NAG Library Routine Document

G08BAF

, often called the null hypothesis, is that the dispersion difference is zero, and this is to be tested against a one- or two-sided alternative hypothesis

H_{1}

(see below).

Both tests are based on the rankings of the sample members within the pooled sample formed by combining both samples. If there is some difference in dispersion, more of the extreme ranks will tend to be found in one sample than in the other.

Let the rank of

x_{i}

be denoted by

r_{i}

, for

i = 1, 2, \dots, n

(a)

Mood's test.

The test statistic

W = \sum_{i = 1}^{n_{1}} {(r_{i} - \frac{n + 1}{2})}^{2}

is found.

W

is the sum of squared deviations from the average rank in the pooled sample. For large

n

W

approaches normality, and so an approximation,

p_{w}

, to the probability of observing

W

not greater than the computed value, may be found.

G08BAF returns

W

and

p_{w}

if Mood's test is selected.

(b)

David's test.

The disadvantage of Mood's test is that it assumes that the means of the two samples are equal. If this assumption is unjustified a high value of

W

could merely reflect the difference in means. David's test reduces this effect by using the variance of the ranks of the first sample about their mean rank, rather than the overall mean rank.

The test statistic for David's test is

V = \frac{1}{n_{1} - 1} \sum_{i = 1}^{n_{1}} {(r_{i} - \bar{r})}^{2}

where

\bar{r} = \frac{\sum_{i = 1}^{n_{1}} r_{i}}{n_{1}} .

For large

n

V

approaches normality, enabling an approximate probability

p_{v}

to be computed, similarly to

p_{w}

G08BAF returns

V

and

p_{v}

if David's test is selected.

Suppose that a significance test of a chosen size

α

is to be performed (i.e.,

α

is the probability of rejecting

H_{0}

when

H_{0}

is true; typically

α

is a small quantity such as

0.05

0.01

The returned value

p

(

= p_{v}

p_{w}

) can be used to perform a significance test, against various alternative hypotheses

H_{1}

, as follows.

(i)	$H_{1}$ : dispersions are unequal. $H_{0}$ is rejected if $2 \times \min (p, 1 - p) < α$ .
(ii)	$H_{1}$ : dispersion of sample $1 >$ dispersion of sample $2$ . $H_{0}$ is rejected if $1 - p < α$ .
(iii)	$H_{1}$ : dispersion of sample $2 >$ dispersion of sample $1$ . $H_{0}$ is rejected if $p < α$ .

4 References

Cooper B E (1975) Statistics for Experimentalists Pergamon Press

5 Parameters

1: $X (N)$ – REAL (KIND=nag_wp) arrayInput

On entry: the first

n_{1}

elements of X must be set to the data values in the first sample, and the next

n_{2}

(

= N - n_{1}

) elements to the data values in the second sample.

2: $N$ – INTEGERInput

On entry: the total of the two sample sizes,

n

(

= n_{1} + n_{2}

Constraint:

N > 2

3: $N1$ – INTEGERInput

On entry: the size of the first sample,

n_{1}

Constraint:

1 < N1 < N

4: $R (N)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the ranks

r_{i}

, assigned to the data values

x_{i}

, for

i = 1, 2, \dots, n

5: $ITEST$ – INTEGERInput

On entry: the test(s) to be carried out.

$ITEST = 0$: Both Mood's and David's tests.
$ITEST = 1$: David's test only.
$ITEST = 2$: Mood's test only.

Constraint:

ITEST = 0

1

2

6: $W$ – REAL (KIND=nag_wp)Output

On exit: Mood's test statistic,

W

, if requested.

7: $V$ – REAL (KIND=nag_wp)Output

On exit: David's test statistic,

V

, if requested.

8: $PW$ – REAL (KIND=nag_wp)Output

On exit: the lower tail probability,

p_{w}

, corresponding to the value of

W

, if Mood's test was requested.

9: $PV$ – REAL (KIND=nag_wp)Output

On exit: the lower tail probability,

p_{v}

, corresponding to the value of

V

, if David's test was requested.

10: $IFAIL$ – 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,

N \leq 2

$IFAIL = 2$

On entry,	$N1 \leq 1$ ,
or	$N1 \geq N$ .

$IFAIL = 3$

On entry,	$ITEST < 0$ ,
or	$ITEST > 2$ .

$IFAIL = - 99$: 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

All computations are believed to be stable. The statistics

V

and

W

should be accurate enough for all practical uses.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by G08BAF is small, and increases with

n

10 Example

This example is taken from page 280 of Cooper (1975). The data consists of two samples of six observations each. Both Mood's and David's test statistics and significances are computed. Note that Mood's statistic is inflated owing to the difference in location of the two samples, the median ranks differing by a factor of two.

NAG Library Routine DocumentG08BAF

▸▿ 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

G08BAF