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Factorial design
The most common design for a
n
way ANOVA is the factorial design.
In a factorial design, there are two ormore experimental factors, each witha given number of levels.
Observations are made for eachcombination of the levels of eachfactor (see example)
In a completely randomized factorialdesign, each experimentally unit israndomly assigned to one of thepossible combination of the existinglevel of the experimental factors.
Example of a factorial design with twofactors (A and B). Each factor has threelevels. y
ijk
represents the
k
th
observation
in the condition defined by the
i
th
level of
factor A and
j
th
level of factor B.
y
33k
y
32k
y
31k
A
Y
23k
y
22k
y
21k
A
y
13k
y
12k
y
11k
A
B
B
B
Factor B
Factor A
Data Analysis (draft) - Gabriel Baud-Bovy
Advantages of the factorial design•
A two-way design enables us to examine the
joint
(or interaction)
effect of the independent variables on the dependent variable. An interaction
means that the effect of one independent variable has
on a dependent variable is not the same for all levels of the otherindependent variable. We cannot get this information by runningseparate one-way analyses.
Factorial design can lead to more powerful test by reducing theerror (within cell) variance. This point will appear clearly when willcompare the result of one-way analyses with the results of a two-way analyses or t-tests.
Interaction plot
An interaction plot represents the meanvalue
m
ij
observed in each one of the
condition of a factorial design.
The Y axis corresponds to the dependent(or criterion) variable. The various level ofone of the two experimental factor arealigned on the X axis. The lines relate themean values that corresponds to thesame level of the second experimentalfactor.
There is an
interaction
between the
factors if the lines are not parallel becausethe effect of one factor depends on thevalue of the other factor.
If the lines are a parallel, the effect of thesecond factor is independent from thevalue of the first factor. In other words,there is no interaction.
A
A
B
B
B
A
A
B
B
B
Exercise.
Make the interaction plots
for the second table. Describe theinteraction (if any).
B
Mean Y
3020 100
A
1.002.
A
Mean Y
30 20 10 0
B
1.002.003.
Data Analysis (draft) - Gabriel Baud-Bovy
-^
The structural model of a two-wayfactorial ANOVA
without interaction
is
-^
In absence of interaction, the mean value μ
ij^
in condition (A
Bi
) depends in aj
additive
manner on the effect of each condition
-^
The complete model of the two-wayfactorial ANOVA iswhere
ij
ij
i^
j^
i•
•j
is the interaction effect. The
interaction effect represents the fact thatthe contribution of one factors depends onthe value of the other factor in a non-additive way.
Structural model (factorial ANOVA)
ijk
j
i
ijk
y
ijk
ij
i
i
ijk
y
Mean
A
A
Mean
B
B
B
y
ijk
be the
k
th
observation of the
i
th
level
of factor A and
j
th
level of factor B.
ij^
be the population mean for the i
th
level of factor A and j
th
level of factor B
(condition A
Bi^
), letj
i•
be the population
mean in condition A
, leti
•j
be the
population mean in condition B
and letj.
be
the grand mean.
i^
i•
- μ
is the effect of
factor A and
j^
•j
- μ
is the effect of
factor B.
j
i
ij
Data Analysis (draft) - Gabriel Baud-Bovy
Exercise.
m
•j
A
A
m
i•
B
B
B
m
ij
m
•j
A
A
m
i•
B
B
B
m
ij
- Compute the main and interaction effects from the mean values (see tables in the
left column). Answer: see tables in the right column.
j
A
A
i
B
B
B
ij
j
A
A
i
B
B
B
ij
Mean values
Table of effects
Data Analysis (draft) - Gabriel Baud-Bovy
Sum of squares
k j i
ij
ijk
k j i
ij
k j i
ijk
m
y
SSE
m
m
SStr
m
y
SST
k j i
j
i
ij
k j i
j
k j i
i
m
m
m
m
SSAB
m
m
SSB
m
m
SSA
-^
Like in the one-way ANOVA, the total sum ofsquares (SST) can be decomposed into a between-groups sum of square (the treatment effect, SStr)and a within-group sum of square (SSE) whichcorresponds to the residual variance:
SST = SStr + SSE
Note that the group (or experimental condition) in afactorial designed is determined by the value of twoor more experimental factors.
