void	nag_chi_sq_2_way_table (Integer nrow, Integer ncol, const Integer nobst[], Integer tdt, double expt[], double chist[], double prob, double chi, double g, double df, NagError *fail)

3

Description

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{array}{l} 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_{r .} \end{array}

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 nag_hypergeom_dist (g01blc). 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.

4

References

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

Kendall M G and Stuart A (1979) The Advanced Theory of Statistics (3 Volumes) (4th Edition) Griffin

5

Arguments

1: $nrow$ – IntegerInput: On entry: the number of rows in the contingency table, $r$ .

Constraint: $nrow \geq 2$ .
2: $ncol$ – IntegerInput: On entry: the number of columns in the contingency table, $c$ .

Constraint: $ncol \geq 2$ .
3: $nobst [nrow \times tdt]$ – const IntegerInput: On entry: the contingency table, $nobst [(i - 1) \times tdt + j - 1]$ must contain $n_{i j}$ , for $i = 1, 2, \dots, r$ and $j = 1, 2, \dots, c$ .

Constraint: $nobst [(i - 1) \times tdt + j - 1] \geq 0$ for $i = 1, 2, \dots, r$ and $j = 1, 2, \dots, c$ .
4: $tdt$ – IntegerInput: On entry: the stride separating matrix column elements in the arrays nobst, expt, chist.

Constraint: $tdt \geq ncol$ .
5: $expt [nrow \times tdt]$ – doubleOutput: On exit: the table of expected values, $expt [(i - 1) \times tdt + j - 1]$ contains $f_{i j}$ , for $i = 1, 2, \dots, r$ and $j = 1, 2, \dots, c$ .
6: $chist [nrow \times tdt]$ – doubleOutput: On exit: the table of $χ^{2}$ contributions, $chist [(i - 1) \times tdt + j - 1]$ contains $\frac{{(n_{i j} - f_{i j})}^{2}}{f_{i j}}$ , for $i = 1, 2, \dots, r$ and $j = 1, 2, \dots, c$ .
7: $prob$ – double *Output: 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.
8: $chi$ – double *Output: On exit: the Pearson $χ^{2}$ statistic.
9: $g$ – double *Output: On exit: the likelihood ratio test statistic.
10: $df$ – double *Output: On exit: the degrees of freedom for the statistics.
11: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tdt = 〈value〉$ while $ncol = 〈value〉$ . These arguments must satisfy $tdt \geq ncol$ .
NE_2D_INT_ARR_ELEM: On entry, $nobst [(〈value〉) \times tdt + 〈value〉] = 〈value〉$ . All elements of this array must be $\geq 0$ .
NE_2D_INT_ARR_ELEMS: On entry, all elements of the array nobst are 0. At least one element of this array must be $> 0$ .
NE_INT_ARG_LT: On entry, $ncol = 〈value〉$ .
Constraint: $ncol \geq 2$ .

On entry, $nrow = 〈value〉$ .
Constraint: $nrow \geq 2$
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_LOW_EXPECTED_FREQ: At least one cell has expected frequency $\leq 0.5$ . The chi-square approximation may be poor.
NE_TABLE_DEGENERATE: On entry, a 2 by 2 table has a row or column with both elements zero, i.e., the table is degenerate.

7

Accuracy

For the accuracy of the probabilities for Fisher's exact test see nag_hypergeom_dist (g01blc).

8

Parallelism and Performance

nag_chi_sq_2_way_table (g11aac) is not threaded in any implementation.

9

Further Comments

Multidimensional contingency tables can be analysed using log-linear models fitted by nag_glm_binomial (g02gbc).

10

Example

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.

NAG Library Function Document

nag_chi_sq_2_way_table (g11aac)

▸▿ 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

1

Purpose

2

Specification