void	g08cgc (Integer nclass, const Integer ifreq[], const double cint[], Nag_Distributions dist, const double par[], Integer npest, const double prob[], double chisq, double p, Integer ndf, double eval[], double chisqi[], NagError fail)

The function may be called by the names: g08cgc, nag_nonpar_test_chisq or nag_chi_sq_goodness_of_fit_test.

3 Description

The

χ^{2}

goodness-of-fit test performed by g08cgc 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

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

, drawn from a random variable

X

, and that the data have been grouped into

k

classes,

\begin{matrix} x \leq c_{1}, \\ c_{i - 1} < x \leq c_{i}, & i = 2, 3, \dots, k - 1, \\ x > c_{k - 1}, \end{matrix}

then the

χ^{2}

goodness-of-fit test statistic is defined by:

X^{2} = \sum_{i = 1}^{k} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

where

O_{i}

is the observed frequency of the

i

th class, and

E_{i}

is the expected frequency of the

i

th class.

The expected frequencies are computed as

E_{i} = p_{i} \times n,

where

p_{i}

is the probability that

X

lies in the

i

th class, that is

\begin{matrix} p_{1} = P (X \leq c_{1}), \\ p_{i} = P (c_{i - 1} < X \leq c_{i}), & i = 2, 3, \dots, k - 1, \\ p_{k} = P (X > c_{k - 1}) . \end{matrix}

These probabilities are either taken from a common probability distribution or are supplied by you. The available probability distributions within this function are:

Normal distribution with mean $μ$ , variance $σ^{2}$ ;
uniform distribution on the interval $[a, b]$ ;
exponential distribution with probability density function $p d f = λ e^{- λ x}$ ;
$χ^{2}$ distribution with $f$ degrees of freedom; and
gamma distribution with $p d f = \frac{x^{α - 1} e^{- x / β}}{Γ (α) β^{α}}$ .

You must supply the frequencies and classes. Given a set of data and classes the frequencies may be calculated using g01aec.

g08cgc returns the

χ^{2}

test statistic,

X^{2}

, together with its degrees of freedom and the upper tail probability from the

χ^{2}

distribution associated with the test statistic. Note that the use of the

χ^{2}

distribution as an approximation to the distribution of the test statistic improves as the expected values in each class increase.

4 References

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

5 Arguments

1: $nclass$ – Integer Input

On entry: the number of classes,

k

, into which the data is divided.

Constraint:

nclass \geq 2

2: $ifreq [nclass]$ – const Integer Input

On entry:

ifreq [i - 1]

must specify the frequency of the

i

th class,

O_{i}

, for

i = 1, 2, \dots, k

Constraint:

ifreq [i - 1] \geq 0

, for

i = 1, 2, \dots, k

3: $cint [nclass - 1]$ – const double Input

On entry:

cint [i - 1]

must specify the upper boundary value for the

i

th class, for

i = 1, 2, \dots, k - 1

Constraints:

$cint [0] < cint [1] < \dots < cint [nclass - 2]$ ;
For the exponential, gamma and $χ^{2}$ distributions $cint [0] \geq 0.0$ .

4: $dist$ – Nag_Distributions Input

On entry: indicates for which distribution the test is to be carried out.

$dist = Nag_Normal$: The Normal distribution is used.
$dist = Nag_Uniform$: The uniform distribution is used.
$dist = Nag_Exponential$: The exponential distribution is used.
$dist = Nag_ChiSquare$: The $χ^{2}$ distribution is used.
$dist = Nag_Gamma$: The gamma distribution is used.
$dist = Nag_UserProb$: You must supply the class probabilities in the array prob.

Constraint:

dist = Nag_Normal

Nag_Uniform

Nag_Exponential

Nag_ChiSquare

Nag_Gamma

Nag_UserProb

5: $par [2]$ – const double Input

On entry: par must contain the arguments of the distribution which is being tested. If you supply the probabilities (i.e.,

dist = Nag_UserProb

) the array par is not referenced.

If a Normal distribution is used then

par [0]

and

par [1]

must contain the mean,

μ

, and the variance,

σ^{2}

, respectively.

If a uniform distribution is used then

par [0]

and

par [1]

must contain the boundaries

a

and

b

respectively.

If an exponential distribution is used then

par [0]

must contain the argument

λ

par [1]

is not used.

If a

χ^{2}

distribution is used then

par [0]

must contain the number of degrees of freedom.

par [1]

is not used.

If a gamma distribution is used

par [0]

and

par [1]

must contain the arguments

α

and

β

respectively.

