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NAG Toolbox

NAG Toolbox: nag_nonpar_test_sign (g08aa)

Purpose

nag_nonpar_test_sign (g08aa) performs the Sign test on two related samples of size nn.

Syntax

[isgn, n1, p, ifail] = g08aa(x, y, 'n', n)
[isgn, n1, p, ifail] = nag_nonpar_test_sign(x, y, 'n', n)

Description

The Sign test investigates the median difference between pairs of scores from two matched samples of size nn, denoted by {xi,yi}{xi,yi}, for i = 1,2,,ni=1,2,,n. The hypothesis under test, H0H0, 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 H1H1 (see below).
nag_nonpar_test_sign (g08aa) computes:
(a) the test statistic SS, which is the number of pairs for which xi < yixi<yi;
(b) the number n1n1 of non-tied pairs (xiyi)(xiyi);
(c) the lower tail probability pp corresponding to SS (adjusted to allow the complement (1p)(1-p) to be used in an upper one tailed or a two tailed test). pp is the probability of observing a value SS if S < (1/2)n1S<12n1, or of observing a value < S<S if S > (1/2)n1S>12n1, given that H0H0 is true. If S = (1/2)n1S=12n1, pp is set to 0.50.5.
Suppose that a significance test of a chosen size αα is to be performed (i.e., αα is the probability of rejecting H0H0 when H0H0 is true; typically αα is a small quantity such as 0.050.05 or 0.010.01). The returned value of pp can be used to perform a significance test on the median difference, against various alternative hypotheses H1H1, as follows
(i) H1H1: median of xx median of yy. H0H0 is rejected if 2 × min (p,1p) < α 2 × min(p,1-p) < α .
(ii) H1H1: median of x > x> median of yy. H0H0 is rejected if p < αp<α.
(iii) H1H1: median of x < x< median of yy. H0H0 is rejected if 1p < α1-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
n, the dimension of the array, must satisfy the constraint n1n1.
x(i)xi and y(i)yi must be set to the iith pair of data values, {xi,yi}{xi,yi}, for i = 1,2,,ni=1,2,,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.)
nn, the size of each sample.
Constraint: n1n1.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     isgn – int64int32nag_int scalar
The Sign test statistic, SS.
2:     n1 – int64int32nag_int scalar
The number of non-tied pairs, n1n1.
3:     p – double scalar
The lower tail probability, pp, corresponding to SS.
4:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:
  ifail = 1ifail=1
On entry,n < 1n<1.
  ifail = 2ifail=2
n1 = 0n1=0, i.e., the samples are identical.

Accuracy

The tail probability, pp, is computed using the relationship between the binomial and beta distributions. For n1 < 120n1<120, pp should be accurate to at least 44 significant figures, assuming that the machine has a precision of 77 or more digits. For n1120n1120, pp should be computed with an absolute error of less than 0.0050.005. 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 nn.

Example

function nag_nonpar_test_sign_example
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];
[isgn, n1, p, ifail] = nag_nonpar_test_sign(x, y)
 

isgn =

                    3


n1 =

                   14


p =

    0.0287


ifail =

                    0


function g08aa_example
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];
[isgn, n1, p, ifail] = g08aa(x, y)
 

isgn =

                    3


n1 =

                   14


p =

    0.0287


ifail =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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