Modified Simes’ critical values under positive dependence

A modification of the critical values of Simes’ test is suggested in this article when the underlying test statistics are multivariate normal with a common non-negative correlation, yielding a more powerful test than the original Simes’ test. A step-up multiple testing procedure with these modified critical values, which is shown to control false discovery rate (FDR), is presented as a modification of the traditional Benjamini–Hochberg (BH) procedure. Simulations were carried out to compare this modified BH procedure with the BH and other modified BH procedures in terms of false non-discovery rate (FNR), 1–FDR–FNR and average power. The present modified BH procedure is observed to perform well compared to others when the test statistics are highly correlated and most of the hypotheses are true.     

Asymptotic comparison of the critical values of step-down and step-up multiple comparison procedures

We consider the problem of comparing step-down and step-up multiple test procedures for testing n hypotheses when independent p-values or independent test statistics are available. The defining critical values of these procedures for independent test statistics are asymptotically equal, which yields a theoretical argument for the numerical observation that the step-up procedure is mostly more powerful than the step-down procedure. The main aim of this paper is to quantify the differences between the critical values more precisely. As a by-product we also obtain more information about the gain when we consider two subsequent steps of these procedures. Moreover, we investigate how liberal the step-up procedure becomes when the step-up critical values are replaced by their step-down counterparts or by more refined approximate values. The results for independent p-values are the basis for obtaining corresponding results when independent real-valued test statistics are at hand. It turns out that the differences of step-down and step-up critical values as well as the differences between subsequent steps tend to zero for many distributions, except for heavy-tailed distributions. The Cauchy distribution yields an example where the critical values of both procedures are nearly linearly increasing in n.     

Multiple Comparisons Controlling Expected Number of False Discoveries

In this paper, it is put forward that the task of designing a procedure for a set of multiple comparisons should be considered as a decision-making under uncertainty. Due to this motivation, for the problem of multiple comparisons, we considered another error rate to be controlled, called PFER (per-family error rate), which requests that the expected number of false rejections of a test procedure should be bounded no more than a prespecified level k. Although PFER was proposed by Tukey in 1953, there is not much studying about it so far. We first present Bonferroni procedure (single-step) and then build two step-up procedures with one having generic critical values and another using critical values in BH (Benjamini and Hochberg) type. These procedures are compared through simulations.     

A generalized step-up-down multiple test procedure

A generalization of step-up and step-down multiple test procedures is proposed. This step-up-down procedure is useful when the objective is to reject a specified minimum number, q, out of a family of k hypotheses. If this basic objective is met at the first step, then it proceeds in a step-down manner to see if more than q hypotheses can be rejected. Otherwise it proceeds in a step-up manner to see if some number less than q hypotheses can be rejected. The usual step-down procedure is the special case where q = 1, and the usual step-up procedure is the special case where q = k. Analytical and numerical comparisons between the powers of the step-up-down procedures with different choices of q are made to see how these powers depend on the actual number of false hypotheses. Examples of application include comparing the efficacy of a treatment to a control for multiple endpoints and testing the sensitivity of a clinical trial for comparing the efficacy of a new treatment with a set of standard treatments.     

Improved bonferroni procedures for testing overall and pairwise homogeneity hypotheses

Simes' (1986) improved Bonferroni test is verified by simulations ^?to control the α-level when testing the overall homogeneity hypothesis with all pairwise t statistics in a balanced parallel group design. Similarly, this result was found to hold (for practical purposes) in various underlying distributions other than the normal and in some unbalanced designs. To allow the use of step-up procedures based on pairwise t statistics, simulations were used to verify that Simes' test, when applied to testing multiple subset homogeneity hypotheses with pairwise t statistics also keeps the level ? α. Some robustness as above was found here too. Tables of the simulation results are provided and an example of a step-up Hommel-Shaffer type procedure with pairwise comparisons is given.     

Sample size determination in step-down and step-up multiple tests for comparing treatments with a control

We address the problem of sample size determination in multiple comparisons of k treatments with a control for step-down and step-up testing, assuming normal data and homogeneous variances. We define power as the probability of correctly rejecting all hypotheses for which the treatment vs. control difference exceeds a specified value. Our paper supplements papers by Hayter and Tamhane (J. Statist. Plann. Inference 27 (1991) 271–290) who solved the problem for one-sided comparisons using the step-down procedure and by Liu (J. Statist. Plann. Inference 62 (1997b) 255–261) who considered the two-sided case using the single-step method. We provide expressions that allow computer evaluation of the power and necessary sample sizes for one- and two-sided tests using either step-down or step-up procedures. Tables are given from which sample sizes to guarantee a specified power can be determined.     

