(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.

g08aff 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 g08acf) or the Mann–Whitney

U

test (see g08ahf) 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: $x (lx)$ – Real (Kind=nag_wp) array Input: On entry: the elements of x must contain the observations in the k samples. The first $l_{1}$ elements must contain the scores in the first sample, the next $l_{2}$ those in the second sample, and so on.
2: $lx$ – Integer Input: On entry: $N$ , the total number of observations.

Constraint: $lx = \sum_{i = 1}^{k} l (i)$ .
3: $l (k)$ – Integer array Input: On entry: $l (i)$ must contain the number of observations $l_{i}$ in sample $i$ , for $i = 1, 2, \dots, k$ .

Constraint: $l (i) > 0$ , for $i = 1, 2, \dots, k$ .
4: $k$ – Integer Input: On entry: $k$ , the number of samples.

Constraint: $k \geq 2$ .
5: $w (lx)$ – Real (Kind=nag_wp) array Workspace
6: $h$ – Real (Kind=nag_wp) Output: On exit: the value of the Kruskal–Wallis test statistic, $H$ .
7: $p$ – Real (Kind=nag_wp) Output: On exit: the approximate significance, $p$ , of the Kruskal–Wallis test statistic.
8: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $k = ⟨ value ⟩$ .
Constraint: $k \geq 2$ .

$ifail = 2$: On entry, $i = ⟨ value ⟩$ and $l (i) = ⟨ value ⟩$ .
Constraint: $l (i) > 0$ .

$ifail = 3$: On entry, $\sum_{i} l (i) = ⟨ value ⟩$ and $lx = ⟨ value ⟩$ .
Constraint: $\sum_{i} l (i) = lx$ .

$ifail = 4$: On entry, all the observations were equal.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

For estimates of the accuracy of the significance

p

, see g01ecf. The

χ^{2}

approximation is acceptable unless

k = 3

and

l_{1}, l_{2}

l_{3} \leq 5

8 Parallelism and Performance

g08aff is not threaded in any implementation.

9 Further Comments

The time taken by g08aff is small, and increases with

N

and

k

k = 2

, the Median test (see g08acf) or the Mann–Whitney

U

test (see g08ahf) is more appropriate.

10 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.

g08af: FL CL CPP AD

NAG FL Interfaceg08aff (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 FL Interface
g08aff (test_kruskal)