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10
OPTIONS LS=80 NODATE PAGENO=1;
TITLE 'Example 11, Y603, Repeated-Measure ANOVA';
*------------------------Repeated Measure Design --------------------------*
* THIS PROBLEM IS TAKEN FROM UNIT 8 OF STATISTICAL DESIGN *
* IN EDUCATIONAL RESEARCH BY C.Y. PENG. THE NATURE OF THIS *
* PROBLEM IS REPEATED MEASURE. PROC CORR is first used to compute *
* the variance-covariance matrix. Then PROC GLM and the REPEATED *
* statements are used to carry out univariate/multivariate analyses of *
* variance of (1) the linear trend, (2) the quadratic trend, and *
* (3) the cubic trend of the repeated scores. *
* In addition, the assumption of sphericity is tested and p-level adjusted
*--------------------------------------------------------------------------*;
DATA ex11;
INPUT a s b1 b2 b3 b4;
y1=(-3)*b1+(-1)*b2+b3+(3)*b4;
y2=b1+(-1)*b2+(-1)*b3+b4;
y3=(-1)*b1+3*b2+(-3)*b3+b4;
LABEL a='Sportiness'
s='Subject id'
b1='First hour response'
b2='Second hour response'
b3='Third hour response'
b4='Fourth hour response'
y1='The linear trend on repeated scores'
y2='The quadratic trend on repeated scores'
y3='The cubic trend on repeated scores';
CARDS;
1 1 3 4 7 7
1 2 6 5 8 8
1 3 3 4 7 9
1 4 3 3 6 8
2 5 1 2 5 10
2 6 2 3 6 10
2 7 2 4 5 9
2 8 2 3 6 11
;
PROC PRINT;
PROC CORR COV NOSIMPLE;
VAR b1 b2 b3 b4;
TITLE2 'the variance-covariance matrix';
RUN;
PROC GLM;
CLASS a;
MODEL b1-b4=a;
REPEATED hour POLYNOMIAL/SUMMARY PRINTM;
TITLE2 'Univariate/multivariate analyses of the repeated design and its assumption';
RUN;
Example 11, Y603, Repeated-Measure ANOVA 1
OBS A S B1 B2 B3 B4 Y1 Y2 Y3
1 1 1 3 4 7 7 15 -1 -5
2 1 2 6 5 8 8 9 1 -7
3 1 3 3 4 7 9 21 1 -3
4 1 4 3 3 6 8 18 2 -4
5 2 5 1 2 5 10 30 4 0
6 2 6 2 3 6 10 27 3 -1
7 2 7 2 4 5 9 22 2 4
8 2 8 2 3 6 11 30 4 0
Example 11, Y603, Repeated-Measure ANOVA 2
the variance-covariance matrix
Correlation Analysis
4 'VAR' Variables: B1 B2 B3 B4
Covariance Matrix DF = 7
B1 B2 B3 B4
B1 2.214285714 1.142857143 1.357142857 -1.142857143 First hour response
B2 1.142857143 0.857142857 0.714285714 -0.714285714 Second hour response
B3 1.357142857 0.714285714 1.071428571 -0.714285714 Third hour response
B4 -1.142857143 -0.714285714 -0.714285714 1.714285714 Fourth hour response
Pearson Correlation Coefficients / Prob > |R| under Ho: Rho=0 / N = 8
B1 B2 B3 B4
B1 1.00000 0.82956 0.88110 -0.58659
First hour response 0.0 0.0108 0.0038 0.1264
B2 0.82956 1.00000 0.74536 -0.58926
Second hour response 0.0108 0.0 0.0338 0.1243
B3 0.88110 0.74536 1.00000 -0.52705
Third hour response 0.0038 0.0338 0.0 0.1795
B4 -0.58659 -0.58926 -0.52705 1.00000
Fourth hour response 0.1264 0.1243 0.1795 0.0
Example 11, Y603, Repeated-Measure ANOVA 3
Univariate/multivariate analyses of the repeated design and its assumption
General Linear Models Procedure
Class Level Information
Class Levels Values
A 2 1 2
Number of observations in data set = 8
Example 11, Y603, Repeated-Measure ANOVA 4
Univariate/multivariate analyses of the repeated design and its assumption
General Linear Models Procedure
Dependent Variable: B1 First hour response
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 8.00000000 8.00000000 6.40 0.0447
Error 6 7.50000000 1.25000000
Corrected Total 7 15.50000000
R-Square C.V. Root MSE B1 Mean
0.516129 40.65578 1.1180340 2.7500000
Source DF Type I SS Mean Square F Value Pr > F
A 1 8.00000000 8.00000000 6.40 0.0447
Source DF Type III SS Mean Square F Value Pr > F
A 1 8.00000000 8.00000000 6.40 0.0447
Example 11, Y603, Repeated-Measure ANOVA 5
Univariate/multivariate analyses of the repeated design and its assumption
General Linear Models Procedure
Dependent Variable: B2 Second hour response
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 2.00000000 2.00000000 3.00 0.1340
Error 6 4.00000000 0.66666667
Corrected Total 7 6.00000000
R-Square C.V. Root MSE B2 Mean
0.333333 23.32847 0.8164966 3.5000000
Source DF Type I SS Mean Square F Value Pr > F
A 1 2.00000000 2.00000000 3.00 0.1340
Source DF Type III SS Mean Square F Value Pr > F
A 1 2.00000000 2.00000000 3.00 0.1340
Example 11, Y603, Repeated-Measure ANOVA 6
Univariate/multivariate analyses of the repeated design and its assumption
General Linear Models Procedure
Dependent Variable: B3 Third hour response