FDR- and FWE-controlling methods using data-driven weights

Weighted methods are an important feature of multiplicity control methods. The weights must usually be chosen a priori, on the basis of experimental hypotheses. Under some conditions, however, they can be chosen making use of information from the data (therefore a posteriori) while maintaining multiplicity control. In this paper we provide: (1) a review of weighted methods for familywise type I error rate (FWE) (both parametric and nonparametric) and false discovery rate (FDR) control; (2) a review of data-driven weighted methods for FWE control; (3) a new proposal for weighted FDR control (data-driven weights) under independence among variables; (4) under any type of dependence; (5) a simulation study that assesses the performance of procedure of point 4 under various conditions.     

Optimally weighted,fixed sequence and gatekeeper multiple testing procedures

We consider a class of closed multiple test procedures indexed by a fixed weight vector. The class includes the Holm weighted step-down procedure, the closed method using the weighted Fisher combination test, and the closed method using the weighted version of Simes’ test. We show how to choose weights to maximize average power, where “average power” is itself weighted by importance assigned to the various hypotheses.Numerical computations suggest that the optimal weights for the multiple test procedures tend to certain asymptotic configurations. These configurations offer numerical justification for intuitive multiple comparisons methods, such as downweighting variables found insignificant in preliminary studies, giving primary variables more emphasis, gatekeeping test strategies, pre-determined multiple testing sequences, and pre-determined sequences of families of tests. We establish that such methods fall within the envelope of weighted closed testing procedures, thus providing a unified view of fixed sequences, fixed sequences of families, and gatekeepers within the closed testing paradigm. We also establish that the limiting cases control the familywise error rate (or FWE), using well-known results about closed tests, along with the dominated convergence theorem.     

A two-step rejection procedure for testing multiple hypotheses

This paper considers p-value based step-wise rejection procedures for testing multiple hypotheses. The existing procedures have used constants as critical values at all steps. With the intention of incorporating the exact magnitude of the p-values at the earlier steps into the decisions at the later steps, this paper applies a different strategy that the critical values at the later steps are determined as functions of the p-values from the earlier steps. As a result, we have derived a new equality and developed a two-step rejection procedure following that. The new procedure is a short-cut of a step-up procedure, and it possesses great simplicity. In terms of power, the proposed procedure is generally comparable to the existing ones and exceptionally superior when the largest p-value is anticipated to be less than 0.5.     

P-value adjustment for multiple binary endpoints

The p-value-based adjustment of individual endpoints and the global test for an overall inference are the two general approaches for the analysis of multiple endpoints. Statistical procedures developed for testing multivariate outcomes often assume that the multivariate endpoints are either independent or normally distributed. This paper presents a general approach for the analysis of multivariate binary data under the framework of generalized linear models. The generalized estimating equations (GEE) approach is applied to estimate the correlation matrix of the test statistics using the identity and exchangeable working correlation matrices with the model-based as well as robust estimators. The objectives of the approaches are the adjustment of p-values of individual endpoints to identify the affected endpoints as well as the global test of an overall effect. A Monte Carlo simulation was conducted to evaluate the overall family wise error (FWE) rates of the single-step down p-value adjustment approach from two adjustment methods to three global test statistics. The p-value adjustment approach seems to control the FWE better than the global approach Applications of the proposed methods are illustrated by analyzing a carcinogenicity experiment designed to study the dose response trend for 10 tumor sites, and a developmental toxicity experiment with three malformation types: external, visceral, and skeletal.     

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.     

Some properties of modified integrated squared error estimators for the stable laws

The estimation procedure of Paulson, Holcomb and Leitch (1975) for the parameters of the stable laws is shown to be similar in spirit to the modified X²minimum procedure. This observation suggests that a class of modified integrated squared error procedures may be developed for the stable laws as well as much more generally. For the stable case, some influence curves, asymptotic covariances, and efficiencies are given, and the robustness of maximum likelihood estimators is discussed.     

On the Critical Value Behaviour of Multiple Decision Procedures

In this paper we investigate the asymptotic critical value behaviour of certain multiple decision procedures as e.g. simultaneous confidence intervals and simultaneous as well as stepwise multiple test procedures. Supposing that n hypotheses or parameters of interest are under consideration we investigate the critical value behaviour when n increases. More specifically, we answer e.g. the question by which amount the lengths of confidence intervals increase when an additional parameter is added to the statistical analysis. Furthermore, critical values of different multiple decision procedures as for instance step-down and step-up procedures will be compared. Some general theoretic results are derived and applied for various distributions.     

