Computational and Mathematical Methods in Medicine

Volume 2018, Article ID 8134132, 8 pages

https://doi.org/10.1155/2018/8134132

## Bayesian Equivalence Testing and Meta-Analysis in Two-Arm Trials with Binary Data

Department of Statistics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2

Correspondence should be addressed to Saman Muthukumarana; ac.abotinamu@anaramukuhtum.namas

Received 15 January 2018; Revised 9 June 2018; Accepted 24 June 2018; Published 8 August 2018

Academic Editor: Kazuhisa Nishizawa

Copyright © 2018 Cynthia Kpekpena and Saman Muthukumarana. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider a Bayesian approach for assessing hypotheses of equivalence in two-arm trials with binary Data. We discuss the development of likelihood, the prior, and the posterior distributions of parameters of interest. We then examine the suitability of a normal approximation to the posterior distribution obtained via a Taylor series expansion. The Bayesian inference is carried out using Markov Chain Monte Carlo (MCMC) methods. We illustrate the methods using actual data arising from two-arm clinical trials on preventing mortality after myocardial infarction.

#### 1. Introduction

Consider a clinical trial where a pharmaceutical company wants to test a new drug against a currently existing drug. Sometimes in these studies, the clinical trial end point may be the success or failure of the treatment. A binary outcome is an outcome whose unit can take on only two possible states “0” and “1.” This success/failure response variable could be heart disease (Yes/No), patient condition (Good/Critical), how often patient feel depressed (Never/Often), and so on. The natural distribution for modeling these types of binary data is the binomial distribution given by

The mean and variance for the binomial random variable are and , respectively. In (1), it is assumed that there are only two outcomes (denoted “success” or “failure”) and a fixed number of trials (*n*). The trials are independent with a constant probability of success.

The main objective of this type of clinical trial is to determine whether there is a significant difference between active treatment (new drug) and reference treatment (current drug). Tests of significance have generally been argued not to be enough. That is, if the value for a test of significance leads to the nonrejection of the null hypothesis, it is not a proof that the null hypothesis holds. The clinician may want to test a null hypothesis of equivalence against an alternative hypothesis that states that there is a sufficient difference between the two drugs.

Equivalence testing is widely used when a choice is to be made between a drug (or a treatment) and an alternative. The term equivalence in the statistical sense is used to mean a weak pattern displayed by the data under study regarding the underlying population distribution. Equivalence tests are designed to show the nonexistence of a relevant difference between two treatments. It is known that Fisher’s one-sided exact test is the same as the test for equivalence in the frequentist approach [1]. This testing procedure is similar to the classical two-sided test procedure but involves an equivalence zone determined by a margin known as equivalence margin ().

The equivalence margin (), which represents a margin of clinical indifference, is usually estimated from previous studies and as such is also based primarily on clinical criteria as well as statistical principle. This margin is influenced by statistical principle but largely dependent on the interest of the experimenter and research questions clinicians wish to answer. As such, the statistical method employed together with the design of the study must be in such a manner that the margin of difference is not too restrictive to capture the bounds of the research question. For a test of equivalence of two binomial proportions, the equivalence margin is discussed in [2].

The frequentist approach to equivalence testing is the two one-sided test (TOST) procedure. By the TOST, equivalence is established at the significance level if a confidence interval for the difference in treatment means is contained within the interval where is the equivalence margin.

The motivation for this paper is based on the fact that for a given disease, there is likely to be many other substitute drugs or new drugs that can be used to treat the patients. But these drugs may not all be at the same cost; some may possibly have adverse side effects, and the method of application could be complex for others. On grounds of these information, we do equivalence testing to see if two different drugs can be regarded as equivalent in terms of their treatment effect. There are a variety of different approaches to this problem as indicated by some recent literature. See Wellek [1], Albert [2], Gamalo et al. [3], Rahardja and Zhao [4] and Zaslavsky [5] for comprehensive details on recent developments. We remark that Gamalo et al. [3] consider a Bayesian approach to proportions along with noninferiority trials. In this paper, we consider a Bayesian approach focusing on equivalence tests. We also construct a simple normal approximation and provide a mechanism for missing data analysis as well.

The remaining sections of this article are organized as follows: In Section 2, Bayesian inferential procedure for binary data is discussed. Section 3 presents a normal approximation to the posterior distribution obtained via a Taylor series expansion. We then examine the suitability of this normal approximation. We discuss a Gibbs sampling mechanism for estimating missing data in Section 4. In Section 5, we analyze a published dataset by Carlin [6] and Yusuf et al. [7]. This dataset consists with 22 treatment-control trials to prevent mortality after myocardial infarction. We conclude with a discussion of the approach in Section 5.

#### 2. Bayesian Inferential Procedure

Let be the number of individuals with positive exposure out of a total of patients in treatment group with proportion . Accordingly, let denote the number of individuals with positive exposure out of a total in the control group with proportion . Then,

The priors on the parameters, and are given by

Then the posterior distributions of and are given by

For Bayesian inference about treatment effect, a test is required to determine whether the posterior probability of treatment proportions and lies within the bounds of the equivalence margin or not. There is therefore, the need to sample from the posterior distribution of . The marginal posteriors of and are Beta distributions, and therefore is not in an analytically tractable form. So, are generated from and independently generated from because and are independent. Then, can be treated as a random sample from .

#### 3. Normal Approximation to the Beta Posterior Distribution

Note that the posterior distributions of and are Beta distributions. By following Kpekpena [8], a normal approximation to posteriors can be obtained using a Taylor series expansion of the Beta distribution. By applying a Taylor series expansion with first three terms, it can be shown that , where

Similarly, the approximation of can also be obtained. The details of this construction are given in the Appendix. We provide some approximations based on this development in Table 1 and Figures 1 and 2. It is clear that the approximation starts to work well for the values of the posterior parameters from and . However, the approximation is not suitable when Beta posterior parameters are less than 10.