Stepwise gatekeeping procedures in clinical trial applications

This paper discusses multiple testing problems in which families of null hypotheses are tested in a sequential manner and each family serves as a gatekeeper for the subsequent families. Gatekeeping testing strategies of this type arise frequently in clinical trials with multiple objectives, e.g., multiple endpoints and/or multiple dose-control comparisons. It is demonstrated in this paper that the parallel gatekeeping procedure of Dmitrienko, Offen and Westfall (2003) admits a simple stepwise representation (n null hypotheses can be tested in n steps rather than 2n steps required in the closed procedure). The stepwise representation considerably simplifies the implementation of gatekeeping procedures in practice and provides an important insight into the nature of gatekeeping inferences. The derived stepwise gatekeeping procedure is illustrated using clinical trial examples.     

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.     

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.     

General multistage gatekeeping procedures

A general multistage (stepwise) procedure is proposed for dealing with arbitrary gatekeeping problems including parallel and serial gatekeeping. The procedure is very simple to implement since it does not require the application of the closed testing principle and the consequent need to test all nonempty intersections of hypotheses. It is based on the idea of carrying forward the Type I error rate for any rejected hypotheses to test hypotheses in the next ordered family. This requires the use of a so-called separable multiple test procedure (MTP) in the earlier family. The Bonferroni MTP is separable, but other standard MTPs such as Holm, Hochberg, Fallback and Dunnett are not. Their truncated versions are proposed which are separable and more powerful than the Bonferroni MTP. The proposed procedure is illustrated by a clinical trial example.     

Gatekeeping testing via adaptive alpha allocation

In a typical clinical trial, there are one or two primary endpoints, and a few secondary endpoints. When at least one primary endpoint achieves statistical significance, there is considerable interest in using results for the secondary endpoints to enhance characterization of the treatment effect. Because multiple endpoints are involved, regulators may require that the familywise type I error rate be controlled at a pre-set level. This requirement can be achieved by using "gatekeeping" methods. However, existing methods suffer from logical oddities such as allowing results for secondary endpoint(s) to impact the likelihood of success for the primary endpoint(s). We propose a novel and easy-to-implement gatekeeping procedure that is devoid of such deficiencies. A real data example and simulation results are used to illustrate efficiency gains of our method relative to existing methods.     

Inference on Multiple Endpoints in Clinical Trials with Adaptive Interim Analyses

Planned interim analyses which permit early stopping or sample size adaption of a trial are desirable for ethical and scientific reasons. Multiple test procedures allow inference about several hypotheses within a single clinical trial. In this paper, a method which combines multiple testing with adaptive interim analyses whilst controlling the experimentwise error rate is proposed. The general closed testing principle, the situation of a priori ordered hypotheses, and application of the Bonferroni-Holm method are considered. The practical application of the method is demonstrated by an example.     

Description of the first two seemingly unrelated Greek Cypriot families with a common C618R RET proto-oncogene mutation

Germ-line mutations of the RET proto-oncogene cause three different cancer syndromes: multiple endocrine neoplasia type 2A (MEN2A), multiple endocrine neoplasia type 2B, and familial medullary thyroid carcinoma (FMTC). The objective of the present study was the clinical and molecular characterization of the first two Greek Cypriot families diagnosed with MEN2A and FMTC. The clinical diagnosis of the probands was based on clinical presentation and supported with laboratory findings (calcitonin and carcinoembryonic antigen tumor marker levels). We screened the RET gene by direct DNA sequencing of exons 10, 11, and 16 using genomic DNA as templates. After identification of the mutation, we also developed the amplification refractory mutation system (ARMS) as an alternative method to direct sequencing for genetic diagnosis of 22 additional individuals from both families. We identified the germ-line missense mutation T --> C of codon 618 of exon 10 (C618R) in the probands of both families. By using ARMS, two members of the MEN2A family and five members of the FMTC family were also found positive for the C618R mutation. These are the first seemingly unrelated families in Cyprus investigated clinically and molecularly in detail and shown to transmit this common RET proto-oncogene mutation.     

A resolution of the ascertainment sampling problem. II. Generalizations and numerical results

The ascertainment problem arises when families are sampled by a nonrandom process and some assumption about this sampling process must be made in order to estimate genetic parameters. Under classical ascertainment assumptions, estimation of genetic parameters cannot be separated from estimation of the parameters of the ascertainment process, so that any misspecification of the ascertainment process causes biases in estimation of the genetic parameters. Ewens and Shute proposed a resolution to this problem, involving conditioning the likelihood of the sample on the part of the data which is "relevant to ascertainment." The usefulness of this approach can only be assessed by examining the properties (in particular, bias and standard error) of the estimates which arise by using it for a wide range of parameter values and family size distributions and then comparing these biases and standard errors with those arising under classical ascertainment procedures. These comparisons are carried out in the present paper, and we also compare the proposed method with procedures which condition on, or ignore, parts of the data.     

Analysis of information content for biological sequences.

