G08AFF : NAG Library, Mark 24

3 Description

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.

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).

5 Parameters

1: X(LX) – REAL (KIND=nag_wp) arrayInput

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 – INTEGERInput

On entry:

N

, the total number of observations.

Constraint:

LX = \sum_{i = 1}^{k} L (i)

3: L(K) – INTEGER arrayInput

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 – INTEGERInput

On entry:

k

, the number of samples.

Constraint:

K \geq 2

5: W(LX) – REAL (KIND=nag_wp) arrayWorkspace

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 – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. 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 < 2

IFAIL = 2

On entry,

L (i) \leq 0

for some

i

i = 1, 2, \dots, k

IFAIL = 3

On entry,

LX \neq \sum_{i = 1}^{k} L (i)

IFAIL = 4

On entry,

all the observations were equal.

NAG Library Routine Document

G08AFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG08AFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G08AFF