-^
The between-group variations (SStr) can themselvesbe decomposed further into a variations that areexplained by factor A (SSA), variations that areexplained by factor B (SSB) and variation that areexplained by the interaction between both factors(SSAB)
SStr = SSA + SSB + SSAB
Data Analysis (draft) - Gabriel Baud-Bovy
If the null hypotheses aretrue, the F ratios follow aFisher distribution with thecorresponding degrees offreedom.In that can be shown thatthe numerator is also anestimate the residualvariance.
F tests
-^
A factorial design aims at answering three different questions:
1. Is there an effect of the first experimental factor?
H
= 0i
y
ijk
j^
ij
ijk
2. Is there an effect of the second experimental factor?
H
= 0j
y
ijk
i^
ij
ijk
3. Is there an interaction?
H
ij^
( y
ijk
i^
j^
ijk
-^
In all cases, the alternative hypothesis is the complete model
H
y
ijk
i^
j^
ij
ijk
-^
The residual variance (within-group variance) for this modelis:
-^
In all cases, the F test is constructed by computing thepercentage of variance that is explained by the parameters ofinterest divided by the residual variance of the more complexmodel.
MSE
n
n
SSAB
F
MSE
n
SSB
F
MSE
n
SSA
F
B
A
AB
B
B
A
A
(^
n n n N n n N
SSE
n
n
n
SSE
MSE
B A B A B A
Data Analysis (draft) - Gabriel Baud-Bovy
[R] Interaction plot
visits<-read.table("visits.dat",header=TRUE)
visits$age<-ordered(visits$age,c("20-29","30-39","40-49",">50"))
interaction.plot(visits$age,visits$disease,visits$duration,
type="b",col=1:4,lty=1,lwd=2,pch=c(15,15,15,15),las=1,
xlab="Age",ylab="Visit duration (min)",trace.label="Disease")
45 40 35 30 25 20
Age
Visit duration (min)
20-
30-
40-
Disease
cancercerebrovascularheartuberculosis
- Lines are not parallel which is the
tell-tale sign of an interaction. Thisplot suggests that the visit timeincrease with the older age groupsfor the cancer and cerebro-vascular diseases while itremained constant for the heartand tuberculosis diseases.
Type I, Type II and Type III sum of squares•
Type I (sequential):
Terms are entered sequentially inthe model.
Type I SS depend on the order inwhich terms are entered in themodel
Type I SS can be added to yield tothe total SS.
Type II (hierarchical)
see textbook
Type III (marginal)
Type III SS correspond to the SSexplained by a term
after
all other
terms have already been included inthe model.
Type III SS do not add.
- The analysis of unbalanced data sets
(different number of observation ineach group) present speficialdifficulties because there are differentways of computing the sum ofsquares. These different wayscorresponding to different hypothesesand, correspondly, the F tests aredifferent.
- The R function anova yields Type I
sum of square.
-^
Most textbooks suggest using theType III sum of squares and manystatistical sofftwares use Type III sumof square as a default but manystasticians think it does not makesense when there are statisticallysignificant interactions.
Overall & Spiegel (1969) Psychol. Bull., 72:311- 322, for a detailed discussion of factorial designs.
Data Analysis (draft) - Gabriel Baud-Bovy
[R] Type III sum of square
The function Anova in the library car compute type II and type IIIsum of squares >
library(car)
Anova(aov(duration~disease+age,v0),type="III")
Anova
Table (Type
III
tests)
Response:
duration
Sum Sq Df
F
value
Pr(>F)
(Intercept) 29261.
2.2e-
age
9.34e-
disease
2.2e-
Residuals
Compare with Type I sum of squares
anova(aov(duration~age+disease,v0))Analysis of Variance TableResponse: duration
Df
Sum Sq Mean Sq F value
Pr(>F)
age
3 1072.
16.532 3.017e-
disease
3 2928.
