Factorial ANOVA
2-Way ANOVA, 3-Way ANOVA,
etc.
Factorial ANOVA
• One-Way ANOVA = ANOVA with one IV
with 1+ levels and one DV
• Factorial ANOVA = ANOVA with 2+ IV’s
and one DV
– Factorial ANOVA Notation:
• 2 x 3 x 4 ANOVA
• The number of numbers = the number of IV’s
• The numbers themselves = the number of levels in
each IV
Factorial ANOVA
•
2 x 3 x 4 ANOVA = an ANOVA with 3 IV’s, one of which has
2 levels, one of which has 3 levels, and the last of which
has 4 levels
• Why use a factorial ANOVA? Why not just use
multiple one-way ANOVA’s?
1. Increased power – with the same sample size and
effect size, a factorial ANOVA is more likely to result
in the rejection of Ho
– aka with equal effect size and probability of rejecting Ho if
it is true (α), you can use fewer subjects (and time and
money)
Factorial ANOVA
• Why use a factorial ANOVA? Why not
just use multiple one-way ANOVA’s?
2. With 3 IV’s, you’d need to run 3 one-way
ANOVA’s, which would inflate your α-level
– However, this could be corrected with a
Bonferroni Correction
3. The best reason is that a factorial ANOVA
can detect interactions, something that
multiple one-way ANOVA’s cannot do
Factorial ANOVA
• Interaction:
– when the effects of one independent variable differ
according to levels of another independent variable
– Ex. We are testing two IV’s, Gender (male and
female) and Age (young, medium, and old) and their
effect on performance
• If males performance differed as a function of age, i.e. males
performed better or worse with age, but females
performance was the same across ages, we would say that
Age and Gender interact, or that we have an Age x Gender
interaction
Factorial ANOVA
• Interaction:
– Presented graphically:
40
• Note how male’s
Performance
•
performance changes as a
function of age while
females does not
Note also that the lines
cross one another, this is
the hallmark of an
interaction, and why
interactions are sometimes
called cross-over or
disordinal interactions
30
20
10
GENDER
Male
0
Female
Young
AGE
Medium
Old
Factorial ANOVA
• Interactions:
– However, it is not necessary that the lines
cross, only that the slopes differ from one
another
• I.e. one line can be flat, and the other sloping
upward, but not cross – this is still an interaction
• See Fig. 17.2 on page 410 in the text for more
examples
Factorial ANOVA
• As opposed to interactions, we have what
are called main effects:
– the effect of an IV independent of any other
IV’s
• This is what we were looking at with one-way
ANOVA’s – if we have a significant main effect of
our IV, then we can say that the mean of at least
one of the groups/levels of that IV is different than
at least one of the other groups/levels
Factorial ANOVA
• Finally, we also have simple effects:
– the effect of one group/level of our IV at one
group/level of another IV
• Using our example earlier of the effects of Gender
(Men/Women) and Age (Young/Medium/Old) on
Performance, to say that young women
outperformed other groups would be to talk about
a simple effect
Factorial ANOVA
• Calculating a Factorial ANOVA:
– First, we have to divide our data into cells
• the data represented by our simple effects
• If we have a 2 x 3 ANOVA, as in our Age and Gender
example, we have 3 x 2 = 6 cells
Young
Medium
Old
Male
Cell #1
Cell #2
Cell #3
Female
Cell #4
Cell #5
Cell #6
Factorial ANOVA
•
Then we calculate means for all of these cells, and for our IV’s across cells
– Mean #1 = Mean for Young Males only
– Mean #2 = Mean for Medium Males only
– Mean #3 = Mean for Old Males
– Mean #4 = Mean for Young Females
– Mean #5 = Mean for Medium Females
– Mean #6 = Mean for Old Females
– Mean #7 = Mean for all Young people (Male and Female)
– Mean #8 = Mean for all Medium people (Male and Female)
– Mean #9 = Mean for all Old people (Male and Female)
– Mean #10 = Mean for all Males (Young, Medium, and Old)
– Mean #11 = Mean for all Females (Young, Medium, and Old)
Young
Medium
Old
Male
Mean #1
Mean #2
Mean #3
Mean #10
Female
Mean #4
Mean #5
Mean #6
Mean #11
Mean #7
Mean #8
Mean #9
Factorial ANOVA
• We then calculate the Grand Mean ( X ..)
– This remains (ΣX)/N, or all of our observations
added together, divided by the number of
observations
• We can also calculate SStotal, which is also
calculated the same as in a one-way
2
ANOVA
X
2

