Assessing noninferiority in a three-arm trial using the Bayesian approach

A dynamic power prior approach to non-inferiority trials for normal means

Non-inferiority trials compare new experimental therapies to standard ones (active control). In these experiments, historical information on the control treatment is often available. This makes Bayesian methodology appealing since it allows a natural way to exploit information from past studies. In the present paper, we suggest the use of previous data for constructing the prior distribution of the control effect parameter. Specifically, we consider a dynamic power prior that possibly allows to discount the level of borrowing in the presence of heterogeneity between past and current control data. The discount parameter of the prior is based on the Hellinger distance between the posterior distributions of the control parameter based, respectively, on historical and current data. We develop the methodology for comparing normal means and we handle the unknown variance assumption using MCMC. We also provide a simulation study to analyze the proposed test in terms of frequentist size and power, as it is usually requested by regulatory agencies. Finally, we investigate comparisons with some existing methods and we illustrate an application to a real case study.

Practical approaches to Bayesian sample size determination in non-inferiority trials with binary outcomes

Bayesian analysis of a non-inferiority trial is advantageous in allowing direct probability statements to be made about the relative treatment difference rather than relying on an arbitrary and often poorly justified non-inferiority margin. When the primary analysis will be Bayesian, a Bayesian approach to sample size determination will often be appropriate for consistency with the analysis. We demonstrate three Bayesian approaches to choosing sample size for non-inferiority trials with binary outcomes and review their advantages and disadvantages. First, we present a predictive power approach for determining sample size using the probability that the trial will produce a convincing result in the final analysis. Next, we determine sample size by considering the expected posterior probability of non-inferiority in the trial. Finally, we demonstrate a precision-based approach. We apply these methods to a non-inferiority trial in antiretroviral therapy for treatment of HIV-infected children. A predictive power approach would be most accessible in practical settings, because it is analogous to the standard frequentist approach. Sample sizes are larger than with frequentist calculations unless an informative analysis prior is specified, because appropriate allowance is made for uncertainty in the assumed design parameters, ignored in frequentist calculations. An expected posterior probability approach will lead to a smaller sample size and is appropriate when the focus is on estimating posterior probability rather than on testing. A precision-based approach would be useful when sample size is restricted by limits on recruitment or costs, but it would be difficult to decide on sample size using this approach alone.

A more powerful test for three-arm non-inferiority via risk difference: Frequentist and Bayesian approaches

Necessity for finding improved intervention in many legacy therapeutic areas are of high priority. This has the potential to decrease the expense of medical care and poor outcomes for many patients. Typically, clinical efficacy is the primary evaluating criteria to measure any beneficial effect of a treatment. Albeit, there could be situations when several other factors (e.g. side-effects, cost-burden, less debilitating, less intensive, etc.) which can permit some slightly less efficacious treatment options favorable to a subgroup of patients. This often leads to non-inferiority (NI) testing. NI trials may or may not include a placebo arm due to ethical reasons. However, when included, the resulting three-arm trial is more prudent since it requires less stringent assumptions compared to a two-arm placebo-free trial. In this article, we consider both Frequentist and Bayesian procedures for testing NI in the three-arm trial with binary outcomes when the functional of interest is risk difference. An improved Frequentist approach is proposed first, which is then followed by a Bayesian counterpart. Bayesian methods have a natural advantage in many active-control trials, including NI trial, as it can seamlessly integrate substantial prior information. In addition, we discuss sample size calculation and draw an interesting connection between the two paradigms.

Bayesian sample size determination in a three-arm non-inferiority trial with binary endpoints

A three-arm non-inferiority trial including a test treatment, a reference treatment, and a placebo is recommended to assess the assay sensitivity and internal validity of a trial when applicable. Existing methods for designing and analyzing three-arm trials with binary endpoints are mainly developed from a frequentist viewpoint. However, these methods largely depend on large sample theories. To alleviate this problem, we propose two fully Bayesian approaches, the posterior variance approach and Bayes factor approach, to determine sample size required in a three-arm non-inferiority trial with binary endpoints. Simulation studies are conducted to investigate the performance of the proposed Bayesian methods. An example is illustrated by the proposed methodologies. Bayes factor method always leads to smaller sample sizes than the posterior variance method, utilizing the historical data can reduce the required sample size, simultaneous test requires more sample size to achieve the desired power than the non-inferiority test, the selection of the hyperparameters has a relatively large effect on the required sample size. When only controlling the posterior variance, the posterior variance criterion is a simple and effective option for obtaining a rough outcome. When conducting a previous clinical trial, it is recommended to use the Bayes factor criterion in practical applications.

