Interpreting results: Comparing two survival curves

Prism compares two survival curves by two methods: the log-rank test (also called the Mantel-Cox test) and the Gehan-Breslow-Wilcoxon test. It doesn't ask for your preference, but always reports both.

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The log-rank (Mantel-Cox) test is the more powerful of the two tests if the assumption of proportional hazards is true. Proportional hazards means that the ratio of hazard functions (deaths per time) is the same at all time points. One example of proportional hazards would be if the control group died at twice the rate as treated group at all time points. Prism actually computes the Mantel-Haenszel method, which is nearly identical to the log-rank method (they differ only in how they deal with two subjects with the same time of death).

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The Gehan-Breslow-Wilcoxon method gives more weight to deaths at early time points. This often makes lots of sense, but the results can be misleading when a large fraction of patients are censored at early time points. In contrast, the log-rank test gives equal weight to all time points. The Gehan-Wilcoxon test does not require a consistent hazard ratio, but does require that one group consistently have a higher risk than the other.

You need to choose which P value to report. Ideally, this choice should be made before you collect and analyze your data.

If in doubt, report the log-rank test (which is more standard) and report the Gehan-Wilcoxon results only if you have a strong reason.

If two survival curves cross, then one group has a higher risk at early time points and the other group has a higher risk at late time points. In this case, neither the log-rank nor the Wilcoxon-Gehan test rests will be very helpful.

The P value tests the null hypothesis that the survival curves are identical in the overall populations. In other words, the null hypothesis is that the treatment did not change survival.

If the null hypothesis is true, what is the probability of randomly selecting subjects whose survival curves are as different (or more so) than was actually observed?

Prism always calculates two-tail P values. If you wish to report a one-tail P value, you must have predicted which group would have the longer median survival before collecting any data. Computing the one-tail P value depends on whether your prediction was correct or not.

•	If your prediction was wrong, the one-tail P value equals 1.0 minus half the two-tail P value. This value will be greater than 0.50, and you must conclude that the survival difference is not statistically significant.

If survival exceeds 50% at the longest time point, then median survival cannot be computed and Prism leaves it blank. Even so, the P values and hazard ratio are still valid.

If the survival curve is horizontal at 50% survival, the median survival is ambiguous, and different programs report median survival differently. Prism reports the average of the first and last times at which survival is 50%.

When comparing two survival curves, Prism also reports the ratio of the median survival times along with its 95% confidence interval. You can be 95% sure that the true ratio of median survival times lies within that range.

This calculation is based on an assumption that is not part of the rest of the survival comparison. The calculation of the 95% CI of ratio of median survivals assumes that the survival curve follows an exponential decay. This means that the chance of dying in a small time interval is the same early in the study and late in the study. If your survival data follow a very different pattern, then the values that Prism reports for the 95% CI of the ratio of median survivals will not be correct.

If you compare two survival curves, Prism reports the hazard ratio and its 95% confidence interval.

Hazard is defined as the slope of the survival curve – a measure of how rapidly subjects are dying. The hazard ratio compares two treatments. If the hazard ratio is 2.0, then the rate of deaths in one treatment group is twice the rate in the other group.

The computation of the hazard ratio assumes that the ratio is consistent over time, and that any differences are due to random sampling. So Prism reports a single hazard ratio, not a different hazard ratio for each time interval. Prism 5 computes the hazard ratio and its confidence interval using the Mantel-Haenszel method. Prism 4 computed the hazard ratio itself (but not the confidence interval) by the log-rank method. The two are usually quite similar.

If the hazard ratio is not consistent over time, the value that Prism reports for the hazard ratio will not be useful. If two survival curves cross, the hazard ratios are certainly not consistent.