X 

N
Factorial ANOVA
• Next we want to calculate our SS terms
for our IV’s, something new to factorial
ANOVA
– SSIV = nxΣ( X IV - X )2
..
• n = number of subjects per group/level of our IV
• x = number of groups/levels in the other IV
Factorial ANOVA
–
SSIV = nxΣ( X IV - X .. )2
1. Subtract the grand mean from each of our levels means
•
•
For SSgender, this would involve subtracting the mean for males
from the grand mean, and the mean for females from the
grand mean
Note: The number of values should equal the number of
levels of your IV
2. Square all of these values
3. Add all of these values up
4. Multiply this number by the number of subjects in each cell
x the number of levels of the other IV
5. Repeat for any IV’s
•
Using the previous example, we would have both SSgender and
SSage
Factorial ANOVA
• Next we want to calculate SScells, which has a
formula similar to SSIV
–
SScells =

n X cell  X ..

2
1. Subtract the grand mean from each of our cell means
•
Note: The number of values should equal the number of cells
2. Square all of these values
3. Add all of these values up
4. Multiply this number by the number of subjects in each cell
Factorial ANOVA
• Now that we have SStotal, the SS’s for our
IV’s, and SScells, we can find SSerror and the
SS for our interaction term, SSint
– SSint = SScells – SSIV1 – SSIV2 – etc…
• Going back to our previous example,
SSint = SScells – SSgender – SSage
– SSerror = SStotal – SScells
Factorial ANOVA
• Similar to a one-way ANOVA, factorial ANOVA
uses df to obtain MS
– dftotal = N – 1
– dfIV = k – 1
• Using the previous example, dfage = 3 (Young/Medium/Old) –
1 = 2 and dfgender = 2 (Male/Female) – 1 = 1
– dfint = dfIV1 x dfIV2 x etc…
• Again, using the previous example, dfint = 2 x 1 = 2
– dferror = dftotal – dfint - dfIV1 – dfIV2 – etc…
Factorial ANOVA
• Factorial ANOVA provides you with Fstatistics for all main effects and
interactions
– Therefore, we need to calculate MS for all of
our IV’s (our main effects) and the interaction
– MSIV = SSIV/dfIV
• We would do this for each of our IV’s
– MSint = SSint/dfint
– MSerror = SSerror/dferror
Factorial ANOVA
• We then divide each of our MS’s by MSerror to
•
obtain our F-statistics
Finally, we compare this with our critical F to
determine if we accept or reject Ho
– All of our main effects and our interaction have their
own critical F’s
– Just as in the one-way ANOVA, use table E.3 or E.4
depending on your alpha level (.05 or .01)
– Just as in the one-way ANOVA, “df numerator” = the
df for the term in question (the IV’s or their
interaction) and “df denominator” = dferror
Factorial ANOVA
• Just like in a one-way ANOVA, a significant
F in factorial ANOVA doesn’t tell you
which groups/levels of your IV’s are
different
– There are several possible ways to determine
where differences lie
Factorial ANOVA
• Multiple Comparison Techniques in Factorial
ANOVA:
1. Several one-way ANOVA’s (as many as there are
IV’s) with their corresponding multiple comparison
techniques
– probably the most common method
– Don’t forget the Bonferroni Method
2. Analysis of Simple Effects
– Calculate MS for each cell/simple effect, obtain an F for
–
each one and determine its associated p-value
See pages 411-413 in your text – you should be familiar
with the theory of the technique, but you will not be asked
to use it on the Final Exam
Factorial ANOVA
• Multiple Comparison Techniques in
Factorial ANOVA:
