A frequentist approach to dynamic borrowing

There has been growing interest in leveraging external control data to augment a randomized control group data in clinical trials and enable more informative decision making. In recent years, the quality and availability of real-world data have improved steadily as external controls. However, information borrowing by directly pooling such external controls with randomized controls may lead to biased estimates of the treatment effect. Dynamic borrowing methods under the Bayesian framework have been proposed to better control the false positive error. However, the numerical computation and, especially, parameter tuning, of those Bayesian dynamic borrowing methods remain a challenge in practice. In this paper, we present a frequentist interpretation of a Bayesian commensurate prior borrowing approach and describe intrinsic challenges associated with this method from the perspective of optimization. Motivated by this observation, we propose a new dynamic borrowing approach using adaptive lasso. The treatment effect estimate derived from this method follows a known asymptotic distribution, which can be used to construct confidence intervals and conduct hypothesis tests. The finite sample performance of the method is evaluated through extensive Monte Carlo simulations under different settings. We observed highly competitive performance of adaptive lasso compared to Bayesian approaches. Methods for selecting tuning parameters are also thoroughly discussed based on results from numerical studies and an illustration example.     

Bayesian data analysis in population ecology: motivations,methods, and benefits

During the 20th century ecologists largely relied on the frequentist system of inference for the analysis of their data. However, in the past few decades ecologists have become increasingly interested in the use of Bayesian methods of data analysis. In this article I provide guidance to ecologists who would like to decide whether Bayesian methods can be used to improve their conclusions and predictions. I begin by providing a concise summary of Bayesian methods of analysis, including a comparison of differences between Bayesian and frequentist approaches to inference when using hierarchical models. Next I provide a list of problems where Bayesian methods of analysis may arguably be preferred over frequentist methods. These problems are usually encountered in analyses based on hierarchical models of data. I describe the essentials required for applying modern methods of Bayesian computation, and I use real-world examples to illustrate these methods. I conclude by summarizing what I perceive to be the main strengths and weaknesses of using Bayesian methods to solve ecological inference problems.     

Bayesian sample size calculations for comparing two strategies in SMART studies

In the management of most chronic conditions characterized by the lack of universally effective treatments, adaptive treatment strategies (ATSs) have grown in popularity as they offer a more individualized approach. As a result, sequential multiple assignment randomized trials (SMARTs) have gained attention as the most suitable clinical trial design to formalize the study of these strategies. While the number of SMARTs has increased in recent years, sample size and design considerations have generally been carried out in frequentist settings. However, standard frequentist formulae require assumptions on interim response rates and variance components. Misspecifying these can lead to incorrect sample size calculations and correspondingly inadequate levels of power. The Bayesian framework offers a straightforward path to alleviate some of these concerns. In this paper, we provide calculations in a Bayesian setting to allow more realistic and robust estimates that account for uncertainty in inputs through the ‘two priors’ approach. Additionally, compared to the standard frequentist formulae, this methodology allows us to rely on fewer assumptions, integrate pre-trial knowledge, and switch the focus from the standardized effect size to the MDD. The proposed methodology is evaluated in a thorough simulation study and is implemented to estimate the sample size for a full-scale SMART of an internet-based adaptive stress management intervention on cardiovascular disease patients using data from its pilot study conducted in two Canadian provinces.     

Incorporating historical information in biosimilar trials: Challenges and a hybrid Bayesian‐frequentist approach

For the approval of biosimilars, it is, in most cases, necessary to conduct large Phase III clinical trials in patients to convince the regulatory authorities that the product is comparable in terms of efficacy and safety to the originator product. As the originator product has already been studied in several trials beforehand, it seems natural to include this historical information into the showing of equivalent efficacy. Since all studies for the regulatory approval of biosimilars are confirmatory studies, it is required that the statistical approach has reasonable frequentist properties, most importantly, that the Type I error rate is controlled—at least in all scenarios that are realistic in practice. However, it is well known that the incorporation of historical information can lead to an inflation of the Type I error rate in the case of a conflict between the distribution of the historical data and the distribution of the trial data. We illustrate this issue and confirm, using the Bayesian robustified meta‐analytic‐predictive (MAP) approach as an example, that simultaneously controlling the Type I error rate over the complete parameter space and gaining power in comparison to a standard frequentist approach that only considers the data in the new study, is not possible. We propose a hybrid Bayesian‐frequentist approach for binary endpoints that controls the Type I error rate in the neighborhood of the center of the prior distribution, while improving the power. We study the properties of this approach in an extensive simulation study and provide a real‐world example.     

