NAG FL Interface
g08aaf (test_​sign)

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1 Purpose

g08aaf performs the Sign test on two related samples of size n.

2 Specification

Fortran Interface
Subroutine g08aaf ( x, y, n, isgn, n1, p, ifail)
Integer, Intent (In) :: n
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: isgn, n1
Real (Kind=nag_wp), Intent (In) :: x(n), y(n)
Real (Kind=nag_wp), Intent (Out) :: p
C Header Interface
#include <nag.h>
void  g08aaf_ (const double x[], const double y[], const Integer *n, Integer *isgn, Integer *n1, double *p, Integer *ifail)
The routine may be called by the names g08aaf or nagf_nonpar_test_sign.

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).
g08aaf 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 × min(p,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: x(n) Real (Kind=nag_wp) array Input
2: y(n) Real (Kind=nag_wp) array Input
On entry: x(i) and y(i) must be set to the ith pair of data values, {xi,yi}, for i=1,2,,n.
3: n Integer Input
On entry: n, the size of each sample.
Constraint: n1.
4: isgn Integer Output
On exit: the Sign test statistic, S.
5: n1 Integer Output
On exit: the number of non-tied pairs, n1.
6: p Real (Kind=nag_wp) Output
On exit: the lower tail probability, p, corresponding to S.
7: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, n=value.
Constraint: n1.
On entry, the samples are identical, i.e., n1=0.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 g01eef.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g08aaf is not threaded in any implementation.

9 Further Comments

The time taken by g08aaf 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 (g08aafe.f90)

10.2 Program Data

Program Data (g08aafe.d)

10.3 Program Results

Program Results (g08aafe.r)