Sample size calculation for the combination test under nonproportional hazards

For sample size calculation in clinical trials with survival endpoints, the logrank test, which is the optimal method under the proportional hazard (PH) assumption, is predominantly used. In reality, the PH assumption may not hold. For example, in immuno-oncology trials, delayed treatment effects are often expected. The sample size without considering the potential violation of the PH assumption may lead to an underpowered study. In recent years, combination tests such as the maximum weighted logrank test have received great attention because of their robust performance in various hazards scenarios. In this paper, we propose a flexible simulation-free procedure to calculate the sample size using combination tests. The procedure extends the Lakatos' Markov model and allows for complex situations encountered in a clinical trial, like staggered entry, dropouts, etc. We evaluate the procedure using two maximum weighted logrank tests, one projection-type test, and three other commonly used tests under various hazards scenarios. The simulation studies show that the proposed method can achieve the target power for all compared tests in most scenarios. The combination tests exhibit robust performance under correct specification and misspecification scenarios and are highly recommended when the hazard-changing patterns are unknown beforehand. Finally, we demonstrate our method using two clinical trial examples and provide suggestions about the sample size calculations under nonproportional hazards.     

Simultaneous Statistical Inference for Epigenetic Data

Epigenetic research leads to complex data structures. Since parametric model assumptions for the distribution of epigenetic data are hard to verify we introduce in the present work a nonparametric statistical framework for two-group comparisons. Furthermore, epigenetic analyses are often performed at various genetic loci simultaneously. Hence, in order to be able to draw valid conclusions for specific loci, an appropriate multiple testing correction is necessary. Finally, with technologies available for the simultaneous assessment of many interrelated biological parameters (such as gene arrays), statistical approaches also need to deal with a possibly unknown dependency structure in the data. Our statistical approach to the nonparametric comparison of two samples with independent multivariate observables is based on recently developed multivariate multiple permutation tests. We adapt their theory in order to cope with families of hypotheses regarding relative effects. Our results indicate that the multivariate multiple permutation test keeps the pre-assigned type I error level for the global null hypothesis. In combination with the closure principle, the family-wise error rate for the simultaneous test of the corresponding locus/parameter-specific null hypotheses can be controlled. In applications we demonstrate that group differences in epigenetic data can be detected reliably with our methodology.     

Testing equality of survival functions of quality-adjusted lifetime

We present a method for comparing the survival functions of quality-adjusted lifetime from two treatments. This test statistic becomes the ordinary log-rank test when quality-adjusted lifetime is the same as the survival time. Simulation experiments are conducted to examine the behavior of our proposed test statistic under both null and alternative hypotheses. In addition, we apply our method to a breast cancer trial for comparing the distribution of quality-adjusted lifetime between two treatment regimes.     

Selecting factors predictive of heterogeneity in multivariate event time data

In multivariate survival analysis, investigators are often interested in testing for heterogeneity among clusters, both overall and within specific classes. We represent different hypotheses about the heterogeneity structure using a sequence of gamma frailty models, ranging from a null model with no random effects to a full model having random effects for each class. Following a Bayesian approach, we define prior distributions for the frailty variances consisting of mixtures of point masses at zero and inverse-gamma densities. Since frailties with zero variance effectively drop out of the model, this prior allocates probability to each model in the sequence, including the overall null hypothesis of homogeneity. Using a counting process formulation, the conditional posterior distributions of the frailties and proportional hazards regression coefficients have simple forms. Posterior computation proceeds via a data augmentation Gibbs sampling algorithm, a single run of which can be used to obtain model-averaged estimates of the population parameters and posterior model probabilities for testing hypotheses about the heterogeneity structure. The methods are illustrated using data from a lung cancer trial.     

