Two-Groups Model Research Articles

Two-Component Mixture Model in the Presence of Covariates

In this article, we study a generalization of the two-groups model in the presence of covariates—a problem that has recently received much attention in the statistical literature due to its applicability in multiple hypotheses testing problems. The model we consider allows for infinite dimensional parameters and offers flexibility in modeling the dependence of the response on the covariates. We discuss the identifiability issues arising in this model and systematically study several estimation strategies. We propose a tuning parameter-free nonparametric maximum likelihood method, implementable via the expectation-maximization algorithm, to estimate the unknown parameters. Further, we derive the rate of convergence of the proposed estimators—in particular we show that the finite sample Hellinger risk for every ‘approximate’ nonparametric maximum likelihood estimator achieves a near-parametric rate (up to logarithmic multiplicative factors). In addition, we propose and theoretically study two ‘marginal’ methods that are more scalable and easily implementable. We demonstrate the efficacy of our procedures through extensive simulation studies and relevant data analyses—one arising from neuroscience and the other from astronomy. We also outline the application of our methods to multiple testing. The companion R package NPMLEmix implements all the procedures proposed in this article.

Blended Learning: A New Approach to Teach Ballet Technique for Undergraduate Students

This study aims to investigate and compare the effect of the Blended Learning (BL) approach to Face to Face (FTF) teaching method on students’ level of performance and sport participation motivation level. It is a qualitative study using the quasi-experimental method and applying the pre-test/post-test two-groups design model. The participants of this study were 40 female undergraduate students of School of Physical Education/University of Jordan. They were beginners enrolled in ballet classes with no previous experience in ballet; 18 students attended the FTF ballet class (control group), and 22 students attended the BL ballet class (experimental group). Skill performance tests and Sport participation Motivation Scale SMS-28 were used to collect the study’s pre and post data. The data were then analyzed using descriptive analysis. The results of this study revealed no significant differences between teaching strategies in relation to students’ level of performance. On the other hand, significant differences occurred in post-tests of students’ motivation level in favor of the BL group. All intrinsic and extrinsic motivation subscales except for amotivation were significantly different in favor of BL classes. In conclusion, BL proved to be an efficient and convenient methodology in terms of enhancing students’ level of performance and boost their motivation for sports participation.

Asymptotic Properties of Bayes Risk of a General Class of Shrinkage Priors in Multiple Hypothesis Testing Under Sparsity

Consider the problem of simultaneous testing for the means of independent normal observations. In this paper, we study some asymptotic optimality properties of certain multiple testing rules induced by a general class of one-group shrinkage priors in a Bayesian decision theoretic framework, where the overall loss is taken as the number of misclassified hypotheses. We assume a two-groups normal mixture model for the data and consider the asymptotic framework adopted in Bogdan et al. (2011) who introduced the notion of asymptotic Bayes optimality under sparsity in the context of multiple testing. The general class of one-group priors under study is rich enough to include, among others, the families of three parameter beta, generalized double Pareto priors, and in particular the horseshoe, the normal–exponential–gamma and the Strawderman–Berger priors. We establish that within our chosen asymptotic framework, the multiple testing rules under study asymptotically attain the risk of the Bayes Oracle up to a multiplicative factor, with the constant in the risk close to the constant in the Oracle risk. This is similar to a result obtained in Datta and Ghosh (2013) for the multiple testing rule based on the horseshoe estimator introduced in Carvalho et al. (2009, 2010). We further show that under very mild assumption on the underlying sparsity parameter, the induced decisions using an empirical Bayes estimate of the corresponding global shrinkage parameter proposed by van der Pas et al. (2014), asymptotically attain the optimal Bayes risk up to the same multiplicative factor. We provide a unifying argument applicable for the general class of priors under study. In the process, we settle a conjecture regarding optimality property of the generalized double Pareto priors made in Datta and Ghosh (2013). Our work also shows that the result in Datta and Ghosh (2013) can be improved further.

A nonparametric empirical Bayes framework for large-scale multiple testing

We propose a flexible and identifiable version of the 2-groups model, motivated by hierarchical Bayes considerations, that features an empirical null and a semiparametric mixture model for the nonnull cases. We use a computationally efficient predictive recursion (PR) marginal likelihood procedure to estimate the model parameters, even the nonparametric mixing distribution. This leads to a nonparametric empirical Bayes testing procedure, which we call PRtest, based on thresholding the estimated local false discovery rates. Simulations and real data examples demonstrate that, compared to existing approaches, PRtest's careful handling of the nonnull density can give a much better fit in the tails of the mixture distribution which, in turn, can lead to more realistic conclusions.

Laplace Approximated EM Microarray Analysis: An Empirical Bayes Approach for Comparative Microarray Experiments

A two-groups mixed-effects model for the comparison of (normalized) microarray data from two treatment groups is considered. Most competing parametric methods that have appeared in the literature are obtained as special cases or by minor modification of the proposed model. Approximate maximum likelihood fitting is accomplished via a fast and scalable algorithm, which we call LEMMA (Laplace approximated EM Microarray Analysis). The posterior odds of treatment $\times$ gene interactions, derived from the model, involve shrinkage estimates of both the interactions and of the gene specific error variances. Genes are classified as being associated with treatment based on the posterior odds and the local false discovery rate (f.d.r.) with a fixed cutoff. Our model-based approach also allows one to declare the non-null status of a gene by controlling the false discovery rate (FDR). It is shown in a detailed simulation study that the approach outperforms well-known competitors. We also apply the proposed methodology to two previously analyzed microarray examples. Extensions of the proposed method to paired treatments and multiple treatments are also discussed.

