Bayesian Analysis of Multiple Hypothesis Testing with Applications to Microarray Experiments

Recently, the field of multiple hypothesis testing has experienced a great expansion, basically because of the new methods developed in the field of genomics. These new methods allow scientists to simultaneously process thousands of hypothesis tests. The frequentist approach to this problem is made by using different testing error measures that allow to control the Type I error rate at a certain desired level. Alternatively, in this article, a Bayesian hierarchical model based on mixture distributions and an empirical Bayes approach are proposed in order to produce a list of rejected hypotheses that will be declared significant and interesting for a more detailed posterior analysis. In particular, we develop a straightforward implementation of a Gibbs sampling scheme where all the conditional posterior distributions are explicit. The results are compared with the frequentist False Discovery Rate (FDR) methodology. Simulation examples show that our model improves the FDR procedure in the sense that it diminishes the percentage of false negatives keeping an acceptable percentage of false positives.     

Bayesian False Discovery Rate Wavelet Shrinkage: Theory and Applications

Statistical inference in the wavelet domain remains a vibrant area of contemporary statistical research because of desirable properties of wavelet representations and the need of scientific community to process, explore, and summarize massive data sets. Prime examples are biomedical, geophysical, and internet related data. We propose two new approaches to wavelet shrinkage/thresholding.

In the spirit of Efron and Tibshirani's recent work on local false discovery rate, we propose Bayesian Local False Discovery Rate (BLFDR), where the underlying model on wavelet coefficients does not assume known variances. This approach to wavelet shrinkage is shown to be connected with shrinkage based on Bayes factors. The second proposal, Bayesian False Discovery Rate (BaFDR), is based on ordering of posterior probabilities of hypotheses on true wavelets coefficients being null, in Bayesian testing of multiple hypotheses.

We demonstrate that both approaches result in competitive shrinkage methods by contrasting them to some popular shrinkage techniques.     

Large-scale multiple testing under dependence

Summary.  The paper considers the problem of multiple testing under dependence in a compound decision theoretic framework. The observed data are assumed to be generated from an underlying two-state hidden Markov model. We propose oracle and asymptotically optimal data-driven procedures that aim to minimize the false non-discovery rate FNR subject to a constraint on the false discovery rate FDR. It is shown that the performance of a multiple-testing procedure can be substantially improved by adaptively exploiting the dependence structure among hypotheses, and hence conventional FDR procedures that ignore this structural information are inefficient. Both theoretical properties and numerical performances of the procedures proposed are investigated. It is shown that the procedures proposed control FDR at the desired level, enjoy certain optimality properties and are especially powerful in identifying clustered non-null cases. The new procedure is applied to an influenza-like illness surveillance study for detecting the timing of epidemic periods.     

Power and stability comparisons of multiple testing procedures with false discovery rate control

High-throughput data analyses are widely used for examining differential gene expression, identifying single nucleotide polymorphisms, and detecting methylation loci. False discovery rate (FDR) has been considered a proper type I error rate to control for discovery-based high-throughput data analysis. Various multiple testing procedures have been proposed to control the FDR. The power and stability properties of some commonly used multiple testing procedures have not been extensively investigated yet, however. Simulation studies were conducted to compare power and stability properties of five widely used multiple testing procedures at different proportions of true discoveries for various sample sizes for both independent and dependent test statistics. Storey's two linear step-up procedures showed the best performance among all tested procedures considering FDR control, power, and variance of true discoveries. Leukaemia and ovarian cancer microarray studies were used to illustrate the power and stability characteristics of these five multiple testing procedures with FDR control.     

A clarifying comparison of methods for controlling the false discovery rate

Traditional multiple hypothesis testing procedures fix an error rate and determine the corresponding rejection region. In 2002 Storey proposed a fixed rejection region procedure and showed numerically that it can gain more power than the fixed error rate procedure of Benjamini and Hochberg while controlling the same false discovery rate (FDR). In this paper it is proved that when the number of alternatives is small compared to the total number of hypotheses, Storey's method can be less powerful than that of Benjamini and Hochberg. Moreover, the two procedures are compared by setting them to produce the same FDR. The difference in power between Storey's procedure and that of Benjamini and Hochberg is near zero when the distance between the null and alternative distributions is large, but Benjamini and Hochberg's procedure becomes more powerful as the distance decreases. It is shown that modifying the Benjamini and Hochberg procedure to incorporate an estimate of the proportion of true null hypotheses as proposed by Black gives a procedure with superior power.     

A note on the adaptive control of false discovery rates

Summary.  The use of a fixed rejection region for multiple hypothesis testing has been shown to outperform standard fixed error rate approaches when applied to control of the false discovery rate. In this work it is demonstrated that, if the original step-up procedure of Benjamini and Hochberg is modified to exercise adaptive control of the false discovery rate, its performance is virtually identical to that of the fixed rejection region approach. In addition, the dependence of both methods on the proportion of true null hypotheses is explored, with a focus on the difficulties that are involved in the estimation of this quantity.     

Operating characteristics and extensions of the false discovery rate procedure

Summary. We investigate the operating characteristics of the Benjamini–Hochberg false discovery rate procedure for multiple testing. This is a distribution-free method that controls the expected fraction of falsely rejected null hypotheses among those rejected. The paper provides a framework for understanding more about this procedure. We first study the asymptotic properties of the `deciding point' D that determines the critical p -value. From this, we obtain explicit asymptotic expressions for a particular risk function. We introduce the dual notion of false non-rejections and we consider a risk function that combines the false discovery rate and false non-rejections. We also consider the optimal procedure with respect to a measure of conditional risk.     

More Powerful Control of the False Discovery Rate Under Dependence

In a breakthrough paper, Benjamini and Hochberg (J Roy Stat Soc Ser B 57:289–300, 1995) proposed a new error measure for multiple testing, the FDR; and developed a distribution-free procedure to control it under independence among the test statistics. In this paper we argue by extensive simulation and theoretical considerations that the assumption of independence is not needed. Along the lines of (Ann Stat 32:1035–1061, 2004b), we moreover provide a more powerful method, that exploits an estimator of the number of false nulls among the tests. We propose a whole family of iterative estimators that prove robust under dependence and independence between the test statistics. These estimators can be used to improve also classical multiple testing procedures, and in general to estimate the weight of a known component in a mixture distribution. Innovations are illustrated by simulations.     

MMCTest—A Safe Algorithm for Implementing Multiple Monte Carlo Tests

Consider testing multiple hypotheses using tests that can only be evaluated by simulation, such as permutation tests or bootstrap tests. This article introduces MMCTest , a sequential algorithm that gives, with arbitrarily high probability, the same classification as a specific multiple testing procedure applied to ideal p‐values. The method can be used with a class of multiple testing procedures that include the Benjamini and Hochberg false discovery rate procedure and the Bonferroni correction controlling the familywise error rate. One of the key features of the algorithm is that it stops sampling for all the hypotheses that can already be decided as being rejected or non‐rejected. MMCTest can be interrupted at any stage and then returns three sets of hypotheses: the rejected, the non‐rejected and the undecided hypotheses. A simulation study motivated by actual biological data shows that MMCTest is usable in practice and that, despite the additional guarantee, it can be computationally more efficient than other methods.     

False discovery rates for large-scale model checking under certain dependence

In many scientific fields, it is interesting and important to determine whether an observed data stream comes from a prespecified model or not, particularly when the number of data streams is of large scale, where multiple hypotheses testing is necessary. In this article, we consider large-scale model checking under certain dependence among different data streams observed at the same time. We propose a false discovery rate (FDR) control procedure to check those unusual data streams. Specifically, we derive an approximation of false discovery and construct a point estimate of FDR. Theoretical results show that, under some mild assumptions, our proposed estimate of FDR is simultaneously conservatively consistent with the true FDR, and hence it is an asymptotically strong control procedure. Simulation comparisons with some competing procedures show that our proposed FDR procedure behaves better in general settings. Application of our proposed FDR procedure is illustrated by the StarPlus fMRI data.     

A Process Monitoring Scheme Controlling False Discovery Rate

A new process monitoring scheme is proposed by using the Storey procedure for controlling the positive false discovery rate in multiple testing. For the 2-span control scheme, it is shown numerically that the proposed method performs better than X-bar chart in terms of the average run length. Some simulations are accomplished to evaluate the performance of the proposed scheme in terms of the average run length and the conditional expected delay. The results are compared with those of the existing monitoring schemes including the X-bar chart. The false discovery rate is also estimated and compared with the target control level.     

