Testing superiority and non-inferiority hypotheses in active controlled clinical trials

Switching between testing for superiority and non-inferiority has been an important statistical issue in the design and analysis of active controlled clinical trial. In practice, it is often conducted with a two-stage testing procedure. It has been assumed that there is no type I error rate adjustment required when either switching to test for non-inferiority once the data fail to support the superiority claim or switching to test for superiority once the null hypothesis of non-inferiority is rejected with a pre-specified non-inferiority margin in a generalized historical control approach. However, when using a cross-trial comparison approach for non-inferiority testing, controlling the type I error rate sometimes becomes an issue with the conventional two-stage procedure. We propose to adopt a single-stage simultaneous testing concept as proposed by Ng (2003) to test both non-inferiority and superiority hypotheses simultaneously. The proposed procedure is based on Fieller's confidence interval procedure as proposed by Hauschke et al. (1999).     

On testing simultaneously non-inferiority in two multiple primary endpoints and superiority in at least one of them

In a clinical trial with an active treatment and a placebo the situation may occur that two (or even more) primary endpoints may be necessary to describe the active treatment's benefit. The focus of our interest is a more specific situation with two primary endpoints in which superiority in one of them would suffice given that non-inferiority is observed in the other. Several proposals exist in the literature for dealing with this or similar problems, but prove insufficient or inadequate at a closer look (e.g. Bloch et al. (2001, 2006) or Tamhane and Logan (2002, 2004)). For example, we were unable to find a good reason why a bootstrap p-value for superiority should depend on the initially selected non-inferiority margins or on the initially selected type I error alpha. We propose a hierarchical three step procedure, where non-inferiority in both variables must be proven in the first step, superiority has to be shown by a bivariate test (e.g. Holm (1979), O'Brien (1984), Hochberg (1988), a bootstrap (Wang (1998)), or L?uter (1996)) in the second step, and then superiority in at least one variable has to be verified in the third step by a corresponding univariate test. All statistical tests are performed at the same one-sided significance level alpha. From the above mentioned bivariate superiority tests we preferred L?uter's SS test and the Holm procedure for the reason that these have been proven to control the type I error strictly, irrespective of the correlation structure among the primary variables and the sample size applied. A simulation study reveals that the performance regarding power of the bivariate test depends to a considerable degree on the correlation and on the magnitude of the expected effects of the two primary endpoints. Therefore, the recommendation of which test to choose depends on knowledge of the possible correlation between the two primary endpoints. In general, L?uter's SS procedure in step 2 shows the best overall properties, whereas Holm's procedure shows an advantage if both a positive correlation between the two variables and a considerable difference between their standardized effect sizes can be expected.     

A mixed approach for proving non-inferiority in clinical trials with binary endpoints

When a new treatment is compared to an established one in a randomized clinical trial, it is standard practice to statistically test for non-inferiority rather than for superiority. When the endpoint is binary, one usually compares two treatments using either an odds-ratio or a difference of proportions. In this paper, we propose a mixed approach which uses both concepts. One first defines the non-inferiority margin using an odds-ratio and one ultimately proves non-inferiority statistically using a difference of proportions. The mixed approach is shown to be more powerful than the conventional odds-ratio approach when the efficacy of the established treatment is known (with good precision) and high (e.g. with more than 56% of success). The gain of power achieved may lead in turn to a substantial reduction in the sample size needed to prove non-inferiority. The mixed approach can be generalized to ordinal endpoints.     

Blinded sample size reestimation in non-inferiority trials with binary endpoints

Sample size calculations in the planning of clinical trials depend on good estimates of the model parameters involved. When the estimates of these parameters have a high degree of uncertainty attached to them, it is advantageous to reestimate the sample size after an internal pilot study. For non-inferiority trials with binary outcome we compare the performance of Type I error rate and power between fixed-size designs and designs with sample size reestimation. The latter design shows itself to be effective in correcting sample size and power of the tests when misspecification of nuisance parameters occurs with the former design.     

Superiority inferences on individual endpoints following noninferiority testing in clinical trials

We consider the problem of drawing superiority inferences on individual endpoints following non-inferiority testing. R?hmel et al. (2006) pointed out this as an important problem which had not been addressed by the previous procedures that only tested for global superiority. R?hmel et al. objected to incorporating the non-inferiority tests in the assessment of the global superiority test by exploiting the relationship between the two, since the results of the latter test then depend on the non-inferiority margins specified for the former test. We argue that this is justified, besides the fact that it enhances the power of the global superiority test. We provide a closed testing formulation which generalizes the three-step procedure proposed by R?hmel et al. for two endpoints. For the global superiority test, R?hmel et al. suggest using the L?uter (1996) test which is modified to make it monotone. The resulting test not only is complicated to use, but the modification does not readily extend to more than two endpoints, and it is less powerful in general than several of its competitors. This is verified in a simulation study. Instead, we suggest applying the one-sided likelihood ratio test used by Perlman and Wu (2004) or the union-intersection t(max) test used by Tamhane and Logan (2004).     

A group sequential type design for three‐arm non‐inferiority trials with binary endpoints

The three‐arm design with a test treatment, an active control and a placebo group is the gold standard design for non‐inferiority trials if it is ethically justifiable to expose patients to placebo. In this paper, we first use the closed testing principle to establish the hierarchical testing procedure for the multiple comparisons involved in the three‐arm design. For the effect preservation test we derive the explicit formula for the optimal allocation ratios. We propose a group sequential type design, which naturally accommodates the hierarchical testing procedure. Under this proposed design, Monte Carlo simulations are conducted to evaluate the performance of the sequential effect preservation test when the variance of the test statistic is estimated based on the restricted maximum likelihood estimators of the response rates under the null hypothesis. When there are uncertainties for the placebo response rate, the proposed design demonstrates better operating characteristics than the fixed sample design.     

Simultaneous confidence regions corresponding to Holm's step-down procedure and other closed-testing procedures

Holm's (1979) step-down multiple-testing procedure (MTP) is appealing for its flexibility, transparency, and general validity, but the derivation of corresponding simultaneous confidence regions has remained an unsolved problem. This article provides such confidence regions. In fact, simultanenous confidence regions are provided for any MTP in the class of short-cut consonant closed-testing procedures based on marginal p -values and weighted Bonferroni tests for intersection hypotheses considered by Hommel, Bretz and Maurer (2007). In addition to Holm's MTP, this class includes the fixed-sequence MTP, recently proposed gatekeeping MTPs, and the fallback MTP. The simultaneous confidence regions are generally valid if underlying marginal p -values and corresponding marginal confidence regions (assumed to be available) are valid. The marginal confidence regions and estimated quantities are not assumed to be of any particular kinds/dimensions. Compared to the rejections made by the MTP for the family of null hypotheses H under consideration, the proposed confidence regions provide extra free information. In particular, with Holm's MTP, such extra information is provided: for all nonrejected H s, in case not all H s are rejected; or for certain (possibly all) H s, in case all H s are rejected. In case not all H s are rejected, no extra information is provided for rejected H s. This drawback seems however difficult to overcome. Illustrations concerning clinical studies are given.     

Testing and estimating the non-disjunction fraction in Meiosis I using reference priors

In this paper we analyze the fraction of non-disjunction in Meiosis I assuming reference (non-informative) priors. We consider Jeffreys's approach to built a non-informative prior (Jeffreys's prior) for the fraction of non-disjunction in Meiosis I. We prove that Jeffreys's prior is a proper distribution. We perform Monte Carlo studies in order to compare Bayes estimates obtained assuming Jeffreys's and uniform priors. We consider full Bayesian significance test (FBST) and Bayes factor (BF) for testing precise hypothesis on the fraction of non-disjunction in Meiosis I. The ultimate goal of this paper is to compare these two test procedures through simulation studies using both prior specifications. An application to Down Syndrome data is also presented.     

Exact Tests using Two Correlated Binomial Variables in Contemporary Cancer Clinical Trials

New therapy strategies for the treatment of cancer are rapidly emerging because of recent technology advances in genetics and molecular biology. Although newer targeted therapies can improve survival without measurable changes in tumor size, clinical trial conduct has remained nearly unchanged. When potentially efficacious therapies are tested, current clinical trial design and analysis methods may not be suitable for detecting therapeutic effects. We propose an exact method with respect to testing cytostatic cancer treatment using correlated bivariate binomial random variables to simultaneously assess two primary outcomes. The method is easy to implement. It does not increase the sample size over that of the univariate exact test and in most cases reduces the sample size required. Sample size calculations are provided for selected designs.     

Asymptotical Tests on the Equivalence,Substantial Difference and Non‐inferiority Problems with Two Proportions

Let d = p₂ ? p₁ be the difference between two binomial proportions obtained from two independent trials. For parameter d, three pairs of hypothesis may be of interest: H₁: d ≤ δ vs. K₁: d > δ; H₂: d ? (δ₁, δ₂) vs. K₂: d ∈ (δ₁, δ₂); and H₃: d ∈ [δ₁, δ₂] vs. K₃: d ? [δ₁, δ₂], where H_i is the null hypothesis and K_i is the alternative hypothesis. These tests are useful in clinical trials, pharmacological and vaccine studies and in statistics generally. The three problems may be investigated by exact unconditional tests when the sample sizes are moderate. Otherwise, one should use approximate (or asymptotical) tests generally based on a Z‐statistics like those suggested in the paper. The article defines a new procedure for testing H₂ or H₃, demonstrates that this is more powerful than tests based on confidence intervals (the classic TOST – two one sided tests – test), defines two corrections for continuity which reduce the liberality of the three tests, and selects the one that behaves better. The programs for executing the unconditional exact and asymptotic tests described in the paper can be loaded at http://www.ugr.es/~bioest/software.htm. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-inferiority of new procedure to standard procedure in stratified matched-pair design