Clinical trials and rare diseases: a way out of a conundrum.

Currently, clinical trials tend to be individually funded and applicants must include a power calculation in their grant request. However, conventional levels of statistical precision are unlikely to be obtainable prospectively if the trial is required to evaluate treatment of a rare disease. This means that clinicians treating such diseases remain in ignorance and must form their judgments solely on the basis of (potentially biased) observational studies experience, and anecdote. Since some unbiased evidence is clearly better than none, this state of affairs should not continue. However, conventional (frequentist) confidence limits are unlikely to exclude a null result, even when treatments differ substantially. Bayesian methods utilise all available data to calculate probabilities that may be extrapolated directly to clinical practice. Funding bodies should therefore fund a repertoire of small trials, which need have no predetermined end, alongside standard larger studies.     

Increasing reproducibility,robustness, and generalizability of biomarker selection from meta-analysis using Bayesian methodology

A major limitation of gene expression biomarker studies is that they are not reproducible as they simply do not generalize to larger, real-world, heterogeneous populations. Frequentist multi-cohort gene expression meta-analysis has been frequently used as a solution to this problem to identify biomarkers that are truly differentially expressed. However, the frequentist meta-analysis framework has its limitations–it needs at least 4–5 datasets with hundreds of samples, is prone to confounding from outliers and relies on multiple-hypothesis corrected p-values. To address these shortcomings, we have created a Bayesian meta-analysis framework for the analysis of gene expression data. Using real-world data from three different diseases, we show that the Bayesian method is more robust to outliers, creates more informative estimates of between-study heterogeneity, reduces the number of false positive and false negative biomarkers and selects more generalizable biomarkers with less data. We have compared the Bayesian framework to a previously published frequentist framework and have developed a publicly available R package for use.     

Exact sample size determination for a single Poisson random sample

Classical power analysis for sample size determination is typically performed in clinical trials. A “hybrid” classical Bayesian or a “fully Bayesian” approach can be alternatively used in order to add flexibility to the design assumptions needed at the planning stage of the study and to explicitly incorporate prior information in the procedure. In this paper, we exploit and compare these approaches to obtain the optimal sample size of a single-arm trial based on Poisson data. We adopt exact methods to establish the rejection of the null hypothesis within a frequentist or a Bayesian perspective and suggest the use of a conservative criterion for sample size determination that accounts for the not strictly monotonic behavior of the power function in the presence of discrete data. A Shiny web app in R has been developed to provide a user-friendly interface to easily compute the optimal sample size according to the proposed criteria and to assure the reproducibility of the results.     

ESTIMATING DIVERSIFICATION RATES: HOW USEFUL ARE DIVERGENCE TIMES?

The dynamics of species diversification rates are a key component of macroevolutionary patterns. Although not absolutely necessary, the use of divergence times inferred from sequence data has led to development of more powerful methods for inferring diversification rates. However, it is unclear what impact uncertainty in age estimates have on diversification rate inferences. Here, we quantify these effects using both Bayesian and frequentist methodology. Through simulation, we demonstrate that adding sequence data results in more precise estimates of internal node ages, but a reasonable approximation of these node ages is often sufficient to approach the theoretical minimum variance in speciation rate estimates. We also find that even crude estimates of divergence times increase the power of tests of diversification rate differences between sister clades. Finally, because Bayesian and frequentist methods provided similar assessments of error, novel Bayesian approaches may provide a useful framework for tests of diversification rates in more complex contexts than are addressed here.     

