$H_{1}$ : $F (x) \neq G (y)$ ; 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 Mann–Whitney

U

test differs from the Median test (see nag_nonpar_test_median (g08ac)) in that the ranking of the individual scores within the pooled sample is taken into account, rather than simply the position of a score relative to the median of the pooled sample. It is therefore a more powerful test if score differences are meaningful.

The test procedure involves ranking the pooled sample, average ranks being used for ties. Let

r_{1 i}

be the rank assigned to

x_{i}

i = 1, 2, \dots, n_{1}

and

r_{2 j}

the rank assigned to

y_{j}

j = 1, 2, \dots, n_{2}

. Then the test statistic

U

is defined as follows;

U = \sum_{i = 1}^{n_{1}} r_{1 i} - \frac{n_{1} (n_{1} + 1)}{2}

U

is also the number of times a score in the second sample precedes a score in the first sample (where we only count a half if a score in the second sample actually equals a score in the first sample).

nag_nonpar_test_mwu (g08ah) returns:

(a)

The test statistic

U

(b)

The approximate Normal test statistic,

z = \frac{U - mean (U) \pm \frac{1}{2}}{\sqrt{var (U)}}

where

mean (U) = \frac{n_{1} n_{2}}{2}

and

var (U) = \frac{n_{1} n_{2} (n_{1} + n_{2} + 1)}{12} - \frac{n_{1} n_{2}}{(n_{1} + n_{2}) (n_{1} + n_{2} - 1)} \times T S

where

T S = \sum_{j = 1}^{τ} \frac{(t_{j}) (t_{j} - 1) (t_{j} + 1)}{12}

τ

is the number of groups of ties in the sample and

t_{j}

is the number of ties in the

j

th group.

Note that if no ties are present the variance of

U

reduces to

\frac{n_{1} n_{2}}{12} (n_{1} + n_{2} + 1)

(c)

An indicator as to whether ties were present in the pooled sample or not.

(d)

The tail probability,

p

, corresponding to

U

(adjusted to allow the complement to be used in an upper one tailed or a two tailed test), depending on the choice of tail, i.e., the choice of alternative hypothesis,

H_{1}

. The tail probability returned is an approximation of

p

is based on an approximate Normal statistic corrected for continuity according to the tail specified. If

n_{1}

and

n_{2}

are not very large an exact probability may be desired. For the calculation of the exact probability see nag_nonpar_prob_mwu_noties (g08aj) (no ties in the pooled sample) or nag_nonpar_prob_mwu_ties (g08ak) (ties in the pooled sample).

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

Conover W J (1980) Practical Nonparametric Statistics Wiley

Neumann N (1988) Some procedures for calculating the distributions of elementary nonparametric teststatistics Statistical Software Newsletter 14(3) 120–126

Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill

Parameters

Compulsory Input Parameters

1: $x (n1)$ – double array

The first vector of observations,

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

2: $y (n2)$ – double array

The second vector of observations.

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

3: $tail$ – string (length ≥ 1)

Indicates the choice of tail probability, and hence the alternative hypothesis.

$tail ='T'$: A two tailed probability is calculated and the alternative hypothesis is $H_{1} : F (x) \neq G (y)$ .
$tail ='U'$: An upper tailed probability is calculated and the alternative hypothesis $H_{1} : F (x) < G (y)$ , i.e., the $x$ 's tend to be greater than the $y$ 's.
$tail ='L'$: A lower tailed probability is calculated and the alternative hypothesis $H_{1} : F (x) > G (y)$ , i.e., the $x$ 's tend to be less than the $y$ 's.

Constraint:

tail ='T'

'U'

'L'

Optional Input Parameters

1: $n1$ – int64int32nag_int scalar: Default: the dimension of the array x.
The size of the first sample, $n_{1}$ .

Constraint: $n1 \geq 1$ .
2: $n2$ – int64int32nag_int scalar: Default: the dimension of the array y.
The size of the second sample, $n_{2}$ .

Constraint: $n2 \geq 1$ .

Output Parameters

1: $u$ – double scalar: The Mann–Whitney rank sum statistic, $U$ .
2: $unor$ – double scalar: The approximate Normal test statistic, $z$ , as described in Description.
3: $p$ – double scalar: The tail probability, $p$ , as specified by the argument tail.
4: $ties$ – logical scalar: Indicates whether the pooled sample contained ties or not. This will be useful in checking which function to use should one wish to calculate an exact tail probability.
$ties = false$ , no ties were present (use nag_nonpar_prob_mwu_noties (g08aj) for an exact probability).

$ties = true$ , ties were present (use nag_nonpar_prob_mwu_ties (g08ak) for an exact probability).
5: $ranks (n1 + n2)$ – double array: Contains the ranks of the pooled sample. The ranks of the first sample are contained in the first n1 elements and those of the second sample are contained in the next n2 elements.
6: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

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

$ifail = 2$

On entry,

tail \neq'T'

'U'

'L'

W $ifail = 3$: The pooled sample values are all the same, that is the variance of $u = 0.0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The approximate tail probability,

p

, returned by nag_nonpar_test_mwu (g08ah) is a good approximation to the exact probability for cases where

\max (n_{1}, n_{2}) \geq 30

and

(n_{1} + n_{2}) \geq 40

. The relative error of the approximation should be less than

10 %

, for most cases falling in this range.

Further Comments

The time taken by nag_nonpar_test_mwu (g08ah) increases with

n_{1}

and

n_{2}

Example

This example performs the Mann–Whitney test on two independent samples of sizes

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. An exact tail probability is also calculated and printed depending on whether ties were found in the pooled sample or not.

Open in the MATLAB editor: g08ah_example

function g08ah_example


fprintf('g08ah 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);
if ties
  fprintf('There are ties in the pooled sample\n\n');
else
  fprintf('There are no ties in the pooled sample\n\n');
end
fprintf('Exact tail probability    = %8.4f\n', pexact);

g08ah 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

There are ties in the pooled sample

Exact tail probability    =   0.0020

PDF version (NAG web site, 64-bit version, 64-bit version)

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