naginterfaces.library.nonpar.test_​kruskal

naginterfaces.library.nonpar.test_kruskal(x, l)

test_kruskal performs the Kruskal–Wallis one-way analysis of variance by ranks on $k$ independent samples of possibly unequal sizes.

For full information please refer to the NAG Library document for g08af

https://www.nag.com/numeric/nl/nagdoc_29.3/flhtml/g08/g08aff.html

Parameters

xfloat, array-like, shape $\left(\text{lx}\right)$

The elements of $x$ must contain the observations in the $\text{k}$ samples. The first $l_1$ elements must contain the scores in the first sample, the next $l_2$ those in the second sample, and so on.

lint, array-like, shape $\left(k\right)$

$l\left[\text{i}\right]$ must contain the number of observations $l_{\text{i}}$ in sample $\text{i}$, for $\text{i}=1,2,\dots ,k$.

Returns

hfloat

The value of the Kruskal–Wallis test statistic, $H$.

pfloat

The approximate significance, $p$, of the Kruskal–Wallis test statistic.

Raises

NagValueError

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 2$.

(errno $2$)

On entry, $i=⟨value⟩$ and $l[i-1]=⟨value⟩$.

Constraint: $l[i-1]>0$.

(errno $3$)

On entry, ${\sum }_i\left(l\right[i-1\left]\right)=⟨value⟩$ and $\text{lx}=⟨value⟩$.

Constraint: ${\sum }_i\left(l\right[i-1\left]\right)=\text{lx}$.

(errno $4$)

On entry, all the observations were equal.

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 Kruskal–Wallis test investigates the differences between scores from $k$ independent samples of unequal sizes, the $i$th sample containing $l_i$ observations. The hypothesis under test, $H_0$, often called the null hypothesis, is that the samples come from the same population, and this is to be tested against the alternative hypothesis $H_1$ that they come from different populations.

The test proceeds as follows:

1. The pooled sample of all the observations is ranked. Average ranks are assigned to tied scores.

2. The ranks of the observations in each sample are summed, to give the rank sums $R_{\text{i}}$, for $\text{i}=1,2,\dots ,k$.

3. The Kruskal–Wallis’ test statistic $H$ is computed as:

   $$H=\frac{12}{N(N+1)}\sum _{i=1}^k\frac{R_i^2}{l_i}-3(N+1)\text{, where}N=\sum _{i=1}^k{l}_i\text{,}$$

   i.e., $N$ is the total number of observations. If there are tied scores, $H$ is corrected by dividing by:

   $$1-\frac{\sum (t^3-t)}{N^3-N}$$

   where $t$ is the number of tied scores in a sample and the summation is over all tied samples.

test_kruskal returns the value of $H$, and also an approximation, $p$, to the probability of a value of at least $H$ being observed, $H_0$ is true. ($H$ approximately follows a ${\chi }_{k-1}^2$ distribution). $H_0$ is rejected by a test of chosen size $\alpha$ if $p<\alpha \text{.}$ The approximation $p$ is acceptable unless $k=3$ and $l_1$, $l_2$ or $l_3\le 5$ in which case tables should be consulted (e.g., O of Siegel (1956)) or $k=2$ (in which case the Median test (see test_median()) or the Mann–Whitney $U$ test (see test_mwu()) is more appropriate).

References

Moore, P G, Shirley, E A and Edwards, D E, 1972, Standard Statistical Calculations, Pitman

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