Wilcoxon signed rank test

History

The test is named for Frank Wilcoxon (1892–1965) who, in a single paper, proposed both it and the rank-sum test for two independent samples (Wilcoxon, 1945). The test was popularized by Sidney Siegel (1956) in his influential textbook on non-parametric statistics. Siegel used the symbol T for a value related to, but not the same as, W . In consequence, the test is sometimes referred to as the Wilcoxon T test, and the test statistic is reported as a value of T

Assumptions

1. Data are paired and come from the same population.
2. Each pair is chosen randomly and independently.
3. The data are measured at least on an ordinal scale (i.e., they cannot be nominal).

Test procedure

Let N be the sample size, i.e., the number of pairs. Thus, there are a total of 2N data points. For pairs i = 1 , . . . , N , let x 1 , i and x 2 , i denote the measurements.

H₀: difference between the pairs follows a symmetric distribution around zero H₁: difference between the pairs does not follow a symmetric distribution around zero.

1. For i = 1 , . . . , N , calculate | x 2 , i − x 1 , i | and sgn ⁡ ( x 2 , i − x 1 , i ) , where sgn is the sign function.
2. Exclude pairs with | x 2 , i − x 1 , i | = 0 . Let N r be the reduced sample size.
3. Order the remaining N r pairs from smallest absolute difference to largest absolute difference, | x 2 , i − x 1 , i | .
4. Rank the pairs, starting with the smallest as 1. Ties receive a rank equal to the average of the ranks they span. Let R i denote the rank.
5. Calculate the test statistic W W = ∑ i = 1 N r [ sgn ⁡ ( x 2 , i − x 1 , i ) ⋅ R i ] , the sum of the signed ranks.
6. Under null hypothesis, W follows a specific distribution with no simple expression. This distribution has an expected value of 0 and a variance of N r ( N r + 1 ) ( 2 N r + 1 ) 6 . W can be compared to a critical value from a reference table. The two-sided test consists in rejecting H 0 if | W | > W c r i t i c a l , N r .
7. As N r increases, the sampling distribution of W converges to a normal distribution. Thus, For N r ≥ 10 , a z-score can be calculated as z = W σ W , σ W = N r ( N r + 1 ) ( 2 N r + 1 ) 6 . To perform a two-sided test, reject H 0 if | z | > z c r i t i c a l . Alternatively, one-sided tests can be performed with either the exact or the approximative distribution. p-values can also be calculated.

Original test

The original Wilcoxon's proposal used a different statistic. Denoted by Siegel as the T statistic, it is the smaller of the two sums of ranks of given sign; in the example given below, therefore, T would equal 3+4+5+6=18. Low values of T are required for significance. As will be obvious from the example below, T is easier to calculate by hand than W and the test is equivalent to the two-sided test described above; however, the distribution of the statistic under H 0 has to be adjusted.

Example

sgn is the sign function, abs is the absolute value, and R i is the rank. Notice that pairs 3 and 9 are tied in absolute value. They would be ranked 1 and 2, so each gets the average of those ranks, 1.5. N r = 10 − 1 = 9 , | W | = | 1.5 + 1.5 − 3 − 4 − 5 − 6 + 7 + 8 + 9 | = 9. | W | < W α = 0.05 ,   9 , two-sided = 35 ∴ fail to reject  H 0 .

Effect size

To compute an effect size for the signed-rank test, one can use the rank correlation.

If the test statistic W is reported, the rank correlation r is equal to the test statistic W divided by the total rank sum S, or r = W/S. Using the above example, the test statistic is W = 9. The sample size of 9 has a total rank sum of S = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) = 45. Hence, the rank correlation is 9/45, so r = 0.20.

If the test statistic T is reported, an equivalent way to compute the rank correlation is with the difference in proportion between the two rank sums, which is the Kerby (2014) simple difference formula. To continue with the current example, the sample size is 9, so the total rank sum is 45. T is the smaller of the two rank sums, so T is 3 + 4 + 5 + 6 = 18. From this information alone, the remaining rank sum can be computed, because it is the total sum S minus T, or in this case 45 - 18 = 27. Next, the two rank-sum proportions are 27/45 = 60% and 18/45 = 40%. Finally, the rank correlation is the difference between the two proportions (.60 minus .40), hence r = .20.

Implementations

ALGLIB includes implementation of the Wilcoxon signed-rank test in C++, C#, Delphi, Visual Basic, etc.

The free statistical software R includes an implementation of the test as wilcox.test(x,y, paired=TRUE), where x and y are vectors of equal length.

GNU Octave implements various one-tailed and two-tailed versions of the test in the wilcoxon_test function.

SciPy includes an implementation of the Wilcoxon signed-rank test in Python

Accord.NET includes an implementation of the Wilcoxon signed-rank test in C# for .NET applications