Decomposing a biological sequence into modular domains is a basic prerequisite to identify functional units in biological molecules. The commonly used segmentation procedures usually have two steps. First, collect and align a set of sequences that are homologous to the target sequence. Then, parse this multiple alignment into several blocks and identify the functionally important ones by using a semi-automatic method, which combines manual analysis and expert knowledge. In this paper, we present a novel exploratory approach to parsing and analyzing such kinds of multiple alignments. It is based on a type of analysis-of-variance (ANOVA) decomposition of the sequence information content. Unlike the traditional change-point method, this approach takes into account not only the composition biases but also the overdispersion effects among the blocks. The new approach is tested on the families of ribosomal proteins and has a promising performance. It is shown that the new approach provides a better way for judging some important residues in these proteins. This allows one to find some subsets of residues, which are critical to these proteins.     

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).     

Alternative Confidence Regions for Bonferroni‐Based Closed‐Testing Procedures that are not Alpha‐Exhaustive

This article complements the results in Guilbaud (Biometrical Journal 2008; 50 :678–692). Simultaneous confidence regions were derived in that article that correspond to any given multiple testing procedure (MTP) in a fairly large class of consonant closed‐testing procedures based on marginal p‐values and weighted Bonferroni tests for intersection hypotheses. This class includes Holm's MTP, the fixed‐sequence MTP, gatekeeping MTPs, fallback MTPs, multi‐stage fallback MTPs, and recently proposed MTPs specified through a graphical representation and associated rejection algorithm. More general confidence regions are proposed in this article. These regions are such that for certain underlying MTPs which are not alpha‐exhaustive, they lead to confidence assertions that may be sharper than rejection assertions for some rejected null hypotheses H when not all Hs are rejected, which is not the case with the previously proposed regions. In fact, various alternative confidence regions may be available for such an underlying MTP. These results are shown through an extension of the previous direct arguments (without invoking the partitioning principle), and under the same general setup; so for instance, estimated quantities and marginal confidence regions are not restricted to be of any particular kinds/dimensions. The relation with corresponding confidence regions of Strassburger and Bretz (Statistics in Medicine 2008; 27 :4914–4927) is described. The results are illustrated with fallback and parallel‐gatekeeping MTPs.     

On stochastic logistic population growth models with immigration and multiple births

This paper develops a stochastic logistic population growth model with immigration and multiple births. The differential equations for the low-order cumulant functions (i.e., mean, variance, and skewness) of the single birth model are reviewed, and the corresponding equations for the multiple birth model are derived. Accurate approximate solutions for the cumulant functions are obtained using moment closure methods for two families of model parameterizations, one for badger and the other for fox population growth. For both model families, the equilibrium size distribution may be approximated well using the Normal approximation, and even more accurately using the saddlepoint approximation. It is shown that in comparison with the corresponding single birth model, the multiple birth mechanism increases the skewness and the variance of the equilibrium distribution, but slightly reduces its mean. Moreover, the type of density-dependent population control is shown to influence the sign of the skewness and the size of the variance.     

Multiple Testing for Identifying Effective and Safe Treatments

In the literature various multiple test procedures to compare k treatments with a control have been investigated. They can be applied to establish either treatment efficacy or treatment safety. In this paper we propose procedures which control the multiple level α with respect to efficacy and safety simultaneously. On the one hand we consider a method with stagewise rejective adjustments of local levels applied to appropriately defined subfamilies of null hypotheses. When order restrictions are assumed to hold among the parameters of interest we can alternatively split the multiple level between the families of efficacy and safety null hypotheses. If either all treatments are declared to be safe or all are declared to be effective then the other family can be tested at the full multiple level α, respectively. The methods are compared in a simulation study.     

Multifactorial inheritance with cultural transmission and assortative mating. III. Family structure and the analysis of separation experiments.

Demographic data about family composition or structure in the United States is reviewed. About 25% of white children and a majority of black children are reared in either broken or extended families, and this must be taken into consideration for valid studies of cultural inheritance. Atypical family structures are described including those in which parents include: biological parents, stepparents, grandparents, uncles, aunts, sibs, foster parents, and their spouses. General formulae for a wide variety of kinship correlations are derived using path analysis. The multifactorial model presented allows for cultural inheritance, polygenic inheritance, correlated sibling environments, and phenotypic assortative mating (as previously described for intact families) plus extensions necessary for the analysis of separation experiments. These extensions allow for variable family structure and differences in parental influence due to separation, age or stage of development of the child, birth order, or type of relationship. Family structure is observed to have a marked effect on familial resemblance. Computer simulation studies demonstrate marked heterogeneity among phenotypic correlations for kinships of the same degree of genetic relationship arising in different family structures. Analyses of multiple types of sibs and other relatives in variable family structures offer great promise for the study of cultural inheritance.     