Causal assessment of surrogacy in a meta-analysis of colorectal cancer trials

When the true end points (T) are difficult or costly to measure, surrogate markers (S) are often collected in clinical trials to help predict the effect of the treatment (Z). There is great interest in understanding the relationship among S, T, and Z. A principal stratification (PS) framework has been proposed by Frangakis and Rubin (2002) to study their causal associations. In this paper, we extend the framework to a multiple trial setting and propose a Bayesian hierarchical PS model to assess surrogacy. We apply the method to data from a large collection of colon cancer trials in which S and T are binary. We obtain the trial-specific causal measures among S, T, and Z, as well as their overall population-level counterparts that are invariant across trials. The method allows for information sharing across trials and reduces the nonidentifiability problem. We examine the frequentist properties of our model estimates and the impact of the monotonicity assumption using simulations. We also illustrate the challenges in evaluating surrogacy in the counterfactual framework that result from nonidentifiability.     

Case–Control Studies of Gene–Environment Interaction: Bayesian Design and Analysis

Summary With increasing frequency, epidemiologic studies are addressing hypotheses regarding gene‐environment interaction. In many well‐studied candidate genes and for standard dietary and behavioral epidemiologic exposures, there is often substantial prior information available that may be used to analyze current data as well as for designing a new study. In this article, first, we propose a proper full Bayesian approach for analyzing studies of gene–environment interaction. The Bayesian approach provides a natural way to incorporate uncertainties around the assumption of gene–environment independence, often used in such an analysis. We then consider Bayesian sample size determination criteria for both estimation and hypothesis testing regarding the multiplicative gene–environment interaction parameter. We illustrate our proposed methods using data from a large ongoing case–control study of colorectal cancer investigating the interaction of N‐acetyl transferase type 2 (NAT2) with smoking and red meat consumption. We use the existing data to elicit a design prior and show how to use this information in allocating cases and controls in planning a future study that investigates the same interaction parameters. The Bayesian design and analysis strategies are compared with their corresponding frequentist counterparts.     

MCMC for hidden Markov models incorporating aggregation of states and filtering

This paper is concerned with the statistical analysis of single ion channel records. Single channels are modelled by using hidden Markov models and a combination of Bayesian statistics and Markov chain Monte Carlo methods. The techniques presented here provide a straightforward generalization to those in Rosales et al. (2001, Biophys. J., 80, 1088–1103), allowing to consider constraints imposed by a gating mechanism such as the aggregation of states into classes. This paper also presents an extension that allows to consider correlated background noise and filtered data, extending the scope of the analysis toward real experimental conditions. The methods described here are based on a solid probabilistic basis and are less computationally intensive than alternative Bayesian treatments or frequentist approaches that consider correlated data.     

Bayesian inference for kappa from single and multiple studies

Cohen's kappa coefficient is a widely popular measure for chance-corrected nominal scale agreement between two raters. This article describes Bayesian analysis for kappa that can be routinely implemented using Markov chain Monte Carlo (MCMC) methodology. We consider the case of m > or = 2 independent samples of measured agreement, where in each sample a given subject is rated by two rating protocols on a binary scale. A major focus here is on testing the homogeneity of the kappa coefficient across the different samples. The existing frequentist tests for this case assume exchangeability of rating protocols, whereas our proposed Bayesian test does not make any such assumption. Extensive simulation is carried out to compare the performances of the Bayesian and the frequentist tests. The developed methodology is illustrated using data from a clinical trial in ophthalmology.     

