Psych 407
Lecture 10
Two factor studies:
ANOVA models II and III
ANOVA Model II Random factor effects
This is a factorial design with two (or more) random effect factors
Model:
Yijk = .+i+j+()ij+ijk
where i, j, ()ij are all random factors
importantl
Psych 407
Lecture 12
Analysis of covariance (ANCOVA)
A technique combining features of ANOVA and regression
Used to reduce error variability
This error variability can be attributed to a quantitative variable
Some of the error variance is not random,
Psych 407
Lecture 5
ANOVA diagnosis
and remedial measures
This is similar to the regression situation
There are three basic assumptions of ANOVA
Independence of error term (residuals)
Normal distribution of the error term
Equal variance for all treat
Psych 407
Lecture 7
Two-factor studies
with equal sample sizes
Examples
The one-factor-at-a-time approach
Problems:
Only explore a selected set of conditions
Interactions cannot be estimated
Full randomization is not possible because the process is seq
Psych 407
Lecture 6
ANOVA Model II
Also called random effect models
Different treatments are considered as random samples from a wider
(possibly infinite) set of possible treatments
E.g., classes in a school, voxels in the brain
The interest is not in
Psych 407
Lecture 3
Alternative formulations of model
Cell mean model
Yij=i+ij
Factor effect model
i = . + (i .)
i = i . ith factor level effect or ith treatment effect
Yij=.+ i +ij
Definition of .
Unweighted mean: . = i /r
Weighted mean: . = wi
Psych 407
Lecture 2
Single factor studies
Single-factor experimental and observational studies
Relation between regression and analysis of variance
(ANOVA)
Both are based on general linear models and in principle
lead to identical results; however ANOV
Psych 407
Lecture 11
Multi-factor studies
Factorial ANOVA may be easily generalized to more than
two factors
In addition to main effects, we have two-factor (first-order)
interactions, three-factor (second-order) interactions, and so on
They can be use
Psych 407
Lecture 8
Two-factor studies:
One case per treatment
In some cases, with factorial studies it is difficult to obtain
more than one observation per treatment level
For example, when the observation is already an aggregate measure
These studies
Psych 407
Lecture 9
Two-factor studies
with unequal sample sizes
The presence of unequal sample sizes destroys the orthogonality of the
ANOVA decomposition
SSTR SSA + SSB + SSAB
This because the vectors of weights associated with each predictor (source
Psych 407
Lecture 13
Nested designs
Distinction between nested and crossed factor
Both involve two (or more) factors
Crossed factors:
the same treatment levels are used for all levels of the
other factors
Nested designs:
different treatment levels a