Transcript BIBD and Adjusted Sums of Squares.pptx

Type I and Type III Sums of Squares
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Confounding in Unbalanced Designs
 When designs are “unbalanced”, typically with missing
values, our estimates of Treatment Effects can be
biased.
 When designs are “unbalanced”, the usual
computation formulas for Sums of Squares can give
misleading results, since some of the variability in the
data can be explained by two or more variables.
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Example BIBD from Hicks
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Type I vs. Type III in partitioning variation
 If an experimental design is not a balanced and complete
factorial design, it is not an orthogonal design.
 If a two factor design is not orthogonal, then the SSModel will
not partition into unique components, i.e., some
components of variation may be explained by either factor
individually (or simultaneously).
 Type I SS are computing according to the order in which
terms are entered in the model.
 Type III SS are computed in an order independent fashion,
i.e. each term gets the SS as though it were the last term
entered for Type I SS.
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Notation for Hicks’ example
There are only two possible factors, Block and Trt.
There are only three possible simple additive models
one could run. In SAS syntax they are:
 Model 1: Model Y=Block;
 Model 2: Model Y=Trt;
 Model 3: Model Y=Block Trt;
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Adjusted SS notation
Each model has its own “Model Sums of Squares”.
These are used to derive the “Adjusted Sums of
Squares”.
 SS(Block)=Model Sums of Squares for Model 1
 SS(Trt)=Model Sums of Squares for Model 2
 SS(Block,Trt)=Model Sums of Squares for Model 3
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The Sums of Squares for Block and Treatment can
be adjusted to remove any possible confounding.
Adjusting Block Sums of Squares for the effect of Trt:
SS(Block|Trt)= SSModel(Block,Trt)- SSModel(Trt)
Adjusting Trt Sums of Squares for the effect of Block:
SS(Trt|Block)= SSModel(Block,Trt)- SSModel(Block)
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From Hicks’ Example
 SS(Block)=100.667
 SS(Trt)=975.333
 SS(Block,Trt)=981.500
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For SAS model Y=Block Trt;
Source
df
Type I SS
Type III SS
Block
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SS(Block)
SS(Block|Trt)
=100.667
=981.50-975.333
SS(Trt|Block)
SS(Trt|Block)
Trt
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=981.50-100.667
=981.50-100.667
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ANOVA Type III and Type I
(Block first term in Model)
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For SAS model Y=Trt Block;
Source
Trt
df
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Type I SS
SS(Trt)
=975.333
Block
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SS(Block|Trt)
=981.50-975.333
Type III SS
SS(Trt|Block)
=981.50-100.667
SS(Block|Trt)
=981.50-975.333
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ANOVA Type III and Type I
(Trt. First term in Model)
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How does variation partition?
SS Total Variation
Block
TRT
Block or Trt
Error
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How this can work-I
Hicks example
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When does case I happen?
In Regression, when two Predictor variables are
positively correlated, either one could explain the “same”
part of the variation in the Response variable. The
overlap in their ability to predict is what is adjusted “out”
of their Sums of Squares.
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Example BIBD
From Montgomery (things can go the other way)
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ANOVA with Adjusted and Unadjusted Sums of
Squares
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Sequential Fit with Block first
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Sequential Fit with Treatment first
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LS Means Plot
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LS Means for Treatment, Tukey HSD
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How this can work- II
Montgomery example
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When does case II happen?
Sometimes two Predictor variables can predict the
Response better in combination than the total of they
might predict by themselves. In Regression this can
occur when Predictor variables are negatively
correlated.
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