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 <nag.h>

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

The routine may be called by the names g08acf or nagf_nonpar_test_median.

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) array Input: 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: $n$ – Integer Input: On entry: the total of the two sample sizes, $n$ ( $= n_{1} + n_{2}$ ).

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

Constraint: $1 \leq n1 < n$ .
4: $w (n)$ – Real (Kind=nag_wp) array Workspace
5: $i1$ – Integer Output: On exit: the number of scores in the first sample which lie below the pooled median, $i_{1}$ .
6: $i2$ – Integer Output: 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: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The probability returned should be accurate enough for practical use.

8 Parallelism and Performance

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

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.

g08ac: FL CL CPP AD PY MB

NAG FL Interfaceg08acf (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

NAG FL Interface
g08acf (test_median)