# NAG Library Routine Document

## 1Purpose

g08aaf performs the Sign test on two related samples of size $n$.

## 2Specification

Fortran Interface
 Subroutine g08aaf ( x, y, n, isgn, n1, p,
 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
#include nagmk26.h
 void g08aaf_ (const double x[], const double y[], const Integer *n, Integer *isgn, Integer *n1, double *p, Integer *ifail)

## 3Description

The Sign test investigates the median difference between pairs of scores from two matched samples of size $n$, denoted by $\left\{{x}_{\mathit{i}},{y}_{\mathit{i}}\right\}$, for $\mathit{i}=1,2,\dots ,n$. 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 $\left({x}_{i}\ne {y}_{i}\right)$; (c) the lower tail probability $p$ corresponding to $S$ (adjusted to allow the complement $\left(1-p\right)$ to be used in an upper one tailed or a two tailed test). $p$ is the probability of observing a value $\text{}\le S$ if $S<\frac{1}{2}{n}_{1}$, or of observing a value $\text{} 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 $\alpha$ is to be performed (i.e., $\alpha$ is the probability of rejecting ${H}_{0}$ when ${H}_{0}$ is true; typically $\alpha$ is a small quantity such as $0.05$ or $0.01$). 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 $x\ne \text{}$ median of $y$. ${H}_{0}$ is rejected if $2×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(p,1-p\right)<\alpha$. (ii) ${H}_{1}$: median of $x>\text{}$ median of $y$. ${H}_{0}$ is rejected if $p<\alpha$. (iii) ${H}_{1}$: median of $x<\text{}$ median of $y$. ${H}_{0}$ is rejected if $1-p<\alpha$.

## 4References

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

## 5Arguments

1:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
2:     $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{x}}\left(\mathit{i}\right)$ and ${\mathbf{y}}\left(\mathit{i}\right)$ must be set to the $\mathit{i}$th pair of data values, $\left\{{x}_{\mathit{i}},{y}_{\mathit{i}}\right\}$, for $\mathit{i}=1,2,\dots ,n$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the size of each sample.
Constraint: ${\mathbf{n}}\ge 1$.
4:     $\mathbf{isgn}$ – IntegerOutput
On exit: the Sign test statistic, $S$.
5:     $\mathbf{n1}$ – IntegerOutput
On exit: the number of non-tied pairs, ${n}_{1}$.
6:     $\mathbf{p}$ – Real (Kind=nag_wp)Output
On exit: the lower tail probability, $p$, corresponding to $S$.
7:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<1$.
${\mathbf{ifail}}=2$
${\mathbf{n1}}=0$, i.e., the samples are identical.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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 ${n}_{1}\ge 120$, $p$ should be computed with an absolute error of less than $0.005$. For further details see g01eef.

## 8Parallelism and Performance

g08aaf is not threaded in any implementation.

The time taken by g08aaf is small, and increases with $n$.

## 10Example

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 $1$ to $5$.

### 10.1Program Text

Program Text (g08aafe.f90)

### 10.2Program Data

Program Data (g08aafe.d)

### 10.3Program Results

Program Results (g08aafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017