Constraints:

if $dist = Nag_Normal$ , $par [1] > 0.0$ ;
if $dist = Nag_Uniform$ , $par [0] < par [1]$ and $par [0] \leq cint [0]$ ;
otherwise $par [1] \geq cint (nclass - 2)$ ;
if $dist = Nag_Exponential$ , $par [0] > 0.0$ ;
if $dist = Nag_ChiSquare$ , $par [0] > 0.0$ ;
if $dist = Nag_Gamma$ , $par [0]$ and $par [1] > 0.0$ .

6: $npest$ – Integer Input

On entry: the number of estimated arguments of the distribution.

Constraint:

0 \leq npest < nclass - 1

7: $prob [nclass]$ – const double Input

On entry: if you are supplying the probability distribution (i.e.,

dist = Nag_UserProb

) then

prob [i - 1]

must contain the probability that

X

lies in the

i

th class.

dist \neq Nag_UserProb

, prob is not referenced.

Constraint: if

dist = Nag_UserProb

prob [i - 1] > 0.0

and

\sum_{i = 1}^{k} prob [i - 1] = 1.0

, for

i = 1, 2, \dots, k

8: $chisq$ – double * Output

On exit: the test statistic,

X^{2}

, for the

χ^{2}

goodness-of-fit test.

9: $p$ – double * Output

On exit: the upper tail probability from the

χ^{2}

distribution associated with the test statistic,

X^{2}

, and the number of degrees of freedom.

10: $ndf$ – Integer * Output

On exit: contains

(nclass - 1 - npest)

, the degrees of freedom associated with the test.

11: $eval [nclass]$ – double Output

On exit:

eval [i - 1]

contains the expected frequency for the

i

th class,

E_{i}

, for

i = 1, 2, \dots, k

12: $chisqi [nclass]$ – double Output

On exit:

chisqi [i - 1]

contains the contribution from the

i

th class to the test statistic, that is

{(O_{i} - E_{i})}^{2} / E_{i}

, for

i = 1, 2, \dots, k

13: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ARRAY_CONS: The contents of array prob are not valid.
Constraint: Sum of $prob [i - 1] = 1$ , for $i = 1, 2, \dots, nclass$ , when $dist = Nag_UserProb$ .
NE_ARRAY_INPUT: On entry, the values provided in par are invalid.
NE_BAD_PARAM: On entry, argument dist had an illegal value.
NE_G08CG_CLASS_VAL: This is a warning that expected values for certain classes are less than $1.0$ . This implies that one cannot be confident that the $χ^{2}$ distribution is a good approximation to the distribution of the test statistic.
NE_G08CG_CONV: The solution obtained when calculating the probability for a certain class for the gamma or $χ^{2}$ distribution did not converge in 600 iterations. The solution may be an adequate approximation.
NE_G08CG_FREQ: An expected frequency is equal to zero when the observed frequency is not.
NE_INT_2: On entry, $npest = 〈value〉$ , $nclass = 〈value〉$ .
Constraint: $0 \leq npest < nclass - 1$ .
NE_INT_ARG_LT: On entry, $nclass = 〈value〉$ .
Constraint: $nclass \geq 2$ .
NE_INT_ARRAY_CONS: On entry, $ifreq [〈value〉] = 〈value〉$ .
Constraint: $ifreq [i - 1] \geq 0$ , for $i = 1, 2, \dots, nclass$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_NOT_STRICTLY_INCREASING: The sequence cint is not strictly increasing $cint [〈value〉] = 〈value〉$ , $cint [〈value〉 - 1] = 〈value〉$ .
NE_REAL_ARRAY_CONS: On entry, $prob [〈value〉] = 〈value〉$ .
Constraint: $prob [i - 1] > 0$ , for $i = 1, 2, \dots, nclass$ , when $dist = Nag_UserProb$ .
NE_REAL_ARRAY_ELEM_CONS: On entry, $cint [0] = 〈value〉$ .
Constraint: $cint [0] \geq 0.0$ , if $dist = Nag_Exponential ‖Nag_ChiSquare‖ Nag_Gamma$ .

7 Accuracy

The computations are believed to be stable.

8 Parallelism and Performance

g08cgc is not threaded in any implementation.

9 Further Comments

The time taken by g08cgc is dependent both on the distribution chosen and on the number of classes,

k

10 Example

The example program applies the

χ^{2}

goodness-of-fit test to test whether there is evidence to suggest that a sample of 100 observations generated by g05sqc do not arise from a uniform distribution

U (0, 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 g01aec.

Interfaces: FL CL AD

NAG CL Interface Introduction

G08 (Nonpar) Chapter Contents

G08 (Nonpar) Chapter Introduction

g08cg: FL CL AD

NAG CL Interfaceg08cgc (test_​chisq)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g08cgc (test_chisq)