A unified sequential test procedure for simultaneous testing the equality of several binomial proportions to a specified standard

In this article, we propose a unified sequentially rejective test procedure for testing simultaneously the equality of several independent binomial proportions to a specified standard. The proposed test procedure is general enough to include some well-known multiple testing procedures such as the Ordinary Bonferroni procedure, Hochberg procedure and Rom procedure. It involves multiple tests of significance based on the simple binomial tests (exact or approximate) which can be easily found in many elementary standard statistics textbooks. Unlike the traditional Chi-square test of the overall hypothesis, the procedure can identify the subset of the binomial proportions, which are different from the prespecified standard with the control of the familywise type I error rate. Moreover, the power computation of the procedure is provided and the procedure is illustrated by two real examples from an ecological study and a carcinogenicity study.     

A Rejection Principle for Sequential Tests of Multiple Hypotheses Controlling Familywise Error Rates

We present a unifying approach to multiple testing procedures for sequential (or streaming) data by giving sufficient conditions for a sequential multiple testing procedure to control the familywise error rate (FWER). Together, we call these conditions a ‘rejection principle for sequential tests’, which we then apply to some existing sequential multiple testing procedures to give simplified understanding of their FWER control. Next, the principle is applied to derive two new sequential multiple testing procedures with provable FWER control, one for testing hypotheses in order and another for closed testing. Examples of these new procedures are given by applying them to a chromosome aberration data set and finding the maximum safe dose of a treatment.     

Evaluations of FWER-controlling methods in multiple hypothesis testing

Simultaneously testing a family of n null hypotheses can arise in many applications. A common problem in multiple hypothesis testing is to control Type-I error. The probability of at least one false rejection referred to as the familywise error rate (FWER) is one of the earliest error rate measures. Many FWER-controlling procedures have been proposed. The ability to control the FWER and achieve higher power is often used to evaluate the performance of a controlling procedure. However, when testing multiple hypotheses, FWER and power are not sufficient for evaluating controlling procedure’s performance. Furthermore, the performance of a controlling procedure is also governed by experimental parameters such as the number of hypotheses, sample size, the number of true null hypotheses and data structure. This paper evaluates, under various experimental settings, the performance of some FWER-controlling procedures in terms of five indices, the FWER, the false discovery rate, the false non-discovery rate, the sensitivity and the specificity. The results can provide guidance on how to select an appropriate FWER-controlling procedure to meet a study’s objective.     

Robust inference for group sequential trials

For ethical reasons, group sequential trials were introduced to allow trials to stop early in the event of extreme results. Endpoints in such trials are usually mortality or irreversible morbidity. For a given endpoint, the norm is to use a single test statistic and to use that same statistic for each analysis. This approach is risky because the test statistic has to be specified before the study is unblinded, and there is loss in power if the assumptions that ensure optimality for each analysis are not met. To minimize the risk of moderate to substantial loss in power due to a suboptimal choice of a statistic, a robust method was developed for nonsequential trials. The concept is analogous to diversification of financial investments to minimize risk. The method is based on combining P values from multiple test statistics for formal inference while controlling the type I error rate at its designated value.This article evaluates the performance of 2 P value combining methods for group sequential trials. The emphasis is on time to event trials although results from less complex trials are also included. The gain or loss in power with the combination method relative to a single statistic is asymmetric in its favor. Depending on the power of each individual test, the combination method can give more power than any single test or give power that is closer to the test with the most power. The versatility of the method is that it can combine P values from different test statistics for analysis at different times. The robustness of results suggests that inference from group sequential trials can be strengthened with the use of combined tests.     

Testing non-inferiority and superiority for two endpoints for several treatments with a control

Some multiple comparison procedures are described for multiple armed studies. The procedures are appropriate for testing all hypotheses for comparing two endpoints and multiple test arms to a single control group, for example three different fixed doses compared to a placebo. The procedure assumes that among the two endpoints, one is designated as a primary endpoint such that for a given treatment arm, no hypothesis for the secondary endpoint can be rejected unless the hypothesis for the primary endpoint was rejected. The procedures described control the family-wise error rate in the strong sense at a specified level α.     

