G08 Chapter Contents
G08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentG08AAF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

G08AAF performs the Sign test on two related samples of size $n$.

## 2  Specification

 SUBROUTINE G08AAF ( X, Y, N, ISGN, N1, P, IFAIL)
 INTEGER N, ISGN, N1, IFAIL REAL (KIND=nag_wp) X(N), Y(N), P

## 3  Description

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 $1-p<\alpha$.

## 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: ${\mathbf{X}}\left(\mathit{i}\right)$ 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:     N – INTEGERInput
On entry: $n$, the size of each sample.
Constraint: ${\mathbf{N}}\ge 1$.
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 $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, 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).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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.

## 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 ${n}_{1}\ge 120$, $p$ should be computed with an absolute error of less than $0.005$. For further details see G01EEF.

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 $1$ to $5$.

### 9.1  Program Text

Program Text (g08aafe.f90)

### 9.2  Program Data

Program Data (g08aafe.d)

### 9.3  Program Results

Program Results (g08aafe.r)