KNR 445
FACTORIAL
ANOVA
Slide 1
Statistically speaking…
 With Factor A, Factor B, and the interaction
A x B, the sums of squares are as follows:
1
2
Recall:
But:
3
SStotal  SSwithin  SSbetween
SSbetween  SSA  SSB  SSAxB
So:
4
SStotal  SSwithin  SSA  SSB  SSAxB
KNR 445
FACTORIAL
ANOVA
Slide 2
1
Statistically speaking…
 These are converted to variance estimates
by dividing by the d. of f.:
Variance for
main effect of
factor B
And:
So…
2
within
S
SSwithin

dfwithin
Variance for
interaction
between A & B
SSAxB
SSA 2 SSB 2
S AxB 
S 
SB 
df
df
df
AxB
A
B
Variance for
2
A
main effect of
factor A
KNR 445
FACTORIAL
ANOVA
Slide 3
Statistically speaking…
 And remember, these are estimates:
2
within
S
   (inherent variance)
estimates
2
1
S   effect of factor A
2
A
estimates
2
S   effect of factor B
2
B
estimates
2
estimates
2
SAxB

 2  the interactio n effect
KNR 445
FACTORIAL
ANOVA
Slide 4
Statistically speaking…
 Or, in another form:
Total sum of squares
Within-groups SS
1
Between-groups SS
SS associated with A
SS associated with B
SS associated with
interaction
KNR 445
FACTORIAL
ANOVA
Slide 5
Statistically speaking…
 So the F-tests are:
For Factor A:
2
A
For Factor B:
S
FA  2
Swithin
2
B
S
FB  2
Swithin
…and for the interaction:
1
FAxB
2
AxB
2
within
S

S
KNR 445
FACTORIAL
ANOVA
Slide 6
Assumptions
Situation
Anxiety
Level
Low
Anxiety
Low
Pressure
Moderat
e
pressure
High
Pressure
M=5
(3, 7)
M=8
(7, 9)
M = 11
(10, 12)
 Similar to 1-way ANOVA
 Assumptions now stated for cells, not groups. It
is assumed that…
1
High
Anxiety
M=4
(3, 5)
M=6
(5, 7)
M=2
(2, 2)
MB1 = 4.5
MB2 = 7
MB3 = 6.5
 Observations are independent (uncorrelated) from
one cell to the next
 2 is the same for all cells (homogeneity of variance)
 Cell populations are normally distributed
 Last 2 are mostly a concern for small sample sizes
MA1 = 8
MA2 = 4
Mtotal =
12
KNR 445
FACTORIAL
ANOVA
Slide 7
Assumptions
 Note – we have been discussing equal cell
1
sizes throughout.
 This is important, as it guarantees
independence of statistical effects in the
analysis
 If cell sizes are unequal, certain adjustments
must be made…
But
These
involve
reporting
different
types
sums
that
is probably
a step
too far
forofthis
of squares (Types
to 3 your
as reported
in SPSS,
class…just
make 1sure
cell sizes
are
Stevens, 1986, 1996)
equal!
KNR 445
FACTORIAL
ANOVA
Slide 8
1
2
Factorial ANOVA in SPSS
 Data considerations
 Dependent variable: interval/ratio, normally
distributed within each cell of the analysis)
 Independent variables: must be discrete
categories
 If the independent variable is continuous, the
number of categories can be created artificially by
doing a median split or quartile split, BUT you’d be
better off with regression
 Must be independent (see slide 28)
KNR 445
FACTORIAL
ANOVA
Slide 9
Factorial ANOVA in SPSS
 Data considerations
1. Each row is 1
person/item
2. There is now more
than one grouping
factor, so “group” is not
a good name for either
3. Dependent
variable
KNR 445
FACTORIAL
ANOVA
Slide 10
Factorial ANOVA in SPSS
 Data considerations
1. You can/should
label both the names
of the variables…
2. And their
individual levels
KNR 445
FACTORIAL
ANOVA
Slide 11
Factorial ANOVA in SPSS
 Performing the analysis
1. The general linear model is the
same family of statistical
techniques as we used in
regression – we could dummy
code these variables and get
exactly the same answer using
regression techniques
2. One dependent
variable implies
univariate analysis
KNR 445
FACTORIAL
ANOVA
Slide 12
Factorial ANOVA in SPSS
1. Here are the variables as listed
in the data file
 Performing the analysis
2. They just need
to be slid over to
the right places
KNR 445
FACTORIAL
ANOVA
Slide 13
Factorial ANOVA in SPSS
 Performing the analysis
2. “Plots” lets you
request a graph of the
2 factors
1. Further
options
KNR 445
FACTORIAL
ANOVA
Slide 14
Factorial ANOVA in SPSS
1.What you see
 Performing the
analysis
on choosing
“plots”
2. Slide over to
the right places
again…
KNR 445
FACTORIAL
ANOVA
Slide 15
Factorial ANOVA in SPSS
 Performing the analysis
1. If you choose the “post hoc”
option, you can specify which
main effects you want post hoc
tests for…but only those. And we
only request for “pressure” here,
because…
KNR 445
FACTORIAL
ANOVA
Slide 16
Factorial ANOVA in SPSS
1. “Options” lets you request
descriptives, effect sizes, power
statistics, and homogeneity tests. It is
the most important box to choose
 Performing the analysis
2. Choose only
those that I have
highlighted
below
KNR 445
FACTORIAL
ANOVA
Slide 17
Factorial ANOVA in SPSS
 Performing the analysis
1. Finally, click “ok”
once you’ve specified
what you want
KNR 445
FACTORIAL
ANOVA
Slide 18
Factorial ANOVA in SPSS
 The output
1. This table shows the
allocation of participants
to levels of each factor
2. This table shows the
descriptive stats – note
the balanced design!
KNR 445
FACTORIAL
ANOVA
Slide 19
Factorial ANOVA in SPSS
 The output
1. Here’s the homogeneity test.
It is not significant, which
means the assumption is met
(no problem).
KNR 445
FACTORIAL
ANOVA
Slide 20
Factorial ANOVA in SPSS
 The output
1. The ANOVA summary table,
showing main effects, interaction,
the degrees of freedom,
significance, and so on
1. Which are
significant?
2. 2 – effect sizes
– cite in write up
(like R2’s for each
effect)
KNR 445
FACTORIAL
ANOVA
Slide 21
1.
Factorial ANOVA in SPSS
 Eta-squared (2) & partial eta-squared.
 2 is exactly like an R2. So it can be treated the same.
From:
Levine &
Hullet,
2002 (see
web site
for full
citation)
SSbetween
 
