Trimmed Weighted Simes' Test for Two One‐Sided Hypotheses With Arbitrarily Correlated Test Statistics

The two‐sided Simes test is known to control the type I error rate with bivariate normal test statistics. For one‐sided hypotheses, control of the type I error rate requires that the correlation between the bivariate normal test statistics is non‐negative. In this article, we introduce a trimmed version of the one‐sided weighted Simes test for two hypotheses which rejects if (i) the one‐sided weighted Simes test rejects and (ii) both p‐values are below one minus the respective weighted Bonferroni adjusted level. We show that the trimmed version controls the type I error rate at nominal significance level α if (i) the common distribution of test statistics is point symmetric and (ii) the two‐sided weighted Simes test at level 2α controls the level. These assumptions apply, for instance, to bivariate normal test statistics with arbitrary correlation. In a simulation study, we compare the power of the trimmed weighted Simes test with the power of the weighted Bonferroni test and the untrimmed weighted Simes test. An additional result of this article ensures type I error rate control of the usual weighted Simes test under a weak version of the positive regression dependence condition for the case of two hypotheses. This condition is shown to apply to the two‐sided p‐values of one‐ or two‐sample t‐tests for bivariate normal endpoints with arbitrary correlation and to the corresponding one‐sided p‐values if the correlation is non‐negative. The Simes test for such types of bivariate t‐tests has not been considered before. According to our main result, the trimmed version of the weighted Simes test then also applies to the one‐sided bivariate t‐test with arbitrary correlation.     

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.     

Minimizing the cost of environmental management decisions by optimizing statistical thresholds

Environmental management decisions are prone to expensive mistakes if they are triggered by hypothesis tests using the conventional Type I error rate (α) of 0.05. We derive optimal α‐levels for decision‐making by minimizing a cost function that specifies the overall cost of monitoring and management. When managing an economically valuable koala population, it shows that a decision based on α = 0.05 carries an expected cost over $5 million greater than the optimal decision. For a species of such value, there is never any benefit in guarding against the spurious detection of declines and therefore management should proceed directly to recovery action. This result holds in most circumstances where the species’ value substantially exceeds its recovery costs. For species of lower economic value, we show that the conventional α‐level of 0.05 rarely approximates the optimal decision‐making threshold. This analysis supports calls for reversing the statistical ‘burden of proof’ in environmental decision‐making when the cost of Type II errors is relatively high.     

Distance-based tests for homogeneity of multivariate dispersions

Summary The traditional likelihood‐based test for differences in multivariate dispersions is known to be sensitive to nonnormality. It is also impossible to use when the number of variables exceeds the number of observations. Many biological and ecological data sets have many variables, are highly skewed, and are zero‐inflated. The traditional test and even some more robust alternatives are also unreasonable in many contexts where measures of dispersion based on a non‐Euclidean dissimilarity would be more appropriate. Distance‐based tests of homogeneity of multivariate dispersions, which can be based on any dissimilarity measure of choice, are proposed here. They rely on the rotational invariance of either the multivariate centroid or the spatial median to obtain measures of spread using principal coordinate axes. The tests are straightforward multivariate extensions of Levene's test, with P‐values obtained either using the traditional F‐distribution or using permutation of either least‐squares or LAD residuals. Examples illustrate the utility of the approach, including the analysis of stabilizing selection in sparrows, biodiversity of New Zealand fish assemblages, and the response of Indonesian reef corals to an El Niño. Monte Carlo simulations from the real data sets show that the distance‐based tests are robust and powerful for relevant alternative hypotheses of real differences in spread.     

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.     

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.     

Formation of truncated peptide by‐products via sequence‐specific formyl group transfer from Trp(For) residues to Nα in the course of Boc‐SPPS

(N^In)‐Formyl protective group of tryptophan has been introduced as a base/nucleophile‐labile protective group. It has long been known that a free Nα‐amino group of the peptide can serve as a nucleophile: an irreversible formyl N^In → NH₂ transfer is consistently observed when deformylation is performed last on an otherwise deprotected peptide that possesses free Nα‐amino group. Obviously, this particular side reaction should be expected any time free amino group is exposed to Trp(For), but, at the best of our knowledge, has never been reported in the course of Boc‐SPPS. In the present communication, we describe a set of appropriately designed model experiments that permitted to detect the title side reaction both in solution and in solid‐phase reactions. We observed intermolecular formyl group transfer with a model compound, Trp(For)‐NH₂. Importantly, we also observed this migration on solid support with the rate roughly estimated to be up to 1% of residues per minute. We also observed that the formyl‐group transfer reaction occurred in a sequence‐dependent manner and was suppressed to a non‐detectable level using ‘in situ neutralization’ technique. Because this side reaction is sequence dependent, there might be situations when the rate of the formation of N_α‐formyl termination by‐products is significant. In other cases, the N_α‐For truncated by‐products would not contaminate the final peptide significantly but still could be a source of microheterogeneity. Copyright © 2013 European Peptide Society and John Wiley & Sons, Ltd.     

Testing Average Equivalence — Finding a Compromise Between Theory and Practice

Recently, Brown , Hwang , and Munk (1998) proposed and unbiased test for the average equivalence problem which improves noticeably in power on the standard two one‐sided tests procedure. Nevertheless, from a practical point of view there are some objections against the use of this test which are mainly adressed to the ‘unusual’ shape of the critical region. We show that every unbiased test has a critical region with such an ‘unusual’ shape. Therefore, we discuss three (biased) modifications of the unbiased test. We come to the conclusion that a suitable modification represents a good compromise between a most powerful test and a test with an appealing shape of its critical region. In order to perform these tests figures are given containing the rejection region. Finally, we compare all tests in an example from neurophysiology. This shows that it is beneficial to use these improved tests instead of the two one‐sided tests procedure.     

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.     

Evaluating the improvement in diagnostic utility from adding new predictors

Multiple diagnostic tests and risk factors are commonly available for many diseases. This information can be either redundant or complimentary. Combining them may improve the diagnostic/predictive accuracy, but also unnecessarily increase complexity, risks, and/or costs. The improved accuracy gained by including additional variables can be evaluated by the increment of the area under (AUC) the receiver‐operating characteristic curves with and without the new variable(s). In this study, we derive a new test statistic to accurately and efficiently determine the statistical significance of this incremental AUC under a multivariate normality assumption. Our test links AUC difference to a quadratic form of a standardized mean shift in a unit of the inverse covariance matrix through a properly linear transformation of all diagnostic variables. The distribution of the quadratic estimator is related to the multivariate Behrens–Fisher problem. We provide explicit mathematical solutions of the estimator and its approximate non‐central F‐distribution, type I error rate, and sample size formula. We use simulation studies to prove that our new test maintains prespecified type I error rates as well as reasonable statistical power under practical sample sizes. We use data from the Study of Osteoporotic Fractures as an application example to illustrate our method.     

Adaptive enrichment designs with a continuous biomarker

A popular design for clinical trials assessing targeted therapies is the two-stage adaptive enrichment design with recruitment in stage 2 limited to a biomarker-defined subgroup chosen based on data from stage 1. The data-dependent selection leads to statistical challenges if data from both stages are used to draw inference on treatment effects in the selected subgroup. If subgroups considered are nested, as when defined by a continuous biomarker, treatment effect estimates in different subgroups follow the same distribution as estimates in a group-sequential trial. This result is used to obtain tests controlling the familywise type I error rate (FWER) for six simple subgroup selection rules, one of which also controls the FWER for any selection rule. Two approaches are proposed: one based on multivariate normal distributions suitable if the number of possible subgroups, k, is small, and one based on Brownian motion approximations suitable for large k. The methods, applicable in the wide range of settings with asymptotically normal test statistics, are illustrated using survival data from a breast cancer trial.     

