naginterfaces.library.nonpar.test_​median

naginterfaces.library.nonpar.test_median(x, n1)

test_median performs the Median test on two independent samples of possibly unequal size.

For full information please refer to the NAG Library document for g08ac

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g08/g08acf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

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.

n1int

The size of the first sample $n_1$.

Returns

i1int

The number of scores in the first sample which lie below the pooled median, $i_1$.

i2int

The number of scores in the second sample which lie below the pooled median, $i_2$.

pfloat

The tail probability $p$ corresponding to the observed dichotomy of the two samples.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $2$)

On entry, $n1=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $1\le n1<n$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

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

$$x_{n_1+1},x_{n_1+2},\dots ,x_n\text{,}$$

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 $\ge$ 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 $n_2/2$.

test_median returns:

1. the frequencies $i_1$ and $i_2$;

2. 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 ${\chi }_1^2$ approximation is used (see stat.contingency_table).

$H_0$ is rejected by a test of chosen size $\alpha$ if $p<\alpha$.

References

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