Analysis of Variance for Standard Designs

1
Analysis of Variance for Standard Designs

• Chapter 15

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Unbiased Estimates
When is true, both
MST and MSE are unbiased estimates of , the
variance of the experimental error. That is,
under Ho, both have a mean value in repeated
sampling, called the expected mean squares,
equal to .
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Expected Mean Squares

• Under ,

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Expected Mean Squares
When is true, both MST and MSE are unbiased
estimates of ,
the variance of the experimental
error.
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Factorial Treatment Structure in a Completely
Randomized Design
A factorial experiment is an experiment in which
the response y is observed at all factor-level
combinations of the independent variables.
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Figure 15.6a Illustration of the Absence of
Interaction in a 2 x 2 Factorial Experiment
Mean response
Factors A and B do not interact
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Figure 15.6b,c Illustration of the Presence of
Interaction in a 2 x 2 Factorial Experiment
Factors A and B interact
Level 1, factor B Level 2, factor B
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Table 15.25 Expected Values for a 2 x 2
Factorial Experiment
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Table 15.26 Expected Values for a 2 x 2
Factorial Experiment, with Replications
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Definition 15.4
Two factors A and B are said to interact if the
difference in mean responses for two levels of
one factor is not constant across levels of the
second factor.
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Profile Plot

• See Figure 15.6
• Used to amplify the notion of interaction when
  no interaction is present, the difference in the
  mean response between two levels of one factor is
  the same for levels of the other factor.

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Table 15.27 AOV Table for a Completely
Randomized Two-Factor Factorial Experiment
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Illustration of Significant, Orderly Interaction
Figure 15.8 Profile plot in which interactions
are present, but interactions are orderly
Level 3, factor B Level 2, factor B Level 1,
factor B
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Illustration of Significant, Disorderly
Interaction
Figure 15.9 Profile plot in which interactions
are present, and interactions are disorderly
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Factorial Treatment Structure in a Randomized
Complete Block Design
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Estimation of Treatment Differences and
Comparisons of Treatment Means
100(1-?) Confidence Interval for the Difference
in Treatment Means
where s? is the square root of MSE in the AOV
table and t?/2 can be obtained from Table 2 in
the Appendix for a ?/2 and the degrees of
freedom for MSE.
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Multiple Comparison Procedures