Performs a Wilcoxon rank sum test.

Synopsis

#include <imsls.h>

float imsls_f_wilcoxon_rank_sum (int n1_observations, float x1[],
int n2_observations, float x2[], ..., 0)

The type double function is imsls_d_wilcoxon_rank_sum.

Required Arguments

int n1_observations   (Input)
Number of observations in the first sample.

float x1[]   (Input)
Array of length n1_observations containing the first sample.

int n2_observations   (Input)
Number of observations in the second sample.

float x2[]   (Input)
Array of length n2_observations containing the second sample.

Return Value

The two-sided p-value for the Wilcoxon rank sum statistic that is computed with average ranks used in the case of ties.

Synopsis with Optional Arguments

#include <imsls.h>

float imsls_f_wilcoxon_rank_sum (int n1_observations, float x1[],
int n2_observations, float x2[],
IMSLS_FUZZ, float fuzz,
IMSLS_STAT, float **stat,
IMSLS_STAT_USER, float stat[],
0)

Optional Arguments

IMSLS_FUZZ, float fuzz   (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: fuzz = 100 × imsls_f_machine(4) × max {|xi1|, |xj2|}

IMSLS_STAT, float **stat   (Output)
Address of a pointer to an internally allocated array of length 10 containing the following statistics:

 

IMSLS_STAT_USER, float stat[]   (Output)
Storage for array stat is provided by the user. See IMSLS_STAT.

Description

Function imsls_f_wilcoxon_rank_sum performs the Wilcoxon rank sum test for identical population distribution functions. The Wilcoxon test is a linear transformation of the Mann-Whitney U test. If the difference between the two populations can be attributed solely to a difference in location, then the Wilcoxon test becomes a test of equality of the population means (or medians) and is the nonparametric equivalent of the two-sample t-test. Function imsls_f_wilcoxon_rank_sum obtains ranks in the combined sample after first eliminating missing values from the data. The rank sum statistic is then computed as the sum of the ranks in the x1 sample. Three methods for handling ties are used. (A tie is counted when two observations are within fuzz of each other.) Method 1 uses the largest possible rank for tied observations in the smallest sample, while Method 2 uses the smallest possible rank for these observations. Thus, the range of possible rank sums is obtained.

Method 3 for handling tied observations between samples uses the average rank of the tied observations. Asymptotic standard normal scores are computed for the W score (based on a variance that has been adjusted for ties) when average ranks are used (see Conover 1980, p. 217), and the probability associated with the two-sided alternative is computed.

Hypothesis Tests

In each of the following tests, the first line gives the hypothesis (and its alternative) under the assumptions 1 to 3 below, while the second line gives the hypothesis when assumption 4 is also true. The rejection region is the same for both hypotheses and is given in terms of Method 3 for handling ties. Another output statistic should be used, (stat[0] or stat[3]), if another method for handling ties is desired.

 

Assumptions

1.    Arguments x1 and x2 contain random samples from their respective populations.

2.         All observations are mutually independent.

3.         The measurement scale is at least ordinal (i.e., an ordering less than, greater than, or equal to exists among the observations).

4.         If f(x) and g(y) are the distribution functions of x and y, then g(y) = f(x + c) for some constant c(i.e., the distribution of y is, at worst, a translation of the distribution of x).

The p-value is calculated using the large-sample normal approximation. This approximate calculation is only valid when the size of one or both samples is greater than 50. For smaller samples, see the exact tables for the Wilcoxon Rank Sum Test.

Examples

Example 1

The following example is taken from Conover (1980, p. 224). It involves the mixing time of two mixing machines using a total of 10 batches of a certain kind of batter, five batches for each machine. The null hypothesis is not rejected at the 5-percent level of significance. The warning error is always printed when one or more ties are detected, unless printing for warning errors is turned off. See function imsls_error_options (Chapter 15, “Utilties”).

#include <imsls.h>

void main()
{
    int    n1_observations = 5;
    int    n2_observations = 5;
    float  x1[5] = {7.3, 6.9, 7.2, 7.8, 7.2};
    float  x2[5] = {7.4, 6.8, 6.9, 6.7, 7.1};
    float  p_value;

    p_value = imsls_f_wilcoxon_rank_sum(n1_observations, x1,
                   n2_observations, x2, 0);
    printf("p-value = %11.4f\n", p_value);

}

Output

*** WARNING Error IMSLS_AT_LEAST_ONE_TIE from imsls_f_wilcoxon_rank_sum.
***         At least one tie is detected between the samples.

p-value =      0.1412

Example 2

The following example uses the same data as the previous example. Now, all the statistics are output in the array stat.

#include <imsls.h>

void main()
{
    int    n1_observations = 5;
    int    n2_observations = 5;
    float  x1[5] = {7.3, 6.9, 7.2, 7.8, 7.2};
    float  x2[5] = {7.4, 6.8, 6.9, 6.7, 7.1};
    float  *stat;
    char   *labels[10] = {"Wilcoxon W statistic ......................",
                     "2*E(W) - W ................................",
                     "p-value ...................................",
                     "Adjusted Wilcoxon statistic ...............",
                     "Adjusted 2*E(W) - W .......................",
                     "Adjusted p-value ..........................",
                     "W statistics for averaged ranks............",
                     "Standard error of W (averaged ranks) ......",
                     "Standard normal score of W (averaged ranks)",
                     "Two-sided p-value of W (averaged ranks ...."};
    imsls_f_wilcoxon_rank_sum(n1_observations, x1,
                   n2_observations, x2,
                   IMSLS_STAT, &stat,
                   0);
    imsls_f_write_matrix("statistics", 10, 1, stat,
                   IMSLS_ROW_LABELS, labels,
                   IMSLS_WRITE_FORMAT, "%7.3f",
                   0);
}

Output

*** WARNING Error IMSLS_AT_LEAST_ONE_TIE from imsls_f_wilcoxon_rank_sum.
***         At least one tie is detected between the samples.

                     statistics
Wilcoxon W statistic ......................   34.000
2*E(W) - W ................................   21.000
p-value ...................................    0.110
Adjusted Wilcoxon statistic ...............   35.000
Adjusted 2*E(W) - W .......................   20.000
Adjusted p-value ..........................    0.075
W statistics for averaged ranks............   34.500
Standard error of W (averaged ranks) ......    4.758
Standard normal score of W (averaged ranks)    1.471
Two-sided p-value of W (averaged ranks ....    0.141

Warning Errors

IMSLS_NOBSX_NOBSY_TOO_SMALL         “n1_observations” = # and “n2_observations” = #. Both sample sizes, “n1_observations” and “n2_observations”, are less than 25. Significance levels should be obtained from tabled values.

IMSLS_AT_LEAST_ONE_TIE                     At least one tie is detected between the samples.

Fatal Errors

IMSLS_ALL_X_Y_MISSING                       Each element of “x1” and/or “x2” is a missing (NaN, Not a Number) value.

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