NAG Library Routine Document

Integer, Intent (In)	::	n
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	isgn, n1
Real (Kind=nag_wp), Intent (In)	::	x(n), y(n)
Real (Kind=nag_wp), Intent (Out)	::	p

C Header Interface

#include <nagmk26.h>

void	g08aaf_ (const double x[], const double y[], const Integer n, Integer isgn, Integer n1, double p, Integer *ifail)

3

Description

The Sign test investigates the median difference between pairs of scores from two matched samples of size

n

, denoted by

\{x_{i}, y_{i}\}

, for

i = 1, 2, \dots, n

. The hypothesis under test,

H_{0}

, often called the null hypothesis, is that the medians are the same, and this is to be tested against a one- or two-sided alternative

H_{1}

(see below).

g08aaf computes:

(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

Arguments

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 argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

$ifail = 2$: On entry, the samples are identical, i.e., $n_{1} = 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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

Parallelism and Performance

g08aaf is not threaded in any implementation.

9

Further Comments

The time taken by g08aaf is small, and increases with

n

10

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 Document

g08aaf (test_sign)

▸▿ 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