Multiple Factor ANOVA: Three-Way Analysis of Variance, Exams of Statistics for Psychologists

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Lecture 33

Multiple Factor ANOVA

STAT 512Spring 2011

Background ReadingKNNL: Chapter 24

Topic Overview

•^

ANOVA with multiple factors

Data for three-way ANOVA

Y, the response variable − Factor A with levels i = 1 to a − Factor B with levels j = 1 to b − Factor C with levels k = 1 to c −

Y

ijkl

is the l

th^

observation in cell (i,j,k), l = 1 to

nijk −

A

balanced design

has n

ijk

= n

Cell Means Model

ijkl

ijk

ijkl

Y^

= μ

• ε

μ^ ijk

is the theoretical mean or expected value of all observations in cell (

i,j

,k

(^

~^

0,

iid ijkl

N

ε^

σ

(^

~^

,

ijkl

ijk

Y^

N^

μ^

σ^

are independent

Estimates

1 1

1

1

,^

,^

,

1

1

1

, ,

, ,^

, ,

1

, , ,

ˆ ˆ^

ˆ^

ˆ

ˆ^

ˆ^

ˆ

ˆ

n

ijk^

ijkl l cn^

an

bn

ij^

ijkl

i k

ijkl

jk^

ijkl

k l^

j l^

i l

acn

bcn

abn

i^

ijkl

j^

ijkl

k^

ijkl

j k l

i k l

i j l

abcn

ijkl Y i j k l Y

Y

Y

Y

Y

Y

Y

μ μ^

μ^

μ

μ^

μ^

μ

μ

= =

=

=

=

=

=

=

∑^ ∑

i^

i^

i

ii^

i i

ii

iii

33-

Factor effects model

(^

)^

(^

)^

(^

)^

(^

ijk^

i^

j^

k^

ijkl

ij^

ik^

jk^

ijk

Y^

= μ + α + β

• γ

• αβ

• αγ

• βγ

• αβγ

• ε

μ is the overall (grand) mean −

,^

, i^

j^

k

α^

β^

γ^

are the main effects of factors A, B,

and C −

(^

)^

(^

)^

(^

,^

,

ij^

ik^

jk

αβ

αγ

βγ

are the two-way (first

order) interactions −

(^

)ijk αβγ

is the three-way (second order)

interaction

Constraints

•^

Usual constraints listed on page 997 – sums of effects for ANY of the indices are zero.Under these,

μ

iii

will be the grand mean.

•^

In SAS, constraints are all set up to compare everything to

μ^ abc

. Thus a factor effect is

zero if it includes any of the “last” levels ofthe factors.

Assumptions

•^

Constancy of variance applies across cells; can do residual plots across treatmentcombinations

-^

For violations, transformations can sometimes be useful; WLS is a standardremedial measure if the error distribution isnormal but the variances are different.

Steps in 3-Factor Analysis

Fit full model and check assumptions

Start with the 3-way interaction anddetermine if it is significant.

If not, may consider pooling. To avoidlikelihood of Type I errors, best to pool onlyin cases where p-value is not close tosignificant.

If 3-way interaction (or multiple 2-wayinteractions) are significant, then analyze thethree factors jointly in terms of

ijk μ

Steps in 3-Factor Analysis (2) 5.^

If only a single two-way interaction issignificant, may again consider pooling, andcan analyze via regular interaction plot. DoNOT pool any term for which higher orderterms are significant.

Can analyze main effects if factor notinvolved in important interaction. May alsobe able to look at main effects if they arelarge compared to the interactions.

Multiple Comparisons

•^

Tukey, Bonferroni, and Scheffe adjustments can be made as before (see page 1017 forappropriate degrees of freedom to use;generally model and/or error).

-^

Can utilize contrasts to study specific questions (should use Scheffe if looking atany unplanned contrasts; Bonferroni isappropriate for contrasts that have beenplanned in advance)

Unequal Cell Sizes

•^

Formulas change a bit as not all of the

ijk n

are the same

-^

Look at Type III SS as well as Type I (the closer the sample sizes are to each other,the less difference there will be).

-^

MUST use LSMeans to do comparisons

Example

•^

Problem 24.6 (alloy.sas)

-^

Studying the effects of three factors on the hardness of an alloy

-^

Factor A: Use of a chemical additive (1 = low amount; 2 = high amount)

-^

Factor B: Temperature (1 = low, 2 = high)

-^

Factor C: Time allowed for process (1 = low, 2 = high)

-^

Three observations per cell, balanced design