Repeated-Measures ANOVA

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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.

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