# NAG FL Interfaceg08baf (test_​mooddavid)

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## 1Purpose

g08baf performs Mood's and David's tests for dispersion differences between two independent samples of possibly unequal size.

## 2Specification

Fortran Interface
 Subroutine g08baf ( x, n, n1, r, w, v, pw, pv,
 Integer, Intent (In) :: n, n1, itest Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: r(n), w, v, pw, pv
#include <nag.h>
 void g08baf_ (const double x[], const Integer *n, const Integer *n1, double r[], const Integer *itest, double *w, double *v, double *pw, double *pv, Integer *ifail)
The routine may be called by the names g08baf or nagf_nonpar_test_mooddavid.

## 3Description

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
 $x1,x2,…,xn1$
and
 $xn1+ 1,xn1+ 2,…,xn, n=n1+n2.$
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 ${x}_{\mathit{i}}$ be denoted by ${r}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
1. (a)Mood's test.
The test statistic $W=\sum _{i=1}^{{n}_{1}}{\left({r}_{i}-\frac{n+1}{2}\right)}^{2}$ 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.
2. (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
 $V=1n1-1 ∑i=1n1 (ri-r¯) 2$
where
 $r¯=∑i= 1n1rin1.$
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 $\alpha$ 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. (i)${H}_{1}$: dispersions are unequal. ${H}_{0}$ is rejected if $2×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(p,1-p\right)<\alpha$.
2. (ii)${H}_{1}$: dispersion of sample $1>\text{}$ dispersion of sample $2$. ${H}_{0}$ is rejected if $1-p<\alpha$.
3. (iii)${H}_{1}$: dispersion of sample $2>\text{}$ dispersion of sample $1$. ${H}_{0}$ is rejected if $p<\alpha$.
Cooper B E (1975) Statistics for Experimentalists Pergamon Press

## 5Arguments

1: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the first ${n}_{1}$ elements of x must be set to the data values in the first sample, and the next ${n}_{2}$ ($\text{}={\mathbf{n}}-{n}_{1}$) elements to the data values in the second sample.
2: $\mathbf{n}$Integer Input
On entry: the total of the two sample sizes, $n$ ($\text{}={n}_{1}+{n}_{2}$).
Constraint: ${\mathbf{n}}>2$.
3: $\mathbf{n1}$Integer Input
On entry: the size of the first sample, ${n}_{1}$.
Constraint: $1<{\mathbf{n1}}<{\mathbf{n}}$.
4: $\mathbf{r}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: the ranks ${r}_{\mathit{i}}$, assigned to the data values ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
5: $\mathbf{itest}$Integer Input
On entry: the test(s) to be carried out.
${\mathbf{itest}}=0$
Both Mood's and David's tests.
${\mathbf{itest}}=1$
David's test only.
${\mathbf{itest}}=2$
Mood's test only.
Constraint: ${\mathbf{itest}}=0$, $1$ or $2$.
6: $\mathbf{w}$Real (Kind=nag_wp) Output
On exit: Mood's test statistic, $W$, if requested.
7: $\mathbf{v}$Real (Kind=nag_wp) Output
On exit: David's test statistic, $V$, if requested.
8: $\mathbf{pw}$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: $\mathbf{pv}$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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n1}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1<{\mathbf{n1}}<{\mathbf{n}}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{itest}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{itest}}=0$, $1$ or $2$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

All computations are believed to be stable. The statistics $V$ and $W$ should be accurate enough for all practical uses.

## 8Parallelism and Performance

g08baf is not threaded in any implementation.

The time taken by g08baf is small, and increases with $n$.

## 10Example

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.

### 10.1Program Text

Program Text (g08bafe.f90)

### 10.2Program Data

Program Data (g08bafe.d)

### 10.3Program Results

Program Results (g08bafe.r)