Estimation of false discovery rates in multiple testing: application to gene microarray data

Testing for significance with gene expression data from DNA microarray experiments involves simultaneous comparisons of hundreds or thousands of genes. If R denotes the number of rejections (declared significant genes) and V denotes the number of false rejections, then V/R, if R > 0, is the proportion of false rejected hypotheses. This paper proposes a model for the distribution of the number of rejections and the conditional distribution of V given R, V / R. Under the independence assumption, the distribution of R is a convolution of two binomials and the distribution of V / R has a noncentral hypergeometric distribution. Under an equicorrelated model, the distributions are more complex and are also derived. Five false discovery rate probability error measures are considered: FDR = E(V/R), pFDR = E(V/R / R > 0) (positive FDR), cFDR = E(V/R / R = r) (conditional FDR), mFDR = E(V)/E(R) (marginal FDR), and eFDR = E(V)/r (empirical FDR). The pFDR, cFDR, and mFDR are shown to be equivalent under the Bayesian framework, in which the number of true null hypotheses is modeled as a random variable. We present a parametric and a bootstrap procedure to estimate the FDRs. Monte Carlo simulations were conducted to evaluate the performance of these two methods. The bootstrap procedure appears to perform reasonably well, even when the alternative hypotheses are correlated (rho = .25). An example from a toxicogenomic microarray experiment is presented for illustration.     

The nonparametric Behrens–Fisher problem in partially complete clustered data

In randomized trials or observational studies involving clustered units, the assumption of independence within clusters is not practical. Existing parametric or semiparametric methods assume specific dependence structures within a cluster. Furthermore, parametric model assumptions may not even be realistic when data are measured in a nonmetric scale as commonly happens, for example, in quality‐of‐life outcomes. In this paper, nonparametric effect‐size measures for clustered data that allow meaningful and interpretable probabilistic comparisons of treatments or intervention programs will be introduced. The dependence among observations within a cluster can be arbitrary. Point estimators along with their asymptotic properties for computing confidence intervals and performing hypothesis test will be discussed. Small sample approximations that retain some of the optimal asymptotic behaviors will be presented. In our setup, some clusters may involve observations coming from both intervention groups (referred to as complete clusters), while others may contain observations from one group only (referred to as incomplete clusters). In deriving the asymptotic theories, we do not impose any relation in the rate of divergence of the numbers of complete and incomplete clusters. Simulations show favorable performance of the methods for arbitrary combinations of complete and incomplete clusters. The developed nonparametric methods are illustrated using data from a randomized trial of indoor wood smoke reduction to improve asthma symptoms and a cluster‐randomized trial for smoking cessation.     

Controlling the false discovery rate with constraints: the Newman-Keuls test revisited

The Newman-Keuls (NK) procedure for testing all pairwise comparisons among a set of treatment means, introduced by Newman (1939) and in a slightly different form by Keuls (1952) was proposed as a reasonable way to alleviate the inflation of error rates when a large number of means are compared. It was proposed before the concepts of different types of multiple error rates were introduced by Tukey (1952a, b; 1953). Although it was popular in the 1950s and 1960s, once control of the familywise error rate (FWER) was accepted generally as an appropriate criterion in multiple testing, and it was realized that the NK procedure does not control the FWER at the nominal level at which it is performed, the procedure gradually fell out of favor. Recently, a more liberal criterion, control of the false discovery rate (FDR), has been proposed as more appropriate in some situations than FWER control. This paper notes that the NK procedure and a nonparametric extension controls the FWER within any set of homogeneous treatments. It proves that the extended procedure controls the FDR when there are well-separated clusters of homogeneous means and between-cluster test statistics are independent, and extensive simulation provides strong evidence that the original procedure controls the FDR under the same conditions and some dependent conditions when the clusters are not well-separated. Thus, the test has two desirable error-controlling properties, providing a compromise between FDR control with no subgroup FWER control and global FWER control. Yekutieli (2002) developed an FDR-controlling procedure for testing all pairwise differences among means, without any FWER-controlling criteria when there is more than one cluster. The empirica example in Yekutieli's paper was used to compare the Benjamini-Hochberg (1995) method with apparent FDR control in this context, Yekutieli's proposed method with proven FDR control, the Newman-Keuls method that controls FWER within equal clusters with apparent FDR control, and several methods that control FWER globally. The Newman-Keuls is shown to be intermediate in number of rejections to the FWER-controlling methods and the FDR-controlling methods in this example, although it is not always more conservative than the other FDR-controlling methods.     

