Tests for equivalence of two survival functions: Alternative to the tests under proportional hazards

Investigating non-inferiority or equivalence in time-to-event data under non-proportional hazards

The classical approach to analyze time-to-event data, e.g. in clinical trials, is to fit Kaplan-Meier curves yielding the treatment effect as the hazard ratio between treatment groups. Afterwards, a log-rank test is commonly performed to investigate whether there is a difference in survival or, depending on additional covariates, a Cox proportional hazard model is used. However, in numerous trials these approaches fail due to the presence of non-proportional hazards, resulting in difficulties of interpreting the hazard ratio and a loss of power. When considering equivalence or non-inferiority trials, the commonly performed log-rank based tests are similarly affected by a violation of this assumption. Here we propose a parametric framework to assess equivalence or non-inferiority for survival data. We derive pointwise confidence bands for both, the hazard ratio and the difference of the survival curves. Further we propose a test procedure addressing non-inferiority and equivalence by directly comparing the survival functions at certain time points or over an entire range of time. Once the model's suitability is proven the method provides a noticeable power benefit, irrespectively of the shape of the hazard ratio. On the other hand, model selection should be carried out carefully as misspecification may cause type I error inflation in some situations. We investigate the robustness and demonstrate the advantages and disadvantages of the proposed methods by means of a simulation study. Finally, we demonstrate the validity of the methods by a clinical trial example.

Equivalence tests before end of follow-up under the class of log transformation model

One important topic in clinical trials is to show that the effects of new and standard treatments are equivalent in terms of clinical relevance. In literature, many equivalence tests based on the maximal difference between two survival functions for the two treatments over the whole time axis have been proposed. However, since survival times can only be observed until the end of follow-up, an equivalence test should be based on a comparison only in the observed time-window dictated by the end of follow-up. In this article, under the class of log transformation model, we propose an asymptotical α-level equivalence test for the difference between two survival functions that only addresses equivalence until the end of follow-up. We demonstrate that the hypothesis of equivalence of two survival functions before the end of follow-up can be formulated as interval-based hypothesis testing which involves the treatment effect parameter. Simulation results indicate that when sample size is sufficiently large the proposed test controls the type I error effectively and performs well at detecting the equivalence. The proposed test is applied to a dataset from veteran's administration lung cancer trial.

Equivalence tests under the Cox-Aalen model and the partly Aalen model

For the equivalence trial with survivor outcomes from two treatment groups, the most popular testing procedure is the test proposed by under the proportional hazards (PH) model. In this article, when the treatment effect is time invariant, we demonstrate that the result under the PH model can be extended to the Cox-Aalen model. When the treatment effect is time-variant, we propose an equivalent test for the differences of two cumulative hazard functions under the partly Aalen model. Simulation studies show that the proposed tests perform well in finite samples. We illustrate the proposed tests using bladder cancer data and the primary biliary cirrhosis (PBC) data.

Testing equivalence of survival before but not after end of follow-up

For equivalence trials with survival outcomes, a popular testing approach is the elegant test for equivalence of two survival functions suggested by Wellek (Biometrics 49: 877-881, 1993). This test evaluates whether or not the difference between the true survival curves is practically irrelevant by specifying an equivalence margin on the hazard ratio under the proportional hazards assumption. However, this approach is based on extrapolating the behavior of the survival curves to the whole time axis, whereas in practice survival times are only observed until the end of follow-up. We propose a modification of Welleks test that only addresses equivalence until end of follow-up and derive the large sample properties of this test. Another issue is the proportional hazards assumption which may not be realistic. If this assumption is violated, one may severely misjudge the actual treatment effect with a hazard ratio quantification and wrongly declare equivalence. We suggest a non-parametric test for assessing survival equivalence within the follow-up period. We derive the large sample properties of this test and provide an approximation to the limiting distribution under some mild assumptions on the functional form of the difference between the two survival curves. Both suggestions are investigated by simulation and applied to a clinical trial on survival of gastric cancer patients.

Tests for equivalence of two survival functions: alternatives to the PH and PO models

For the equivalence trial with survivor outcomes from two treatment groups, the most popular testing procedure is the extension of log-rank test under proportional hazards (PH) model. In literature, an alternative test has been proposed under the  proportional odds (PO) survival model. In practice, both PH and PO hazards assumptions can be violated. One alternative is the log transformation model other than the PH or PO models. Another alternative model is the partly Aalen's additive risk model which allows the influence of some covariates to vary over time. In this article, we propose equivalence tests for the difference of two survival functions under the class of log transformation model and that of two cumulative hazard functions under the partly Aalen's model. Simulation studies demonstrate that the proposed tests perform well in finite samples.