Semiparametric Bayesian approach to assess non-inferiority with assay sensitivity in a three-arm trial with normally distributed endpoints

The non-inferiority (NI) trial is designed to show that an experimental treatment is not worse than an active reference by more than a pre-specified margin. Traditional NI trials do not include a placebo for ethical reasons; however, three-arm NI trials consisting of placebo, reference, and experimental treatment, can test the NI of experimental treatment to the reference while assessing the superiority of the reference over placebo. Assay sensitivity (AS) of a clinical trial is defined as its ability to distinguish between an effective and ineffective treatment and has been used to assess the superiority of the reference over placebo. Bayesian approaches have been predominantly used in clinical trials, particularly in NI trials. Most previous Bayesian approaches have focused on parametric priors of treatment effects. Restriction to parametric priors can mislead investigators into an inappropriate illusion of posterior certainty, leading to misleading decisions and inference. In this paper, we develop a novel semiparametric Bayesian approach to simultaneously assess NI of experimental treatment over the reference and AS of the reference over placebo in a three-arm trial with normally distributed endpoints. We use Dirichlet process priors to specify the priors of treatment effects. A Markov chain Monte Carlo algorithm is developed to calculate the posterior probability for assessing NI and AS. Simulation studies show that our proposed method is comparable to, or better than, the frequentist approach and parametric Bayesian methods in terms of the ability of controlling the type I errors and empirical statistical powers for testing NI. Data from two real trials are illustrated by the proposed methods. We recommend the usage of the proposed method in a three-arm trial.

The authors are grateful to the Editor, an Associate Editor and two anonymous referees for their constructive suggestions that greatly improved this manuscript. This work was supported by grants from the National Natural Science Foundation of China (No. 12271472), and the National Key R &D Program of China (No. 102022YFA1003701).

Authors and Affiliations

1. Yunnan Key Laboratory of Statistical Modeling and Data Analysis, Yunnan University, Kunming, 650091, People’s Republic of China

   Niansheng Tang & Fan Liang

2. Department of Community Health Sciences, University of Manitoba, Winnipeg, R3E 0W3, Canada

   Depeng Jiang

Corresponding author

Correspondence to Niansheng Tang.

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Appendix A Conditional distributions required for implementing Gibbs sampler

Steps (i)-(vi): Conditional distributions related to joint posterior distribution of \(\mu _R\) and \(\mu _P\)

Step (i). Conditional distribution of \(\mu _{kz}\) given \(Z_k,\psi _{kz},\mu _{0kz},\phi _{0kz}\) is

where \(\psi _{k\mu }=(G\psi _{kz}^{-1}+\phi _{0kz}^{-1})^{-1}\), \(\mu _{k\mu }=\psi _{k\mu }\{\phi _{0kz}^{-1}\mu _{0kz}+\psi _{kz}^{-1}\sum _{g=1}^GZ_{kg}\}\).

Step (ii). Conditional distribution of \(\psi _{kz}\) given \(Z_k,\mu _{kz}, c_{k1}\) and \(c_{k2}\) is

where \(Z_{kg}\) is the value of \(Z_k\) associated with cluster g.

Step (iii). Conditional distribution \(\tau _k\) given \((\pi _k,a_{k1},a_{k2})\) is

where \(v_{kg}^*\) is the random weight sampled from beta distribution, and is drawn within Step (iv).

Step (iv). As \(\pi _k\) given \(\tau _k\) is independent of \(Z_k\) given \((X_k,\sigma _k^2)\), the conditional distribution \(f(\pi _k,Z_k\mid \mathcal {L}_k,\mu _{kz},\psi _{kz},\tau _k,{\sigma _k^2},X_k)\) is proportional to \(f(\pi _k\mid \mathcal {L}_k,\tau _k)f(Z_k\mid \mathcal {L}_k\), \(\mu _{kz},\psi _{kz},\sigma _k^2,X_k)\). Thus, the conditional distribution of \((\pi _k,Z_k)\) can be decomposed into two independent components to be derived separately.

Conditional distribution \(f(\pi _k\mid \mathcal {L}_k,\tau _k)\). It can be shown that conditional distribution \(f(\pi _k\mid \mathcal {L}_k,\tau _k)\) is a generalized Dirichlet distribution:

where \(a_{kg}^*=1+d_{kg}\), \(b_{kg}^*=\tau _k+\sum _{j=g+1}^{G}d_{kj}\) for \(g=1,\dots ,G-1\), and \(d_{kg}\) is the number of \(\mathcal {L}_{ki}\)s whose value equals g. Sampling from \(f(\pi _k\mid \mathcal {L}_k,\tau _k)\) can be implemented as follows. First, \(v_{kg}^*\) is drawn from a Beta\((a_{kg}^*,b_{kg}^*)\) distribution. Next, \(\pi _{kg}\)’s are obtained as

Conditional distribution \(f(Z_k\mid \mathcal {L}_k,\mu _{kz},\psi _{kz},{\sigma _k^2},X_k)\) can be decomposed as

which leads to

where \(\psi _{kL}=(\psi _{kz}^{-1}+n_k\sigma _k^{-1})^{-1}\), and \(\mu _{kL}=\psi _{kL}(\sigma _{kz}^{-1}\mu _{kz}+\sigma _k^{-1}\sum {X_{ki}})\).

Step (v). The conditional distribution \(f(\mathcal {L}_{k}\mid \pi _k,Z_k,X_k)\) is given by

where \(\pi _{kg}^*\) is proportional to \(\pi _{kg}\prod _{i=1}^np\left( X_{ki}\mid \mu _k=Z_{k{\mathcal {L}_k}},\sigma _k^2\right)\), and \(\pi _{kg}\)’s are available from step (iv).

Step (vi). Conditional distribution \(f(\sigma _k^2\mid Z_{k\mathcal {L}_k},X_k,\mathcal {G}_{k1},\mathcal {G}_{k2})\) is

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