Multiple testing with minimal assumptions

Resampling-based multiple testing methods that control the Familywise Error Rate in the strong sense are presented. It is shown that no assumptions whatsoever on the data-generating process are required to obtain a reasonably powerful and flexible class of multiple testing procedures. Improvements are obtained with mild assumptions. The methods are applicable to gene expression data in particular, but more generally to any multivariate, multiple group data that may be character or numeric. The role of the disputed "subset pivotality" condition is clarified.     

Testing Hypotheses about Regression Parameters,When the Error Term Is Heteroscedastic

The paper considers methods for testing H₀: β₁ = … = β_p = 0, where β₁, … ,β_p are the slope parameters in a linear regression model with an emphasis on p = 2. It is known that even when the usual error term is normal, but heteroscedastic, control over the probability of a type I error can be poor when using the conventional F test in conjunction with the least squares estimator. When the error term is nonnormal, the situation gets worse. Another practical problem is that power can be poor under even slight departures from normality. Liu and Singh (1997) describe a general bootstrap method for making inferences about parameters in a multivariate setting that is based on the general notion of depth. This paper studies the small-sample properties of their method when applied to the problem at hand. It is found that there is a practical advantage to using Tukey's depth versus the Mahalanobis depth when using a particular robust estimator. When using the ordinary least squares estimator, the method improves upon the conventional F test, but practical problems remain when the sample size is less than 60. In simulations, using Tukey's depth with the robust estimator gave the best results, in terms of type I errors, among the five methods studied.     

Power Estimation for Two-Sample Tests Using Importance and Antithetic Resampling

Collings and Hamilton (1988), described a uniform bootstrap method that is applied on observed or pilot data in order to approximate the power of the two-sample Wilcoxon test for location shift alternatives. In this paper we demonstrate how importance and antithetic resampling can be used to substantially reduce the amount of computation needed to approximate the power of the two-sample tests for location shift and scale alternatives. Importance and antithetic bootstrap resampling methods are applied to simulated data of different sample sizes from a variety of distributions as well as to data from the Iowa 65+ Rural Health Study. Also, a suggestion is given for using a combination of importance and antithetic resampling for approximating the power of two-sample tests.     

Testing and estimation of proportion (or risk) ratio under the matched‐pair design with multiple binary endpoints

The proportion ratio (PR) of responses between an experimental treatment and a control treatment is one of the most commonly used indices to measure the relative treatment effect in a randomized clinical trial. We develop asymptotic and permutation‐based procedures for testing equality of treatment effects as well as derive confidence intervals of PRs for multivariate binary matched‐pair data under a mixed‐effects exponential risk model. To evaluate and compare the performance of these test procedures and interval estimators, we employ Monte Carlo simulation. When the number of matched pairs is large, we find that all test procedures presented here can perform well with respect to Type I error. When the number of matched pairs is small, the permutation‐based test procedures developed in this paper is of use. Furthermore, using test procedures (or interval estimators) based on a weighted linear average estimator of treatment effects can improve power (or gain precision) when the treatment effects on all response variables of interest are known to fall in the same direction. Finally, we apply the data taken from a crossover clinical trial that monitored several adverse events of an antidepressive drug to illustrate the practical use of test procedures and interval estimators considered here.     

Euclidean distance matrix analysis: a coordinate-free approach for comparing biological shapes using landmark data.

For problems of classification and comparison in biological research, the primary focus is on the similarity of forms. A biological form can be conveniently defined as consisting of size and shape. Several approaches for comparing biological shapes using landmark data are available. Lele (1991a) critically discusses these approaches and proposes a new method based on the Euclidean distance matrix representation of the form of an object. The purpose of this paper is to extend this new methodology to the comparison of groups of objects. We develop the statistical versions of various concepts introduced by Lele (1991a) and use them for developing statistical procedures for testing the hypothesis of shape difference between biological forms. We illustrate the use of this method by studying morphological differences between normal children and those affected with Crouzon and Apert syndromes and craniofacial sexual dimorphism in Cebus apella.