Applied Statistics and Probability for Engineers

15-5 NONPARAMETRIC METHODS IN THE ANALYSIS OF VARIANCE 589

15-5 NONPARAMETRIC METHODS IN THE ANALYSIS

OF VARIANCE

15-5.1 Kruskal-Wallis Test

The single-factor analysis of variance model developed in Chapter 13 for comparing a

population means is

(15-9)

In this model, the error terms ijare assumed to be normally and independently distributed with

mean zero and variance ^2. The assumption of normality led directly to the F-test described in

Chapter 13. The Kruskal-Wallis test is a nonparametric alternative to the F-test; it requires only

that the ijhave the same continuous distribution for all factor levels i 1, 2, ...,a.

Suppose that is the total number of observations. Rank all Nobservations

from smallest to largest, and assign the smallest observation rank 1, the next smallest

rank 2,...,and the largest observation rank N. If the null hypothesis

is true, the Nobservations come from the same distribution, and all possible assignments of

the Nranks to the asamples are equally likely, we would expect the ranks 1, 2,... , Nto be

mixed throughout the asamples. If, however, the null hypothesis H 0 is false, some samples

will consist of observations having predominantly small ranks, while other samples will con-

sist of observations having predominantly large ranks. Let Rijbe the rank of observation Yij,

and let Ri. and. denote the total and average of the niranks in the ith treatment. When the

null hypothesis is true,

and

The Kruskal-Wallis test statistic measures the degree to which the actual observed average

ranks. differ from their expected value (N1)2. If this difference is large, the null

hypothesis H 0 is rejected. The test statistic is

Ri

E 1 Ri. 2 

1

ni^ a

ni

j 1

E 1 Rij 2 

N 1

2

E 1 Rij 2 

N 1

2

Ri

H 0 :  1  2 .. .a

N gai 1 ni

Yijiij e

i1, 2,... , a

j1, 2,... , ni

H (15-10)

12

N 1 N 12

(^) a
a
i 1
ni aRi.
N 1
2
b
2
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