NAG Toolbox: nag_nonpar_test_chisq (g08cg)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{ifreq}\left(\mathbf{nclass}\right)$ – int64int32nag_int array

$\mathbf{ifreq}\left(\mathit{i}\right)$ must specify the frequency of the $\mathit{i}$th class, $O_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$.

Constraint: $\mathbf{ifreq}\left(\mathit{i}\right)\ge 0$, for $\mathit{i}=1,2,\dots ,k$.

2:     $\mathrm{cb}\left(\mathbf{nclass}-1\right)$ – double array

$\mathbf{cb}\left(\mathit{i}\right)$ must specify the upper boundary value for the $\mathit{i}$th class, for $\mathit{i}=1,2,\dots ,k-1$.

Constraint: $\mathbf{cb}\left(1\right)<\mathbf{cb}\left(2\right)<\cdots <\mathbf{cb}\left(\mathbf{nclass}-1\right)$. For the exponential, gamma and ${\chi }^2$-distributions $\mathbf{cb}\left(1\right)\ge 0.0$.

3:     $\mathrm{dist}$ – string (length ≥ 1)

Indicates for which distribution the test is to be carried out.

$\mathbf{dist}=\text{'N'}$

The Normal distribution is used.

$\mathbf{dist}=\text{'U'}$

The uniform distribution is used.

$\mathbf{dist}=\text{'E'}$

The exponential distribution is used.

$\mathbf{dist}=\text{'C'}$

The ${\chi }^2$-distribution is used.

$\mathbf{dist}=\text{'G'}$

The gamma distribution is used.

$\mathbf{dist}=\text{'A'}$

You must supply the class probabilities in the array prob.

Constraint: $\mathbf{dist}=\text{'N'}$, $\text{'U'}$, $\text{'E'}$, $\text{'C'}$, $\text{'G'}$ or $\text{'A'}$.

4:     $\mathrm{par}\left(2\right)$ – double array

Must contain the parameters of the distribution which is being tested. If you supply the probabilities (i.e., $\mathbf{dist}=\text{'A'}$) the array par is not referenced.

If a Normal distribution is used then $\mathbf{par}\left(1\right)$ and $\mathbf{par}\left(2\right)$ must contain the mean, $\mu$, and the variance, ${\sigma }^2$, respectively.

If a uniform distribution is used then $\mathbf{par}\left(1\right)$ and $\mathbf{par}\left(2\right)$ must contain the boundaries $a$ and $b$ respectively.

If an exponential distribution is used then $\mathbf{par}\left(1\right)$ must contain the parameter $\lambda$. $\mathbf{par}\left(2\right)$ is not used.

If a ${\chi }^2$-distribution is used then $\mathbf{par}\left(1\right)$ must contain the number of degrees of freedom. $\mathbf{par}\left(2\right)$ is not used.

If a gamma distribution is used $\mathbf{par}\left(1\right)$ and $\mathbf{par}\left(2\right)$ must contain the parameters $\alpha$ and $\beta$ respectively.

Constraints:

• if $\mathbf{dist}=\text{'N'}$, $\mathbf{par}\left(2\right)>0.0$;
• if $\mathbf{dist}=\text{'U'}$, $\mathbf{par}\left(1\right)<\mathbf{par}\left(2\right)$ and $\mathbf{par}\left(1\right)\le \mathbf{cb}\left(1\right)$ and $\mathbf{par}\left(2\right)\ge \mathbf{cb}\left(\mathbf{nclass}-1\right)$;
• if $\mathbf{dist}=\text{'E'}$, $\mathbf{par}\left(1\right)>0.0$;
• if $\mathbf{dist}=\text{'C'}$, $\mathbf{par}\left(1\right)>0.0$;
• if $\mathbf{dist}=\text{'G'}$, $\mathbf{par}\left(1\right)>0.0$ and $\mathbf{par}\left(2\right)>0.0$.

5:     $\mathrm{npest}$ – int64int32nag_int scalar

The number of estimated parameters of the distribution.

Constraint: $0\le \mathbf{npest}<\mathbf{nclass}-1$.

6:     $\mathrm{prob}\left(\mathbf{nclass}\right)$ – double array

If you are supplying the probability distribution (i.e., $\mathbf{dist}=\text{'A'}$) then $\mathbf{prob}\left(i\right)$ must contain the probability that $X$ lies in the $i$th class.

If $\mathbf{dist}\ne \text{'A'}$, prob is not referenced.

Constraint: if $\mathbf{dist}=\text{'A'}$, ${\displaystyle \sum _{i=1}^k}\mathbf{prob}\left(i\right)=1.0$, $\mathbf{prob}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,k$.

Optional Input Parameters

1:     $\mathrm{nclass}$ – int64int32nag_int scalar

Default: the dimension of the arrays ifreq, prob. (An error is raised if these dimensions are not equal.)

$k$, the number of classes into which the data is divided.

Constraint: $\mathbf{nclass}\ge 2$.

Output Parameters

1:     $\mathrm{chisq}$ – double scalar

The test statistic, $X^2$, for the ${\chi }^2$ goodness-of-fit test.

2:     $\mathrm{p}$ – double scalar

The upper tail probability from the ${\chi }^2$-distribution associated with the test statistic, $X^2$, and the number of degrees of freedom.

