NAG Library Routine Document
g08aaf (test_sign)
1
Purpose
g08aaf performs the Sign test on two related samples of size $n$.
2
Specification
Fortran Interface
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 $\left\{{x}_{\mathit{i}},{y}_{\mathit{i}}\right\}$, 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 twosided 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 nontied pairs $\left({x}_{i}\ne {y}_{i}\right)$; 
(c) 
the lower tail probability $p$ corresponding to $S$ (adjusted to allow the complement $\left(1p\right)$ 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 $\text{}<S$ 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
$\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\times \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(p,1p\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 $1p<\alpha $. 
4
References
Siegel S (1956) Nonparametric Statistics for the Behavioral Sciences McGraw–Hill
5
Arguments
 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 nontied 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).
6
Error 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.
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.
8
Parallelism and Performance
g08aaf is not threaded in any implementation.
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
$1$ to
$5$.
10.1
Program Text
Program Text (g08aafe.f90)
10.2
Program Data
Program Data (g08aafe.d)
10.3
Program Results
Program Results (g08aafe.r)