g08ak:: Nonparametric Statistics (NAG Toolbox)

nag_nonpar_prob_mwu_ties (g08ak) computes the exact tail probability for the Mann–Whitney

U

test statistic (calculated by nag_nonpar_test_mwu (g08ah) and returned through the argument u) using a method based on an algorithm developed by Neumann (1988), for the case where there are ties in the pooled sample.

The Mann–Whitney

U

test investigates the difference between two populations defined by the distribution functions

F (x)

and

G (y)

respectively. The data consist of two independent samples of size

n_{1}

and

n_{2}

, denoted by

x_{1}, x_{2}, \dots, x_{n_{1}}

and

y_{1}, y_{2}, \dots, y_{n_{2}}

, taken from the two populations.

The hypothesis under test,

H_{0}

, often called the null hypothesis, is that the two distributions are the same, that is

F (x) = G (x)

, and this is to be tested against an alternative hypothesis

H_{1}

which is

$H_{1}$ : $F (x) \neq G (y)$ ; or
$H_{1}$ : $F (x) < G (y)$ , i.e., the $x$ 's tend to be greater than the $y$ 's; or
$H_{1}$ : $F (x) > G (y)$ , i.e., the $x$ 's tend to be less than the $y$ 's,

using a two tailed, upper tailed or lower tailed probability respectively. You select the alternative hypothesis by choosing the appropriate tail probability to be computed (see the description of argument tail in Arguments).

Note that when using this test to test for differences in the distributions one is primarily detecting differences in the location of the two distributions. That is to say, if we reject the null hypothesis

H_{0}

in favour of the alternative hypothesis

H_{1}

F (x) > G (y)

we have evidence to suggest that the location, of the distribution defined by

F (x)

, is less than the location of the distribution defined by

G (y)

The value of

p

can be used to perform a significance test on the null hypothesis

H_{0}

against the alternative hypothesis

H_{1}

. Let

α

be the size of the significance test (that is

α

is the probability of rejecting

H_{0}

when

H_{0}

is true). If

p < α

then the null hypothesis is rejected. Typically

α

might be

0.05

0.01

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example

This example finds the Mann–Whitney test statistic, using nag_nonpar_test_mwu (g08ah) for two independent samples of size

16

and

23

respectively. This is used to test the null hypothesis that the distributions of the two populations from which the samples were taken are the same against the alternative hypothesis that the distributions are different. The test statistic, the approximate Normal statistic and the approximate two tail probability are printed. nag_nonpar_prob_mwu_ties (g08ak) is then called to obtain the exact two tailed probability. The exact probability is also printed.

function g08ak_example


fprintf('g08ak example results\n\n');

x = [13;  6; 12;  7; 12;  7; 10;  7;
     10;  7; 16;  7; 10;  8;  9;  8];
y = [17;  6; 10;  8; 15;  8; 15; 10;
     15; 10; 14; 10; 14; 11; 14; 11;
     13; 12; 13; 12; 13; 12; 12];
n1 = int64(numel(x));
n2 = int64(numel(y));

fprintf('Mann-Whitney U test\n\n');
fprintf('Sample size of group 1 = %5d\n', n1);
fprintf('Sample size of group 2 = %5d\n\n', n2);
fprintf('Data values\n');
fprintf('\n     Group 1 ');
for j = 1:floor(n1/8)
  i1 = (j-1)*8 + 1;
  i2 = min(n1,i1+7);
  fprintf('%4.0f',x(i1:i2));
  fprintf('\n             ');
end
fprintf('\n     Group 2 ');
for j = 1:floor(n2/8)
  i1 = (j-1)*8 + 1;
  i2 = min(n2,i1+7);
  fprintf('%4.0f',y(i1:i2));
  fprintf('\n             ');
end

% Perform test
tail = 'Lower-tail';
[u, unor, p, ties, ranks, ifail] =  ...
  g08ah(x, y, tail);

% Calculate exact probabilities
if ties
  [pexact, ifail] = g08ak( ...
                      n1, n2, tail, ranks, u);
else
  [pexact, ifail] = g08aj( ...
                      n1, n2, tail, u);
end
   
fprintf('\nTest statistic            = %8.4f\n', u);
fprintf('Normalized test statistic = %8.4f\n', unor);
fprintf('Approx. tail probability  = %8.4f\n\n', p);

fprintf('\nRanks\n\n');
fprintf('     Group 1 ');
for j = 1:floor(n1/8)
  i1 = (j-1)*8 + 1;
  i2 = min(n1,i1+7);
  fprintf('%5.1f',ranks(i1:i2));
  fprintf('\n             ');
end
fprintf('\n     Group 2 ');
for j = 1:floor(n2/8)
  i1 = (j-1)*8 + 1;
  i2 = min(n2,i1+7);
  fprintf('%5.1f',ranks(i1+n1:i2+n1));
  fprintf('\n             ');
end

fprintf('\nExact tail probability    = %8.4f\n', pexact);

g08ak example results

Mann-Whitney U test

Sample size of group 1 =    16
Sample size of group 2 =    23

Data values

     Group 1   13   6  12   7  12   7  10   7
               10   7  16   7  10   8   9   8
             
     Group 2   17   6  10   8  15   8  15  10
               15  10  14  10  14  11  14  11
               13  12  13  12  13  12  12
             
Test statistic            =  86.0000
Normalized test statistic =  -2.8039
Approx. tail probability  =   0.0025


Ranks

     Group 1  29.5  1.5 24.5  5.0 24.5  5.0 16.0  5.0
              16.0  5.0 38.0  5.0 16.0  9.5 12.0  9.5
             
     Group 2  39.0  1.5 16.0  9.5 36.0  9.5 36.0 16.0
              36.0 16.0 33.0 16.0 33.0 20.5 33.0 20.5
              29.5 24.5 29.5 24.5 29.5 24.5 24.5
             
Exact tail probability    =   0.0020

On entry,	$n1 < 1$ ,
or	$n2 < 1$ .

NAG Toolbox: nag_nonpar_prob_mwu_ties (g08ak)

▸▿ Contents

Purpose

Syntax

Description