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# NAG Toolbox: nag_nonpar_test_sign (g08aa)

## Purpose

nag_nonpar_test_sign (g08aa) performs the Sign test on two related samples of size $n$.

## Syntax

[isgn, n1, p, ifail] = g08aa(x, y, 'n', n)
[isgn, n1, p, ifail] = nag_nonpar_test_sign(x, y, 'n', n)

## 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).
nag_nonpar_test_sign (g08aa) 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 $1-p<\alpha$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
2:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
${\mathbf{x}}\left(\mathit{i}\right)$ 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 $\mathit{i}=1,2,\dots ,n$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the arrays x, y. (An error is raised if these dimensions are not equal.)
$n$, the size of each sample.
Constraint: ${\mathbf{n}}\ge 1$.

### Output Parameters

1:     $\mathrm{isgn}$int64int32nag_int scalar
The Sign test statistic, $S$.
2:     $\mathrm{n1}$int64int32nag_int scalar
The number of non-tied pairs, ${n}_{1}$.
3:     $\mathrm{p}$ – double scalar
The lower tail probability, $p$, corresponding to $S$.
4:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 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.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## 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 nag_stat_prob_beta (g01ee).

The time taken by nag_nonpar_test_sign (g08aa) is small, and increases with $n$.

## 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 $5$.
```function g08aa_example

fprintf('g08aa example results\n\n');

x = [4; 4; 5; 5; 3; 2; 5; 3; 1; 5; 5; 5; 4; 5; 5; 5; 5];
y = [2; 3; 3; 3; 3; 3; 3; 3; 2; 3; 2; 2; 5; 2; 5; 3; 1];

fprintf('Sign test\n\n')
fprintf('Data values\n\n');
fprintf('%3.0f',x);
fprintf('\n')
fprintf('%3.0f',y);
fprintf('\n\n')

[isgn, n1, p, ifail] = g08aa( ...
x, y);

fprintf('Test statistic   %5d\n', isgn);
fprintf('Observations     %5d\n', n1);
fprintf('Lower tail prob. %5.3f\n', p);

```
```g08aa example results

Sign test

Data values

4  4  5  5  3  2  5  3  1  5  5  5  4  5  5  5  5
2  3  3  3  3  3  3  3  2  3  2  2  5  2  5  3  1

Test statistic       3
Observations        14
Lower tail prob. 0.029
```

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