New approaches for testing non-inferiority for three-arm trials with Poisson distributed outcomes

With the availability of limited resources, innovation for improved statistical method for the design and analysis of randomized controlled trials (RCTs) is of paramount importance for newer and better treatment discovery for any therapeutic area. Although clinical efficacy is almost always the primary evaluating criteria to measure any beneficial effect of a treatment, there are several important other factors (e.g., side effects, cost burden, less debilitating, less intensive, etc.), which can permit some less efficacious treatment options favorable to a subgroup of patients. This leads to non-inferiority (NI) testing. The objective of NI trial is to show that an experimental treatment is not worse than an active reference treatment by more than a pre-specified margin. Traditional NI trials do not include a placebo arm for ethical reason; however, this necessitates stringent and often unverifiable assumptions. On the other hand, three-arm NI trials consisting of placebo, reference, and experimental treatment, can simultaneously test the superiority of the reference over placebo and NI of experimental treatment over the reference. In this article, we proposed both novel Frequentist and Bayesian procedures for testing NI in the three-arm trial with Poisson distributed count outcome. RCTs with count data as the primary outcome are quite common in various disease areas such as lesion count in cancer trials, relapses in multiple sclerosis, dermatology, neurology, cardiovascular research, adverse event count, etc. We first propose an improved Frequentist approach, which is then followed by it's Bayesian version. Bayesian methods have natural advantage in any active-control trials, including NI trial when substantial historical information is available for placebo and established reference treatment. In addition, we discuss sample size calculation and draw an interesting connection between the two paradigms.

Confidence limits for the averted infections ratio estimated via the counterfactual placebo incidence rate

Objectives

The averted infections ratio (AIR) is a novel measure for quantifying the preservation-of-effect in active-control non-inferiority clinical trials with a time-to-event outcome. In the main formulation, the AIR requires an estimate of the counterfactual placebo incidence rate. We describe two approaches for calculating confidence limits for the AIR given a point estimate of this parameter, a closed-form solution based on a Taylor series expansion (delta method) and an iterative method based on the profile-likelihood.

Methods

For each approach, exact coverage probabilities for the lower and upper confidence limits were computed over a grid of values of (1) the true value of the AIR (2) the expected number of counterfactual events (3) the effectiveness of the active-control treatment.

Results

Focussing on the lower confidence limit, which determines whether non-inferiority can be declared, the coverage achieved by the delta method is either less than or greater than the nominal coverage, depending on the true value of the AIR. In contrast, the coverage achieved by the profile-likelihood method is consistently accurate.

Conclusions

The profile-likelihood method is preferred because of better coverage properties, but the simpler delta method is valid when the experimental treatment is no less effective than the control treatment. A complementary Bayesian approach, which can be applied when the counterfactual incidence rate can be represented as a prior distribution, is also outlined.

Simultaneous confidence interval for assessing non-inferiority with assay sensitivity in a three-arm trial with binary endpoints

A three-arm trial including an experimental treatment, an active reference treatment and a placebo is often used to assess the non-inferiority (NI) with assay sensitivity of an experimental treatment. Various hypothesis-test-based approaches via a fraction or pre-specified margin have been proposed to assess the NI with assay sensitivity in a three-arm trial. There is little work done on confidence interval in a three-arm trial. This paper develops a hybrid approach to construct simultaneous confidence interval for assessing NI and assay sensitivity in a three-arm trial. For comparison, we present normal-approximation-based and bootstrap-resampling-based simultaneous confidence intervals. Simulation studies evidence that the hybrid approach with the Wilson score statistic performs better than other approaches in terms of empirical coverage probability and mesial-non-coverage probability. An example is used to illustrate the proposed approaches.

Sequential parallel comparison design for "gold standard" noninferiority trials with a prespecified margin

Three-arm noninferiority trials (involving an experimental treatment, a reference treatment, and a placebo)-called the "gold standard" noninferiority trials-are conducted in patients with mental disorders whenever feasible, but often fail to show superiority of the experimental treatment and/or the reference treatment over the placebo. One possible reason is that some of the patients receiving the placebo show apparent improvement in the clinical condition. An approach to addressing this problem is the use of the sequential parallel comparison design (SPCD). Nonetheless, the SPCD has not yet been discussed in relation to gold standard noninferiority trials. In this article, our aim was to develop a hypothesis-testing method and its corresponding sample size calculation method for gold standard noninferiority trials with the SPCD. In a simulation, we show that the proposed hypothesis-testing method achieves the nominal type I error rate and power and that the proposed sample size calculation method has adequate power accuracy.

Bayesian Approach for Assessing Non-inferiority in Three-arm Trials for Risk Ratio and Odds Ratio

In this paper we consider three-arm non-inferiority (NI) trial that includes an experimental, a reference, and a placebo arm. While for binary outcomes the risk difference (RD) is the most common and well explored functional form for testing efficacy (or effectiveness), recent FDA guideline suggested other measures such as relative risk (RR) and odds ratio (OR) on the basis of which NI of an experimental treatment can be claimed. However, developing test based on these different functions of binary outcomes are challenging since the construction and interpretation of NI margin for such functions are not trivial extensions of RD based approach. Recently, we have proposed Frequentist approaches for testing NI for these functionals. In this article we further develop Bayesian approaches for testing NI based on effect retention approach for RR and OR. Bayesian paradigm provides a natural path to integrate historical trials' information, as well as it allows the usage of patients'/clinicians' opinions as prior information via sequential learning. In addition we discuss, in detail, the sample size/power calculation which could be readily used while designing such trials in practice.

Bayes factors for superiority, non-inferiority, and equivalence designs

Background

In clinical trials, study designs may focus on assessment of superiority, equivalence, or non-inferiority, of a new medicine or treatment as compared to a control. Typically, evidence in each of these paradigms is quantified with a variant of the null hypothesis significance test. A null hypothesis is assumed (null effect, inferior by a specific amount, inferior by a specific amount and superior by a specific amount, for superiority, non-inferiority, and equivalence respectively), after which the probabilities of obtaining data more extreme than those observed under these null hypotheses are quantified by p-values. Although ubiquitous in clinical testing, the null hypothesis significance test can lead to a number of difficulties in interpretation of the results of the statistical evidence.

Methods

We advocate quantifying evidence instead by means of Bayes factors and highlight how these can be calculated for different types of research design.

Results

We illustrate Bayes factors in practice with reanalyses of data from existing published studies.

Conclusions

Bayes factors for superiority, non-inferiority, and equivalence designs allow for explicit quantification of evidence in favor of the null hypothesis. They also allow for interim testing without the need to employ explicit corrections for multiple testing.

Approaches for testing noninferiority in two-arm trials for risk ratio and odds ratio

For an existing established drug regimen, active control trials are defacto standard due to ethical reason as well as for clinical equipoise. However, when superiority claim of a new drug against the active control is unlikely to be successful, researchers often address the issue in terms of noninferiority (NI), provided the experimental drug demonstrates the evidence of other benefits beyond efficacy. Such trials aim to demonstrate that an experimental treatment is non-inferior to an existing comparator by not more than a pre-specified margin. The issue of choosing such a margin is complex. In this article, two-arm NI trials with binary outcomes are considered when margin is defined in terms of relative risk or odds ratio. A Frequentist test based on proposed NI margin is developed first. Since two-arm NI trials without placebo arm are dependent upon historical information, in order to make accurate and meaningful interpretation of their results, a Bayesian approach is developed next. Bayesian approach is flexible to incorporate the available information from the historical trial. The operating characteristics of the proposed methods are studied in terms of power and sample size for varying design factors. A clinical trial data is reanalyzed to study the properties of the proposed approach.

