	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

nag_nonpar_test_median (g08ac) 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 nag_stat_contingency_table (g01af)).

H_{0}

is rejected by a test of chosen size

α

p < α

References

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

Parameters

Compulsory Input Parameters

1: $x (n)$ – double array: 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: $n1$ – int64int32nag_int scalar: The size of the first sample $n_{1}$ .

Constraint: $1 \leq n1 < n$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array x.
The total of the two sample sizes, $n$ ( $= n_{1} + n_{2}$ ).

Constraint: $n \geq 2$ .

Output Parameters

1: $i1$ – int64int32nag_int scalar: The number of scores in the first sample which lie below the pooled median, $i_{1}$ .
2: $i2$ – int64int32nag_int scalar: The number of scores in the second sample which lie below the pooled median, $i_{2}$ .
3: $p$ – double scalar: The tail probability $p$ corresponding to the observed dichotomy of the two samples.
4: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,

n < 2

$ifail = 2$

On entry,	$n1 < 1$ ,
or	$n1 \geq n$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The probability returned should be accurate enough for practical use.

Further Comments

The time taken by nag_nonpar_test_median (g08ac) is small, and increases with

n

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.

Open in the MATLAB editor: g08ac_example

function g08ac_example


fprintf('g08ac example results\n\n');

x = [13;  6; 12;  7; 12;  7; 10;  7;
     10;  7; 10;  7; 10;  8;  9;  8;
     17;  6; 16;  8; 15;  8; 15; 10;
     15; 10; 14; 10; 14; 11; 14; 11;
     13; 12; 13; 12; 13; 12; 12];

n  = numel(x);
n1 = int64(16);
n2 = n-n1;

fprintf('Median test\n\n');
fprintf('Data values\n\n');

fprintf('     Group 1 ');
for j = 1:floor(n1/8)
  i1 = (j-1)*8 + 1;
  i2 = min(n1,i1+7);
  fprintf('%4.0f',x(i1:i2));
  fprintf('\n             ');
end
fprintf('\n     Group 2 ');
for j = 1:floor(n2/8)
  i1 = (j-1)*8 + 1;
  i2 = min(n2,i1+7);
  fprintf('%4.0f',x(i1+n1:i2+n1));
  fprintf('\n             ');
end

[i1, i2, p, ifail] = g08ac( ...
                            x, n1);

fprintf('\n%6d scores below median in group 1\n',i1);
fprintf('%6d scores below median in group 2\n\n',i2);
fprintf('     Significance %10.5f\n',p);

g08ac example results

Median test

Data values

     Group 1   13   6  12   7  12   7  10   7
               10   7  10   7  10   8   9   8
             
     Group 2   17   6  16   8  15   8  15  10
               15  10  14  10  14  11  14  11
               13  12  13  12  13  12  12
             
    13 scores below median in group 1
     6 scores below median in group 2

     Significance    0.00088

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox