(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 $1 - p < α$ .

References

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

Parameters

Compulsory Input Parameters

1: $x (n)$ – double array
2: $y (n)$ – double array: $x (i)$ and $y (i)$ must be set to the $i$ th pair of data values, $\{x_{i}, y_{i}\}$ , for $i = 1, 2, \dots, n$ .

Optional Input Parameters

1: $n$ – 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: $n \geq 1$ .

Output Parameters

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

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,

n < 1

$ifail = 2$: $n1 = 0$ , i.e., the samples are identical.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: 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} \geq 120

p

should be computed with an absolute error of less than

0.005

. For further details see nag_stat_prob_beta (g01ee).

Further Comments

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

5

Open in the MATLAB editor: g08aa_example

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

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox