nag_sign_test (g08aac) (PDF version)
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NAG 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 entry, n=value.
Constraint: n1.

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  Parallelism and Performance

Not applicable.

9  Further Comments

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

10  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.

10.1  Program Text

Program Text (g08aace.c)

10.2  Program Data

Program Data (g08aace.d)

10.3  Program Results

Program Results (g08aace.r)

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014