Statistical Inference: A Short Course

Appendix 11.C Wilcoxon Signed Rank Test When Sampling from Two Dependent Populations: Paired Comparisons

As in Sections 2 and 5, let us assume that a paired experiment yields the n pairs of observations (X₁,Y₁), . . . , (X_n,Y_n), where X_i and Y_i, i = 1, . . . , n, are members of the same pair (X_i, Y_i), with X_i drawn from population 1 and Y_i drawn from population 2. For the ith pair (X_i,Y_i), let D_i = X_i − Y_i, i = 1,. . ., n, where D_i is the ith observation on the random variable D. It is assumed that the population probability distributions are continuous, with the measurements taken on an interval or ratio scale since both the signs of the D_i's as well as their ranks will be utilized. Moreover, the D_i's are taken to be independent random variables that follow a distribution that is symmetrical about a common median.

A comparison of the members of each pair (X_i,Y_i) will render a “+” sign, a “−” sign, or a zero value. In this regard, when the value of an element from population 1 exceeds the value of its paired element from population 2, we will assign a “+” sign to the pair (X_i,Y_i) and D_i > 0. If the value of an element from population 1 falls short of the value of its paired counterpart from population 2, then the pair is assigned a “−” sign and D_i < 0. Ties obviously occur if D_i = 0, in which case the pair (X_i,Y_i) producing the tie is eliminated from the sample.

The Wilcoxon signed rank test involving matched pairs is constructed so as to test the null hypothesis H₀: the population ...