Data from Table 19.4, page 439
data table19_4; input s a b seeds @@; datalines; 1 1 1 13 1 1 2 14 1 1 3 17 1 1 4 20 2 1 1 10 2 1 2 11 2 1 3 15 2 1 4 14 3 1 1 13 3 1 2 19 3 1 3 18 3 1 4 22 4 1 1 4 4 1 2 12 4 1 3 14 4 1 4 16 5 2 1 5 5 2 2 13 5 2 3 21 5 2 4 24 6 2 1 8 6 2 2 18 6 2 3 25 6 2 4 27 7 2 1 14 7 2 2 19 7 2 3 26 7 2 4 26 8 2 1 12 8 2 2 24 8 2 3 29 8 2 4 29 9 3 1 13 9 3 2 24 9 3 3 28 9 3 4 32 10 3 1 9 10 3 2 22 10 3 3 22 10 3 4 24 11 3 1 14 11 3 2 22 11 3 3 28 11 3 4 28 12 3 1 8 12 3 2 18 12 3 3 27 12 3 4 29 ; run; data table19_4w; input s a b1 b2 b3 b4; datalines; 1 1 13 14 17 20 2 1 10 11 15 14 3 1 13 19 18 22 4 1 4 12 14 16 5 2 5 13 21 24 6 2 8 18 25 27 7 2 14 19 26 26 8 2 12 24 29 29 9 3 13 24 28 32 10 3 9 22 22 24 11 3 14 22 28 28 12 3 8 18 27 29 ; run;
Table 19.4, page 439. Numerical Example of a Mixed Ax(BxS) design
NOTE1: The parenthesis ( ) denote nesting. In this example, subjects
are nested within Factor A.
NOTE2: The proc glm for the narrow data set is a lot more complicated
than either the proc glm on the wide data set or the proc mixed on
on the narrow data set.
proc glm data = table19_4; class s a b; model seeds = a|b|s(a) /ss3; test h = a e = s(a); test h = b e = b*s(a); test h = a*b e = s*b(a); run; quit;
The GLM Procedure
Class Level Information
Class Levels Values
s 12 1 2 3 4 5 6 7 8 9 10 11 12
a 3 1 2 3
b 4 1 2 3 4
Number of observations 48
Dependent Variable: seeds
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 47 2419.000000 51.468085 . .
Error 0 0.000000 .
Corrected Total 47 2419.000000
R-Square Coeff Var Root MSE seeds Mean
1.000000 . . 18.75000
Source DF Type III SS Mean Square F Value Pr > F
a 2 458.000000 229.000000 . .
b 3 1405.500000 468.500000 . .
a*b 6 155.500000 25.916667 . .
s(a) 9 301.000000 33.444444 . .
s*b(a) 27 99.000000 3.666667 . .
Tests of Hypotheses Using the Type III MS for s(a) as an Error Term
Source DF Type III SS Mean Square F Value Pr > F
a 2 458.0000000 229.0000000 6.85 0.0156
Tests of Hypotheses Using the Type III MS for s*b(a) as an Error Term
Source DF Type III SS Mean Square F Value Pr > F
b 3 1405.500000 468.500000 127.77 <.0001
a*b 6 155.500000 25.916667 7.07 0.0001
proc glm data = table19_4w; class a; model b1 b2 b3 b4 = a / ss3; repeated b 4; run; quit;
[b-level output omitted]
The GLM Procedure
Repeated Measures Analysis of Variance
Repeated Measures Level Information
Dependent Variable b1 b2 b3 b4
Level of b 1 2 3 4
Manova Test Criteria and Exact F Statistics for the Hypothesis of no b Effect
H = Type III SSCP Matrix for b
E = Error SSCP Matrix
S=1 M=0.5 N=2.5
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.03191968 70.77 3 7 <.0001
Pillai's Trace 0.96808032 70.77 3 7 <.0001
Hotelling-Lawley Trace 30.32863944 70.77 3 7 <.0001
Roy's Greatest Root 30.32863944 70.77 3 7 <.0001
Manova Test Criteria and F Approximations for the Hypothesis of no b*a Effect
H = Type III SSCP Matrix for b*a
E = Error SSCP Matrix
S=2 M=0 N=2.5
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda 0.17277151 3.28 6 14 0.0314
Pillai's Trace 0.92601669 2.30 6 16 0.0859
Hotelling-Lawley Trace 4.21620600 4.73 6 7.7895 0.0251
Roy's Greatest Root 4.07592236 10.87 3 8 0.0034
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.
Tests of Hypotheses for Between Subjects Effects Source DF Type III SS Mean Square F Value Pr > F a 2 458.0000000 229.0000000 6.85 0.0156 Error 9 301.0000000 33.4444444
Univariate Tests of Hypotheses for Within Subject Effects
Adj Pr > F
Source DF Type III SS Mean Square F Value Pr > F G - G H - F
b 3 1405.500000 468.500000 127.77 <.0001 <.0001 <.0001
b*a 6 155.500000 25.916667 7.07 0.0001 0.0006 0.0001
Error(b) 27 99.000000 3.666667
Greenhouse-Geisser Epsilon 0.7903
Huynh-Feldt Epsilon 1.3301
proc mixed data = table19_4; class s a b ; model seeds = a | b; repeated / subject = s type = cs; run; quit;
The Mixed Procedure
Model Information
Data Set WORK.TABLE19_4
Dependent Variable seeds
Covariance Structure Compound Symmetry
Subject Effect s
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Between-Within
Class Level Information
Class Levels Values
s 12 1 2 3 4 5 6 7 8 9 10 11 12
a 3 1 2 3
b 4 1 2 3 4
Dimensions
Covariance Parameters 2
Columns in X 20
Columns in Z 0
Subjects 12
Max Obs Per Subject 4
Observations Used 48
Observations Not Used 0
Total Observations 48
Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 205.48514864
1 1 185.46871848 0.00000000
Convergence criteria met.
Covariance Parameter Estimates
Cov Parm Subject Estimate
CS s 7.4444
Residual 3.6667
Fit Statistics
-2 Res Log Likelihood 185.5
AIC (smaller is better) 189.5
AICC (smaller is better) 189.8
BIC (smaller is better) 190.4
Null Model Likelihood Ratio Test
DF Chi-Square Pr > ChiSq
1 20.02 <.0001
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
a 2 9 6.85 0.0156
b 3 27 127.77 <.0001
a*b 6 27 7.07 0.0001