The averted infections ratio: a novel measure of effectiveness of experimental HIV pre-exposure prophylaxis agents

Tenofovir disoproxil fumarate combined with emtricitabine is a highly effective oral pre-exposure prophylaxis (PrEP) agent for preventing the acquisition of HIV. This effectiveness has consequences for the design and analysis of trials assessing experimental PrEP regimens, which now generally include an active-control tenofovir disoproxil fumarate plus emtricitabine group, rather than a placebo group, as a comparator. Herein, we describe major problems in the interpretation of the primary measure of effectiveness proposed for these trials, namely the ratio of HIV incidence in the experimental agent group to that in the active-control group. We argue that valid interpretation requires an assumption about one of two parameters: either the incidence among trial participants had they not received PrEP or the effectiveness of tenofovir disoproxil fumarate plus emtricitabine within the trial. However, neither parameter is directly observed because of the absence of a no-treatment group, thus requiring the use of external evidence or subjective judgment. We propose an alternative measure of effectiveness based on the concept of averted infections, which incorporates one of these parameters. The measure is simple to interpret, has clinical and public health relevance, and is a natural preservation-of-effect criterion for assessing statistical non-inferiority. Its adoption could also allow the use of smaller sample sizes, currently a major barrier to the assessment of experimental PrEP regimens.

Bayesian approach for assessing noninferiority in a three-arm trial with binary endpoint

With the recent advancement in many therapeutic areas, quest for better and enhanced treatment options is ever increasing. While the "efficacy" metric plays the most important role in this development, emphasis on other important clinical factors such as less intensive side effects, lower toxicity, ease of delivery, and other less debilitating factors may result in the selection of treatment options, which may not beat current established treatment option in terms efficacy, yet prove to be desirable for subgroups of patients. The resultant clinical trial by means of which one establishes such slightly less efficacious treatment is known as noninferiority (NI) trial. Noninferiority trials often involve an active established comparator arm, along with a placebo and an experimental treatment arm, resulting into a 3-arm trial. Most of the past developments in a 3-arm NI trial consider defining a prespecified fraction of unknown effect size of reference drug, i.e., without directly specifying a fixed NI margin. However, in some recent developments, more direct approach is being considered with prespecified fixed margin, albeit in the frequentist setup. In this article, we consider Bayesian implementation of such trial when primary outcome of interest is binary. Bayesian paradigm is important, as it provides a path to integrate historical trials and current trial information via sequential learning. We use several approximation-based and 2 exact fully Bayesian methods to evaluate the feasibility of the proposed approach. Finally, a clinical trial example is reanalyzed to demonstrate the benefit of the proposed approach.

A more efficient three-arm non-inferiority test based on pooled estimators of the homogeneous variance

Hida and Tango established a statistical testing framework for the three-arm non-inferiority trial including a placebo with a pre-specified non-inferiority margin to overcome the shortcomings of traditional two-arm non-inferiority trials (such as having to choose the non-inferiority margin). In this paper, we propose a new method that improves their approach with respect to two aspects. We construct our testing statistics based on the best unbiased pooled estimators of the homogeneous variance; and we use the principle of intersection-union tests to determine the rejection rule. We theoretically prove that our test is better than that of Hida and Tango for large sample sizes. Furthermore, when that sample size was small or moderate, our simulation studies showed that our approach performed better than Hida and Tango's. Although both controlled the type I error rate, their test was more conservative and the statistical power of our test was higher.

Some practical considerations in three-arm non-inferiority trial design

Non-inferiority trials aim to demonstrate whether an experimental therapy is not unacceptably worse than an active reference therapy already in use. When applicable, a three-arm non-inferiority trial, including an experiment therapy, an active reference therapy, and a placebo, is often recommended to assess assay sensitivity and internal validity of a trial. In this paper, we share some practical considerations based on our experience from a phase III three-arm non-inferiority trial. First, we discuss the determination of the total sample size and its optimal allocation based on the overall power of the non-inferiority testing procedure and provide ready-to-use R code for implementation. Second, we consider the non-inferiority goal of 'capturing all possibilities' and show that it naturally corresponds to a simple two-step testing procedure. Finally, using this two-step non-inferiority testing procedure as an example, we compare extensively commonly used frequentist p -value methods with the Bayesian posterior probability approach. Copyright © 2016 John Wiley & Sons, Ltd.

