NAG CL Interface
g08aac (test_​sign)

1 Purpose

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

2 Specification

#include <nag.h>
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.

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).
g08aac computes:
  1. (a)the test statistic S, which is the number of pairs for which xi<yi;
  2. (b)the number n1 of non-tied pairs xiyi;
  3. (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
  1. (i)H1: median of x median of y. H0 is rejected if 2 × minp,1-p < α .
  2. (ii)H1: median of x> median of y. H0 is rejected if p<α.
  3. (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: n Integer Input
On entry: n, the size of each sample.
Constraint: n1.
2: x[n] const double Input
3: y[n] const double Input
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: 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, n1.
7: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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 g01eec.

8 Parallelism and Performance

g08aac is not threaded in any implementation.

9 Further Comments

The time taken by 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)