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 <nag.h>

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

The routine may be called by the names g08aaf or nagf_nonpar_test_sign.

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) array Input
2: $y (n)$ – Real (Kind=nag_wp) array Input: 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: $n$ – Integer Input: On entry: $n$ , the size of each sample.

Constraint: $n \geq 1$ .
4: $isgn$ – Integer Output: On exit: the Sign test statistic, $S$ .
5: $n1$ – Integer Output: 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: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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

g08aa: FL CL CPP AD

NAG FL Interfaceg08aaf (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

NAG FL Interface
g08aaf (test_sign)