nested two-way ANOVA

68 Pages·2006·0.11 MB·English

O. Nested Designs
Introduction
In the CRDs we have seen thus far, there are randomly
assigned experimental units at every combination of
the experimental factors. In these designs, the levels
of any one factor appear identically at every level of
every other factor.
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Example: Kidney Dialysis Study
Moderate weight gain has the same meaning for those
with long treatment duration as for those with short
treatment duration.
Such factors are crossed.
Sometimes factor levels are nested, not crossed. If B
is nested within A, this means that each level of factor
A will contain different levels of factor B.
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Example: Tutoring Effectiveness
Kaplan is a commercial tutoring company. Students
needing help preparing for the GRE sign up for classes.
Kaplan has three locations near the U of M: Stadium
Village, Dinkeytown, and Falcon Heights. It needs to
assign students to instructors and locations. With 3
locations and 3 instructors, write down a design that
shows locations crossed with instructors.
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Next, write down a design that shows instructors nested
within location.
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Fixed or random:
Instructors?
Locations?
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Example: Food Safety Study
Researchers are interested in the shelf life of certain
nutrients contained in baby food. One study will
examine Vitamin A in pureed carrots. 5 of the 12
brands of baby food available regionally are randomly
chosen, and 6 jars of each of the five brands are
obtained (presumably selected randomly). 3 samples
of 5 grams each are taken from each of the 30 jars
and tested for Vitamin A content.
What are the two factors in this study?
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Are they crossed or nested?
Fixed or random: brands?
jars?
This type of design is called a subsampling design.
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Nested Design with Fixed Effects
Y = µ + α + β + (cid:15)
ijk i j(i) ijk
∼iid 2
(cid:15) N(0, σ )
ijk e
∀
with constraints α = 0 and β = 0 i for
Pi i Pj j(i)
i = 1, . . . , a, j = 1, . . . , b, and k = 1, . . . , r.
Note that for any of the models we’ve seen this semester,
we could have denoted (cid:15) as (cid:15) .
ijk k(ij)
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E[Y ] =
ijk
Var[Y ] =
ijk
Why is there no AB interaction?
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