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.     

Statistical analysis of microarray data: a Bayesian approach

The potential of microarray data is enormous. It allows us to monitor the expression of thousands of genes simultaneously. A common task with microarray is to determine which genes are differentially expressed between two samples obtained under two different conditions. Recently, several statistical methods have been proposed to perform such a task when there are replicate samples under each condition. Two major problems arise with microarray data. The first one is that the number of replicates is very small (usually 2-10), leading to noisy point estimates. As a consequence, traditional statistics that are based on the means and standard deviations, e.g. t-statistic, are not suitable. The second problem is that the number of genes is usually very large (approximately 10,000), and one is faced with an extreme multiple testing problem. Most multiple testing adjustments are relatively conservative, especially when the number of replicates is small. In this paper we present an empirical Bayes analysis that handles both problems very well. Using different parametrizations, we develop four statistics that can be used to test hypotheses about the means and/or variances of the gene expression levels in both one- and two-sample problems. The methods are illustrated using experimental data with prior knowledge. In addition, we present the result of a simulation comparing our methods to well-known statistics and multiple testing adjustments.     

An estimator of number of species from quadrat sampling

We consider the problem of estimating the number of distinct species S in a study area from the recorded presence or absence of species in each of a sample of quadrats. A generalized jackknife estimator of S is derived, along with an estimate of its variance. It is compared with the jackknife estimator for S proposed by Heltshe and Forrester and the empirical Bayes estimator of Mingoti and Meeden. We show that the empirical Bayes estimator has the form of a generalized jackknife estimator under a specific model for species distribution. We compare the new estimators of S to the empirical Bayes estimator via simulation. We characterize circumstances under which each is superior.     

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.     

Practical Use of MCMC Methods: Lessons from a Case Study

This paper uses the analysis of a data set to examine a number of issues in Bayesian statistics and the application of MCMC methods. The data concern the selectivity of fishing nets and logistic regression is used to relate the size of a fish to the probability it will be retained or escape from a trawl net. Hierarchical models relate information from different trawls and posterior distributions are determined using MCMC. Centring data is shown to radically reduce autocorrelation in chains and Rao‐Blackwellisation and chain‐thinning are found to have little effect on parameter estimates. The results of four convergence diagnostics are compared and the sensitivity of the posterior distribution to the prior distribution is examined using a novel method. Nested models are fitted to the data and compared using intrinsic Bayes factors, pseudo‐Bayes factors and credible intervals.     

Bayesian Estimation Using Ranked Set Sampling

Bayesian Estimation of the parameter of a distribution is considered using Ranked set sampling (RSS). It is shown that for at least one RSS plan, the Bayes estimator has smaller Bayes risk than the Bayes estimator using simple random sampling (SRS). Furthermore, for exponential family with conjugate prior, the Bayes estimator of the mean using balanced RSS dominates, in terms of its Bayes risk, the Bayes estimator of the mean using SRS. This procedure is used to estimate the average Milk yield of four hundreds and two sheep. The empirical efficiency supports the theoretical findings.     

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.     

An empirical bayes adjustment to increase the sensitivity of detecting differentially expressed genes in microarray experiments

MOTIVATION: Detection of differentially expressed genes is one of the major goals of microarray experiments. Pairwise comparison for each gene is not appropriate without controlling the overall (experimentwise) type 1 error rate. Dudoit et al. have advocated use of permutation-based step-down P-value adjustments to correct the observed significance levels for the individual (i.e. for each gene) two sample t-tests. RESULTS: In this paper, we consider an ANOVA formulation of the gene expression levels corresponding to multiple tissue types. We provide resampling-based step-down adjustments to correct the observed significance levels for the individual ANOVA t-tests for each gene and for each pair of tissue type comparisons. More importantly, we introduce a novel empirical Bayes adjustment to the t-test statistics that can be incorporated into the step-down procedure. Using simulated data, we show that the empirical Bayes adjustment improved the sensitivity of detecting differentially expressed genes up to 16%, while maintaining a high level of specificity. This adjustment also reduces the false non-discovery rate to some degree at the cost of a modest increase in the false discovery rate. We illustrate our approach using a human colon cancer dataset consisting of oligonucleotide arrays of normal, adenoma and carcinoma cells. The number of genes with differential expression level declared statistically significant was about 50 when comparing normal to adenoma cells and about five when comparing adenoma to carcinoma cells. This list includes genes previously known to be associated with colon cancer as well as some novel genes. AVAILABILITY: R code for the empirical Bayes adjustment and step-down P-value calculation via resampling are available from the supplementary web-site. Supplementary information: http://www.mathstat.gsu.edu/~matsnd/EB/supp.htm     

医学实验中常见设计类型的辨析及统计方法的合理选用

目的帮助医学研究工作者提高对常见统计应用识别能力,提高对统计学的正确利用率。方法：明确阐述秒种常见医学实验设计类型的异同点和应用场合,揭示常见统计应用错误的实质和危害性。结果：给出不适合用ｔ检验取代方差分析的充足理由和正确运用统计学的基本要领。结论：要提高实际工作者的统计学应用水平。需要经常对其进行统计培养,积极引导与统计学工作者开展科研协作。     

Identifying sites under positive selection with uncertain parameter estimates.

