Repeated-Measures ANOVA
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Transcript Repeated-Measures ANOVA
REPEATED-MEASURES ANOVA
Old and New
One-Way ANOVA looks at differences between
different samples exposed to different
manipulations (or different samples that may
come from different groups) within a single
factor.
That is, 1 factor, k levels k separate groups are
compared.
NEW: Often, we can answer the same research
questions looking at just 1 sample that is
exposed to different manipulations within a
factor.
That is, 1 factor, k levels 1 sample is compared
across k conditions.
Multiple measurements for
each sample.
That is, a single sample is measured on the same
dependent variable once for each condition
(level) within a factor.
Investigate development over time
(Quazi-Independent: time 1, time 2, time 3)
Chart learning (manipulate different levels of practice)
(Independent: 1 hour, 4 hours, 7 hours)
Compare different priming effects in a LDT
(Independent: Forward, backward, no-prime, non-word)
Simply examine performance under different
conditions with the same individuals.
(Independent: suspect, equal, detective, audio format)
Extending t-tests
T-test
ANOVA cousin
Comparing two
Comparing more than two
independent samples?
Independent-samples t-test!
independent samples
within a single factor?
One-way ANOVA!
Comparing the dependent
Comparing the dependent
measures of a single
sample under two different
conditions?
measures of a single
sample under more than
two different conditions?
Related- (or dependent- or
Repeated-Measures
paired-) sample t-test!
ANOVA!
R-M ANOVA
Like the Related-samples t, repeated measures
ANOVA is more powerful because we eliminate
individual differences from the equation.
In a One-way ANOVA, the F-ratio is calculated
using the variance from three sources:
F = Treatment(group) Effect + Individual differences +
Experimenter error/Individual differences + Experimenter
error.
The denominator represents “random” error and we
do not know how much was from ID and EE.
This is error we expect from chance.
Why R-M ANOVA is COOL…
With a Repeated-Measure ANOVA, we can
measure and eliminate the variability due to
individual differences!!!!
So, the F ratio is conceptually calculated using
the variance from two sources:
F = Treatment(group) Effect + Experimenter error/
Experimenter error.
The denominator represents truly random error that
we cannot directly measure…the leftovers.
This is error we expect just from chance.
What does this mean for our F-value?
R-M vs. One-Way: Power
All else being equal, Repeated-Measures
ANOVAs will be more powerful because they
have a smaller error term bigger Fs.
Let me demonstrate, if you will.
Assume treatment variance = 10, experimental-
error variance = 1, and individual difference
variance = 1000.
In a One-Way ANOVA, F conceptually = (10 + 1 +
1000)/(1 + 1000) = 1.01
In a Repeated-Measures ANOVA, F conceptually =
(10 + 1)/(1) = 11
What is a R-M ANOVA doing?
Again, SStotal = SSbetween + Sswithin
Difference: We break the SSwithin into two parts:
SSwithin/error = SSsubjects/individual differences + SSerror
To get SSerror we subtract SSsubjects from SSwithin/error.
This time, we truly have SSerror , or random variability.
Other than breaking the SSwithin into two
components and subtracting out SSsubjects,
repeated measures ANOVA is similar to OneWay ANOVA.
Let’s learn through example.
A Priming Experiment!
10 participants engage in an LDT, within
which they are exposed to 4 different types of
word pairs. RT to click Word or Non-Word
recorded and averaged for each pair type.
Forward-Prime pairs (e.g., baby-stork)
Backward-Prime pairs (e.g., stork-baby)
Unrelated Pairs (e.g., glass-apple)
Non-Word Pairs (e.g., door-blug)
Hypotheses
For the overall RM ANOVA:
Ho: µf = µb = µu= µn
Ha: At least one treatment mean is different from
another.
Specifically:
Ho: µf < µb
Ho: µf and µb < µu and µn
Ho: µu < µn
The Data
Part
Forward
Backward
Unrelated
Nonword
Sum
Part.
