Example 30.9: Analyzing a Doubly-multivariate Repeated Measures Design

The GLM Procedure

Example 30.9: Analyzing a Doubly-multivariate Repeated Measures Design

This example shows how to analyze a doubly-multivariate 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 follow-up). 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 Doubly-multivariate Repeated Measures Design

The GLM Procedure

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 follow-up measurements within each response. The multivariate tests for within-subject 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: Within-subject 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
Hotelling-Lawley 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 ResponseTreatment Effect H = Type III SSCP Matrix for ResponseTreatment 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
Hotelling-Lawley 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 ResponseTime Effect H = Type III SSCP Matrix for ResponseTime 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
Hotelling-Lawley 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 ResponseTimeTreatment Effect H = Type III SSCP Matrix for ResponseTimeTreatment 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
Hotelling-Lawley 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 treatment-by-time 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