Repeated Measures
Analysis and MANOVA
Repeated Measures
•
•
•
Often times were are interested in comparing
treatments where the response of interest is measured
at several points over time.
For example, we might compare a drug that is
supposed to reduce cholesterol to placebo where
cholesterol is measured every two months over a 12month period.
This differs from a block design approach in that we
are interested in comparing comparing treatment
groups not just looking at whether there has been a
change over time a SINGLE group of subjects, i.e.
H o : Time1  Time2   TimeT
Repeated Measures
Hypothetical Cholesterol Study
•
Group
DRUG
Placebo
Initial
2 mo.
4 mo.
6 mo.
8 mo.
10 mo.
12 mo.
The individual(s) who measure cholesterol levels at
each follow-up are blind to which group the subjects
are in.
Questions
Interest:
This
study of
would
be conducted as a randomized double-blind.
1) Is there a change in the cholesterol levels of subjects over time, i.e. is there
a TIME EFFECT? (within-subjects effect)
2) Is there a TREATMENT EFFECT? (between-subjects effect)
3) Is the effect of TIME the same for both TREATMENTS?
(within-subjects effect)
 THIS IS OF PRIMARY INTEREST!
Profile Plots Illustrating the
Questions of Interest
TIME EFFECT ONLY
Cholesterol Level (mg/dl)
Cholesterol Levels for both groups decreased
over TIME however the decrease appears to be
the same for both treatment groups, i.e. there is
NO TREATMENT effect nor a
TIME*TREATMENT interaction.
treatment
placebo
0
2
4
6
8
TIME (months)
10
12
Profile Plots Illustrating the
Questions of Interest
TIME and TREATMENT EFFECT
Cholesterol Level (mg/dl)
Cholesterol Levels for both groups decreased
over TIME and the trend over time was the
same for both groups, however the decrease
for those receiving the drug was larger, i.e.
there is a TREATMENT EFFECT.
treatment
placebo
0
2
4
6
8
TIME (months)
10
12
Profile Plots Illustrating the
Questions of Interest
TIME*TREATMENT INTERACTION
Cholesterol Level (mg/dl)
Here the effect of time is
NOT the same for both
groups. Thus we say that
there is TIME and
TREATMENT
interaction.
treatment
placebo
0
2
4
6
8
TIME (months)
10
12
Example 1: Homeopathy vs. placebo
in treating pain after surgery
Day of surgery
Mean pain
assessments by
visual analogue
scales (VAS)
Days 1-7 after surgery
(morning and evening)
Copyright ©1995 BMJ Publishing Group Ltd.
Lokken, P. et al. BMJ 1995;310:1439-1442
Example 2: Divalproex vs. placebo
for treating bipolar depression
Davis et al. “Divalproex in the treatment of bipolar depression: A placebo controlled study.” J
Affective Disorders 85 (2005) 259-266.
Example 3: Pint of milk vs. control on
bone acquisition in adolescent females
Mean (w/ SE bars) percentage
increases in total body bone
mineral and bone density over
18 months.
P-values are for the
differences
between groups by
repeated measures
analysis of variance
Copyright ©1997 BMJ Publishing Group Ltd.
Cadogan, J. et al. BMJ 1997;315:1255-1260
Example: Two treatment groups with four
measurements taken over equally spaced time
intervals (e.g., A = treatment B = placebo)
id
1
2
3
4
5
6
group
time1 time2 time3 time4
A
A
A
B
B
B
31
24
14
38
25
30
29
28
20
34
29
28
15
20
28
30
25
16
26
32
30
34
29
34
Hypothetical data from Twisk, chapter 3, page 40, table 3.7
Profile plots by group
B
A
Mean profile plots by group
B
A
Questions of Interest
1.
Overall, are there significant differences between TIME
points?
From plots it looks like some differences over time, in
particular times 3 and 4 look different.
2.
Do the two groups differ at any time points, i.e. is there
a TREATMENT effect?
From plots it looks like the groups differ at baseline and there
are some difference everywhere else.
3.
Do the two groups differ in their responses over time,
i.e is there a TIME*TREATMENT interaction?
Their response profiles looks similar over time, though A and
B are closer by the end.
Two Main Methods of Analyses
•
•
Option 1: Use repeated-measures
ANOVA using the “Univariate”
approach (restrictive assumptions)
Option 2: “Multivariate” ANOVA
approach, i.e. MANOVA
( less restrictive assumptions)
Assumptions
1.
