Nonparametric non-inferiority analyses in the three-arm design with active control and placebo

Power of the Wilcoxon-Mann-Whitney test for non-inferiority in the presence of death-censored observations

In clinical trials with patients in a critical state, death may preclude measurement of a quantitative endpoint of interest, and even early measurements, for example for intention-to-treat analysis, may not be available. For example, a non-negligible proportion of patients with acute pulmonary embolism will die before 30 day measurements on the efficacy of thrombolysis can be obtained. As excluding such patients may introduce bias, alternative analyses, and corresponding means for sample size calculation are needed. We specifically consider power analysis in a randomized clinical trial setting in which the goal is to demonstrate noninferiority of a new treatment as compared to a reference treatment. Also, a nonparametric approach may be needed due to the distribution of the quantitative endpoint of interest. While some approaches have been developed in a composite endpoint setting, our focus is on the continuous endpoint affected by death-related censoring, for which no approach for noninferiority is available. We propose a solution based on ranking the quantitative outcome and assigning worst rank scores to the patients without quantitative outcome because of death. Based on this, we derive power formulae for a noninferiority test in the presence of death-censored observations, considering settings with and without ties. The approach is illustrated for an exemplary clinical trial in pulmonary embolism. The results there show a substantial effect of death on power, also depending on differential effects in the two trial arms. Therefore, use of the proposed formulae is advisable whenever there is death to be expected before measurement of a quantitative primary outcome of interest.

Design and analysis of a 3-arm noninferiority trial with a prespecified margin for the hazard ratio

A 3-arm trial design that includes an experimental treatment, an active reference treatment, and a placebo is useful for assessing the noninferiority of an experimental treatment. The inclusion of a placebo arm enables the assessment of assay sensitivity and internal validation, in addition to the testing of the noninferiority of the experimental treatment compared with the reference treatment. In 3-arm noninferiority trials, various statistical test procedures have been considered to evaluate the following 3 hypotheses: (i) superiority of the experimental treatment over the placebo, (ii) superiority of the reference treatment over the placebo, and (iii) noninferiority of the experimental treatment compared with the reference treatment. However, hypothesis (ii) can be insufficient and may not accurately assess the assay sensitivity for the noninferiority of the experimental treatment compared with the reference treatment. Thus, demonstrating that the superiority of the reference treatment over the placebo is greater than the noninferiority margin (the nonsuperiority of the reference treatment compared with the placebo) can be necessary. Here, we propose log-rank statistical procedures for evaluating data obtained from 3-arm noninferiority trials to assess assay sensitivity with a prespecified margin Δ. In addition, we derive the approximate sample size and optimal allocation required to minimize the total sample size and that of the placebo treatment sample size, hierarchically.

Study duration for three-arm non-inferiority survival trials designed for accrual by cohorts

Study planning is particularly complex for survival trials because it usually involves an accrual period and a continued observation period after accrual closure. The three-arm clinical trial design, which includes a test treatment, an active reference, and a placebo control, is the gold standard design for the assessment of non-inferiority. The existing statistical methods of calculating minimal sample size for non-inferiority trials with three-arm design and survival-type endpoints cannot take into consideration the accrual rate of patients to the trial, the length of accrual period, the length of continued observation period after accrual closure, and unbalanced allocation of the total sample size. The purpose of this paper is to develop a statistical method, which allows for all these sources of variability for planning non-inferiority trials with the gold standard design for censored, exponentially distributed time-to-event data. The proposed method is based on the assumption of exponentially distributed failure times and a non-inferiority test formulated in terms of the retention of effect hypotheses. It can be used to calculate the duration of accrual required to assure a desired power for non-inferiority trials with active and placebo control. We illustrate the use of the method by considering a randomized, active- and placebo-controlled trial in depression associated with Parkinson's disease. We then explore the validity of the proposed method by simulation studies. An R-language program for the implementation of the proposed algorithm is provided as supplementary material.

Correcting for non-compliance in randomized non-inferiority trials with active and placebo control using structural models

The three-arm clinical trial design, which includes a test treatment, an active reference, and placebo control, is the gold standard for the assessment of non-inferiority. In the presence of non-compliance, one common concern is that an intent-to-treat (ITT) analysis (which is the standard approach to non-inferiority trials), tends to increase the chances of erroneously concluding non-inferiority, suggesting that the per-protocol (PP) analysis may be preferable for non-inferiority trials despite its inherent bias. The objective of this paper was to develop statistical methodology for dealing with non-compliance in three-arm non-inferiority trials for censored, time-to-event data. Changes in treatment were here considered the only form of non-compliance. An approach using a three-arm rank preserving structural failure time model and G-estimation analysis is here presented. Using simulations, the impact of non-compliance on non-inferiority trials was investigated in detail using ITT, PP analyses, and the present proposed method. Results indicate that the proposed method shows good characteristics, and that neither ITT nor PP analyses can always guarantee the validity of the non-inferiority conclusion. A Statistical Analysis System program for the implementation of the proposed test procedure is available from the authors upon request.

Three-arm noninferiority trials with a prespecified margin for inference of the difference in the proportions of binary endpoints

The design of a three-arm trial including the experimental treatment, an active reference treatment, and a placebo is recommended as a useful approach to the assessment of noninferiority of the experimental treatment. The inclusion of the placebo arm enables the assessment of assay sensitivity and internal validation, in addition to testing the noninferiority of the experimental treatment to the reference. Generally, the acceptable noninferiority margin Δ has been defined as the maximum clinically irrelevant difference between treatments in many two-arm noninferiority trials. However, many articles have considered a design in which the noninferiority margin Δ is relatively defined as a prespecified fraction f of the unknown effect size of the reference treatment. Therefore, these methods cannot be applied to cases where the margin is defined as a prespecified difference between treatments. In this article, we propose score-based statistical procedures for a three-arm noninferiority trial with a prespecified margin Δ for inference of the difference in the proportions of binary endpoints. In addition, we derive the approximate sample size and optimal allocation to minimize the total sample size and that of the placebo arm. A randomized controlled trial on major depressive disorder based on the difference in the proportions of remission is used to demonstrate our proposed method.

Group sequential designs for three-arm 'gold standard' non-inferiority trials with fixed margin

In the recent years there have been numerous publications on the design and the analysis of three-arm 'gold standard' noninferiority trials. Whenever feasible, regulatory authorities recommend the use of such three-arm designs including a test treatment, an active control, and a placebo. Nevertheless, it is desirable in many respects, for example, ethical reasons, to keep the placebo group size as small as possible. We first give a short overview on the fixed sample size design of a three-arm noninferiority trial with normally distributed outcomes and a fixed noninferiority margin. An optimal single stage design is derived that should serve as a benchmark for the group sequential designs proposed in the main part of this work. It turns out, that the number of patients allocated to placebo is substantially low for the optimal design. Subsequently, approaches for group sequential designs aiming to further reduce the expected sample sizes are presented. By means of choosing different rejection boundaries for the respective null hypotheses, we obtain designs with quite different operating characteristics. We illustrate the approaches via numerical calculations and a comparison with the optimal single stage design. Furthermore, we derive approximately optimal boundaries for different goals, for example, to reduce the overall average sample size. The results show that the implementation of a group sequential design further improves the optimal single stage design. Besides cost and time savings, the possible early termination of the placebo arm is a key advantage that could help to overcome ethical concerns.

Multiple comparisons to both a negative and a positive control

This paper addresses multiple comparisons in the presence of both a negative and a positive control. The methodology of the three-arm trial is extended to the case of many experimental treatment arms or different doses of a compound. In contrast to the classic three-arm trial, the focus is on the family-wise error type I. Normally distributed data with either homogeneous or heterogeneous group variances are considered. Explicit criteria for an optimal allocation are proposed. Depending on the pattern of heterogeneity, remarkably unbalanced designs are power-optimal. As an example, the method will be applied to a toxicological experiment.