Codon-based substitution models are routinely used to measure selective pressures acting on protein-coding genes. To this effect, the nonsynonymous to synonymous rate ratio (dN/dS = omega) is estimated. The proportion of amino-acid sites potentially under positive selection, as indicated by omega > 1, is inferred by fitting a probability distribution where some sites are permitted to have omega > 1. These sites are then inferred by means of an empirical Bayes or by a Bayes empirical Bayes approach that, respectively, ignores or accounts for sampling errors in maximum-likelihood estimates of the distribution used to infer the proportion of sites with omega > 1. Here, we extend a previous full-Bayes approach to include models with high power and low false-positive rates when inferring sites under positive selection. We propose some heuristics to alleviate the computational burden, and show that (i) full Bayes can be superior to empirical Bayes when analyzing a small data set or small simulated data, (ii) full Bayes has only a small advantage over Bayes empirical Bayes with our small test data, and (iii) Bayesian methods appear relatively insensitive to mild misspecifications of the random process generating adaptive evolution in our simulations, but in practice can prove extremely sensitive to model specification. We suggest that the codon model used to detect amino acids under selection should be carefully selected, for instance using Akaike information criterion (AIC).     

Parametric and nonparametric FDR estimation revisited

Nonparametric and parametric approaches have been proposed to estimate false discovery rate under the independent hypothesis testing assumption. The parametric approach has been shown to have better performance than the nonparametric approaches. In this article, we study the nonparametric approaches and quantify the underlying relations between parametric and nonparametric approaches. Our study reveals the conservative nature of the nonparametric approaches, and establishes the connections between the empirical Bayes method and p-value-based nonparametric methods. Based on our results, we advocate using the parametric approach, or directly modeling the test statistics using the empirical Bayes method.     

Differential expression analysis of multifactor RNA-Seq experiments with respect to biological variation

A flexible statistical framework is developed for the analysis of read counts from RNA-Seq gene expression studies. It provides the ability to analyse complex experiments involving multiple treatment conditions and blocking variables while still taking full account of biological variation. Biological variation between RNA samples is estimated separately from the technical variation associated with sequencing technologies. Novel empirical Bayes methods allow each gene to have its own specific variability, even when there are relatively few biological replicates from which to estimate such variability. The pipeline is implemented in the edgeR package of the Bioconductor project. A case study analysis of carcinoma data demonstrates the ability of generalized linear model methods (GLMs) to detect differential expression in a paired design, and even to detect tumour-specific expression changes. The case study demonstrates the need to allow for gene-specific variability, rather than assuming a common dispersion across genes or a fixed relationship between abundance and variability. Genewise dispersions de-prioritize genes with inconsistent results and allow the main analysis to focus on changes that are consistent between biological replicates. Parallel computational approaches are developed to make non-linear model fitting faster and more reliable, making the application of GLMs to genomic data more convenient and practical. Simulations demonstrate the ability of adjusted profile likelihood estimators to return accurate estimators of biological variability in complex situations. When variation is gene-specific, empirical Bayes estimators provide an advantageous compromise between the extremes of assuming common dispersion or separate genewise dispersion. The methods developed here can also be applied to count data arising from DNA-Seq applications, including ChIP-Seq for epigenetic marks and DNA methylation analyses.     

Bayes and Empirical Bayes Estimators of Abundance and Density from Spatial Capture-Recapture Data

In capture-recapture and mark-resight surveys, movements of individuals both within and between sampling periods can alter the susceptibility of individuals to detection over the region of sampling. In these circumstances spatially explicit capture-recapture (SECR) models, which incorporate the observed locations of individuals, allow population density and abundance to be estimated while accounting for differences in detectability of individuals. In this paper I propose two Bayesian SECR models, one for the analysis of recaptures observed in trapping arrays and another for the analysis of recaptures observed in area searches. In formulating these models I used distinct submodels to specify the distribution of individual home-range centers and the observable recaptures associated with these individuals. This separation of ecological and observational processes allowed me to derive a formal connection between Bayes and empirical Bayes estimators of population abundance that has not been established previously. I showed that this connection applies to every Poisson point-process model of SECR data and provides theoretical support for a previously proposed estimator of abundance based on recaptures in trapping arrays. To illustrate results of both classical and Bayesian methods of analysis, I compared Bayes and empirical Bayes esimates of abundance and density using recaptures from simulated and real populations of animals. Real populations included two iconic datasets: recaptures of tigers detected in camera-trap surveys and recaptures of lizards detected in area-search surveys. In the datasets I analyzed, classical and Bayesian methods provided similar – and often identical – inferences, which is not surprising given the sample sizes and the noninformative priors used in the analyses.     

