This paper presents a simple and widely ap- plicable multiple test procedure of the sequentially rejective type, i.e. hypotheses are rejected one at a tine until no further rejections can be done. It is shown that the test has a prescribed level of significance protection against error of the first kind for any combination of true hypotheses. The power properties of the test and a number of possible applications are also discussed.

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A Simple Sequentially Rejective Multiple Test Procedure

Introduction General Concepts Error rates and general concepts Construction methods Methods based on Bonferroni's inequality Methods based on Simes' inequality Multiple Comparisons in Parametric Models General linear models Extensions to general parametric models The multcomp package Applications Multiple comparisons with a control All pairwise comparisons Dose response analyses Variable selection in regression models Simultaneous confidence bands Multiple comparisons under heteroscedasticity Multiple comparisons in logistic regression models Multiple comparisons in survival models Multiple comparisons in mixed-effects models Further Topics Resampling-based multiple comparison procedures Group sequential and adaptive designs Combining multiple comparisons with modeling Bibliography Index

Multiple Comparisons Using R

We consider the problem of simultaneously testing k ≥ 2 hypotheses on parameters θ1, …, θk . In a typical application the θ's may be a set of contrasts, for instance, a set of orthogonal contrasts among population means or a set of differences between k treatment means and a standard treatment mean. It is assumed that least squares estimators 1, …, k are available that are jointly normally distributed with a common variance (known up to a scalar, namely the error variance σ 2) and a common known correlation. An independent χ 2-distributed unbiased estimator of σ 2 is also assumed to be available. We propose a step-up multiple test procedure for this problem which tests the t statistics for the k hypotheses in order starting with the least significant one and continues as long as an acceptance occurs. (By contrast, the step-down approach, which is usually used, starts with the most significant and continues as long as a rejection occurs.) Critical constants required by this step-up procedure to co...

http://users.iems.northwestern.edu/~ajit/papers/14)%20Step-up%20Procedure.pdf

A Step-Up Multiple Test Procedure

The multiple-comparison procedure originally proposed by R. A. Fisher (1935) for the 1-way analysis of variance context has several desirable properties when K (the number of groups) is equal to 3. In this article, the logic of the procedure is described in conjunction with those properties. A discussion follows of how the Fisher procedure can be similarly applied in a number of other K=3 (and, more generally, 2-degree-of-freedom) hypothesis-testing situations. Finally, the Fisher logic is combined with recent sequential applications of the Bonferroni inequality to illustrate the utility and versatility of that combination for the applied researcher

A controlled, powerful multiple-comparison strategy for several situations.

The basic ideas of multiple testing are outlined and the problem of how to control the probability of erroneous decisions is discussed. The main emphasis is on the concept of the multiple level of significance (controlling the experiment, or family error in the strong sense) which can be achieved by applying the principle of closed tests. Various practical situations encountered in multiple testing in clinical trials are considered: more than one end point; more than two treatments, such as comparisons with a single control, comparisons using ordered alternatives, all pairwise comparisons and contrast methods; and more than one trial. Tests based on global statistics, the union intersection principle and other criteria are discussed. The application of the multiple test concept in sequential sampling is investigated. Finally some comments are made on multiple power, multiple confidence intervals and directed decisions.

Multiple testing in clinical trials.

In this paper we discuss two types of roles for comparing several processes or populations with a standard. The rules are given in a general setting and comparisons of the rules are provided. Details of construction and modifications of the rules are given when each of the processes is characterized by two types of random variables x and y having exponential distributions with mans λ and μ respectively, and processes are compared in terms of θ = λ/μ.

Some selection rules for comparing p processes with a standard

SUMMARY Suppose that we have a sample from a bivariate normal population with variances 0-2, 0-2 and correlation coefficient p, but some observations on one of the two variables are missing. Based on these data, we wish to test the hypothesis of equality of means of the two variables. In this paper, classes of test procedures are given for when the alternative is one-sided and for when it is two-sided. The tests satisfy the property that the probability of type I error does not exceed a preassigned level a when p is negative, and when p is positive but small. The test that is to be recommended in a particular situation depends on what information, if any, is available about the nuisance parameters. Comparisons of the tests with the paired t test and the Mehta & Gurland (1969, 1973) test are also included.

On testing equality of means of correlated variables with incomplete data

Consider that we have a collection of k populations π1, π2…,πk. The quality of the ith population is characterized by a real parameter θi and the population is to be designated as superior or inferior depending on how much the θi differs from θmax = max{θ1, θ2,…,θk}. From the set {π1, π2,…,πk}, we wish to select the subset of superior populations. In this paper we devise rules of selection which have the property that their selected set excludes all the inferior populations with probability at least 1−α, where a is a specified number.

Some subset selection problems

It is known that in the random effect model, exact F-tests are not obtainable from the analysis of variance table for testing the main effects in a three-way layout. Similarly exact F-tests are not obtainable for the main effects and certain interactions in higher-way layouts. In this paper level-α: tests obtainable from the analysis of variance table are given for testing these effects. Tables are provided to illustrate the size function and the power function of the tests.

Umesh D. Naik

Papers

Some selection rules for comparing p processes with a standard

On testing equality of means of correlated variables with incomplete data

Some subset selection problems

On Tests of Main Effects and Interactions in Higher-Way Layouts in the Analysis of variance random Effects Model

On Some Procedures for Testing a Linear Relation Among Variances