Power and Sample Size
When we begin to design an experiment, it is useful to have some
idea in advance how many experimental units will be required to
obtain the information we desire. Obviously, typically, more
experimental units cost more money, in the
Transformations
When data do not form to model assumptions all is not
(necessarily) lost.
Transformations
When data do not form to model assumptions all is not
(necessarily) lost.
It is possible sometimes to transform the response so that after
transforma
Assessing Model Assumptions
Except at the very outset when we discussed (briey)
randomization tests, we have been relying on our model
assumptions to test hypotheses and construct CIs.
Assessing Model Assumptions
Except at the very outset when we discusse
Dunnets Procedure
A specic set of pairwise comparisons of interest are all treatment
means compared to a control of all compared to the best.
Best could be either smallest or largest treatment mean
depending on the context.
Dunnets Procedure
A specic set
Bonferroni
We observe that in general
P (A1 A2 An ) P (A1 ) + P (A2 ) + + P (An )
Bonferroni
We observe that in general
P (A1 A2 An ) P (A1 ) + P (A2 ) + + P (An )
Consider the events Ai as corresponding to Type I errors for
hypotheses H0i .
Bonferroni
We
Multiple Testing
Repeated hypothesis testing is likely to result in rejected true
hypotheses.
Multiple Testing
Repeated hypothesis testing is likely to result in rejected true
hypotheses.
Consider a sequence of 100 independent tests of a null hypotheses
H
Contrasts
Lets assume that we have a one-way CRD for which we have
completed an ANOVA table and found signicant evidence for
diering treatment means, i.e. for some treatment eects which
are dierent from zero.
Contrasts
Lets assume that we have a one-way C
Odds and Ends
The model comparison performed in a CRD is a special case of a
more general procedure. We can think of linear models explaining
data as being arranged in an hierarchy:
Odds and Ends
The model comparison performed in a CRD is a special case o
Constructing Interval Estimates
From the parameter estimates we can also construct condence
intervals for the various parameters.
Constructing Interval Estimates
From the parameter estimates we can also construct condence
intervals for the various paramet
Estimating Parameters, Contd
By way of summary: Our model parameters are the overall mean
estimated by
=
yij /N = y ,
i
j
Estimating Parameters, Contd
By way of summary: Our model parameters are the overall mean
estimated by
=
yij /N = y ,
i
j
the i th
Estimating Parameters, Contd
By way of summary: Our model parameters are the overall mean
estimated by
=
yij /N = y ,
i
j
Estimating Parameters, Contd
By way of summary: Our model parameters are the overall mean
estimated by
=
yij /N = y ,
i
j
the i th
Specifying the Model/ More Notation
In terms of notation, Model 1 is
yij N (i , 2 ).
Specifying the Model/ More Notation
In terms of notation, Model 1 is
yij N (i , 2 ).
This equivalent to:
yij = i +
ij ,
ij
N (0, 2 ) all i , j .
Specifying the Model/ Mo
Completely Randomized Design
The simplest experimental design and analysis is a completely
randomized design (CRD) of a single factor experiment.
Completely Randomized Design
The simplest experimental design and analysis is a completely
randomized design
A Little Review
Before jumping o into ANOVA, we review a little.
A Little Review
Before jumping o into ANOVA, we review a little.
The single most important distribution in probability and statistics
is the family of normal distributions.
A Little Review
B
Math 487/587- Midterm Solutions
There are fty points possible on the exam with points per problem as indicated. Please work the problems in the order given in your bluebook. Viel
Glck!
u
1. A one-factor CRD experiment with g = 4 levels and n = 12 observat