2.
3.
4.
Both repeated-measures ANOVA and MANOVA
assume that time intervals are equally spaced.
Both methods assume response is normally
distributed, but both approaches are robust
against violations of normality.
Repeated-measures ANOVA sphericity, or
compound symmetry (see next slide)
Both approaches require complete data for all
subjects, i.e. no missing observations for any
subjects.
Sphericity/Compound Symmetry
Repeated Measures ANOVA required compound
symmetry which is:
(a) The variances of the response variable must be
the same at each time point
(b) The correlation between repeated measurements
are equal, regardless of the time interval between
measurements.
This assumption can be tested using Mauchly’s
Sphericity Test.
(a) Variances at each time point
(visually)
Does variance look equal across time points?
Looks like most variability is at Time1 and least at Time 4.
(b) Correlation across time points
Time 1
Time 2
Time 3
Time 4
Thus
for
this
example
we
Time 1
1.00000
0.94035
-0.14150
0.28445
might
conclude
that
the
Time 2
0.94035
1.00000
-0.02819
0.26921
conditions
for compound
Time 3
-0.14150
-0.02819
1.00000
0.27844
symmetry
are
NOT
met.
Time
4
0.28445
0.26921
0.27844
1.00000
Certainly does not look like we have equal correlations!
Time 1 and Time 2 are highly positively correlated, but Time 1
and Time 3 are negatively correlated!
JMP and SPSS
•
•
•
Both software packages will give the results
using the two approaches.
Therefore you really don’t need to give much
thought as to which method to use. Simply
run a complete analyses and compare &
contrast the results.
Report MANOVA results when compound
symmetry is suspect.
Hypothetical Example in JMP
Treatment Group
(A or B)
Response measured at four
equally spaced time points in
separate columns (Time 1 =
baseline).
Hypothetical Example in JMP
IMPORTANT: Be sure to change
Personality to MANOVA
Put the columns of repeated
measurements in the Y box
Put treatment
effect “Group” in
the model effects
box.
Hypothetical Example in JMP
Check this box so JMP gives the
Repeated Measure ANOVA
(univariate) results as well.
You can change the name to
whatever, it is set to be
“Time” by default. If you
were doing a cross-over
experiment where the
repeated measures were
different treatments you enter
that here instead.
Hypothetical Example in JMP
(MANOVA Approach)
The MANOVA approach
effect tests for TIME and
TIME*TREATMENT
interaction are boxed in the
output shown. From this we
see that both TIME
(p = .3287) and
There
is no evidence of a significant
TIME*TREATMENT
treatment
effect
= .1408).
(p = .8932)
are (p
not
significant at
HOWEVER, you should always look
the a  .05 level.
for a Time*Treatment interaction first!!!
Mean Profile Plots
There does not appear to a large time
effect and response varies very similarly
over time for both treatments, i.e. there
does not appear to be a interaction.
The separation between the profiles
suggests a possible treatment effect.
Hypothetical Example in JMP
(Repeated Measures ANOVA)
The Univariate Repeated
Measures ANOVA output is also
contained on the right. The
Mauchly’s Sphericity Test result
yields (p = .2967) which does not
suggest a problem with sphericity.
However, the two methods used to
adjust for a lack of sphericity,
Greenhouse-Geisser Epsilon and
Huyn-Feldt
Epsilon
both
Treatment effect
test for(which
this approach
should
1 if(psphericity
isn’t
is the same
= .1408), but
we need to
consider thesuggest
Time*Treatment
problem),
otherwise.
interaction first!!!
Hypothetical Example in JMP
(Repeated Measures ANOVA)
From this we see that both effects:
TIME
G-G e = .486, p = .1743
H-F e = .885, p = .1283
&
TIME*TREATMENT
G-G e = 0.486, p = .6954
H-F e = 0.886, p = .8118
are not significant at the a = .05
level.
Summary of Findings
1. Overall, are there significant differences between TIME
points? NO
From MANOVA results the time effect was not
significant (p = .3287).
2. Do the two groups differ at any time points, i.e. is there a
TREATMENT effect? NO
From MANOVA or Repeated Measures ANOVA the
treatment effect (p = .1408)
3. Do the two groups differ in their responses over time,
i.e
is there a TIME*TREATMENT interaction? NO
From MANOVA the time*treatment effect is also not
significant (p = .8932).
