naginterfaces.library.nonpar.test_​chisq

naginterfaces.library.nonpar.test_chisq(ifreq, cb, dist, par, npest, prob)

test_chisq computes the test statistic for the ${\chi }^2$ goodness-of-fit test for data with a chosen number of class intervals.

For full information please refer to the NAG Library document for g08cg

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g08/g08cgf.html

Parameters

ifreqint, array-like, shape $\left(\text{nclass}\right)$

$ifreq\left[\text{i}\right]$ must specify the frequency of the $\text{i}$th class, $O_{\text{i}}$, for $\text{i}=1,2,\dots ,k$.

cbfloat, array-like, shape $(\text{nclass}-1)$

$cb[\text{i}-1]$ must specify the upper boundary value for the $\text{i}$th class, for $\text{i}=1,2,\dots ,k-1$.

diststr, length 1

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

$dist=\text{'N'}$

The Normal distribution is used.

$dist=\text{'U'}$

The uniform distribution is used.

$dist=\text{'E'}$

The exponential distribution is used.

$dist=\text{'C'}$

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

$dist=\text{'G'}$

The gamma distribution is used.

$dist=\text{'A'}$

You must supply the class probabilities in the array $prob$.

parfloat, array-like, shape $\left(2\right)$

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

If a Normal distribution is used then $par\left[0\right]$ and $par\left[1\right]$ must contain the mean, $\mu$, and the variance, ${\sigma }^2$, respectively.

If a uniform distribution is used then $par\left[0\right]$ and $par\left[1\right]$ must contain the boundaries $a$ and $b$ respectively.

If an exponential distribution is used then $par\left[0\right]$ must contain the parameter $\lambda$. $par\left[1\right]$ is not used.

If a ${\chi }^2$-distribution is used then $par\left[0\right]$ must contain the number of degrees of freedom. $par\left[1\right]$ is not used.

If a gamma distribution is used $par\left[0\right]$ and $par\left[1\right]$ must contain the parameters $\alpha$ and $\beta$ respectively.

npestint

The number of estimated parameters of the distribution.

probfloat, array-like, shape $\left(\text{nclass}\right)$

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

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

Returns

chisqfloat

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

pfloat

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

ndfint

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

efreqfloat, ndarray, shape $\left(\text{nclass}\right)$

$efreq[\text{i}-1]$ contains the expected frequency for the $\text{i}$th class, $E_{\text{i}}$, for $\text{i}=1,2,\dots ,k$.

chisqifloat, ndarray, shape $\left(\text{nclass}\right)$

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

Raises

NagValueError

(errno $1$)

On entry, $\text{nclass}=⟨value⟩$.

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

(errno $2$)

On entry, $dist=⟨value⟩$.

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

(errno $3$)

On entry, $npest=⟨value⟩$.

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

(errno $4$)

On entry, $i=⟨value⟩$ and $ifreq[i-1]=⟨value⟩$.

Constraint: $ifreq[i-1]\ge 0$.

(errno $5$)

On entry, $i=⟨value⟩$, $cb[i-2]=⟨value⟩$ and $cb[i-1]=⟨value⟩$.

Constraint: $cb[i-2]<cb[i-1]$.

(errno $6$)

On entry, $cb\left[0\right]=⟨value⟩$.

Constraint: $cb\left[0\right]\ge 0.0$.

(errno $7$)

On entry, $par\left[0\right]=⟨value⟩$ and $par\left[1\right]=⟨value⟩$.

Constraint: for the gamma distribution, $par\left[0\right]>0.0$ and $par\left[1\right]>0.0$.

(errno $7$)

On entry, $par\left[0\right]=⟨value⟩$.

Constraint: for the ${\chi }^2$ distribution, $par\left[0\right]>0.0$.

(errno $7$)

On entry, $par\left[0\right]=⟨value⟩$.

Constraint: for the exponential distribution, $par\left[0\right]>0.0$.

(errno $7$)

On entry, $par\left[0\right]=⟨value⟩$ and $par\left[1\right]=⟨value⟩$.

Constraint: for the uniform distribution, $par\left[0\right]<par\left[1\right]$, $par\left[0\right]\le cb\left[0\right]$ and $par\left[1\right]\ge cb[\text{nclass}-2]$.

(errno $7$)

On entry, $par\left[1\right]=⟨value⟩$.

Constraint: for the Normal distribution, $par\left[1\right]>0.0$.

(errno $8$)

On entry, ${\sum }_i\left(prob\right[i\left]\right)=⟨value⟩$.

Constraint: ${\sum }_i\left(prob\right[i\left]\right)=1.0$.

(errno $8$)

On entry, $i=⟨value⟩$ and $prob[i-1]=⟨value⟩$.

Constraint: $prob>0.0$

(errno $9$)

An expected frequency equals zero, when the observed frequency was not.

Warns

NagAlgorithmicWarning

(errno $10$)

At least one class has an expected frequency less than $1$.

(errno $11$)

The solution has failed to converge whilst computing the expected values. The returned solution may be an adequate approximation.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

The ${\chi }^2$ goodness-of-fit test performed by test_chisq 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 has been grouped into $k$ classes,

$$\begin{array}{c}\begin{array}{cc}x\le c_1\text{,}& \\ c_{i-1}<x\le c_i\text{,}& i=2,3,\dots ,k-1\text{,}\\ x>c_{k-1}\text{,}& \end{array}\end{array}$$

then the ${\chi }^2$ goodness-of-fit test statistic is defined by

$$X^2=\sum _{i=1}^k\frac{{(O_i-E_i)}^2}{E_i}\text{,}$$

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

$$E_i=p_i\times n\text{,}$$

where $p_i$ is the probability that $X$ lies in the $i$th class, that is

$$\begin{array}{c}\begin{array}{cc}{p}_1=P(X\le c_1)\text{,}& \\ p_i=P(c_{i-1}<X\le c_i)\text{,}& i=2,3,\dots ,k-1\text{,}\\ p_k=P(X>c_{k-1})\text{.}& \end{array}\end{array}$$

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 $\mu$, variance ${\sigma }^2$;

uniform distribution on the interval $[a,b]$;

exponential distribution with probability density function $\left(pdf\right)=\lambda e^{-\lambda x}$;

${\chi }^2$-distribution with $f$ degrees of freedom; and

gamma distribution with $pdf=\frac{x^{\alpha -1}{e}^{-x/\beta }}{\Gamma \left(\alpha \right){\beta }^{\alpha }}$.

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

test_chisq returns the ${\chi }^2$ test statistic, $X^2$, together with its degrees of freedom and the upper tail probability from the ${\chi }^2$-distribution associated with the test statistic. Note that the use of the ${\chi }^2$-distribution as an approximation to the distribution of the test statistic improves as the expected values in each class increase.

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