STAT 705 Chapter 19: Two-way ANOVA

38 Pages·2015·0.29 MB·Dutch

by  Timothy Hanson

STAT 705 Chapter 19: Two-way ANOVA
Timothy Hanson
DepartmentofStatistics,UniversityofSouthCarolina
Stat705: DataAnalysisII
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Two-way ANOVA
Material covered in Sections 19.2–19.4, but a bit differently.
Have two factors, A and B.
Levels of A indexed by i = 1,...,a.
Levels of B indexed by j = 1,...,b.
Number sampled when A = i and B = j is n . If data are
ij
balanced then n = n for all i,j.
ij
n = (cid:80)a (cid:80)b n . If balanced, n = nab.
T i=1 j=1 ij T
Y is kth replicate of factor A = i & B = j.
ijk
Let µ = E{Y }.
ij ijk
Have ab different means.
Factor B
Factor A 1 2 ··· b
1 µ µ ··· µ
11 12 1b
2 µ µ ··· µ
21 22 2b
... ... ... ... ...
a µ µ ··· µ
a1 a2 1b
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Two-way ANOVA
Model is written
Y = µ +(cid:15) , (cid:15) i∼id N(0,σ2).
ijk ij ijk ijk
Most general, least restrictive case is when each combination of
levels (i,j) has its own distinct mean. We’ll look at special cases
that impose structure on {µ }.
ij
I Y = µ+(cid:15)
ijk ijk
II Y = µ+α +(cid:15)
ijk i ijk
III Y = µ+β +(cid:15)
ijk j ijk
IV Y = µ+α +β +(cid:15)
ijk i j ijk
V Y = µ+α +β +(αβ) +(cid:15)
ijk i j ij ijk
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I. µ = µ, neither A nor B important
ij
When a = b = 2, means are
FactorA
FactorB 1 2
1 µ µ
2 µ µ
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I. µ = µ, neither A nor B important
ij
If this model fits, you’re done! Nothing to look at.
Overall µ is estimated by µˆ = Y¯ .
•••
Fit in SAS proc glm as model response = ;
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II. µ = µ + α , only A important
ij i
When a = b = 2, means are
FactorA
FactorB 1 2
1 µ+α1 µ+α2
2 µ+α1 µ+α2
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II. µ = µ + α , only A important
ij i
If this fits, have oneway model in A. Interested in L = (cid:80)a c α .
i=1 i i
SAS sets µˆ = Y¯ and αˆ = Y¯ −Y¯ .
a•• i i•• a••
Fit in SAS proc glm as model response = A;
Can get pairwise differences in factor A levels via, e.g. lsmeans A
/ pdiff adj=tukey;
General contrasts available in estimate or lsmestimate.
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III. µ = µ + β , only B important
ij j
When a = b = 2, means are
FactorA
FactorB 1 2
1 µ+β1 µ+β1
2 µ+β2 µ+β2
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III. µ = µ + β , only B important
ij j
If this fits, have oneway model in B. Interested in L = (cid:80)b c β .
j=1 j j
SAS sets µˆ = Y¯ and βˆ = Y¯ −Y¯ .
•b• j •j• •b•
Fit in SAS proc glm as model response = B;
Can get pairwise differences in factor B levels via, e.g. lsmeans B
/ pdiff adj=tukey;
General contrasts available in estimate or lsmestimate.
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IV. µ = µ + α + β , A and B additive
ij i j
When a = b = 2, means are
FactorA
FactorB 1 2
1 µ+α1+β1 µ+α2+β1
2 µ+α1+β2 µ+α2+β2
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