Hypothetical Example in SPSS
Subject ID, group and a numeric group (A = 1, B = 2)
identifier are entered. The repeated measurements over
time are entered in separate columns.
Hypothetical Example in SPSS
Select Analyze > General Linear
Model > Repeated Measures…
You are first expected to enter a
name for the within-subject
factor (usually Time) and the
number of levels it has (4 in this
case) then click Add.
Hypothetical Example in SPSS
Treatment identifier
goes here.
Set up Mean Profile Plots
by clicking here.
Mean Profile Plots in SPSS
Hypothetical Example in SPSS
(MANOVA Results and Sphericity Test)
Time effect is not significant
(p = .329) & the Time*Treatment interaction
is not significant either (p = .893).
Mauchly’s Sphericity Test is not significant (p = .324)
Hypothetical Example in SPSS
(Repeated Measures ANOVA)
Time (G-G e, p = .174) and
Time*Treatment (G-G e, p = .695)
interaction effects are not significant.
These tests are only relevant
if there is a significant Time
effect. They look at whether
the time trend is linear,
quadratic, or cubic in nature.
Hypothetical Example in SPSS
There is not a significant treatment effect (p = .141). This
test result is the same regardless of approach used for the
within-subject effects. Not of primary interest.
Other Ways to Test for Changes Over
Time
•
•
•
Compare all means relative to baseline, i.e.
2 vs. 1, 3 vs. 1, 4 vs. 1, etc.
Compare all means to the one previous, i.e.
2 vs. 1, 3 vs. 2, 4 vs. 3, etc.
Compare each mean to the mean of the others
1 vs. (2  3  4 ) / 3, 2 vs. (1  3  4 ) / 3, etc.
•
Compare each mean to the mean of
subsequent times
1 vs. (2  3  4 ) / 3 , 2 vs. (3  4 ) / 2, etc.
Other Ways to Test for Changes Over
Time (JMP)
After choosing Mean option
Check this box so the results of
each comparison are given.
Repeated Measures – this is the
standard, it allows us to test for
Treatment, Time, and interaction
effects. Same as running Sum
and Contrast, try it!
Polynomial – tests for linear,
quadratic, cubic time trends.
Helmert – each mean vs. mean
of ones following it.
Mean – each mean vs. mean of
other time points.
Notice that when you mouse over
option it tells what it will do.
Other Ways to Test for Changes Over
Time (JMP ~ Mean Option)
The mean at baseline (1) is not
significantly mean of the rest (p = .987).
The mean at time 2 (2) is not
significantly mean of the rest (p = .414).
The nature of the difference between
baseline mean and the mean of the rest
does not depend on treatment received
(p = .503), i.e. no interaction.
The nature of the difference between
mean at time 2 and the mean of the rest
does not depend on treatment received
(p = .970), i.e. no interaction.
Other Ways to Test for Changes Over
Time (SPSS)
Deviation – each mean vs. mean
of the rest
Simple – each mean vs. baseline
Difference – each mean vs.
mean of previous.
Helmert – each mean vs. mean
of following time periods.
Select desired contrast
and then click Change
button to make the
change.
Click here to define
alternative mean
comparisons to
make.
Repeated – each mean vs.
previous mean.
Polynomial – linear, quadratic,
and cubic time trends.
Other Ways to Test for Changes Over
Time (SPSS ~ Deviation Contrast)
The mean at time 2 (2) is not
significantly mean of the rest (p = .414).
The nature of the difference between
mean at time 2 and the mean of the rest
does not depend on treatment received
(p = .970), i.e. no interaction.
Replacing Missing Values
•
•
•
Often times in measuring subjects over
the course of time they drop out of the
study for whatever reason.
A standard/simple approach is to fill
missing values with the last observed
value. This is called “Last Observation
Carried Forward (LOCF)”.
Many more complicated schemes exist!!
Example: Last Observation Carried
Forward (LOCF)
Subject
0 Months 2 Months 4 Months 6 Months
1
20
13
2
21
21
20
19
3
19
18
10
6
4
30
25
23
= missing
value
Example: LOCF (cont’d)
Last Observation Carried Forward
Subject 0 Month 2 Month 4 Month 6 Month
1
20
16
2
21
21
20
19
3
19
18
10
6
4
32
25
23
16
32
16
Repeated Measures Example 2
From Daniels book with imputation for both SPSS
and JMP