Sample Sizes for Comparisons of k Treatments with a Control Based on Different Definitions of the Power

The determination of sample sizes for the comparison of k treatments against a control by means of the test of Dunnett (1955, 1964) as well as by means of the multiple t-test will be considered. The power in multiple comparisons can be defined in different ways, see Hochberg and Tamhane (1987). We will derive formulas for the per-pair power, the any-pair power and the all-pairs power for both one- and two-sided comparisons. Tables will be provided that allow sample sizes to be determined for preassigned values of the power.     

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.     

k‐FWER Control without p ‐value Adjustment,with Application to Detection of Genetic Determinants of Multiple Sclerosis in Italian Twins

Summary We show a novel approach for k‐FWER control which does not involve any correction, but only testing the hypotheses along a (possibly data‐driven) order until a suitable number of p‐values are found above the uncorrected α level. p‐values can arise from any linear model in a parametric or nonparametric setting. The approach is not only very simple and computationally undemanding, but also the data‐driven order enhances power when the sample size is small (and also when k and/or the number of tests is large). We illustrate the method on an original study about gene discovery in multiple sclerosis, in which were involved a small number of couples of twins, discordant by disease. The methods are implemented in an R package (someKfwer ), freely available on CRAN.     

Simultaneous Comparisons of Multiple Treatments to Two (or More) Control

DUNNETT (1955) developed a procedure simultaneously comparing k treatments to one control with an exact overall type I error of α when all sampling distributions are normal. Sometimes it is desirable to compare k treatments to m≧2 controls, in particular to two controls. For instance, several new therapies (e.g., pain relievers) could be compared to two standard therapies (e.g., Aspirin and Tylenol). Alternatively, a standard therapy could be very expensive, difficult to apply and/or have bad side effects, making it useful to compare each new therapy to both standard therapy and no therapy (Placebo). Dunnett's method is expanded here to give comparisons of mean values for k treatments to mean values for m≧2 controls at an exact overall type I error of α when all sampling distributions are normal. Tabled values needed to make exact simultaneous comparisons at α = .05 are given for m = 2. An application is made to an example from the literature.     

Adaptive patient enrichment designs in therapeutic trials

The utility of clinical trial designs with adaptive patient enrichment is investigated in an adequate and well‐controlled trial setting. The overall treatment effect is the weighted average of the treatment effects in the mutually exclusive subsets of the originally intended entire study population. The adaptive enrichment approaches permit assessment of treatment effect that may be applicable to specific nested patient (sub)sets due to heterogeneous patient characteristics and/or differential response to treatment, e.g. a responsive patient subset versus a lack of beneficial patient subset, in all patient (sub)sets studied. The adaptive enrichment approaches considered include three adaptive design scenarios: (i) total sample size fixed and with futility stopping, (ii) sample size adaptation and futility stopping, and (iii) sample size adaptation without futility stopping. We show that regardless of whether the treatment effect eventually assessed is applicable to the originally studied patient population or only to the nested patient subsets; it is possible to devise an adaptive enrichment approach that statistically outperforms one‐size‐fits‐all fixed design approach and the fixed design with a pre‐specified multiple test procedure. We emphasize the need of additional studies to replicate the finding of a treatment effect in an enriched patient subset. The replication studies are likely to need fewer number of patients because of an identified treatment effect size that is larger than the diluted overall effect size. The adaptive designs, when applicable, are along the line of efficiency consideration in a drug development program.     

Within‐population variation in social strategies characterize the social and mating system of an Australian lizard,Egernia whitii

The lizard genus Egernia has been suggested as an excellent model system for examining the evolution of sociality as it exhibits considerable diversity in social organization both between and within species. To date the majority of work examining the factors responsible for the evolution of sociality within Egernia has advocated a broad scale approach; identifying the social structure of specific species or populations and comparing the degree of sociality between them. However, we argue that significant advancements could also be gained by examining variation in social strategies within populations. Here we integrate a detailed, 3‐year, field‐based examination of social spacing and juvenile dispersal with molecular analyses of paternity to determine the social and mating system of a Tasmanian population of White's skink (Egernia whitii). We show that E. whitii live in small stable family groups consisting of an adult male, his female partner(s), as well as juvenile or sub‐adults individuals. In addition, while the mating system is characterized by considerable genetic monogamy, extra‐pair fertilizations are relatively common, with 34% of litters containing offspring sired by males from outside the social group. We also show that traits related to social organization (social group composition, group size, stability and the level of extra‐pair paternity) vary both between and within individuals. We suggest that ecological factors, such as habitat saturation, quality and availability, play a key role in maintaining between individual variation in social strategies, and that examining these individual level processes will allow us to more clearly understand variation in sociality among species.     