45.142 < 2.2e-
Residuals 71 1535.
anova(aov(duration~disease+age,v0))Analysis of Variance TableResponse: duration
Df
Sum Sq Mean Sq F value
Pr(>F)
disease
3 2839.
43.767 3.964e-
age
3 1161.
17.908 9.339e-
Residuals 71 1535.
Data Analysis (draft) - Gabriel Baud-Bovy
Repeated-measure designsThis example of one-wayrepeated measure ANOVAshows only small differencesbetween treatments and largedifference between subjects.In the repeated-measureANOVA, we neglect thevariations between subjectsand consider only thevariation for each treatmentwithin each subject.
-^
In repeated-measure designs, severalobservations are made on the same experimentalunits. For a example, one of the most commonresearch paradigm is that where subjects areobserved at several different point in time (e.g.,before and after treatment, longitudinal studies).
-^
In repeated measure design, it is important todistinguish between-subject and within-subjectfactors.
Within-subject factors
are variables (like
time or treatment or repetition) that identify thedifferences between conditions or treamentsthat have been assigned to each subject.-
Between-subject factors
are varables (like
age or sex or group) that identify differencesbetween the subjects.
Data Analysis (draft) - Gabriel Baud-Bovy
Statistical approaches•^
There are three approaches to repeated-measure designs: 1.
The univariate approach:
This approach uses the classic univariate F test of
the ANOVA. However, the data must satisfy the so-called sphericity conditionin addition of the usual assumptions for the test to be valid. It is possible toadjust degrees of freedom to account for possible violation of the sphericityassumption.
The multivariate approach:
This sphericity condition does not need to be
satisfied. However, this approach requires a larger number of observation(number of subjects must be larger than number of experimental conditions)and, in general, is less powerful than the univariate approach.
The linear mixed model approach:
This approach is probably the best
approach from a theoretical point of view but it is quite complex.
-^
References: Keselman, H. J., Algina, J., & Kowalchuk, R. K. (2001). The analysis of repeatedmeasures designs: a review. British Journal of Mathematical and Statistical Psychology, 54, 1-20.
The univariate approach•
In the previous examples of ANOVAs, we have assumed that theobservations between experimental conditions are uncorrelated (orindependent). This assumption is valid if different subjects are used indifferent experimental conditions. However, this assumption is no morevalid if the same subjects are used in several (or all) experimentalcondition because better subjects in one condition are also likely toperform better in the other conditions.
In the repeated-measure ANOVA, the data must also satisfy the so-called sphericity
(or
circularity
)^
condition or the
compound symmetry
condition in addition of the usual assumptions (independence,homogeneity of the variances, and normality).
The compound symmetry condition is a stronger assumption than thesphericity condition.
The sphericity condition needs to apply only to within-subject factors. It isautomatically satisfied if the within-subject factor has only two levels.
Data Analysis (draft) - Gabriel Baud-Bovy
Adjusting of the degrees of freedom•^
While tests for the sphericity or compound symmetry exist (e.g. Mauchly’s test),they are not very reliable because they are quite sensitive to deviations of thenormality assumption.
-^
A better approach is to
adjust the degrees of freedom
in order to make the tests of
the repeated measure ANOVA more conservative. Several correction factors exist:Greenhouse-Geisser (1959), Huynh-Feldt (1990) and a lower-bound value which ismost conservative (see relevant literature for more details). SPSS will automaticallycompute the value of these factors.
-^
To adjust the F test, it is necessary to multiply the two degrees of freedom of the Fdistribution by the correction factor. Since the value of the correction factor issmaller than 1, this will decrease the degrees of freedom of the F distribution andmake, in general, the test more conservative.
