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Unformatted text preview: ciﬁcally as
oneway ANOVA because there was one factor with k levels
(treatments).
Suppose there are two factors, say A and B ; factor A has a
levels and factor B has b levels.
Each combination of levels of factor and levels of factor B is
called a treatment. So there are a · b treatments.
Suppose n (equal number of) experimental units (replications)
are to be used for each of the a · b treatments.
This does not always have to be the case, but we will assume
it is for this section. See the example later for unequal
replications.
The treatments are randomly assigned to the experimental
units in such a way that n experimental units receive each of
the treatments.
Chapter 12: Multisample Inference
Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Example: Genetics
A biologist assayed the activity of the enzyme
mannose6phosphate isomerase (MPI) in the Platorchestia
platensis, a species of sand ﬂea, an amphipod crustacean that lives
on beaches. There are three genotypes at the locus for MPI,
Mpiﬀ , Mpifs , and Mpiss , and the researcher wanted to know
whether the genotypes had diﬀerent activity. Because the
researcher didn’t know whether sex would aﬀect activity, she also
recorded the sex. Each amphipod was lyophilized, weighed, and
homogenized; then MPI activity of the soluble portion was assayed.
MPI activity in ∆ O.D. units/sec/mg dry weight were computed
for each amphipod. Is there sexgenotype interaction on the
activity of the enzyme? Is there a sex eﬀect on activity? Is there a
genotype eﬀect on activity?
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Data from TwoWay layout in CRD
1
1 Factor A 2 a
Mean y111
y112
y11.
¯
...
y11n
y211
y212
y21.
¯
...
y21n
...
ya11
. . . ya1.
¯
ya1n
y.1.
¯ Chapter 12: Multisample Inference Factor B
2
...
y121
y122
y12.
¯
...
y12n
y221
y222
y22.
¯
...
y22n
...
ya21
. . . ya2.
¯
ya2n
y.2.
¯ ... ...
...
...
... Stat 491: Biostatistics b
y1b1
y1b2
y1b.
¯
...
y1bn
y2b1
y2b2
y2b.
¯
...
y2bn
...
yab1
. . . yab.
¯
yabn
y.b.
¯ Mean
y1..
¯ y2..
¯
...
ya..
¯
y...
¯ Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Model
The model for the observation yijk is:
yijk = µ + τi + βj + γij + εijk
where
µ:
τi :
βj :
γij : is the overall mean (unknown constant).
is an eﬀect due to the i th level of factor A (unknown constant).
is an eﬀect due to the j th level of factor B (unknown constant).
is an eﬀect due to the interaction between i th level of factor A
and the j th level of factor B (unknown constants).
εijk : is the error associated with the k th experimental unit receiving
the i th level of factor A and j th level of factor B . εijk are
2
indep. normally distributed with mean 0 and variance σε . τ s and β s are known as main eﬀects and γ s are known as
interaction eﬀects.
Chapter...
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This note was uploaded on 02/03/2014 for the course STAT 491 taught by Professor Solomonharrar during the Fall '12 term at Montana.
 Fall '12
 SolomonHarrar
 Statistics, Biostatistics

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