1
.2
.1
.4
.7
1.4
2
.5
.3
.8
.9
2.5
3
.4
.3
.6
.8
2.1
ƩF2= 1.36
4
.4
.2
.8
.9
2.3
ƩB2= .58
5
.6
.4
.8
.8
2.6
ƩU2= 3.82
6
.3
.3
.5
.8
1.9
ƩN2= 6.63
7
.1
.1
.5
.7
1.4
ƩP2= 385
8
.2
.1
.6
.9
1.8
ƩXt2= 12.39
9
.3
.2
.4
.7
1.6
10
.4
.2
.6
.9
2.1
Ʃ
3.4
2.2
6
8.1
19.7
All = 19.7
Formula for SStotal
Ian’s SSt =
X
2
TOT
( X TOT )
2
N TOT
Remember, this is the sum of the squared deviations
from the grand mean.
So, SStotal = 12.39– (19.72/40)
= 12.39 – 9.70225
= 2.68775
Importantly, SStotal = SSwithin/error + SSbetween
The Data
Part
Forward
Backward
Unrelated
Nonword
Sum
Part.
1
.2
.1
.4
.7
1.4
2
.5
.3
.8
.9
2.5
3
.4
.3
.6
.8
2.1
ƩF2= 1.36
4
.4
.2
.8
.9
2.3
ƩB2= .58
5
.6
.4
.8
.8
2.6
ƩU2= 3.82
6
.3
.3
.5
.8
1.9
ƩN2= 6.63
7
.1
.1
.5
.7
1.4
ƩP2= 385
8
.2
.1
.6
.9
1.8
ƩXt2= 12.39
9
.3
.2
.4
.7
1.6
10
.4
.2
.6
.9
2.1
Ʃ
3.4
2.2
6
8.1
19.7
All = 19.7
Within-Groups Sum of Squares:
SSwithin/error
SSwithin/error = the sum of each SS with each
group/condition.
Measures variability within each condition, then
adds them together.
( X 1
2
( X 1 )
n1
2
) ( X 2
2
( X 2 )
n2
2
) ... ( X k
2
( X k )
2
)
nk
So, SSwithin/error =
(1.36– ([3.4]2/10)) + (.58– ([2.2]2/10)) + (3.82– ([6]2/10)
+ (6.63– ([8.1]2/10)
= (1.36– 1.156) + (.58– .484) + (3.82– 3.6) + (6.63– 6.561)
= .204+ .096+ .22+ .069= .589
The Data
Part
Forward
Backward
Unrelated
Nonword
Sum
Part.
1
.2
.1
.4
.7
1.4
2
.5
.3
.8
.9
2.5
3
.4
.3
.6
.8
2.1
ƩF2= 1.36
4
.4
.2
.8
.9
2.3
ƩB2= .58
5
.6
.4
.8
.8
2.6
ƩU2= 3.82
6
.3
.3
.5
.8
1.9
ƩN2= 6.63
7
.1
.1
.5
.7
1.4
ƩP2= 385
8
.2
.1
.6
.9
1.8
ƩXt2= 12.39
9
.3
.2
.4
.7
1.6
10
.4
.2
.6
.9
2.1
Ʃ
3.4
2.2
6
8.1
19.7
All = 19.7
Breaking up SSwithin/error
We must find SSSUBJECTS and subtract that
from total within variance to get SSERROR
SSSUBJECTS =
K is generic for the number of conditions, as usual.
SSSUBJECTS = (1.42/4 +2.52/4 +2.12/4 +2.32/4 +2.62/4
+1.92/4 +1.42/4 +1.82/4 +1.62/4 +2.12/4 +) – 19.72/40
=
.49+1.5625+1.1025+1.3225+1.69+.9025+.49+.81+.6
4+1.1025) -9.70225 = 10.1125-9.70225
=.41025
Now for SSerror
SSerror = SSwithin/error – SSsubjects
SSerror = .589 - .41025 = .17875 or .179
Weeeeeeeeee! We have pure randomness!
The Data
Part
Forward
Backward
Unrelated
Nonword
Sum
Part.
1
.2
.1
.4
.7
1.4
2
.5
.3
.8
.9
2.5
3
.4
.3
.6
.8
2.1
ƩF2= 1.36
4
.4
.2
.8
.9
2.3
ƩB2= .58
5
.6
.4
.8
.8
2.6
ƩU2= 3.82
6
.3
.3
.5
.8
1.9
ƩN2= 6.63
7
.1
.1
.5
.7
1.4
ƩP2= 385
8
.2
.1
.6
.9
1.8
ƩXt2= 12.39
9
.3
.2
.4
.7
1.6
10
.4
.2
.6
.9
2.1
Ʃ
3.4
2.2
6
8.1
19.7
All = 19.7
SSbetween-group:
The (same) Formula:
( X 1 )
2
n1
( X 2 )
n2
2
...
( X k )
nk
2
( X TOT )
2
N TOT
So = SSbetween =
[((3.4)2/10) + ((2.2)2/10) + ((6)2/10) + ((8.1)2/10)]
– 19.72/40
= (1.156+ .484+ 3.6+ 6.561) – 9.70225
= 11.801 – 9.70225 = 2.09875 or 2.099
Getting Variance from SS
Need? …DEGREES OF FREEDOM!
K=4
Ntotal = total number of scores = 40 (4x10)
DfTOTAL = Ntotal – 1 = 39
DfBETWEEN/GROUP = k – 1 = 3
Dfwithin/error= N – K = 40 – 4 = 36
Dfsubjects = s-1 = 10-1 (where s is the # of subjects) = 9
Dferror = (k-1)(s-1) = (3)(9) = 27
OR
Dfwithin/error - Dfsubjects = 36 – 9 = 27
Mean Squared
(deviations from the
mean)
We want to find the average squared deviations
from the mean for each type of variability.
To get an average, you divide by n in some form
(or k which is n of groups) and do a little
correction with “-1.”
That is, you use df.
MSbetween/group =
MSwithin/error =
SS between
df between
SS ERROR
df ERROR
= 2.099/3 = .7
= .179/27 = .007
How do we interpret these MS
MS error is an estimate of population
variance.
Or, variability due to ___________?
F?
F = MSMS
BET
= .7/.007 = 105.671
ERROR
(looks like it should be 100, but there were
rounding issues due to very small numbers.
OK, what is Fcrit? Do we reject the Null??
Pros and Cons
Advantages
Each participant serves as own control.
Do not need as many participants as one-way. Why? More power,
smaller error term.
Great for investigating trends and changes.
Disadvantages
Practice effects (learning)
Carry-over effects (bias)
Demand characteristics (more exposure, more time to think
about meaning of the experiment).
Control
Counterbalancing
Time (greater spacing…but still have implicit memory).
Cover Stories
Sphericity
Levels of our IV are not independent
same participants are in each level (condition).
Our conditions are dependent, or related.
We want to make sure all conditions are equally
related to one another, or equally dependent.
We look at the variance of the differences
between every pair of conditions, and assume
these variances are the same.
If these variances are equal, we have Sphericity
More Sphericity
Testing for Sphericity
Mauchly’s test
Significant, no sphericity, NS… Sphericity!
If no sphericity, we must engage in a correction of the F-ratio.
Actually, we alter the degrees of freedom associated with the
F-ratio.
Four types of correction (see book)
Estimate sphericity from 0 (no sphericity) to 1 (sphericity)
Greenhouse-Geiser (1/k-1)
Huynh-Feldt
MANOVA (assumes measures are independent)
non-parametric, rank-based Friedman test (one-factor only)
Symmetry
Effect Sizes
The issue is not entirely settled. Still some
debate and uncertainty on how to best
measure effect sizes given the different
possible error terms.
ω2 = See book for equation.
Specific tests
Can use Tukey post-hoc for exploration
Can use planned comparisons if you have a
priori predictions.
Sphericity not an issue
Contrast Formula
Same as one way, except error term is different
Contrasts
Some in SPSS:
Difference: Each level of a factor is compared to the mean
of the previous level
Helmert: Each level of a factor is compared to the mean of
the next level
Polynomial: orthogonal polynomial contrasts
Simple: Each level of a factor is compared to the last level
Specific:
GLM forward backward unrelate nonword
/WSFACTOR = prime 4 special (1 1 1 1
-1 -1 1 1
-1 1 0 0
0 0 -1 1)
2+ Within Factors
Set up.
Have participants run on tread mill for 30min.
Within-subject factors:
Factor A
Measure Fatigue every 10min, 3 time points.
Factor B
Do this once after drinking water, and again
(different day) after drinking new sports drink.
3 (time) x 2 (drink) within-subject design.
Much is the same, much
different…
We have 2 factors (A and B) and an interaction
between A and B.
These are within-subjects factors
All participants go through all the levels of each factor.
Again, we will want to find SS for the factors and
interaction, and eventually the respective MS as
well.
Again, this will be very similar to a one-way
ANOVA.
Like a 1-factor RM ANOVA, we will also compute
SSsubject so we can find SSerror.
What is very different?
Again, we can parse up SS w/e into SSsubject and
SSerror.
NEW: We will do this for each F we
calculate.
For each F, we will calculate:
SS Effect ; SS Subject (within that effect) ; and SS Error
What IS SSError for each effect?
(We will follow the logic of factorial ANOVAS)
What are the SSErrors now?
Lets start with main effects.
Looking at Factor A, we have
Variability due to Factor A: SSFactor A
Variability due to individual differences.
How do we measure that?
By looking at the variability due to a main effect of
Participant (i.e., Subject): SSSubject (within Factor A)
Variability just due to error.
How do we calculate that!?!?!?
Think about the idea that we actually have 2 factors here,
Factor A and Subject.
The Error is in the
INTERACTIONS with “Subject.”
For FFactor A (Time)
SSAS is the overall variability looking at Factor A
and Subjects in Factor A (collapsing across Drink).
To find SSerror for the FFactor A (Time) Calculate:
SSAS; SSFactor A ; SSSubject (within Factor A)
SSerror is: SSAS - (SSFactor A + SSSubject (within Factor A))
That is, SSerror for FFactor A (Time)is SS A*subj !!!!!!
Which measures variability within factor A due to the
different participants (i.e., error)
The Same for Factor B.
For FFactor B (Drink)
SSAS is the overall variability looking at Factor B
and Subjects in Factor B (collapsing across Time).
To find SSerror for the FFactor B (Drink) Calculate:
SSBS; SSFactor B ; SSSubject (within Factor B)
SSerror is: SSBS - (SSFactor B + SSSubject (within Factor B))
That is, SSerror for FFactor B (Drink)is SS B*subj !!!!!!
SS B*subj: Which measures variability within factor
B due to the different participants (i.e., error)
Similar for AxB Interaction
For FAxB
SSBetween groups is the overall variability due to
Factor A, Factor B, and Subjects.
To find SSerror for the FAxB Calculate:
SSBetween; SSFactor A ; SSFactor B ; SSSubject
SSerror is: SSBetween - (SSFactor A + SSFactor B + SSSubject)
That is, SSerror for FAxB is SS A*B*subj !!!!!!
SS A*B*subj: measures variability within the AxB interaction
due to the different participants (i.e., error)
We are finely chopping SSW/E
SSSub
(A)
SSSub
(B)
SSSub
(AxB)
Getting to F
Factor A (Time)
SSA = 91.2; dfA = (kA – 1) = 3 – 1 = 2
SSError(A*S) = 16.467; dfError(A*S) = (kA – 1)(s – 1) = (3-
1)(10-1) = 18
So, MSA = 91.2/ 2 = 45.6
So, MSError(A*S) = 16.467/ 18 = .915
FA = 45.6/.915 = 49.846
Snapshot of other Fs
Factor B (Drink)
dfB = (kB – 1) = 2 – 1 = 1
dfError(B*S) = (kB – 1)(s – 1) = (2-1)(10-1) = 9
AxB (Time x Drink)
dfAxB = (kA – 1)(kB – 1) = 2 x 1= 2
dfError(A*B*S) = (kA – 1)(kB – 1)(s – 1) = 2 x 1 x 9 = 18
Use SS’s to calculate the respective MSeffect
and MSerror for the other main effect and the
Interaction F-values.