NAG Library Routine Document

G08AGF

The Wilcoxon one-sample signed rank test may be used to test whether a particular sample came from a population with a specified median. It is assumed that the population distribution is symmetric. The data consists of a single sample of

n

observations denoted by

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

. This sample may arise from the difference between pairs of observations from two matched samples of equal size taken from two populations, in which case the test may be used to test whether the median of the first population is the same as that of the second population.

The hypothesis under test,

H_{0}

, often called the null hypothesis, is that the median is equal to some given value

(X_{med})

, and this is to be tested against an alternative hypothesis

H_{1}

which is

$H_{1}$ : population median $\neq X_{med}$ ; or
$H_{1}$ : population median $> X_{med}$ ; or
$H_{1}$ : population median $< X_{med}$ ,

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 5).

The Wilcoxon test differs from the Sign test (see G08AAF) in that the magnitude of the scores is taken into account, rather than simply the direction of such scores.

The test procedure is as follows

(a)	For each $x_{i}$ , for $i = 1, 2, \dots, n$ , the signed difference $d_{i} = x_{i} - X_{med}$ is found, where $X_{med}$ is a given test value for the median of the sample.
(b)	The absolute differences $\|d_{i}\|$ are ranked with rank $r_{i}$ and any tied values of $\|d_{i}\|$ are assigned the average of the tied ranks. You may choose whether or not to ignore any cases where $d_{i} = 0$ by removing them before or after ranking (see the description of the parameter ZER in Section 5).
(c)	The number of nonzero $d_{i}$ is found.
(d)	To each rank is affixed the sign of the $d_{i}$ to which it corresponds. Let $s_{i} = sign (d_{i}) r_{i}$ .
(e)	The sum of the positive-signed ranks, $W = \sum_{s_{i} > 0} s_{i} = \sum_{i = 1}^{n} \max (s_{i}, 0.0)$ , is calculated.

G08AGF returns

(a)

the test statistic

W

;

(b)

the number

n_{1}

of nonzero

d_{i}

;

(c)

the approximate Normal test statistic

z

, where

z = \frac{(W - \frac{n_{1} (n_{1} + 1)}{4}) - sign (W - \frac{n_{1} (n_{1} + 1)}{4}) \times \frac{1}{2}}{\sqrt{\frac{1}{4} \sum_{i = 1}^{n} s_{i}^{2}}};

(d)

the tail probability,

p

, corresponding to

W

, depending on the choice of the alternative hypothesis,

H_{1}

n_{1} \leq 80

p

is computed exactly; otherwise, an approximation to

p

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

The value of

p

can be used to perform a significance test on the median against the alternative hypothesis. 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

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 Parameters

1: N – INTEGERInput

On entry:

n

, the size of the sample.

Constraint:

N \geq 1

2: X(N) – REAL (KIND=nag_wp) arrayInput

On entry: the sample observations,

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

3: XME – REAL (KIND=nag_wp)Input

On entry: the median test value,

X_{med}

4: TAIL – CHARACTER(1)Input

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

$TAIL ='T'$: A two tailed probability is calculated and the alternative hypothesis is $H_{1}$ : population median $\neq X_{med}$ .
$TAIL ='U'$: An upper tailed probability is calculated and the alternative hypothesis is $H_{1}$ : population median $> X_{med}$ .
$TAIL ='L'$: A lower tailed probability is calculated and the alternative hypothesis is $H_{1}$ : population median $< X_{med}$ .

Constraint:

TAIL ='T'

'U'

'L'

5: ZER – CHARACTER(1)Input

On entry: indicates whether or not to include the cases where

d_{i} = 0.0

in the ranking of the

d_{i}

's.

$ZER ='Y'$: All $d_{i} = 0.0$ are included when ranking.
$ZER ='N'$: All $d_{i} = 0.0$ , are ignored, that is all cases where $d_{i} = 0.0$ are removed before ranking.

Constraint:

ZER ='Y'

'N'

6: W – REAL (KIND=nag_wp)Output

On exit: the Wilcoxon rank sum statistic,

W

, being the sum of the positive ranks.

7: WNOR – REAL (KIND=nag_wp)Output

On exit: the approximate Normal test statistic,

z

, as described in Section 3.

8: P – REAL (KIND=nag_wp)Output

On exit: the tail probability,

p

, as specified by the parameter TAIL.

9: N1 – INTEGEROutput

On exit: the number of nonzero

d_{i}

's,

n_{1}

10: WRK( $3 \times N$ ) – REAL (KIND=nag_wp) arrayWorkspace

11: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$TAIL \neq'T'$ , $'U'$ or $'L'$ .
or	$ZER \neq'Y'$ or $'N'$ .

$IFAIL = 2$

On entry,

N < 1

$IFAIL = 3$: The whole sample is identical to the given median test value.

7 Accuracy

The approximation used to calculate

p

when

n_{1} > 80

will return a value with a relative error of less than

10 %

for most cases. The error may increase for cases where there are a large number of ties in the sample.

8 Further Comments

The time taken by G08AGF increases with

n_{1}

, until

n_{1} > 80

, from which point on the approximation is used. The time decreases significantly at this point and increases again modestly with

n_{1}

for

n_{1} > 80

9 Example

This example performs the Wilcoxon signed rank test on two matched samples of size

8

, taken from two populations. The distribution of the differences between pairs of observations from the two populations is assumed to be symmetric. The test is used to test whether the medians of the two distributions of the populations are equal or not. The test statistic, the approximate Normal statistic and the two tailed probability are computed and printed.

NAG Library Routine DocumentG08AGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G08AGF