Empirical Bayes analysis of single nucleotide polymorphisms

Background  

An important goal of whole-genome studies concerned with single nucleotide polymorphisms (SNPs) is the identification of SNPs associated with a covariate of interest such as the case-control status or the type of cancer. Since these studies often comprise the genotypes of hundreds of thousands of SNPs, methods are required that can cope with the corresponding multiple testing problem. For the analysis of gene expression data, approaches such as the empirical Bayes analysis of microarrays have been developed particularly for the detection of genes associated with the response. However, the empirical Bayes analysis of microarrays has only been suggested for binary responses when considering expression values, i.e. continuous predictors.     

Resampling-based empirical Bayes multiple testing procedures for controlling generalized tail probability and expected value error rates: focus on the false discovery rate and simulation study

This article proposes resampling-based empirical Bayes multiple testing procedures for controlling a broad class of Type I error rates, defined as generalized tail probability (gTP) error rates, gTP (q,g) = Pr(g (V(n),S(n)) > q), and generalized expected value (gEV) error rates, gEV (g) = E [g (V(n),S(n))], for arbitrary functions g (V(n),S(n)) of the numbers of false positives V(n) and true positives S(n). Of particular interest are error rates based on the proportion g (V(n),S(n)) = V(n) /(V(n) + S(n)) of Type I errors among the rejected hypotheses, such as the false discovery rate (FDR), FDR = E [V(n) /(V(n) + S(n))]. The proposed procedures offer several advantages over existing methods. They provide Type I error control for general data generating distributions, with arbitrary dependence structures among variables. Gains in power are achieved by deriving rejection regions based on guessed sets of true null hypotheses and null test statistics randomly sampled from joint distributions that account for the dependence structure of the data. The Type I error and power properties of an FDR-controlling version of the resampling-based empirical Bayes approach are investigated and compared to those of widely-used FDR-controlling linear step-up procedures in a simulation study. The Type I error and power trade-off achieved by the empirical Bayes procedures under a variety of testing scenarios allows this approach to be competitive with or outperform the Storey and Tibshirani (2003) linear step-up procedure, as an alternative to the classical Benjamini and Hochberg (1995) procedure.     

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.     

Leveraging Multi-ethnic Evidence for Mapping Complex Traits in Minority Populations: An Empirical Bayes Approach

Elucidating the genetic basis of complex traits and diseases in non-European populations is particularly challenging because US minority populations have been under-represented in genetic association studies. We developed an empirical Bayes approach named XPEB (cross-population empirical Bayes), designed to improve the power for mapping complex-trait-associated loci in a minority population by exploiting information from genome-wide association studies (GWASs) from another ethnic population. Taking as input summary statistics from two GWASs—a target GWAS from an ethnic minority population of primary interest and an auxiliary base GWAS (such as a larger GWAS in Europeans)—our XPEB approach reprioritizes SNPs in the target population to compute local false-discovery rates. We demonstrated, through simulations, that whenever the base GWAS harbors relevant information, XPEB gains efficiency. Moreover, XPEB has the ability to discard irrelevant auxiliary information, providing a safeguard against inflated false-discovery rates due to genetic heterogeneity between populations. Applied to a blood-lipids study in African Americans, XPEB more than quadrupled the discoveries from the conventional approach, which used a target GWAS alone, bringing the number of significant loci from 14 to 65. Thus, XPEB offers a flexible framework for mapping complex traits in minority populations.     

Cognition of and Demand for Education and Teaching in Medical Statistics in China: A Systematic Review and Meta-Analysis

BackgroundAlthough a substantial number of studies focus on the teaching and application of medical statistics in China, few studies comprehensively evaluate the recognition of and demand for medical statistics. In addition, the results of these various studies differ and are insufficiently comprehensive and systematic.ObjectivesThis investigation aimed to evaluate the general cognition of and demand for medical statistics by undergraduates, graduates, and medical staff in China.MethodsWe performed a comprehensive database search related to the cognition of and demand for medical statistics from January 2007 to July 2014 and conducted a meta-analysis of non-controlled studies with sub-group analysis for undergraduates, graduates, and medical staff.ResultsThere are substantial differences with respect to the cognition of theory in medical statistics among undergraduates (73.5%), graduates (60.7%), and medical staff (39.6%). The demand for theory in medical statistics is high among graduates (94.6%), undergraduates (86.1%), and medical staff (88.3%). Regarding specific statistical methods, the cognition of basic statistical methods is higher than of advanced statistical methods. The demand for certain advanced statistical methods, including (but not limited to) multiple analysis of variance (ANOVA), multiple linear regression, and logistic regression, is higher than that for basic statistical methods. The use rates of the Statistical Package for the Social Sciences (SPSS) software and statistical analysis software (SAS) are only 55% and 15%, respectively.ConclusionThe overall statistical competence of undergraduates, graduates, and medical staff is insufficient, and their ability to practically apply their statistical knowledge is limited, which constitutes an unsatisfactory state of affairs for medical statistics education. Because the demand for skills in this area is increasing, the need to reform medical statistics education in China has become urgent.