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Test for Main Effect of Factor A
×
Factor B Interaction
MS AB
MS E
Test statistic: F 0 , ( a 1)( b 1) , ab ( n 1) =
with a numerator degree of freedom
equal to (a
1)( b
1) and a denominator degree of freedom equal to ab(n
-1).
H 0
hypothesis rejection region:
F 0 , ( a 1)( b 1) , ab ( n 1)
F α, ( a 1)( b 1) , ab ( n 1)
with a numerator degree of freedom equal to (a
1)( b
1) and a de-
nominator degree of freedom equal to ab(n
1)
The interaction null hypothesis is tested first by computing the Fisher F test of
the mean square for interaction with the mean square for error. If the test results in
nonrejection of the null hypothesis, then proceed to test the main effects of the factors.
If the test results in a rejection of the null hypothesis, then we conclude that the two
factors interact in the mean response ( y ). If the test of interaction is significant, then
a multiple comparison method such as Tukey's grouping procedure can be used to
compare any or all pairs of the treatment means.
Next, test the two null hypotheses that the mean response is the same at each level
of factor A and factor B by computing the Fisher F test of the mean square for each
factor main effect of the mean square for error. If one or both tests result in rejection
of the null hypothesis, then we conclude that the factor affects the mean response
( y ). If both tests result in nonrejection, then an apparent contradiction has occurred.
Although the treatment means apparently differ, the interaction and main effect tests
have not supported that result. Further experimentation is advised. If the test for one
or both main effects is significant, then a multiple comparison is needed, such as the
Tukey grouping procedure, to compare the pairs of the means corresponding with the
levels of the significant factor(s).
The results and data analysis methods discussed can be extended to the general
case in which there are a levels of factor A, b levels of factor B, c levels of factor C, and
so on arranged in a factorial experiment. There will be abc
n total number of trials
if there are n replicates. Clearly, the number of trials needed to run the experiment will
increase quickly with the increase in the number of factors and the number of levels.
In practical application, we rarely use a general full factorial experiment for more
than two factors. Two-level factorial experiments are the most popular experimental
methods.
...
REFERENCES
El-Haik, Basem, S. (2005), Axiomatic Quality: Integrating Axiomatic Design with Six-Sigma,
Reliability, and Quality , Wiley-Interscience, New York.
El-Haik, Basem S., and Mekki, K (2008), Medical Device Design for Six Sigma: A Road Map
for Safety and Effectiveness , 1st Ed., Wiley-Interscience, New York.
El-Haik, Basem S., and Roy, D. (2005), Service Design for Six Sigma: A Roadmap for Excel-
lence , Wiley-Interscience, New York.
Halstead, M. H. (1977), Elements of Software Science , Elsevier, Amsterdam, The Netherlands.
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