Example 30.7: 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 30.7.1
through Output 30.7.7:
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 LogHistamine0LogHistamine5 =
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.
The output is displayed in Output 30.7.1 through Output 30.7.7.
Output 30.7.1: Summary Information on Groups
Class Level Information 
Class 
Levels 
Values 
Drug 
2 
Morphine Trimethaphan 
Depleted 
2 
N Y 
Number of observations 
16 
NOTE: 
Observations with missing values will not be included in this analysis. Thus, only 15 observations can be used in this analysis. 


The "Repeated Measures Level Information" table
gives information on the repeated measures effect; it is
displayed in Output 30.7.2.
In this example, the withinsubject (withindog)
effect is Time, which has the levels 0, 1, 3, and 5.
Output 30.7.2: 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 withinsubject effects and related interactions are displayed
in Output 30.7.3.
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 30.7.3: Multivariate Tests of WithinSubject Effects
The GLM Procedure 
Repeated Measures Analysis of Variance 
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 
HotellingLawley 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 
HotellingLawley 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 
HotellingLawley 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 
HotellingLawley Trace 
4.15915732 
12.48 
3 
9 
0.0015 
Roy's Greatest Root 
4.15915732 
12.48 
3 
9 
0.0015 

Output 30.7.4 displays tests of hypotheses
for betweensubject (betweendog) effects.
This section tests the hypotheses that the different
Drugs, Depleteds, and their interactions have no effects on the
dependent variables, while ignoring the withindog effects.
From this analysis, there is a
significant betweendog effect for Depleted (pvalue=0.0229).
The interaction and the main effect
for Drug are not significant
(pvalues=0.1734 and 0.1281, respectively).
Output 30.7.4: Tests of BetweenSubject 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 withinsubject
(withindog) effects and related interactions are
displayed in Output 30.7.6.
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 30.7.5) should
be examined.
If the sphericity test is rejected, use
the adjusted GG or HF probabilities.
See the "Repeated Measures Analysis of Variance" section for more information.
Output 30.7.5: Sphericity Test
The GLM Procedure 
Repeated Measures Analysis of Variance 
Sphericity Tests 
Variables 
DF 
Mauchly's Criterion 
ChiSquare 
Pr > ChiSq 
Transformed Variates 
5 
0.1752641 
16.930873 
0.0046 
Orthogonal Components 
5 
0.1752641 
16.930873 
0.0046 

Output 30.7.6: Univariate Tests of WithinSubject 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 




GreenhouseGeisser Epsilon 
0.5694 
HuynhFeldt Epsilon 
0.8475 

Output 30.7.7 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 30.7.7: Tests of BetweenSubject 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 



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