NAG Library Routine Document

G08AAF

(a)	the test statistic $S$ , which is the number of pairs for which $x_{i} < y_{i}$ ;
(b)	the number $n_{1}$ of non-tied pairs $(x_{i} \neq y_{i})$ ;
(c)	the lower tail probability $p$ corresponding to $S$ (adjusted to allow the complement $(1 - p)$ to be used in an upper one tailed or a two tailed test). $p$ is the probability of observing a value $\leq S$ if $S < \frac{1}{2} n_{1}$ , or of observing a value $< S$ if $S > \frac{1}{2} n_{1}$ , given that $H_{0}$ is true. If $S = \frac{1}{2} n_{1}$ , $p$ is set to $0.5$ .

Suppose that a significance test of a chosen size

α

is to be performed (i.e.,

α

is the probability of rejecting

H_{0}

when

H_{0}

is true; typically

α

is a small quantity such as

0.05

0.01

). The returned value of

p

can be used to perform a significance test on the median difference, against various alternative hypotheses

H_{1}

, as follows

(i)	$H_{1}$ : median of $x \neq$ median of $y$ . $H_{0}$ is rejected if $2 \times \min (p, 1 - p) < α$ .
(ii)	$H_{1}$ : median of $x >$ median of $y$ . $H_{0}$ is rejected if $p < α$ .
(iii)	$H_{1}$ : median of $x <$ median of $y$ . $H_{0}$ is rejected if $1 - p < α$ .

4 References

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

5 Parameters

1: X(N) – REAL (KIND=nag_wp) arrayInput
2: Y(N) – REAL (KIND=nag_wp) arrayInput: On entry: $X (i)$ and $Y (i)$ must be set to the $i$ th pair of data values, $\{x_{i}, y_{i}\}$ , for $i = 1, 2, \dots, n$ .
3: N – INTEGERInput: On entry: $n$ , the size of each sample.
Constraint: $N \geq 1$ .
4: ISGN – INTEGEROutput: On exit: the Sign test statistic, $S$ .
5: N1 – INTEGEROutput: On exit: the number of non-tied pairs, $n_{1}$ .
6: P – REAL (KIND=nag_wp)Output: On exit: the lower tail probability, $p$ , corresponding to $S$ .
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 < 1

$IFAIL = 2$: $N1 = 0$ , i.e., the samples are identical.

7 Accuracy

The tail probability,

p

, is computed using the relationship between the binomial and beta distributions. For

n_{1} < 120

p

should be accurate to at least

4

significant figures, assuming that the machine has a precision of

7

or more digits. For

n_{1} \geq 120

p

should be computed with an absolute error of less than

0.005

. For further details see G01EEF.

8 Further Comments

The time taken by G08AAF is small, and increases with

n

9 Example

This example is taken from page 69 of Siegel (1956). The data relates to ratings of ‘insight into paternal discipline’ for

17

sets of parents, recorded on a scale from

1

5

NAG Library Routine DocumentG08AAF

+− Contents

1 Purpose

2 Specification

3 Description