Moving beyond noninformative priors: why and how to choose weakly informative priors in Bayesian analyses

Throughout the last two decades, Bayesian statistical methods have proliferated throughout ecology and evolution. Numerous previous references established both philosophical and computational guidelines for implementing Bayesian methods. However, protocols for incorporating prior information, the defining characteristic of Bayesian philosophy, are nearly nonexistent in the ecological literature. Here, I hope to encourage the use of weakly informative priors in ecology and evolution by providing a ‘consumer's guide’ to weakly informative priors. The first section outlines three reasons why ecologists should abandon noninformative priors: 1) common flat priors are not always noninformative, 2) noninformative priors provide the same result as simpler frequentist methods, and 3) noninformative priors suffer from the same high type I and type M error rates as frequentist methods. The second section provides a guide for implementing informative priors, wherein I detail convenient ‘reference’ prior distributions for common statistical models (i.e. regression, ANOVA, hierarchical models). I then use simulations to visually demonstrate how informative priors influence posterior parameter estimates. With the guidelines provided here, I hope to encourage the use of weakly informative priors for Bayesian analyses in ecology. Ecologists can and should debate the appropriate form of prior information, but should consider weakly informative priors as the new ‘default’ prior for any Bayesian model.     

Integrating biological knowledge into variable selection: an empirical Bayes approach with an application in cancer biology

ABSTRACT: BACKGROUND: An important question in the analysis of biochemical data is that of identifying subsets of molecular variables that may jointly influence a biological response. Statistical variable selection methods have been widely used for this purpose. In many settings, it may be important to incorporate ancillary biological information concerning the variables of interest. Pathway and network maps are one example of a source of such information. However, although ancillary information is increasingly available, it is not always clear how it should be used nor how it should be weighted in relation to primary data. RESULTS: We put forward an approach in which biological knowledge is incorporated using informative prior distributions over variable subsets, with prior information selected and weighted in an automated, objective manner using an empirical Bayes formulation. We employ continuous, linear models with interaction terms and exploit biochemically-motivated sparsity constraints to permit exact inference. We show an example of priors for pathway- and network-based information and illustrate our proposed method on both synthetic response data and by an application to cancer drug response data. Comparisons are also made to alternative Bayesian and frequentist penalised-likelihood methods for incorporating network-based information. CONCLUSIONS: The empirical Bayes method proposed here can aid prior elicitation for Bayesian variable selection studies and help to guard against mis-specification of priors. Empirical Bayes, together with the proposed pathway-based priors, results in an approach with a competitive variable selection performance. In addition, the overall procedure is fast, deterministic, and has very few user-set parameters, yet is capable of capturing interplay between molecular players. The approach presented is general and readily applicable in any setting with multiple sources of biological prior knowledge.     

Bayesian Inference for the Ratio of the Means of Two Normal Populations with Unequal Variances

The problem of making inferences about the ratio of two normal means has been addressed, both from the frequentist and Bayesian perspectives, by several authors. Most of this work is concerned with the homoscedastic case. In contrast, the situation where the variances are not equal has received little attention. Cox (1985) deals, within the frequentist framework, with a model where the variances are related to the means. His results are mainly based on Fieller's theorem whose drawbacks are well known. In this paper we present a Bayesian analysis of this model and discuss some related problems. An agronomical example is used throughout to illustrate the methods.     

生物群落多样性的测度方法 V.生物群落物种数目的估计方法

群落的物种数目,即物种丰富度,是最古老、同时也是最基本的一个多样性概念,从对它的估计中可以得到关于物种灭绝速率方面的信息,这对生物多样性保护是非常重要的。已经提出了很多方法来估计群落中的物种数目,这些方法可以分为两大类,即基于理论抽样的方法和基于数据分析的方法。前者包括经典估计方法和贝叶斯估计方法;后者包括对数正态分布的积分方法、再抽样方法和种－面积曲线的外推方法。发现：（１）有些方法适用于动物群落,如大多数基于理论抽样的方法;有些方法则适用于植物群落,如大多数基于数据分析的方法;（２）这些方法还没有经过全面而系统地比较;（３）还没有一个普遍认为比较好的方法。因此,建议采用野外调查与模拟研究相结合的方法对各种估计方法进行系统地评价。     

A Bayesian basket trial design that borrows information across strata based on the similarity between the posterior distributions of the response probability

