x1 <- c(18, 21, 19, 25, 27, 23, 22)
x2 <- c(14, 15, 17, 16, 19, 18, 15)
wilcox.test(x1, x2)
Wilcoxon rank sum exact test
data: x1 and x2
W = 47, p-value = 0.003
alternative hypothesis: true location shift is not equal to 08 Rank-based tests
9 Rank-based tests
Hypothesis tests that are based on knowledge of the probability distributions (e.g. normal or binomial) that the data follow are known as parametric tests. When data do not meet the parametric test assumptions, we can use non-parametric tests, also called distribution free tests, that replace the data with their ranks.
Rank-based tests are often useful when the data are skewed, contain outliers, or are measured on an ordinal scale.
Instead of analyzing the observed values directly, rank-based tests are based on the ranks of the observations. Large values receive high ranks and small values receive low ranks.
Rank-based tests that are commonly used as alternatives to t-tests are:
- one-sample or paired t-test \(\rightarrow\) Wilcoxon signed-rank test
- two-sample t-test \(\rightarrow\) Mann–Whitney test / Wilcoxon rank-sum test
9.1 Two independent samples
For comparing two independent groups when a two-sample t-test is not appropriate, a rank-based alternative is the Mann–Whitney test, also known as the Wilcoxon rank-sum test.
Suppose we have two independent samples from two populations and want to investigate whether the groups differ in location.
The hypotheses may be written as
\[H_0: \text{the two groups come from the same distribution}\] \[H_1: \text{the two groups differ in location}\]
In practice, the test is often interpreted as a test of whether one group has a higher median than the other, but this interpretation is only valid if the distribution shapes in the two groups are similar.
The test is performed by pooling all observations from both groups, ranking them from smallest to largest, and comparing the sums of ranks between the groups. A significant Mann-Whitney U test means that scores in one group generally rank higher than scores in the other.
In R, the Mann–Whitney / Wilcoxon rank-sum test can be performed using the function wilcox.test with two independent samples.
9.2 Paired samples
For paired data, a rank-based alternative to the paired t-test is the Wilcoxon signed-rank test. This test can also be used for the one sample setting, where one sample is compared to a known location parameter.
As for the paired t-test, the analysis is based on the within-pair differences. However, instead of using the observed differences directly, the Wilcoxon signed-rank test uses the ranks of the absolute differences together with their signs.
The hypotheses are commonly expressed as
\[H_0: \text{the distribution of within-pair differences is centered at 0}\] \[H_1: \text{the distribution of within-pair differences is not centered at 0}\]
In R, the Wilcoxon signed-rank test can be performed using wilcox.test with paired=TRUE.
before <- c(132, 128, 136, 141, 130, 125, 138, 129)
after <- c(127, 125, 130, 136, 128, 121, 134, 126)
wilcox.test(after, before, paired=TRUE)
Wilcoxon signed rank exact test
data: after and before
V = 0, p-value = 0.008
alternative hypothesis: true location shift is not equal to 0