– In addition, interactions must be decomposed
to determine what they mean
• A significant interaction between two variables
means that one IV’s value changes as a function of
the other, but gives no specific information
• The most simple and common method of
interpreting interactions is to look at a graph
40
30
Performance
20
10
GENDER
Male
0
Female
Young
Medium
Old
AGE
• Interpreting Interactions:
– In the example above, you can see that for Males, as age
increases, Performance increases, whereas for Females there is
no relation between Age and Performance
– To interpret an interaction, we graph the DV on the y-axis, place
one IV on the x-axis, and define the lines by the other IV
• You may have to try switching the IV’s if you don’t get a nice
interaction pattern the first time
Factorial ANOVA
• Effect Size in Factorial ANOVA:
– η2 (eta squared) = SSIV/SStotal (for any of the
IV’s) or SSint/SStotal (for the interaction)
• tells you the percent of variability in the DV
accounted for by the IV/interaction
• like the one-way ANOVA, very easily computed and
commonly used, but also very biased – don’t ever
use it
Factorial ANOVA
• Effect Size in Factorial ANOVA:
– ω2 (omega squared) = SS IV  df IV MS error
SS int  df int MS error
• or
SS total  MS error
SS total  MS error
• also provides an estimate of the percent of
variability in the DV accounted for by the
IV/interaction, but is not biased
Factorial ANOVA
• Effect Size in Factorial ANOVA:
– Cohen’s d = X  X
1
2
sp
• the two means can be between two IV’s, or between two
groups/levels within an IV, depending on what you want to
estimate
• Reminder: Cohen’s conventions for d – small = .3, medium =
.5, large = .8
– Your text says that d = .5 corresponds to a large effect (pg.
415), but is mistaken – check the Cohen article on the top of
pg. 157
Factorial ANOVA
• Example #1:
– Remember the example we used in one-way ANOVA
of the study by Eysenck (1974) looking at the effects
of Age/Depth of Recall on Memory Performance?
Recall how I said that although 2 IV’s were used it
was appropriate for a one-way ANOVA because the
IV’s were mushed-together. Now we will explore the
same data with the IV’s unmushed.
• DV = Memory Performance
• 2 IV’s = Age – 2 levels (Young and Old); Depth of Recall – 5
levels/conditions (Counting, Rhyming, Adjective, Imagery, &
Intentional)
• 2 x 5 Factorial ANOVA = 10 cells
Old
Young
Counting
Rhyming
Adjective
Imagery
Intentional
9
7
11
12
10
8
9
13
11
19
6
6
8
16
14
8
6
6
11
5
10
6
14
9
10
4
11
11
23
11
6
6
13
12
14
5
3
13
10
15
7
8
10
19
11
7
7
11
11
11
8
10
14
20
21
6
7
11
16
19
4
8
18
16
17
6
10
14
15
15
7
4
13
18
22
6
7
22
16
16
5
10
17
20
22
7
6
16
22
22
9
7
12
14
18
7
7
11
19
21
Factorial ANOVA
• 10 cells
• Red = means of entire levels of IV’s
Counting Rhyming Adjective
Imagery Intentional Mean
Old
7.0
6.9
11.0
13.4
12.0
10.06
Young
6.5
7.6
14.8
17.6
19.3
13.16
Mean
6.75
7.25
12.9
15.5
15.65
11.61
Factorial ANOVA
–
–
–
–
–
dftotal = N – 1 = 100 – 1 = 99
dfage = k – 1 = 2 – 1 = 1
dfcondition = 5 – 1 = 4
dfint = dfage x dfcondition = 4 x 1 = 4
dferror = dftotal – dfage – dfcondition - dfint = 99 – 4 – 4 – 1
= 90
• Critical F’s:
– For Age – F.05(1, 90) = 3.96
– For Condition – F.05(4, 90) = 2.49
– For the Age x Condition Interaction - F.05(4, 90) =
2.49
Factorial ANOVA
• SStotal =
2



X
X 2 
N
= 16,147 – 11612/100
= 2667.79
• Grand Mean = ΣX/N = 1161/100 = 11.61
• SSage = nc X  X 2

age
..