On the power of some binomial modifications of the Bonferroni multiple test

Widely used in testing statistical hypotheses, the Bonferroni multiple test has a rather low power that entails a high risk to accept falsely the overall null hypothesis and therefore to not detect really existing effects. We suggest that when the partial test statistics are statistically independent, it is possible to reduce this risk by using binomial modifications of the Bonferroni test. Instead of rejecting the null hypothesis when at least one of n partial null hypotheses is rejected at a very high level of significance (say, 0.005 in the case of n = 10), as it is prescribed by the Bonferroni test, the binomial tests recommend to reject the null hypothesis when at least k partial null hypotheses (say, k = [n/2]) are rejected at much lower level (up to 30-50%). We show that the power of such binomial tests is essentially higher as compared with the power of the original Bonferroni and some modified Bonferroni tests. In addition, such an approach allows us to combine tests for which the results are known only for a fixed significance level. The paper contains tables and a computer program which allow to determine (retrieve from a table or to compute) the necessary binomial test parameters, i.e. either the partial significance level (when k is fixed) or the value of k (when the partial significance level is fixed).     

Extending conventional priors for testing general hypotheses in linear models

We consider that observations come from a general normal linearmodel and that it is desirable to test a simplifying null hypothesisabout the parameters. We approach this problem from an objectiveBayesian, model-selection perspective. Crucial ingredients forthis approach are ‘proper objective priors’ to beused for deriving the Bayes factors. Jeffreys-Zellner-Siow priorshave good properties for testing null hypotheses defined byspecific values of the parameters in full-rank linear models.We extend these priors to deal with general hypotheses in generallinear models, not necessarily of full rank. The resulting priors,which we call ‘conventional priors’, are expressedas a generalization of recently introduced ‘partiallyinformative distributions’. The corresponding Bayes factorsare fully automatic, easily computed and very reasonable. Themethodology is illustrated for the change-point problem andthe equality of treatments effects problem. We compare the conventionalpriors derived for these problems with other objective Bayesianproposals like the intrinsic priors. It is concluded that bothpriors behave similarly although interesting subtle differencesarise. We adapt the conventional priors to deal with nonnestedmodel selection as well as multiple-model comparison. Finally,we briefly address a generalization of conventional priors tononnormal scenarios.     

Using Weighted Kaplan-Meier Statistics in Nonparametric Comparisons of Paired Censored Survival Outcomes

This research introduces methods for nonparametric testing of weighted integrated survival differences in the context of paired censored survival designs. The current work extends work done by Pepe and Fleming (1989, Biometrics 45, 497-507), which considered similar test statistics directed toward independent treatment group comparisons. An asymptotic closed-form distribution of the proposed family of tests is presented, along with variance estimates constructed under null and alternative hypotheses using nonparametric maximum likelihood estimates of the closed-form quantities. The described method allows for additional information from individuals with no corresponding matched pair member to be incorporated into the test statistic in sampling scenarios where singletons are not prone to selection bias. Simulations presented over a range of potential dependence in the paired censored survival data demonstrate substantial power gains associated with taking into account the dependence structure. Consequences of ignoring the paired nature of the data include overly conservative tests in terms of power and size. In fact, simulation results using tests for independent samples in the presence of positive correlation consistently undershot both size and power targets that would have been attained in the absence of correlation. This additional worrisome effect on operating characteristics highlights the need for accounting for dependence in this popular family of tests.     

Sample size and power for the weighted log-rank test and Kaplan-Meier based tests with allowance for nonproportional hazards

Asymptotic distributions under alternative hypotheses and their corresponding sample size and power equations are derived for nonparametric test statistics commonly used to compare two survival curves. Test statistics include the weighted log-rank test and the Wald test for difference in (or ratio of) Kaplan-Meier survival probability, percentile survival, and restricted mean survival time. Accrual, survival, and loss to follow-up are allowed to follow any arbitrary continuous distribution. We show that Schoenfeld's equation—often used by practitioners to calculate the required number of events for the unweighted log-rank test—can be inaccurate even when the proportional hazards (PH) assumption holds. In fact, it can mislead one to believe that 1:1 is the optimal randomization ratio (RR), when actually power can be gained by assigning more patients to the active arm. Meaningful improvements to Schoenfeld's equation are made. The present theory should be useful in designing clinical trials, particularly in immuno-oncology where nonproportional hazards are frequently encountered. We illustrate the application of our theory with an example exploring optimal RR under PH and a second example examining the impact of delayed treatment effect. A companion R package npsurvSS is available for download on CRAN.     