Exact Multiple Comparisons of Three or More Regression Lines: Pairwise Comparisons and Comparisons with a Control

The problem of finding exact simultaneous confidence bounds for differences in regression models for k groups via the union‐intersection method is considered. The error terms are taken to be iid normal random variables. Under an assumption slightly more general than having identical design matrices for each of the k groups, it is shown that an existing probability point for the multivariate studentized range can be used to find the necessary probability point for pairwise comparisons of regression models. The resulting methods can be used with simple or multiple regression. Under a weaker assumption on the k design matrices that allows more observations to be taken from the control group than from the k‐1 treatment groups, a method is developed for computing exact probability points for comparing the simple linear regression models of the k‐1 groups to that of the control. Within a class of designs, the optimal design for comparisons with a control takes the square root of (k‐1) times as many observations from the control than from each treatment group. The simultaneous confidence bounds for all pairwise differences and for comparisons with a control are much narrower than Spurrier's intervals for all contrasts of k regression lines.     

Challenges and regulatory experiences with non‐inferiority trial design without placebo arm

For a non‐inferiority trial without a placebo arm, the direct comparison between the test treatment and the selected positive control is in principle the only basis for statistical inference. Therefore, evaluating the test treatment relative to the non‐existent placebo presents extreme challenges and requires some kind of bridging from the past to the present with no current placebo data. For such inference based partly on an indirect bridging manipulation, fixed margin method and synthesis method are the two widely discussed methods in the recent literature. There are major differences in statistical inference paradigm between the two methods. The fixed margin method employs the historical data that assess the performances of the active control versus a placebo to guide the selection of the non‐inferiority margin. Such guidance is not part of the ultimate statistical inference in the non‐inferiority trial. In contrast, the synthesis method connects the historical data to the non‐inferiority trial data for making broader inferences relating the test treatment to the non‐existent current placebo. On the other hand, the type I error rate associated with the direct comparison between the test treatment and the active control cannot shed any light on the appropriateness of the indirect inference for faring the test treatment against the non‐existent placebo. This work explores an approach for assessing the impact of potential bias due to violation of a key statistical assumption to guide determination of the non‐inferiority margin.     

Oxytetracycline age validation of an adult shortfin mako shark Isurus oxyrinchus after 6 years at liberty

This study presents findings on an oxytetracycline injected adult male shortfin mako Isurus oxyrinchus recaptured in waters off of southern California after 6 years at liberty. During the period at liberty, the vertebral band‐pair deposition rate was validated at one per year. This result indicates that from a time at or near sexual maturity, male I. oxyrinchus in the north‐east Pacific Ocean exhibit a band‐pair deposition rate of one band pair per year, while deposition rates for juveniles in the area have been validated at two band pairs per year.     

Nonparametric All‐Pairs Multiple Comparisons

Nonparametric all‐pairs multiple comparisons based on pairwise rankings can be performed in the one‐way design with the Steel‐Dwass procedure. To apply this test, Wilcoxon's rank sum statistic is calculated for all pairs of groups; the maximum of the rank sums is the test statistic. We provide exact calculations of the asymptotic critical values (and P‐values, respectively) even for unbalanced designs. We recommend this asymptotic method whenever large sample sizes are present. For small sample sizes we recommend the use of the new statistic according to Baumgartner , Weiss , and Schindler (1998, Biometrics 54 , 1129–1135) instead of Wilcoxon's rank sum for the multiple comparisons. We show that the resultant procedure can be less conservative and, according to simulation results, more powerful than the original Steel‐Dwass procedure. We illustrate the methods with a practical data set.     

Testing for Treatment Effects on Subsets of Endpoints

Multiple endpoints are tested to assess an overall treatment effect and also to identify which endpoints or subsets of endpoints contributed to treatment differences. The conventional p‐value adjustment methods, such as single‐step, step‐up, or step‐down procedures, sequentially identify each significant individual endpoint. Closed test procedures can also detect individual endpoints that have effects via a step‐by‐step closed strategy. This paper proposes a global‐based statistic for testing an a priori number, say, r of the k endpoints, as opposed to the conventional approach of testing one (r = 1) endpoint. The proposed test statistic is an extension of the single‐step p‐value‐based statistic based on the distribution of the smallest p‐value. The test maintains strong control of the FamilyWise Error (FWE) rate under the null hypothesis of no difference in any (sub)set of r endpoints among all possible combinations of the k endpoints. After rejecting the null hypothesis, the individual endpoints in the sets that are rejected can be tested further, using a univariate test statistic in a second step, if desired. However, the second step test only weakly controls the FWE. The proposed method is illustrated by application to a psychosis data set.     

