Type I and Type III Sums of Squares in Experimental Design
Type I and Type III Sums of Squares
1
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.
2
Example BIBD
from Hicks
3
What would be a simple linear model and what does it
say about any treatment mean comparison?
Notation for layout and model.
4
Suppose we want to compare treatment 1 to
treatment 2?
The model says that if we compare averages, then we get
a possible bias due to block.
5
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.
6
If BIBD, design is unbalanced and some variation
may be explained by either factor.
If we use a Venn diagram:
7
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;
8
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
9
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)
10
From Hicks’ Example
SS
(Block)
=100.667
SS
(Trt)
=975.333
SS
(Block,Trt)
=981.500
11
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
12
ANOVA Type III and Type I
(Block first term in Model)
13
For SAS model Y=Trt Block;
Source df Type I SS Type III SS
Trt 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
14
ANOVA Type III and Type I
(Trt. First term in Model)
15
How does variation partition?
16
How this can work-
I
Hicks example
17
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.
18
Example BIBD
From Montgomery (things can go the other way)
19
ANOVA with Adjusted and Unadjusted Sums of
Squares
20
Sequential Fit with Block first
21
Sequential Fit with Treatment first
22
LS Means Plot
23
LS Means for Treatment, Tukey HSD
24
How this can work- II
Montgomery example
25
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.
26
Exploring the significance of Type I and Type III sums of squares in unbalanced experimental designs, highlighting the potential biases in treatment effect estimates and the differences in partitioning variation based on the order of terms entered in the model.
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Presentation Transcript
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. 2
What would be a simple linear model and what does it say about any treatment mean comparison? Notation for layout and model. 4
Suppose we want to compare treatment 1 to treatment 2? The model says that if we compare averages, then we get a possible bias due to block. 5
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. 6
If BIBD, design is unbalanced and some variation may be explained by either factor. If we use a Venn diagram: 7
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; 8
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 9
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) 10
From Hicks Example SS(Block)=100.667 SS(Trt)=975.333 SS(Block,Trt)=981.500 11
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 12
ANOVA Type III and Type I (Block first term in Model) 13
For SAS model Y=Trt Block; Source df Type I SS Type III SS Trt 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 14
ANOVA Type III and Type I (Trt. First term in Model) 15
How does variation partition? SS Total Variation Block TRT Block or Trt Error 16
How this can work-I Hicks example 17
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. 18
Example BIBD From Montgomery (things can go the other way) 19
ANOVA with Adjusted and Unadjusted Sums of Squares 20
How this can work- II Montgomery example 25
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. 26
