Factorial Experiments
When something we wish to predict depends on two or more factors what we do is fit a multilinear
model i.e. make predictions of the form
ˆ
y
i
=
b
p
x
p,i
+
b
p

1
x
p

1
,i
+
· · ·
+
b
1
x
1
,i
+
b
0
by calculating
b
p
,
b
p

1
,
. . . , b
0
to minimise
∑
n
i
=1
(
y
i

ˆ
y
i
)
2
by using some software package. We now
have to deal with the possibility that different factors interact as their levels are changed.
To fit any model we need data and this should come from an experiment which is properly
designed
i.e. the values of the
x
j,i
must be chosen appropriately.
An experiment in which several
controllable factors may affect the results should use a
factorial design
– all possible combinations of
factor levels (the different possible values of the
x
j,i
) should be investigated in the
replicates
(indexed
by
i
) of the experiment.
The purpose of the experiment is to discover how varying the levels of
controllable factors
in any combination
affects output. Where possible more than one trial should
be run at each selected combination of factor levels – this replication makes possible the estimation
of the variation caused by uncontrolled factors. Combinations should be tested in
randomised order
to avoid confounding with uncontrolled factors. Designs that do some fraction e.g. one half or one
sixteenth of the total number of combinations exist and are called fractional factorial designs but
there won’t be time to discuss them in this set of lectures.
Consider the following experiments in which there are two factors each of which are tested at two
levels. When factor levels are not numerical we modify our prediction equation to
ˆ
y
i
=
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 Spring '10
 Various
 Statistics

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