nag_kruskal_wallis_test (g08afc) (PDF version)
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NAG C Library Manual

# NAG Library Function Documentnag_kruskal_wallis_test (g08afc)

## 1  Purpose

nag_kruskal_wallis_test (g08afc) performs the Kruskal–Wallis one-way analysis of variance by ranks on $k$ independent samples of possibly unequal sizes.

## 2  Specification

 #include #include
 void nag_kruskal_wallis_test (Integer k, const Integer l[], const double x[], Integer lx, double *h, double *p, NagError *fail)

## 3  Description

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:
 $H = 12 N N+1 ∑ i=1 k R i 2 l i - 3 N+1 , where N = ∑ i=1 k l i ,$
i.e., $N$ is the total number of observations. If there are tied scores, $H$ is corrected by dividing by:
 $1 - ∑ t 3 - t N 3 - N$
where $t$ is the number of tied scores in a group and the summation is over all tied groups.
nag_kruskal_wallis_test (g08afc) 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 nag_median_test (g08acc)) or the Mann–Whitney $U$ test (see nag_mann_whitney (g08amc)) is more appropriate).

## 4  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

## 5  Arguments

1:     kIntegerInput
On entry: the number of samples, $k$.
Constraint: ${\mathbf{k}}\ge 2$.
2:     l[k]const IntegerInput
On entry: ${\mathbf{l}}\left[\mathit{i}-1\right]$ must contain the number of observations ${l}_{\mathit{i}}$ in sample $\mathit{i}$, for $\mathit{i}=1,2,\dots ,k$.
Constraint: ${\mathbf{l}}\left[\mathit{i}-1\right]>0$, for $\mathit{i}=1,2,\dots ,k$.
3:     x[lx]const doubleInput
On entry: the elements of x must contain the observations in the k groups. The first ${l}_{1}$ elements must contain the scores in the first group, the next ${l}_{2}$ those in the second group, and so on.
4:     lxIntegerInput
On entry: the total number of observations, $N$.
Constraint: ${\mathbf{lx}}={\sum }_{i=1}^{k}{\mathbf{l}}\left[i-1\right]$.
5:     hdouble *Output
On exit: the value of the Kruskal–Wallis test statistic, $H$.
6:     pdouble *Output
On exit: the approximate significance, $p$, of the Kruskal–Wallis test statistic.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_ARRAY_CONS
The contents of array l are not valid.
Constraint: ${\mathbf{l}}\left[\mathit{i}-1\right]>0$, for $\mathit{i}=1,2,\dots ,k$.
NE_INT
On entry, ${\mathbf{lx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lx}}={\sum }_{\mathit{i}=1}^{k}{\mathbf{l}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,k$.
NE_INT_ARG_LT
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 2$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_X_IDEN
On entry, all elements of x are equal.

## 7  Accuracy

For estimates of the accuracy of the significance $p$, see nag_prob_chi_sq (g01ecc). The ${\chi }^{2}$ approximation is acceptable unless $k=3$ and ${l}_{1},{l}_{2}$ or ${l}_{3}\le 5$.

## 8  Further Comments

The time taken by nag_kruskal_wallis_test (g08afc) is small, and increases with $N$ and $k$.
If $k=2$, the Median test (see nag_median_test (g08acc)) or the Mann–Whitney $U$ test (see nag_mann_whitney (g08amc)) is more appropriate.

## 9  Example

This example is taken from Moore et al. 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.

### 9.1  Program Text

Program Text (g08afce.c)

### 9.2  Program Data

Program Data (g08afce.d)

### 9.3  Program Results

Program Results (g08afce.r)

nag_kruskal_wallis_test (g08afc) (PDF version)
g08 Chapter Contents
g08 Chapter Introduction
NAG C Library Manual