Basket trials simultaneously evaluate the effect of one or more drugs on a defined biomarker, genetic alteration, or molecular target in a variety of disease subtypes, often called strata. A conventional approach for analyzing such trials is an independent analysis of each of the strata. This analysis is inefficient as it lacks the power to detect the effect of drugs in each stratum. To address these issues, various designs for basket trials have been proposed, centering on designs using Bayesian hierarchical models. In this article, we propose a novel Bayesian basket trial design that incorporates predictive sample size determination, early termination for inefficacy and efficacy, and the borrowing of information across strata. The borrowing of information is based on the similarity between the posterior distributions of the response probability. In general, Bayesian hierarchical models have many distributional assumptions along with multiple parameters. By contrast, our method has prior distributions for response probability and two parameters for similarity of distributions. The proposed design is easier to implement and less computationally demanding than other Bayesian basket designs. Through a simulation with various scenarios, our proposed design is compared with other designs including one that does not borrow information and one that uses a Bayesian hierarchical model.     

Evaluating the Impact of Prior Assumptions in Bayesian Biostatistics

A common concern in Bayesian data analysis is that an inappropriately informative prior may unduly influence posterior inferences. In the context of Bayesian clinical trial design, well chosen priors are important to ensure that posterior-based decision rules have good frequentist properties. However, it is difficult to quantify prior information in all but the most stylized models. This issue may be addressed by quantifying the prior information in terms of a number of hypothetical patients, i.e., a prior effective sample size (ESS). Prior ESS provides a useful tool for understanding the impact of prior assumptions. For example, the prior ESS may be used to guide calibration of prior variances and other hyperprior parameters. In this paper, we discuss such prior sensitivity analyses by using a recently proposed method to compute a prior ESS. We apply this in several typical settings of Bayesian biomedical data analysis and clinical trial design. The data analyses include cross-tabulated counts, multiple correlated diagnostic tests, and ordinal outcomes using a proportional-odds model. The study designs include a phase I trial with late-onset toxicities, a phase II trial that monitors event times, and a phase I/II trial with dose-finding based on efficacy and toxicity.     

Hierarchical commensurate and power prior models for adaptive incorporation of historical information in clinical trials

Bayesian clinical trial designs offer the possibility of a substantially reduced sample size, increased statistical power, and reductions in cost and ethical hazard. However when prior and current information conflict, Bayesian methods can lead to higher than expected type I error, as well as the possibility of a costlier and lengthier trial. This motivates an investigation of the feasibility of hierarchical Bayesian methods for incorporating historical data that are adaptively robust to prior information that reveals itself to be inconsistent with the accumulating experimental data. In this article, we present several models that allow for the commensurability of the information in the historical and current data to determine how much historical information is used. A primary tool is elaborating the traditional power prior approach based upon a measure of commensurability for Gaussian data. We compare the frequentist performance of several methods using simulations, and close with an example of a colon cancer trial that illustrates a linear models extension of our adaptive borrowing approach. Our proposed methods produce more precise estimates of the model parameters, in particular, conferring statistical significance to the observed reduction in tumor size for the experimental regimen as compared to the control regimen.     

Bayesian statistics for parasitologists

Bayesian statistical methods are increasingly being used in the analysis of parasitological data. Here, the basis of differences between the Bayesian method and the classical or frequentist approach to statistical inference is explained. This is illustrated with practical implications of Bayesian analyses using prevalence estimation of strongyloidiasis and onchocerciasis as two relevant examples. The strongyloidiasis example addresses the problem of parasitological diagnosis in the absence of a gold standard, whereas the onchocerciasis case focuses on the identification of villages warranting priority mass ivermectin treatment. The advantages and challenges faced by users of the Bayesian approach are also discussed and the readers pointed to further directions for a more in-depth exploration of the issues raised. We advocate collaboration between parasitologists and Bayesian statisticians as a fruitful and rewarding venture for advancing applied research in parasite epidemiology and the control of parasitic infections.