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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_nonpar_test_kruskal (g08af)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_nonpar_test_kruskal (g08af) performs the Kruskal–Wallis one-way analysis of variance by ranks on k independent samples of possibly unequal sizes.


[h, p, ifail] = g08af(x, l, 'lx', lx, 'k', k)
[h, p, ifail] = nag_nonpar_test_kruskal(x, l, 'lx', lx, 'k', k)


The Kruskal–Wallis test investigates the differences between scores from k independent samples of unequal sizes, the ith sample containing li observations. The hypothesis under test, H0, often called the null hypothesis, is that the samples come from the same population, and this is to be tested against the alternative hypothesis H1 that they come from different populations.
The test proceeds as follows:
(a) The pooled sample of all the observations is ranked. Average ranks are assigned to tied scores.
(b) The ranks of the observations in each sample are summed, to give the rank sums Ri, for i=1,2,,k.
(c) The Kruskal–Wallis' test statistic H is computed as:
H=12NN+1 i=1kRi2li-3N+1,   where ​N=i=1kli,  
i.e., N is the total number of observations. If there are tied scores, H is corrected by dividing by:
1-t3-t N3-N  
where t is the number of tied scores in a sample and the summation is over all tied samples.
nag_nonpar_test_kruskal (g08af) returns the value of H, and also an approximation, p, to the probability of a value of at least H being observed, H0 is true. (H approximately follows a χk-12 distribution). H0 is rejected by a test of chosen size α if p<α. The approximation p is acceptable unless k=3 and l1, l2 or l35 in which case tables should be consulted (e.g., O of Siegel (1956)) or k=2 (in which case the Median test (see nag_nonpar_test_median (g08ac)) or the Mann–Whitney U test (see nag_nonpar_test_mwu (g08ah)) is more appropriate).


Moore P G, Shirley E A and Edwards D E (1972) Standard Statistical Calculations Pitman
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill


Compulsory Input Parameters

1:     xlx – double array
The elements of x must contain the observations in the k samples. The first l1 elements must contain the scores in the first sample, the next l2 those in the second sample, and so on.
2:     lk int64int32nag_int array
li must contain the number of observations li in sample i, for i=1,2,,k.
Constraint: li>0, for i=1,2,,k.

Optional Input Parameters

1:     lx int64int32nag_int scalar
Default: the dimension of the array x.
N, the total number of observations.
Constraint: lx=i=1kli.
2:     k int64int32nag_int scalar
Default: the dimension of the array l.
k, the number of samples.
Constraint: k2.

Output Parameters

1:     h – double scalar
The value of the Kruskal–Wallis test statistic, H.
2:     p – double scalar
The approximate significance, p, of the Kruskal–Wallis test statistic.
3:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
On entry,k<2.
On entry,li0 for some i, i=1,2,,k.
On entry,lxi=1kli.
On entry,all the observations were equal.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


For estimates of the accuracy of the significance p, see nag_stat_prob_chisq (g01ec). The χ2 approximation is acceptable unless k=3 and l1,l2 or l35.

Further Comments

The time taken by nag_nonpar_test_kruskal (g08af) is small, and increases with N and k.
If k=2, the Median test (see nag_nonpar_test_median (g08ac)) or the Mann–Whitney U test (see nag_nonpar_test_mwu (g08ah)) is more appropriate.


This example is taken from Moore et al. (1972). There are 5 groups of sizes 5, 8, 6, 8 and 8. The data represent the weight gain, in pounds, of pigs from five different litters under the same conditions.
function g08af_example

fprintf('g08af example results\n\n');

x = [23; 27; 26; 19; 30;
     29; 25; 33; 36; 32; 28; 30; 31;
     38; 31; 28; 35; 33; 36;
     30; 27; 28; 22; 33; 34; 34; 32;
     31; 33; 31; 28; 30; 24; 29; 30];
l = [int64(5);8;6;8;8];

fprintf('Kruskal-Wallis test\n\n');
fprintf('Data values\n\n');
fprintf('  Group    Observations');

ix = 1;
for j = 1:numel(l) 
  fprintf('\n%5d    ', j);
  fprintf('%4.0f', x(ix:ix+l(j)-1));
  ix = ix + l(j);

[h, p, ifail] = g08af(x, l);

fprintf('\n\nTest statistic        %8.3f\n', h);
fprintf('Degrees of freedom    %4d\n', numel(l)-1);
fprintf('Significance          %8.3f\n', p);

g08af example results

Kruskal-Wallis test

Data values

  Group    Observations
    1      23  27  26  19  30
    2      29  25  33  36  32  28  30  31
    3      38  31  28  35  33  36
    4      30  27  28  22  33  34  34  32
    5      31  33  31  28  30  24  29  30

Test statistic          10.537
Degrees of freedom       4
Significance             0.032

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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