Efficient Stratified Testing Procedure for a False Discovery Rate

The false discovery rate (FDR) has become a popular error measure in the large-scale simultaneous testing. When data are collected from heterogenous sources and form grouped hypotheses testing, it may be beneficial to use the distinct feature of groups to conduct the multiple hypotheses testing. We propose a stratified testing procedure that uses different FDR levels according to the stratification features based on p-values. Our proposed method is easy to implement in practice. Simulations studies show that the proposed method produces more efficient testing results. The stratified testing procedure minimizes the overall false negative rate (FNR) level, while controlling the overall FDR. An example from a type II diabetes mice study further illustrates the practical advantages of this new approach.     

Performance of nonparametric multiple comparison tests under heteroscedasticity,dependency, and skewed error distribution

In this article, an extensive Monte Carlo simulation study is conducted to evaluate and compare nonparametric multiple comparison tests under violations of classical analysis of variance assumptions. Simulation space of the Monte Carlo study is composed of 288 different combinations of balanced and unbalanced sample sizes, number of groups, treatment effects, various levels of heterogeneity of variances, dependence between subgroup levels, and skewed error distributions under the single factor experimental design. By this large simulation space, we present a detailed analysis of effects of the violations of assumptions on the performance of nonparametric multiple comparison tests in terms of three error and four power measures. Observations of this study are beneficial to decide the optimal nonparametric test according to requirements and conditions of undertaken experiments. When some of the assumptions of analysis of variance are violated and number of groups is small, use of stepwise Steel-Dwass procedure with Holm's approach is appropriate to control type I error at a desired level. Dunn's method should be employed for greater number of groups. When subgroups are unbalanced and number of groups is small, Nemenyi's procedure with Duncan's approach produces high power values. Conover's procedure successfully provides high power values with a small number of unbalanced groups or with a greater number of balanced or unbalanced groups. At the same time, Conover's procedure is unable to control type I error rates.     

Tests for Coefficients in High‐dimensional Additive Hazard Models

We consider hypothesis testing problems for low‐dimensional coefficients in a high dimensional additive hazard model. A variance reduced partial profiling estimator (VRPPE) is proposed and its asymptotic normality is established, which enables us to test the significance of each single coefficient when the data dimension is much larger than the sample size. Based on the p‐values obtained from the proposed test statistics, we then apply a multiple testing procedure to identify significant coefficients and show that the false discovery rate can be controlled at the desired level. The proposed method is also extended to testing a low‐dimensional sub‐vector of coefficients. The finite sample performance of the proposed testing procedure is evaluated by simulation studies. We also apply it to two real data sets, with one focusing on testing low‐dimensional coefficients and the other focusing on identifying significant coefficients through the proposed multiple testing procedure.     

Adaptive graph‐based multiple testing procedures

Multiple testing procedures defined by directed, weighted graphs have recently been proposed as an intuitive visual tool for constructing multiple testing strategies that reflect the often complex contextual relations between hypotheses in clinical trials. Many well‐known sequentially rejective tests, such as (parallel) gatekeeping tests or hierarchical testing procedures are special cases of the graph based tests. We generalize these graph‐based multiple testing procedures to adaptive trial designs with an interim analysis. These designs permit mid‐trial design modifications based on unblinded interim data as well as external information, while providing strong family wise error rate control. To maintain the familywise error rate, it is not required to prespecify the adaption rule in detail. Because the adaptive test does not require knowledge of the multivariate distribution of test statistics, it is applicable in a wide range of scenarios including trials with multiple treatment comparisons, endpoints or subgroups, or combinations thereof. Examples of adaptations are dropping of treatment arms, selection of subpopulations, and sample size reassessment. If, in the interim analysis, it is decided to continue the trial as planned, the adaptive test reduces to the originally planned multiple testing procedure. Only if adaptations are actually implemented, an adjusted test needs to be applied. The procedure is illustrated with a case study and its operating characteristics are investigated by simulations. Copyright © 2014 John Wiley & Sons, Ltd.