Wilcoxon signed-rank testAdvance Statistics

Joshua BatallaMP-Industrial

Introduction of the statistical concept

• The test is named for Frank Wilcoxon (1892–1965)

• The Wilcoxon Signed Ranks test is designed to test a hypothesis about the location (median) of a population distribution. It often involves the use of matched pairs, for example, before and after data, in which case it tests for a median difference of zero.

• The Wilcoxon Signed Ranks test does not require the assumption that the population is normally distributed

Uses of Wilcoxon signed rank test

• You use the Wilcoxon signed-rank test when there are two nominal variables and onemeasurement variable. One of the nominal variables has only two values, such as "before" and "after," and the other nominal variable often represents individuals. This is the non-parametric analogue to the paired t-test, and should be used if the distribution of differences between pairs may be non-normally distributed.

Requirements• 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

FormulaLet be the sample size, the number of pairs. Thus, there are a total of 2N data points. For ,

let and denote the measurements.

H0: median difference between the pairs is zero

H1: median difference is not zero.

1. For , calculate and , where is

the sign function.

2. Exclude pairs with . Let be the reduced sample size.

3. Order the remaining pairs from smallest absolute difference to largest absolute

difference, .

4. 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

Formula

, the absolute value of the sum of the signed ranks.

1. 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 .

Sample Application

Wilcoxon test Worked Example:

In order to investigate whether adults report verballypresented material more accurately from their right than from their left ear, a dichotic listening task was carried out. The data were found to be positively skewed.