Permutation tests for random effects in linear mixed models

Inference regarding the inclusion or exclusion of random effects in linear mixed models is challenging because the variance components are located on the boundary of their parameter space under the usual null hypothesis. As a result, the asymptotic null distribution of the Wald, score, and likelihood ratio tests will not have the typical χ(2) distribution. Although it has been proved that the correct asymptotic distribution is a mixture of χ(2) distributions, the appropriate mixture distribution is rather cumbersome and nonintuitive when the null and alternative hypotheses differ by more than one random effect. As alternatives, we present two permutation tests, one that is based on the best linear unbiased predictors and one that is based on the restricted likelihood ratio test statistic. Both methods involve weighted residuals, with the weights determined by the among- and within-subject variance components. The null permutation distributions of our statistics are computed by permuting the residuals both within and among subjects and are valid both asymptotically and in small samples. We examine the size and power of our tests via simulation under a variety of settings and apply our test to a published data set of chronic myelogenous leukemia patients.     

Change-plane analysis for subgroup detection with a continuous treatment

Detecting and characterizing subgroups with differential effects of a binary treatment has been widely studied and led to improvements in patient outcomes and population risk management. Under the setting of a continuous treatment, however, such investigations remain scarce. We propose a semiparametric change-plane model and consequently a doubly robust test statistic for assessing the existence of two subgroups with differential treatment effects under a continuous treatment. The proposed testing procedure is valid when either the baseline function for the covariate effects or the generalized propensity score function for the continuous treatment is correctly specified. The asymptotic distributions of the test statistic under the null and local alternative hypotheses are established. When the null hypothesis of no subgroup is rejected, the change-plane parameters that define the subgroups can be estimated. This paper provides a unified framework of the change-plane method to handle various types of outcomes, including the exponential family of distributions and time-to-event outcomes. Additional extensions with nonparametric estimation approaches are also provided. We evaluate the performance of our proposed methods through extensive simulation studies under various scenarios. An application to the Health Effects of Arsenic Longitudinal Study with a continuous environmental exposure of arsenic is presented.     

Using the noninformative families in family-based association tests: a powerful new testing strategy

For genetic association studies with multiple phenotypes, we propose a new strategy for multiple testing with family-based association tests (FBATs). The strategy increases the power by both using all available family data and reducing the number of hypotheses tested while being robust against population admixture and stratification. By use of conditional power calculations, the approach screens all possible null hypotheses without biasing the nominal significance level, and it identifies the subset of phenotypes that has optimal power when tested for association by either univariate or multivariate FBATs. An application of our strategy to an asthma study shows the practical relevance of the proposed methodology. In simulation studies, we compare our testing strategy with standard methodology for family studies. Furthermore, the proposed principle of using all data without biasing the nominal significance in an analysis prior to the computation of the test statistic has broad and powerful applications in many areas of family-based association studies.     

Parametric and semiparametric approaches to testing for seasonal trend in serial count data

We present two tests for seasonal trend in monthly incidence data. The first approach uses a penalized likelihood to choose the number of harmonic terms to include in a parametric harmonic model (which includes time trends and autogression as well as seasonal harmonic terms) and then tests for seasonality using a parametric bootstrap test. The second approach uses a semiparametric regression model to test for seasonal trend. In the semiparametric model, the seasonal pattern is modeled nonparametrically, parametric terms are included for autoregressive effects and a linear time trend, and a parametric bootstrap test is used to test for seasonality. For both procedures, a null distribution is generated under a null Poisson model with time trends and autoregression parameters.We apply the methods to skin melanoma incidence rates collected by the surveillance, epidemiology, and end results (SEER) program of the National Cancer Institute, and perform simulation studies to evaluate the type I error rate and power for the two procedures. These simulations suggest that both procedures are alpha-level procedures. In addition, the harmonic model/bootstrap test had similar or larger power than the semiparametric model/bootstrap test for a wide range of alternatives, and the harmonic model/bootstrap test is much easier to implement. Thus, we recommend the harmonic model/bootstrap test for the analysis of seasonal incidence data.     

A Likelihood Ratio Test for the Nonparametric Behrens‐Fisher Problem

The nonparametric Behrens‐Fisher hypothesis is the most appropriate null hypothesis for the two‐sample comparison when one does not wish to make restrictive assumptions about possible distributions. In this paper, a numerical approach is described by which the likelihood ratio test can be calculated for the nonparametric Behrens‐Fisher problem. The approach taken here effectively reduces the number of parameters in the score equations to one by using a recursive formula for the remaining parameters. The resulting single dimensional problem can be solved numerically. The power of the likelihood ratio test is compared by simulation to that of a generalized Wilcoxon test of Brunner and Munzel. The tests have similar power for all alternatives considered when a simulated null distribution is used to generate cutoff values for the tests. The methods are illustrated on data on shoulder pain from a clinical trial.     

The two-sample problem with induced dependent censorship

Induced dependent censorship is a general phenomenon in health service evaluation studies in which a measure such as quality-adjusted survival time or lifetime medical cost is of interest. We investigate the two-sample problem and propose two classes of nonparametric tests. Based on consistent estimation of the survival function for each sample, the two classes of test statistics examine the cumulative weighted difference in hazard functions and in survival functions. We derive a unified asymptotic null distribution theory and inference procedure. The tests are applied to trial V of the International Breast Cancer Study Group and show that long duration chemotherapy significantly improves time without symptoms of disease and toxicity of treatment as compared with the short duration treatment. Simulation studies demonstrate that the proposed tests, with a wide range of weight choices, perform well under moderate sample sizes.     

Screening for partial conjunction hypotheses

SUMMARY: We consider the problem of testing for partial conjunction of hypothesis, which argues that at least u out of n tested hypotheses are false. It offers an in-between approach to the testing of the conjunction of null hypotheses against the alternative that at least one is not, and the testing of the disjunction of null hypotheses against the alternative that all hypotheses are not null. We suggest powerful test statistics for testing such a partial conjunction hypothesis that are valid under dependence between the test statistics as well as under independence. We then address the problem of testing many partial conjunction hypotheses simultaneously using the false discovery rate (FDR) approach. We prove that if the FDR controlling procedure in Benjamini and Hochberg (1995, Journal of the Royal Statistical Society, Series B 57, 289-300) is used for this purpose the FDR is controlled under various dependency structures. Moreover, we can screen at all levels simultaneously in order to display the findings on a superimposed map and still control an appropriate FDR measure. We apply the method to examples from microarray analysis and functional magnetic resonance imaging (fMRI), two application areas where the need for partial conjunction analysis has been identified.     

The problem of a covariate-time qualitative interaction in a survival study

We introduce a test for the equality of two survival distributions against the specific alternative of crossing hazards. Although this kind of alternative is somewhat rare, designing a test specifically aimed at detecting such departures from the null hypothesis in this direction leads to powerful procedures, upon which we can call in those few cases where such departures are suspected. Furthermore, the proposed test and an approximate version of the test are seen to suffer only moderate losses in power, when compared with their optimal counterparts, should the alternative be one of proportional hazards. Our interest in the problem is motivated by clinical studies on the role of acute graft versus host disease as a risk factor in leukemic children and we discuss the analysis of this study in detail. The model we use in this work is a special case of the one introduced by Anderson and Senthilselvan (1982. Applied Statistics 31, 44-51). We propose overcoming an inferential problem stemming from their model by using the methods of Davies (1977, Biometrika 64, 247-254; 1987, Biometrika 74, 33-43) backed up by resampling techniques. We also look at an approach relying directly on resampling techniques. The distributional aspects of this approach under the null hypothesis are interesting but, practically, its behaviour is such that its use cannot be generally recommended. Outlines of the necessary asymptotic theory are presented and for this we use the tools of martingale theory.     

