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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_stat_contingency_table (g01af)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_stat_contingency_table (g01af) 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.

Syntax

[nobs, num, pred, chis, p, npos, ndf, m1, n1, ifail] = g01af(nobs, 'm', m, 'n', n, 'num', num)

[nobs, num, pred, chis, p, npos, ndf, m1, n1, ifail] = nag_stat_contingency_table(nobs, 'm', m, 'n', n, 'num', num)

Note: the interface to this routine has changed since earlier releases of the toolbox:

At Mark 23:	num was made optional (default 0)
At Mark 22:	m was made optional

Description

The data consist of the frequencies for the two-way classification, denoted by

n_{i j}

, for

i = 1, 2, \dots, m

and

j = 1, 2, \dots, n

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}

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}

References

None.

Parameters

Compulsory Input Parameters

1: $nobs (ldnob, n)$ – int64int32nag_int array: ldnob, the first dimension of the array, must satisfy the constraint $ldnob \geq m$ .
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.

Constraint: $nobs (i, j) \geq 0$ , for $i = 1, 2, \dots, m - 1$ and $j = 1, 2, \dots, n - 1$ .

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the array nobs.
$m + 1$ , one more than the number of rows of the frequency matrix.

Constraint: $m > 2$ .
2: $n$ – int64int32nag_int scalar: Default: the second dimension of the array nobs.
$n + 1$ , one more than the number of columns of the frequency matrix.

Constraint: $n > 2$ .
3: $num$ – int64int32nag_int scalar: Default: $0$
The value assigned to num must determine whether automatic ‘shrinkage’ is required when any $r_{i j} < 1$ , as outlined in Description (i).
If $num = 1$ , shrinkage is required, otherwise shrinkage is not required.

Output Parameters

1: $nobs (ldnob, n)$ – int64int32nag_int array

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 Description).
$nobs (i, n + 1)$ , 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.

2: $num$ – int64int32nag_int scalar

Default:

0

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.

3: $pred (ldpred, n)$ – double array

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.

4: $chis$ – double scalar

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.

5: $p (21)$ – double array

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.

6: $npos$ – int64int32nag_int scalar

p (npos)

holds the probability associated with the given table of frequencies.

7: $ndf$ – int64int32nag_int scalar

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

8: $m1$ – int64int32nag_int scalar

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

m_{1}

9: $n1$ – int64int32nag_int scalar

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

n_{1}

10: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: 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	$ldnob < m$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The method used is believed to be stable.

Further Comments

The time taken by nag_stat_contingency_table (g01af) 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.

Example

In the example program, NPROB determines the number of two-way classifications to be analysed. For each classification the frequencies are read, nag_stat_contingency_table (g01af) 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.

Open in the MATLAB editor: g01af_example

function g01af_example


fprintf('g01af example results\n\n');

nr = 2;
nc = 3;
nobs = zeros(nr+1,nc+1,'int64');
nobs(1:nr,1:nc) = [ 86,  51, 13;
		   130, 115, 41];

[nobs, num, pred, chis, p, npos, ndf, m1, n1, ifail] = ...
  g01af(nobs);

if (m1~=nr)
fprintf('Number of rows reduced from %2d to %2d\n', nr, m1);
end
if (n1~=nc)
  fprintf('Number of rows reduced from %2d to %2d\n', nc, n1);
end
fprintf('\nTable of observed frequencies\n\n');
fprintf('                            total\n');
for j = 1:m1
  fprintf('%8s',' ');
  fprintf('%5d',nobs(j,1:n1));
  fprintf('%8d\n',nobs(j,n1+1));
end
fprintf('\n%8s','total');
fprintf('%5d',nobs(m1+1,1:n1));
fprintf('%8d\n',nobs(m1+1,n1+1));

fprintf('\n\nTable of expected frequencies\n\n');
for j = 1:m1
  fprintf('%8s',' ');
  fprintf('%5d',int64(pred(j,1:n1)));
  fprintf('\n');
end

fprintf('\nChi-squared        = %7.3f\n', chis);
fprintf('Degrees of freedom = %4d\n', ndf);



function g01af_table(m1,n1,obs)
  fprintf('                            total\n');
  for j = 1:m1
    fprintf('%8s',' ');
    fprintf('%5d',obs(j,1:n1));
    fprintf('%8d\n',obs(j,n1+1));
  end
  fprintf('\n%8s','total');
  fprintf('%5d',obs(m1+1,1:n1));
  fprintf('%8d\n',obs(m1+1,n1+1));

g01af example results


Table of observed frequencies

                            total
           86   51   13     150
          130  115   41     286

   total  216  166   54     436


Table of expected frequencies

           74   57   19
          142  109   35

Chi-squared        =   6.352
Degrees of freedom =    2

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox