Correlated-Samples
ANOVA
The Univariate Approach
An ANOVA Factor Can Be
• Independent Samples
– Between Subjects
• Correlated Samples
– Within Subjects, Repeated Measures
– Randomized Blocks, Split Plot
• Matched Pairs if k = 2
The Design
•
•
•
•
DV = cumulative duration of headaches
Factor 1 = Weeks
Factor 2 = Subjects (crossed with weeks)
The first two weeks represent a baseline
period.
• The remaining three weeks are the
treatment weeks.
• The treatment was designed to reduce
headaches.
The Data
Subject
Wk1
Wk2
Wk3
Wk4
Wk5
1
2
3
4
5
6
7
8
9
21
20
17
25
30
19
26
17
26
22
19
15
30
27
27
16
18
24
8
10
5
13
13
8
5
8
14
6
4
4
12
8
7
2
1
8
6
4
5
17
6
4
5
5
9
Crossed and Nested Factors
• Subjects is crossed with Weeks here – we
have score for each subject at each level
of Week.
• That is, we have a Weeks x Subjects
ANOVA.
• In independent samples ANOVA subjects
is nested within the other factor
– If I knew the subject ID, I would know which
treatment e got.
Order Effects
• Suppose the within-subjects effect was
dose of drug given (0, 5, 10 mg)
• DV = score on reaction time task.
• All subject tested first at 0 mg, second at 5
mg, and thirdly at 10 mg
• Are observed differences due to dose of
drug or the effect of order
• Practice effects and fatigue effects
Complete Counterbalancing
• There are k! possible orderings of the
treatments.
• Run equal numbers of subjects in each of
the possible orderings.
• Were k = 5, that would be 120 different
orderings.
Asymmetrical Transfer
• We assume that the effect of A preceding
B is the same as the effect of B preceding
A.
• Accordingly, complete counterbalancing
will cancel out any order effects
• If there is asymmetrical transfer, it will not.
Incomplete Counterbalancing
• Each treatment occurs once in each
ordinal position.
• Latin Square
ABCDE
EABCD
DEABC
CDEAB
BCDEA
Power
• If the correlations between conditions are
positive and substantial, power will be
greater than with the independent samples
designs
• Even though error df will be reduced
• Because we are able to remove subject
effects from the error term
• Decreasing the denominator of the F ratio.
Reducing Extraneous
Variance
• Matched pairs, randomized blocks, splitplot.
• Repeated measures or within-subjects.
• Variance due to the blocking variable is
removed from error variance.
Blocks
Error
Error
Treatm ent
Blocks
Treatm ent
Partitioning the SS
• The sum of all 5 x 9 = 45 squared scores
is 11,060.
• The correction for the mean, CM, is
(596)2 / 45 = = 7893.69.
• The total SS is then 11,060 - 7893.69 =
3166.31.
SS TOT   Y
2

( Y )
N
2
SSweeks
• From the marginal totals for week we
compute the SS for the main effect of
Week as: (2012+ 1982+ 842+ 522+ 612) /
9 - 7893.69 = 2449.20.
• Wj is the sum of scores for the jth week.
SS weeks 
W j
n
2
 CM
SSSubjects
• From the subject totals, the SS for
subjects is: (632+ 572+ ...... + 812) /
5 - 7893.69 = 486.71.
• S is the sum of score for one subject
SS subjects 
S
n
2
i
 CM
SSerror
• We have only one score in each of the 5
weeks x 9 subjects = 45 cells.
• So the traditional within-cells error variance
does not exist.
• The appropriate error term is the Subjects x
Weeks Interaction.
• SSSubjects x Weeks = SStotal – SSsubjects – SS weeks
• = 3166.31 - 486.71 - 2449.2 = 230.4.
df, MS, F, p
• The df are computed as usual in a factorial
ANOVA -- (s-1) = (9-1) = 8 for Subjects,
(w-1) = (5-1) = 4 for Week, and 8 x 4 = 32
for the interaction.
• The F(4, 32) for the effect of Week is then
(2449.2/4) / (230.4/32) = 612.3/7.2 =
85.04, p < .01.
Assumptions
• Normality
• Homogeneity of Variance
• Sphericity
– For each (ij) pair of levels of the Factor
– Compute (Yi  Yj) for each subject
– The standard deviation of these difference
scores is constant – that is, you get the same
SD regardless of which pair of levels you
select.
Sphericity
• Test it with Mauchley’s criterion
• Correct for violation of sphericity by using
a procedure that adjust downwards the df
• Or by using a procedure that does not
assume sphericity.
Mixed Designs
• You may have one or more correlated
ANOVA factors and one or more
independent ANOVA factors
Multiple Comparisons
• You can employ any of the procedures
that we earlier applied with independent
samples ANOVA.
• Example: I want to compare the two
baseline weeks with the three treatment
weeks.
• The means are (201 + 198)/18 = 22.17 for
baseline, (84 + 52 + 61)/27 = 7.30 for
treatment.
t
t 
Mi  M j
MS error
 1
1 



n

n
j 
 i

22 . 17  7 . 30
1 
 1
7 . 20 


27 
 18
• The 7.20 is the MSE from the overall
analysis.
• df = 32, from the overall analysis
• p < .01
 18 . 21
Controlling FW
• Compute q  t 2
– And use it for Tukey or related procedure
• Or apply a Bonferroni or Sidak procedure
• For example, Week 2 versus Week 3
• t = (22-9.33)/SQRT(7.2(1/9 + 1/9)) =
10.02, q = 10.02 * SQRT(2) = 14.16.
• For Tukey, with r = 5 levels, and 32 df,
critical q.01 = 5.05
Heterogenity of Variance
• If suspected, use individual error terms for
a posteriori comparisons
– Error based only on the two levels being
compared.
– For Week 2 versus Week 3, t(8) = 10.75, q(8)
= 15.2
– Notice the drop in df
SAS
•
•
•
•
•
WS-ANOVA.sas
Proc Anova;
Class subject week;
Model duration = subject week;
SAS will use SSerror =
SStotal – SSsubjects – SSweeks
Source
DF
subject
week
8
4
Source
DF
Model
12
Error
32
Corrected 44
Total
Anova
SS
486.7111
2449.200
Mean
F Value
Square
60.83888 8.45
612.3000 85.04
Pr > F
Sum of
Squares
2935.91111
230.40000
3166.31111
Mean
F Value
Square
244.65925 33.98
7.200000
Pr > F
<.0001
<.0001
<.0001
Data in Multivariate Setup
data ache; input subject week1-week5;
d23 = week2-week3; cards;
1 21 22 8 6 6
2 20 19 10 4 4
3 17 15 5 4 5
4 25 30 13 12 17
5 30 27 13 8 6
6 19 27 8 7 4
And data for three more subjects
Week 2 versus Week 3
• proc anova; model week2 week3 = /
nouni; repeated week 2 / nom;
Source
DF
week
1
Error(week) 8
Anova
SS
722.0000
50.00000
Mean
F Value
Square
722.0000 115.52
6.250000
Pr > F
<.0001
• The value of F here is just the square of
the value of t, 10.75, reported on Slide 23,
with an individual error term.
proc means mean t prt;
var d23 week1-week5;
Proc Anova;
Model week1-week5 = / nouni;
Repeated week 5 profile / summary printe;
Sphericity Tests
Variables
DF
Orthogonal
9
Components
Mauchly's Chi-Square Pr > ChiSq
Criterion
0.2823546 8.1144619 0.5227
We retain the null that there is sphericity.
Univariate Tests of Hypotheses
for Within Subject Effects
Source
DF
Anova
SS
week
4
Error(week) 32
2449.2
230.40
Mean F
Pr > F Adj Pr > F
Square Value
G-G H-F
612.30 85.04 <.0001 <.0001 <.0001
7.2000
Greenhouse-Geisser Epsilon 0.6845
Huynh-Feldt Epsilon
1.0756
Epsilon
• Used to correct for lack of sphericity
• Multiply both numerator and denominator
df by epsilon.
• For example: Degrees of freedom were
adjusted according to Greenhouse and
Geisser to correct for violation of the
assumption of sphericity. Duration of
headches changed significantly across the
weeks, F(2.7, 21.9) = 85.04, MSE = 7.2, p
< .001.
Which Epsilon to Use?
• The G-G correction is more conservative
(less power) than the H-F correction.
• If both the G-G and the H-F are near or
above .75, it is probably best to use the
H-F.
Profile Analysis
• Compares each level with the next level,
using individual error.
• Look at the output.
– Week 1 versus Week 2, p = .85
– Week 2 versus Week 3, p < .001
– Week 3 versus Week 4, p = .002
– Week 4 versus Week 5, p = .29
Multivariate Analysis
MANOVA Test Criteria and Exact F Statistics for the
Hypothesis of no week Effect
Statistic
Value
F
Num Den
Value DF
DF
86.39 4
5
Pr > F
0.98573
69.1126
86.39 4
86.39 4
5
5
<.0001
<.0001
Roy's Greatest 69.1126
Root
86.39 4
5
<.0001
Wilks' Lambda 0.01426
Pillai's Trace
HotellingLawley Trace
<.0001
Strength of Effect
• 2 = SSweeks / SStotal = 2449.2/3166.3 = .774
• Alternatively, if we remove from the
denominator variance due to subject,

2
partial

SS Conditions
SS Conditions
 SS Error


2449 . 2
2449 . 2  230 . 4 
 . 914
Higher-Order Mixed or
Repeated Univariate Models
• If the effect contains only betweensubjects factors, the error term is
Subjects(nested within one or more
factors).
• For any effect that includes one or
more within-subjects factors the error
term is the interaction between
Subjects and those one or more withinsubjects factors.
AxBxS Two-Way Repeated
Measures
CLASS A B S; MODEL Y=A|B|S;
TEST H=A E=AS;
TEST H=B E=BS;
TEST H=AB E=ABS;
MEANS A|B;
Ax(BxS) Mixed (B Repeated)
CLASS A B S; MODEL Y=A|B|S(A);
TEST H=A E=S(A);
TEST H=B AB E=BS(A);
MEANS A|B;
AxBx(CxS) Three-Way Mixed
(C Repeated)
CLASS A B C S;
MODEL Y=A|B|C|S(A B);
TEST H=A B AB E=S(A B);
TEST H=C AC BC ABC E=CS(A B);
MEANS A|B|C;
Ax(BxCxS) Mixed
(B and C Repeated)
CLASS A B C S;
MODEL Y=A|B|C|S(A);
TEST H=A E=S(A);
TEST H=B AB E=BS(A);
TEST H=C AC E=CS(A);
TEST H=BC ABC E=BCS(A);
MEANS A|B|C;
AxBxCxS All Within
CLASS A B C S; MODEL Y=A|B|C|S;
TEST H=A E=AS;
TEST H=B E=BS;
TEST H=C E=CS;
TEST H=AB E=ABS;
TEST H=AC E=ACS;
TEST H=BC E=BCS;
TEST H=ABC E=ABCS;
MEANS A|B|C;