Comparison of Repeated
Measures and Covariance
Analysis for
Pretest-Posttest Data
-By Chunmei Zhou
Introduction



We develop a comparison of repeated measures and
covariance analysis for pretest-posttest data.
We consider a study design for which subjects are
randomized to a drug or placebo group and measured for
systolic blood pressure before and after receiving the
treatment.
To develop models to assess the effect of treatment on
SBP, we first consider baseline blood pressure as a
covariate. As an alternative, the baseline and final blood
pressure can be considered to be repeated measures.
Data Description

We simulate data for a pretest-posttest study for
comparison of covariance analysis and repeated measures.

We assume the response variable to be normally
distributed, the sample size is 30, the average response for
baseline is 175 and the treatment effect is –5.
data sbp;
%let basemean=175; *overall average response for baseline;
%let nsub=30;
*Number of subjects;
%let err_v=100;
*Residual error variance at a time point;
%let tsubv=100;
*Treatment by subject variance;
%let subv=200;
*Variance of the subject effects;
%let nrep=1000;
*Number of replications of the simulation;
%let effp1=0;
*Placebo effect;
%let effp2=-5;
*Treatment effect;
do trial=1 to &nrep;
do subj=1 to ⊄
m=&basemean;
sub=sqrt(&subv)*rannor(3345);
do time=1 to 2;
v=sqrt(&err_v)*rannor(3345);
if subj<=&nsub/2 then do;
trt=1;
if time=1 then y=m+sub+v;
if time=2 then y=m+sub+&effp1+v;
end;
if subj>&nsub/2 then do;
trt=2;
if time=1 then y=m+sub+v;
if time=2 then y=m+sub+&effp2+v+sqrt(&tsubv)*rannor(3345);
end;
output;
end;
end;
Covariance Model Fitting

We use placebo as reference group to fit the covariance
model using the following statement:
proc mixed data=a;
by trial;
class subj tref;
model sbp2=sbp1 tref/s;
make 'solutionf' out=est1;
run;
Covariance Model Fitting
Covariance Parameter Estimates
Cov Parm
Estimate
Residual
207.11
Solution for Fixed Effects
Effect
Intercept
sbp1
tref
tref
tref
0
1
Estimate
66.6827
0.6280
-5.9770
0
Standard
Error
31.0513
0.1834
5.3231
.
DF
27
27
27
.
t Value
2.15
3.42
-1.12
.
Pr > |t|
0.0409
0.0020
0.2714
.
Repeated-Measures Model Fitting

we fit repeated measures model using placebo and pretest
as reference groups. We also create a group variable to
have value of 1 for placebo group at post test, 2 for
treatment group at post test, and 3 for pretest.
proc mixed data=b;
by trial;
class subj tref timref grp;
model y=tref timref tref*timref/s;
repeated /group=grp;
random subj;
make 'solutionf' out=est2;
run;

Repeated-Measures Model Fitting
Covariance Parameter Estimates
Cov Parm
subj
Residual
Residual
Residual
Group
grp 1
grp 2
grp 3
Estimate
170.26
72.3522
259.29
49.7515
Repeated-Measures Model Fitting
Solution for Fixed Effects
Effect
Intercept
tref
tref
timref
timref
tref*timref
tref*timref
tref*timref
tref*timref
tref timref
0
1
0
0
1
1
0
1
0
1
0
1
Estimate
168.13
4.6322
0
4.1336
0
-7.7003
0
0
0
Standard
Error
3.8298
5.4162
.
2.8531
.
5.3612
.
.
.
DF t Value Pr >|t|
28
43.90
<.0001
28
0.86
0.3997
.
.
.
28
1.45
0.1585
.
.
.
28
-1.44
0.1620
.
.
.
.
.
.
.
.
.
Comparison of Two Approaches

To compare the two approaches, we set different
parameters for three variances (subject variance, residual
error variance and treatment by subject variance) to
generate data, and then run the simulation 1000 times and
get estimates of treatment effect each time from each
methods.

However, we could not get model fitting results for the
total 1000 simulations from repeated-measures model
because of infinite likelihood or convergence failure for
some trials.
Table1. Comparison of Estimates for treatment effect from two
models with 1000 trials with sample size 30, treatment effect –5
and different variance compontents
Method
Covariance
Sub. var.
Res. Var.
Trt.*Sub.
Var.
No. of Trials
Mean of Est.
of trt effect
Std. Dev. of
Estimate
200
100
1000
1000
-4.66
9.20
877
-4.66
9.27
1000
-4.97
5.30
911
-4.99
5.70
1000
-4.94
4.14
962
-4.92
4.39
1000
-4.95
5.07
962
-4.98
5.69
1000
-5.00
6.74
965
-5.05
7.65
1000
-5.06
9.21
970
-5.13
10.51
Repeat measures
Covariance
200
100
100
Repeat measures
Covariance
100
50
100
Repeat measures
Covariance
100
100
100
Repeat measures
Covariance
200
200
100
Repeat measures
Covariance
Repeat measures
400
400
100
Covariance Model
Covariance Model
Repeated-Measures
15
10
Repeated Measures
25
20
15
5
0
-5
10
5
0
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
-35
-35
-40
Figure1. Box Plots of Estimates for treatment effect
from two models with 1000 trials with treatment effect
–5, subject variance 200, residual error variance 200
and treatment by subject variance 100
Figure2. Box Plots of Estimates for treatment effect
from two models with 1000 trials with treatment
effect –5, subject variance 400, residual error var
400 and treatment by subject variance 100
Discussion

The repeated-measures analysis of variance and analysis of covariance
are two common approaches for analysis of pretest-posttest data. They
are closely related but assumptions for the analysis and variance
estimates for the parameters are differ.

Our analysis results suggest the covariance analysis method may
improve the precision of the estimates of the treatment effects
compared to repeated measures analysis.

But we should know the assumption that there is a linear relationship
between pre- and post-treatment values may not be true. If this were
the case, fitting a baseline covariate could lead to less precise results.
