NAG Library Function Document

nag_kruskal_wallis_test (g08afc)

+− 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

1 Purpose

nag_kruskal_wallis_test (g08afc) performs the Kruskal–Wallis one-way analysis of variance by ranks on

k

independent samples of possibly unequal sizes.

2 Specification

#include <nag.h>

#include <nagg08.h>

void	nag_kruskal_wallis_test (Integer k, const Integer l[], const double x[], Integer lx, double h, double p, NagError *fail)

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 group and the summation is over all tied groups.

nag_kruskal_wallis_test (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 nag_median_test (g08acc)) or the Mann–Whitney

U

test (see nag_mann_whitney (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 – IntegerInput: On entry: the number of samples, $k$ .
Constraint: $k \geq 2$ .
2: l[k] – const IntegerInput: 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 doubleInput: 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 – IntegerInput: 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 3.6 in the Essential Introduction).

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 nag_prob_chi_sq (g01ecc). 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 nag_kruskal_wallis_test (g08afc) is small, and increases with

N

and

k

k = 2

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

U

test (see nag_mann_whitney (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.

NAG Library Function Documentnag_kruskal_wallis_test (g08afc)