Information Technology Reference
In-Depth Information
19.3.4
Two-Way ANOVA
≥
Recently, Nakama et al. [14, 15] have established factorial (
m
-way with
m
2)
ANOVA for fuzzy data. A factorial layout is designed to statistically examine the
effects of two or more factors that each involve at least two levels. For concreteness,
we will describe two-way ANOVA.
Two-way ANOVA statistically determines wether two factors affect a response
significantly. For example, suppose that we measure a response to two different
drugs, drugs 1 and 2, in both men and women. Here the two factors are drug treat-
ment and gender. In statistics, the two factors are described as independent vari-
ables, and the response is described as the dependent variable of this experiment.
Two-way ANOVA tests the following null hypotheses:
(a) Drug treatment has no effect on the response.
(b) Gender has no effect on the response.
(c) There is no interaction between drug treatment and gender in affecting the re-
sponse.
By (c), we mean that the effect of drug treatment does not depend on gender. The
following is an example of interaction between the two factors: For men, the re-
sponse is increased by drug 1 but decreased by drug 2, whereas for women, the
response is decreased by drug 1 but increased by drug 2.
Suppose that we examine the effects of factors 1 and 2 that have
I
and
J
levels,
respectively. In the example described above, drug treatment has two levels (drugs
1 and 2), and gender also has two levels (male and female). Let
X
ijk
denote a fuzzy
random variable that represents the
k
th observed value of the dependent variable
measured under level
i
of factor 1 and under level
j
of factor 2. In two-way ANOVA
(and more generally factorial ANOVA), an unbalanced design may introduce artifi-
cial differential effects of one factor (or of interactions) on the marginal means of
the other factor. Thus we assume that for each level of factor 1 and for each level
of factor 2, there are
K
observations (thus 1
≤
k
≤
K
). In two-way ANOVA, we
consider the following additive statistical model:
X
ijk
=
μ
+
α
i
+
β
j
+
γ
ij
+
ε
ijk
,
(19.1)
where
ε
ijk
represents the random component of the model. In (19.1),
α
i
denotes the
main effect of level
i
of factor 1 on the dependent variable, and
β
j
denotes the main
effect of level
j
of factor 2. The interaction between the two factors is quantified by
γ
ij
, which denotes the effect of concurrently having both level
i
of factor 1 and level
j
of factor 2.
Using this notation, the null hypotheses corresponding to (a)-(c) can be ex-
pressed as follows:
H
(
1
)
0
:
Factor 1 has no effect:
α
1
=
α
2
=
···
=
α
I
.
H
(
2
)
0
:
Factor 2 has no effect:
β
1
=
β
2
=
···
=
β
J
.
H
(
1
,
2
)
0
:
There is no interaction between factors 1 and 2:
γ
1
,
1
=
γ
1
,
2
=
···
=
γ
IJ
.
