Factorial ANOVA

1
Factorial ANOVA

• (what do I do when Im manipulating too many
  categorical variables?)

2
Nomenclature
Variable (Factor) 2 B

j1
j2
jb
i1
Variable (Factor) 1 A
i2

ia
Thus
is the average of all data in the 2nd row
is the variance of the data in cell (2,3), etc.
3
ANOVA stuff

• main effects
• what is the effect of A (say) ignoring B entirely
  (collapsing across B)?
• use all data, but forget to label the B factor
• simple effects
• what is the effect of A at one value of B?
• interaction
• does the simple effect of A depend on B?

4
main effects
5
main effects II
6
main effects III
7
interactions
8
interactions II
9
interaction III
10
partitioning the variances
total
treatment (cells)
error
main A
main B
AB inter.
and obviously (hopefully) these are additive
11
calculations I
total
treatment (cells)
df N-1
error
(variance of all data relative to the grand mean)
main A
main B
AB inter.
12
calculations II
total
treatment (cells)
error
df ab-1
main A
main B
AB inter.
(variance of the cell means relative to the grand
mean, weighted by n)
13
calculations III
total
treatment (cells)
df a-1
error
main A
AB inter.
main B
df b-1
14
calculations IV
total
df dftot - dfA - dfB - dfAB ab(n-1)
treatment (cells)
error
main A
main B
AB inter.
df dfA x dfB (a-1)(b-1)
15
The Model
one-way ANOVA
Xij m (mj-m) eij
m tj eij
factorial ANOVA
Xijk m ai bj abij eijk
where ai mAi - m, bj mBj - m and
abij m - mAi - mBj mij (which might not be
so obvious)
16
The Model II
Xijk m ai bj abij eijk
m (mAi - m) (mBj - m) (m - mAi - mBj
mij) eijk
m (mAi - m) (mBj - m) (m - mAi - mBj
mij) eijk
mij eijk
which must be true
17
power
(as before, think before you crunch numbers)
and
one-way ANOVA
note similarity of f to F, but with sample size
factored out
18
power II
main effects
19
power III
interaction
20
effect sizes

• calculate h2 (as before)
• calculate w2 (as per the textbook)
• think about your data with respect to their
  variability and the unit of measurement

21
SPSS example
performance ( of errors) as a function of task
and smoking status

• task
• pattern recognition (find a target on screen)
• cognitive (recall a previously read passage)
• sensory-motor (driving simulation)
• smoking status
• non-smoker
• smoker on 3 hour abstention
• smoker smoking

22
graph the data (cant wait)
23
useful visualization (not to publish)
24
and in publishable form
25
checking the distributions
26
and box plots
27
at last, the topic of the lecture
28
relevant ANOVA table
29
(No Transcript)
30
about that F issue
or
?
Does
31
a little more on F
First, express the variance of two points re.
their separation, d (seems silly, I know)
32
a little more on F (cont.)
Now compute F for two groups (my way)
or, rearranging
taking the square root gives
(which is Students t)