completely randomized design reviews for later topics reviews for later topics –model...
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Completely Randomized Design
Reviews for later topics– Model parameterization
(estimability)– Contrasts (power analysis)
Analysis with contrasts– Orthogonal polynomial contrasts– Polynomial goodness-of-fit
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Completely Randomized Design
Cell means model:
2,0~,
,,1,,,1,
Niid
njaiY
ijij
iijiij
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Effects Model
Yij ˜ . i ij
Possible constraints : ii1
a
0 or a 0
ij independent, ij ~ N 0, 2
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GLM for Effects Model
aa an
a
n
n
a
an
a
n
n
Y
Y
Y
Y
Y
Y
1
2
21
11
11
1
2
1
.
1
2
21
1
11
22
1 ~
1111
1111
0101
0101
0011
0011
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CRD Contrasts
Balanced case (ni=n)
-A linear combination L has the form:
-A contrast is a linear combination with the additional constraint: 0
1
a
i ic
i
a
iicL
1
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Cotton Fiber Example
Treatment--% cotton by weight (15%, 20%, 25%, 30%, 35%)
Response--Tensile strength
Montgomery, D. (2005) Design and Analysis of Experiments, 6th Ed. Wiley, NY.
51
51
,,
,,
kkk cc
c
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Cotton Fiber Example
c 1 1, 1,0,0,0 L1 c1
' 1 2
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Cotton Fiber Example
ˆ L 1 c 1 ˆ y 1. y 2.
ˆ V ˆ L 1 ˆ 2
nc1i
2
i1
a
SSL1
ˆ L 12
1
nc1i
2
i1
a
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Contrast Test Statistic
T ˆ L 1
ˆ n
c1i2
i1
a
~ tn . a,
T 2 SSL1
ˆ 2F ~ F1,n . a
Under Ho:L1=0,
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Unbalanced CRD Contrast SS
5
1
2
2ˆ
i i
iL
nc
LSS
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Orthogonality
Contrasts are orthogonal if, for contrasts L1 and L2, we have
)caseunbalanced(0
)casebalanced(0
21
21
i
ii
ii
n
cccc
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Orthogonality
The usual a-1 ANOVA contrasts are not orthogonal (though columns are linearly independent)
Orthogonality implies effect estimates are unaffected by presence/absence of other model terms
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Orthogonality
Sums of squares for orthogonal contrasts are additive, allowing treatment sums of squares to be partitioned
Mathematically attractive, though not all contrasts will be interesting to the researcher
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Cotton Fiber Example
Two sets of covariates (orthogonal and non-orthogonal) to test for linear and quadratic terms
Term Orth. SS Non-Orth SS
L 33.6 33.6
L|Q 33.6 364.0
Q 343.2 12.8
Q|L 343.2 343.2
L & Q 376.8 376.8
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Cotton Fiber Example
For Orthogonal SS, L&Q=L+Q; Q=Q|L; L=L|Q
For Nonorthogonal SS, L&Q=L+Q|L=Q+L|QTerm Orth. SS Non-Orth SS
L 33.6 33.6
L|Q 33.6 364.0
Q 343.2 12.8
Q|L 343.2 343.2
L & Q 376.8 376.8
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Orthogonal polynomial contrasts
Require quantitative factors Equal spacing of factor levels (d)
Equal ni
Usually, only the linear and quadratic contrasts are of interest
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Orthogonal polynomial contrasts
Cotton Fiber Example
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Orthogonal polynomial contrasts
Cotton Fiber Example
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Orthogonal polynomial contrasts
F 33.62 343.21 /2
8.0623.38 (p .0001)
F 64.98 33.95 /2
8.066.137 (p .0084)
Cotton Fiber ExampleIs a L+Q model better than an intercept model?Is a L+Q model not as good as a cell means model? (Lack of Fit test)
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Orthogonal polynomial contrasts
Yandell has an interesting approach to reconstructing these tests– Construct the first (linear) term– Include a quadratic term that is neither
orthogonal, nor a contrast– Do not construct higher-order contrasts
at all– Use a Type I analysis for testing