TESTING FOR PHYLOGENETIC SIGNAL IN BIOLOGICAL TRAITS: THE UBIQUITY OF CROSS‐PRODUCT STATISTICS

To evaluate rates of evolution, to establish tests of correlation between two traits, or to investigate to what degree the phylogeny of a species assemblage is predictive of a trait value so‐called tests for phylogenetic signal are used. Being based on different approaches, these tests are generally thought to possess quite different statistical performances. In this article, we show that the Blomberg et al. K and K*, the Abouheif index, the Moran's I, and the Mantel correlation are all based on a cross‐product statistic, and are thus all related to each other when they are associated to a permutation test of phylogenetic signal. What changes is only the way phylogenetic and trait similarities (or dissimilarities) among the tips of a phylogeny are computed. The definitions of the phylogenetic and trait‐based (dis)similarities among tips thus determines the performance of the tests. We shortly discuss the biological and statistical consequences (in terms of power and type I error of the tests) of the observed relatedness among the statistics that allow tests for phylogenetic signal. Blomberg et al. K* statistic appears as one on the most efficient approaches to test for phylogenetic signal. When branch lengths are not available or not accurate, Abouheif's C_mean statistic is a powerful alternative to K*.     

Evidence of genetic monogamy in the lemur Indri (Indri indri)

Monogamy is a rare strategy among mammals but relatively common among primates. The study of the evolution of monogamy in mammals and primates is lacking empirical studies that assess the relationship between a pair‐living social organization and genetic monogamy. Sexual or genetic monogamy can only be assessed by performing molecular analyses and investigating rates of extra‐pair paternity (EPP). Studying the occurrence of EPP can provide valuable insights into reproductive strategies and their adaptive value. The indri is a pair‐living primate that lives in stable groups. Their social units are composed of the reproductive pair and up to four more individuals, but extra‐pair copulation (EPC) can occur. This raises the question of whether this event may or may not lead to EPP. Here, we investigated whether a pair‐living social organization corresponds to genetic monogamy in indris (Indri indri). We analyzed the paternity of 12 offspring from seven pairs using a set of six microsatellite loci on fecal samples (mean number of alleles 11.7 ± 1.8 (mean ± standard deviation). We found that in 92% of cases the genetic profile of the offspring matched the paired male of the group for all the loci considered. In the only case of paternity mismatch, the paternity assignment remained inconclusive. Our results show that I. indri genetic monogamy is the norm and supports the hypothesis that pair‐living social organization is associated with low EPP rate. Also, our results are in contrast with the hypothesis of infertility as a reason to engage in EPC for this species.     

Combining Global and Marginal Tests to Compare Two Treatments on Multiple Endpoints

We consider the problem of comparing two treatments on multiple endpoints where the goal is to identify the endpoints that have treatment effects, while controlling the familywise error rate. Two current approaches for this are (i) applying a global test within a closed testing procedure, and (ii) adjusting individual endpoint p‐values for multiplicity. We propose combining the two current methods. We compare the combined method with several competing methods in a simulation study. It is concluded that the combined approach maintains higher power under a variety of treatment effect configurations than the other methods and is thus more power‐robust.     

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.     

Testing for Homogeneity in Meta‐Analysis I. The One‐Parameter Case: Standardized Mean Difference

Summary Meta‐analysis seeks to combine the results of several experiments in order to improve the accuracy of decisions. It is common to use a test for homogeneity to determine if the results of the several experiments are sufficiently similar to warrant their combination into an overall result. Cochran’s Q statistic is frequently used for this homogeneity test. It is often assumed that Q follows a chi‐square distribution under the null hypothesis of homogeneity, but it has long been known that this asymptotic distribution for Q is not accurate for moderate sample sizes. Here, we present an expansion for the mean of Q under the null hypothesis that is valid when the effect and the weight for each study depend on a single parameter, but for which neither normality nor independence of the effect and weight estimators is needed. This expansion represents an order O(1/n) correction to the usual chi‐square moment in the one‐parameter case. We apply the result to the homogeneity test for meta‐analyses in which the effects are measured by the standardized mean difference (Cohen’s d‐statistic). In this situation, we recommend approximating the null distribution of Q by a chi‐square distribution with fractional degrees of freedom that are estimated from the data using our expansion for the mean of Q. The resulting homogeneity test is substantially more accurate than the currently used test. We provide a program available at the Paper Information link at the Biometrics website http://www.biometrics.tibs.org for making the necessary calculations.     

