Truncated Simes tests

It is known that the one‐sided Simes’ test controls the error rate if the underlying distribution is multivariate totally positive of order 2 (MTP2), but not in general. The two‐sided test also controls the error rate when the coordinate absolute value has an MTP2 distribution, which holds more generally. We prove mathematically that when the coordinate absolute value controls the error rate at level 2α, then certain kinds of truncated Simes’ tests also control the one‐sided error rate at level α. We also compare the closure of the truncated tests with the Holms, Hochberg, and Hommel procedures in many scenarios when the test statistics are multivariate normal.     

Tests for Genetic Differentiation

When testing for genetic differentiation the joint null hypothesis that there is no allele frequency difference at any locus is of interest. Common approaches to test this hypothesis are based on the summation of χ² statistics over loci and on the Bonferroni correction, respectively. Here, we also consider the Simes adjustment and a recently proposed truncated product method (TPM) to combine P‐values. The summation and the TPM (using a relatively large truncation point) are powerful when there are differences in many or all loci. The Simes adjustment, however, is powerful when there are differences regarding one or a few loci only. As a compromise between the different approaches we introduce a combination between the Simes adjustment and the TPM, i.e. the joint null hypothesis is rejected if at least one of the two methods, Simes and TPM, is significant at the α/2‐level. Simulation results indicate that this combination is a robust procedure with high power over the different types of alternatives.     

On generalized Simes critical constants

We consider the problem treated by Simes of testing the overall null hypothesis formed by the intersection of a set of elementary null hypotheses based on ordered p‐values of the associated test statistics. The Simes test uses critical constants that do not need tabulation. Cai and Sarkar gave a method to compute generalized Simes critical constants which improve upon the power of the Simes test when more than a few hypotheses are false. The Simes constants can be viewed as the first order (requiring solution of a linear equation) and the Cai‐Sarkar constants as the second order (requiring solution of a quadratic equation) constants. We extend the method to third order (requiring solution of a cubic equation) constants, and also offer an extension to an arbitrary kth order. We show by simulation that the third order constants are more powerful than the second order constants for testing the overall null hypothesis in most cases. However, there are some drawbacks associated with these higher order constants especially for , which limits their practical usefulness.     

Adaptive designs with arbitrary dependence structure

Adaptive designs were originally developed for independent and uniformly distributed p‐values. There are trial settings where independence is not satisfied or where it may not be possible to check whether it is satisfied. In these cases, the test statistics and p‐values of each stage may be dependent. Since the probability of a type I error for a fixed adaptive design depends on the true dependence structure between the p‐values of the stages, control of the type I error rate might be endangered if the dependence structure is not taken into account adequately. In this paper, we address the problem of controlling the type I error rate in two‐stage adaptive designs if any dependence structure between the test statistics of the stages is admitted (worst case scenario). For this purpose, we pursue a copula approach to adaptive designs. For two‐stage adaptive designs without futility stop, we derive the probability of a type I error in the worst case, that is for the most adverse dependence structure between the p‐values of the stages. Explicit analytical considerations are performed for the class of inverse normal designs. A comparison with the significance level for independent and uniformly distributed p‐values is performed. For inverse normal designs without futility stop and equally weighted stages, it turns out that correcting for the worst case is too conservative as compared to a simple Bonferroni design.     

Graphical approaches for multiple comparison procedures using weighted Bonferroni, Simes, or parametric tests

The confirmatory analysis of pre-specified multiple hypotheses has become common in pivotal clinical trials. In the recent past multiple test procedures have been developed that reflect the relative importance of different study objectives, such as fixed sequence, fallback, and gatekeeping procedures. In addition, graphical approaches have been proposed that facilitate the visualization and communication of Bonferroni-based closed test procedures for common multiple test problems, such as comparing several treatments with a control, assessing the benefit of a new drug for more than one endpoint, combined non-inferiority and superiority testing, or testing a treatment at different dose levels in an overall and a subpopulation. In this paper, we focus on extended graphical approaches by dissociating the underlying weighting strategy from the employed test procedure. This allows one to first derive suitable weighting strategies that reflect the given study objectives and subsequently apply appropriate test procedures, such as weighted Bonferroni tests, weighted parametric tests accounting for the correlation between the test statistics, or weighted Simes tests. We illustrate the extended graphical approaches with several examples. In addition, we describe briefly the gMCP package in R, which implements some of the methods described in this paper.     