SStotal
2
2.
 As the number of factors in the equation increases, so the SStotal
will increase
 So the 2 of any one factor will diminish with the introduction
of other factors
 All 2 in the analysis will sum to 1 or less (as per R2)
KNR 445
FACTORIAL
ANOVA
Slide 22
Factorial ANOVA in SPSS
 Eta-squared (2) & partial eta-squared.
 Partial 2 is calculated so that it does not shrink
with the introduction of more factors:
From:
Levine &
Hullet,
2002 (see
web site
for full
citation)
SSbetween
 
SSbetween  SSerror
1.
2
 For 1-way ANOVA, identical to 2. For multi-factorial
2.
designs, can be drastically different
 Partial 2 can sum to more than one when there are several
strong effects in the analysis.
KNR 445
FACTORIAL
ANOVA
Slide 23
Factorial ANOVA in SPSS
 Eta-squared (2) & partial eta-squared.
 SPSS reports partial 2
1.
 What does this imply?
From:
Levine &
Hullet,
2002 (see
web site
for full
citation)
 Partial is still ok as a measure of effect size, and is accepted by
journals.
 But you should be aware that it overestimates effect size when
sample size is small (it is a “biased estimator”)
 The best estimates are omega () and epsilon ()(neither of
which are available in SPSS, of course) – each is unbiased, and
so doesn’t change with sample size
 So, just keep using partial eta squared, but report it as
that, and be aware of the bias.
KNR 445
FACTORIAL
ANOVA
Slide 24
Factorial ANOVA in SPSS
 And now back to...The output
2. If we’d had sig main effects and an
interaction, we’d report the means for the
main effects but examine whether they are
superseded by the interaction (in which case
the main effects are not genuine)
1. Here, we have a
significant interaction…
and that’s all
KNR 445
FACTORIAL
ANOVA
Slide 25
Factorial ANOVA in SPSS
 The output
1. Now the post-hoc
tests on the
pressure main
effect. Ignore, as
the main effect was
not significant.
KNR 445
FACTORIAL
ANOVA
Slide 26
Factorial ANOVA in SPSS
 The output
1. Now the plot
of the cell means
making up the
interaction.
KNR 445
FACTORIAL
ANOVA
Slide 27
Factorial ANOVA in SPSS
 Next:
1. Next…
 Follow-up tests
 What to do if the homogeneity assumption is
not met
 What to do if you have multiple measures on
each participant(repeated measures!)