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Two-Factor ANOVA
Outline
Basic logic of a two-factor ANOVA
 Recognizing and interpreting main &
interaction effects
 F-ratios
 How to compute & interpret a two-way
ANOVA
 Assumptions
 Extension of Factorial ANOVA

Factorial Designs
Move beyond the one-way ANOVA to designs
that have 2+ IVs
 The variables can have unique effects or can
combine with other variables to have a
combined effect

Why Should We Use a Factorial
Design?

We can examine the influence that each
factor by itself has on a behaviour, as well
as the influence that combining these
factors has on the behaviour

Can be efficient and cost-effective
Interpretation of Factorial
Designs
Two Kinds of Information:
1.
Main effect of an IV
– Effect that one IV has independently
of the effect of the other IV
– Design with 2 IVs, there are 2 main
effects (one for each IV):

Main Effect of Factor A (1st IV): Overall difference among the levels of A
collapsing across the levels of B.

Main Effect of Factor B (2nd IV): Overall difference among the levels of B
collapsing across the levels of A.
Interpretation of Factorial
Designs
Two Kinds of Information:
2.
Interaction
– Represent how independent variables work
together to influence behavior
– The relationship between one factor and the
DV change with, or depends on, the level of
the other factor that is present
– The influence of changing one factor is NOT
the same for each level of the other factor
Two-Way ANOVA
F= variance between groups
variance within groups
 In a 2-way ANOVA, there are 3 F-ratios:
1. Main effect for Factor A
2. Main effect for Factor B
3. Interaction A x B
Guidelines for the Analysis of a
Factorial Design
First determine whether the interaction between
the independent variables is statistically
significant.
– If the interaction is statistically significant,
identify the source of the interaction by
examining the simple main effects
– Main effects should be interpreted cautiously
whenever an interaction is present in an
experiment
 Then examine whether the main effects of each
independent variable are statistically significant.

Analysis of Main Effects
When a statistically significant main effect
has only 2 levels, the nature of the
relationship is determined in the same
manner as for the independent samples ttest
 When a main effect has 3 or more levels,
the nature of the relationship is
determined using a Tukey HSD test

Effect Size

Three different values of ŋ2 are computed

ŋ2 for Factor A =
SSA_______
SStotal – SSB - SSAxB

ŋ2 for Factor B =
SSB_______
SStotal – SSA - SSAxB

ŋ2 for Factor AxB =
SSAxB______
SStotal – SSA - SSB
Effect Size – alternate formulas

2

2

2
A
SS A

SS A  SS within
B
SS B

SS B  SS within
AxB
SS AxB

SS AxB  SS within
Assumptions
The observations within each sample must
be independent
 DV is measured on an interval or ratio
scale
 The populations from which the samples
are selected have must have equal
variances
 The populations for which the samples are
selected must be normally distributed

Calculating 2 Factor Between
Subjects Design ANOVA by hand
Influence of a specific hormone on eating behaviour
 IV (A): Gender

– Males
– Females

IV (B): Drug Dose
– No drug
– Small dose
– Large dose

DV: Eating consumption over a 48-hour period
The Data ….
Factor B – Amount of drug
No drug
Factor A - Gender
Male
Female
Small dose
Large dose
1
7
3
6
7
1
1
11
1
1
4
6
1
6
4
0
0
0
3
0
2
7
0
0
5
5
0
5
0
3
Homogeneity of variance
=
s2 largest =
s2 smallest
 Satisfied
or violated???
Step 1: State the Hypotheses

Main Effect for Factor A

Main Effect for Factor B
Step 1: State the Hypotheses

Interaction between dosage & gender
Step 2: Compute df
Double Check:
dftotal= dfbetween + dfwithin
Step 3: Determine F-critical
Use the F distribution table
F Critical (df effect, df within)
 Using  = .05

Step 4: Calculate SS
SSTOTAL = 2 – G2
N
Step 4:
Calculate SS
SSBETWEEN Tx =  T2 – G2
n
N
Step 4: Calculate SS
SSWITHIN TX =  SS
inside each treatment
Double Check SS
SSTotal  SS Within Tx 
SS BetweenTx
SS for Factor A
SS A =  Trow2 – G2
nrow
N
SS for Factor B
SS B =  TColumn2 – G2
nColumn
N
SS for Interaction
SS AxB  SS Between  SS A  SS B
Step 5: Calculate MS for
Factor A
MSA = SSA
dfA
Step 5: Calculate MS for
Factor B
MSB = SSB
dfB
Step 5: Calculate MS for
Interaction
MSAxB = SSAxB
dfAxB
Step 5: Calculate MS Within
Treatments
MSwithin = SSwithin
dfwithin
Step 6: Calculate F ratios –
Factor A
MS A
F
MS within
Step 6: Calculate F ratios –
Factor B
MS B
F
MS within
Step 6: Calculate F ratios –
Interaction
MS AxB
F
MS within
Step 7: Summary Table
Source
Between Tx
Factor A
Factor B
Interaction
Within Tx
Total
SS
df
MS
F
Extension of Factorial ANOVA


1 factor is between subject & 1 factor is
within subject
e.g.: pre-post-control design
– All subjects are given a pre-test and a
post-test
– Participants divided into two groups
– Experimental group vs. control group
2 x 3 mixed design
Group
Therapy
Time
Control
Between-Subjects
Before
After
3 mos. after
Within-Subjects