Type I and Type III Sums of Squares

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.

Example BIBD from Hicks

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 SS Model 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.

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;

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

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) = SS Model(Block,Trt) - SS Model(Trt) Adjusting Trt Sums of Squares for the effect of Block: SS (Trt|Block) = SS Model(Block,Trt) - SS Model(Block)

From Hicks’ Example SS (Block) = SS (Trt) = SS (Block,Trt) =

For SAS model Y=Block Trt; Source df Type I SS Type III SS Block 3 SS (Block) SS (Block|Trt) = = Trt 3 SS (Trt|Block) SS (Trt|Block) = =

ANOVA Type III and Type I (Block first term in Model)

For SAS model Y=Trt Block; Source df Type I SS Type III SS Trt 3 SS (Trt) SS (Trt|Block) = = Block 3 SS (Block|Trt) SS (Block|Trt) = =

ANOVA Type III and Type I (Trt. First term in Model)

How does variation partition?

How this can work-I Hicks example

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.

Example BIBD From Montgomery (things can go the other way)

ANOVA with Adjusted and Unadjusted Sums of Squares

Sequential Fit with Block first

Sequential Fit with Treatment first

LS Means Plot

LS Means for Treatment, Tukey HSD

How this can work- II Montgomery example

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.