= (10)(5)[(10.06 – 11.61)2 + (13.16
– 11.61)2 = 240.25
Factorial ANOVA
• SScondition = naX condition  X .. 
2
= (10)(2)[(6.75 – 11.61)2 +
(7.25 – 11.61)2 + (12.9 – 11.61)2
+ (15.5 – 11.61)2 + (15.65 –
11.61)2 = 1514.94
Factorial ANOVA

• SScells = n X cell  X ..
•

2
= 10 [(7.0 – 11.61)2 + (6.9 –
11.61)2 + (11.0 – 11.61)2 + (13.4 –
11.61)2 + (12.0 – 11.61)2 + (6.5 –
11.61)2 + (7.6 – 11.61)2 + (14.8 – 11.61)2 +
(17.6 – 11.61)2 + (19.3 – 11.61)2 = 1945.49
SSint = SScells – SSage – SScondition = 1945.49 –
240.25 – 1514.94 = 190.30
Factorial ANOVA
• SSerror = SStotal – SScells = 2667.79 –
1945.49 = 722.30
• MSage = 240.25/1 = 240.25
• MScondition = 1514.94/4 = 378.735
• MSint = 190.30/4 = 47.575
• MSerror = 722.30/90 = 8.026
Factorial ANOVA
• F (Age) = 240.25/8.026 = 29.94
– Critical F.05(1, 90) = 3.96
• F (Condition) = 378.735/8.026 = 47.19
– Critical F.05(4, 90) = 2.49
• F (Interaction) = 47.575/8.026 = 5.93
– Critical F.05(4, 90) = 2.49
• All 3 F’s are significant, therefore we can
reject Ho in all cases
Factorial ANOVA
• Example #2:
– The previous example used data from Eysenck’s
(1974) study of the effects of age and various
conditions on memory performance. Another aspect
of this study manipulated depth of processing more
directly by placing the participants into conditions that
directly elicited High or Low levels of processing. Age
was maintained as a variable and was subdivided into
Young and Old groups. The data is as follows:
Factorial ANOVA
•
•
•
•
Young/Low: 8 6 4 6 7 6 5 7 9 7
Young/High: 21 19 17 15 22 16 22 22 18 21
Old/Low: 9 8 6 8 10 4 6 5 7 7
Old/High: 10 19 14 5 10 11 14 15 11 11
1.
2.
3.
4.
5.
Get into groups of 2 or more
Identify the IV’s and the DV’s, and the number of levels of each
Identify the number of cells
Calculate the various df’s and the critical F’s
Calculate the various F’s [two main effects (one for each IV) and
one interaction]
Determine the effect sizes (Cohen’s d) for the F-statistics that
you’ve obtained
6.
Factorial ANOVA
Descriptive Statistics
Dependent Variable: MEMPERF
• IV = Age (2
•
•
•
•
levels) and
Condition (2
levels)
2 x 2 ANOVA =
4 cells
dage = .70
dcondition = 1.82
dint = .80
Between-Subjects Factors
AGE
CONDITIO
.00
1.00
.00
1.00
Value Label
Young
Old
Low
High
AGE
Young
N
20
20
20
20
Old
Total
CONDITIO
Low
High
Total
Low
High
Total
Low
High
Total
Mean
6.5000
19.3000
12.9000
7.0000
12.0000
9.5000
6.7500
15.6500
11.2000
Std. Deviation
1.43372
2.66875
6.88935
1.82574
3.74166
3.84571
1.61815
4.90193
5.76995
Tests of Between-Subjects Effects
Dependent Variable: MEMPERF
Source
Corrected Model
Intercept
AGE
CONDITIO
AGE *
CONDITIO
Error
Total
Corrected Total
Type III Sum
of Squares
df
a
Mean Square
F
Sig.
Partial Eta
Squared
1059.800
3
353.267
53.301
.000
.816
5017.600
115.600
792.100
1
1
1
5017.600
115.600
792.100
757.056
17.442
119.512
.000
.000
.000
.955
.326
.769
152.100
1
152.100
22.949
.000
.389
238.600
6316.000
1298.400
36
40
39
6.628
a. R Squared = .816 (Adjusted R Squared = .801)
N
10
10
20
10
10
20
20
20
40