Elastic priors to dynamically borrow information from historical data in clinical trials

Use of historical data and real-world evidence holds great potential to improve the efficiency of clinical trials. One major challenge is to effectively borrow information from historical data while maintaining a reasonable type I error and minimal bias. We propose the elastic prior approach to address this challenge. Unlike existing approaches, this approach proactively controls the behavior of information borrowing and type I errors by incorporating a well-known concept of clinically significant difference through an elastic function, defined as a monotonic function of a congruence measure between historical data and trial data. The elastic function is constructed to satisfy a set of prespecified criteria such that the resulting prior will strongly borrow information when historical and trial data are congruent, but refrain from information borrowing when historical and trial data are incongruent. The elastic prior approach has a desirable property of being information borrowing consistent, that is, asymptotically controls type I error at the nominal value, no matter that historical data are congruent or not to the trial data. Our simulation study that evaluates the finite sample characteristic confirms that, compared to existing methods, the elastic prior has better type I error control and yields competitive or higher power. The proposed approach is applicable to binary, continuous, and survival endpoints.     

The incorporation of historical control information in tests of proportions: simulation study of Tarone's procedure

Empirical Bayes procedures for incorporating historical control information in bioassay carcinogenesis studies are receiving attention in the literature. In general, the empirical Bayes methods fail to take into account the error in estimating the parameters of a prior distribution. The implications of this are studied for the beta prior of Tarone (1982, Biometrics 38, 215-220). Using simulations, we show that the skewness in the maximum likelihood estimators for the parameters of the beta prior increases the false positive rate in the test of dose-related trend.     

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.     

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.     

An empirical Bayes method for updating inferences in analysis of quantitative trait loci using information from related genome scans

Individual genome scans for quantitative trait loci (QTL) mapping often suffer from low statistical power and imprecise estimates of QTL location and effect. This lack of precision yields large confidence intervals for QTL location, which are problematic for subsequent fine mapping and positional cloning. In prioritizing areas for follow-up after an initial genome scan and in evaluating the credibility of apparent linkage signals, investigators typically examine the results of other genome scans of the same phenotype and informally update their beliefs about which linkage signals in their scan most merit confidence and follow-up via a subjective-intuitive integration approach. A method that acknowledges the wisdom of this general paradigm but formally borrows information from other scans to increase confidence in objectivity would be a benefit. We developed an empirical Bayes analytic method to integrate information from multiple genome scans. The linkage statistic obtained from a single genome scan study is updated by incorporating statistics from other genome scans as prior information. This technique does not require that all studies have an identical marker map or a common estimated QTL effect. The updated linkage statistic can then be used for the estimation of QTL location and effect. We evaluate the performance of our method by using extensive simulations based on actual marker spacing and allele frequencies from available data. Results indicate that the empirical Bayes method can account for between-study heterogeneity, estimate the QTL location and effect more precisely, and provide narrower confidence intervals than results from any single individual study. We also compared the empirical Bayes method with a method originally developed for meta-analysis (a closely related but distinct purpose). In the face of marked heterogeneity among studies, the empirical Bayes method outperforms the comparator.     

Borrowing information across genes and experiments for improved error variance estimation in microarray data analysis

Statistical inference for microarray experiments usually involves the estimation of error variance for each gene. Because the sample size available for each gene is often low, the usual unbiased estimator of the error variance can be unreliable. Shrinkage methods, including empirical Bayes approaches that borrow information across genes to produce more stable estimates, have been developed in recent years. Because the same microarray platform is often used for at least several experiments to study similar biological systems, there is an opportunity to improve variance estimation further by borrowing information not only across genes but also across experiments. We propose a lognormal model for error variances that involves random gene effects and random experiment effects. Based on the model, we develop an empirical Bayes estimator of the error variance for each combination of gene and experiment and call this estimator BAGE because information is Borrowed Across Genes and Experiments. A permutation strategy is used to make inference about the differential expression status of each gene. Simulation studies with data generated from different probability models and real microarray data show that our method outperforms existing approaches.     

Establishing Informative Prior for Gene Expression Variance from Public Databases

Identifying differential expressed genes across various conditions or genotypes is the most typical approach to studying the regulation of gene expression. An estimate of gene-specific variance is often needed for the assessment of statistical significance in most differential expression (DE) detection methods, including linear models (e.g., for transformed and normalized microarray data) and generalized linear models (e.g., for count data in RNAseq). Due to a common limit in sample size, the variance estimate is often unstable in small experiments. Shrinkage estimates using empirical Bayes methods have proven useful in improving the variance estimate, hence improving the detection of DE. The most widely used empirical Bayes methods borrow information across genes within the same experiments. In these methods, genes are considered exchangeable or exchangeable conditioning on expression level. We propose, with the increasing accumulation of expression data, borrowing information from historical data on the same gene can provide better estimate of gene-specific variance, thus further improve DE detection. Specifically, we show that the variation of gene expression is truly gene-specific and reproducible between different experiments. We present a new method to establish informative gene-specific prior on the variance of expression using existing public data, and illustrate how to shrink the variance estimate and detect DE. We demonstrate improvement in DE detection under our strategy compared to leading DE detection methods.     

Quantification of prior impact in terms of effective current sample size

Bayesian methods allow borrowing of historical information through prior distributions. The concept of prior effective sample size (prior ESS) facilitates quantification and communication of such prior information by equating it to a sample size. Prior information can arise from historical observations; thus, the traditional approach identifies the ESS with such a historical sample size. However, this measure is independent of newly observed data, and thus would not capture an actual “loss of information” induced by the prior in case of prior-data conflict. We build on a recent work to relate prior impact to the number of (virtual) samples from the current data model and introduce the effective current sample size (ECSS) of a prior, tailored to the application in Bayesian clinical trial designs. Special emphasis is put on robust mixture, power, and commensurate priors. We apply the approach to an adaptive design in which the number of recruited patients is adjusted depending on the effective sample size at an interim analysis. We argue that the ECSS is the appropriate measure in this case, as the aim is to save current (as opposed to historical) patients from recruitment. Furthermore, the ECSS can help overcome lack of consensus in the ESS assessment of mixture priors and can, more broadly, provide further insights into the impact of priors. An R package accompanies the paper.     

Hierarchical Bayes models for cDNA microarray gene expression

cDNA microarrays are used in many contexts to compare mRNA levels between samples of cells. Microarray experiments typically give us expression measurements on 1000-20 000 genes, but with few replicates for each gene. Traditional methods using means and standard deviations to detect differential expression are not satisfactory in this context. A handful of alternative statistics have been developed, including several empirical Bayes methods. In the present paper we present two full hierarchical Bayes models for detecting gene expression, of which one (D) describes our microarray data very well. We also compare the full Bayes and empirical Bayes approaches with respect to model assumptions, false discovery rates and computer running time. The proposed models are compared to existing empirical Bayes models in a simulation study and for a set of data (Yuen et al., 2002), where 27 genes have been categorized by quantitative real-time PCR. It turns out that the existing empirical Bayes methods have at least as good performance as the full Bayes ones.     

Accounting for multiplicity in the evaluation of "signals" obtained by data mining from spontaneous report adverse event databases

Surveillance of drug products in the marketplace continues after approval, to identify rare potential toxicities that are unlikely to have been observed in the clinical trials carried out before approval. This surveillance accumulates large numbers of spontaneous reports of adverse events along with other information in spontaneous report databases. Recently developed empirical Bayes and Bayes methods provide a way to summarize the data in these databases, including a quantitative measure of the strength of the reporting association between the drugs and the events. Determining which of the particular drug-event associations, of which there may be many tens of thousands, are real reporting associations and which random noise presents a substantial problem of multiplicity because the resources available for medical and epidemiologic followup are limited. The issues are similar to those encountered with the evaluation of microarrays, but there are important differences. This report compares the application of a standard empirical Bayes approach with micorarray-inspired methods for controlling the False Discovery Rate, and a new Bayesian method for the resolution of the multiplicity problem to a relatively small database containing about 48,000 reports. The Bayesian approach appears to have attractive diagnostic properties in addition to being easy to interpret and implement computationally.     

