Multifactor Analysis of VarianceISE 500
University of Southern California11/14/2013
Outline
• Single-factor ANOVA• Two-factor ANOVA Randomized block experiment
• Three-factor ANOVA
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
Single-factor ANOVA
• Factor 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
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
/F MSTr MSE
0 1 2: .... IH
2
1(x x )
/ ( 1)1
I
iiJ
MSTr SSTr II
2
1 1(x x )
/ I(J I)(J 1)
I J
ij ii jJ
MSE SSEI
SST SSTr SSE
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.
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
Two-factor ANOVA
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:
ij i j ijX
1
0I
ii
1
0J
jj
2~ N(0, )ij
i j
ˆˆˆ i i j jX X X X X
Two-factor ANOVA
main effects for factor A
Interaction parameters
i
j
ijmain effects for factor B
Multiple replicates
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.
Two-factor ANOVA
• Sum of squares
Two-factor ANOVA
• F test
Two-factor ANOVA
• ANOVA table for previous example
0.05,2,6 05.14, AF H not rejected
0.05,3,6 04.76, BF H rejected
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
0.05,4,6 4.9, w 4.9 (0.01447) / 3 0.34Q
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
Randomized block experiment
• Block:• 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
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
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)
Randomized block experiment
• ANOVA• Same procedures as two-factor ANOVA
• Block difference is significant• Tukey method is applied to identify significant pair of
treatments
0.05,4,12 03.26, 95.57 3.26,AF f H rejected
Three-Factor ANOVA
• Extension of two factor ANOVA• Linear model of three factor layout:
• Two-factor interactions
• Three-factor interactions, ,AB AC BC
ij ik jk
ijk
Three-Factor ANOVA• Estimation:
Three-Factor ANOVA
• Test of hypotheses:
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
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
Latin square design
• Examples of Latin square design
I=J=3 I=J=4
Latin square design
Reference
[1]http://en.wikipedia.org/wiki/Analysis_of_variance
[2] Jay L. Devore. Probability and Statistics for Engineering and the Sciences, eighth edition
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