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BIBD and Adjusted Sums of SquaresType I and Type III Sums of Squares

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Confounding in Unbalanced DesignsWhen 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 3 SS(Block) SS(Block|Trt)

=100.667 =981.50-975.333

Trt 3 SS(Trt|Block) SS(Trt|Block)

=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 df Type I SS Type III SSTrt 3 SS(Trt) SS(Trt|Block)

=975.333 =981.50-100.667

Block 3 SS(Block|Trt) SS(Block|Trt)

=981.50-975.333 =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-IHicks 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 BIBDFrom 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- IIMontgomery 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.