TEMPORAL VARIATION IN ALLELE FREQUENCIES: TESTING THE RIGHT HYPOTHESIS

Although standard statistical tests (such as contingency chi-square or G tests) are not well suited to the analysis of temporal changes in allele frequencies, they continue to be used routinely in this context. Because the null hypothesis stipulated by the test is violated if samples are temporally spaced, the true probability of a significant test statistic will not equal the nominal α level, and conclusions drawn on the basis of such tests can be misleading. A generalized method, applicable to a wide variety of organisms and sampling schemes, is developed here to estimate the probability of a significant test statistic if the only forces acting on allele frequencies are stochastic ones (i.e., sampling error and genetic drift). Results from analyses and simulations indicate that the rate at which this probability increases with time is determined primarily by the ratio of sample size to effective population size. Because this ratio differs considerably among species, the seriousness of the error in using the standard test will also differ. Bias is particularly strong in cases in which a high percentage of the total population can be sampled (for example, endangered species). The model used here is also applicable to the analysis of parent-offspring data and to comparisons of replicate samples from the same generation. A generalized test of the hypothesis that observed changes in allele frequency can be satisfactorily explained by drift follows directly from the model, and simulation results indicate that the true α level of this adjusted test is close to the nominal one under most conditions.     

Optimizing α for better statistical decisions: A case study involving the pace‐of‐life syndrome hypothesis

Setting optimal significance levels that minimize Type I and Type II errors allows for more transparent and well‐considered statistical decision making compared to the traditional α = 0.05 significance level. We use the optimal α approach to re‐assess conclusions reached by three recently published tests of the pace‐of‐life syndrome hypothesis, which attempts to unify occurrences of different physiological, behavioral, and life history characteristics under one theory, over different scales of biological organization. While some of the conclusions reached using optimal α were consistent to those previously reported using the traditional α = 0.05 threshold, opposing conclusions were also frequently reached. The optimal α approach reduced probabilities of Type I and Type II errors, and ensured statistical significance was associated with biological relevance. Biologists should seriously consider their choice of α when conducting null hypothesis significance tests, as there are serious disadvantages with consistent reliance on the traditional but arbitrary α = 0.05 significance level.     

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.     

Confidence Interval for Rate Ratio in a 2 × 2 Table with Structural Zero: An Application in Assessing False-Negative Rate Ratio When Combining Two Diagnostic Tests

Summary . In this article, we consider problems with correlated data that can be summarized in a 2 × 2 table with structural zero in one of the off‐diagonal cells. Data of this kind sometimes appear in infectious disease studies and two‐step procedure studies. Lui (1998, Biometrics 54, 706–711) considered confidence interval estimation of rate ratio based on Fieller‐type, Wald‐type, and logarithmic transformation statistics. We reexamine the same problem under the context of confidence interval construction on false‐negative rate ratio in diagnostic performance when combining two diagnostic tests. We propose a score statistic for testing the null hypothesis of nonunity false‐negative rate ratio. Score test–based confidence interval construction for false‐negative rate ratio will also be discussed. Simulation studies are conducted to compare the performance of the new derived score test statistic and existing statistics for small to moderate sample sizes. In terms of confidence interval construction, our asymptotic score test–based confidence interval estimator possesses significantly shorter expected width with coverage probability being close to the anticipated confidence level. In terms of hypothesis testing, our asymptotic score test procedure has actual type I error rate close to the pre‐assigned nominal level. We illustrate our methodologies with real examples from a clinical laboratory study and a cancer study.     

Analysis of chromosomal structural variation in patients with congenital left‐sided cardiac lesions

Background: We sought to characterize the landscape of structural variation associated with the subset of congenital cardiac defects characterized by left‐sided obstruction. Methods: Cases with left‐sided cardiac defects (LSCD) and pediatric controls were uniformly genotyped and assessed for copy number variant (CNV) calls. Significance testing was performed to ascertain differences in overall CNV incidence, and for CNV enrichment of specific genes and gene functions in LSCD cases relative to controls. Results: A total of 257 cases of European descent and 962 ethnically matched, disease‐free pediatric controls were included. Although there was no difference in CNV rate between cases and controls, a significant enrichment in rare LSCD CNVs was detected overall (p = 7.30 × 10^?3, case/control ratio = 1.26) and when restricted either to deletions (p = 7.58 × 10^?3, case/control ratio = 1.20) or duplications (3.02 × 10^?3, case/control ratio = 1.43). Neither gene‐based, functional nor knowledge‐based analyses identified genes, loci or pathways that were significantly enriched in cases as compared to controls when appropriate corrections for multiple tests were applied. However, several genes of interest were identified by virtue of their association with cardiac development, known human conditions, or reported disruption by CNVs in other patient cohorts. Conclusion: This study examines the largest cohort to date with LSCD for structural variation. These data suggest that CNVs play a role in disease risk and identify numerous genes disrupted by CNVs of potential disease relevance. These findings further highlight the genetic heterogeneity and complexity of these disorders. Birth Defects Research (Part A) 100:951–964, 2014. © 2014 Wiley Periodicals, Inc.     

UMP(U)‐Tests for a Binomial Parameter: A Paradox

We consider uniformly most powerful (UMP) as well as uniformly most powerful unbiased (UMPU) tests and their non‐randomized versions for certain hypotheses concerning a binomial parameter. It will be shown that the power function of a UMP(U)‐test based on sample size n can coincide on the entire parameter space with the power function of the corresponding test based on sample size n + 1. A complete characterization of this paradox will be derived. Apart some exceptional cases for two‐sided tests and equivalence tests the paradox appears if and only if a test based on sample size n is non‐randomized.     

Assessing Equivalence Tests with Respect to their Expected p‐Value

Monte‐Carlo simulation methods are commonly used for assessing the performance of statistical tests under finite sample scenarios. They help us ascertain the nominal level for tests with approximate level, e.g. asymptotic tests. Additionally, a simulation can assess the quality of a test on the alternative. The latter can be used to compare new tests and established tests under certain assumptions in order to determinate a preferable test given characteristics of the data. The key problem for such investigations is the choice of a goodness criterion. We expand the expected p‐value as considered by Sackrowitz and Samuel‐Cahn (1999) to the context of univariate equivalence tests. This presents an effective tool to evaluate new purposes for equivalence testing because of its independence of the distribution of the test statistic under null‐hypothesis. It helps to avoid the often tedious search for the distribution under null‐hypothesis for test statistics which have no considerable advantage over yet available methods. To demonstrate the usefulness in biometry a comparison of established equivalence tests with a nonparametric approach is conducted in a simulation study for three distributional assumptions.     

Evaluating Co‐primary Endpoints Collectively in Clinical Trials

Often a treatment is assessed by co‐primary endpoints so that a comprehensive picture of the treatment effect can be obtained. Co‐primary endpoints can be different medical assessments angled at different aspects of a disease, therefore, are used collectively to strengthen evidence for the treatment effect. It is common sense that if a treatment is ineffective, the chance to show that the treatment is effective in all co‐primary endpoints should be small. Therefore, it may not be necessary to require all the co‐primary endpoints to be statistically significant at the 1‐sided 0.025 level to control the error rate of wrongly approving an ineffective treatment. Rather it is reasonable to allow certain variation for the p ‐values within a range close to 0.025. In this paper, statistical methods are developed to derive decision rules to evaluate co‐primary endpoints collectively. The decision rules control the error rate of wrongly accepting an ineffective treatment at the level of 0.025 for a study and the error rate at a slightly higher level for a treatment that works for all the co‐primary endpoints except perhaps one. The decision rules also control the error rates for individual endpoints. Potential applications in clinical trials are presented (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Merosesquiterpenoids and Ten‐Membered Macrolides from a Soft Coral‐Derived Lophiostoma sp. Fungus

One new merosesquiterpenoid, craterellin D ( 1 ), along with one known analog, craterellin A ( 2 ), and five known ten‐membered macrolides, 3 – 7 , were isolated from a soft coral‐derived Lophiostoma sp. fungus. The absolute configuration of 1 was established based on biogenetic consideration with the co‐isolated analog 2 , whose configuration was determined by modified Mosher's method and single‐crystal X‐ray diffraction analysis using CuK_α radiation. The absolute configuration of 3 was determined by X‐ray diffraction analysis using CuK_α radiation. Compounds 2 and 3 showed antibacterial activities against Bacillus cereus with a MIC value of 3.12 μM .