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 4.50000000 4.50000000 9.00 0.0240
Error 6 3.00000000 0.50000000
Corrected Total 7 7.50000000
R-Square C.V. Root MSE B3 Mean
0.600000 11.31371 0.7071068 6.2500000
Source DF Type I SS Mean Square F Value Pr > F
A 1 4.50000000 4.50000000 9.00 0.0240
Source DF Type III SS Mean Square F Value Pr > F
A 1 4.50000000 4.50000000 9.00 0.0240
Example 11, Y603, Repeated-Measure ANOVA 7
Univariate/multivariate analyses of the repeated design and its assumption
General Linear Models Procedure
Dependent Variable: B4 Fourth hour response
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 8.00000000 8.00000000 12.00 0.0134
Error 6 4.00000000 0.66666667
Corrected Total 7 12.00000000
R-Square C.V. Root MSE B4 Mean
0.666667 9.072184 0.8164966 9.0000000
Source DF Type I SS Mean Square F Value Pr > F
A 1 8.00000000 8.00000000 12.00 0.0134
Source DF Type III SS Mean Square F Value Pr > F
A 1 8.00000000 8.00000000 12.00 0.0134
Example 11, Y603, Repeated-Measure ANOVA 8
Univariate/multivariate analyses of the repeated design and its assumption
General Linear Models Procedure
Repeated Measures Analysis of Variance
Repeated Measures Level Information
Dependent Variable B1 B2 B3 B4
Level of HOUR 1 2 3 4
HOUR.N represents the nth degree polynomial contrast for HOUR
M Matrix Describing Transformed Variables
B1 B2 B3 B4
HOUR.1 -.6708203932 -.2236067977 0.2236067977 0.6708203932
HOUR.2 0.5000000000 -.5000000000 -.5000000000 0.5000000000
HOUR.3 -.2236067977 0.6708203932 -.6708203932 0.2236067977
Manova Test Criteria and Exact F Statistics for the Hypothesis of no HOUR Effect
H = Type III SS&CP Matrix for HOUR E = Error SS&CP Matrix
S=1 M=0.5 N=1
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.02747678 47.1925 3 4 0.0014
Pillai's Trace 0.97252322 47.1925 3 4 0.0014
Hotelling-Lawley Trace 35.39436620 47.1925 3 4 0.0014
Roy's Greatest Root 35.39436620 47.1925 3 4 0.0014
Manova Test Criteria and Exact F Statistics for
the Hypothesis of no HOUR*A Effect
H = Type III SS&CP Matrix for HOUR*A E = Error SS&CP Matrix
S=1 M=0.5 N=1
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.14430894 7.9061 3 4 0.0371
Pillai's Trace 0.85569106 7.9061 3 4 0.0371
Hotelling-Lawley Trace 5.92957746 7.9061 3 4 0.0371
Roy's Greatest Root 5.92957746 7.9061 3 4 0.0371
Example 11, Y603, Repeated-Measure ANOVA 9
Univariate/multivariate analyses of the repeated design and its assumption
General Linear Models Procedure
Repeated Measures Analysis of Variance
Tests of Hypotheses for Between Subjects Effects
Source DF Type III SS Mean Square F Value Pr > F
A 1 3.12500000 3.12500000 2.00 0.2070
Error 6 9.37500000 1.56250000
Example 11, Y603, Repeated-Measure ANOVA 10
Univariate/multivariate analyses of the repeated design and its assumption
General Linear Models Procedure
Repeated Measures Analysis of Variance
Univariate Tests of Hypotheses for Within Subject Effects
Source: HOUR
Adj Pr > F
DF Type III SS Mean Square F Value Pr > F G - G H - F
3 194.50000000 64.83333333 127.89 0.0001 0.0001 0.0001
Source: HOUR*A
Adj Pr > F
DF Type III SS Mean Square F Value Pr > F G - G H - F
3 19.37500000 6.45833333 12.74 0.0001 0.0019 0.0002
Source: Error(HOUR)
DF Type III SS Mean Square
18 9.12500000 0.50694444
Greenhouse-Geisser Epsilon = 0.5841
Huynh-Feldt Epsilon = 0.9432
Example 11, Y603, Repeated-Measure ANOVA 11
Univariate/multivariate analyses of the repeated design and its assumption
General Linear Models Procedure
Repeated Measures Analysis of Variance
Analysis of Variance of Contrast Variables
HOUR.N represents the nth degree polynomial contrast for HOUR
Contrast Variable: HOUR.1
Source DF Type III SS Mean Square F Value Pr > F
MEAN 1 184.90000000 184.90000000 182.62 0.0001
A 1 13.22500000 13.22500000 13.06 0.0112
Error 6 6.07500000 1.01250000
Contrast Variable: HOUR.2
Source DF Type III SS Mean Square F Value Pr > F
MEAN 1 8.00000000 8.00000000 25.60 0.0023
A 1 3.12500000 3.12500000 10.00 0.0195
Error 6 1.87500000 0.31250000
Contrast Variable: HOUR.3
Source DF Type III SS Mean Square F Value Pr > F
MEAN 1 1.60000000 1.60000000 8.17 0.0289
A 1 3.02500000 3.02500000 15.45 0.0077
Error 6 1.17500000 0.19583333
Comments: peng@indiana.edu
Dr. Peng's Home Page: Dr.
Chao-Ying Joanne Peng
Copyright
1999, The Trustees of Indiana
University
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