(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 $(x_{i} \neq y_{i})$ ;
(c)	the lower tail probability $p$ corresponding to $S$ (adjusted to allow the complement $(1 - p)$ to be used in an upper one tailed or a two tailed test). $p$ is the probability of observing a value $\leq S$ if $S < \frac{1}{2} n_{1}$ , or of observing a value $< S$ 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

α

is to be performed (i.e.,

α

is the probability of rejecting

H_{0}

when

H_{0}

is true; typically

α

is a small quantity such as

0.05

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 $x \neq$ median of $y$ . $H_{0}$ is rejected if $2 \times \min (p, 1 - p) < α$ .
(ii)	$H_{1}$ : median of $x >$ median of $y$ . $H_{0}$ is rejected if $p < α$ .
(iii)	$H_{1}$ : median of $x <$ median of $y$ . $H_{0}$ is rejected if $1 - p < α$ .

4 References

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

5 Arguments

1: $n$ – IntegerInput: On entry: $n$ , the size of each sample.

Constraint: $n \geq 1$ .
2: $x [n]$ – const doubleInput
3: $y [n]$ – const doubleInput: On entry: $x [i - 1]$ and $y [i - 1]$ must be set to the $i$ th pair of data values, $\{x_{i}, y_{i}\}$ , for $i = 1, 2, \dots, n$ .
4: $s$ – Integer *Output: On exit: the Sign test statistic, $S$ .
5: $p$ – double *Output: On exit: the lower tail probability, $p$ , corresponding to $S$ .
6: $non_tied$ – Integer *Output: On exit: the number of non-tied pairs, $n_{1}$ .
7: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

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} \geq 120

p

should be computed with an absolute error of less than

0.005

nag_sign_test (g08aac) is not threaded in any implementation.

The time taken by nag_sign_test (g08aac) is small, and increases with

n

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

5