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 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 ofno Response EffectH = Type III SSCP Matrix for ResponseE = 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 ofno Response*Treatment EffectH = Type III SSCP Matrix for Response*TreatmentE = 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 ofno Response*Time EffectH = Type III SSCP Matrix for Response*TimeE = 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 theHypothesis of no Response*Time*Treatment EffectH = Type III SSCP Matrix for Response*Time*TreatmentE = 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 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, whereH = Type III SSCP Matrix for InterceptE = 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 ofNo Overall Intercept Effecton the Variables Defined by the M Matrix TransformationH = Type III SSCP Matrix for InterceptE = 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

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 follow-up 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 pre-versus-post 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

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