Hommel's procedure in linear time

Hommel's and Hochberg's procedures for familywise error control are both derived as shortcuts in a closed testing procedure with the Simes local test. Hommel's shortcut is exact but takes quadratic time in the number of hypotheses. Hochberg's shortcut takes only linear time after the P‐values are sorted, but is conservative. In this paper, we present an exact shortcut in linear time on sorted P‐values, combining the strengths of both procedures. The novel shortcut also applies to a robust variant of Hommel's procedure that does not require the assumption of the Simes inequality.     

On weighted Hochberg procedures

We consider different ways of constructing weighted Hochberg-typestep-up multiple test procedures including closed proceduresbased on weighted Simes tests and their conservative step-upshort-cuts, and step-up counterparts of two weighted Holm procedures.It is shown that the step-up counterparts have some seriouspitfalls such as lack of familywise error rate control and lackof monotonicity in rejection decisions in terms of p-values.Therefore an exact closed procedure appears to be the best alternative,its only drawback being lack of simple stepwise structure. Aconservative step-up short-cut to the closed procedure may beused instead, but with accompanying loss of power. Simulationsare used to study the familywise error rate and power propertiesof the competing procedures for independent and correlated p-values.Although many of the results of this paper are negative, theyare useful in highlighting the need for caution when procedureswith similar pitfalls may be used.     

General multistage gatekeeping procedures

A general multistage (stepwise) procedure is proposed for dealing with arbitrary gatekeeping problems including parallel and serial gatekeeping. The procedure is very simple to implement since it does not require the application of the closed testing principle and the consequent need to test all nonempty intersections of hypotheses. It is based on the idea of carrying forward the Type I error rate for any rejected hypotheses to test hypotheses in the next ordered family. This requires the use of a so-called separable multiple test procedure (MTP) in the earlier family. The Bonferroni MTP is separable, but other standard MTPs such as Holm, Hochberg, Fallback and Dunnett are not. Their truncated versions are proposed which are separable and more powerful than the Bonferroni MTP. The proposed procedure is illustrated by a clinical trial example.     

Controlling False Discoveries in Multidimensional Directional Decisions,with Applications to Gene Expression Data on Ordered Categories

Summary Microarray gene expression studies over ordered categories are routinely conducted to gain insights into biological functions of genes and the underlying biological processes. Some common experiments are time‐course/dose‐response experiments where a tissue or cell line is exposed to different doses and/or durations of time to a chemical. A goal of such studies is to identify gene expression patterns/profiles over the ordered categories. This problem can be formulated as a multiple testing problem where for each gene the null hypothesis of no difference between the successive mean gene expressions is tested and further directional decisions are made if it is rejected. Much of the existing multiple testing procedures are devised for controlling the usual false discovery rate (FDR) rather than the mixed directional FDR (mdFDR), the expected proportion of Type I and directional errors among all rejections. Benjamini and Yekutieli (2005, Journal of the American Statistical Association 100, 71–93) proved that an augmentation of the usual Benjamini–Hochberg (BH) procedure can control the mdFDR while testing simple null hypotheses against two‐sided alternatives in terms of one‐dimensional parameters. In this article, we consider the problem of controlling the mdFDR involving multidimensional parameters. To deal with this problem, we develop a procedure extending that of Benjamini and Yekutieli based on the Bonferroni test for each gene. A proof is given for its mdFDR control when the underlying test statistics are independent across the genes. The results of a simulation study evaluating its performance under independence as well as under dependence of the underlying test statistics across the genes relative to other relevant procedures are reported. Finally, the proposed methodology is applied to a time‐course microarray data obtained by Lobenhofer et al. (2002, Molecular Endocrinology 16, 1215–1229). We identified several important cell‐cycle genes, such as DNA replication/repair gene MCM4 and replication factor subunit C2, which were not identified by the previous analyses of the same data by Lobenhofer et al. (2002) and Peddada et al. (2003, Bioinformatics 19, 834–841). Although some of our findings overlap with previous findings, we identify several other genes that complement the results of Lobenhofer et al. (2002) .     

Using Discrete Distributions to Compute Powers for Goodness-of-Fit Test Statistics in a One-Way Multinomial Setting

Both the Bionomial and Poisson distributions are employed in this study to compute approximate powers for goodness-of-fit test statistics. The procedure adopted involves simulating 1000 samples from each of these distributions. These samples are then employed to compute both the randomized nominal critical values and estimated powers. The type I error rates returned from the use of the randomized critical levels Cα fall within the acceptable regions. We illustrate the use of the procedure with the Pearson's X² test statistic and show that this can readily be extended to any of the other well known goodness-of-fit test statistics.     

Weighted Logrank Testing Procedures for the Simple Tree Alternatives

Weighted logrank testing procedures for comparing r treatments with a control when some of the data are randomly censored are discussed. Four kinds of test statistics for the simple tree alternatives are considered. The weighted logrank statistics based on pairwise ranking scheme is proposed and the covariances of the test statistics are explicitly obtained. This class of test statistics can be viewed as the general statistics of constructing the test procedures for various order restricted alternatives by modifying weights. Four kinds of weighted logrank tests are illustrated with an example. Simulation studies are performed to compare the sizes and the powers of the considered tests with the other.     

Testing Equality of Correlated Means in the Presence of Unequal Variances and Missing Values

Several statistics are proposed for testing the hypothesis of equality of the means of bivariate normal distribution with unknown variances and correlation coefficient when observations are missing on both variatea. The null distributions of the statistics are approximated by well-known distributions. The empirical sizes and powers of the statistics are computed and compared with paired t test and some of the known statistics based on available data. The comparisons support the use of two of the statistics proposed in this paper.     

An Adaptive Location‐Scale Test

Increasing locations are often accompanied by an increase in variability. In this case apparent heteroscedasticity can indicate that there are treatment effects and it is appropriate to consider an alternative involving differences in location as well as in scale. As a location‐scale test the sum of a location and a scale test statistic can be used. However, the power can be raised through weighting the sum. In order to select values for this weighting an adaptive design with an interim analysis is proposed: The data of the first stage are used to calculate the weights and with the second stage's data a weighted location‐scale test is carried out. The p‐values of the two stages are combined through Fisher's combination test. With a Lepage‐type location‐scale test it is illustrated that the resultant adaptive test can be more powerful than the ‘optimum’ test with no interim analysis. The principle to calculate weights, which cannot be reasonably chosen a priori, with the data of the first stage may be useful for other tests which utilize weighted statistics, too. Furthermore, the proposed test is illustrated with an example from experimental ecology.     

Alternative Confidence Regions for Bonferroni‐Based Closed‐Testing Procedures that are not Alpha‐Exhaustive

This article complements the results in Guilbaud (Biometrical Journal 2008; 50 :678–692). Simultaneous confidence regions were derived in that article that correspond to any given multiple testing procedure (MTP) in a fairly large class of consonant closed‐testing procedures based on marginal p‐values and weighted Bonferroni tests for intersection hypotheses. This class includes Holm's MTP, the fixed‐sequence MTP, gatekeeping MTPs, fallback MTPs, multi‐stage fallback MTPs, and recently proposed MTPs specified through a graphical representation and associated rejection algorithm. More general confidence regions are proposed in this article. These regions are such that for certain underlying MTPs which are not alpha‐exhaustive, they lead to confidence assertions that may be sharper than rejection assertions for some rejected null hypotheses H when not all Hs are rejected, which is not the case with the previously proposed regions. In fact, various alternative confidence regions may be available for such an underlying MTP. These results are shown through an extension of the previous direct arguments (without invoking the partitioning principle), and under the same general setup; so for instance, estimated quantities and marginal confidence regions are not restricted to be of any particular kinds/dimensions. The relation with corresponding confidence regions of Strassburger and Bretz (Statistics in Medicine 2008; 27 :4914–4927) is described. The results are illustrated with fallback and parallel‐gatekeeping MTPs.     