Rejoinder: Microarrays, Empirical Bayes and the Two-Groups Model

The Fisher–Neyman–Pearson theory of hypothesis testing was a triumph of mathematical elegance and practical utility. It was never designed, though, to handle 10,000 tests at once, and one can see contemporary statisticians struggling to develop theories appropriate to our new scientific environment. This paper is part of that effort: starting from just the two-groups model (2.1), it aims to show Bayesian and frequentist ideas merging into a practical framework for largescale simultaneous testing. False discovery rates, Benjamini and Hochberg’s influential contribution to modern statistical theory, is the main methodology featured in the paper, but I really was not trying to sell any specific technology as the final word. In fact, the discussants offer an attractive menu of alternatives. It is still early in the large-scale hypothesis testing story, and I expect, and hope for, major developments in both theory and practice. The central issue, as Carl Morris makes clear, is the combination of information from a collection of more or less similar sources, for example from the expression levels of different genes in a microarray study. Crucial questions revolve around the comparability and relevance of the various sources, as well as the proper choice of a null distribution. Technical issues such as the exact control of Type I errors are important as well, but, in my opinion, have played too big a role in the microarray literature. The discussions today are an appealing mixture of technical facility and big-picture thinking. They are substantial essays in their own right, and I will be able to respond here to only a few of the issues raised. I once wrote, about the jackknife, that good simple ideas are our most precious intellectual commodity. False discovery rates fall into that elite category. The two-groups model is used here to unearth the Bayesian roots of Benjamini and Hochberg’s originally frequentist construction. In a Bayesian framework it is natural

Microarrays, Empirical Bayes and the Two-Groups Model

The classic frequentist theory of hypothesis testing developed by Neyman, Pearson and Fisher has a claim to being the twentieth century’s most influential piece of applied mathematics. Something new is happening in the twenty-first century: high-throughput devices, such as microarrays, routinely require simultaneous hypothesis tests for thousands of individual cases, not at all what the classical theory had in mind. In these situations empirical Bayes information begins to force itself upon frequentists and Bayesians alike. The two-groups model is a simple Bayesian construction that facilitates empirical Bayes analysis. This article concerns the interplay of Bayesian and frequentist ideas in the two-groups setting, with particular attention focused on Benjamini and Hochberg’s False Discovery Rate method. Topics include the choice and meaning of the null hypothesis in large-scale testing situations, power considerations, the limitations of permutation methods, significance testing for groups of cases (such as pathways in microarray studies), correlation effects, multiple confidence intervals and Bayesian competitors to the two-groups model.

Comment: Microarrays, Empirical Bayes and the Two-Groups Model

Efron has given us a comprehensive and thoughtful review of his approach to large-scale testing stemming from the challenges of analyzing microarray data. Addressing the microarray challenge right from the emergence of the technology, and adapting the point of view on multiple testing that emphasizes the false discovery rate, Efron’s contributions in both fields have been immense. In the discussed paper he reviews philosophy, motivation, methodologies and even practicalities, and along this process gives us a view of the field of statistics from an eagle’s eye. A thorough discussion of such a work is a major undertaking. Instead, I shall first comment on five issues and then discuss new directions for research on largescale multiple inference that bear on Efron’s review. The scope of the review challenges a discussant to try and address some of the issues raised from a broader point of view. I shall give it a try.

Comment: Microarrays, Empirical Bayes and the Two-Groups Model

Through his various examples, Professor Efron makes a convincing case that cutting-edge science requires methods for detecting multiple non-nulls. These methods must be straightforward to implement, but perhaps more importantly statisticians need to be able to justify them unambiguously. Efron's Empirical Bayes approach is certainly computationally efficient, but we feel the rationale for making each of his steps is unattractively ad hoc. This concern is practical, not philosophical; Efron's criterion for choice of tuning parameters seems to be that they look believable. In less expert hands, this approach seems to introduce a lot of leeway for practitioners to simply tune away until they get the results they want. In an attempt to address this problem, we will de scribe an approach developed in a fully model-based framework. As with locfdr, the calculations are fast, but our whole analysis derives from clear up-front statements about what the analysis is trying to achieve, and the modeling assumptions made. The results look reassuringly similar to Professor Efron's. We hope this will be helpful for understanding the current paper, and in making a contribution to this general field. We begin by following Efron in placing the local false discovery rate, fdr(z), as the primary focus of the analysis, and exploit the fact that it can offer a neat pa rameterization of the two-part model. If the marginal, mixture density for the z-values is

Comment: Microarrays, Empirical Bayes and the Two-Groups Model

Abstract. Brad Efron’s paper has inspired a return to the ideas be-hind Bayes, frequency and empirical Bayes. The latter preferably wouldnot be limited to exchangeable models for the data and hyperparam-eters. Parallels are revealed between microarray analyses and profilingof hospitals, with advances suggesting more decision modeling for geneidentification also. Then good multilevel and empirical Bayes modelsfor random effects should be sought when regression toward the meanis anticipated. Key words and phrases: Bayes, frequency, interval estimation, ex-changeable, general model, random effects.1. FREQUENCY, BAYES, EMPIRICAL BAYESAND A GENERAL MODELBrad Efron’s two-groups approach and the empir-ical null (“null” refers to a distribution, not to ahypothesis) extension of his local fdr addresses test-ing many hypotheses simultaneously, with model-ing enabled by the repeated presence of many simi-lar problems. He assumes two-level models for ran-dom effects, developing theory by drawing on andcombining ideas from frequency, Bayesian and em-pirical Bayesian perspectives. The last half-centuryin statistics has seen exciting developments frommany perspectives for simultaneous estimation ofrandom effects, but there has been little explicit par-allel work on the complementary problem of hypoth-esis testing. That changes in Brad’s paper,especiallyfor testing many hypotheses when exchangeabilityrestrictions are plausible.“Empirical Bayes” is in the paper’s title, said inSection 3 to be a “bipolar” methodology that draws