naginterfaces.library.stat.contingency_​table

naginterfaces.library.stat.contingency_table(nobs, num=0)

contingency_table performs the analysis of a two-way $r\times c$ contingency table or classification. If $r=c=2$, and the total number of objects classified is $40$ or fewer, then the probabilities for Fisher’s exact test are computed. Otherwise, a test statistic is computed (with Yates’ correction when $r=c=2$), which under the assumption of no association between the classifications has approximately a chi-square distribution with $(r-1)\times (c-1)$ degrees of freedom.

For full information please refer to the NAG Library document for g01af

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01aff.html

Parameters

nobsint, array-like, shape $(m,n)$

The elements $nobs[\text{i}-1,\text{j}-1]$, for $\text{j}=1,2,\dots ,n$, for $\text{i}=1,2,\dots ,m$, must contain the frequencies for the two-way classification. The $(m+1)$th row and the $(n+1)$th column of $nobs$ need not be set.

numint, optional

The value assigned to $num$ must determine whether automatic ‘shrinkage’ is required when any $r_{ij}<1$, as outlined in Notes(1).

If $num=1$, shrinkage is required, otherwise shrinkage is not required.

Returns

nobsint, ndarray, shape $(m,n)$

Contains the following information:

$nobs[\text{i}-1,\text{j}-1]$, for $\text{j}=1,2,\dots ,n_1$, for $\text{i}=1,2,\dots ,m_1$, contain the frequencies for the two-way classification after ‘shrinkage’ has taken place (see Notes).

$nobs[\text{i}-1,n]$, for $\text{i}=1,2,\dots ,m_1$, contain the total frequencies in the remaining rows, $R_i$.

$nobs[m,\text{j}-1]$, for $\text{j}=1,2,\dots ,n_1$, contain the total frequencies in the remaining columns, $C_j$.

$nobs[m,n]$, contains the total frequency, $T$.

If any ‘shrinkage’ has occurred, all other cells contain no useful information.

numint

When Fisher’s exact test for a $2\times 2$ classification is used then $num$ contains the number of elements used in the array $p$, otherwise $num$ is set to zero.

predfloat, ndarray, shape $(m,n)$

The elements $pred[i-1,j-1]$, where $i=1,2,\dots ,m1$ and $j=1,2,\dots ,n1$ contain the expected frequencies, $r_{ij}$ corresponding to the observed frequencies $nobs[i-1,j-1]$, except in the case when Fisher’s exact test for a $2\times 2$ classification is to be used, when $pred$ is not used. No other elements are utilized.

chisfloat

The value of the test statistic, ${\chi }^2$, except when Fisher’s exact test for a $2\times 2$ classification is used in which case it is unspecified.

pfloat, ndarray, shape $\left(21\right)$

The first $num$ elements contain the probabilities associated with the various possible frequency tables, $P_{\text{r}}$, for $\text{r}=0,1,\dots ,R_1$, the remainder are unspecified.

nposint

$p[npos-1]$ holds the probability associated with the given table of frequencies.

ndfint

The value of $ndf$ gives the number of degrees of freedom for the chi-square distribution, $(m_1-1)\times (n_1-1)$; when Fisher’s exact test is used $ndf=1$.

m1int

The number of rows of the two-way classification, after any ‘shrinkage’, $m_1$.

n1int

The number of columns of the two-way classification, after any ‘shrinkage’, $n_1$.

Raises

NagValueError

(errno $1$)

The number of rows or columns of $nobs$ is less than $2$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>2$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m>2$.

(errno $2$)

At least one frequency is negative, or all frequencies are zero.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

The data consist of the frequencies for the two-way classification, denoted by $n_{\text{i}\text{j}}$, for $\text{j}=1,2,\dots ,n$, for $\text{i}=1,2,\dots ,m$ with $m,n>1$.

A check is made to see whether any row or column of the matrix of frequencies consists entirely of zeros, and if so, the matrix of frequencies is reduced by omitting that row or column. Suppose the final size of the matrix is $m_1\times n_1$ ($m_1,n_1>1$), and let

$R_{\text{i}}={\sum }_{j=1}^{n_1}{n}_{\text{i}j}$, the total frequency for the $\text{i}$th row, for $\text{i}=1,2,\dots ,m_1$,

$C_{\text{j}}={\sum }_{i=1}^{m_1}{n}_{i\text{j}}$, the total frequency for the $\text{j}$th column, for $\text{j}=1,2,\dots ,n_1$, and

$T={\sum }_{i=1}^{m_1}{R}_i={\sum }_{j=1}^{n_1}{C}_j$, the total frequency.

There are two situations:

1. If $m_1>2$ and/or $n_1>2$, or $m_1=n_1=2$ and $T>40$, then the matrix of expected frequencies, denoted by $r_{ij}$, for $i=1,2,\dots ,m_1$ and $j=1,2,\dots ,n_1$, and the test statistic, ${\chi }^2$, are computed, where

   $$r_{ij}=R_i{C}_j/T\text{,}i=1,2,\dots ,m_1\text{;}j=1,2,\dots ,n_1$$

   and

   $${\chi }^2=\sum _{i=1}^{m_1}\sum _{j=1}^{n_1}{[|r_{ij}-n_{ij}|-Y]}^2/r_{ij}\text{,}$$

   where

   $$\begin{array}{c}Y=\{\begin{array}{c}\frac{1}{2}\text{if}{m}_1=n_1=2\\ 0\text{otherwise}\end{array}\end{array}$$

   is Yates’ correction for continuity.

   Under the assumption that there is no association between the two classifications, ${\chi }^2$ will have approximately a chi-square distribution with $(m_1-1)\times (n_1-1)$ degrees of freedom.

   An option exists which allows for further ‘shrinkage’ of the matrix of frequencies in the case where $r_{ij}<1$ for the ($i,j$)th cell. If this is the case, then row $i$ or column $j$ will be combined with the adjacent row or column with smaller total. Row $i$ is selected for combination if $R_i\times m_1\le C_j\times n_1$. This ‘shrinking’ process is continued until $r_{ij}\ge 1$ for all cells ($i,j$).

2. If $m_1=n_1=2$ and $T\le 40$, the probabilities to enable Fisher’s exact test to be made are computed.

   The matrix of frequencies may be rearranged so that $R_1$ is the smallest marginal (i.e., column and row) total, and $C_2\ge C_1$. Under the assumption of no association between the classifications, the probability of obtaining $r$ entries in cell $(1,1)$ is computed where

   $$P_{r+1}=\frac{R_1!R_2!C_1!C_2!}{T!r!(R_1-r)!(C_1-r)!(T-C_1-R_1+r)!}\text{,}r=0,1,\dots ,R_1\text{.}$$

   The probability of obtaining the table of given frequencies is returned. A test of the assumption against some alternative may then be made by summing the relevant values of $P_r$.