1. Multifactor Analysis of Variance ISE 500 University of Southern California 11/14/2013

2. Outline Single-factor ANOVA Two-factor ANOVA Three-factor ANOVA
Randomized block experiment
Three-factor ANOVA

3. Single-factor ANOVA Analysis of Variance(ANOVA)[1]:
A collection of statistical models used to analyze the differences between group means and their associated procedures
ANOVA provides a statistical test of whether or not the means of several groups are equal
Observed variance in a particular variable is partitioned into components attributable to different sources of variation

4. Single-factor ANOVA Factor Levels Treatment
Variable that is studied in the experiment, e.g. single variable: temperature
Levels
In order to study the effect of a factor on the response, two or more values of factors are used
Treatment
Combination of factor levels

5. Single-factor ANOVA Hypotheses:
Test statistic for single-factor ANOVA is:
Treatment sum of squares(SSTr) is also called between-treatment sum of squares
Error sum of squares(SSE): within-treatment sum of squares
No difference btw treatment
SST is a measure of the total variation in the data—the sum of all squared deviations about the grand mean
SSTr is the amount of variation (between rows) that can be explained by possible differences in the treatment
SSE measures variation that would be present (within rows) whether H0 is true or false, and is thus the part of total variation
that is unexplained by the status of H0.

6. Two-factor ANOVA
In many experimental situations, there are two or more factors that are of simultaneous interest.
Use I to denote the number of levels of the first factor (A) and J to denote the number of levels of the second factor (B).
IJ different treatments: there are IJ possible combinations consisting of one level of factor A and one of factor B.

7. Two-factor ANOVA Example
Compare three different brands of pens and four different wash treatments with respect to their ability to remove marks on a particular type of fabric
The lower the value, the more marks were removed

8. Two-factor ANOVA

9. Two-factor ANOVA Linear model for two-way layout is:
is the true grand mean (mean response averaged over all levels of both factors)
is the effect of factor A at level i (measured as a deviation from ), and is the effect of factor B at level j.
Unbiased (and maximum likelihood) estimators for these parameters are:
No interaction effect

10. Two-factor ANOVA Multiple replicates main effects for factor A
No interaction effect
main effects for factor B
Interaction parameters

11. Two-factor ANOVA Test hypotheses
1. Different levels of factor A have no effect on true average response.
2. There is no factor B effect.

12. Two-factor ANOVA
Sum of squares

13. Two-factor ANOVA
F test

14. Two-factor ANOVA
ANOVA table for previous example

15. Two-factor ANOVA Multiple comparisons Tukey method
Find pairs of sample means differ less than w
E.g. significant differences among the four washing treatments
Washing treatment 1 appears to differ significantly from the other three treatments

16. Randomized block experiment
Single-factor experiment:
Test the effects of treatments, experimental units are assigned to treatments randomly
Heterogeneous units may affect the observed responses
E.g: apply drugs to patients: males and females
Variation exist in males and females would affect the assessment of drug effects

17. Randomized block experiment
A group of homogeneous units
e.g. males, females
For blocking to be effective, units should be arranged so that:
Within-block variation is much smaller than between-block variation

18. Randomized block experiment
Paired comparison is a special case of randomized block design
Similar to two-factor experiment:
One treatment factor: with k levels
One block factor: each block has size of k
Within each block, all treatments are assigned to k units randomly

19. Randomized block experiment
Compare the annual power consumption for five different brands of dehumidifier
Power consumption depends on the prevailing humidity level
Resulting observations (annual kWh)

20. Randomized block experiment
ANOVA
Same procedures as two-factor ANOVA
Block difference is significant
Tukey method is applied to identify significant pair of treatments
Difference btw RBD and Two – way layout
In two-way layout: IJ units of each replicate are randomly assigned to IJ level combinations of two factors. More replicates
In RBD, no block x treatment interaction, single replicate: the interaction sum of squares and the residual sum of squares are identical

21. Three-Factor ANOVA Extension of two factor ANOVA
Linear model of three factor layout:
Two-factor interactions
Three-factor interactions

22. Three-Factor ANOVA
Estimation:

23. Three-Factor ANOVA
Test of hypotheses:

24. Latin square design
Complete layout: at least one observation for each treatment. E.g: factor A,B and C with I,J and K levels, total IJK observations
This size is either impracticable because of cost, time, or space constraints or literally impossible
Incomplete layout: A three-factor experiment in which fewer than IJK observations

25. Latin square design
All two- and three-factor interaction effects are assumed absent
Levels of factor A and B: I =J = K
Levels of factor A are identified with the rows of a two-way table
Levels of B with the columns of the table
Every level of factor C appears exactly once in each row and exactly once in each column

26. Latin square design
Examples of Latin square design
I=J=3
I=J=4

27. Latin square design

28. Reference [1]http://en.wikipedia.org/wiki/Analysis_of_variance
[2] Jay L. Devore. Probability and Statistics for Engineering and the Sciences, eighth edition