C. One-way ANOVA Models (continued)
4
Diagnosis of Model Assumptions
4.1
Model Assumptions, revisited
The following assumptions (in descending order of importance) are made to the one-way xedeffects model (in Equations (1) or (2):
Randomness observations
F. Variance Component Models
1
Random Effects One-Way ANOVA Models.
1.1
Example
We try to make inference about the product characteristics of machines of a factory. However, the
number of machines is relatively large (say 20 or above) and, due to time and
3
Unreplicated Two-Way Fixed-Effect ANOVA Models
Each cell has n = 1 observation.
3.1
When Both Factors Are of Interest
Assume both factors are of primary interest.
3.1.1
Interaction is Present
i. i.d
yij = + i + j + ( )ij + ij , i = 1, , a, j = 1, , b, i
D. Two-Way Fixed-Effects ANOVA Model
1
Completely Crossed Designs Versus Nested Designs
Now consider experiments that involve two factors, say factor A at a levels, and factor B at b
levels.
1.1
Completely Crossed Designs
When each of the a levels of A is
11
Welch t Statistic
The Welch, Patnaik, and Satterthwaite two-moment approximation is used to approximate the
distribution of a linear combination of independent central chi-square distributions Y = k ai Xi
i=1
using a single central chi-square r.v. 2 (r
Stat664
About Expected Mean Squares
The discussions below are for Balanced Multi-Way Mixed-Effects Models.
Unrestricted Models
Any random effect (including any interaction that involves xed effects) has its elements sampled
from normal population with zer
4
4.1
The Need for Transformation and How to Do It
Toxic Agents Example
The following experiment and data were from Statistics for Experimenters by Box, Hunter, and
Hunter (a.k.a. BH 2 ) and can be found in
http:/www.statsci.org/data/general/poison.txt
Th
E. Multi-Way Fixed-Effects ANOVA Model
1
Balanced Three-Way Fixed-Effects ANOVA Model
Completely crossed three-way design has row factor A at a levels, column factor B at b levels, and
depth factor C at c levels where n (replicates) experimental units are
9
Follow-up Study
If the hypothesis of equal means is rejected then its imperative to nd out how treatment means
differ. Multiple questions like
Does 1 differ from 2 ?
Does 1 differ from 3 ?
etc.
will be asked. Multiple decisions are then to be made. A
A. Theory Review and Preparation
1
(Nondegenerate) Normal Random Variates and Distributions
1. Univariate normal X N (, 2 )
(a) p.d.f.
1
( x ) 2
f ( x) =
exp
2 2
2
1
1
1
= (2 ) 2 ( 2 ) 2 exp (x ) ( 2 )1 (x ) , x R
2
(b) m.g.f.
2 t2
2
1
= exp t + t ( 2
C. One-way ANOVA Models
1
Completely Randomized Experiments
An experiment is to be designed and conducted to compare k treatments with respective sample
sizes ni , i = 1, , k and total sample size of N = k ni .
i=1
1.1
Selection of Experimental Units
From
7
Sample Size Determination for One-factor Balanced Design
Let
1
=
k
=
k
n
k
(i )2
i=1
A trial-and-error procedure is employed to recommend a common sample size n where values of
i s are specied for which one desires a high power of rejecting equal means
G. Nested Designs
1
Introduction.
1.1
Denition (Nested Factors)
When each level of one factor B is associated with one and only one level of another factor A, we
say that B is nested within factor A.
1.2
Notation
Use B A to denote that B is nested within