its standard error, and a p-value from a published hazard ratio. Enter the hazard ratio and either its confidence interval or its standard error on the log scale.
How to use the hazard ratio calculator
Most survival papers report a hazard ratio with a 95 percent confidence interval but no standard error. This calculator works backward from what the paper gives you.
- 1. Enter the hazard ratio. Type the published point estimate, for example 0.70.
- 2. Add the interval or the standard error. Enter the lower and upper confidence limits if that is what the paper reports, or the standard error directly if you have it.
- 3. Match the confidence level. Tell the calculator whether the reported interval is 90, 95, or 99 percent so the back-calculation uses the right multiplier.
- 4. Read the recovered statistics. You get the log hazard ratio, the standard error on the log scale, a p-value, and a clean recomputed interval, all ready to paste into the generic-effect input of a pooling tool.
A worked example
Say a trial reports a hazard ratio of 0.70 with a 95 percent confidence interval of 0.55 to 0.89. The calculator takes natural logs of the bounds, so the log scale interval runs from about -0.60 to -0.12. The standard error of the log hazard ratio is the width of that interval divided by twice 1.96, which comes to roughly 0.12.
With the log hazard ratio of about -0.36 and a standard error of 0.12, the test statistic is around -2.9, giving a p-value near 0.004. The hazard ratio is significantly below 1, so the treatment slows the event rate by about 30 percent. Those two recovered numbers, the log hazard ratio and its standard error, are precisely what a pooled analysis needs from each survival study.
Getting a hazard ratio ready for meta-analysis
Survival studies almost never report the standard error of the log hazard ratio directly, yet that is exactly what a meta-analysis needs. This calculator back-calculates it from the confidence interval: it takes the natural log of the upper and lower bounds, finds the distance between them, and divides by twice the appropriate multiplier for your confidence level. The result is the standard error on the log scale, which you can paste straight into the generic-effect input of a pooling tool. The same procedure works for any ratio measure reported only as an estimate with an interval.
For the meaning behind the numbers, the guide to interpreting hazard ratios explains why a hazard ratio is not the same as a risk ratio and how to read it against survival curves. Once you have the log hazard ratio and its standard error for every study, combine them in the pool the log hazard ratios and present the pooled hazard ratio as a forest plot of the pooled result. If your outcomes are simple events rather than time-to-event data, the relative risk calculator is the right tool instead.
Common mistakes to avoid
- Reading a hazard ratio as a risk ratio. A hazard ratio acts on the event rate at each instant, not on the cumulative probability over the whole study, so a hazard ratio of 2 does not mean twice as many people will ever have the event.
- Recovering the standard error from the wrong level. If a paper reports a 90 percent interval and you treat it as 95 percent, the standard error is wrong and the pooled result drifts. Always match the reported confidence level.
- Pooling on the raw scale. Hazard ratios, like other ratios, must be combined on the log scale. Feed the log hazard ratio and its standard error into the pooling tool, not the raw ratio.
- Ignoring the proportional-hazards assumption. A single hazard ratio assumes the rate ratio is constant over time. If survival curves cross, one summary hazard ratio can be misleading no matter how cleanly it pools.
Pooling hazard ratios across survival studies?
Extracting hazard ratios consistently, handling studies that report them differently, and pooling them correctly is delicate work. A methodologist can take the extraction and the analysis off your plate and write it up for peer review.
Get help with your analysisFrequently asked questions
How do you interpret hazard ratio results?
A hazard ratio compares the rate at which events happen in two groups at any given moment over the follow-up period. A value of 1 means the event rates are the same, above 1 means the event happens faster in the first group, and below 1 means it happens more slowly. Because it is a rate ratio rather than a risk ratio, it describes the instantaneous hazard rather than the cumulative probability, so it is best read alongside survival curves. The confidence interval tells you whether the difference in rates is statistically reliable.
What does a hazard ratio of 1.5 mean?
A hazard ratio of 1.5 means that, at any point during follow-up, the event is occurring about 50 percent faster in the first group than in the comparison group. In a survival study of a harmful outcome this signals worse prognosis for that group; in a study of a desirable event such as recovery it would signal a benefit. As with all ratio measures, the confidence interval decides credibility: if it stretches below 1, the elevated hazard could be a chance result.
What does a hazard ratio of 2 mean?
A hazard ratio of 2 means the event rate is twice as high in the first group at any instant over the study. It is a strong effect, but it does not mean members of that group are twice as likely to ever experience the event, because the hazard ratio acts on the rate over time rather than the final cumulative risk. Over a short follow-up the difference in actual event counts can be modest even when the hazard ratio is large, which is why the absolute numbers and the survival curves matter.
Is a high or low hazard ratio good?
It depends entirely on the outcome. For a harmful event such as death or relapse, a hazard ratio below 1 is good because the treatment slows the event rate, while a value above 1 is bad. For a desirable event such as discharge or pregnancy, the reverse holds. The reference value of 1 always means no difference, so the direction of benefit is read relative to that, and the confidence interval shows whether the apparent benefit or harm is statistically secure.
Is a hazard ratio the same as relative risk?
No, although they are often close. Relative risk compares the cumulative probability of an event by the end of the study, while a hazard ratio compares the instantaneous event rate at each moment during follow-up. When events are rare and the follow-up is short, the two are numerically similar, but for common events over long follow-up they can differ noticeably. The hazard ratio is the natural measure for time-to-event data analysed with a Cox model.
Can a hazard ratio be negative?
No. A hazard ratio is a ratio of two rates, both of which are positive, so it can never be below zero. It ranges from just above zero upward, with 1 marking no difference. What can be negative is the log hazard ratio, which is simply the natural logarithm of the hazard ratio and turns values below 1 into negative numbers. That log scale is the one used for pooling.