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.

/pdf/a-simple-sequentially-rejective-multiple-test-procedure-458rh2uo9s.pdf

A Simple Sequentially Rejective Multiple Test Procedure

INTRODUCTION ..................................................................................................................... 561 ORGANIZING CO CEPTS ..................................................................................................... 564 Primary Hypotheses, Closure, Hierarchical Sets, and Minimal Hypotheses ...................... 564 Families ................................................................................................................................ 565 Type 1 Error Control ............................................................................................................ 566 Power ................................................................................................................................... 567 P-Values and Adjusted P-Values ......................................................................................... 568 Closed Test Procedures ....................................................................................................... 569 METHODS BA ED ON ORDERED P-VALUES ................................................................... 569 Methods Based on the First-Order Bonferroni Inequality .................................................. 569 Methods Based on the Simes Equality ................................................................................. 570 Modifications for Logically Related Hypotheses ................................................................. 571 Methods Controlling the False Discovery Rate ................................................................... 572 COMPARING NORMALLY DISTRIBUTED M ANS ......................................................... 573 OTHER ISSUES ........................................................................................................................ 575 Tests vs Confidence I tervals ............................................................................................... 575 Directional vs Nondirectional Inference ............................................................................. 576 Robustness ............................................................................................................................ 577 Others ........................................................................ ....................................................... 578 CONCLUSION .......................................................................................................................... 580

Multiple Hypothesis Testing

The authors discuss the problems of describing geographic variation data and develop statistical methods for categorizing sets of populations sampled from different localities. The general approach of the simultaneous test procedures, available with a variety of statistical tests and for continuous as well as for categorical data, is employed with these techniques. Geographical regions are defined as sets of connected localities, with connectedness being defined geometrically. Maximal acceptahle connected sets of localities (defined as regions) or coarsest acceptable connected partitions of the entire set of localities are found by these procedures. These are illustrated with several examples. I. AIMS AND PURPOSES The primary aim of geographic variation anialysis in biological systematics is the description and summarization of patterns of variation and covariation of characteristics of organisms that are distributed over an area. Such analyses are generally applied to species populations to study the variation of diverse characters. The most frequently studied characters are morphological, but recently there have also been such studies of biochemical, physiological, behavioral, cytological, immunological, as well as genetic characters. The basis for studies in geographic variation rests on the existence of populations of comparable organisms at a number of localities in the area under study. Comparisons of these populations are made in terms of one or more observable characters and the analyses relate these comparisons to differences in location. Data for studies in geographic variation consist of samples taken from the populations at a given number of localities, with a set of characteristics observed for each organism sampled. Summary statistics may be computed for each sample, as, for example, means and standard deviations for single measurable characteristics, correlations for pairs of such variables, etc. A summary of the data would then consist of a list of localities, each accompanied by its set of summary statistics for observed characteristics. In most cases, the infornation in such a list would be difficult to grasp and much would be gained by plotting the statistics on a map (or several maps) according to their location. Such graphic representation often reveals a good deal about the geographic variation pattern involved. A step beyond mere description of the pattern of variation of the characteristics of organisms is categorization. Usually, one prefers to group together localities that are geographically adjacent and whose populations are similar in their characteristics. This may be desired merely for purposes of simplification and summarization, or for the formal or semi-formal recognition of a population or series of populations in terms of the Linnean system. The study of patterns of geographic variation will often lead to causal ana:lysis. One may attempt to interpret the variational and correlational patterns of a species as adaptations to variation in known environmental factors, such as climatic, topographic, or edaphic variables. Other possible causes of variation may be differences in associated species populations, such as host plants, parasites, predators, etc. Marked and abrupt changes in characters between close localities may be related to abrupt changes in the above factors, to strong barriers to dispersal of the organisms or to secondary zones of intergradation between allopatrically differentiated populations. Unusual patterns of

A New Statistical Approach to Geographic Variation Analysis

We consider the problem of model selection and accounting for model uncertainty in high-dimensional contingency tables, motivated by expert system applications. The approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic P values leading to the selection of a single model; inference is then conditional on the selected model. The sampling properties of such a strategy are complex, and the failure to take account of model uncertainty leads to underestimation of uncertainty about quantities of interest. In principle, a panacea is provided by the standard Bayesian formalism that averages the posterior distributions of the quantity of interest under each of the models, weighted by their posterior model probabilities. Furthermore, this approach is optimal in the sense of maximizing predictive ability. But this has not been used in practice, because computing the posterior model probabilities is hard and the number of models is very large (often greater than 1...

Model Selection and Accounting for Model Uncertainty in Graphical Models Using Occam's Window

On closed testing procedures with special reference to ordered analysis of variance

1. Introduction and summary. When a hypothesis is tested by a significanice test and is not rejected, it is generally agreed that all hypotheses implied by that hypothesis (its "components") must also be considered as non-rejected. However, when the hypothesis is rejected the question remains as to which components may also be rejected. Various writers have given attention to this question and have proposed a variety of multiple comparisons methods based either on tests of each one of the components or on simultaneous confidence bounds on parametric functions related to the various hypotheses. An approach to such methods, apparently originally due to Tukey [27], is to test each component hypothesis by comparing its statistic with the a level critical value of the statistic for the overall hypothesis. This is called a Simultaneous Test Procedure (STP for short) as all hypotheses may be tested simultaneously and without reference to one another. An STP involves no stepwise testing of the kind employed by some other methods of multiple comparisons for means, in which subsets are tested for equality only if they are contained in sets which have already been found significant. (See 13], [4], [10], [18]). A general formalization of STP's is attempted in this paper. Section 2 introduces the requisite concepts of families of hypotheses and the implication relations between them, as well as the monotonicity relations between the related statistics. Section 3 defines STP's and shows conditions for coherence and consonance of their decisions, these properties being that hypothesis implication relations are preserved in the decisions of the STP. Section 4 discusses comparison of various STP's for the same hypotheses and shows the advantages of the unionintersection type of statistics and of reducing the family of hypotheses tested as much as possible. Section 5 translates all these results to simultaneous confidence statements after introducing the definitions necessary to allow such translation. The analogy between simultaneous test and confidence methods is of special importance as it brings a wide spectrum of methods within this framework, most of which was originally formulated in confidence region terms. This covers the original work of Tukey [27] and Scheff6 [25] and continues with that of Roy and his associates [21] and most recently Krishnaiah [12], [13]. A general discussion of this confidence approach has been given by Aitchison [1) since the first draft of the present paper. In view of the close analogies pointed out in Section 5, it is

/pdf/simultaneous-test-procedures-some-theory-of-multiple-4bu26i2zgh.pdf

Simultaneous test procedures-some theory of multiple comparisons'

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

K. R. Gabriel

Author Tools

Chat about Author

Papers

Simultaneous test procedures-some theory of multiple comparisons'