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).     

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).     

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.     

A class of goodness of fit tests for a copula based on bivariate right-censored data

The copula of a bivariate distribution, constructed by making marginal transformations of each component, captures all the information in the bivariate distribution about the dependence between two variables. For frailty models for bivariate data the choice of a family of distributions for the random frailty corresponds to the choice of a parametric family for the copula. A class of tests of the hypothesis that the copula is in a given parametric family, with unspecified association parameter, based on bivariate right censored data is proposed. These tests are based on first making marginal Kaplan-Meier transformations of the data and then comparing a non-parametric estimate of the copula to an estimate based on the assumed family of models. A number of options are available for choosing the scale and the distance measure for this comparison. Significance levels of the test are found by a modified bootstrap procedure. The procedure is used to check the appropriateness of a gamma or a positive stable frailty model in a set of survival data on Danish twins.     

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.     

Procedures for two-sample comparisons with multiple endpoints controlling the experimentwise error rate

Clinical trials are often concerned with the comparison of two treatment groups with multiple endpoints. As alternatives to the commonly used methods, the T2 test and the Bonferroni method, O'Brien (1984, Biometrics 40, 1079-1087) proposes tests based on statistics that are simple or weighted sums of the single endpoints. This approach turns out to be powerful if all treatment differences are in the same direction [compare Pocock, Geller, and Tsiatis (1987, Biometrics 43, 487-498)]. The disadvantage of these multivariate methods is that they are suitable only for demonstrating a global difference, whereas the clinician is further interested in which specific endpoints or sets of endpoints actually caused this difference. It is shown here that all tests are suitable for the construction of a closed multiple test procedure where, after the rejection of the global hypothesis, all lower-dimensional marginal hypotheses and finally the single hypotheses are tested step by step. This procedure controls the experimentwise error rate. It is just as powerful as the multivariate test and, in addition, it is possible to detect significant differences between the endpoints or sets of endpoints.     

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.     

Analysis of Group Randomized Trials with Multiple Binary Endpoints and Small Number of Groups

The group randomized trial (GRT) is a common study design to assess the effect of an intervention program aimed at health promotion or disease prevention. In GRTs, groups rather than individuals are randomized into intervention or control arms. Then, responses are measured on individuals within those groups. A number of analytical problems beset GRT designs. The major problem emerges from the likely positive intraclass correlation among observations of individuals within a group. This paper provides an overview of the analytical method for GRT data and applies this method to a randomized cancer prevention trial, where multiple binary primary endpoints were obtained. We develop an index of extra variability to investigate group-specific effects on response. The purpose of the index is to understand the influence of individual groups on evaluating the intervention effect, especially, when a GRT study involves a small number of groups. The multiple endpoints from the GRT design are analyzed using a generalized linear mixed model and the stepdown Bonferroni method of Holm.     

Robust estimation and inference for bivariate line-fitting in allometry

In allometry, bivariate techniques related to principal component analysis are often used in place of linear regression, and primary interest is in making inferences about the slope. We demonstrate that the current inferential methods are not robust to bivariate contamination, and consider four robust alternatives to the current methods -- a novel sandwich estimator approach, using robust covariance matrices derived via an influence function approach, Huber's M-estimator and the fast-and-robust bootstrap. Simulations demonstrate that Huber's M-estimators are highly efficient and robust against bivariate contamination, and when combined with the fast-and-robust bootstrap, we can make accurate inferences even from small samples.     

Modeling continuous covariates with a “spike” at zero: Bivariate approaches

In epidemiology and clinical research, predictors often take value zero for a large amount of observations while the distribution of the remaining observations is continuous. These predictors are called variables with a spike at zero. Examples include smoking or alcohol consumption. Recently, an extension of the fractional polynomial (FP) procedure, a technique for modeling nonlinear relationships, was proposed to deal with such situations. To indicate whether or not a value is zero, a binary variable is added to the model. In a two stage procedure, called FP‐spike, the necessity of the binary variable and/or the continuous FP function for the positive part are assessed for a suitable fit. In univariate analyses, the FP‐spike procedure usually leads to functional relationships that are easy to interpret. This paper introduces four approaches for dealing with two variables with a spike at zero (SAZ). The methods depend on the bivariate distribution of zero and nonzero values. Bi‐Sep is the simplest of the four bivariate approaches. It uses the univariate FP‐spike procedure separately for the two SAZ variables. In Bi‐D3, Bi‐D1, and Bi‐Sub, proportions of zeros in both variables are considered simultaneously in the binary indicators. Therefore, these strategies can account for correlated variables. The methods can be used for arbitrary distributions of the covariates. For illustration and comparison of results, data from a case‐control study on laryngeal cancer, with smoking and alcohol intake as two SAZ variables, is considered. In addition, a possible extension to three or more SAZ variables is outlined. A combination of log‐linear models for the analysis of the correlation in combination with the bivariate approaches is proposed.     

