Statistics for Health Research
Simple Repeated
measures
Peter T. Donnan
Professor of Epidemiology and Biostatistics
Objectives of session
• Understand what is meant by
repeated measures
• Be able to set out data in required
format
• Carry out simple analyses in SPSS
• Interpret the output
Repeated Measures
Repeated Measures arise when:
• Measuring the same experimental
unit (cell, rat, patient) on a
number of occasions
• Standard analysis of variance not
valid as assumes independent
measures
• Essentially measurements are
paired or correlated
Examples of Repeated
Measures
• Measuring glucose uptake by cells
at different time points, under
different stimuli, etc.
• Measuring cholesterol in a
randomised controlled trial of a
new statin at 3, 6 and 12 months
• Implementing weight loss
intervention and measuring weight
at different time points
Analysis of Repeated
Measures
• T-test at each time point very common –
•
•
•
•
•
multiple t-tests
Least valid analysis!
Primary hypothesis is usually a single test of
overall effect
By testing at each time point we are
increasing the type I error
P=0.05 means that we would reject the null
hypothesis incorrectly on 1 in 20 occasions
If we keep testing we will eventually find a
significant result!
Repeated Measures
Could perform t-test at each time point
Analysis of Repeated
Measures
• Sometimes account for multiple testing by
•
•
•
•
adjusting p-value i.e. 0.05/k with k tests
Known as the Bonferoni correction
Multiple tests assumes that aim of study is
to show significant difference at every time
point
Most studies aim to show OVERALL
difference between treatments and /or
reaching therapeutic target quicker
PRIMARY HYPOTHESIS IS GLOBAL
Simple general approach
• Basically just an extension of
analysis of variance (ANOVA)
• Pairing or matching of
measurements on same unit needs
to be taken into account
• Method is General Linear Model
for continuous measures and
adjusts tests for correlation
Organisation of data (1)
Generally each unit in one row and repeated
measures in separate columns
Unit
1
Score 1
2.8
Score2
3.1
Score3
4.1
2
5.6
5.7
5.1
3
4.3
4.1
5.4
….
Organisation of data (2)
Note most other programs and later analyses
require ONE row per measurement
Unit
Score
Time
1
2.8
1
1
3.1
2
1
4.1
3
2
5.6
1
2
5.7
2
2
5.1
3
3
4.3
1
Etc…….
Glucose uptake of two cell types (liver and
muscle). Each cell challenged with four
different ‘treatments’
Basal
Insulin
Palmitate
12.41
19.37
5.00
14.95
15.84
14.20
Insul+Palm
Cell
type
Exper.
9.88
1
1
10.08
8.99
1
1
19.04
9.63
11.28
1
1
11.96
16.78
13.16
12.46
1
2
..
…..
…
…
1
2
Data given in ‘Glucose uptake.sav’
Note: cell type is a fixed BETWEEN CELL factor
‘Treatments’ are REPEATED WITHIN CELL factors
Repeated Measures in
SPSS
• Simplest method in SPSS is:
• General Linear Model
Repeated Measures
• Note many other methods in
SPSS – Mixed Models described
on day 4
Repeated Measures in SPSS:
Set factor and number of levels
Within subject
factor
Within subject
factor levels
Within subject
factor name
Repeated Measures in SPSS:
Enter columns of repeated measures
Use arrow to
enter each
repeated
measure column
Between subject
factor column
Repeated Measures in SPSS:
Select options
Use arrow to
select display
of means and
Bonferroni
corrected
comparisons
Select other
options
Repeated Measures in SPSS:
Select options
Select a
plot of
means of
each
within
subject
treatment
Repeated Measures in SPSS:
Output - Mean glucose uptake
Estimates
Means for
four
treatments
and 95%
CI
Measure: treat
f actor1
1
2
3
4
Mean
9.617
13.026
7.538
8.420
Std. Error
.911
1.155
.525
.685
95% Conf idence Interval
Lower Bound Upper Bound
7.732
11.502
10.636
15.415
6.453
8.623
7.004
9.837
1 = Basal; 2 = Insulin;
3 = Palmitate; 4 = Insulin+Palmitate
Repeated Measures in SPSS:
Output – Plot of Mean glucose uptake
Basal
Insulin
Palmitate
Insulin+Palmitate
Repeated Measures in SPSS:
Output – Comparisons of Mean
glucose uptake
Comparison
of means
with
Bonferroni
correction
Pairwise Comparison s
Measure: treat
(I) f actor1
1
2
3
4
(J) f actor1
2
3
4
1
3
4
1
2
4
1
2
3
Mean
Dif f erence
(I-J)
-3.409*
2.079
1.196
3.409*
5.488*
4.605*
-2.079
-5.488*
-.882
-1.196
-4.605*
.882
St d. Error
.637
.723
.873
.637
1.013
1.015
.723
1.013
.741
.873
1.015
.741
a
Sig.
.000
.051
1.000
.000
.000
.001
.051
.000
1.000
1.000
.001
1.000
95% Conf idence Interv al f or
a
Dif f erence
Lower Bound
Upper Bound
-5.249
-1.569
-.007
4.164
-1.325
3.717
1.569
5.249
2.563
8.413
1.677
7.534
-4.164
.007
-8.413
-2.563
-3.022
1.257
-3.717
1.325
-7.534
-1.677
-1.257
3.022
Based on estimated marginal means
*. The mean dif f erence is signif icant at the . 05 lev el.
a. Adjustment f or multiple comparisons: Bonf erroni.
