Chapter 15 Wilcoxon signed-rank test

The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used to compare two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ (i.e. it is a paired difference test). It can be used as an alternative to the paired Student’s t-test (also known as “t-test for matched pairs” or “t-test for dependent samples”) when the distribution of the difference between two samples’ means cannot be assumed to be normally distributed. A Wilcoxon signed-rank test can be used to determine whether two dependent samples were selected from populations having the same distribution. wiki

Two data samples are matched if they come from repeated observations of the same subject. Using the Wilcoxon Signed-Rank Test, we can decide whether the corresponding data population distributions are identical without assuming them to follow the normal distribution.

Example In the built-in data set named immer, the barley yield in years 1931 and 1932 of the same field are recorded. The yield data are presented in the data frame columns Y1 and Y2.

library(MASS)
head(immer)

##   Loc Var    Y1    Y2
## 1  UF   M  81.0  80.7
## 2  UF   S 105.4  82.3
## 3  UF   V 119.7  80.4
## 4  UF   T 109.7  87.2
## 5  UF   P  98.3  84.2
## 6   W   M 146.6 100.4

Problem Without assuming the data to have normal distribution, test at .05 significance level if the barley yields of 1931 and 1932 in data set immer have identical data distributions.

Solution The null hypothesis is that the barley yields of the two sample years are identical populations. To test the hypothesis, we apply the wilcox.test function to compare the matched samples. For the paired test, we set the “paired” argument as TRUE. As the p-value turns out to be 0.005318, and is less than the .05 significance level, we reject the null hypothesis.

wilcox.test(immer$Y1, immer$Y2, paired=TRUE)

## Warning in wilcox.test.default(immer$Y1, immer$Y2, paired = TRUE): cannot compute exact p-
## value with ties

## 
##  Wilcoxon signed rank test with continuity correction
## 
## data:  immer$Y1 and immer$Y2
## V = 368.5, p-value = 0.005318
## alternative hypothesis: true location shift is not equal to 0

Answer At .05 significance level, we conclude that the barley yields of 1931 and 1932 from the data set immer are nonidentical populations.

Source Wilcoxon Signed-Rank Test