7.
Product and Process Comparisons
7.4.
Comparisons based on data from more than two processes
7.4.3.
Are the means equal?
7.4.3.7.
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The two-way ANOVA
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Definition of a factorial experiment
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The 2-way ANOVA is probably the most popular layout in the
Design of Experiments. To begin with,
let us define a factorial experiment:
An experiment that utilizes every combination of factor levels
as treatments is called a factorial experiment.
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Model for the two-way factorial experiment
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In a factorial experiment with factor A at a levels and factor
B at b levels, the model for the
general layout can be written as
where  is the
overall mean response,
 i
is the effect due to the i-th level of factor A,
 j
is the effect due to the j-th level of factor B and
 ij
is the effect due to any interaction between the i-th level of A
and the j-th level of B.
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Fixed factors and fixed effects models
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At this point, consider the levels of factor A and of factor B chosen
for the experiment to be the only levels of interest to the experimenter
such as predetermined levels for temperature settings or the length of
time for process step. The factors A and B are said to be fixed
factors and the model is a fixed-effects model. Random
actors will be discussed later.
When an a x b factorial experiment is conducted with an
equal number of observations per treatment combination, the total
(corrected) sum of squares is partitioned as:
SS(total) = SS(A) + SS(B) + SS(AB) + SSE
where AB represents the interaction between A and B.
For reference, the formulas for the sums of squares are:
![SS(A) = r*b*SUM[i=1 to a][(ybar(i.) - ybar(..))^2;
SS(B) = r*a*SUM[j=1 to b][(Ybar(j.) - ybar(..))^2;
SS(AB) = r*SUM[j=1 to b][SUM[i=1 to a]
[(ybar(ij) - ybar(i) - ybar(j) + ybar(..))^2]];
SSE = SUM[k=1 to r][SUM[j=1 to b][SUM[i=1 to a][(y(ijk) - ybar(ij.))^2;
SS(Total) = SUM[k=1 to r][SUM[j=1 to b][SUM[i=1 to a]
[(y(ijk) - ybar(...))^2](eqns/twoss.gif)
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The breakdown of the total (corrected for the mean) sums
of squares
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The resulting ANOVA table for an a x b factorial
experiment is
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| Source |
SS |
df |
MS |
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| Factor A |
SS(A) |
(a - 1) |
MS(A) = SS(A)/(a-1) |
| Factor B |
SS(B) |
(b - 1) |
MS(B) = SS(B)/(b-1) |
| Interaction AB |
SS(AB) |
(a-1)(b-1) |
MS(AB)=
SS(AB)/(a-1)(b-1) |
| Error |
SSE |
(N - ab) |
SSE/(N - ab) |
| Total (Corrected) |
SS(Total) |
(N - 1) |
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| The ANOVA
table can be used to test
hypotheses
about the effects and interactions |
The various hypotheses
that can be tested using this ANOVA table concern whether the different
levels of Factor A, or Factor B, really make a difference in the response,
and whether the AB interaction is significant (see previous discussion
of
ANOVA hypotheses). |
