NAG Library Routine Document

G08AFF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

1 Purpose

G08AFF performs the Kruskal–Wallis one-way analysis of variance by ranks on

k

independent samples of possibly unequal sizes.

2 Specification

SUBROUTINE G08AFF (

X, LX, L, K, W, H, P, IFAIL)

INTEGER	LX, L(K), K, IFAIL
REAL (KIND=nag_wp)	X(LX), W(LX), H, P

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).

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 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.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction 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

Not applicable.

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.

NAG Library Routine DocumentG08AFF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Library Routine Document

G08AFF