# NAG Library Routine Document

## 1Purpose

g08aff performs the Kruskal–Wallis one-way analysis of variance by ranks on $k$ independent samples of possibly unequal sizes.

## 2Specification

Fortran Interface
 Subroutine g08aff ( x, lx, l, k, w, h, p,
 Integer, Intent (In) :: lx, l(k), k Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(lx) Real (Kind=nag_wp), Intent (Out) :: w(lx), h, p
#include <nagmk26.h>
 void g08aff_ (const double x[], const Integer *lx, const Integer l[], const Integer *k, double w[], double *h, double *p, Integer *ifail)

## 3Description

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:
(a) The pooled sample of all the observations is ranked. Average ranks are assigned to tied scores.
(b) The ranks of the observations in each sample are summed, to give the rank sums ${R}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,k$.
(c) The Kruskal–Wallis' test statistic $H$ is computed as:
 $H=12NN+1 ∑i=1kRi2li-3N+1, where ​N=∑i=1kli,$
i.e., $N$ is the total number of observations. If there are tied scores, $H$ is corrected by dividing by:
 $1-∑t3-t N3-N$
where $t$ is the number of tied scores in a sample and the summation is over all tied samples.
g08aff 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 g08acf) or the Mann–Whitney $U$ test (see g08ahf) is more appropriate).

## 4References

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

## 5Arguments

1:     $\mathbf{x}\left({\mathbf{lx}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the elements of x must contain the observations in the 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.
2:     $\mathbf{lx}$ – IntegerInput
On entry: $N$, the total number of observations.
Constraint: ${\mathbf{lx}}=\sum _{i=1}^{k}{\mathbf{l}}\left(i\right)$.
3:     $\mathbf{l}\left({\mathbf{k}}\right)$ – Integer arrayInput
On entry: ${\mathbf{l}}\left(\mathit{i}\right)$ must contain the number of observations ${l}_{\mathit{i}}$ in sample $\mathit{i}$, for $\mathit{i}=1,2,\dots ,k$.
Constraint: ${\mathbf{l}}\left(\mathit{i}\right)>0$, for $\mathit{i}=1,2,\dots ,k$.
4:     $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of samples.
Constraint: ${\mathbf{k}}\ge 2$.
5:     $\mathbf{w}\left({\mathbf{lx}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
6:     $\mathbf{h}$ – Real (Kind=nag_wp)Output
On exit: the value of the Kruskal–Wallis test statistic, $H$.
7:     $\mathbf{p}$ – Real (Kind=nag_wp)Output
On exit: the approximate significance, $p$, of the Kruskal–Wallis test statistic.
8:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . 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  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  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, $i=〈\mathit{\text{value}}〉$ and ${\mathbf{l}}\left(i\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{l}}\left(i\right)>0$.
${\mathbf{ifail}}=3$
On entry, ${\sum }_{i}{\mathbf{l}}\left(i\right)=〈\mathit{\text{value}}〉$ and ${\mathbf{lx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\sum }_{i}{\mathbf{l}}\left(i\right)={\mathbf{lx}}$.
${\mathbf{ifail}}=4$
On entry, all the observations were equal.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

For estimates of the accuracy of the significance $p$, see g01ecf. The ${\chi }^{2}$ approximation is acceptable unless $k=3$ and ${l}_{1},{l}_{2}$ or ${l}_{3}\le 5$.

## 8Parallelism and Performance

g08aff is not threaded in any implementation.

The time taken by g08aff is small, and increases with $N$ and $k$.
If $k=2$, the Median test (see g08acf) or the Mann–Whitney $U$ test (see g08ahf) is more appropriate.

## 10Example

This example is taken from Moore et al. (1972). There are $5$ groups of sizes $5$, $8$, $6$, $8$ and $8$. The data represent the weight gain, in pounds, of pigs from five different litters under the same conditions.

### 10.1Program Text

Program Text (g08affe.f90)

### 10.2Program Data

Program Data (g08affe.d)

### 10.3Program Results

Program Results (g08affe.r)