On the False Discovery Rate and Expected Type I Errors

The paper is concerned with expected type I errors of some stepwise multiple test procedures based on independent p‐values controlling the so‐called false discovery rate (FDR). We derive an asymptotic result for the supremum of the expected type I error rate(EER) when the number of hypotheses tends to infinity. Among others, it will be shown that when the original Benjamini‐Hochberg step‐up procedure controls the FDR at level α, its EER may approach a value being slightly larger than α/4 when the number of hypotheses increases. Moreover, we derive some least favourable parameter configuration results, some bounds for the FDR and the EER as well as easily computable formulae for the familywise error rate (FWER) of two FDR‐controlling procedures. Finally, we discuss some undesirable properties of the FDR concept, especially the problem of cheating.     

Inference in response-adaptive clinical trials when the enrolled population varies over time

A common assumption of data analysis in clinical trials is that the patient population, as well as treatment effects, do not vary during the course of the study. However, when trials enroll patients over several years, this hypothesis may be violated. Ignoring variations of the outcome distributions over time, under the control and experimental treatments, can lead to biased treatment effect estimates and poor control of false positive results. We propose and compare two procedures that account for possible variations of the outcome distributions over time, to correct treatment effect estimates, and to control type-I error rates. The first procedure models trends of patient outcomes with splines. The second leverages conditional inference principles, which have been introduced to analyze randomized trials when patient prognostic profiles are unbalanced across arms. These two procedures are applicable in response-adaptive clinical trials. We illustrate the consequences of trends in the outcome distributions in response-adaptive designs and in platform trials, and investigate the proposed methods in the analysis of a glioblastoma study.     

Exact calculations of average power for the Benjamini-Hochberg procedure

Exact analytic expressions are developed for the average power of the Benjamini and Hochberg false discovery control procedure. The result is based on explicit computation of the joint probability distribution of the total number of rejections and the number of false rejections, and expressed in terms of the cumulative distribution functions of the p-values of the hypotheses. An example of analytic evaluation of the average power is given. The result is confirmed by numerical experiments and applied to a meta-analysis of three clinical studies in mammography.     

Propensity score methods for time-dependent cluster confounding

In observational studies, subjects are often nested within clusters. In medical studies, patients are often treated by doctors and therefore patients are regarded as nested or clustered within doctors. A concern that arises with clustered data is that cluster-level characteristics (e.g., characteristics of the doctor) are associated with both treatment selection and patient outcomes, resulting in cluster-level confounding. Measuring and modeling cluster attributes can be difficult and statistical methods exist to control for all unmeasured cluster characteristics. An assumption of these methods however is that characteristics of the cluster and the effects of those characteristics on the outcome (as well as probability of treatment assignment when using covariate balancing methods) are constant over time. In this paper, we consider methods that relax this assumption and allow for estimation of treatment effects in the presence of unmeasured time-dependent cluster confounding. The methods are based on matching with the propensity score and incorporate unmeasured time-specific cluster effects by performing matching within clusters or using fixed- or random-cluster effects in the propensity score model. The methods are illustrated using data to compare the effectiveness of two total hip devices with respect to survival of the device and a simulation study is performed that compares the proposed methods. One method that was found to perform well is matching within surgeon clusters partitioned by time. Considerations in implementing the proposed methods are discussed.     

Imputation strategies for missing continuous outcomes in cluster randomized trials

In cluster randomized trials, intact social units such as schools, worksites or medical practices - rather than individuals themselves - are randomly allocated to intervention and control conditions, while the outcomes of interest are then observed on individuals within each cluster. Such trials are becoming increasingly common in the fields of health promotion and health services research. Attrition is a common occurrence in randomized trials, and a standard approach for dealing with the resulting missing values is imputation. We consider imputation strategies for missing continuous outcomes, focusing on trials with a completely randomized design in which fixed cohorts from each cluster are enrolled prior to random assignment. We compare five different imputation strategies with respect to Type I and Type II error rates of the adjusted two-sample t -test for the intervention effect. Cluster mean imputation is compared with multiple imputation, using either within-cluster data or data pooled across clusters in each intervention group. In the case of pooling across clusters, we distinguish between standard multiple imputation procedures which do not account for intracluster correlation and a specialized procedure which does account for intracluster correlation but is not yet available in standard statistical software packages. A simulation study is used to evaluate the influence of cluster size, number of clusters, degree of intracluster correlation, and variability among cluster follow-up rates. We show that cluster mean imputation yields valid inferences and given its simplicity, may be an attractive option in some large community intervention trials which are subject to individual-level attrition only; however, it may yield less powerful inferences than alternative procedures which pool across clusters especially when the cluster sizes are small and cluster follow-up rates are highly variable. When pooling across clusters, the imputation procedure should generally take intracluster correlation into account to obtain valid inferences; however, as long as the intracluster correlation coefficient is small, we show that standard multiple imputation procedures may yield acceptable type I error rates; moreover, these procedures may yield more powerful inferences than a specialized procedure, especially when the number of available clusters is small. Within-cluster multiple imputation is shown to be the least powerful among the procedures considered.     

Distinguishing between selective sweeps and demography using DNA polymorphism data

In 2002 Kim and Stephan proposed a promising composite-likelihood method for localizing and estimating the fitness advantage of a recently fixed beneficial mutation. Here, we demonstrate that their composite-likelihood-ratio (CLR) test comparing selective and neutral hypotheses is not robust to undetected population structure or a recent bottleneck, with some parameter combinations resulting in a false positive rate of nearly 90%. We also propose a goodness-of-fit test for discriminating rejections due to directional selection (true positive) from those due to population and demographic forces (false positives) and demonstrate that the new method has high sensitivity to differentiate the two classes of rejections.     

Spatial Cluster Detection for Repeatedly Measured Outcomes while Accounting for Residential History

Spatial cluster detection has become an important methodology in quantifying the effect of hazardous exposures. Previous methods have focused on cross‐sectional outcomes that are binary or continuous. There are virtually no spatial cluster detection methods proposed for longitudinal outcomes. This paper proposes a new spatial cluster detection method for repeated outcomes using cumulative geographic residuals. A major advantage of this method is its ability to readily incorporate information on study participants relocation, which most cluster detection statistics cannot. Application of these methods will be illustrated by the Home Allergens and Asthma prospective cohort study analyzing the relationship between environmental exposures and repeated measured outcome, occurrence of wheeze in the last 6 months, while taking into account mobile locations.     

On small‐sample inference in group randomized trials with binary outcomes and cluster‐level covariates

Group randomized trials (GRTs) randomize groups, or clusters, of people to intervention or control arms. To test for the effectiveness of the intervention when subject‐level outcomes are binary, and while fitting a marginal model that adjusts for cluster‐level covariates and utilizes a logistic link, we develop a pseudo‐Wald statistic to improve inference. Alternative Wald statistics could employ bias‐corrected empirical sandwich standard error estimates, which have received limited attention in the GRT literature despite their broad utility and applicability in our settings of interest. The test could also be carried out using popular approaches based upon cluster‐level summary outcomes. A simulation study covering a variety of realistic GRT settings is used to compare the accuracy of these methods in terms of producing nominal test sizes. Tests based upon the pseudo‐Wald statistic and a cluster‐level summary approach utilizing the natural log of observed cluster‐level odds worked best. Due to weighting, some popular cluster‐level summary approaches were found to lead to invalid inference in many settings. Finally, although use of bias‐corrected empirical sandwich standard error estimates did not consistently result in nominal sizes, they did work well, thus supporting the applicability of marginal models in GRT settings.     

