wilcoxonSignRank¶

Performs a Wilcoxon signed rank test.

Synopsis¶

wilcoxonSignRank (x)

Required Arguments¶

float x[] (Input): Array of length nObservations containing the data.

Return Value¶

An array of length two containing the values described below.

The asymptotic probability of not exceeding the standardized (to an asymptotic variance of 1.0) minimum of (W+, W-) using method 1 under the null hypothesis that the distribution is symmetric about 0.0.

And, the asymptotic probability of not exceeding the standardized (to an asymptotic variance of 1.0) minimum of (W+, W-) using method 2 under the null hypothesis that the distribution is symmetric about 0.0.

Optional Arguments¶

fuzz, float (Input)

Nonnegative constant used to determine ties in computing ranks in the combined samples. A tie is declared when two observations in the combined sample are within fuzz of each other.

Default value for fuzz is 0.0.

stat (Output)

An array of length 10 containing the following statistics:

Row	Statistics
0	The positive rank sum, W+, using method 1.
1	The absolute value of the negative rank sum, W-, using method 1.
2	The standardized (to an asymptotic variance of 1.0) minimum of (W+, W-) using method 1.
3	The asymptotic probability of not exceeding `stat[2]` under the null hypothesis that the distribution is symmetric about 0.0.
4	The positive rank sum, W+, using method 2.
5	The absolute value of the negative rank sum, W-, using method 2.
6	The standardized (to an asymptotic variance of 1.0) minimum of (W+, W-) using method 2.
7	The asymptotic probability of not exceeding `stat[6]` under the null hypothesis that the distribution is symmetric about 0.0.
8	The number of zero observations.
9	The total number of observations that are tied, and that are not within fuzz of zero.

nMissing (Output): Number of missing values in y.

Description¶

Function wilcoxonSignRank performs a Wilcoxon signed rank test of symmetry about zero. In one sample, this test can be viewed as a test that the population median is zero. In matched samples, a test that the medians of the two populations are equal can be computed by first computing difference scores. These difference scores would then be used as input to wilcoxonSignRank. A general reference for the methods used is Conover (1980).

Function wilcoxonSignRank computes statistics for two methods for handling zero and tied observations. In the first method, observations within fuzz of zero are not counted, and the average rank of tied observations is used. (Observations within fuzz of each other are said to be tied.) In the second method, observations within fuzz of zero are randomly assigned a positive or negative sign, and the ranks of tied observations are randomly permuted.

The W+ and W- statistics are computed as the sums of the ranks of the positive observations and the sum of the ranks of the negative observations, respectively. Asymptotic probabilities are computed using standard methods (see, e.g., Conover 1980, page 282).

The W+ and W- statistics may be used to test the following hypotheses about the median, M. In deciding whether to reject the null hypothesis, use the bracketed statistic if method 2 for handling ties is preferred. Possible null hypotheses and alternatives are given as follows:

\(H_0 : M\leq 0\) \(H_1 : M>0\)

Reject if stat[0] [or stat[4]] is too large.
\(H_0 : M\geq 0\) \(H_1 : M<0\)

Reject if stat[1] [or stat[5]] is too large.
\(H_0 : M=0\) \(H_1 : M\neq 0\)

Reject if stat[2][or stat[6]] is too small. Alternatively, if an asymptotic test is desired, reject if 2 * stat[3] [or 2 * stat[7]] is less than the significance level.

Tabled values of the test statistic can be found in the references. If possible, tabled values should be used. If the number of nonzero observations is too large, then the asymptotic probabilities computed by wilcoxonSignRank can be used.

The assumptions required for the hypothesis tests are as follows:

The distribution of each \(X_i\) is symmetric.
The \(X_i\) are mutually independent.
All \(X_i\)’s have the same median.
An ordering of the observations exists (i.e., \(X_1>X_2\) and \(X_2>X_3\) implies that \(X_1>X_3\)).

If other assumptions are made, related hypotheses that are more (or less) restrictive can be tested.

Example¶

This example illustrates the application of the Wilcoxon signed rank test to a test on a difference of two matched samples (matched pairs) {X1 = 223, 216, 211, 212, 209, 205, 201; and X2 = 208, 205, 202, 207, 206, 204, 203}. A test that the median difference is 10.0 (rather than 0.0) is performed by subtracting 10.0 from each of the differences prior to calling wilcoxonSignRank. As can be seen from the output, the null hypothesis is rejected. The warning error will always be printed when the number of observations is 50 or less unless printing is turned off for warning errors.

from __future__ import print_function
from numpy import *
from pyimsl.stat.wilcoxonSignRank import wilcoxonSignRank

stat = []
nmiss = []
fuzz = .0001
x = array([-25., -21., -19., -15., -13., -11., -8.])

result = wilcoxonSignRank(x,
                          nMissing=nmiss,
                          fuzz=fuzz,
                          stat=stat)

print("\nStatistic\t\t\tMethod 1\tMethod 2")
print("W+\t\t\t\t %3.0f\t\t %3.0f" % (stat[0], stat[4]))
print("W-\t\t\t\t %3.0f\t\t %3.0f" % (stat[1], stat[5]))
print("Standardized Minimum\t\t%6.4f\t\t%6.4f" % (stat[2], stat[6]))
print("p-value\t\t\t\t %6.4f\t\t %6.4f\n" % (stat[3], stat[7]))
print("Number of zeros\t\t\t%3.0f" % stat[8])
print("Number of ties\t\t\t%3.0f" % stat[9])
print("Number of missing\t\t  %d" % nmiss[0])

Output¶

***
*** Warning error issued from IMSL function wilcoxonSignRank:
*** "n_observations" = 7.  The number of observations is less than 50, and exact tables should be referenced for probabilities.
***

Statistic			Method 1	Method 2
W+				   0		   0
W-				  28		  28
Standardized Minimum		-2.3664		-2.3664
p-value				 0.0090		 0.0090

Number of zeros			  0
Number of ties			  0
Number of missing		  0