naginterfaces.library.nonpar.test_​mooddavid

naginterfaces.library.nonpar.test_mooddavid(x, n1, itest)

test_mooddavid performs Mood’s and David’s tests for dispersion differences between two independent samples of possibly unequal size.

For full information please refer to the NAG Library document for g08ba

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g08/g08baf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

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.

n1int

The size of the first sample, $n_1$.

itestint

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.

Returns

rfloat, ndarray, shape $\left(n\right)$

The ranks $r_{\text{i}}$, assigned to the data values $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

wfloat

Mood’s test statistic, $W$, if requested.

vfloat

David’s test statistic, $V$, if requested.

pwfloat

The lower tail probability, $p_w$, corresponding to the value of $W$, if Mood’s test was requested.

pvfloat

The lower tail probability, $p_v$, corresponding to the value of $V$, if David’s test was requested.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>2$.

(errno $2$)

On entry, $n1=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $1<n1<n$.

(errno $3$)

On entry, $itest=⟨value⟩$.

Constraint: $itest=0$, $1$ or $2$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

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

$$x_{n_1+1},x_{n_1+2},\dots ,x_n\text{,}n=n_1+n_2\text{.}$$

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 $x_{\text{i}}$ be denoted by $r_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

1. 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.

   test_mooddavid returns $W$ and $p_w$ if Mood’s test is selected.

2. 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

   $$V=\frac{1}{n_1-1}\sum _{i=1}^{n_1}{(r_i-\overline{r})}^2$$

   where

   $$\overline{r}=\frac{{\sum }_{i=1}^{n_1}{r}_i}{n_1}\text{.}$$

   For large $n$, $V$ approaches normality, enabling an approximate probability $p_v$ to be computed, similarly to $p_w$.

   test_mooddavid returns $V$ and $p_v$ if David’s test is selected.

Suppose that a significance test of a chosen size $\alpha$ is to be performed (i.e., $\alpha$ is the probability of rejecting $H_0$ when $H_0$ is true; typically $\alpha$ is a small quantity such as $0.05$ or $0.01$).

The returned value $p$ ($=p_v$ or $p_w$) can be used to perform a significance test, against various alternative hypotheses $H_1$, as follows.

1. $H_1$: dispersions are unequal. $H_0$ is rejected if $2\times min(p,1-p)<\alpha$.

2. $H_1$: dispersion of sample $1>$ dispersion of sample $2$. $H_0$ is rejected if $1-p<\alpha$.

3. $H_1$: dispersion of sample $2>$ dispersion of sample $1$. $H_0$ is rejected if $p<\alpha$.

References

Cooper, B E, 1975, Statistics for Experimentalists, Pergamon Press