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2016 | OriginalPaper | Buchkapitel

2. Completely Randomized Data

verfasst von : Kenneth J. Berry, Paul W. Mielke Jr., Janis E. Johnston

Erschienen in: Permutation Statistical Methods

Verlag: Springer International Publishing

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Abstract

Chapter 2 introduces a generalized Minkowski distance function that is the basis for a set of multi-response permutation procedures for univariate and multivariate completely randomized data. Multi-response permutation procedures constitute a class of permutation methods for one or more response measurements that are designed to distinguish possible differences among two or more groups. The multi-response permutation procedures provide a synthesizing foundation for a variety of statistical tests and measures developed in successive chapters.

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Fußnoten
1
The 1976 paper by Mielke, Berry, and Johnson was the first published account of MRPP [300]. Previously, Mielke utilized MRPP in a study sponsored by the National Communicable Disease Center that involved comparisons of proportional contributions of five plague organism protein bands based on electrophoresis measurements obtained from samples of organisms associated with distinct geographical regions.
 
2
The Journal für die Reine und Angewandte Mathematik was founded by August Leopold Crelle in 1826. It continues today, although it is more popularly known as Crelle’s Journal.
 
3
A distance function is a metric if it satisfies three properties given by (1) \(\Delta (x,y) \geq 0\) and \(\Delta (x,x) = 0\), i.e., the distance is positive between two different points and is equal to zero from any point to itself; (2) the distance is symmetric: \(\Delta (x,y) = \Delta (y,x)\), i.e., the distance between points x and y is the same in either direction; and (3) the triangle inequality is satisfied: \(\Delta (x,y) \leq \Delta (x,z) + \Delta (z,y)\), i.e., the distance between any two points is the shortest distance along any path.
 
4
Multi-response permutation procedures also provide for a group of unclassified response measurement scores such as might result from a survey with question choices that include “none of the above” or “not applicable.” See, for example, a 1983 article on lead concentrations in inner-city soils by Mielke, Anderson, Berry, Mielke, Chaney, and Leech [302] and a discussion by Mielke and Berry in 2007 [297, pp. 35–40].
 
5
A sufficient condition for a permutation statistical test is the exchangeability of the random variables. Sequences that are independent and identically distributed (i.i.d.) are always exchangeable, but so is sampling without replacement from a finite population. However, while i.i.d. implies exchangeability, exchangeability does not imply i.i.d. [150, 168, 217].
 
6
As will be shown in Chap. 3, \(\mathfrak{R}\) may also be interpreted as a chance-corrected measure of effect size.
 
7
Each element of the S matrix is constructed from two corresponding elements in the \(\hat{\boldsymbol{\Sigma }}\) matrices, weighted by the degrees of freedom, i.e., n − 1. For example, the first element of the S matrix is \(0.5333 = [(4 - 1)(0.6667) + (3 - 1)(0.3333)]/(4 + 3 - 2)\).
 
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Metadaten
Titel
Completely Randomized Data
verfasst von
Kenneth J. Berry
Paul W. Mielke Jr.
Janis E. Johnston
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-28770-6_2