FACTORIAL DESIGNS
• What is a factorial design?
• Why use it?
• When should it be used?
FACTORIAL DESIGNS
• What is a factorial design?
Two or more ANOVA factors are combined in
a single study: eg. Treatment (experimental
or control) and Gender (male or female).
Each combination of treatment and gender
are present as a group in the design.
FACTORIAL DESIGNS
• Why use it?
• In social science research, we often
hypothesize the potential for a specific
combination of factors to produce effects
different from the average effects- thus, a
treatment might work better for girls than
boys. This is termed an INTERACTION
FACTORIAL DESIGNS
• Why use it?
• Power is increased for all statistical tests by
combining factors, whether or not an
interaction is present. This can be seen by
the Venn diagram for factorial designs
SSDependent Variable
Treatment
SSe
SSTG
SST
SSG
Gender
Treatment x Gender
Fig. 10.6: Venn diagram for balanced two factor ANOVA design
FACTORIAL DESIGN
• When should it be used?
• Almost always in educational and
psychological research when there are
characteristics of subjects/participants that
would reduce variation in the dependent
variable, aid explanation, or contribute to
interaction
TYPES OF FACTORS
• FIXED- all population levels are present in
the design (eg. Gender, treatment condition,
ethnicity, size of community, etc.)
• RANDOM- the levels present in the design
are a sample of the population to be
generalized to (eg. Classrooms, subjects,
teacher, school district, clinic, etc.)
GRAPHICALLY
REPRESENTING A DESIGN
Factor B
B1
Factor
A1
2
4
B2
B3
B3
B4
A
A2
Two-dimensional representation of 2 x 4 factorial design
GRAPHICALLY REPRESENTING A DESIGN
Factor
B1
Factor B
1
3
B2
B3
B4
A1
C1
A
A2
Factor C
Factor
A1
A
C2
A2
Table 10.1: Two-dimensional representation of 2 x 4 factorial design
Three-dimensional representation of 2 x 4 x3 factorial design
LINEAR MODEL
yijk =  + i + j + ij + eijk
where  = population mean for populations of all subjects,
called the grand mean,
i = effect of group i in factor 1 (Greek letter nu),
j = effect of group j in factor 2 (Greek letter omega),
ij = effect of the combination of group i in factor 1 and
group j in factor 2,
eijk = individual subject k’s variation not accounted for by any
of the effects above
Interaction Graph
y
Suzy’s predicted
score; she is in E
Effect of being in
Experimental
group
Effect of
being a girl
Effect of being a
girl in
Experimental
group
mean
0
Effect of
being a boy
Effect of not being
a girl in
Experimental
group
Effect of being in
Control group
INTERACTION
y
y
level 2 of
Factor K
M
E
A
N
S
level 1 of
Factor K
M
E
A
N
S
level 2 of
Factor K
level 1 of
Factor K
L1
L
2
Factor L
Ordinal Interaction
L3
L1
L
2
L
3
Factor L
Disordinal Interaction
Fig. 10.4: Graphs of ordinal and disordinal interactions
INTERACTION
20
M
E
A
N
S
15
10
Boys
5
Girls
0
Treatment 1
Treatment 2
Gender
Disordinal interaction for 2 x 2 treatment by gender design
ANOVA TABLE
• SUMMARY OF INFORMATION:
SOURCE DF
Independent
variable
or factor
Degrees
of freedom
SS MS F
Sum of Mean Fisher
Squares Square statistic
E(MS)
Expected mean
square (sampling
theory)
PATH DIAGRAM
• EACH EFFECT IS REPRESENTED BY A
SINGLE DEGREE OF FREEDOM PATH
• IF THE DESIGN IS BALANCED (EQUAL
SAMPLE SIZE) ALL PATHS ARE
INDEPENDENT
• EACH FACTOR HAS AS MANY PATHS
AS DEGREES OF FREEDOM,
REPRESENTING POC’S
A1
eijk
A2
B1
yijk
B2
AB2,2
AB1,1
:
AB1,2
AB2,1
SEM representation of balanced factorial
3 x 3 Treatment (A) by Ethnicity (B) ANOVA
Contrasts in Factorial Designs
• Contrasts on main effects as in 1 way
ANOVA: POCs or post hoc
• Interaction contrasts are possible: are
differences between treatments across
groups (or interaction within part of the
design) significant? eg. Is the Treatmentcontrol difference the same for Whites as
for African-Americans (or Hispanics)?
