Example 30.9: Analyzing a Doublymultivariate Repeated Measures Design
This example shows how to analyze a doublymultivariate repeated
measures design by using PROC GLM with an IDENTITY factor in the
REPEATED statement. Note that this differs from previous releases of
PROC GLM, in which you had to use a MANOVA statement to get a doubly
repeated measures analysis.
Two responses, Y1 and Y2, are each measured three times for each subject
(pretreatment, posttreatment, and in a later followup). Each
subject receives one of three treatments; A, B, or the control. In
PROC GLM, you use a REPEATED factor of type IDENTITY to identify the
different responses and another repeated factor to identify the
different measurement times. The repeated measures analysis includes
multivariate tests for time and treatment main effects, as well as
their interactions, across responses.
The following statements produce Output 30.9.1 through Output 30.9.3.
data Trial;
input Treatment $ Repetition PreY1 PostY1 FollowY1
PreY2 PostY2 FollowY2;
datalines;
A 1 3 13 9 0 0 9
A 2 0 14 10 6 6 3
A 3 4 6 17 8 2 6
A 4 7 7 13 7 6 4
A 5 3 12 11 6 12 6
A 6 10 14 8 13 3 8
B 1 9 11 17 8 11 27
B 2 4 16 13 9 3 26
B 3 8 10 9 12 0 18
B 4 5 9 13 3 0 14
B 5 0 15 11 3 0 25
B 6 4 11 14 4 2 9
Control 1 10 12 15 4 3 7
Control 2 2 8 12 8 7 20
Control 3 4 9 10 2 0 10
Control 4 10 8 8 5 8 14
Control 5 11 11 11 1 0 11
Control 6 1 5 15 8 9 10
;
proc glm data=Trial;
class Treatment;
model PreY1 PostY1 FollowY1
PreY2 PostY2 FollowY2 = Treatment / nouni;
repeated Response 2 identity, Time 3;
run;
Output 30.9.1: A Doublymultivariate Repeated Measures Design
Class Level Information 
Class 
Levels 
Values 
Treatment 
3 
A B Control 
Number of observations 
18 

The levels of the repeated factors are displayed in Output 30.9.2.
Note that RESPONSE is 1 for all the Y1 measurements and 2 for all
the Y2 measurements, while the three levels of Time identify the
pretreatment, posttreatment, and followup measurements within
each response. The multivariate tests for withinsubject effects
are displayed in Output 30.9.3.
Output 30.9.2: Repeated Factor Levels
The GLM Procedure 
Repeated Measures Analysis of Variance 
Repeated Measures Level Information 
Dependent Variable 
PreY1 
PostY1 
FollowY1 
PreY2 
PostY2 
FollowY2 
Level of Response 
1 
1 
1 
2 
2 
2 
Level of Time 
1 
2 
3 
1 
2 
3 

Output 30.9.3: Withinsubject Tests
The GLM Procedure 
Repeated Measures Analysis of Variance 
Manova Test Criteria and Exact F Statistics for the Hypothesis of no Response Effect H = Type III SSCP Matrix for Response E = Error SSCP Matrix S=1 M=0 N=6 
Statistic 
Value 
F Value 
Num DF 
Den DF 
Pr > F 
Wilks' Lambda 
0.02165587 
316.24 
2 
14 
<.0001 
Pillai's Trace 
0.97834413 
316.24 
2 
14 
<.0001 
HotellingLawley Trace 
45.17686368 
316.24 
2 
14 
<.0001 
Roy's Greatest Root 
45.17686368 
316.24 
2 
14 
<.0001 
Manova Test Criteria and F Approximations for the Hypothesis of no Response*Treatment Effect H = Type III SSCP Matrix for Response*Treatment E = Error SSCP Matrix S=2 M=0.5 N=6 
Statistic 
Value 
F Value 
Num DF 
Den DF 
Pr > F 
Wilks' Lambda 
0.72215797 
1.24 
4 
28 
0.3178 
Pillai's Trace 
0.27937444 
1.22 
4 
30 
0.3240 
HotellingLawley Trace 
0.38261660 
1.31 
4 
15.818 
0.3074 
Roy's Greatest Root 
0.37698780 
2.83 
2 
15 
0.0908 
NOTE: 
F Statistic for Roy's Greatest Root is an upper bound. 

NOTE: 
F Statistic for Wilks' Lambda is exact. 

Manova Test Criteria and Exact F Statistics for the Hypothesis of no Response*Time Effect H = Type III SSCP Matrix for Response*Time E = Error SSCP Matrix S=1 M=1 N=5 
Statistic 
Value 
F Value 
Num DF 
Den DF 
Pr > F 
Wilks' Lambda 
0.14071380 
18.32 
4 
12 
<.0001 
Pillai's Trace 
0.85928620 
18.32 
4 
12 
<.0001 
HotellingLawley Trace 
6.10662362 
18.32 
4 
12 
<.0001 
Roy's Greatest Root 
6.10662362 
18.32 
4 
12 
<.0001 
Manova Test Criteria and F Approximations for the Hypothesis of no Response*Time*Treatment Effect H = Type III SSCP Matrix for Response*Time*Treatment E = Error SSCP Matrix S=2 M=0.5 N=5 
Statistic 
Value 
F Value 
Num DF 
Den DF 
Pr > F 
Wilks' Lambda 
0.22861451 
3.27 
8 
24 
0.0115 
Pillai's Trace 
0.96538785 
3.03 
8 
26 
0.0151 
HotellingLawley Trace 
2.52557514 
3.64 
8 
15 
0.0149 
Roy's Greatest Root 
2.12651905 
6.91 
4 
13 
0.0033 
NOTE: 
F Statistic for Roy's Greatest Root is an upper bound. 

