Chapter 6: Nonparametric Statistics


Performs a Kruskal-Wallis test for identical population medians.


#include <imsls.h>

float *imsls_f_kruskal_wallis_test (int n_groups, int ni[],
float y[], ..., 0)

The type double function is imsls_d_kruskal_wallis_test.

Required Arguments

int n_groups   (Input)
Number of groups.

int ni[]   (Input)
Array of length n_groups containing the number of responses for each of the n_groups groups.

float y[]   (Input)
Array  of length ni[0] + ... + ni[n_groups-1] that contains the responses for each of the n_groups groups. y must be sorted by group, with the ni[0] observations in group 1 coming first, the ni[1] observations in group two coming second, and so on.

Return Value

Array of length 4 containing the Kruskal-Wallis statistics.

I               stat[I]

0              Kruskal-Wallis H statistic.

1              Asymptotic probability of a larger H under the null hypothesis of identical population medians.

2              H corrected for ties.

3              Asymptotic probability of a larger H (corrected for ties) under the null hypothesis of identical populations

Synopsis with Optional Arguments

#include <imsls.h>

float *imsls_f_kruskal_wallis_test (int n_groups, int ni, float y[],
IMSLS_FUZZ, float fuzz,
IMSLS_RETURN_USER, float stat[],

Optional Arguments

IMSLS_FUZZ, float fuzz   (Input)
Constant used to determine ties in y.  If (after sorting)
|y[i] y[i + 1]| is less than or equal to fuzz, then a tie
is counted. fuzz must be nonnegative.

IMSLS_RETURN_USER, float stat[]  (Output)
User defined array for storage of Kruskal-Wallis statistics.


The function imsls_f_kruskal_wallis_test generalizes the Wilcoxon two-sample test computed by routine imsls_f_wilcoxon_rank_sum to more than two populations. It computes a test statistic for testing that the population distribution functions in each of K populations are identical. Under appropriate assumptions, this is a nonparametric analogue of the one-way analysis of variance. Since more than two samples are involved, the alternative is taken as the analogue of the usual analysis of variance alternative, namely that the populations are not identical.

The calculations proceed as follows: All observations are ranked regardless of the population to which they belong. Average ranks are used for tied observations (observations within fuzz of each other). Missing observations (observations equal to NaN, not a number) are not included in the ranking. Let Ri denote the sum of the ranks in the i-th population. The test statistic H is defined as:


where N is the total of the sample sizes, ni is the number of observations in the i-th sample, and S2 is computed as the (bias corrected) sample variance of the Ri.

The null hypothesis is rejected when stat[3] (or stat[1]) is less than the significance level of the test. If the null hypothesis is rejected, then the procedures given in Conover (1980, page 231) may be used for multiple comparisons. The routine imsls_f_kruskal_wallis_test computes asymptotic probabilities using the chi-squared distribution when the number of groups is 6 or greater, and a Beta approximation (see Wallace 1959) when the number of groups is 5 or less. Tables yielding exact probabilities in small samples may be obtained from Owen (1962).


The following example is taken from Conover (1980, page 231). The data represents the yields per acre of four different methods for raising corn. Since H = 25.5, the four methods are clearly different. The warning error is always printed when the Beta approximation is used, unless printing for warning errors is turned off.


#include <imsls.h>


int main()


       int ngroup = 4, ni[] = {9, 10, 7, 8};

       float y[] = {83., 91., 94., 89., 89., 96., 91., 92., 90., 91., 90.,

              81., 83., 84., 83., 88., 91., 89., 84., 101., 100., 91.,

              93., 96., 95., 94., 78., 82., 81., 77., 79., 81., 80.,


       float fuzz = .001, stat[4];

       char *rlabel[] = {"H (no ties)    =",

                            "Prob (no ties) =",

                            "H (ties)       =",

                            "Prob (ties)    ="};


       imsls_f_kruskal_wallis_test(ngroup, ni, y,

              IMSLS_FUZZ, fuzz,

              IMSLS_RETURN_USER, stat,



       imsls_f_write_matrix(" ", 4, 1, stat,

              IMSLS_ROW_LABELS, rlabel,




*** WARNING  ERROR  from imsls_kruskal_wallis_test.  The chi-squared degrees ***   of freedom are less than 5, so the Beta approximation is used.


H (no ties)    =     25.46

Prob (no ties) =      0.00

H (ties)       =     25.63

Prob (ties)    =      0.00

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