We consider the statistical testing for non-inferiority of a new treatment compared with the standard one under matched-pair setting in a stratified study or in several trials. A non-inferiority test based on the efficient scores and a Mantel-Haenszel (M-H) like procedure with restricted maximum likelihood estimators (RMLEs) of nuisance parameters and their corresponding sample size formulae are presented. We evaluate the above tests and the M-H type Wald test in level and power. The stratified score test is conservative and provides the best power. The M-H like procedure with RMLEs gives an accurate level. However, the Wald test is anti-conservative and we suggest caution when it is used. The unstratified score test is not biased but it is less powerful than the stratified score test when base-line probabilities related to strata are not the same. This investigation shows that the stratified score test possesses optimum statistical properties in testing non-inferiority. A common difference between two proportions across strata is the basic assumption of the stratified tests, we present appropriate tests to validate the assumption and related remarks.     

Optimal multiple testing and design in clinical trials

A central goal in designing clinical trials is to find the test that maximizes power (or equivalently minimizes required sample size) for finding a false null hypothesis subject to the constraint of type I error. When there is more than one test, such as in clinical trials with multiple endpoints, the issues of optimal design and optimal procedures become more complex. In this paper, we address the question of how such optimal tests should be defined and how they can be found. We review different notions of power and how they relate to study goals, and also consider the requirements of type I error control and the nature of the procedures. This leads us to an explicit optimization problem with objective and constraints that describe its specific desiderata. We present a complete solution for deriving optimal procedures for two hypotheses, which have desired monotonicity properties, and are computationally simple. For some of the optimization formulations this yields optimal procedures that are identical to existing procedures, such as Hommel's procedure or the procedure of Bittman et al. (2009), while for other cases it yields completely novel and more powerful procedures than existing ones. We demonstrate the nature of our novel procedures and their improved power extensively in a simulation and on the APEX study (Cohen et al., 2016).     

Testing hybridization hypotheses and evaluating the evolutionary potential of hybrids in mangrove plant species

Natural hybridization is of marked importance from global to local biological diversity. In mangroves, species ranges overlap extensively with one another and species share a long overlap of flowering time. Although hybridization has been suggested, patterns of hybridization and the evolutionary potential of hybrids are not yet fully understood. This study provides molecular evidence for the parental origins and status of hybrids in the dominant mangrove genus Rhizophora based on comparisons of chloroplast and nuclear phylogenies and estimations of genetic relatedness and structure from inter‐simple sequence repeat (ISSR) markers. Phylogenetic analyses indicate that almost all species can act as maternal parents to hybrids and that hybridization can be bidirectional. Bayesian analyses indicate that hybrids are simple F₁s, and no trace of backcrossing was detected within populations. Hybridization, for the most part, occurs almost only locally and dispersal of hybrid individuals is limited beyond the hybrid sites.     

Testing "species pair" hypotheses: evolutionary processes in the lichen-forming species complex Porpidia flavocoerulescens and Porpidia melinodes

Pairs of taxa are commonly found in lichen-forming ascomycetes that differ primarily in their reproductive modes: one taxon reproduces sexually, the other vegetatively. The evolutionary processes underlying such "species pairs" are unknown. The species pair formed by Porpidia flavocoerulescens (sexual) and Porpidia melinodes (vegetative) was chosen to investigate four previously proposed hypotheses. These hypotheses posit that species pairs are either two monophyletic, independently evolving species with contrasting reproductive mode; a single outcrossing species polymorphic with regard to its reproductive modes; a sexual mother lineage frequently giving rise to asexual spin-offs; or a complex of cryptic species. The phylogenetic patterns observed within the species pair in the present study were analyzed using stringent hypothesis testing and visualizations of relationships and conflict based on tree and network reconstructions. DNA sequences at the three analyzed loci revealed the same four to five deeply divergent lineages. A detailed analysis of DNA-sequence variability revealed closely linked gene loci, but high levels of conflict within each of the gene fragments, as well as between observed genetic lineages. The observed patterns of phylogenetic relationships, linkage, and conflict are not congruent with any of the previously proposed species pair hypotheses. Rather, it is proposed that the observed results can be explained by conflicting reproductive and nutritional requirements imposed by an obligate symbiotic lifestyle. These interacting constraints produce recurring selective sweeps within predominantly vegetatively reproducing lineages and are the main forces that shape the evolution within the investigated species pair.