NAG Library Routine Document

G08ALF

), where the observed response can take only one of two possible values; for example a treatment may result in a ‘success’ or ‘failure’. The data is recorded as either

1

0

to represent this dichotomization.

The use of this ‘randomized block design’ allows the effect of differences between the blocks to be separated from the differences between the treatments. The test assumes that the blocks were randomly selected from all possible blocks and that the result may be one of two possible outcomes common to all treatments within blocks.

The null and alternative hypotheses to be tested may be stated as follows.

$H_{0}$ :	the treatments are equally effective, that is the probability of obtaining a $1$ within a block is the same for each treatment.
$H_{1}$ :	there is a difference between the treatments, that is the probability of obtaining a $1$ is not the same for different treatments within blocks.

The data is often represented in the form of a table with the

n

rows representing the blocks and the

k

columns the treatments. Let

R_{i}

represent the row totals, for

i = 1, 2, \dots, n

, and

C_{j}

represent the column totals, for

j = 1, 2, \dots, k

. Let

x_{i j}

represent the response or result where

x_{i j} = 0 ​ or ​ 1

	Treatments
Blocks	1	2		$k$	Row Totals
1	$x_{11}$	$x_{12}$	$\dots$	$x_{1 k}$	$R_{1}$
2	$x_{21}$	$x_{22}$	$\dots$	$x_{2 k}$	$R_{2}$
$⋮$			$⋮$		$⋮$
$n$	$x_{n 1}$	$x_{n 2}$	$\dots$	$x_{n k}$	$R_{n}$
Column Totals	$C_{1}$	$C_{2}$		$C_{k}$	$N = Grand Total$

p_{i j} = \Pr (x_{i j} = 1)

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, k

, then the hypotheses may be restated as follows

$H_{0}$ :	$p_{i 1} = p_{i 2} = \dots = p_{i k}$ , for each $i = 1, 2, \dots, n$ .
$H_{1}$ :	$p_{i j} \neq p_{i k}$ , for some $j$ and $k$ , and for some $i$ .

The test statistic is defined as

Q = \frac{k (k - 1) \sum_{j = 1}^{k} {(C_{j} - \frac{N}{k})}^{2}}{\sum_{i = 1}^{n} R_{i} (k - R_{i})} .

When the number of blocks,

n

, is large relative to the number of treatments,

k

Q

has an approximate

χ^{2}

-distribution with

k - 1

degrees of freedom. This is used to find the probability,

p

, of obtaining a statistic greater than or equal to the computed value of

Q

. Thus

p

is the upper tail probability associated with the computed value of

Q

, where the

χ^{2}

-distribution is used to approximate the true distribution of

Q

4 References

Conover W J (1980) Practical Nonparametric Statistics Wiley

Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill

5 Parameters

1: N – INTEGERInput: On entry: $n$ , the number of blocks.
Constraint: $N \geq 2$ .
2: K – INTEGERInput: On entry: $k$ , the number of treatments.
Constraint: $K \geq 2$ .
3: X(LDX,K) – REAL (KIND=nag_wp) arrayInput: On entry: the matrix of observed zero-one data. $X (i, j)$ must contain the value $x_{i j}$ , for $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, k$ .
Constraint: $X (i, j) = 0.0$ or $1.0$ , for $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, k$ .
4: LDX – INTEGERInput: On entry: the first dimension of the array X as declared in the (sub)program from which G08ALF is called.
Constraint: $LDX \geq N$ .
5: Q – REAL (KIND=nag_wp)Output: On exit: the value of the Cochran $Q$ -test statistic.
6: PROB – REAL (KIND=nag_wp)Output: On exit: the upper tail probability, $p$ , associated with the Cochran $Q$ -test statistic, that is the probability of obtaining a value greater than or equal to the observed value (the output value of Q).
7: 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:

$IFAIL = 1$

On entry,	$N < 2$ ,
or	$K < 2$ ,
or	$LDX < N$ .

$IFAIL = 2$

On entry,

X (i, j) \neq 0.0

1.0

for some

i

and

j

i = 1, 2, \dots, n

and

j = 1, 2, \dots, k

$IFAIL = 3$: The approximation process used to calculate the tail probability has failed to converge. The result returned in PROB may still be a reasonable approximation.

7 Accuracy

The use of the

χ^{2}

-distribution as an approximation to the true distribution of the Cochran

Q

-test statistic improves as

k

increases and as

n

increases relative to

k

. This approximation should be a reasonable one when the total number of observations left, after omitting those rows containing all

0

1

, is greater than about

25

and the number of rows left is larger than

5

8 Further Comments

None.

9 Example

The following example is taken from page 201 of Conover (1980). The data represents the success of three basketball enthusiasts in predicting the outcome of

12

collegiate basketball games, selected at random, using

1

for successful prediction of the outcome and

0

for unsuccessful prediction. This data is read in and the Cochran

Q

-test statistic and its corresponding upper tail probability are computed and printed.

NAG Library Routine DocumentG08ALF

+− Contents

1 Purpose

2 Specification

3 Description