g08af:: Nonparametric Statistics (NAG Toolbox)

The Kruskal–Wallis test investigates the differences between scores from

k

independent samples of unequal sizes, the

i

th sample containing

l_{i}

observations. The hypothesis under test,

H_{0}

, often called the null hypothesis, is that the samples come from the same population, and this is to be tested against the alternative hypothesis

H_{1}

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

R_{i}

, for

i = 1, 2, \dots, k

(c)

The Kruskal–Wallis' test statistic

H

is computed as:

H = \frac{12}{N (N + 1)} \sum_{i = 1}^{k} \frac{R_{i}^{2}}{l_{i}} - 3 (N + 1),   where ​ N = \sum_{i = 1}^{k} l_{i},

i.e.,

N

is the total number of observations. If there are tied scores,

H

is corrected by dividing by:

1 - \frac{\sum (t^{3} - t)}{N^{3} - 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,

H_{0}

is true. (

H

approximately follows a

χ_{k - 1}^{2}

distribution).

H_{0}

is rejected by a test of chosen size

α

p < α .

The approximation

p

is acceptable unless

k = 3

and

l_{1}

l_{2}

l_{3} \leq 5

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).

References

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

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Errors or warnings detected by the function:

Accuracy

For estimates of the accuracy of the significance

p

, see nag_stat_prob_chisq (g01ec). The

χ^{2}

approximation is acceptable unless

k = 3

and

l_{1}, l_{2}

l_{3} \leq 5

Further Comments

The time taken by nag_nonpar_test_kruskal (g08af) is small, and increases with

N

and

k

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.

Example

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);
end

[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

NAG Toolbox: nag_nonpar_test_kruskal (g08af)

▸▿ Contents

Purpose

Syntax

Description