nag_sign_test (g08aac) (PDF version)
g08 Chapter Contents
g08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_sign_test (g08aac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_sign_test (g08aac) performs the Sign test on two related samples of size n.

2  Specification

#include <nag.h>
#include <nagg08.h>
void  nag_sign_test (Integer n, const double x[], const double y[], Integer *s, double *p, Integer *non_tied, NagError *fail)

3  Description

The Sign test investigates the median difference between pairs of scores from two matched samples of size n, denoted by xi,yi, for i=1,2,,n. The hypothesis under test, H0, 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 H1 (see below).
nag_sign_test (g08aac) computes:
(a) the test statistic S, which is the number of pairs for which xi<yi;
(b) the number n1 of non-tied pairs xiyi;
(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 S if S<12n1, or of observing a value <S if S>12n1, given that H0 is true. If S=12n1, 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 H0 when H0 is true; typically α 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 H1, as follows
(i) H1: median of x median of y. H0 is rejected if 2 × minp,1-p < α .
(ii) H1: median of x> median of y. H0 is rejected if p<α.
(iii) H1: median of x< median of y. H0 is rejected if 1-p<α.

4  References

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

5  Arguments

1:     nIntegerInput
On entry: n, the size of each sample.
Constraint: n1.
2:     x[n]const doubleInput
3:     y[n]const doubleInput
On entry: x[i-1] and y[i-1] must be set to the ith pair of data values, xi,yi, for i=1,2,,n.
4:     sInteger *Output
On exit: the Sign test statistic, S.
5:     pdouble *Output
On exit: the lower tail probability, p, corresponding to S.
6:     non_tiedInteger *Output
On exit: the number of non-tied pairs, n1.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On exit, the number of non_tied pairs, non_tied=0 , i.e., the samples are identical.
On entry, n=value.
Constraint: n1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7  Accuracy

The tail probability, p, is computed using the relationship between the binomial and beta distributions. For n1<120, p should be accurate to at least 4 significant figures, assuming that the machine has a precision of 7 or more digits. For n1120, p should be computed with an absolute error of less than 0.005. For further details see nag_prob_beta_dist (g01eec).

8  Further Comments

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

9  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.

9.1  Program Text

Program Text (g08aace.c)

9.2  Program Data

Program Data (g08aace.d)

9.3  Program Results

Program Results (g08aace.r)

nag_sign_test (g08aac) (PDF version)
g08 Chapter Contents
g08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012