
Announcements this week: factorial ANOVAs (ch 14), correlation
(ch
15)
HW 5 due today, HW 5R due Thurs 11/6 Quiz 6 fillin question
regraded today/
tomorrow (Quiz 5 almost done)
Prelim 2 next week: Wed., Nov 12 focus on ttests, oneway
ANOVAs (both types),
twoway ANOVAs (this week), some estimation and signal
detection
1
1Monday, November 3, 14

Test multiple independent variables (factors) at multiple
levels
Factor 1: Dosage (2 levels: 1 mg, 3 mg)Factor 2: Age (2 levels:
students and faculty)
2
What is a factorial design?
What is the effect of caffeine on reaction time?
Caffeine might influence people differently (age, lifetime
exposure): we can test dosage in combination
with age
2Monday, November 3, 14

Test multiple independent variables (factors) at multiple
levels
Factor 1: Dosage (2 levels: 1 mg, 3 mg)Factor 2: Age (2 levels:
students and faculty)
3
What is a factorial design?
0
15
30
45
60
75
90
1 mg 3 mg
Effect of caffeine and age on reaction timestudentsfaculty
Dosage
Rea
ctio
n tim
e
3Monday, November 3, 14

can look for relationships between the variables
The variables interact:As caffeine dosage increases, reaction
time decreases more for faculty than for students
4
Advantages over repeated twolevel designs?
Dosage
Rea
ctio
n tim
e
0
15
30
45
60
75
90
1 mg 3 mg
Effect of caffeine and age on reaction timestudentsfaculty
4Monday, November 3, 14

Sampling from how many populations?
Dosage
Age
Student Faculty
Low
High
Factorial designs
5
5Monday, November 3, 14

Sampling from how many populations?
Student Faculty
Dosage
Age
Student Faculty
Factorial designs
6
6Monday, November 3, 14

Sampling from how many populations?
Low High
Dosage
Age
Low
High
Factorial designs
7
7Monday, November 3, 14

0.7 0.6
0.8 0.2Dosage
Age
Student Faculty
Low
High
Multiple populations?
8
To see if there is an effect of just one variable, collapse
(average) across the other variable
8Monday, November 3, 14

Main effect of Dosage? Collapse across Age
0.70 0.60
0.80 0.20
.65
.5Dosage
Age
Student Faculty
Low
High
Main effects
9
9Monday, November 3, 14

Main effect of Age? Collapse across Dosage
0.70 0.60
0.80 0.20
.75 .4
Dosage
Age
Student Faculty
Low
High
Main effects
10
10Monday, November 3, 14

Identifying relationships among variables Interaction: when the
effect of one variable
depends on the level of another variable
Does the relationship between the reaction times observed in
high and low caffeine dosages depend on age?
Interactions
11
11Monday, November 3, 14

Interaction?
12
Relationship between variables:Moving from 1mg to 3mg, the
student reaction time increases slightly, but faculty reaction time
decreases.
0
15
30
45
60
75
90
1 mg 3 mg
Effect of caffeine and age on reaction timestudentsfaculty
Dosage
Rea
ctio
n tim
e
12Monday, November 3, 14

No Can have an interaction without main effects Can also have
main effects without an
interaction
Are main effects a prerequisite for an interaction?
13
13Monday, November 3, 14

2882
Dosage
AgeStudent Faculty
LowHigh
Interaction without main effects
5
55 5
14
Relationship between variables:Moving from low to high, reaction
time for students increases, while faculty reaction time
decreases.
0
2
4
6
8
low high
studentsfaculty
Dosage
Rea
ctio
n tim
e
14Monday, November 3, 14

10662
Dosage
AgeStudent Faculty
Low
High
Main effects without interaction
4
84 8
15
Relationship between variables:Moving from 1 mg to 3mg, reaction
time for students and faculty increases by the same amount.
Dosage
Rea
ctio
n tim
e
0
2
4
6
8
10
low high
studentsfaculty
15Monday, November 3, 14

ANOVA answers two questions: Do the different levels of a factor
represent
real differences in the dependent variable?
Is there an interaction between the factors?
ANOVA: a statistical test of main effects and interactions
16
16Monday, November 3, 14

Sum of squares Sum of the variances from the grand mean for each
level of a factor Degrees of freedom Number of independent
observations Mean square Mean deviation from the grand mean of each
observation in a factor Error Tendency for scores to vary from the
overall mean
Essential parts of an ANOVA
17
17Monday, November 3, 14

Variance (within a factor)
errorFratio =
Main effects
18
18Monday, November 3, 14

Fratio: If betweengroup differences equal withingroup
differences: H0 true
If betweengroup differences are larger than withingroup
differences: H0 false
Main effects: Hypotheses
19
19Monday, November 3, 14

All the variance must be accounted for Any variance not due to
the main effects or
error is due to an interaction of the factors
Interactions: the variance left over
20
20Monday, November 3, 14

Example:
TwoFactor ANOVA
70 degrees 80 degrees 90 degrees
30% humidity M = 85 M = 80 M = 75 M = 80
70% humidity M = 75 M = 70 M = 65 M = 70
M = 80 M = 75 M = 70
21
21Monday, November 3, 14

Twofactor ANOVA will do three things:
 Examine differences in sample means for humidity (factor
A)
 Examine differences in sample means for temperature (factor
B)
 Examine differences in sample means for combinations of
humidity and temperature (factor A and B).
Three Fratios.
TwoFactor ANOVA
22
22Monday, November 3, 14

Main effect for humidity (Factor A)Main effect for temperature
(Factor B)
The differences among the levels of one factor are referred to
as the main effect of that factor.
Main Effects and Interactions
An example: 70 degrees 80 degrees 90 degrees
30% humidity M = 85 M = 80 M = 75 M = 80
70% humidity M = 75 M = 70 M = 65 M = 70
M = 80 M = 75 M = 70
23
23Monday, November 3, 14

Evaluation of main effects two out of three hypothesis tests in
twofactor ANOVA.
Factor A (humidity  2 levels):Hypotheses:
H0: A1 = A2
H1: A1 A2
Main Effects and Interactions
F = variance between means (Factor A)variance expected by
chance/error
24
24Monday, November 3, 14

Factor B (temperature  3 levels):
Hypotheses:
H0: B1 = B2 = B3
H1: At least one is different.
Fratio:
Main Effects and Interactions
F = variance between means (Factor B)variance expected by
chance/error
25
25Monday, November 3, 14

All the variance must be accounted for
Any variance not due to the main effects or error is due to an
interaction of the factors
Interactions: the variance left over
26
26Monday, November 3, 14

variance not explained by main effectsvariance expected by
chance/error
Interaction Hypotheses
H0: There is no interaction between factors A and B. (all mean
differences are explained by main effects)
H1: There is an interaction between factors A and B
Main Effects and Interactions
F = 27
27Monday, November 3, 14

28
In a graph, lines that are nonparallel indicate the presence of
an interaction between two factors.
Main Effects and Interactions
28Monday, November 3, 14

29
Twofactor ANOVA consists of three hypothesis tests. The
outcomes of these tests are totally independent.
All combinations of outcomes are possible:
2 main effects and interaction
1 main effect and interaction
2 main effects and no interaction
1 main effect and no interaction
interaction but no main effects
no main effects, no interaction
Main Effects and Interactions
29Monday, November 3, 14

30
Twofactor ANOVA hypothesis test
Step 1: State hypotheses
Step 2: Determine critical region (critical F values)
Step 3: Calculate Fratios
Step 4: Make decision
30Monday, November 3, 14

31
Notation and Formulas
Three hypothesis tests three Fratios four variances.
Schematic view:
31Monday, November 3, 14

32
Notation and Formulas
32Monday, November 3, 14

33
Stage 1:
Total variability:
SStotal = X2 G2
N
dftotal = N1
Notation and Formulas
33Monday, November 3, 14

34
Stage 1:
Betweentreatments variability:
dfbetween treatments = number of cells 1
SSbetween treatments = T2
nG2
N
Notation and Formulas
34Monday, November 3, 14