– May be planned or post hoc
CT
1
CT
Ry.T
2
yijk
eijk
T
yijk
G1
Ry.G
eijk
Ry.TG
G
CTG
1
CTG
2
Orthogonal contrast path diagram
TxG
Generalized effect path diagram
Two path diagrams for a 3 x 2 Treatment by Gender balanced factorial design
UNEQUAL GROUP SAMPLE SIZES
• Unequal sample sizes induce overlap in the
estimation of sum of squares, estimates of
treatment effects
• No single estimate of effect or SS is correct,
but different methods result in different
effects
• Two approaches: parameter estimates or
group mean estimates
UNEQUAL GROUP SAMPLE SIZES
• Proportional design: main effects sample
sizes are proportional:
M F
–
–
–
–
Experimental-Male n=20 E 20
Experimental-Female n=30 C 10
Control- Male
n=10
Control-Female
n=15
30
15
• Disproportional: no proportionality across
cells
SST T
SST T
e
SSe
SSTGGT
SSe
e
SSG G
SSTGTG
SSG G
Unbalanced factorial design
Unbalanced factorial
design with
proportional marginal
sample sizes
Venn diagrams for disproportional and proportional unbalanced designs
ASSUMPTIONS
• NORMALITY
– Robust with respect to normality and skewness
with equal sample sizes, simulations may be
useful in other cases
• HOMOGENEOUS VARIANCES
– problem if unequal sample sizes: small groups
with large variances cause high Type I error
rates
• INDEPENDENT ERRORS: subjects’ scores
do not depend on each other
– always a problem if violated in multiple testing
GRAPHING INTERACTIONS
• Graph means for groups:
– horizontal axis represents one factor
– construct separate connected lines for each
crossing factor group
– construct multiple graphs for 3 way or higher
interactions
GRAPHING INTERACTIONS
O
u
females
t
c
o
males
m
e
c
e1
Treatment groups
e2
EXPECTED MEAN SQUARES
• E(MS) = expected average value for a mean
square computed in an ANOVA based on
sampling theory
• Two conditions: null hypothesis E(MS) and
alternative hypothesis E(MS)
– null hypothesis condition gives us the basis to
test the alternative hypothesis contribution
(effect of factor or interaction)
EXPECTED MEAN SQUARES
• 1 Factor design:
Source
E(MS)
Treatment A
2e + n2A
error
2e (sampling variation)
Thus F=MS(A)/MS(e) tests to see if
Treatment A adds variation to what might be
expected from usual sampling variability of
subjects. If the F is large, 2A  0.
EXPECTED MEAN SQUARES
• Factorial design (AxB):
Source
E(MS)
Treatment A
2e + (1-b/B)n2AB + nb2A
error
2e (sampling variation)
Thus F=MS(A)/MS(e) does not test to see if
Treatment A adds variation to what might be
expected from usual sampling variability of
subjects unless b=B or 2AB = 0 .
If b (number of levels in study) = B (number in the
population, factor is FIXED; else RANDOM
EXPECTED MEAN SQUARES
• Factorial design (AxB):
Source
E(MS)
Treatment A
2e + (1-b/B)n2AB + nb2A
AxB
2e + (1-b/B)n2AB
error
2e (sampling variation)
If 2AB = 0 , and B is random, then
F = MS(A) / MS(AB) gives the correct test of
the A effect.
EXPECTED MEAN SQUARES
• Factorial design (AxB):
Source
E(MS)
Treatment A
2e + (1-b/B)n2AB + nb2A
AB
2e + (1-b/B)n2AB
error
2e (sampling variation)
If instead we test F = MS(AB)/MS(e) and it is
nonsignificant, then 2AB = 0 and we can test
F = MS(A) / MS(e)
*** More power since df= a-1, df(error) instead of
df = a-1, (a-1)*(b-1)
Source
df
Expected mean square
A
I-1
2e + n2AB + nJ2A
B
J-1
2e + n2AB + nI2B
AB
(I-1)(J-1)
error
N-IJK
2e + n2AB
2e
Table 10.3: Expected mean square table for I x J random factorial design
Source
df
Expected mean square
A (fixed)
I-1
2e + n2AB + nJ2A
B (random)
J-1
2e + nI2B
AB
(I-1)(J-1)
2e + n2AB
error
N-IJK
2e
Table 10.5: Expected mean square table for I x J mixed model factorial design
Mixed and Random Design Tests
• General principle: look for denominator E(MS)
with same form as numerator E(MS) without the
effect of interest:
F = 2effect + other variances /other variances
• Try to eliminate interactions not important to the
study, test with MS(error) if possible
n
-
D
I
q
S
d
S
S
u
F
i
f
g
I
H
n
4
1
4
7
1
a
E
0
1
0
P
H
4
1
4
0
0
b
E
4
1
4
S
H
0
1
0
2
0
b
E
4
1
4
P
H
4
1
4
5
0
c
*
E
9
0
3
a
M
b
M
c
M
NOTE: SPSS tests parameter effects, not mean effects; thus,
SCHOOL should be tested with MS(SCHOOL)/MS(Error),
which gives F=1.532, df=1,40, still not significant
PLOT OF INTERACTION OF SCHOOL AND
PROGRAM ON SOCIAL SKILLS
Estimated Marginal Means of SOCLPOST
12
Estimated Marginal Means
11
10
9
PROGRAM
8
1.00
7
2.00
3
SCHOOL
5