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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_nonpar_test_mwu (g08ah)

## Purpose

nag_nonpar_test_mwu (g08ah) performs the Mann–Whitney U$U$ test on two independent samples of possibly unequal size.

## Syntax

[u, unor, p, ties, ranks, ifail] = g08ah(x, y, tail, 'n1', n1, 'n2', n2)
[u, unor, p, ties, ranks, ifail] = nag_nonpar_test_mwu(x, y, tail, 'n1', n1, 'n2', n2)

## Description

The Mann–Whitney U$U$ test investigates the difference between two populations defined by the distribution functions F(x)$F\left(x\right)$ and G(y)$G\left(y\right)$ respectively. The data consist of two independent samples of size n1${n}_{1}$ and n2${n}_{2}$, denoted by x1,x2,,xn1${x}_{1},{x}_{2},\dots ,{x}_{{n}_{1}}$ and y1,y2,,yn2${y}_{1},{y}_{2},\dots ,{y}_{{n}_{2}}$, taken from the two populations.
The hypothesis under test, H0${H}_{0}$, often called the null hypothesis, is that the two distributions are the same, that is F(x) = G(x)$F\left(x\right)=G\left(x\right)$, and this is to be tested against an alternative hypothesis H1${H}_{1}$ which is
• H1${H}_{1}$: F(x)G(y)$F\left(x\right)\ne G\left(y\right)$; or
• H1${H}_{1}$: F(x) < G(y)$F\left(x\right), i.e., the x$x$'s tend to be greater than the y$y$'s; or
• H1${H}_{1}$: F(x) > G(y)$F\left(x\right)>G\left(y\right)$, i.e., the x$x$'s tend to be less than the y$y$'s,
using a two tailed, upper tailed or lower tailed probability respectively. You select the alternative hypothesis by choosing the appropriate tail probability to be computed (see the description of parameter tail in Section [Parameters]).
Note that when using this test to test for differences in the distributions one is primarily detecting differences in the location of the two distributions. That is to say, if we reject the null hypothesis H0${H}_{0}$ in favour of the alternative hypothesis H1${H}_{1}$: F(x) > G(y)$F\left(x\right)>G\left(y\right)$ we have evidence to suggest that the location, of the distribution defined by F(x)$F\left(x\right)$, is less than the location, of the distribution defined by G(y)$G\left(y\right)$.
The Mann–Whitney U$U$ 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 r1i${r}_{1i}$ be the rank assigned to xi${x}_{i}$, i = 1,2,,n1$i=1,2,\dots ,{n}_{1}$ and r2j${r}_{2j}$ the rank assigned to yj${y}_{j}$, j = 1,2,,n2$j=1,2,\dots ,{n}_{2}$. Then the test statistic U$U$ is defined as follows;
 n1 U = ∑ r1i − (n1(n1 + 1))/2 i = 1
$U=∑i=1n1r1i-n1(n1+1)2$
U$U$ 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$U$.
(b) The approximate Normal test statistic,
 z = (U − mean(U) ± (1/2))/(sqrt(var(U))) $z=U-mean(U)±12 var(U)$
where
 mean(U) = (n1n2)/2 $mean(U)=n1n22$
and
 var(U) = (n1n2(n1 + n2 + 1))/12 − (n1n2)/((n1 + n2)(n1 + n2 − 1)) × TS $var(U)=n1n2(n1+n2+1)12-n1n2 (n1+n2)(n1+n2-1) ×TS$
where
 τ TS = ∑ ((tj)(tj − 1)(tj + 1))/12 j = 1
$TS=∑j= 1 τ(tj)(tj- 1)(tj+ 1)12$
τ$\tau$ is the number of groups of ties in the sample and tj${t}_{j}$ is the number of ties in the j$j$th group.
Note that if no ties are present the variance of U$U$ reduces to (n1n2)/12(n1 + n2 + 1)$\frac{{n}_{1}{n}_{2}}{12}\left({n}_{1}+{n}_{2}+1\right)$.
(c) An indicator as to whether ties were present in the pooled sample or not.
(d) The tail probability, p$p$, corresponding to U$U$ (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, H1${H}_{1}$. The tail probability returned is an approximation of p$p$ is based on an approximate Normal statistic corrected for continuity according to the tail specified. If n1${n}_{1}$ and n2${n}_{2}$ 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$p$ can be used to perform a significance test on the null hypothesis H0${H}_{0}$ against the alternative hypothesis H1${H}_{1}$. Let α$\alpha$ be the size of the significance test (that is, α$\alpha$ is the probability of rejecting H0${H}_{0}$ when H0${H}_{0}$ is true). If p < α$p<\alpha$ then the null hypothesis is rejected. Typically α$\alpha$ might be 0.05$0.05$ or 0.01$0.01$.

## 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
n1, the dimension of the array, must satisfy the constraint n11${\mathbf{n1}}\ge 1$.
The first vector of observations, x1,x2,,xn1${x}_{1},{x}_{2},\dots ,{x}_{{n}_{1}}$.
2:     y(n2) – double array
n2, the dimension of the array, must satisfy the constraint n21${\mathbf{n2}}\ge 1$.
The second vector of observations. y1,y2,,yn2${y}_{1},{y}_{2},\dots ,{y}_{{n}_{2}}$.
3:     tail – string (length ≥ 1)
Indicates the choice of tail probability, and hence the alternative hypothesis.
tail = 'T'${\mathbf{tail}}=\text{'T'}$
A two tailed probability is calculated and the alternative hypothesis is H1 : F(x)G(y)${H}_{1}:F\left(x\right)\ne G\left(y\right)$.
tail = 'U'${\mathbf{tail}}=\text{'U'}$
An upper tailed probability is calculated and the alternative hypothesis H1 : F(x) < G(y)${H}_{1}:F\left(x\right), i.e., the x$x$'s tend to be greater than the y$y$'s.
tail = 'L'${\mathbf{tail}}=\text{'L'}$
A lower tailed probability is calculated and the alternative hypothesis H1 : F(x) > G(y)${H}_{1}:F\left(x\right)>G\left(y\right)$, i.e., the x$x$'s tend to be less than the y$y$'s.
Constraint: tail = 'T'${\mathbf{tail}}=\text{'T'}$, 'U'$\text{'U'}$ or 'L'$\text{'L'}$.

### Optional Input Parameters

1:     n1 – int64int32nag_int scalar
Default: The dimension of the array x.
The size of the first sample, n1${n}_{1}$.
Constraint: n11${\mathbf{n1}}\ge 1$.
2:     n2 – int64int32nag_int scalar
Default: The dimension of the array y.
The size of the second sample, n2${\mathit{n}}_{2}$.
Constraint: n21${\mathbf{n2}}\ge 1$.

wrk

### Output Parameters

1:     u – double scalar
The Mann–Whitney rank sum statistic, U$U$.
2:     unor – double scalar
The approximate Normal test statistic, z$z$, as described in Section [Description].
3:     p – double scalar
The tail probability, p$p$, as specified by the parameter 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${\mathbf{ties}}=\mathbf{false}$, no ties were present (use nag_nonpar_prob_mwu_noties (g08aj) for an exact probability).
ties = true${\mathbf{ties}}=\mathbf{true}$, ties were present (use nag_nonpar_prob_mwu_ties (g08ak) for an exact probability).
5:     ranks(n1 + n2${\mathbf{n1}}+{\mathbf{n2}}$) – 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
${\mathrm{ifail}}={\mathbf{0}}$ 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${\mathbf{ifail}}=1$
 On entry, n1 < 1${\mathbf{n1}}<1$, or n2 < 1${\mathbf{n2}}<1$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, tail ≠ 'T'${\mathbf{tail}}\ne \text{'T'}$, 'U'$\text{'U'}$ or 'L'$\text{'L'}$.
W ifail = 3${\mathbf{ifail}}=3$
The pooled sample values are all the same, that is the variance of u = 0.0${\mathbf{u}}=0.0$.

## Accuracy

The approximate tail probability, p$p$, returned by nag_nonpar_test_mwu (g08ah) is a good approximation to the exact probability for cases where max (n1,n2)30$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({n}_{1},{n}_{2}\right)\ge 30$ and (n1 + n2)40$\left({n}_{1}+{n}_{2}\right)\ge 40$. The relative error of the approximation should be less than 10%$10%$, for most cases falling in this range.

The time taken by nag_nonpar_test_mwu (g08ah) increases with n1${n}_{1}$ and n2${n}_{2}$.

## Example

```function nag_nonpar_test_mwu_example
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];
tail = 'Lower-tail';
[u, unor, p, ties, ranks, ifail] = nag_nonpar_test_mwu(x, y, tail)
```
```

u =

86

unor =

-2.8039

p =

0.0025

ties =

1

ranks =

29.5000
1.5000
24.5000
5.0000
24.5000
5.0000
16.0000
5.0000
16.0000
5.0000
38.0000
5.0000
16.0000
9.5000
12.0000
9.5000
39.0000
1.5000
16.0000
9.5000
36.0000
9.5000
36.0000
16.0000
36.0000
16.0000
33.0000
16.0000
33.0000
20.5000
33.0000
20.5000
29.5000
24.5000
29.5000
24.5000
29.5000
24.5000
24.5000

ifail =

0

```
```function g08ah_example
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];
tail = 'Lower-tail';
[u, unor, p, ties, ranks, ifail] = g08ah(x, y, tail)
```
```

u =

86

unor =

-2.8039

p =

0.0025

ties =

1

ranks =

29.5000
1.5000
24.5000
5.0000
24.5000
5.0000
16.0000
5.0000
16.0000
5.0000
38.0000
5.0000
16.0000
9.5000
12.0000
9.5000
39.0000
1.5000
16.0000
9.5000
36.0000
9.5000
36.0000
16.0000
36.0000
16.0000
33.0000
16.0000
33.0000
20.5000
33.0000
20.5000
29.5000
24.5000
29.5000
24.5000
29.5000
24.5000
24.5000

ifail =

0

```