Example. RQ data set•
Mauchly’s test
is sued to to check if sphericity is statisfied
mauchly.test(fit,idata=idata,X=~1)
Mauchly's
test
of
sphericity
Contrasts
orthogonal
to
data:
SSD
matrix
from
aov(formula
cbind(rq.0,
rq.3,
rq.7)
data
=^
rq.w)
W
p-value
-^
Multivariate tests
do not assume sphericity
anova(fit,idata=idata,X=~1,test="Pillai") Analysis
of
Variance
Table
Contrasts
orthogonal
to
Df
Pillai
approx
F
num
Df
den
Df
Pr(>F)
(Intercept)
Residuals
Argument
test="Spherical"
gives access to alternative multivariate tests (
"Wilks",
"Hotelling-Lawley", "Roy",
"Spherical")
Data Analysis (draft) - Gabriel Baud-Bovy
The dataset contains the absolute thresholds of 6 subjects (3 males and 3 females).Each subject performed 10 trials with one of two possible different starting values(start=0 and 10).
We want to test whether there is a difference between the thresholds of the twosexes and iIf the threshold depend on the starting values
One between-subject factor
(sex) and
one within-subject factor
(start)
Example. Threshold dataset
Example. Threshold dataset
define
threshold
dataset
(wide
format)
th.w<-data.frame(
sex=factor(c("M","F","M","F","M","F")),y=matrix(
c(5.4,3.9,5.8,4.9,5.2,3.9,3.9,6.3,5.9,5.3,
4.2,5.5,4.8,5.5,5.0,6.1,5.4,6.0,4.0,7.7,3.6,3.1,4.5,4.9,5.1,5.1,6.2,4.9,5.4,4.5,4.6,6.1,5.6,2.8,5.9,4.5,5.5,3.0,5.7,8.1,4.0,4.9,4.9,4.5,5.5,3.7,6.2,5.2,3.6,6.3,4.3,6.1,4.3,5.3,4.3,4.9,5.9,4.9,6.0,7.6),nrow=6,byrow=TRUE)
define
within-subject
factors
idata<-data.frame(
half=factor(rep(1:2,each=5),start=factor(rep(c(0,10),5)))
reshape
into
long
format
th.l<-reshape(th.w,
varying=paste(“y",1:10,sep="."),v.names="y",idvar="su",timevar="trial",direction="long")
add
start
value
th.l$start<-ifelse(th.l$trial%%2,0,10)#
reorder
data
th.l<-th.l[order(th.l$su,th.l$trial),c("su","sex","trial","start","y")]#
define
factors
th.l$su<-factor(th.l$su)th.l$sex<-factor(th.l$sex)th.l$start<-factor(th.l$start)
Data Analysis (draft) - Gabriel Baud-Bovy
Example. Threshold data set^ >
summary(aov(y~sex*start+Error(su/start),th.l)) Error:
su
Df
Sum
Sq
Mean
Sq
F
value
Pr(>F)
sex
Residuals
Error:
su:start
Df
Sum
Sq
Mean
Sq
F
value
Pr(>F)
start
sex:start
Residuals
Error:
Within
Df
Sum
Sq
Mean
Sq
F
value
Pr(>F)
Residuals
compute
the
means
across
repetition
for
each
starting
values
th.l0<-aggregate(th.l[,"y",drop=FALSE],th.l[,c("su","sex","start")],mean)
th.l0<-th.l0[order(th.l0$su),]
repeated
measure
ANOVA
summary(aov(y~sex*start+Error(su/start),th.l0)) Error:
su
Df
Sum
Sq
Mean
Sq
F
value
Pr(>F)
sex
Residuals
Error:
su:start
Df
Sum
Sq
Mean
Sq
F
value
Pr(>F)
start
sex:start
Residuals
-^
Within group variability is not takeninto acount in the test of withingsubject factors
-^
=> ANOVA can be done with meanvalues
-^
Adjustement of dofs is necessaryonly if the number of dofs > 1.
read.table("elashof.dat",header=TRUE)
head(ela.l)group
su
drug
dose
dv
ela.l$group<-factor(ela.l$group)
ela.l$su<-factor(ela.l$su)
ela.l$drug<-factor(ela.l$drug)
ela.l$dose<-factor(ela.l$dose)
ela.w<-data.frame(
group=rep(1:2,each=8),matrix(ela.l$dv,nrow=16,byrow=T))
names(ela.w)<-c("group",outer(c("v1","v2"),1:3,paste,sep=""))
idata<-data.frame(
drug=factor(rep(1:2,each=3)),dose=factor(rep(1:3,2)))
ela.w$group<-factor(ela.w$group)
head(ela.w)
group su v11 v21 v31 v12 v22 v
Example. The Elashoff dataset
- The questions of interest are: Will the drug be differentially effective for different groups?
