Wilcoxon Signed Rank Test

The Wilcoxon signed-rank test can be used to test for the median of a single population or to test matched-pairs data for a common median. In the case of matched pairs, the test reduces to testing the single population of paired differences for a median of 0. The test assumes that the underlying population is symmetric.

The Wilcoxon test allows tied values. The test statistic is adjusted for differences of zero using a method suggested by Pratt. See Lehmann and D’Abrera (2006), Pratt (1959), and Cureton (1967).

Testing for the Median of a Single Population

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There are N observations:

X1, X2,..., XN

•

The null hypothesis is:

H0: median = m

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The differences between observations and the hypothesized value m are calculated as follows:

Dj = Xj - m

Testing for the Equality of Two Population Medians with Matched Pairs Data

A special case of the Wilcoxon signed-rank test is applied to matched-pairs data.

•

There are N pairs of observations from two populations:

X1, X2,..., XN and Y1, Y2,..., YN

•

The null hypothesis is:

H0: medianX - Y = 0

•

The differences between pairs of observations are calculated as follows:

Dj = Xj -Yj

Wilcoxon Signed-Rank Test Statistic

The test statistic is based on the sum of the signed ranks. Signed ranks are defined as follows:

•

The absolute values of the differences,

, are ranked from smallest to largest.

•

The ranks start with the value 1, even if there are differences of zero.

•

When there are tied absolute differences, they are assigned the average, or midrank, of the ranks of the observations.

Denote the rank or midrank for a difference

by Rj. Define the signed rank for

as follows:

•

If the difference

is positive, the signed rank is Rj.

•

If the difference

is zero, the signed rank is 0.

•

If the difference

is negative, the signed rank is -Rj.

The signed-rank statistic is computed as follows:

Define the following:

is the number of signed ranks that equal zero

R+ is the sum of the positive signed ranks

Then the following holds:

Wilcoxon Signed-Rank Test P-Values

For

, exact p-values are calculated.

For N > 20, a Student’s t approximation to the statistic defined below is used. Note that a correction for ties is applied. See Iman (1974) and Lehmann and D’Abrera (2006).

Under the null hypothesis, the mean of W is zero. The variance of W is given by the following:

The last summation in the expression for Var(W) is a correction for ties. The notation di for i > 0 represents the number of values in the ith group of nonzero signed ranks. (If there are no ties for a given signed rank, then di = 1 and the summand is 0.)

The statistic t given by the following has an approximate t distribution with N - 1 degrees of freedom: