C. One-way ANOVA Models (continued)
Diagnosis of Model Assumptions
Model Assumptions, revisited
The following assumptions (in descending order of importance) are made to the one-way xedeffects model (in Equations (1) or (2):
F. Variance Component Models
Random Effects One-Way ANOVA Models.
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
Unreplicated Two-Way Fixed-Effect ANOVA Models
Each cell has n = 1 observation.
When Both Factors Are of Interest
Assume both factors are of primary interest.
Interaction is Present
yij = + i + j + ( )ij + ij , i = 1, , a, j = 1, , b, i
D. Two-Way Fixed-Effects ANOVA Model
Completely Crossed Designs Versus Nested Designs
Now consider experiments that involve two factors, say factor A at a levels, and factor B at b
Completely Crossed Designs
When each of the a levels of A is
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
using a single central chi-square r.v. 2 (r
About Expected Mean Squares
The discussions below are for Balanced Multi-Way Mixed-Effects Models.
Any random effect (including any interaction that involves xed effects) has its elements sampled
from normal population with zer
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
E. Multi-Way Fixed-Effects ANOVA Model
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
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 ?
will be asked. Multiple decisions are then to be made. A
A. Theory Review and Preparation
(Nondegenerate) Normal Random Variates and Distributions
1. Univariate normal X N (, 2 )
( x ) 2
f ( x) =
= (2 ) 2 ( 2 ) 2 exp (x ) ( 2 )1 (x ) , x R
= exp t + t ( 2
C. One-way ANOVA Models
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 .
Selection of Experimental Units
Sample Size Determination for One-factor Balanced Design
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
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.
Use B A to denote that B is nested within