Nonparametric Analysis of Ordered Categorical Data in Designs with Longitudinal Observations and Small Sample Sizes

For designs with longitudinal observations of ordered categorical data, a nonparametric model is considered where treatment effects and interactions are defined by means of the marginal distributions. These treatment effects are estimated consistently by ranking methods. The hypotheses in this nonparametric setup are formulated by means of the distribution functions. The asymptotic distribution of the estimators for the nonparametric effects are given under the hypotheses. For small samples, a rather accurate approximation is suggested. A clinical trial with ordered categorical data is used to motivate the ideas and to explain the procedures which are extensions of the Wilcoxon‐Mann‐Whitney test to factorial designs with longitudinal observations. The application of the procedures requires only some trivial regularity assumptions.     

Comparing Trials with Multiple Outcomes: The Multivariate One-Sided Hypothesis with Unknown Covariances

The paper deals with a problem arising for tests in clinical trials. The outcomes of a standard and a new treatment to be compared are multivariate normally distributed with common but unknown covariance matrix. Under the null hypothesis the means of the outcomes are equal, under the alternative the new treatment is assumed to be superior, i.e. the means are larger without further quantification. For known covariance matrix there is a variety of tests for this problem. Some of these procedures can be extended to the case of unknown covariances if one is willing to accept a bias. There is, however, also an efficient unbiased test. The paper contains some numerical comparisons of these different procedures and takes a look on the minimax properties of the unbiased test.     

The second generation of the International Equine Gene Mapping Workshop half-sibling linkage map

A low-density, male-based linkage map was constructed as one of the objectives of the International Equine Gene Mapping Workshop. Here we report the second generation map based on testing 503 half-sibling offspring from 13 sire families for 344 informative markers using the CRIMAP program. The multipoint linkage analysis localized 310 markers (90%) with 257 markers being linearly ordered. The map included 34 linkage groups representing all 31 autosomes and spanning 2262 cM with an average interval between loci of 10.1 cM. This map is a milestone in that it is the first map with linkage groups assigned to each of the 31 automosomes and a single linkage group to all but three chromosomes.     

A detailed genetic map of the long arm of chromosome 11

We describe 14 new restriction fragment length polymorphisms, corresponding to 13 loci on the long arm of chromosome 11. A detailed genetic map of chromosome 11q has been constructed from these and other loci (a total of 31 loci) typed in 59 reference families. The 23 most informative markers were selected to establish a map with a strongly supported order; regional localizations are provided for eight other markers. The loci span 88 cM in males and 148 cM in females and form a dense continuum on 11q. These ordered polymorphic markers will be of help in studying the genes responsible for several diseases that have been localized to this region, including genes responsible for multiple endocrine neoplasia type I (MEN1), ataxia telangiectasia (AT), tuberous sclerosis (TSC), and some forms of asthma and rhinitis.     

Evolutionary rate heterogeneity in proteins with long disordered regions

The dominant view in protein science is that a three-dimensional (3-D) structure is a prerequisite for protein function. In contrast to this dominant view, there are many counterexample proteins that fail to fold into a 3-D structure, or that have local regions that fail to fold, and yet carry out function. Protein without fixed 3-D structure is called intrinsically disordered. Motivated by anecdotal accounts of higher rates of sequence evolution in disordered protein than in ordered protein we are exploring the molecular evolution of disordered proteins. To test whether disordered protein evolves more rapidly than ordered protein, pairwise genetic distances were compared between the ordered and the disordered regions of 26 protein families having at least one member with a structurally characterized region of disorder of 30 or more consecutive residues. For five families, there were no significant differences in pairwise genetic distances between ordered and disordered sequences. The disordered region evolved significantly more rapidly than the ordered region for 19 of the 26 families. The functions of these disordered regions are diverse, including binding sites for protein, DNA, or RNA and also including flexible linkers. The functions of some of these regions are unknown. The disordered regions evolved significantly more slowly than the ordered regions for the two remaining families. The functions of these more slowly evolving disordered regions include sites for DNA binding. More work is needed to understand the underlying causes of the variability in the evolutionary rates of intrinsically ordered and disordered protein.     

The clinical and screening age-at-onset distribution for the MEN-2 syndrome.

The decision to screen for multiple endocrine neoplasia type 2 (MEN-2) is generally based on family history, the rationale for this approach being the presumed 100% penetrance of the disease. To determine the validity of this presumption we have estimated--by applying modifications of the life-table method--the clinical and screening age-at-diagnosis distributions for MEN-2, using families from the Cancer Research Campaign Medullary Thyroid Cancer Register and one large American family. The clinical penetrance of MEN-2 is shown to be incomplete, an estimated 41% of gene carriers not presenting with symptoms by age 70 on the basis of clinical history. Screening by the standard tests for detecting the earliest manifestations of the syndrome increases the penetrance to an estimated 93% by age 31. There is no evidence of a difference in the age-at-diagnosis distributions between maternal and paternal transmission, or among different families, but there is some suggestion of an earlier onset of medullary thyroid cancer in female gene carriers, and of a tendency of pheochromocytoma to cluster in families. These results can be used to calculate risks to relatives of affected individuals, which in turn can be used to guide decisions on which individuals to screen.