NOTE: 
F Statistic for Wilks' Lambda is exact. 


The table for Response*Treatment tests for an overall treatment
effect across the two responses; likewise, the tables for
Response*Time and Response*Treatment*Time test for time and the
treatmentbytime interaction, respectively. In this case, there is a
strong main effect for time and possibly for the interaction, but not
for treatment.
In previous releases (before the IDENTITY transformation was
introduced), in order to perform a doubly repeated measures analysis,
you had to use a MANOVA statement with a customized transformation
matrix M. You might still want to use this approach to see details of
the analysis, such as the univariate ANOVA for each transformed
variate. The following statements demonstrate this approach by using
the MANOVA statement to test for the overall main effect of time and
specifying the SUMMARY option.
proc glm data=Trial;
class Treatment;
model PreY1 PostY1 FollowY1
PreY2 PostY2 FollowY2 = Treatment / nouni;
manova h=intercept m=prey1  posty1,
prey1  followy1,
prey2  posty2,
prey2  followy2 / summary;
run;
The M matrix used to perform the test for time effects is displayed in
Output 30.9.4, while the results of the multivariate test are given in
Output 30.9.5. Note that the test results are the same as for the
Response*Time effect in Output 30.9.3.
Output 30.9.4: M Matrix to Test for Time Effect (Repeated Measure)
The GLM Procedure 
Multivariate Analysis of Variance 
M Matrix Describing Transformed Variables 

PreY1 
PostY1 
FollowY1 
PreY2 
PostY2 
FollowY2 
MVAR1 
1 
1 
0 
0 
0 
0 
MVAR2 
1 
0 
1 
0 
0 
0 
MVAR3 
0 
0 
0 
1 
1 
0 
MVAR4 
0 
0 
0 
1 
0 
1 

Output 30.9.5: Tests for Time Effect (Repeated Measure)
The GLM Procedure 
Multivariate Analysis of Variance 
Characteristic Roots and Vectors of: E Inverse * H, where H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix Variables have been transformed by the M Matrix 
Characteristic Root 
Percent 
Characteristic Vector V'EV=1 
MVAR1 
MVAR2 
MVAR3 
MVAR4 
6.10662362 
100.00 
0.00157729 
0.04081620 
0.04210209 
0.03519437 
0.00000000 
0.00 
0.00796367 
0.00493217 
0.05185236 
0.00377940 
0.00000000 
0.00 
0.03534089 
0.01502146 
0.00283074 
0.04259372 
0.00000000 
0.00 
0.05672137 
0.04500208 
0.00000000 
0.00000000 
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Intercept Effect on the Variables Defined by the M Matrix Transformation H = Type III SSCP Matrix for Intercept E = Error SSCP Matrix S=1 M=1 N=5 
Statistic 
Value 
F Value 
Num DF 
Den DF 
Pr > F 
Wilks' Lambda 
0.14071380 
18.32 
4 
12 
<.0001 
Pillai's Trace 
0.85928620 
18.32 
4 
12 
<.0001 
HotellingLawley Trace 
6.10662362 
18.32 
4 
12 
<.0001 
Roy's Greatest Root 
6.10662362 
18.32 
4 
12 
<.0001 

The SUMMARY option in the MANOVA statement creates an ANOVA table for
each transformed variable as defined by the M matrix. MVAR1 and MVAR2
contrast the pretreatment measurement for Y1 with the posttreatment
and followup measurements for Y1, respectively; MVAR3 and MVAR4 are
the same contrasts for Y2.
Output 30.9.6 displays these univariate ANOVA tables and shows that
the contrasts are all strongly significant except for the preversuspost
difference for Y2.
Output 30.9.6: Summary Output for the Test for Time Effect
The GLM Procedure 
Multivariate Analysis of Variance 
Dependent Variable: MVAR1 
Source 
DF 
Type III SS 
Mean Square 
F Value 
Pr > F 
Intercept 
1 
512.0000000 
512.0000000 
22.65 
0.0003 
Error 
15 
339.0000000 
22.6000000 


The GLM Procedure 
Multivariate Analysis of Variance 
Dependent Variable: MVAR2 
Source 
DF 
Type III SS 
Mean Square 
F Value 
Pr > F 
Intercept 
1 
813.3888889 
813.3888889 
32.87 
<.0001 
Error 
15 
371.1666667 
24.7444444 


The GLM Procedure 
Multivariate Analysis of Variance 
Dependent Variable: MVAR3 
Source 
DF 
Type III SS 
Mean Square 
F Value 
Pr > F 
Intercept 
1 
68.0555556 
68.0555556 
3.49 
0.0814 
Error 
15 
292.5000000 
19.5000000 


The GLM Procedure 
Multivariate Analysis of Variance 
Dependent Variable: MVAR4 
Source 
DF 
Type III SS 
Mean Square 
F Value 
Pr > F 
Intercept 
1 
800.0000000 
800.0000000 
26.43 
0.0001 
Error 
15 
454.0000000 
30.2666667 



Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.