An Objective and Robust Bayes Factor for the Hypothesis Test One Sample and Two Population Means.

It has been over 100 years since the discovery of one of the most fundamental statistical tests: the Student's t test. However, reliable conventional and objective Bayesian procedures are still essential for routine practice. In this work, we proposed an objective and robust Bayesian approach for hypothesis testing for one-sample and two-sample mean comparisons when the assumption of equal variances holds. The newly proposed Bayes factors are based on the intrinsic and Berger robust prior. Additionally, we introduced a corrected version of the Bayesian Information Criterion (BIC), denoted BIC-TESS, which is based on the effective sample size (TESS), for comparing two population means. We studied our developed Bayes factors in several simulation experiments for hypothesis testing. Our methodologies consistently provided strong evidence in favor of the null hypothesis in the case of equal means and variances. Finally, we applied the methodology to the original Gosset sleep data, concluding strong evidence favoring the hypothesis that the average sleep hours differed between the two treatments. These methodologies exhibit finite sample consistency and demonstrate consistent qualitative behavior, proving reasonably close to each other in practice, particularly for moderate to large sample sizes.

A decision-theoretic approach to Bayesian clinical trial design and evaluation of robustness to prior-data conflict.

Bayesian clinical trials allow taking advantage of relevant external information through the elicitation of prior distributions, which influence Bayesian posterior parameter estimates and test decisions. However, incorporation of historical information can have harmful consequences on the trial's frequentist (conditional) operating characteristics in case of inconsistency between prior information and the newly collected data. A compromise between meaningful incorporation of historical information and strict control of frequentist error rates is therefore often sought. Our aim is thus to review and investigate the rationale and consequences of different approaches to relaxing strict frequentist control of error rates from a Bayesian decision-theoretic viewpoint. In particular, we define an integrated risk which incorporates losses arising from testing, estimation, and sampling. A weighted combination of the integrated risk addends arising from testing and estimation allows moving smoothly between these two targets. Furthermore, we explore different possible elicitations of the test error costs, leading to test decisions based either on posterior probabilities, or solely on Bayes factors. Sensitivity analyses are performed following the convention which makes a distinction between the prior of the data-generating process, and the analysis prior adopted to fit the data. Simulation in the case of normal and binomial outcomes and an application to a one-arm proof-of-concept trial, exemplify how such analysis can be conducted to explore sensitivity of the integrated risk, the operating characteristics, and the optimal sample size, to prior-data conflict. Robust analysis prior specifications, which gradually discount potentially conflicting prior information, are also included for comparison. Guidance with respect to cost elicitation, particularly in the context of a Phase II proof-of-concept trial, is provided.

Power gains by using external information in clinical trials are typically not possible when requiring strict type I error control.

In the era of precision medicine, novel designs are developed to deal with flexible clinical trials that incorporate many treatment strategies for multiple diseases in one trial setting. This situation often leads to small sample sizes in disease-treatment combinations and has fostered the discussion about the benefits of borrowing of external or historical information for decision-making in these trials. Several methods have been proposed that dynamically discount the amount of information borrowed from historical data based on the conformity between historical and current data. Specifically, Bayesian methods have been recommended and numerous investigations have been performed to characterize the properties of the various borrowing mechanisms with respect to the gain to be expected in the trials. However, there is common understanding that the risk of type I error inflation exists when information is borrowed and many simulation studies are carried out to quantify this effect. To add transparency to the debate, we show that if prior information is conditioned upon and a uniformly most powerful test exists, strict control of type I error implies that no power gain is possible under any mechanism of incorporation of prior information, including dynamic borrowing. The basis of the argument is to consider the test decision function as a function of the current data even when external information is included. We exemplify this finding in the case of a pediatric arm appended to an adult trial and dichotomous outcome for various methods of dynamic borrowing from adult information to the pediatric arm. In conclusion, if use of relevant external data is desired, the requirement of strict type I error control has to be replaced by more appropriate metrics.

Addressing potential prior-data conflict when using informative priors in proof-of-concept studies.

Bayesian methods are increasingly used in proof-of-concept studies. An important benefit of these methods is the potential to use informative priors, thereby reducing sample size. This is particularly relevant for treatment arms where there is a substantial amount of historical information such as placebo and active comparators. One issue with using an informative prior is the possibility of a mismatch between the informative prior and the observed data, referred to as prior-data conflict. We focus on two methods for dealing with this: a testing approach and a mixture prior approach. The testing approach assesses prior-data conflict by comparing the observed data to the prior predictive distribution and resorting to a non-informative prior if prior-data conflict is declared. The mixture prior approach uses a prior with a precise and diffuse component. We assess these approaches for the normal case via simulation and show they have some attractive features as compared with the standard one-component informative prior. For example, when the discrepancy between the prior and the data is sufficiently marked, and intuitively, one feels less certain about the results, both the testing and mixture approaches typically yield wider posterior-credible intervals than when there is no discrepancy. In contrast, when there is no discrepancy, the results of these approaches are typically similar to the standard approach. Whilst for any specific study, the operating characteristics of any selected approach should be assessed and agreed at the design stage; we believe these two approaches are each worthy of consideration.

GENERALIZED DOUBLE PARETO SHRINKAGE.

We propose a generalized double Pareto prior for Bayesian shrinkage estimation and inferences in linear models. The prior can be obtained via a scale mixture of Laplace or normal distributions, forming a bridge between the Laplace and Normal-Jeffreys' priors. While it has a spike at zero like the Laplace density, it also has a Student's t-like tail behavior. Bayesian computation is straightforward via a simple Gibbs sampling algorithm. We investigate the properties of the maximum a posteriori estimator, as sparse estimation plays an important role in many problems, reveal connections with some well-established regularization procedures, and show some asymptotic results. The performance of the prior is tested through simulations and an application.