1. STAT 312 11.1 - Two-Factor ANOVA with Kij = 1
Chapter 11 - Multifactor Analysis of Variance
Introduction
Two-Factor ANOVA with Kij = 1
Two-Factor ANOVA with Kij > 1
Three-Factor ANOVA
n Factorial Experiments

2. One-Way ANOVA
Example: Different levels of a treatment
(quick review)
k groups
“Grand Mean”
“residuals”
Each Yij response is assumed to be a normally-distributed random variable, centered around the group mean i, with the same variance .

3. One-Way ANOVA
Example: Different levels of a treatment
(quick review)
k groups
“Grand Mean”
“residuals”
Each residual εij is therefore assumed to be a normally-distributed random variable, centered around 0, and with the same variance .
Each Yij response is assumed to be a normally-distributed random variable, centered around the group mean i, with the same variance .

4. (Linear Regression form)
One-Way ANOVA
Example: Different levels of a treatment
(quick review)
k groups
“Grand Mean”
“residuals”
ANOVA Model
(Linear Regression form)

5. How are the i estimated?
One-Way ANOVA
Example: Different levels of a treatment
(quick review)
k groups
“Grand Mean”
“residuals”
How are the i estimated?

6. k groups One-Way ANOVA Example: Different levels of a treatment
(quick review)
k groups
“Grand Mean”
“residuals”
Many generalizations of ANOVA exist, depending on application, assumptions, design, etc. (Covered in remainder of Ch 11.)
Example: Different levels of two treatments

7. Suppose there are two factors, A and B.
Two-Way ANOVA
Example: Different levels of two treatments
Suppose there are two factors, A and B.
Factor A has levels i = 1, 2, 3,…, I, and B has levels j = 1, 2, 3,…, J.
“Grand Mean”
ij = mean response of Yij when
Factor A is fixed at level i,
Factor B is fixed at level j. 1 2 3 … J .  I
“residuals”

8. Suppose there are two factors, A and B.
Two-Way ANOVA
Example: Different levels of two treatments
Suppose there are two factors, A and B.
Factor A has levels i = 1, 2, 3,…, I, and B has levels j = 1, 2, 3,…, J.
“Grand Mean”
ij = mean response of Yij when
Factor A is fixed at level i,
Factor B is fixed at level j.
“Additive Model”
“residuals”
i measures the effect of Factor A on the response Yij, while holding level j fixed.
j measures the effect of Factor B on the response Yij, while holding level i fixed.

9. ANOVA Table SSTot = SSA + SSB + SSErr dfTot = dfA + dfB + dfErr SourceMS
F-ratio
p-value
Factor A
I – 1
SSA
MSA pA
Factor B
J – 1
SSB
MSB pB
Error
(I – 1)(J – 1)
SSErr
MSErr
Total
IJ – 1
SSTot –

10. Suppose there are two factors, A and B.
Two-Way ANOVA
Example: Different levels of two treatments
Suppose there are two factors, A and B.
Factor A has levels i = 1, 2, 3,…, I, and B has levels j = 1, 2, 3,…, J.
“Grand Mean”
ij = mean response when
Factor A is fixed at level i,
Factor B is fixed at level j.
“Additive Model”
i measures the effect of Factor A on the response Yij, while holding level j fixed.
j measures the effect of Factor B on the response Yij, while holding level i fixed.
“residuals”

11. Suppose there are two factors, A and B.
Two-Way ANOVA
Example: Different levels of two treatments
Suppose there are two factors, A and B.
Factor A has levels i = 1, 2, 3,…, I, and B has levels j = 1, 2, 3,…, J.
“Grand Mean”
ij = mean response when
Factor A is fixed at level i,
Factor B is fixed at level j.
“Interaction term”
“residuals”
i measures the effect of Factor A on the response Yij, while holding level j fixed.
j measures the effect of Factor B on the response Yij, while holding level i fixed.