nag_nonpar_test_mooddavid (g08ba) performs Mood's and David's tests for dispersion differences between two independent samples of possibly unequal size.

Syntax

[r, w, v, pw, pv, ifail] = g08ba(x, n1, itest, 'n', n)

[r, w, v, pw, pv, ifail] = nag_nonpar_test_mooddavid(x, n1, itest, 'n', n)

Description

Mood's and David's tests investigate the difference between the dispersions of two independent samples of sizes

n_{1}

and

n_{2}

, denoted by

x_{1}, x_{2}, \dots, x_{n_{1}}

and

x_{n_{1} + 1}, x_{n_{1} + 2}, \dots, x_{n}, n = n_{1} + n_{2} .

The hypothesis under test,

H_{0}

, 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.

nag_nonpar_test_mooddavid (g08ba) 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}

nag_nonpar_test_mooddavid (g08ba) 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 $p < α$ .

References

Cooper B E (1975) Statistics for Experimentalists Pergamon Press

Parameters

Compulsory Input Parameters

1: $x (n)$ – double array

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: $n1$ – int64int32nag_int scalar

The size of the first sample,

n_{1}

Constraint:

1 < n1 < n

3: $itest$ – int64int32nag_int scalar

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

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array x.
The total of the two sample sizes, $n$ ( $= n_{1} + n_{2}$ ).

Constraint: $n > 2$ .

Output Parameters

1: $r (n)$ – double array: The ranks $r_{i}$ , assigned to the data values $x_{i}$ , for $i = 1, 2, \dots, n$ .
2: $w$ – double scalar: Mood's test statistic, $W$ , if requested.
3: $v$ – double scalar: David's test statistic, $V$ , if requested.
4: $pw$ – double scalar: The lower tail probability, $p_{w}$ , corresponding to the value of $W$ , if Mood's test was requested.
5: $pv$ – double scalar: The lower tail probability, $p_{v}$ , corresponding to the value of $V$ , if David's test was requested.
6: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

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.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

All computations are believed to be stable. The statistics

V

and

W

should be accurate enough for all practical uses.

Further Comments

The time taken by nag_nonpar_test_mooddavid (g08ba) is small, and increases with

n

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.

Open in the MATLAB editor: g08ba_example

function g08ba_example


fprintf('g08ba example results\n\n');

x = [6;  9; 12;  4; 10; 11;  8;  1;  3;  7;  2;  5];

n1 = int64(6);
fprintf('Mood''s test and David''s test\n\n');
fprintf('Data values\n\n');
fprintf('Group 1\n');
fprintf('%4.0f',x(1:n1));
fprintf('\n\nGroup 2\n');
fprintf('%4.0f',x(n1+1:end));
fprintf('\n\n');

itest = int64(0);
[r, w, v, pw, pv, ifail] = g08ba( ...
                                  x, n1, itest);

fprintf('    Mood''s measure  = %8.3f   Significance = %8.4f\n', w, pw);
fprintf('    David''s measure = %8.3f   Significance = %8.4f\n', v, pv);

g08ba example results

Mood's test and David's test

Data values

Group 1
   6   9  12   4  10  11

Group 2
   8   1   3   7   2   5

    Mood's measure  =   75.500   Significance =   0.5830
    David's measure =    9.467   Significance =   0.1986

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox