Exact Permutation/Randomization Tests Algorithms
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Exact Permutation/Randomization Tests Algorithms Subir Ghosh1 Accepted: 8 September 2020 © Grace Scientific Publishing 2020
Abstract Fisher (The design of experiments, Oliver & Boyd, London, 1935) described the exact permutation and randomization tests for comparative experiments without assuming normality or any particular probability distribution. While having this as an attractive feature, the computational challenge was a disadvantage at that time but not now with modern computers. This paper introduces a permutation/randomization data algorithm to generate the permutation/randomization distributions under the null hypotheses for calculating the P-values. The properties of permutation/randomization data matrices developed by algorithms following the proposed mathematical processes are derived. Two illustrative examples demonstrate the usefulness of the proposed computational methods. Keywords Block design · Completely randomized design · Hypothesis testing · Paired data · Permutation · Randomization · Randomized control trial · Two-sample data
1 Introduction The randomization test for paired comparison of two population distributions shares a rich history starting from Fisher [15] with the permutation test for comparison of two populations by using the data obtained from two independent samples (see Yates [34]). The work of Pitman [26–28]; Welch [33]; Wald & Wolfowitz [32]; Hoeffding [17]; Kempthorne [18]; Box & Anderson [4] Tukey [30, 31]; Kempthorne & Doerfler [19]; Rao [29]; Lehmann [20]; Basu [1]; Boik [3]; Edgington & Onghena, P. [11]; Ludbrook and Dudley [21]; Calinsky and Kageyama [6, 7]; Ernst [13]; Good [16]; Manly [22]; Mielke and Berry [23]; David [8]; Pesarin and Salmaso
Part of special issue guest edited by Dieter Rasch, Jürgen Pilz, and Subir Ghosh—Advances in Statistical and Simulation Methods. * Subir Ghosh [email protected] 1
Department of Statistics, University of California, Riverside, CA 92521, USA
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Journal of Statistical Theory and Practice
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[25]; Ferron and Levin [14]; Dugard [9]; Efron and Hastie [12]; Berry, Johnston, and Mielke [2]; Onghena [24]; and many others enriched the understanding of the randomization/permutation tests. The application areas include Bio-statistics and Bio-informatics, Computer Science and Engineering, Economics, Education, Psychology, and Sociology. Section 2 provides the background information on the permutation matrices and the exchangeable condition on a vector of random variables. The main idea behind the algorithm for the paired comparison randomization test in Sect. 3 is to start with the vector Y d of n paired differences. Then, the task is to develop a randomization matrix R(n) with n rows and 2n columns to obtain the randomization distribution of Y d satisfying the exchangeable condition that the probability distribution of the columns of R(n) ⊙Y d to be the same under the null hypothesis, where the Schur product ⊙ is defined in Definition 3. The algorithm identifies the randomizatio
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