# Testing equivalence between two survival curves

What is the standard procedure to test equivalence (not difference) between two survival curves? Is it available in R?

One way, I think, would be to calculate the 90% confidence interval for the hazard ratio and see if it lies within the equivalence region ($\delta_{low}$, $\delta_{upp}$). Would it be correct? Is it the standard way to test equivalence between two survival curves?

Your basic idea is correct. Testing equivalence (or perhaps more common, in clinical applications, non-inferiority) of survival curves is covered nicely in this freely available review.

If your data satisfy the proportional hazards assumption for Cox regression, then you can test whether the confidence interval for the ratio of two hazard ratios (or the difference between their Cox regression coefficients) falls completely within the equivalence region that you have set. Quoting from the above review:

One could conclude noninferiority if the (1 − 2α) level confidence interval for the relative risk is entirely below γ0. Similarly, one could conclude equivalence if the (1 − 2α) level confidence interval is entirely between 1/γ0 and γ0.

Here, $\alpha$ is the choice for Type I error rate and $\gamma 0$ is the ratio of hazard ratios deemed acceptable for non-inferiority. Should your data not satisfy the proportional hazards assumption, the cited review covers how to test for non-inferiority or equivalence at specific time points.

Perhaps even more important to consider is whether you will have enough power to provide a useful test of equivalence. Survival data sets with small numbers of events can have very wide confidence intervals for hazard ratios, so that there is a risk of deeming 2 survival curves "equivalent" only if you are willing to accept a very wide equivalence region. Think carefully about how you or others will apply your findings of statistical equivalence.

• Thank you very much for your input (+1, and I will probably accept it shortly). You have raised a good point. My concern comes from the fact that the power for such a test is very low, except if the number of events is very very large or if the equivalence region is very wide. – user7064 Nov 22 '16 at 15:35

Cox's Proportional Hazards Model is usually used as equivalence tests for two survival curves.well you need to look for sample size before going to test for equivalence.

There is a package 'survival' in R , that you can use. I have no idea on how your data looks.probably, looking at datasets(kidney,logan,nwtco,heart and many more) in 'survival' package might help you.

well there is another test log rank that you can use. let me explain you the difference between cox and log rank

Cox Model ,provides the primary information desired from a survival analysis, hazard ratios and adjusted survival curves, with a minimum number of assumptions. The long rank analysis answers the question of whether the two arms of a trial were different enough to be statistically significant. It place no conditions or assumptions on this analysis and ignores the fact that even the most well run clinical trials have imbalances between arms.you always want to see a Cox analysis produce a p-value equal to or lower than that of the log rank analysis of the survival data.When the Cox analysis produces a higher p-value - particularly a p-value greater than p=0.05 - then you have problems. Log rank is also considered by some to be the more pure measure of statistical significance since the Cox analysis depends on a qualitative judgment i.e weather there is increase or decrease of survival

• How is the Cox model used to test equivalence? I know the survival package. As far as I know, however, there is no method for equivalence testing. The word "equivalence" does not appear in the link you provide... I am sorry but I do not see how this answers my question... – user7064 Nov 22 '16 at 13:01
• The Cox analysis answers a slightly different question: If we mathematically control for all imbalances between the arms, were the two arms of the trial different enough to be statistically significant. It is a subtle, but important, difference. – GD_N Nov 22 '16 at 13:39
• @GN_D: I understand the difference you mention. My concern, however, is to test equivalence, not difference. Hence my point that your post does not answer the question... sorry – user7064 Nov 22 '16 at 13:41
• well have add more information and edited my answer for you. hope it helps you! @user7064 – GD_N Nov 22 '16 at 13:48
• Sorry but I do not think you understand my concern. I know the log-rank test and Cox regression which both assess difference. However, I want to establish equivalence. In any case, thank you for your input. – user7064 Nov 22 '16 at 13:53