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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_nonpar_test_chisq (g08cg)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_nonpar_test_chisq (g08cg) computes the test statistic for the χ2 goodness-of-fit test for data with a chosen number of class intervals.


[chisq, p, ndf, eval, chisqi, ifail] = g08cg(ifreq, cb, dist, par, npest, prob, 'nclass', nclass)
[chisq, p, ndf, eval, chisqi, ifail] = nag_nonpar_test_chisq(ifreq, cb, dist, par, npest, prob, 'nclass', nclass)


The χ2 goodness-of-fit test performed by nag_nonpar_test_chisq (g08cg) is used to test the null hypothesis that a random sample arises from a specified distribution against the alternative hypothesis that the sample does not arise from the specified distribution.
Given a sample of size n, denoted by x1,x2,,xn, drawn from a random variable X, and that the data has been grouped into k classes,
xc1, ci-1<xci, i=2,3,,k-1, x>ck-1,  
then the χ2 goodness-of-fit test statistic is defined by
X2=i=1k Oi-Ei 2Ei,  
where Oi is the observed frequency of the ith class, and Ei is the expected frequency of the ith class.
The expected frequencies are computed as
where pi is the probability that X lies in the ith class, that is
p1=PXc1, pi=Pci-1<Xci, i=2,3,,k-1, pk=PX>ck-1.  
These probabilities are either taken from a common probability distribution or are supplied by you. The available probability distributions within this function are:
You must supply the frequencies and classes. Given a set of data and classes the frequencies may be calculated using nag_stat_frequency_table (g01ae).
nag_nonpar_test_chisq (g08cg) returns the χ2 test statistic, X2, together with its degrees of freedom and the upper tail probability from the χ2-distribution associated with the test statistic. Note that the use of the χ2-distribution as an approximation to the distribution of the test statistic improves as the expected values in each class increase.


Conover W J (1980) Practical Nonparametric Statistics Wiley
Kendall M G and Stuart A (1973) The Advanced Theory of Statistics (Volume 2) (3rd Edition) Griffin
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill


Compulsory Input Parameters

1:     ifreqnclass int64int32nag_int array
ifreqi must specify the frequency of the ith class, Oi, for i=1,2,,k.
Constraint: ifreqi0, for i=1,2,,k.
2:     cbnclass-1 – double array
cbi must specify the upper boundary value for the ith class, for i=1,2,,k-1.
Constraint: cb1<cb2<<cbnclass-1. For the exponential, gamma and χ2-distributions cb10.0.
3:     dist – string (length ≥ 1)
Indicates for which distribution the test is to be carried out.
The Normal distribution is used.
The uniform distribution is used.
The exponential distribution is used.
The χ2-distribution is used.
The gamma distribution is used.
You must supply the class probabilities in the array prob.
Constraint: dist='N', 'U', 'E', 'C', 'G' or 'A'.
4:     par2 – double array
Must contain the parameters of the distribution which is being tested. If you supply the probabilities (i.e., dist='A') the array par is not referenced.
If a Normal distribution is used then par1 and par2 must contain the mean, μ, and the variance, σ2, respectively.
If a uniform distribution is used then par1 and par2 must contain the boundaries a and b respectively.
If an exponential distribution is used then par1 must contain the parameter λ. par2 is not used.
If a χ2-distribution is used then par1 must contain the number of degrees of freedom. par2 is not used.
If a gamma distribution is used par1 and par2 must contain the parameters α and β respectively.
  • if dist='N', par2>0.0;
  • if dist='U', par1<par2 and par1cb1 and par2cbnclass-1;
  • if dist='E', par1>0.0;
  • if dist='C', par1>0.0;
  • if dist='G', par1>0.0 and par2>0.0.
5:     npest int64int32nag_int scalar
The number of estimated parameters of the distribution.
Constraint: 0npest<nclass-1.
6:     probnclass – double array
If you are supplying the probability distribution (i.e., dist='A') then probi must contain the probability that X lies in the ith class.
If dist'A', prob is not referenced.
Constraint: if dist='A', i=1kprobi=1.0, probi>0.0, for i=1,2,,k.

Optional Input Parameters

1:     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: nclass2.

Output Parameters

1:     chisq – double scalar
The test statistic, X2, for the χ2 goodness-of-fit test.
2:     p – double scalar
The upper tail probability from the χ2-distribution associated with the test statistic, X2, and the number of degrees of freedom.
3:     ndf int64int32nag_int scalar
Contains nclass-1-npest, the degrees of freedom associated with the test.
4:     evalnclass – double array
evali contains the expected frequency for the ith class, Ei, for i=1,2,,k.
5:     chisqinclass – double array
chisqii contains the contribution from the ith class to the test statistic, that is, Oi-Ei 2/Ei, for i=1,2,,k.
6:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_nonpar_test_chisq (g08cg) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

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

On entry,nclass<2.
On entry,dist is invalid.
On entry,npest<0,
On entry,ifreqi<0.0 for some i, for i=1,2,,k.
On entry, the elements of cb are not in ascending order. That is, cbicbi-1 for some i, for i=2,3,,k-1.
On entry, dist='E', 'C' or 'G' and cb1<0.0. No negative class boundary values are valid for the exponential, gamma or χ2-distributions.
On entry,the values provided in par are invalid.
On entry,with dist='A', probi0.0 for some i, for i=1,2,,k,
An expected frequency is equal to zero when the observed frequency was not.
W  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 χ2-distribution is a good approximation to the distribution of the test statistic.
W  ifail=11
The solution obtained when calculating the probability for a certain class for the gamma or χ2-distribution did not converge in 600 iterations. The solution may be an adequate approximation.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


The computations are believed to be stable.

Further Comments

The time taken by nag_nonpar_test_chisq (g08cg) is dependent both on the distribution chosen and on the number of classes, k.


This example applies the χ2 goodness-of-fit test to test whether there is evidence to suggest that a sample of 100 randomly generated observations do not arise from a uniform distribution U0,1. The class intervals are calculated such that the interval 0,1 is divided into five equal classes. The frequencies for each class are calculated using nag_stat_frequency_table (g01ae).
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');

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

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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