NAG Library Routine Document

Integer, Intent (In)	::	n, n1
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	i1, i2
Real (Kind=nag_wp), Intent (In)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	w(n), p

C Header Interface

#include <nagmk26.h>

void	g08acf_ (const double x[], const Integer n, const Integer n1, double w[], Integer i1, Integer i2, double p, Integer ifail)

3

Description

The Median test investigates the difference between the medians of two independent samples of sizes

n_{1}

and

n_{2}

, denoted by:

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

and

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

where

n = n_{1} + n_{2}

The hypothesis under test,

H_{0}

, often called the null hypothesis, is that the medians are the same, and this is to be tested against the alternative hypothesis

H_{1}

that they are different.

The test proceeds by forming a

2 \times 2

frequency table, giving the number of scores in each sample above and below the median of the pooled sample:

	Sample 1	Sample 2	Total
Scores $<$ pooled median	$i_{1}$	$i_{2}$	$i_{1} + i_{2}$
Scores $\geq$ pooled median	$n_{1} - i_{1}$	$n_{2} - i_{2}$	$n - (i_{1} + i_{2})$
Total	$n_{1}$	$n_{2}$	$n$

Under the null hypothesis,

H_{0}

, we would expect about half of each group's scores to be above the pooled median and about half below, that is, we would expect

i_{1}

, to be about

n_{1} / 2

and

i_{2}

to be about

n_{2} / 2

g08acf returns:

(a)	the frequencies $i_{1}$ and $i_{2}$ ;
(b)	the probability, $p$ , of observing a table at least as ‘extreme’ as that actually observed, given that $H_{0}$ is true. If $n < 40$ , $p$ is computed directly (‘Fisher's exact test’); otherwise a $χ_{1}^{2}$ approximation is used (see g01aff).

H_{0}

is rejected by a test of chosen size

α

p < α

4

References

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

5

Arguments

1: $x (n)$ – Real (Kind=nag_wp) arrayInput: On entry: the first $n_{1}$ elements of x must be set to the data values in the first sample, and the next $n_{2}$ ( $= n - n_{1}$ ) elements to the data values in the second sample.
2: $n$ – IntegerInput: On entry: the total of the two sample sizes, $n$ ( $= n_{1} + n_{2}$ ).

Constraint: $n \geq 2$ .
3: $n1$ – IntegerInput: On entry: the size of the first sample $n_{1}$ .

Constraint: $1 \leq n1 < n$ .
4: $w (n)$ – Real (Kind=nag_wp) arrayWorkspace
5: $i1$ – IntegerOutput: On exit: the number of scores in the first sample which lie below the pooled median, $i_{1}$ .
6: $i2$ – IntegerOutput: On exit: the number of scores in the second sample which lie below the pooled median, $i_{2}$ .
7: $p$ – Real (Kind=nag_wp)Output: On exit: the tail probability $p$ corresponding to the observed dichotomy of the two samples.
8: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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, $n = 〈value〉$ .
Constraint: $n \geq 2$ .

$ifail = 2$: On entry, $n1 = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 \leq n1 < n$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The probability returned should be accurate enough for practical use.

8

Parallelism and Performance

g08acf is not threaded in any implementation.

9

Further Comments

The time taken by g08acf is small, and increases with

n

10

Example

This example is taken from page 112 of Siegel (1956). The data relate to scores of ‘oral socialisation anxiety’ in

39

societies, which can be separated into groups of size

16

and

23

on the basis of their attitudes to illness.

NAG Library Routine Document

g08acf (test_median)

▸▿ 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

1

Purpose

2

Specification