test on two independent samples of possibly unequal size and calculates the exact probability for the Mann–Whitney rank sum test statistic for the case where there are either ties or no ties in the samples pooled together.

2 Specification

#include <nag.h>

void	g08amc (Integer n1, const double x[], Integer n2, const double y[], Nag_TailProbability tail, Nag_CompProb exact, double u, double z, double p, NagError fail)

The function may be called by the names: g08amc, nag_nonpar_mann_whitney or nag_mann_whitney.

3 Description

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 Section 5).

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 g08acc) 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}

, for

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

and

r_{2 j}

the rank assigned to

y_{j}

, for

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).

g08amc 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 $U$ , depending on the choice of tail, i.e., the choice of alternative hypothesis, $H_{1}$ . An exact probability or a normal approximation may be selected using the exact option. For large values of $n_{1}$ and $n_{2}$ the normal approximation should be adequate while the exact computation may require extensive calculation, particularly in cases where there are ties in the observations. For small samples the exact probability should be used.
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 $0.05$ or 0.01.

4 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

5 Arguments

1: $n1$ – Integer Input

On entry: the number of non-tied pairs,

n_{1}

Constraint:

n1 \geq 1

2: $x [n1]$ – const double Input

On entry: the first vector of observations.

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

3: $n2$ – Integer Input

On entry: the size of the second sample,

n_{2}

Constraint:

n2 \geq 1

4: $y [n2]$ – const double Input

On entry: the second vector of observations.

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

5: $tail$ – Nag_TailProbability Input

On entry: indicates the choice of tail probability, and hence the alternative hypothesis.

$tail = Nag_TwoTail$: A two tailed probability is calculated and the alternative hypothesis is $H_{1} : F (x) \neq G (y)$ .
$tail = Nag_UpperTail$: 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.
$tail = Nag_LowerTail$: 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 = Nag_TwoTail

Nag_UpperTail

Nag_LowerTail

6: $exact$ – Nag_CompProb Input

On entry: indicates if exact probability,

p

, is to be computed.

$exact = Nag_CompProbExact$: Compute the exact probability.
$exact = Nag_CompProbApprox$: Compute the approximate probability.

Constraint:

exact = Nag_CompProbExact

Nag_CompProbApprox

7: $u$ – double * Output

On exit: the Mann–Whitney rank sum statistic,

U

8: $z$ – double * Output

On exit: the approximate Normal test statistic,

z

, as described in Section 3.

9: $p$ – double * Output

On exit: the exact tail probability,

p

, as specified by the argument tail.

10: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument exact had an illegal value.

On entry, argument tail had an illegal value.
NE_G08AH_SAMP_IDEN: The pooled samples are all the same, that is the variance of $U = 0.0$ .
NE_INT_3: On entry, n1 and n2 require allocated work length exceeding integer limit.
NE_INT_ARG_LT: On entry, n1 must not be less than 1: $n1 = ⟨ value ⟩$ .

On entry, n2 must not be less than 1: $n2 = ⟨ value ⟩$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_MANN_WHITNEY_STAT: The computed Mann–Whitney statistic, $U = ⟨ value ⟩$ . If the exact tail probability is to be computed, then $U \geq 0$ .

7 Accuracy

The approximate tail probability,

p

, returned by g08amc 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 percent, for most cases falling in this range.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g08amc is not threaded in any implementation.

9 Further Comments

The time taken by g08amc increases with

n_{1}

and

n_{2}

and the product

n_{1} n_{2}

. Note that when exact probability is requested and ties are present, the amount of interrnal workspace required becomes very large for even moderate sizes of

n_{1}

and

n_{2}

10 Example

The example program 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.

g08am: FL CL CPP AD PY MB

NAG CL Interfaceg08amc (mann_​whitney)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g08amc (mann_whitney)