kruskalWallisTest

Performs a Kruskal-Wallis test for identical population medians.

Synopsis

kruskalWallisTest (ni, y)

Required Arguments

int ni[] (Input)

Array of length nGroups containing the number of responses for each of the nGroups groups.

float y[] (Input)

Array of length ni[0] + ... + ni[nGroups-1] that contains the responses for each of the nGroups groups. y must be sorted by group, with the ni[0] observations in group 1 coming first, the ni[1] observations in group two coming second, and so on.

Return Value

Array of length 4 containing the Kruskal-Wallis statistics.

i	stat[i]
0	Kruskal-Wallis H statistic.
1	Asymptotic probability of a larger H under the null hypothesis of identical population medians.
2	H corrected for ties.
3	Asymptotic probability of a larger H (corrected for ties) under the null hypothesis of identical populations.

Optional Arguments

fuzz, float (Input)Constant used to determine ties in y. If (after sorting) |y[i] – y[i + 1]| is less than or equal to fuzz, then a tie is counted. fuzz must be nonnegative.

Description

The function kruskalWallisTest generalizes the Wilcoxon two-sample test computed by routine wilcoxonRankSum to more than two populations. It computes a test statistic for testing that the population distribution functions in each of K populations are identical. Under appropriate assumptions, this is a nonparametric analogue of the one-way analysis of variance. Since more than two samples are involved, the alternative is taken as the analogue of the usual analysis of variance alternative, namely that the populations are not identical.

The calculations proceed as follows: All observations are ranked regardless of the population to which they belong. Average ranks are used for tied observations (observations within fuzz of each other). Missing observations (observations equal to NaN, not a number) are not included in the ranking. Let \(R_i\) denote the sum of the ranks in the i-th population. The test statistic H is defined as:

\[H = \frac{1}{S^2} \sum_{i=1}^{K} \left(\frac{R_i^2}{n_i} - \frac{N(N+1)^2}{4}\right)\]

where N is the total of the sample sizes, \(n_i\) is the number of observations in the i-th sample, and \(S^2\) is computed as the (bias corrected) sample variance of the \(R_i\).

The null hypothesis is rejected when stat[3] (or stat[1]) is less than the significance level of the test. If the null hypothesis is rejected, then the procedures given in Conover (1980, page 231) may be used for multiple comparisons. The routine kruskalWallisTest computes asymptotic probabilities using the chi-squared distribution when the number of groups is 6 or greater, and a Beta approximation (see Wallace 1959) when the number of groups is 5 or less. Tables yielding exact probabilities in small samples may be obtained from Owen (1962).

Example

The following example is taken from Conover (1980, page 231). The data represents the yields per acre of four different methods for raising corn. Since \(H=25.5\), the four methods are clearly different. The warning error is always printed when the Beta approximation is used, unless printing for warning errors is turned off.

from numpy import *
from pyimsl.stat.kruskalWallisTest import kruskalWallisTest
from pyimsl.stat.writeMatrix import writeMatrix

ni = [9, 10, 7, 8]
y = array([83., 91., 94., 89., 89., 96., 91., 92., 90., 91., 90.,
           81., 83., 84., 83., 88., 91., 89., 84., 101., 100., 91.,
           93., 96., 95., 94., 78., 82., 81., 77., 79., 81., 80.,
           81.])
fuzz = .001
stat = []
rlabel = ["H (no ties)    =",
          "Prob (no ties) =",
          "H (ties)       =",
          "Prob (ties)    ="]

stat = kruskalWallisTest(ni, y, fuzz=fuzz)

writeMatrix(" ", stat, rowLabels=rlabel, column=True)

Output

***
*** Warning error issued from IMSL function kruskalWallisTest:
*** The chi-squared degrees of freedom are less than 5, so the Beta approximation is used.
***
 
               
H (no ties)    =        25.46
Prob (no ties) =         0.00
H (ties)       =        25.63
Prob (ties)    =         0.00