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.     

Improved methods for the isolation of individual and clustered mitotic chromosomes

We have optimized procedures for the isolation of mitotic chromosomes from tissue culture cells. The first procedure is a rapid method for obtaining individual, structurally intact chromosomes suitable for analysis by electron microscopy. Further purification of these on Percoll gradients results in chromosomes free of cytoplasmic contamination, allowing biochemical characterization of the structural proteins and enzymatic activities intrinsic to mitotic chromosomes. A third procedure permits efficient, large-scale purification of chromosomes clustered together, referred to as a chromosomal cluster. The use of EDTA-containing polyamine buffers minimizes modifications of proteins and DNA during isolation and maintains the integrity of the chromosomal structure. The conditions which lead to the isolation of chromosomal clusters, as opposed to individual chromosomes, have been analyzed. Comparison of the gei patterns of proteins derived from individual chromosomes, as compared to clusters, identifies additional proteins in the latter pattern. These proteins could be involved in maintaining interchromosomal organization or positioning in the metaphase cell.     

On phylogenetic tests of irreversible evolution

"Dollo's law" states that, following loss, a complex trait cannot reevolve in an identical manner. Although the law has previously fallen into disrepute, it has only recently been challenged with statistical phylogenetic methods. We employ simulation studies of an irreversible binary character to show that rejections of Dollo's law based on likelihood-ratio tests of transition rate constraints or on reconstructions of ancestral states are frequently incorrect. We identify two major causes of errors: incorrect assignment of root state frequencies, and neglect of the effect of the character state on rates of speciation and extinction. Our findings do not necessarily overturn the conclusions of phylogenetic studies claiming reversals, but we demonstrate devastating flaws in the methods that are the foundation of all such studies. Furthermore, we show that false rejections of Dollo's law can be reduced by the use of appropriate existing models and model selection procedures. More powerful tests of irreversibility require data beyond phylogenies and character states of extant taxa, and we highlight empirical work that incorporates additional information.     

Clustering threshold gradient descent regularization: with applications to microarray studies

MOTIVATION: An important goal of microarray studies is to discover genes that are associated with clinical outcomes, such as disease status and patient survival. While a typical experiment surveys gene expressions on a global scale, there may be only a small number of genes that have significant influence on a clinical outcome. Moreover, expression data have cluster structures and the genes within a cluster have correlated expressions and coordinated functions, but the effects of individual genes in the same cluster may be different. Accordingly, we seek to build statistical models with the following properties. First, the model is sparse in the sense that only a subset of the parameter vector is non-zero. Second, the cluster structures of gene expressions are properly accounted for. RESULTS: For gene expression data without pathway information, we divide genes into clusters using commonly used methods, such as K-means or hierarchical approaches. The optimal number of clusters is determined using the Gap statistic. We propose a clustering threshold gradient descent regularization (CTGDR) method, for simultaneous cluster selection and within cluster gene selection. We apply this method to binary classification and censored survival analysis. Compared to the standard TGDR and other regularization methods, the CTGDR takes into account the cluster structure and carries out feature selection at both the cluster level and within-cluster gene level. We demonstrate the CTGDR on two studies of cancer classification and two studies correlating survival of lymphoma patients with microarray expressions. AVAILABILITY: R code is available upon request. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

Sample size considerations for stepped wedge designs with subclusters

The stepped wedge cluster randomized trial (SW-CRT) is an increasingly popular design for evaluating health service delivery or policy interventions. An essential consideration of this design is the need to account for both within-period and between-period correlations in sample size calculations. Especially when embedded in health care delivery systems, many SW-CRTs may have subclusters nested in clusters, within which outcomes are collected longitudinally. However, existing sample size methods that account for between-period correlations have not allowed for multiple levels of clustering. We present computationally efficient sample size procedures that properly differentiate within-period and between-period intracluster correlation coefficients in SW-CRTs in the presence of subclusters. We introduce an extended block exchangeable correlation matrix to characterize the complex dependencies of outcomes within clusters. For Gaussian outcomes, we derive a closed-form sample size expression that depends on the correlation structure only through two eigenvalues of the extended block exchangeable correlation structure. For non-Gaussian outcomes, we present a generic sample size algorithm based on linearization and elucidate simplifications under canonical link functions. For example, we show that the approximate sample size formula under a logistic linear mixed model depends on three eigenvalues of the extended block exchangeable correlation matrix. We provide an extension to accommodate unequal cluster sizes and validate the proposed methods via simulations. Finally, we illustrate our methods in two real SW-CRTs with subclusters.     

The Cultural Dynamics of Copycat Suicide

The observation that suicides sometimes cluster in space and/or time has led to suggestions that these clusters are caused by the social learning of suicide-related behaviours, or “copycat suicides”. Point clusters are clusters of suicides localised in both time and space, and have been attributed to direct social learning from nearby individuals. Mass clusters are clusters of suicides localised in time but not space, and have been attributed to the dissemination of information concerning celebrity suicides via the mass media. Here, agent-based simulations, in combination with scan statistic methods for detecting clusters of rare events, were used to clarify the social learning processes underlying point and mass clusters. It was found that social learning between neighbouring agents did generate point clusters as predicted, although this effect was partially mimicked by homophily (individuals preferentially assorting with similar others). The one-to-many transmission dynamics characterised by the mass media were shown to generate mass clusters, but only where social learning was weak, perhaps due to prestige bias (only copying prestigious celebrities) and similarity bias (only copying similar models) acting to reduce the subset of available models. These findings can help to clarify and formalise existing hypotheses and to guide future empirical work relating to real-life copycat suicides.     

Wrangling Phosphoproteomic Data to Elucidate Cancer Signaling Pathways

The interpretation of biological data sets is essential for generating hypotheses that guide research, yet modern methods of global analysis challenge our ability to discern meaningful patterns and then convey results in a way that can be easily appreciated. Proteomic data is especially challenging because mass spectrometry detectors often miss peptides in complex samples, resulting in sparsely populated data sets. Using the R programming language and techniques from the field of pattern recognition, we have devised methods to resolve and evaluate clusters of proteins related by their pattern of expression in different samples in proteomic data sets. We examined tyrosine phosphoproteomic data from lung cancer samples. We calculated dissimilarities between the proteins based on Pearson or Spearman correlations and on Euclidean distances, whilst dealing with large amounts of missing data. The dissimilarities were then used as feature vectors in clustering and visualization algorithms. The quality of the clusterings and visualizations were evaluated internally based on the primary data and externally based on gene ontology and protein interaction networks. The results show that t-distributed stochastic neighbor embedding (t-SNE) followed by minimum spanning tree methods groups sparse proteomic data into meaningful clusters more effectively than other methods such as k-means and classical multidimensional scaling. Furthermore, our results show that using a combination of Spearman correlation and Euclidean distance as a dissimilarity representation increases the resolution of clusters. Our analyses show that many clusters contain one or more tyrosine kinases and include known effectors as well as proteins with no known interactions. Visualizing these clusters as networks elucidated previously unknown tyrosine kinase signal transduction pathways that drive cancer. Our approach can be applied to other data types, and can be easily adopted because open source software packages are employed.