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 41.9.1 through Output 41.9.3.
options ls=96; 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;
Class Level Information | ||
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Class | Levels | Values |
Treatment | 3 | A B Control |
Number of Observations Read | 18 |
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Number of Observations Used | 18 |
The levels of the repeated factors are displayed in Output 41.9.2. 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 41.9.3.
Repeated Measures Level Information | ||||||
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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 |
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 |
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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 Response*Treatment Effect H = Type III SSCP Matrix for Response*Treatment E = Error SSCP Matrix S=2 M=-0.5 N=6 |
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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 Response*Time Effect H = Type III SSCP Matrix for Response*Time E = Error SSCP Matrix S=1 M=1 N=5 |
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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 Response*Time*Treatment Effect H = Type III SSCP Matrix for Response*Time*Treatment E = Error SSCP Matrix S=2 M=0.5 N=5 |
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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 41.9.4, while the results of the multivariate test are given in Output 41.9.5. Note that the test results are the same as for the Response*Time effect in Output 41.9.3.
M Matrix Describing Transformed Variables | ||||||
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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 |
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 |
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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 |
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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 41.9.6 displays these univariate ANOVA tables and shows that the contrasts are all strongly significant except for the pre-versus-post difference for Y2.
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 |
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 |
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 |
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 |