3:     $\mathrm{ndf}$ – int64int32nag_int scalar

Contains $\left(\mathbf{nclass}-1-\mathbf{npest}\right)$, the degrees of freedom associated with the test.

4:     $\mathrm{eval}\left(\mathbf{nclass}\right)$ – double array

$\mathbf{eval}\left(\mathit{i}\right)$ contains the expected frequency for the $\mathit{i}$th class, $E_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$.

5:     $\mathrm{chisqi}\left(\mathbf{nclass}\right)$ – double array

$\mathbf{chisqi}\left(\mathit{i}\right)$ contains the contribution from the $\mathit{i}$th class to the test statistic, that is, ${\left(O_{\mathit{i}}-E_{\mathit{i}}\right)}^2/E_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$.

6:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{ifail}=1$

On entry,

$\mathbf{nclass}<2$.

   $\mathbf{ifail}=2$

On entry,

dist is invalid.

   $\mathbf{ifail}=3$

On entry,	$\mathbf{npest}<0$,
or	$\mathbf{npest}\ge \mathbf{nclass}-1$.

   $\mathbf{ifail}=4$

On entry,

$\mathbf{ifreq}\left(\mathit{i}\right)<0.0$ for some $\mathit{i}$, for $\mathit{i}=1,2,\dots ,k$.

   $\mathbf{ifail}=5$

On entry, the elements of cb are not in ascending order. That is, $\mathbf{cb}\left(\mathit{i}\right)\le \mathbf{cb}\left(\mathit{i}-1\right)$ for some $\mathit{i}$, for $\mathit{i}=2,3,\dots ,k-1$.

   $\mathbf{ifail}=6$

On entry, $\mathbf{dist}=\text{'E'}$, $\text{'C'}$ or $\text{'G'}$ and $\mathbf{cb}\left(1\right)<0.0$. No negative class boundary values are valid for the exponential, gamma or ${\chi }^2$-distributions.

   $\mathbf{ifail}=7$

On entry,

the values provided in par are invalid.

   $\mathbf{ifail}=8$

On entry,	with $\mathbf{dist}=\text{'A'}$, $\mathbf{prob}\left(i\right)\le 0.0$ for some $i$, for $i=1,2,\dots ,k$,
or	${\displaystyle \sum _{i=1}^k}\mathbf{prob}\left(i\right)\ne 1.0$.

   $\mathbf{ifail}=9$

An expected frequency is equal to zero when the observed frequency was not.

W  $\mathbf{ifail}=10$

This is a warning that expected values for certain classes are less than $1.0$. This implies that we cannot be confident that the ${\chi }^2$-distribution is a good approximation to the distribution of the test statistic.

W  $\mathbf{ifail}=11$

The solution obtained when calculating the probability for a certain class for the gamma or ${\chi }^2$-distribution did not converge in $600$ iterations. The solution may be an adequate approximation.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: g08cg_example

function g08cg_example


fprintf('g08cg example results\n\n');

x = [ 0.59 0.23 0.76 0.96 0.20 0.91 0.29 0.22 0.36 0.81 ...
      0.91 0.80 0.17 0.82 0.07 0.74 0.15 0.91 0.26 0.98 ...
      0.59 0.34 0.28 0.95 0.33 0.42 0.72 0.35 0.86 0.22 ...
      0.15 0.39 0.32 0.82 0.13 0.48 0.46 0.74 0.99 0.26 ...
      0.04 0.21 0.04 0.24 0.56 0.36 0.48 0.53 1.00 0.58 ...
      0.50 0.41 0.03 0.38 0.89 0.40 0.66 0.79 0.34 0.94 ...
      0.49 0.12 0.24 0.05 1.00 0.29 0.67 0.29 0.75 0.81 ...
      0.45 0.21 0.51 0.68 0.78 0.20 0.23 0.57 0.25 0.48 ...
      0.96 0.33 0.48 0.55 0.04 0.48 0.42 0.11 0.38 0.73 ...
      0.91 0.45 0.59 0.97 0.27 0.27 0.25 0.99 0.99 0.80];

cb     = [0.2;     0.4;     0.6;     0.8;    1.0 ];
nclass = int64(5);

% Produce frequency table
[~, ifreq, ~, ~, ifail] = ...
  g01ae( ...
         nclass, x, 'cb', cb);

% Test parameters
dist   = 'Uniform';
npest  = int64(0);
par    = [0;  1];
prob   = zeros(nclass,1);

% Perform Chi^2 test
[chisq, p, ndf, eval, chisqi, ifail] = ...
  g08cg( ...
         ifreq, cb, dist, par, npest, prob, 'nclass', nclass);

fprintf('Chi-squared test statistic   = %10.4f\n', chisq);
fprintf('Degrees of freedom.          = %5d\n', ndf);
fprintf('Significance level           = %10.4f\n\n', p);
fprintf('The contributions to the test statistic are :-\n');
disp(chisqi');

g08cg example results

Chi-squared test statistic   =    14.2000
Degrees of freedom.          =     4
Significance level           =     0.0067

The contributions to the test statistic are :-
    3.2000    6.0500    0.4500    4.0500    0.4500