These
SAS statistics tutorials briefly explain the use and
interpretation of standard statistical analysis techniques for Medical,
Pharmaceutical, Clinical Trials, Marketing or Scientific Research. The examples
include how-to instructions for SAS Software.
Repeated
Measures
are observations taken from the same or related subjects over time or in
differing circumstances. Examples would be weight loss or reaction to a drug
over time. When there are two repeated measures, the analysis of the data
becomes a paired t-test (as discussed earlier). When there are three or more
repeated measures, the analysis is a repeated measures analysis of variance.
As in the Independent GROUPS ANOVA procedure, you will usually perform the
analysis in two steps. First an analysis of variance will determine if there
is a difference in means across time. If a difference is evident, then
multiple comparisons may be performed to determine where the differences lie.
NOTE: This
analysis is also called a within-subjects or treatment-by-subject design. Some
call it a “Single-factor experiment having repeated measures on the same
element.”
|
Note: |
|
A repeated
measures analysis may be performed using PROC ANOVA, PROC GLM, or PROC
MIXED. In this discussion, PROC GLM will be used. The syntax used for the
other procedures is similar, but each procedure offers a different set of
options and capabilities. There are also a number of other Repeated
Measures that will discussed in a different tutorial. |
The hypotheses
being tested with a repeated measures ANOVA is:
Ho: There is no difference among means of the
groups (repeated measures).
Ha: There is a difference among means of the
groups.
The data in the
following example are repeated measures of reaction times of five persons
after being treated with four drugs in randomized order. (This type of data
may come from a crossover experimental design.) The data are as follows:
Subj
Drug1 Drug2 Drug3 Drug4
1 31 29
17 35
2 15 17
11 23
3 25 21
19 31
4 35 35
21 45
5 27 27 15
31
If you data are
in this form, you must first restructure the data into this format:
Subj Drug Time
1
1 31
1
2 29
1
3 17
etc.
. .
5
1 7
5
2 27
5
3 15
5
4 31
The following
code restructures the data into the format needed for the analysis. Note that
the DRUG variable here goes from 1 to 4 “DO DRUG =1 to 4;” representing
the number of repeated measures in the data.
DATA
STUDY;
SUBJ+1;
DO
DRUG =1
to
4;
INPUT
OBS @;
OUTPUT;
END;
DATALINES;
31 29 17 35
15 17 11 23
25 21 19 31
35 35 21 45
27 27 15 31
PROC
GLM
DATA=STUDY;
CLASS
SUBJ DRUG;
MODEL
OBS= SUBJ DRUG;
MEANS
DRUG/TUKEY;
TITLE
'Repeated Measures ANOVA;
RUN;
|
Source |
DF |
Sum of Squares |
Mean Square |
F Value |
Pr > F |
|
Model |
7 |
1331.800000 |
190.257143 |
25.03 |
<.0001 |
|
Error |
12 |
91.200000 |
7.600000 |
|
|
|
Corrected Total |
19 |
1423.000000 |
|
|
|
|
R-Square |
Coeff Var |
Root MSE |
OBS Mean |
|
0.935910 |
10.81102 |
2.756810 |
25.50000 |
|
Source |
DF |
Type III SSu |
Mean Square |
F Value |
Pr > F |
|
SUBJ |
4 |
648.0000000 |
162.0000000 |
21.32 |
<.0001 |
|
DRUG |
3 |
683.8000000 |
227.9333333 |
29.99 |
<.0001 |
u The “Type III
SS” analysis of variance table (DRUG line,) reports a p-value of p < 0.0001.
This gives evidence to reject the null hypothesis that there is no difference
in the drugs. Since there is a difference in drugs, a multiple comparison
test is performed. The results of that test age presented in the next few
tables:
|
Alpha |
0.05 |
|
Error Degrees of Freedom |
12 |
|
Error Mean Square |
7.6 |
|
Critical Value of
Studentized Range |
4.19852 |
|
Minimum Significant Difference |
5.1763 |
|
Means with the same
letter are not significantly different.
v |
|
Tukey Grouping |
Mean |
N |
DRUG |
|
A |
33.000 |
5 |
4 |
|
|
|
|
|
|
B |
26.600 |
5 |
1 |
|
B |
|
|
|
|
B |
25.800 |
5 |
2 |
|
|
|
|
|
|
C |
16.600 |
5 |
3 |
v
The Tukey multiple comparison test for DRUGS indicates that the time to relief
for DRUG 3 is significantly lower than for all other drugs. There is no
statistical difference between drugs 2 and 1, and DRUG 4 has the highest time
to relief for all drugs tested.
ALTERNATIVE METHOD:
If your data are already in the form of the second table above, your code
would be as follows (PROCGLM2a.SAS). The results are the same.
DATA
STUDY;
INPUT
SUBJ DRUG OBS;
DATALINES;
1 1 31
1 2 29
1 3 17
1 4 35
2 1 15
…etc
5 3 15
5 4 31
;
run;
ODS
RTF;
ODS
GRAPHICS
ON;
PROC
GLM
DATA=STUDY;
CLASS
SUBJ DRUG;
MODEL
OBS= SUBJ DRUG;
MEANS
DRUG/TUKEY;
TITLE
'Repeated Measures ANOVA’;
RUN;
ODS
RTF
CLOSE;
ODS
GRAPHICS
OFF;
This discussion only covered the case of a
“one-way repeated measures analysis.” There are a number of more complex
repeated measures analysis of variance designs that will be discussed later.
End of tutorial
See
http://www.stattutorials.com/SAS
