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