A computational example with a nested design
Suppose we had two teaching methods, and four teachers were randomly assigned so that two
used each method. In the exclusive school where they teach, there are only two students per class.
Suppose the nal test
1
Analysis for Factorial Treatment Designs
Understanding the results from analyses of factorial treatment designs is
aided by recalling the types of contrasts being tested. Sometimes it is easiest
to think in terms of the cell means model:
yijk =
ij
+ eij
1
Incomplete Block Designs
Incomplete block designs are needed when we cannot t all levels of a treatment into our blocks. The
book has a good example where the treatment levels are temperatures for tomato seed germination.
1.1
Balanced Incomplete Block D
Lack of independence for observations sharing a block
In the completely randomized design, Yij =
between observations Yij and Yij 0 is:
cov(Yij ; Yij 0 ) = E(Yij Yij 0 )
= E[( +
2
= E(
=
2
i
+
+2
i
+ eij , the covariance
E(Yij )E(Yij 0 )
+ eij )( +
i
i
+
A brief introduction to maximum likelihood
The key idea behind the method of maximum likelihood is that we start
with a probability distribution that we believe is appropriate for our data,
then we change the focus from calculating a probability of an obs
Confounded Block Designs
In previous chapters we encountered incomplete block designs for a single factor, and one
approach (BIBD) dealt with this problem by retaining the same amount of information about each
pair of treatment levels. In Chapters 11 and
Treatment Designs with Nested or Nested and Crossed Factors
Designs with nested factors, in which the levels of one factor (B) are not identical across all
levels of another factor (A) occur commonly in many types of research. The model for a nested
desig
Modeling Covariance Structure in Repeated Measures
Pros: Can be more efficient, can yield useful models
Cons: Modeling process must be monitored, problems can occur for some data sets
The goal is usually to find a reasonable covariance model, not necessar
1
1.1
Chapter 3: More on Treatment Comparisons
The need to control error rates with multiple comparisons
As shown in the text, when a set of n hypotheses are tested where each has a Type I error rate of C (the
comparisonwise error rate), then an upper lim
1
Relatives of the Split-plot
Here are listed three of the most common designs that are related to the split-plot design.
1.1
The Split-Split Plot
In this design, after performing a second randomization of the split-plot levels, you again split
each split
Treatment structure equivalence shown with contrasts
As mentioned in class, a factorial treatment structure with a levels of
factor A and b levels of factor B (a two way ANOVA) can equivalently be
seen as a completely randomized structure (one way ANOVA)
1
Chapter 5: Further topics in Random eects models
1.1
The Intraclass Correlation Coe cient
One measure of interest in a random eects model is the ratio of the variance of the random eect
to the total variance of the response. This quantity is called the
1
Chapter 1: Research Design Principles
The legacy of Sir Ronald A. Fisher.
Fisher three fundamental principles: local control, replication, and randomization.
s
2
Chapter 2: Completely Randomized Design
2.1
The cell means model
The rst model we consider
Factorial Treatment Designs: Random and Mixed Models
In Chapter 6 we considered factorial treatment designs for xed eect models. In this chapter
we expand the use of factorial treatment designs to cases where each factor is random (Random
Eects Factorial)
Randomized Complete Block Design Topics; Latin Square Designs
Some issues that arise with RCB designs:
The additivity assumption
The text discusses Tukey one-degree of freedom test for nonadditivity. This test detects
s
a particular type of nonadditivity
Split-Plot Designs
The basic split plot design with two factors is characterized by having
a dierent size of experimental unit for each of the two treatment factors.
Also, there are two distinct randomizations, one for each size of unit. When
examining da
Approximate F tests
In the factorial treatment design with three random factors, we have
E(M SC ) =
2
+r
2
abc
2
ac
+ rb
+ ra
2
bc
+ rab
2
c
To test the null hypothesis H0 : 2 = 0; we would like to have a mean square
c
with expectation 2 + r 2 + rb 2 + ra