A simple and flexible Holm gatekeeping procedure

Major objectives of a clinical trial are commonly stated in a hierarchical order as primary and secondary. The parallel gatekeeping testing strategy provides an opportunity to assess secondary objectives when all or partial primary objectives are achieved. The current available gatekeeping procedures have different pros and cons so users either need to justify the assumption associated with some procedures or tolerate suboptimal power performance of other procedures. By applying the Holm test with a flexible alpha splitting technique, we propose a procedure which (1) is powerful for assessing the primary objectives, (2) can be used when no assumption can be made on the dependency structure of test statistics, and (3) has the full flexibility to allocate user-preferred alpha to assess the secondary objectives based on the number of primary objectives achieved. A real clinical trial example is used for illustration of the proposed procedure.     

Categorical variables with many categories are preferentially selected in bootstrap‐based model selection procedures for multivariable regression models

Automated variable selection procedures, such as backward elimination, are commonly employed to perform model selection in the context of multivariable regression. The stability of such procedures can be investigated using a bootstrap‐based approach. The idea is to apply the variable selection procedure on a large number of bootstrap samples successively and to examine the obtained models, for instance, in terms of the inclusion of specific predictor variables. In this paper, we aim to investigate a particular important problem affecting this method in the case of categorical predictor variables with different numbers of categories and to give recommendations on how to avoid it. For this purpose, we systematically assess the behavior of automated variable selection based on the likelihood ratio test using either bootstrap samples drawn with replacement or subsamples drawn without replacement from the original dataset. Our study consists of extensive simulations and a real data example from the NHANES study. Our main result is that if automated variable selection is conducted on bootstrap samples, variables with more categories are substantially favored over variables with fewer categories and over metric variables even if none of them have any effect. Importantly, variables with no effect and many categories may be (wrongly) preferred to variables with an effect but few categories. We suggest the use of subsamples instead of bootstrap samples to bypass these drawbacks.     

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

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

Testing marginal homogeneity against stochastic order in multivariate ordinal data

Summary .  Many assessment instruments used in the evaluation of toxicity, safety, pain, or disease progression consider multiple ordinal endpoints to fully capture the presence and severity of treatment effects. Contingency tables underlying these correlated responses are often sparse and imbalanced, rendering asymptotic results unreliable or model fitting prohibitively complex without overly simplistic assumptions on the marginal and joint distribution. Instead of a modeling approach, we look at stochastic order and marginal inhomogeneity as an expression or manifestation of a treatment effect under much weaker assumptions. Often, endpoints are grouped together into physiological domains or by the body function they describe. We derive tests based on these subgroups, which might supplement or replace the individual endpoint analysis because they are more powerful. The permutation or bootstrap distribution is used throughout to obtain global, subgroup, and individual significance levels as they naturally incorporate the correlation among endpoints. We provide a theorem that establishes a connection between marginal homogeneity and the stronger exchangeability assumption under the permutation approach. Multiplicity adjustments for the individual endpoints are obtained via stepdown procedures, while subgroup significance levels are adjusted via the full closed testing procedure. The proposed methodology is illustrated using a collection of 25 correlated ordinal endpoints, grouped into six domains, to evaluate toxicity of a chemical compound.     

Flexible designs by adaptive plans of generalized Pocock- and O'Brien-Fleming-type and by self-designing clinical trials

Flexible designs are provided by adaptive planning of sample sizes as well as by introducing the weighted inverse normal combining method and the generalized inverse chi-square combining method in the context of conducting trials consecutively step by step. These general combining methods allow quite different weighting of sequential study parts, also in a completely adaptive way, based on full information from unblinded data in previously performed stages. So, in reviewing some basic developments of flexible designing, we consider a generalizing approach to group sequentially performed clinical trials of Pocock-type, of O'Brien-Fleming-type, and of Self-designing-type. A clinical trial may be originally planned either to show non-inferiority or superiority. The proposed flexible designs, however, allow in each interim analysis to change the planning from showing non-inferiority to showing superiority and vice versa. Several examples of clinical trials with normal and binary outcomes are worked out in detail. We demonstrate the practicable performance of the discussed approaches, confirmed in an extensive simulation study. Our flexible designing is a useful tool, provided that a priori information about parameters involved in the trial is not available or subject to uncertainty.     

Diagnostic potential of tear proteomic patterns in Sjögren's syndrome

Histological and functional changes of the lacrimal gland might be reflected in proteomic patterns in tear fluids. In this study, we carried out a determination of the disease biomarkers in tear fluid for Sj?gren's syndrome (SS) and a performance of noninvasive diagnostic test based on the proteomic patterns. Thirty-one SS patients and 57 control subjects were enrolled to this study. Their details were 23 cases with primary SS, 8 with secondary SS, 14 with dry eyes, 22 with miscellaneous ocular diseases, and 21 of healthy volunteers. Protein profiling in tear fluids was identified by surface enhanced laser desorption/ionization time-of-flight mass spectrometry (SELDI-TOF-MS). Multiple protein changes were reproducibly detected in the primary SS group, including 10 potential novel biomarkers. Seven of the biomarkers (2094, 2743, 14191, 14702, 16429, 17453, 17792 m/z) were down-regulated and 3 biomarkers (3483, 4972, 10860 m/z) were up-regulated in primary SS group, comparing to the protein profiles of control subjects. When cutoff value of SS down-score was set less than 0.5, this result yielded 87% sensitivity and 100% specificity. The positive predictive value for this sample set was 100%. There was a significant inverse correlation between SS down-scores and epithelial damages of the ocular surface in primary SS patients. These findings support the potential of proteomic pattern technology in tear fluids as the noninvasive diagnostic test for primary SS.     

Sample size determination in clinical trials with multiple co‐primary endpoints including mixed continuous and binary variables

In the field of pharmaceutical drug development, there have been extensive discussions on the establishment of statistically significant results that demonstrate the efficacy of a new treatment with multiple co‐primary endpoints. When designing a clinical trial with such multiple co‐primary endpoints, it is critical to determine the appropriate sample size for indicating the statistical significance of all the co‐primary endpoints with preserving the desired overall power because the type II error rate increases with the number of co‐primary endpoints. We consider overall power functions and sample size determinations with multiple co‐primary endpoints that consist of mixed continuous and binary variables, and provide numerical examples to illustrate the behavior of the overall power functions and sample sizes. In formulating the problem, we assume that response variables follow a multivariate normal distribution, where binary variables are observed in a dichotomized normal distribution with a certain point of dichotomy. Numerical examples show that the sample size decreases as the correlation increases when the individual powers of each endpoint are approximately and mutually equal.     

双歧杆菌对迁延性及慢性腹泻患儿部分胃肠激素的影响

目的：探讨双歧杆菌对迁延性及慢性腹泻患儿血清胃泌素(GAS)、血浆胃动素(MOT)和血浆生长抑素(SS)水平的影响。方法：应用放免法测定20例用双歧杆菌和19例未用微生态制剂治疗的迁延性及慢性腹泻患儿治疗前和治疗7d后空腹及餐后1h的血清GAS、血浆MOT和血浆SS水平,结果与20例正常儿童比较,并分析三种胃肠激素间的相关性。结果：治疗前两组病例空腹和餐后的血清GAS、血浆MOT和血浆SS水平高于正常儿童。除空腹未用微生态制剂病例的血清GAS水平外,治疗前两组病例血中的三种胃肠激素水平高于治疗7d后。治疗7d后双歧杆菌治疗病例空腹和餐后的血清GAS和血浆MOT水平低于未用微生态制剂病例,且血清GAS水平与正常儿童差异无显著性。治疗7d后血浆SS水平三组间差异无显著性。另两组病例治疗前空腹和餐后三种胃肠激素间均无关联,治疗7d后的双歧杆菌治疗病例和正常儿童GAS与SS和MOT与SS则呈正相关。结论：双歧杆菌制剂对迁延性及慢性腹泻儿童胃肠激素的分泌有调理作用,应用双歧杆菌治疗的患儿血清GAS、血浆MOT和血浆SS水平较未用微生态制剂的患儿更快恢复正常。     

A pivotal registration phase III, multicenter, randomized tuberculosis controlled trial: design issues and lessons learnt from the Gatifloxacin for TB (OFLOTUB) project

ABSTRACT: BACKGROUND: There have been no major advances in tuberculosis (TB) drug development since the first East African/British Medical Research Council short course chemotherapy trial 35 years ago. Since then, the landscape for conducting TB clinical trials has profoundly changed with the emergence of HIV infection, the spread of resistant TB bacilli strains, recent advances in mycobacteriological capacity, and drug discovery. As a consequence questions have arisen on the most appropriate approach to design and conduct current TB trials. To highlight key issues discussed: Is a superiority, equivalence, or non-inferiority design most appropriate? What should be the primary efficacy outcome? How to consider re-infections in the definition of the outcome? What is the optimal length of patient follow-up? Is blinding appropriate when treatment duration in test arm is shorter? What are the appropriate assumptions for sample size calculation? METHODS: Various drugs are currently in the development pipeline. We are presenting in this paper the design of the most recently completed phase III TB trial, the OFLOTUB project, which is the pivotal trial of a registration portfolio for a gatifloxacin-containing TB regimen. It is a randomized, open-label, multicenter, controlled trial aiming to evaluate the efficacy and safety of a gatifloxacin-containing 4-month regimen (trial registration: ClinicalTrial.gov database: NCT00216385). RESULTS: In the light of the recent scientific and regulatory discussions, we discuss some of the design issues in TB clinical trials and more specifically the reasons that guided our choices, in order to best answer the trial objectives, while at the same time satisfying regulatory authority requirements. CONCLUSION: When shortening TB treatment, we are advocating for a non-inferiority, non-blinded design, with a composite unfavorable endpoint assessed 12 months post treatment completion, and added trial procedures specifically aiming to: (1) minimize endpoint unavailability; and (2) distinguish between relapse and re-infection.