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We remember the cell means model:
𝑌𝑖𝑗 = 𝜇𝑖 + 𝜀𝑖𝑗
The error terms 𝜀𝑖𝑗 are assumed to:

Be normally distributed

Be independent

Have constant variance
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The error terms are estimated with the residuals.
𝑒𝑖𝑗 = 𝑌𝑖𝑗 − 𝑌𝑖∙ = 𝑌𝑖𝑗 − 𝑌𝑖∙
The residuals should be plotted…

… against fitted values

… in a normal probability plot

… in a dot plot

… in a sequence plot (if ordered observations)
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We can also calculate:

Semistudentized residuals
𝑒𝑖𝑗
∗
𝑒𝑖𝑗 =
𝑀𝑆𝐸
Studentized residuals
𝑒𝑖𝑗
𝑒𝑖𝑗
𝑟𝑖𝑗 =
=
𝑠 𝑒𝑖𝑗
𝑀𝑆𝐸(𝑛𝑖 − 1)
𝑛𝑖

Studentized deleted residuals

𝑡𝑖𝑗 = 𝑒𝑖𝑗
𝑛𝑇 − 𝑟 − 1
1
2
𝑆𝑆𝐸 1 −
− 𝑒𝑖𝑗
𝑛𝑖
2
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When the ANOVA model assumptions are violated, a
transformation of the response can be useful.

𝜎𝑖2 proportional to 𝜇𝑖 :
𝑌′ = 𝑌

𝜎𝑖 proportional to 𝜇𝑖 :
𝑌 ′ = log 𝑌

𝜎𝑖 proportional to
𝜇𝑖2 :

Response is a proportion:
𝑌′
=
1
𝑌
𝑌 ′ = 2𝑎𝑟𝑐𝑠𝑖𝑛 𝑌
In practice: test which transformation that’s useful.
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In two-way ANOVA, one more factor is added. The
advantage in doing so is:

Time-saving studying two factors at the same time

Interactions can be found
In this course, we’re only studying two-way ANOVA
with equal sample sizes.
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Cell means model
𝑌𝑖𝑗𝑘 = 𝜇𝑖𝑗 + 𝜀𝑖𝑗𝑘
Factor effects model
𝑌𝑖𝑗𝑘 = 𝜇∙∙ + 𝛼𝑖 + 𝛽𝑗 + 𝛼𝛽
𝑖𝑗
+ 𝜀𝑖𝑗𝑘
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Two-way ANOVA is calculated in different ways
whether the factor effects are fixed or random.


Fixed
We only want to draw conclusions about the
observed factor levels
Random
The factor levels are a random sample of a big
number of factor levels
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
Chapter 18.1, 18.3, 18.5 (not Box-Cox)

Start reading chapter 19
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