Is the effectiveness of the drugs dependent on the dose level? Is the effectiveness of thedrug dependent on the does level and the group?
-^
The dataset is in long-format (ela.l)
-^
The datset is balanced.
-^
Define factors
-^
reshape in wide format (ela.w)
-^
define within-subject factors (idata)
- The Elashoff dataset (Stevens, Table 13.10): Two groups of eight subjects were given
three different doses of two drugs. This experimental design has
two within-subject
factors
(dose and drug) and
one between-subject factor
(group).
Data Analysis (draft) - Gabriel Baud-Bovy
Example. The Elashoff dataset^ >
fit<-aov(dv~groupdrugdose+Error(su/(drug*dose)),ela.l)
summary(fit)
Error: su
Df Sum Sq Mean Sq F
value
Pr(>F)
group
Residuals
Error: su:drug
Df
Sum
Sq Mean
Sq F value
Pr(>F)
drug
group:drug
Residuals
Error: su:dose
Df
Sum
Sq Mean
Sq F value
Pr(>F)
dose
379.39 36.5097 1.580e-08 ***
group:dose
Residuals
Error: su:drug:dose
Df
Sum Sq
Mean Sq
F value
Pr(>F)
drug:dose
group:drug:dose
Residuals
-^
Let us initially assume the sphericitycondition.
-^
This analysis indicates a statisticallysignificant effect of the drug(F(1,14)=13.001, P=0.003).
-^
The significant interactionsDRUG*GP (F(1,14)=12.163,P=0.04)indicates that the effect of the drug isdifferent for each group.
-^
The only other significant effect isthe dose main effect(F(2,28)=36.510, P<0.001).
-^
The main effect of group is difficult to interpret because there is also statistically significantGROUP*DRUG interaction.)
-^
The is a significant difference in theresponses of the two groups(F(1,14)=7.092,P=0.019)
Example. The Elashoff dataset
- The first plot shows that the average
value of the response increases withthe dose. The absence of interactionbetween the DOSE and the DRUGor the GROUP factor is independentfrom these factors.
- The interaction plot shows that the
response to the second drug is muchlarger for the second than for thefirst group.
- Note that the main effect of group is
misleading in this case. It is a side-effect of the observation that theresponse is much higher in thesecond group for the second drug.
28 26 24 22
Group
1
2
Drug
21
dose main effect group:drug interaction
27262524232221
Dose
Data Analysis (draft) - Gabriel Baud-Bovy
Example. The Elashoff dataset
- Tests corresponding to sequential (Type 1) SS. >
anova(fit,idata=idata,M=~drug,X=~1,test="Spherical") Contrasts
orthogonal
to
Contrasts
spanned
by
~drug
Greenhouse-Geisser
epsilon:
Huynh-Feldt
epsilon:
Df
F
num
Df
den
Df
Pr(>F)
G-G
Pr
H-F
Pr
(Intercept)
group
Residuals
anova(fit,idata=idata,M=~drug+dose,X=~drug,test="Spherical") Contrasts
orthogonal
to
~drug
Contrasts
spanned
by
~drug
dose
Greenhouse-Geisser
epsilon:
Huynh-Feldt
epsilon:
Df
F
num
Df
den
Df
Pr(>F)
G-G
Pr
H-F
Pr
(Intercept)
1.580e-
9.785e-
1.705e-
group
Residuals
anova(fit,idata=idata,M=~drug:dose,X=~drug+dose,test="Spherical") Contrasts
orthogonal
to
~drug
dose
Contrasts
spanned
to
~drug:dose
Greenhouse-Geisser
epsilon:
Huynh-Feldt
epsilon:
Df
F
num
Df
den
Df
Pr(>F)
G-G
Pr
H-F
Pr
(Intercept)
group
Residuals
With this dataset, there is nodifference between Type Iand Type III SS.