Page's sequential procedure for change-point detection in time series regression

In a variety of different settings cumulative sum (CUSUM) procedures have been applied for the sequential detection of structural breaks in the parameters of stochastic models. Yet their performance depends strongly on the time of change and is best under early change scenarios. For later changes their finite sample behavior is rather questionable. We therefore propose modified CUSUM procedures for the detection of abrupt changes in the regression parameter of multiple time series regression models, that show a higher stability with respect to the time of change than ordinary CUSUM procedures. The asymptotic distributions of the test statistics and the consistency of the procedures are provided. In a simulation study it is shown that the proposed procedures behave well in finite samples. Finally the procedures are applied to a set of capital asset pricing data related to the Fama–French extension of the CAPM.     

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.     

Closure properties of classes of multiple testing procedures

Statistical discoveries are often obtained through multiple hypothesis testing. A variety of procedures exists to evaluate multiple hypotheses, for instance the ones of Benjamini–Hochberg, Bonferroni, Holm or Sidak. We are particularly interested in multiple testing procedures with two desired properties: (solely) monotonic and well-behaved procedures. This article investigates to which extent the classes of (monotonic or well-behaved) multiple testing procedures, in particular the subclasses of so-called step-up and step-down procedures, are closed under basic set operations, specifically the union, intersection, difference and the complement of sets of rejected or non-rejected hypotheses. The present article proves two main results: First, taking the union or intersection of arbitrary (monotonic or well-behaved) multiple testing procedures results in new procedures which are monotonic but not well-behaved, whereas the complement or difference generally preserves neither property. Second, the two classes of (solely monotonic or well-behaved) step-up and step-down procedures are closed under taking the union or intersection, but not the complement or difference.     

Nonparametric multiple comparison procedures for unbalanced two-way layouts

In this paper, we consider nonparametric multiple comparison procedures for unbalanced two-way factorial designs under a pure nonparametric framework. For multiple comparisons of treatments versus a control concerning the main effects or the simple factor effects, the limiting distribution of the associated rank statistics is proven to satisfy the multivariate totally positive of order two condition. Hence, asymptotically the proposed Hochberg procedure strongly controls the familywise type I error rate for the simultaneous testing of the individual hypotheses. In addition, we propose to employ Shaffer's modified version of Holm's stepdown procedure to perform simultaneous tests on all pairwise comparisons regarding the main or simple factor effects and to perform simultaneous tests on all interaction effects. The logical constraints in the corresponding hypothesis families are utilized to sharpen the rejective thresholds and improve the power of the tests.     

Step-Down Multiple Comparison Procedures Using Medians and Permutation Tests

Richter and McCann (2007 Richter , S. J. , McCann , M. H. ( 2007 ). Multiple comparisons using medians and permutation tests . Journal of Modern Applied Statistical Methods 6 ( 2 ): 399 – 412 . [Google Scholar]) presented a median-based multiple comparison procedure for assessing evidence of group location differences. The sampling distribution was based on the permutation distribution of the maximum median difference among all pairs, and provides strong control of the FWE. This idea is extended to develop a step-down procedure for comparing group locations. The new step-down procedure exploits logical dependencies between pairwise hypotheses and provides greater power than the single-step procedure, while still maintaining strong FWE control. The new procedure can also be a more powerful alternative to existing methods based on means, especially for heavy-tailed distributions.     

STEPWISE TESTS WHEN THE TEST STATISTICS ARE INDEPENDENT

Consider the problem of simultaneously testing a nonhierarchical finite family of hypotheses based on independent test statistics. A general stepwise test is defined, of which the well known step-down and step-up tests are special cases. The step-up test is shown to dominate the other stepwise tests, including the step-down test, for situations of practical importance. When testing against two-sided alternatives, it is pointed out that if the step-up test is augmented to include directional decisions then the augmented step-up test controls the type I and III familywise error jointly at the original level q. The definition of the adjusted p values for the step-up test is justified. The results are illustrated by a numerical example.     

A modified random-effect procedure for combining risk difference in sets of 2×2 tables from clinical trials

Summary Meta-analyses of sets of clinical trials often combine risk differences from several 2×2 tables according to a random-effects model. The DerSimonian-Laird random-effects procedure, widely used for estimating the populaton mean risk difference, weights the risk difference from each primary study inversely proportional to an estimate of its variance (the sum of the between-study variance and the conditional within-study variance). Because those weights are not independent of the risk differences, however, the procedure sometimes exhibits bias and unnatural behavior. The present paper proposes a modified weighting scheme that uses the unconditional within-study variance to avoid this source of bias. The modified procedure has variance closer to that available from weighting by ideal weights when such weights are known. We studied the modified procedure in extensive simulation experiments using situations whose parameters resemble those of actual studies in medical research. For comparison we also included two unbiased procedures, the unweighted mean and a sample-size-weighted mean; their relative variability depends on the extent of heterogeneity among the primary studies. An example illustrates the application of the procedures to actual data and the differences among the results. This research was supported by Grant HS 05936 from the Agency for Health Care Policy and Research to Harvard University.