Wilcoxon signed-rank test

The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test used when comparing two related samples or repeated measurements on a single sample to assess whether their population mean ranks differ. It is a paired difference test used in situations similar to the Student's t-test for paired samples, but does not require the assumption of normality.

Overview

The Wilcoxon signed-rank test is named after Frank Wilcoxon, who proposed it in a single paper along with the Wilcoxon rank-sum test in 1945. The test is applicable for situations where the data is measured at least on an ordinal scale (i.e., the data can be ranked). It is widely used in situations where the data does not meet the assumptions necessary for the t-test, particularly the assumption of the normal distribution of differences.

Test Procedure

The procedure involves several steps:

1. Pair the observations from two samples or related observations from a single sample.
2. Calculate the differences between pairs, ignoring pairs with zero difference.
3. Rank the absolute values of the differences.
4. Assign signs to the ranks based on the sign of the differences (positive or negative).
5. Sum the ranks for the positive differences and the ranks for the negative differences.
6. The test statistic is the smaller of the two sums of ranks.

The null hypothesis for the test is that the median difference between the pairs of observations is zero. Depending on the alternative hypothesis chosen (two-tailed, left-tailed, or right-tailed), the sums of ranks are compared to critical values from the distribution of the sum of ranks, which can be approximated by a normal distribution for large samples.

Assumptions

The Wilcoxon signed-rank test assumes that:

• The data are paired and come from the same population.
• Each pair is chosen randomly and independently.
• The data are measured at least on an ordinal scale.
• The differences are symmetrically distributed about the median.

Applications

The Wilcoxon signed-rank test is used in various fields including medicine, psychology, and other social sciences, where the measurement scales are ordinal or when normality cannot be assumed. It is particularly useful in studies involving before-and-after measurements, matched participant studies, or repeated measurements on the same units.

Comparison with Other Tests

The Wilcoxon signed-rank test is often compared to the Student's t-test for paired samples. While the t-test is parametric and assumes normally distributed differences, the Wilcoxon test does not assume normality and is thus more robust in the presence of outliers or non-normal distributions. However, the t-test is generally more powerful for data that do actually follow a normal distribution.

Limitations

The main limitation of the Wilcoxon signed-rank test is its requirement for the data to be paired and matched, which is not always feasible in practical situations. Additionally, the test may lose power if there are many tied ranks or zero differences.

See Also

• Non-parametric statistics
• Mann-Whitney U test
• Kruskal-Wallis test