NAG Library Routine Document

G01AFF

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}

n_{1}

(

m_{1}, n_{1} > 1

), and let

$R_{i} = \sum_{j = 1}^{n_{1}} n_{i j}$ , the total frequency for the $i$ th row, for $i = 1, 2, \dots, m_{1}$ ,
$C_{j} = \sum_{i = 1}^{m_{1}} n_{i j}$ , the total frequency for the $j$ th column, for $j = 1, 2, \dots, n_{1}$ , and
$T = \sum_{i = 1}^{m_{1}} R_{i} = \sum_{j = 1}^{n_{1}} C_{j}$ , the total frequency.

There are two situations:

(i)

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_{i j}

, for

i = 1, 2, \dots, m_{1}

and

j = 1, 2, \dots, n_{1}

, and the test statistic,

χ^{2}

, are computed, where

r_{i j} = R_{i} C_{j} / T, i = 1, 2, \dots, m_{1}; j = 1, 2, \dots, n_{1}

and

χ^{2} = \sum_{i = 1}^{m_{1}} \sum_{j = 1}^{n_{1}} {[|r_{i j} - n_{i j}| - Y]}^{2} / r_{i j},

where

Y = \{\begin{cases} \frac{1}{2} if ​ m_{1} = n_{1} = 2 \\ 0 otherwise \end{cases}

is Yates' correction for continuity.

Under the assumption that there is no association between the two classifications,

χ^{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_{i j} < 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} \leq C_{j} \times n_{1}

. This ‘shrinking’ process is continued until

r_{i j} \geq 1

for all cells (

i, j

(ii)

m_{1} = n_{1} = 2

and

T \leq 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} \geq 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)!}, r = 0, 1, \dots, R_{1} .

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}

4 References

None.

5 Parameters

1: LDNOB – INTEGERInput

On entry: the first dimension of the array NOBS as declared in the (sub)program from which G01AFF is called.

Constraint:

LDNOB \geq M

2: LDPRED – INTEGERInput

On entry: the first dimension of the array PRED as declared in the (sub)program from which G01AFF is called.

Constraint:

LDPRED \geq M

3: M – INTEGERInput

On entry:

m + 1

, one more than the number of rows of the frequency matrix.

Constraint:

M > 2

4: N – INTEGERInput

On entry:

n + 1

, one more than the number of columns of the frequency matrix.

Constraint:

N > 2

5: NOBS(LDNOB,N) – INTEGER arrayInput/Output

On entry: the elements

NOBS (i, j)

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

, 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.

On exit: contains the following information:

$NOBS (i, j)$ , for $i = 1, 2, \dots, m_{1}$ and $j = 1, 2, \dots, n_{1}$ , contain the frequencies for the two-way classification after ‘shrinkage’ has taken place (see Section 3).
$NOBS (i, n + 1)$ , for $i = 1, 2, \dots, m_{1}$ , contain the total frequencies in the remaining rows, $R_{i}$ .
$NOBS (m + 1, j)$ , for $j = 1, 2, \dots, n_{1}$ , contain the total frequencies in the remaining columns, $C_{j}$ .
$NOBS (m + 1, n + 1)$ , contains the total frequency, $T$ .

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

Constraint:

NOBS (i, j) \geq 0

, for

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

and

j = 1, 2, \dots, N - 1

6: NUM – INTEGERInput/Output

On entry: the value assigned to NUM must determine whether automatic ‘shrinkage’ is required when any

r_{i j} < 1

, as outlined in Section 3 (i).

NUM = 1

, shrinkage is required, otherwise shrinkage is not required.

On exit: 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.

7: PRED(LDPRED,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the elements

PRED (i, j)

, where

i = 1, 2, \dots, M1

and

j = 1, 2, \dots, N1

contain the expected frequencies,

r_{i j}

corresponding to the observed frequencies

NOBS (i, j)

, 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.

8: CHIS – REAL (KIND=nag_wp)Output

On exit: the value of the test statistic,

χ^{2}

, except when Fisher's exact test for a

2 \times 2

classification is used in which case it is unspecified.

9: P( $21$ ) – REAL (KIND=nag_wp) arrayOutput

P is used only when Fisher's exact test for a

2 \times 2

classification is to be used.

On exit: the first NUM elements contain the probabilities associated with the various possible frequency tables,

P_{r}

, for

r = 0, 1, \dots, R_{1}

, the remainder are unspecified.

10: NPOS – INTEGEROutput

NPOS is used only when Fisher's exact test for a

2 \times 2

classification is to be used.

On exit:

P (NPOS)

holds the probability associated with the given table of frequencies.

11: NDF – INTEGEROutput

On exit: 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

12: M1 – INTEGEROutput

On exit: the number of rows of the two-way classification, after any ‘shrinkage’,

m_{1}

13: N1 – INTEGEROutput

On exit: the number of columns of the two-way classification, after any ‘shrinkage’,

n_{1}

14: IFAIL – INTEGERInput/Output

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, if you are not familiar with this parameter, the recommended value is

0

. 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).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

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

Errors or warnings detected by the routine:

$IFAIL = 1$: The number of rows or columns of NOBS is less than $2$ , possibly after shrinkage.

$IFAIL = 2$: At least one frequency is negative, or all frequencies are zero.

$IFAIL = 4$

On entry,	$LDPRED < M$ ,
or	$LDNOB < M$ .

7 Accuracy

The method used is believed to be stable.

8 Further Comments

The time taken by G01AFF will increase with M and N, except when Fisher's exact test is to be used, in which case it increases with size of the marginal and total frequencies.

If, on exit,

NUM > 0

, or alternatively NDF is

1

and

NOBS (M, N) \leq 40

, the probabilities for use in Fisher's exact test for a

2 \times 2

classification will be calculated, and not the test statistic with approximately a chi-square distribution.

9 Example

In the example program, NPROB determines the number of two-way classifications to be analysed. For each classification the frequencies are read, G01AFF called, and information given on how much ‘shrinkage’ has taken place. If Fisher's exact test is to be used, the given frequencies and the array of probabilities associated with the possible frequency tables are printed. Otherwise, if the chi-square test is to be used, the given and expected frequencies, and the test statistic with its degrees of freedom are printed. In the example, there is one

2 \times 3

classification, with shrinkage not requested.

NAG Library Routine DocumentG01AFF

+− 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

G01AFF