C532: ANOVA & Regression Modeling
Nov 29, 2011
Y. Cao
1
Topics:
learn about
factorial design
and
two-way ANOVA model
.
Datasets
: np423aug2007.sav, C530_clbp.sav
Two way
ANOVA, we mean that we have
2 explanatory variables
of categorical
type, x
1
and x
2.
A
factorial design
, by definition, we mean that it’s an experiment designed to explore
the effects of several (2 or more) treatment factors on a single outcome (or dependent
as in SPSS) variable.
Each factor (because it affects the outcome so we call it a
factor
) takes a small
number of different values (typically 2, 3 or 4), which may or may not be quantitative
and if it is NOT quantitative we call the values of the factor (variable)
levels
of the
factor.
The
main effects
(because it is the primary factors in the study) compare the mean
values of the observations at the different levels of a factor, while the
interaction
combines effects of two variables.
The equation may look like
The sum of separate effects of the two variables in an interaction is
not
equivalent to
that of the combined effects of them. The main effect is the mean difference at different
levels for the outcome. In some experiments, the difference in outcome between the
levels of one factor is not the same at all levels of the other factor(s). When this
happens, we say that there is an interaction between the factors.
A significant interaction (with p value < 0.05 for the term) may mask the significance of
main effects if there is one. Therefore, if there is an interaction, the main effects of the
factors involved in the interaction may not so meaningful. We cannot report it with clear
explanations, that is, we do not report the main effects under this situation!
Based on these definitions (you can see that mathematics is something that consists of
a series of definitions!), if the interaction has significant affect on the outcome then it
makes the explanation of individual main effects harder or even impossible. That is,
the two factors (variables) only affect the outcome “simultaneously” but not separable!
Usually, we use two-way interaction like x
1
*x
2
for our design but theoretically we can
have three way interaction like x
1
*x
2
*x
3
or even high order interactions.

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