On Testing Equality of Means of Correlated Variates with Incomplete Data

A statistic is proposed for testing the hypothesis of equality of the means of a bivariate normal distribution with unknown common variance and correlation coefficient when observations are missing on one of the variates. Expressions for the second and fourth central moments of the statistic are obtained. These moments are used to approximate the distribution of the statistic by a Student's t distribution under the null hypothesis. The powers of the test are computed and compared with those of the conventional paired t and the other known statistics.     

A Modified Chi-squared Approach for Fitting WEIBULL Models to Synthetic Life Tables1

WEIBULL models are fitted to synthetic life table data by applying weighted least squares analysis to log log functions which are constructed from appropriate underlying contingency tables. As such, the resulting estimates and test statistics are based on the linearized minimum modified X²₁-criterion and thus have satisfactory properties in moderately large samples. The basic methodology is illustrated in terms of an example which is bivariate in the sense of involving two simultaneous, but non-competing, vital events. For this situation, the estimation of WEIBULL model parameters is described for both marginal as well as certain conditional distributions either individually or jointly.     

Simultaneous Non‐inferiority Test of Sensitivity and Specificity for Two Diagnostic Procedures in the Presence of a Gold Standard

Sensitivity and specificity have traditionally been used to assess the performance of a diagnostic procedure. Diagnostic procedures with both high sensitivity and high specificity are desirable, but these procedures are frequently too expensive, hazardous, and/or difficult to operate. A less sophisticated procedure may be preferred, if the loss of the sensitivity or specificity is determined to be clinically acceptable. This paper addresses the problem of simultaneous testing of sensitivity and specificity for an alternative test procedure with a reference test procedure when a gold standard is present. The hypothesis is formulated as a compound hypothesis of two non‐inferiority (one‐sided equivalence) tests. We present an asymptotic test statistic based on the restricted maximum likelihood estimate in the framework of comparing two correlated proportions under the prospective and retrospective sampling designs. The sample size and power of an asymptotic test statistic are derived. The actual type I error and power are calculated by enumerating the exact probabilities in the rejection region. For applications that require high sensitivity as well as high specificity, a large number of positive subjects and a large number of negative subjects are needed. We also propose a weighted sum statistic as an alternative test by comparing a combined measure of sensitivity and specificity of the two procedures. The sample size determination is independent of the sampling plan for the two tests.     

On Testing Equality of Means of Correlated Variates with Missing Data on One Response

A statistic is proposed for testing the hypothesis of equality of the means of a bivariate normal distribution with unknown common variance and correlation coefficient when observations are missing on one of the variates. The distribution of the statistic is approximated by a normal distribution under the null hypothesis. The empirical powers of the statistic are computed and compared with those of the conventional paired t and the other known statistics. The power comparisons support the use of the proposed test.     

Multiple Comparisons Among Dependent Groups Based on a Modified One‐Step M‐Estimator

Currently, among multiple comparison procedures for dependent groups, a bootstrap‐t with a 20% trimmed mean performs relatively well in terms of both Type I error probabilities and power. However, trimmed means suffer from two general concerns described in the paper. Robust M‐estimators address these concerns, but now no method has been found that gives good control over the probability of a Type I error when sample sizes are small. The paper suggests using instead a modified one‐step M‐estimator that retains the advantages of both trimmed means and robust M‐estimators. Yet another concern is that the more successful methods for trimmed means can be too conservative in terms of Type I errors. Two methods for performing all pairwise multiple comparisons are considered. In simulations, both methods avoid a familywise error (FWE) rate larger than the nominal level. The method based on comparing measures of location associated with the marginal distributions can have an actual FWE that is well below the nominal level when variables are highly correlated. However, the method based on difference scores performs reasonably well with very small sample sizes, and it generally performs better than any of the methods studied in Wilcox (1997b).     

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.