# Nonparametric Procedures

Introduction:

Let's start with this quote from M. Hollander and D. A. Wolfe, from their book "Nonparametric Statistical Methods(2nd Ed., Wiley Series Probability and Statistics,1999): Nonparametric procedures are statistical procedures having desirable properties under relatively mild assumptions regarding the underlying population from which the data are obtained.

These methods have some of the following advantages:

1. Few assumptions on the population under study are required, in particular, they work well the normality assumption fails.
2. The are straightforward since they are mostly based on ranks of the data.
3. They are robust, in the sense that presence of outlying observations mildy if not affect the results.

One major drawback is that they do not provide insight into the sample size effect.

Wilcoxon-signed-Rank procedure:

This procedure can be thought of as an alternative to the parametric one sample t-test, paired t-test or matched pairs t-test, when the assumption of normality cannot be guaranteed. This test was first proposed by Frank Wilcoxon in his famous paper "Individual comparisons by ranking methods", Biometrics Bulletin 1 (6): 80-83 (1945).

 R SAS Minitab

Wilcoxon-Mann-Whithney (or Rank-Sum) procedure:

This procedure can be thought of as an alternative to two-sample t-test. The test was first proposed by Frank Wilcoxon in the same paper in which he introduced the wilcoxon signed rank, but a complete analysis was provided by  Henry Mann, and his student Donald Ransum Whithney in their paper "On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other", Ann. Stat. 1 (18):50-60 (1947).

 R SAS Minitab

Kruskall-Wallis procedure:

It is  a nonparametric (or distribution free) procedure considered as alternative to the One-Factor Anova (when the number of treatments is at least equal to 3), derived by William Kruskal and W. Allen  Wallis (Use of Rank in one-Criterion Analysis of variance, JASA,47 (260): 583–621,1952).
This procedure uses ranks of observations instead of actual observations and tests if samples are from the same populations ( or if at least two treatment effects are equal).

 R SAS Minitab

Friedman (or Friedman-Kendal-Babington-Smith) procedure:

It is a nonparametric procedure that can be considered as an alternative to the two-way Anova or more broadly an alternative to the Anova of randomized block designs. This procedure was first developed by Milton Friedman in his paper  "The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance" (Jasa, Vol. 32, Issue 200, 1937).

 R SAS Minitab