G11AAF : NAG Library, Mark 24

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

G11AAF computes

χ^{2}

statistics for a two-way contingency table. For a

2 \times 2

table with a small number of observations exact probabilities are computed.

For a set of

n

observations classified by two variables, with

r

and

c

levels respectively, a two-way table of frequencies with

r

rows and

c

columns can be computed.

\begin{matrix} n_{11} & n_{12} & \dots & n_{1 c} & n_{1 .} \\ n_{21} & n_{22} & \dots & n_{2 c} & n_{2 .} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ n_{r 1} & n_{r 2} & \dots & n_{r c} & n_{r .} \\ n_{. 1} & n_{. 2} & \dots & n_{. c} & n \end{matrix}

To measure the association between the two classification variables two statistics that can be used are, the Pearson

χ^{2}

statistic,

\sum_{i = 1}^{r} \sum_{j = 1}^{c} \frac{{(n_{i j} - f_{i j})}^{2}}{f_{i j}}

, and the likelihood ratio test statistic,

2 \sum_{i = 1}^{r} \sum_{j = 1}^{c} n_{i j} \times \log (n_{i j} / f_{i j})

, where

f_{i j}

are the fitted values from the model that assumes the effects due to the classification variables are additive, i.e., there is no association. These values are the expected cell frequencies and are given by

f_{i j} = n_{i .} n_{. j} / n .

Under the hypothesis of no association between the two classification variables, both these statistics have, approximately, a

χ^{2}

-distribution with

(c - 1) (r - 1)

degrees of freedom. This distribution is arrived at under the assumption that the expected cell frequencies,

f_{i j}

, are not too small. For a discussion of this point see Everitt (1977). He concludes by saying, ‘... in the majority of cases the chi-square criterion may be used for tables with expectations in excess of

0.5

in the smallest cell’.

In the case of the

2 \times 2

table, i.e.,

c = 2

and

r = 2

, the

χ^{2}

approximation can be improved by using Yates' continuity correction factor. This decreases the absolute value of

(n_{i j} - f_{i j})

\frac{1}{2}

. For

2 \times 2

tables with a small value of

n

the exact probabilities from Fisher's test are computed. These are based on the hypergeometric distribution and are computed using G01BLF. A two tail probability is computed as

\min (1, 2 p_{u}, 2 p_{l})

, where

p_{u}

and

p_{l}

are the upper and lower one-tail probabilities from the hypergeometric distribution.

Everitt B S (1977) The Analysis of Contingency Tables Chapman and Hall

Kendall M G and Stuart A (1973) The Advanced Theory of Statistics (Volume 2) (3rd Edition) Griffin

On entry:

r

, the number of rows in the contingency table.

Constraint:

NROW \geq 2

On entry:

c

, the number of columns in the contingency table.

Constraint:

NCOL \geq 2

On entry: the contingency table

NOBS (i, j)

must contain

n_{i j}

, for

i = 1, 2, \dots, r

and

j = 1, 2, \dots, c

Constraint:

NOBS (i, j) \geq 0

, for

i = 1, 2, \dots, r

and

j = 1, 2, \dots, c

On entry: the first dimension of the arrays NOBS, EXPT and CHIST as declared in the (sub)program from which G11AAF is called.

Constraint:

LDNOBS \geq NROW

On exit: the table of expected values.

EXPT (i, j)

contains

f_{i j}

, for

i = 1, 2, \dots, r

and

j = 1, 2, \dots, c

On exit: the table of

χ^{2}

contributions.

CHIST (i, j)

contains

\frac{{(n_{i j} - f_{i j})}^{2}}{f_{i j}}

, for

i = 1, 2, \dots, r

and

j = 1, 2, \dots, c

On exit: if

c = 2

r = 2

and

n \leq 40

then PROB contains the two tail significance level for Fisher's exact test, otherwise PROB contains the significance level from the Pearson

χ^{2}

statistic.

On exit: the Pearson

χ^{2}

statistic.

On exit: the likelihood ratio test statistic.

On exit: the degrees of freedom for the statistics.

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 0

on exit, the recommended value is

- 1

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Note: G11AAF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

At least one cell has expected frequency,

f_{i j}

\leq 0.5

. The

χ^{2}

approximation may be poor.

For the accuracy of the probabilities for Fisher's exact test see G01BLF.

The routine G01AFF allows for the automatic amalgamation of rows and columns. In most circumstances this is not recommended; see Everitt (1977).

Multidimensional contingency tables can be analysed using log-linear models fitted by G02GBF.

The data below, taken from Everitt (1977), is from

141

patients with brain tumours. The row classification variable is the site of the tumour: frontal lobes, temporal lobes and other cerebral areas. The column classification variable is the type of tumour: benign, malignant and other cerebral tumours.

\begin{array}{r} 23 & 9 & 6 & 38 \\ 21 & 4 & 3 & 28 \\ 34 & 24 & 17 & 75 \\ 78 & 37 & 26 & 141 \end{array}

The data is read in and the statistics computed and printed.

On entry,	$NROW < 2$ ,
or	$NCOL < 2$ ,
or	$LDNOBS < NROW$ .

NAG Library Routine Document

G11AAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG11AAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G11AAF