Inference in ecology and evolution

Most ecologists and evolutionary biologists continue to rely heavily on null hypothesis significance testing, rather than on recently advocated alternatives, for inference. Here, we briefly review null hypothesis significance testing and its major alternatives. We identify major objectives of statistical analysis and suggest which analytical approaches are appropriate for each. Any well designed study can improve our understanding of biological systems, regardless of the inferential approach used. Nevertheless, an awareness of available techniques and their pitfalls could guide better approaches to data collection and broaden the range of questions that can be addressed. Although we should reduce our reliance on significance testing, it retains an important role in statistical education and is likely to remain fundamental to the falsification of scientific hypotheses.     

TESTING FOR UNEQUAL AMOUNTS OF EVOLUTION IN A CONTINUOUS CHARACTER ON DIFFERENT BRANCHES OF A PHYLOGENETIC TREE USING LINEAR AND SQUARED-CHANGE PARSIMONY: AN EXAMPLE USING LESSER ANTILLEAN ANOLIS LIZARDS

Although a large body of work investigating tests of correlated evolution of two continuous characters exists, hypotheses such as character displacement are really tests of whether substantial evolutionary change has occurred on a particular branch or branches of the phylogenetic tree. In this study, we present a methodology for testing such a hypothesis using ancestral character state reconstruction and simulation. Furthermore, we suggest how to investigate the robustness of the hypothesis test by varying the reconstruction methods or simulation parameters. As a case study, we tested a hypothesis of character displacement in body size of Caribbean Anolis lizards. We compared squared-change, weighted squared-change, and linear parsimony reconstruction methods, gradual Brownian motion and speciational models of evolution, and several resolution methods for linear parsimony. We used ancestor reconstruction methods to infer the amount of body size evolution, and tested whether evolutionary change in body size was greater on branches of the phylogenetic tree in which a transition from occupying a single-species island to a two-species island occurred. Simulations were used to generate null distributions of reconstructed body size change. The hypothesis of character displacement was tested using Wilcoxon Rank-Sums. When tested against simulated null distributions, all of the reconstruction methods resulted in more significant P-values than when standard statistical tables were used. These results confirm that P-values for tests using ancestor reconstruction methods should be assessed via simulation rather than from standard statistical tables. Linear parsimony can produce an infinite number of most parsimonious reconstructions in continuous characters. We present an example of assessing the robustness of our statistical test by exploring the sample space of possible resolutions. We compare ACCTRAN and DELTRAN resolutions of ambiguous character reconstructions in linear parsimony to the most and least conservative resolutions for our particular hypothesis.     

False discovery rate control in two-stage designs

ABSTRACT: BACKGROUND: For gene expression or gene association studies with a large number of hypotheses the number of measurements per marker in a conventional single-stage design is often low due to limited resources. Two-stage designs have been proposed where in a first stage promising hypotheses are identified and further investigated in the second stage with larger sample sizes. For two types of two-stage designs proposed in the literature we derive multiple testing procedures controlling the False Discovery Rate (FDR) demonstrating FDR control by simulations: designs where a fixed number of top-ranked hypotheses are selected and designs where the selection in the interim analysis is based on an FDR threshold. In contrast to earlier approaches which use only the second-stage data in the hypothesis tests (pilot approach), the proposed testing procedures are based on the pooled data from both stages (integrated approach). Results: For both selection rules the multiple testing procedures control the FDR in the considered simulation scenarios. This holds for the case of independent observations across hypotheses as well as for certain correlation structures. Additionally, we show that in scenarios with small effect sizes the testing procedures based on the pooled data from both stages can give a considerable improvement in power compared to tests based on the second-stage data only. Conclusion: The proposed hypothesis tests provide a tool for FDR control for the considered two-stage designs. Comparing the integrated approaches for both selection rules with the corresponding pilot approaches showed an advantage of the integrated approach in many simulation scenarios.