Bayesian methods for a three-state model for rodent carcinogenicity studies

The objective of a chronic rodent bioassay is to assess the impact of a chemical compound on the development of tumors. However, most tumor types are not observable prior to necropsy, making direct estimation of the tumor incidence rate problematic. In such cases, estimation can proceed only if the study incorporates multiple interim sacrifices or we make use of simplified parametric or nonparametric models. In addition, it is widely accepted that other factors, such as weight, can be related to both dose level and tumor onset, confounding the association of interest. However, there is not typically enough information in the current study to assess such effects. The addition of historical data can help alleviate this problem. In this article, we propose a novel Bayesian semiparametric model for the analysis of data from rodent carcinogenicity studies. We develop informative prior distributions for covariate effects through the use of historical control data and outline a Gibbs sampling scheme. We implement the model by analyzing data from a National Toxicology Program chronic rodent bioassay.     

A bayesian approach to the design of phase II clinical trials

A new strategy for the design of Phase II clinical trials is presented which utilizes the information provided by the prior distribution of the response rate, the costs of treating a patient, and the losses or gains resulting from the decisions taken at the completion of the study. A risk function is derived from which one may determine the optimal Bayes sampling plan. The decision theoretic/Bayesian approach is shown to provide a formal justification for the sample sizes often used in practice and shows the conditions under which such sample sizes are clearly inappropriate.     

Putting background information about relative risks into conjugate prior distributions

In Bayesian and empirical Bayes analyses of epidemiologic data, the most easily implemented prior specifications use a multivariate normal distribution for the log relative risks or a conjugate distribution for the discrete response vector. This article describes problems in translating background information about relative risks into conjugate priors and a solution. Traditionally, conjugate priors have been specified through flattening constants, an approach that leads to conflicts with the true prior covariance structure for the log relative risks. One can, however, derive a conjugate prior consistent with that structure by using a data-augmentation approximation to the true log relative-risk prior, although a rescaling step is needed to ensure the accuracy of the approximation. These points are illustrated with a logistic regression analysis of neonatal-death risk.     

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.     

Preoperative statin therapy and infectious complications in cardiac surgery

AimTo assess whether preoperative statin therapy is associated with the risk of postoperative infection in patients undergoing cardiac surgery.Methods520 patients undergoing cardiac surgery in 2010 were retrospectively examined. Data regarding statin and antibiotic use prior to and after surgery were available from the hospital pharmacy information system. Cultures and clinical data of patients on postoperative antibiotics other than standard prophylactic therapy were studied to identify postoperative infections up to 30 days from day of surgery.Results370 (71.2 %) patients were on preoperative statin therapy. Overall, 82 patients (15.8 %) suffered from postoperative infection of which 11 were surgical site infections. In multivariable regression analysis, statin therapy was associated with a reduced risk of postoperative infection (adjusted odds ratio: 0.329, 95 %: CI 0.19–0.57; P < 0.001).ConclusionsPreoperative statin use was associated with a considerable reduced risk of postoperative infections following cardiac surgery. Randomised controlled trials are required to clarify the role of statin therapy in the prevention of postoperative infections.     

On the relevance of prognostic information for clinical trials: A theoretical quantification

The question of how individual patient data from cohort studies or historical clinical trials can be leveraged for designing more powerful, or smaller yet equally powerful, clinical trials becomes increasingly important in the era of digitalization. Today, the traditional statistical analyses approaches may seem questionable to practitioners in light of ubiquitous historical prognostic information. Several methodological developments aim at incorporating historical information in the design and analysis of future clinical trials, most importantly Bayesian information borrowing, propensity score methods, stratification, and covariate adjustment. Adjusting the analysis with respect to a prognostic score, which was obtained from some model applied to historical data, received renewed interest from a machine learning perspective, and we study the potential of this approach for randomized clinical trials. In an idealized situation of a normal outcome in a two-arm trial with 1:1 allocation, we derive a simple sample size reduction formula as a function of two criteria characterizing the prognostic score: (1) the coefficient of determination R² on historical data and (2) the correlation ρ between the estimated and the true unknown prognostic scores. While maintaining the same power, the original total sample size n planned for the unadjusted analysis reduces to $(1-R^2{\rho }^2)\times n$ in an adjusted analysis. Robustness in less ideal situations was assessed empirically. We conclude that there is potential for substantially more powerful or smaller trials, but only when prognostic scores can be accurately estimated.     

