# NAG CL Interfaceg08aac (test_​sign)

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## 1Purpose

g08aac performs the Sign test on two related samples of size $n$.

## 2Specification

 #include
 void g08aac (Integer n, const double x[], const double y[], Integer *s, double *p, Integer *non_tied, NagError *fail)
The function may be called by the names: g08aac, nag_nonpar_test_sign or nag_sign_test.

## 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).
g08aac computes:
1. (a)the test statistic $S$, which is the number of pairs for which ${x}_{i}<{y}_{i}$;
2. (b)the number ${n}_{1}$ of non-tied pairs $\left({x}_{i}\ne {y}_{i}\right)$;
3. (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
1. (i)${H}_{1}$: median of $x\ne \text{}$ median of $y$. ${H}_{0}$ is rejected if $2×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(p,1-p\right)<\alpha$.
2. (ii)${H}_{1}$: median of $x>\text{}$ median of $y$. ${H}_{0}$ is rejected if $p<\alpha$.
3. (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: $\mathbf{n}$Integer Input
On entry: $n$, the size of each sample.
Constraint: ${\mathbf{n}}\ge 1$.
2: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
3: $\mathbf{y}\left[{\mathbf{n}}\right]$const double Input
On entry: ${\mathbf{x}}\left[\mathit{i}-1\right]$ and ${\mathbf{y}}\left[\mathit{i}-1\right]$ must be set to the $\mathit{i}$th pair of data values, $\left\{{x}_{\mathit{i}},{y}_{\mathit{i}}\right\}$, for $\mathit{i}=1,2,\dots ,n$.
4: $\mathbf{s}$Integer * Output
On exit: the Sign test statistic, $S$.
5: $\mathbf{p}$double * Output
On exit: the lower tail probability, $p$, corresponding to $S$.
6: $\mathbf{non_tied}$Integer * Output
On exit: the number of non-tied pairs, ${n}_{1}$.
7: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 1$.

## 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 g01eec.

## 8Parallelism and Performance

g08aac is not threaded in any implementation.

The time taken by g08aac 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 (g08aace.c)

### 10.2Program Data

Program Data (g08aace.d)

### 10.3Program Results

Program Results (g08aace.r)