# Nonparametric Inference Using Randomization and Permutation Reference Distribution and Their Monte-Carlo Approximation

Mara Tableman

Spring 2009

12-2009

Paper

## Department

Mathematics and Statistics

English

## DOI

10.15760/etd.7798

## Physical Description

1 online resource (30 pages)

## Abstract

The purpose of this paper is to explain the concepts and replicate some results in the article Permutation Methods: A Basis for Exact Inference by Michael D. Ernst (2004). In the paper, we discuss how to perform nonparametric inferences using randomization and permutation reference distributions. We clearly explain the differences and similarities between the two distributions and show how to use Monte-Carlo sampling to approximate them. In the process, we introduced and proved two theorems that are the basis for some of the inferences. Specifically, one of the theorems describes the exact p-value of a two-sample test as a left-continuous decreasing step function of the treatment effect [triangle], which enables us to construct a confidence interval for [triangle]. This description appears to be new and the proof is independently derived. The second theorem is a basis for constructing a confidence interval for this exact p-value based on binomial distribution which results from the Monte- Carlo approximation. This theorem was given as an exercise in a book by Casella and Berger (C-B, 2002, Exercise 9.21, page 451). In addition, we introduce two self-written R functions, pval and cint, and demonstrate how to use them to replicate all the results in the article. Readers will see that, in some situations, the two functions are more versitile than standard packages such as Resampling Stat (Resampling Stats, Inc., 2003) used by Higgins (2004) in his book on the topic of nonparametric method and StatXact (Cytel Software Corporation, 2003) used by Ernst in his article. The functions will soon be put in a standard R package for free distribution to be used in classroom, study, and research.

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