Assessing Differential Expression in Two-Color Microarrays: A Resampling-Based Empirical Bayes Approach

Microarrays are widely used for examining differential gene expression, identifying single nucleotide polymorphisms, and detecting methylation loci. Multiple testing methods in microarray data analysis aim at controlling both Type I and Type II error rates; however, real microarray data do not always fit their distribution assumptions. Smyth''s ubiquitous parametric method, for example, inadequately accommodates violations of normality assumptions, resulting in inflated Type I error rates. The Significance Analysis of Microarrays, another widely used microarray data analysis method, is based on a permutation test and is robust to non-normally distributed data; however, the Significance Analysis of Microarrays method fold change criteria are problematic, and can critically alter the conclusion of a study, as a result of compositional changes of the control data set in the analysis. We propose a novel approach, combining resampling with empirical Bayes methods: the Resampling-based empirical Bayes Methods. This approach not only reduces false discovery rates for non-normally distributed microarray data, but it is also impervious to fold change threshold since no control data set selection is needed. Through simulation studies, sensitivities, specificities, total rejections, and false discovery rates are compared across the Smyth''s parametric method, the Significance Analysis of Microarrays, and the Resampling-based empirical Bayes Methods. Differences in false discovery rates controls between each approach are illustrated through a preterm delivery methylation study. The results show that the Resampling-based empirical Bayes Methods offer significantly higher specificity and lower false discovery rates compared to Smyth''s parametric method when data are not normally distributed. The Resampling-based empirical Bayes Methods also offers higher statistical power than the Significance Analysis of Microarrays method when the proportion of significantly differentially expressed genes is large for both normally and non-normally distributed data. Finally, the Resampling-based empirical Bayes Methods are generalizable to next generation sequencing RNA-seq data analysis.     

Bayes empirical bayes inference of amino acid sites under positive selection

Codon-based substitution models have been widely used to identify amino acid sites under positive selection in comparative analysis of protein-coding DNA sequences. The nonsynonymous-synonymous substitution rate ratio (d(N)/d(S), denoted omega) is used as a measure of selective pressure at the protein level, with omega > 1 indicating positive selection. Statistical distributions are used to model the variation in omega among sites, allowing a subset of sites to have omega > 1 while the rest of the sequence may be under purifying selection with omega < 1. An empirical Bayes (EB) approach is then used to calculate posterior probabilities that a site comes from the site class with omega > 1. Current implementations, however, use the naive EB (NEB) approach and fail to account for sampling errors in maximum likelihood estimates of model parameters, such as the proportions and omega ratios for the site classes. In small data sets lacking information, this approach may lead to unreliable posterior probability calculations. In this paper, we develop a Bayes empirical Bayes (BEB) approach to the problem, which assigns a prior to the model parameters and integrates over their uncertainties. We compare the new and old methods on real and simulated data sets. The results suggest that in small data sets the new BEB method does not generate false positives as did the old NEB approach, while in large data sets it retains the good power of the NEB approach for inferring positively selected sites.     

Exploration of empirical Bayes hierarchical modeling for the analysis of genome-wide association study data

In the analysis of genome-wide association (GWA) data, the aim is to detect statistical associations between single nucleotide polymorphisms (SNPs) and the disease or trait of interest. These SNPs, or the particular regions of the genome they implicate, are then considered for further study. We demonstrate through a comprehensive simulation study that the inclusion of additional, biologically relevant information through a 2-level empirical Bayes hierachical model framework offers a more robust method of detecting associated SNPs. The empirical Bayes approach is an objective means of analyzing the data without the need for the setting of subjective parameter estimates. This framework gives more stable estimates of effects through a reduction of the variability in the usual effect estimates. We also demonstrate the consequences of including additional information that is not informative and examine power and false-positive rates. We apply the methodology to a number of genome-wide association (GWA) data sets with the inclusion of additional biological information. Our results agree with previous findings and in the case of one data set (Crohn's disease) suggest an additional region of interest.