Introduction to Probability and Statistics for Engineers and Scientists

10.5Two-Factor Analysis of Variance: Testing Hypotheses 461

We base our test of the null hypothesisH 0 that there is no row effect, on the ratio of
the two estimators ofσ^2. Specifically, we use the test statistic

TS=

SSr/(m−1)

SSe/(n−1)(m−1)

Because the estimators can be shown to be independent whenH 0 is true, it follows that
the significance levelαtest is to

reject H 0 if TS≥Fm−1,(n−1)(m−1),α

do not reject H 0 otherwise

Alternatively, the test can be performed by calculating thep-value. If the value of the test
statistic isv, then thep-value is given by

p-value=P{Fm−1,(n−1)(m−1)≥v}

A similar test can be derived for testing the null hypothesis that there is no column
effect — that is, that all theβjare equal to 0. The results are summarized in Table 10.3.
Program 10.5 will do the computations and give thep-value.

TABLE 10.3 Two-Factor ANOVA

Sum of Squares Degrees of Freedom

Row SSr=n∑mi= 1 (Xi.−X..)^2 m− 1
Column SSc=∑nj= 1 (X.j−X..)^2 n− 1
Error SSe=

∑m

i= 1

∑n
j= 1 (Xij−Xi.−X.j+X..)^2 (n−1)(m−1)
LetN=(n– 1)(m–1)
Null Test Significance p-value if
Hypothesis Statistic LevelαTest TS=v

Allαi= 0

SSr/(m−1)

SSe/N Reject if P{Fm−1,N≥v}

TS≥Fm−1,N,α

Allβj= 0 SSSSc/(n−1)
e/N

Reject if P{Fn−1,N≥v}

TS≥Fn−1,N,α

EXAMPLE 10.5a The following data* represent the number of different macroinvertebrate
species collected at 6 stations, located in the vicinity of a thermal discharge, from 1970 to
1977.

• Taken from Wartz and Skinner, “A 12 year macroinvertebrate study in the vicinity of 2 thermal discharges to the
  Susquehanna River near York, Haven, PA.”Jour. of Testing and Evaluation. Vol. 12. No. 3, May 1984, 157–163.