The GLM Procedure

## Example 39.7 Repeated Measures Analysis of Variance

This example uses data from Cole and Grizzle (1966) to illustrate a commonly occurring repeated measures ANOVA design. Sixteen dogs are randomly assigned to four groups. (One animal is removed from the analysis due to a missing value for one dependent variable.) Dogs in each group receive either morphine or trimethaphan (variable Drug) and have either depleted or intact histamine levels (variable Depleted) before receiving the drugs. The dependent variable is the blood concentration of histamine at 0, 1, 3, and 5 minutes after injection of the drug. Logarithms are applied to these concentrations to minimize correlation between the mean and the variance of the data.

The following SAS statements perform both univariate and multivariate repeated measures analyses and produce Output 39.7.1 through Output 39.7.7.

```   data dogs;
input Drug \$12. Depleted \$ Histamine0 Histamine1
Histamine3 Histamine5;
LogHistamine0=log(Histamine0);
LogHistamine1=log(Histamine1);
LogHistamine3=log(Histamine3);
LogHistamine5=log(Histamine5);
datalines;
Morphine      N  .04  .20  .10  .08
Morphine      N  .02  .06  .02  .02
Morphine      N  .07 1.40  .48  .24
Morphine      N  .17  .57  .35  .24
Morphine      Y  .10  .09  .13  .14
Morphine      Y  .12  .11  .10   .
Morphine      Y  .07  .07  .06  .07
Morphine      Y  .05  .07  .06  .07
Trimethaphan  N  .03  .62  .31  .22
Trimethaphan  N  .03 1.05  .73  .60
Trimethaphan  N  .07  .83 1.07  .80
Trimethaphan  N  .09 3.13 2.06 1.23
Trimethaphan  Y  .10  .09  .09  .08
Trimethaphan  Y  .08  .09  .09  .10
Trimethaphan  Y  .13  .10  .12  .12
Trimethaphan  Y  .06  .05  .05  .05
;
```
```   proc glm;
class Drug Depleted;
model LogHistamine0--LogHistamine5 =
Drug Depleted Drug*Depleted / nouni;
repeated Time 4 (0 1 3 5) polynomial / summary printe;
run;
```

The NOUNI option in the MODEL statement suppresses the individual ANOVA tables for the original dependent variables. These analyses are usually of no interest in a repeated measures analysis. The POLYNOMIAL option in the REPEATED statement indicates that the transformation used to implement the repeated measures analysis is an orthogonal polynomial transformation, and the SUMMARY option requests that the univariate analyses for the orthogonal polynomial contrast variables be displayed. The parenthetical numbers (0 1 3 5) determine the spacing of the orthogonal polynomials used in the analysis.

Output 39.7.1 Summary Information about Groups
The GLM Procedure

Class Level Information
Class Levels Values
Drug 2 Morphine Trimethaphan
Depleted 2 N Y

 Number of Observations Read 16 15

The "Repeated Measures Level Information" table gives information about the repeated measures effect; it is displayed in Output 39.7.2. In this example, the within-subject (within-dog) effect is Time, which has the levels 0, 1, 3, and 5.

Output 39.7.2 Repeated Measures Levels
The GLM Procedure
Repeated Measures Analysis of Variance

Repeated Measures Level Information
Dependent Variable LogHistamine0 LogHistamine1 LogHistamine3 LogHistamine5
Level of Time 0 1 3 5

The multivariate analyses for within-subject effects and related interactions are displayed in Output 39.7.3. For the example, the first table displayed shows that the TIME effect is significant. In addition, the Time*Drug*Depleted interaction is significant, as shown in the fourth table. This means that the effect of Time on the blood concentration of histamine is different for the four Drug*Depleted combinations studied.

Output 39.7.3 Multivariate Tests of Within-Subject Effects
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no Time Effect
H = Type III SSCP Matrix for Time
E = Error SSCP Matrix

S=1 M=0.5 N=3.5
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.11097706 24.03 3 9 0.0001
Pillai's Trace 0.88902294 24.03 3 9 0.0001
Hotelling-Lawley Trace 8.01087137 24.03 3 9 0.0001
Roy's Greatest Root 8.01087137 24.03 3 9 0.0001

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no Time*Drug Effect
H = Type III SSCP Matrix for Time*Drug
E = Error SSCP Matrix

S=1 M=0.5 N=3.5
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.34155984 5.78 3 9 0.0175
Pillai's Trace 0.65844016 5.78 3 9 0.0175
Hotelling-Lawley Trace 1.92774470 5.78 3 9 0.0175
Roy's Greatest Root 1.92774470 5.78 3 9 0.0175

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no Time*Depleted Effect
H = Type III SSCP Matrix for Time*Depleted
E = Error SSCP Matrix

S=1 M=0.5 N=3.5
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.12339988 21.31 3 9 0.0002
Pillai's Trace 0.87660012 21.31 3 9 0.0002
Hotelling-Lawley Trace 7.10373567 21.31 3 9 0.0002
Roy's Greatest Root 7.10373567 21.31 3 9 0.0002

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no Time*Drug*Depleted Effect
H = Type III SSCP Matrix for Time*Drug*Depleted
E = Error SSCP Matrix

S=1 M=0.5 N=3.5
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.19383010 12.48 3 9 0.0015
Pillai's Trace 0.80616990 12.48 3 9 0.0015
Hotelling-Lawley Trace 4.15915732 12.48 3 9 0.0015
Roy's Greatest Root 4.15915732 12.48 3 9 0.0015

Output 39.7.4 displays tests of hypotheses for between-subject (between-dog) effects. This section tests the hypotheses that the different Drugs, Depleteds, and their interactions have no effects on the dependent variables, while ignoring the within-dog effects. From this analysis, there is a significant between-dog effect for Depleted (-value=0.0229). The interaction and the main effect for Drug are not significant (-values=0.1734 and 0.1281, respectively).

Output 39.7.4 Tests of Between-Subject Effects
The GLM Procedure
Repeated Measures Analysis of Variance
Tests of Hypotheses for Between Subjects Effects

Source DF Type III SS Mean Square F Value Pr > F
Drug 1 5.99336243 5.99336243 2.71 0.1281
Depleted 1 15.44840703 15.44840703 6.98 0.0229
Drug*Depleted 1 4.69087508 4.69087508 2.12 0.1734
Error 11 24.34683348 2.21334850

Univariate analyses for within-subject (within-dog) effects and related interactions are displayed in Output 39.7.6. The results for this example are the same as for the multivariate analyses; this is not always the case. In addition, before the univariate analyses are used to make conclusions about the data, the result of the sphericity test (requested with the PRINTE option in the REPEATED statement and displayed in Output 39.7.5) should be examined. If the sphericity test is rejected, use the adjusted G-G or H-F probabilities. See the section Repeated Measures Analysis of Variance for more information.

Output 39.7.5 Sphericity Test
Sphericity Tests
Variables DF Mauchly's Criterion Chi-Square Pr > ChiSq
Transformed Variates 5 0.1752641 16.930873 0.0046
Orthogonal Components 5 0.1752641 16.930873 0.0046

Output 39.7.6 Univariate Tests of Within-Subject Effects
The GLM Procedure
Repeated Measures Analysis of Variance
Univariate Tests of Hypotheses for Within Subject Effects

Source DF Type III SS Mean Square F Value Pr > F Adj Pr > F
G - G H - F
Time 3 12.05898677 4.01966226 53.44 <.0001 <.0001 <.0001
Time*Drug 3 1.84429514 0.61476505 8.17 0.0003 0.0039 0.0008
Time*Depleted 3 12.08978557 4.02992852 53.57 <.0001 <.0001 <.0001
Time*Drug*Depleted 3 2.93077939 0.97692646 12.99 <.0001 0.0005 <.0001
Error(Time) 33 2.48238887 0.07522391

 Greenhouse-Geisser Epsilon 0.5694 0.8475

Output 39.7.7 is produced by the SUMMARY option in the REPEATED statement. If the POLYNOMIAL option is not used, a similar table is displayed using the default CONTRAST transformation. The linear, quadratic, and cubic trends for Time, labeled as ‘Time_1’, ‘Time_2’, and ‘Time_3’, are displayed, and in each case, the Source labeled ‘Mean’ gives a test for the respective trend.

Output 39.7.7 Tests of Between-Subject Effects for Transformed Variables
The GLM Procedure
Repeated Measures Analysis of Variance
Analysis of Variance of Contrast Variables

Time_N represents the nth degree polynomial contrast for Time

Contrast Variable: Time_1

Source DF Type III SS Mean Square F Value Pr > F
Mean 1 2.00963483 2.00963483 34.99 0.0001
Drug 1 1.18069076 1.18069076 20.56 0.0009
Depleted 1 1.36172504 1.36172504 23.71 0.0005
Drug*Depleted 1 2.04346848 2.04346848 35.58 <.0001
Error 11 0.63171161 0.05742833

Contrast Variable: Time_2

Source DF Type III SS Mean Square F Value Pr > F
Mean 1 5.40988418 5.40988418 57.15 <.0001
Drug 1 0.59173192 0.59173192 6.25 0.0295
Depleted 1 5.94945506 5.94945506 62.86 <.0001
Drug*Depleted 1 0.67031587 0.67031587 7.08 0.0221
Error 11 1.04118707 0.09465337

Contrast Variable: Time_3

Source DF Type III SS Mean Square F Value Pr > F
Mean 1 4.63946776 4.63946776 63.04 <.0001
Drug 1 0.07187246 0.07187246 0.98 0.3443
Depleted 1 4.77860547 4.77860547 64.94 <.0001
Drug*Depleted 1 0.21699504 0.21699504 2.95 0.1139
Error 11 0.80949018 0.07359002

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