Extra‐pair mating in a socially monogamous and paternal mouth‐brooding cardinalfish

Many vertebrates form monogamous pairs to mate and care for their offspring. However, genetic tools have increasingly shown that offspring often arise from matings outside of the monogamous pair bond. Social monogamy is relatively common in coral reef fishes, but there have been few studies that have confirmed monogamy or extra‐pair reproduction, either for males or for females. Here, long‐term observations and genetic tools were applied to examine the parentage of embryos in a paternally mouth‐brooding cardinalfish, Sphaeramia nematoptera. Paternal care in fishes, such as mouth‐brooding, is thought to be associated with a high degree of confidence in paternity. Two years of observations confirmed that S. nematoptera form long‐term pair bonds within larger groups. However, genetic parentage revealed extra‐pair mating by both sexes. Of 105 broods analysed from 64 males, 30.1% were mothered by a female that was not the partner and 11.5% of broods included eggs from two females. Despite the high paternal investment associated with mouth‐brooding, 7.6% of broods were fertilized by two males. Extra‐pair matings appeared to be opportunistic encounters with individuals from outside the immediate group. We argue that while pair formation contributes to group cohesion, both males and females can maximize lifetime reproductive success by taking advantage of extra‐pair mating opportunities.     

Oxytocin and pair compatibility in adult male rhesus macaques (Macaca mulatta)

Pair housing is considered one of the best ways of promoting psychological wellbeing for caged macaques. However, incompatible partnerships can result in stress or aggression. Though previous studies have analyzed the role of variables such as age, weight, gender, and temperament on pair compatibility, few have examined the relationship between physiological parameters and pair compatibility. Oxytocin is known to promote prosocial nonsexual behavior in various primate species and may serve as an indicator of pair compatibility. In this study, we examined the association between peripheral oxytocin levels and prosocial behaviors in isosexual pairs of male rhesus macaques. We hypothesized that animals that demonstrated high levels of prosocial behaviors would have higher oxytocin levels than those showing low levels of the behavior. In addition, to elucidate the relationship between oxytocin and compatibility, we compared peripheral oxytocin between the highly affiliative animals and single‐housed males identified as having multiple unsuccessful pair attempts with multiple partners. We collected plasma oxytocin on 40 pairs of monkeys that had lived together for at least 1 month and 20 single‐housed animals. Further, we simultaneously collected behavioral data on the pairs, recording prosocial interactions (e.g., groom, play). Oxytocin varied among individuals, but was highly correlated between members of a pair (r = 0.58, p < .001). Additionally, prosocial behavior was positively correlated with plasma oxytocin (r = 0.38, p < .02). However, contrary to our expectations, oxytocin did not differ between single and highly affiliative pair‐housed animals (F_(1,38) = 0.71, p = .40). Our results suggest that oxytocin may be associated with the quality of isosexual pairs of male macaques. More work is needed to determine the nature of this relationship.     

Experimental evidence that brighter males sire more extra‐pair young in tree swallows

Across taxa, extra‐pair mating is widespread among socially monogamous species, but few studies have identified male ornamental traits associated with extra‐pair mating success, and even fewer studies have experimentally manipulated male traits to determine whether they are related directly to paternity. As a consequence, there is little experimental evidence to support the widespread hypothesis that females choose more ornamented males as extra‐pair mates. Here, we conducted an experimental study of the relationship between male plumage colour and fertilization success in tree swallows (Tachycineta bicolor), which have one of the highest levels of extra‐pair mating in birds. In this study, we experimentally dulled the bright blue plumage on the back of males (with nontoxic ink markers) early in the breeding season prior to most mating. Compared with control males, dulled males sired fewer extra‐pair young, and, as a result, fewer young overall. Among untreated males, brighter blue males also sired more extra‐pair young, and in paired comparisons, extra‐pair sires had brighter blue plumage than the within‐pair male they cuckolded. These results, together with previous work on tree swallows, suggest that extra‐pair mating behaviour is driven by benefits to both males and females.     

Tables and Applications of the Bonferroni t-Statistics: A Revision of Dunn's Simultaneous t-Tests

Tables for Bonferroni-adjusted significance levels of Student's t are provided for r = 3(1)30(5)50 etc. and alphas of 0.05, 0.01 and 0.001. The tables are suggested for various applications, for example in replacing analysis of variance of k samples by r simultaneous t tests. Use of the tables is shown by numerical examples. The adjustment of alpha for improving the efficiency of testing is made by WILKINSON for orthogonal comparisons and by HOLM for nonorthogonal comparisons.