Non-parametric assessment of non-inferiority with censored data

Non-inferiority trials with time-to-event data: clarifying the impact of censoring

In non-inferiority (NI) trials with time-to-event data, different types and patterns of censoring may occur, but their impact on trial results is not entirely clear. We investigated the influence of informative and non-informative censoring by conducting extensive simulation studies under the assumption that the NI margin is defined as a maximum acceptable hazard ratio and scenarios typically observed in recent NI trials. We found that while non-informative censoring tends to only affect the power, informative censoring can impact the treatment effect estimates, type I error rate, and power. The magnitude of these effects depends on the between-group differences in the failure and informative censoring risks, as well as the correlation between censoring and failure times, among other factors. The adverse impact of informative censoring was generally decreased with larger NI margins.

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.

Bisphosphonates and mortality: confounding in observational studies?

Numerous observational studies suggest that bisphosphonates reduce mortality. This study showed that bisphosphonate use is associated with lower mortality within days of treatment, although the association was not significant until the second week. Such an early association is consistent with confounding, although an early treatment effect cannot be ruled out.

INTRODUCTION

The purpose of this study was to examine whether confounding explains why numerous observational studies show that bisphosphonate use is associated with lower mortality. To this end, we examined how soon after treatment initiation a lower mortality rate can be observed. We hypothesized that, due to confounding, the association would be observed immediately.

METHODS

This was a retrospective cohort study of hip fracture patients discharged from Swedish hospitals between 1 July 2006 and 31 December 2015. The data covered 260,574 hip fracture patients and were obtained from the Swedish Hip Fracture Register and national registers. Of the 260,574 patients, 49,765 met all eligibility criteria and 10,178 were pair matched (bisphosphonate users to controls) using time-dependent propensity scores. The matching variables were age, sex, diagnoses, prescription medications, type of hip fracture, type of surgical procedure, known or suspected dementia, and physical functioning status.

RESULTS

Over a median follow-up of 2.8 years, 2922 of the 10,178 matched patients died. The mortality rate was 7.9 deaths per 100 person-years in bisphosphonate users and 9.4 deaths in controls, which corresponded to a 15% lower mortality rate in bisphosphonate users (hazard ratio 0.85, 95% confidence interval 0.79-0.91). The risk of death was lower in bisphosphonate users from day 6 of treatment, although the association was not significant until the second week.

CONCLUSION

Bisphosphonate use was associated with lower mortality within days of treatment initiation. This finding is consistent with confounding, although an early treatment effect cannot be ruled out.

NONPARAMETRIC TESTING FOR MULTIPLE SURVIVAL FUNCTIONS WITH NON-INFERIORITY MARGINS

New nonparametric tests for the ordering of multiple survival functions are developed with the possibility of right censorship taken into account. The motivation comes from non-inferiority trials with multiple treatments. The proposed tests are based on nonparametric likelihood ratio statistics, which are known to provide more powerful tests than Wald-type procedures, but in this setting have only been studied for pairs of survival functions or in the absence of censoring. We introduce a novel type of pool adjacent violator algorithm that leads to a complete solution of the problem. The limit distributions can be expressed as weighted sums of squares involving projections of certain Gaussian processes onto the given ordered alternative. A simulation study shows that the new procedures have superior power to a competing combined-pairwise Cox model approach. We illustrate the proposed methods using data from a three-arm non-inferiority trial.

Planning and evaluating clinical trials with composite time-to-first-event endpoints in a competing risk framework

Composite endpoints combine several events of interest within a single variable. These are often time-to-first-event data, which are analyzed via survival analysis techniques. To demonstrate the significance of an overall clinical benefit, it is sufficient to assess the test problem formulated for the composite. However, the effect observed for the composite does not necessarily reflect the effects for the components. Therefore, it would be desirable that the sample size for clinical trials using composite endpoints provides enough power not only to detect a clinically relevant superiority for the composite but also to address the components in an adequate way. The single components of a composite endpoint assessed as time-to-first-event define competing risks. We consider multiple test problems based on the cause-specific hazards of competing events to address the problem of analyzing both a composite endpoint and its components. Thereby, we use sequentially rejective test procedures to reduce the power loss to a minimum. We show how to calculate the sample size for the given multiple test problem by using a simply applicable simulation tool in SAS. Our ideas are illustrated by two clinical study examples.

Showing that the race model inequality is not violated

When participants are asked to respond in the same way to stimuli from different sources (e.g., auditory and visual), responses are often observed to be substantially faster when both stimuli are presented simultaneously (redundancy gain). Different models account for this effect, the two most important being race models and coactivation models. Redundancy gains consistent with the race model have an upper limit, however, which is given by the well-known race model inequality (Miller, 1982). A number of statistical tests have been proposed for testing the race model inequality in single participants and groups of participants. All of these tests use the race model as the null hypothesis, and rejection of the null hypothesis is considered evidence in favor of coactivation. We introduce a statistical test in which the race model prediction is the alternative hypothesis. This test controls the Type I error if a theory predicts that the race model prediction holds in a given experimental condition.

Assessing non-inferiority with time-to-event data via the method of non-parametric covariance

Non-parametric methods have been well recognised as useful tools for time-to-event (survival) data analysis because they provide valid statistical inference with few assumptions. Tangen and Koch have proposed the use of the method of non-parametric covariance for time-to-event data in a traditional superiority setting. In this article, we extended their method to assess non-inferiority of two treatments. To evaluate this non-parametric method versus the classical semi-parametric Cox proportional hazards regression model, simulations in terms of the Type 1 error rate and power were performed and compared. The results showed that the two methods were generally comparable regarding the Type 1 error rate when adjustment for the covariates correlated with the survival time was made. In the non-inferiority setting, the covariate-adjusted non-parametric analysis was shown to always increase power. However, this was not necessarily the case for the adjusted Cox model where results were inconsistent to those seen in the superiority setting. For illustration, an application of the proposed non-parametric method to a trial involving pemetrexed, a recently approved drug for first-line treatment of non-small cell lung cancer, is included.