(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 group and the summation is over all tied groups.

g08afc 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 g08acc) or the Mann–Whitney

U

test (see g08amc) is more appropriate).

4 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

5 Arguments

1: $k$ – Integer Input: On entry: the number of samples, $k$ .

Constraint: $k \geq 2$ .
2: $l [k]$ – const Integer Input: On entry: $l [i - 1]$ must contain the number of observations $l_{i}$ in sample $i$ , for $i = 1, 2, \dots, k$ .

Constraint: $l [i - 1] > 0$ , for $i = 1, 2, \dots, k$ .
3: $x [lx]$ – const double Input: On entry: the elements of x must contain the observations in the k groups. The first $l_{1}$ elements must contain the scores in the first group, the next $l_{2}$ those in the second group, and so on.
4: $lx$ – Integer Input: On entry: the total number of observations, $N$ .

Constraint: $lx = \sum_{i = 1}^{k} l [i - 1]$ .
5: $h$ – double * Output: On exit: the value of the Kruskal–Wallis test statistic, $H$ .
6: $p$ – double * Output: On exit: the approximate significance, $p$ , of the Kruskal–Wallis test statistic.
7: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_ARRAY_CONS: The contents of array l are not valid.
Constraint: $l [i - 1] > 0$ , for $i = 1, 2, \dots, k$ .
NE_INT: On entry, $lx = ⟨ value ⟩$ .
Constraint: $lx = \sum_{i = 1}^{k} l [i - 1]$ , for $i = 1, 2, \dots, k$ .
NE_INT_ARG_LT: On entry, $k = ⟨ value ⟩$ .
Constraint: $k \geq 2$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_X_IDEN: On entry, all elements of x are equal.

7 Accuracy

For estimates of the accuracy of the significance

p

, see g01ecc. The

χ^{2}

approximation is acceptable unless

k = 3

and

l_{1}, l_{2}

l_{3} \leq 5

8 Parallelism and Performance

g08afc is not threaded in any implementation.

9 Further Comments

The time taken by g08afc is small, and increases with

N

and

k

k = 2

, the Median test (see g08acc) or the Mann–Whitney

U

test (see g08amc) is more appropriate.

10 Example

This example is taken from Moore et al. 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.

g08af: FL CL CPP AD

NAG CL Interfaceg08afc (test_​kruskal)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g08afc (test_kruskal)