Bayesian approach for assessing non-inferiority in a three-arm trial with pre-specified margin

Non-inferiority trials are becoming increasingly popular for comparative effectiveness research. However, inclusion of the placebo arm, whenever possible, gives rise to a three-arm trial which has lesser burdensome assumptions than a standard two-arm non-inferiority trial. Most of the past developments in a three-arm trial consider defining a pre-specified fraction of unknown effect size of reference drug, that is, without directly specifying a fixed non-inferiority margin. However, in some recent developments, a more direct approach is being considered with pre-specified fixed margin albeit in the frequentist setup. Bayesian paradigm provides a natural path to integrate historical and current trials' information via sequential learning. In this paper, we propose a Bayesian approach for simultaneous testing of non-inferiority and assay sensitivity in a three-arm trial with normal responses. For the experimental arm, in absence of historical information, non-informative priors are assumed under two situations, namely when (i) variance is known and (ii) variance is unknown. A Bayesian decision criteria is derived and compared with the frequentist method using simulation studies. Finally, several published clinical trial examples are reanalyzed to demonstrate the benefit of the proposed procedure.

Testing non-inferiority of a new treatment in three-arm clinical trials with binary endpoints

BACKGROUND

A two-arm non-inferiority trial without a placebo is usually adopted to demonstrate that an experimental treatment is not worse than a reference treatment by a small pre-specified non-inferiority margin due to ethical concerns. Selection of the non-inferiority margin and establishment of assay sensitivity are two major issues in the design, analysis and interpretation for two-arm non-inferiority trials. Alternatively, a three-arm non-inferiority clinical trial including a placebo is usually conducted to assess the assay sensitivity and internal validity of a trial. Recently, some large-sample approaches have been developed to assess the non-inferiority of a new treatment based on the three-arm trial design. However, these methods behave badly with small sample sizes in the three arms. This manuscript aims to develop some reliable small-sample methods to test three-arm non-inferiority.

METHODS

Saddlepoint approximation, exact and approximate unconditional, and bootstrap-resampling methods are developed to calculate p-values of the Wald-type, score and likelihood ratio tests. Simulation studies are conducted to evaluate their performance in terms of type I error rate and power.

RESULTS

Our empirical results show that the saddlepoint approximation method generally behaves better than the asymptotic method based on the Wald-type test statistic. For small sample sizes, approximate unconditional and bootstrap-resampling methods based on the score test statistic perform better in the sense that their corresponding type I error rates are generally closer to the prespecified nominal level than those of other test procedures.

CONCLUSIONS

Both approximate unconditional and bootstrap-resampling test procedures based on the score test statistic are generally recommended for three-arm non-inferiority trials with binary outcomes.

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.

Bayesian approach to non-inferiority trials for normal means

Regulatory framework recommends that novel statistical methodology for analyzing trial results parallels the frequentist strategy, e.g. the new method must protect type-I error and arrive at a similar conclusion. Keeping these in mind, we construct a Bayesian approach for non-inferiority trials with normal response. A non-informative prior is assumed for the mean response of the experimental treatment and Jeffrey's prior for its corresponding variance when it is unknown. The posteriors of the mean response and variance of the treatment in historical trials are then assumed as priors for its corresponding parameters in the current trial, where that treatment serves as the active control. From these priors, a Bayesian decision criterion is derived to determine whether the experimental treatment is non-inferior to the active control. This criterion is evaluated and compared with the frequentist method using simulation studies. Results show that both Bayesian and frequentist approaches perform alike, but the Bayesian approach has a higher power when the variances are unknown. Both methods also arrive at the same conclusion of non-inferiority when applied on two real datasets. A major advantage of the proposed Bayesian approach lies in its ability to provide posterior probabilities for varying effect sizes of the experimental treatment over the active control.