1 = Basal; 2 = Insulin;
3 = Palmitate; 4 = Insulin+Palmitate
Repeated Measures:
Conclusion
• Energy intake significantly higher
with insulin compared to all other
treatments
• Addition of palmitate removes
this effect
Mauchley’s Sphericity test
• Sphericity refers to the issue
of the similarity (homogeneity) of
variances in the differences between
treatments
Think of it as an extension to assumption
in ANOVA of similar variances
• It is an assumption of SPSS Repeated
Measures i.e.
s2a-b ~ s2a-c ~ s2a-d ~ s2b-c ~ s2b-d ~ s2c-d
Meeting conditions of repeated
measures: Mauchly Sphericity
Test
P-value for
test of
Sphericity
Mauchly's Test of Sphericityb
Measure: treat
Epsilon
Within Subjects Ef fect Mauchly 's W
f actor1
.544
Approx.
Chi-Square
13.208
df
5
Sig.
.022
Greenhous
e-Geisser
.739
a
Huynh-Feldt
.822
Lower-bound
.333
Tests the null hypothesis that the error covariance matrix of the orthonormalized transf ormed dependent variables is
proportional to an identity matrix.
a. May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are display ed in
the Tests of Within-Subjects Ef fects table.
b.
Design: Intercept
Within Subjects Design: f actor1
Significant so need to correct F test
by multiplying degrees of freedom by
Greenhouse-Geisser epsilon
Meeting conditions of repeated
measures: Use corrected p-value
if significant non-sphericity
Output
gives four
different
tests
Tests of Within-Subjects Effects
Measure: treat
Source
f actor1
Error(f actor1)
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Type III Sum
of Squares
416.883
416.883
416.883
416.883
592.991
592.991
592.991
592.991
df
3
2.218
2.465
1.000
69
51.012
56.697
23.000
Mean Square
138.961
187.963
169.115
416.883
8.594
11.625
10.459
25.782
F
16.169
16.169
16.169
16.169
Overall test of differences between
treatments within subjects: Use
Greenhouse-Geisser corrected p-value
Sig.
.000
.000
.000
.001
Alternatives
• When Sphericity is not met an
alternative to the correction
factors is to use MANOVA
• Unfortunately this has less power
than the Repeated Measures
analysis demonstrated and so should
generally be avoided
Repeated Measures in SPSS:
Output – Pedometer Trial
• Randomised controlled trial in
sedentary elderly women
• Three groups Pedometer+advice,
Advice only, Control
• Physical activity measured on three
occasions
• 1 – baseline;
• 2 - 3 mnths;
• 3 – 9 mnths
Repeated Measures in SPSS:
Output – Pedometer Trial
Pedometer Group * factor1
Measure: Physical Activity
Pedometer
factor1 Mean
Std. Error
95%CI
group
Lower Upper
.00
1
126792
4665 117578 136006
2
133619
5089 123568 143669
3
128766
4969 118952 138580
1.00
1
141138a
6520 128262 154015
2
147371a
7111 133326 161417
3
136059a
6944 122344 149773
a. Based on modified population marginal mean.
0 = Control; 1 = Pedometer+advice
Factor 1 – time baseline, 3 mnths, 9 mnths
Repeated Measures in SPSS:
Output – Plot of Mean Activity over time
Repeated Measures in SPSS:
Output – Tests of significance
Not quite significant!
Repeated Measures:
Conclusion
• Activity
increased with
pedometer +
advice but rise
was greatest in
Advice only
group
Repeated Measures:
Conclusion
• Simple repeated measures is
•
•
•
•
useful analysis for overall effect
Avoid multiple testing at each
time point
Check assumption of Sphericity
Use adjusted Greenhouse-Geisser
or Huynh-Feldt adjustment if
sphericity not met
Later demonstrate mixed model
References
Field A. A bluffers guide to …Sphericity.
J Educational Statistics 13(3): 215-226.
Pallant J. SPSS Survival Manual 3rd ed, Open University Press,
2007.
Field A. Discovering Statistics using SPSS for Windows. Sage
publications, London, 2000.
Foster JJ. Data Analysis using SPSS for Windows (Versions 8 –
10). Sage publications, London, 2001.
Puri BK. SPSS in practice. An illustrated guide. Arnold, London,
2002.
Repeated Measures:
Practical in SPSS
• Previous analysis lumped all cells
•
•
•
•
together
Two types: liver and muscle
1) Repeat the analysis separately
for each cell type
2) Then compare results from two
types in single analysis
Is cell type within subject or
between subject factor?
Repeated Measures:
Practical in SPSS
• Hint - To do separate analysis by
cell type use:
• Data
Select Cases
(If celltype = 1 or 2)
OR
• Data
Split file
(compare groups by celltype)
Repeated Measures: Practical in SPSS
SPSS Study database.sav
• Trial of pedometers in elderly
•
•
•
•
•
sedentary women
Try repeated measures of
Accelerometer trial data
Baseline, 3 months and 9 months
AccelVM1a, AccelVM2, AccelVM3
Trial arms Ran_grp (1,2,3)
Try adjusting for Age,
StairsDifficult, SIMD