The Wilcoxon signed-rank test is a non-parametricstatistical hypothesis test used when comparing 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, t-test for matched pairs, or the t-test for dependent samples when the population cannot be assumed to be normally distributed.^{[1]}
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).^{[2]} The test was popularized by Siegel (1956)^{[3]} in his influential text book on non-parametric statistics. Siegel used the symbol T for a value related to, but not the same as, . 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[edit]
Data are paired and come from the same population.
Each pair is chosen randomly and independent.
The data are measured at least on an ordinal scale, but need not be normal.
The distribution of the differences is symmetric around the median.^{[citation needed]}
Test procedure[edit]
Let be the sample size, the number of pairs. Thus, there are a total of 2N data points. For , let and denote the measurements.
H_{0}: median difference between the pairs is zero
Exclude pairs with . Let be the reduced sample size.
Order the remaining pairs from smallest absolute difference to largest absolute difference, .
Rank the pairs, starting with the smallest as 1. Ties receive a rank equal to the average of the ranks they span. Let denote the rank.
Calculate the test statistic
, the absolute value of the sum of the signed ranks.
As increases, the sampling distribution of converges to a normal distribution. Thus,
For , a z-score can be calculated as .
If then reject
For , is compared to a critical value from a reference table.^{[1]}
If then reject
Alternatively, a p-value can be calculated from enumeration of all possible combinations of given .
The T statistic used by Siegel is the smaller of 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.
Example[edit]
1
125
110
1
15
2
115
122
–1
7
3
130
125
1
5
4
140
120
1
20
5
140
140
0
6
115
124
–1
9
7
140
123
1
17
8
125
137
–1
12
9
140
135
1
5
10
135
145
–1
10
order by absolute difference
5
140
140
0
3
130
125
1
5
1.5
1.5
9
140
135
1
5
1.5
1.5
2
115
122
–1
7
3
–3
6
115
124
–1
9
4
–4
10
135
145
–1
10
5
–5
8
125
137
–1
12
6
–6
1
125
110
1
15
7
7
7
140
123
1
17
8
8
4
140
120
1
20
9
9
is the sign function, is the absolute value, and 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.
Sign test (Like Wilcoxon test, but without the assumption of symmetric distribution of the differences around the median, and without using the magnitude of the difference)