Data from table 24.1, page 532.
data table24_1; input a b c y @@; datalines; 1 1 1 16 1 2 1 11 1 3 1 10 2 1 1 12 2 2 1 10 2 3 1 10 1 1 1 14 1 2 1 9 1 3 1 12 2 1 1 11 2 2 1 12 2 3 1 11 1 1 2 15 1 2 2 10 1 3 2 12 2 1 2 7 2 2 2 9 2 3 2 9 1 1 2 11 1 2 2 8 1 3 2 10 2 1 2 9 2 2 2 7 2 3 2 7 1 1 3 13 1 2 3 10 1 3 3 11 2 1 3 8 2 2 3 7 2 3 3 8 1 1 3 11 1 2 3 8 1 3 3 9 2 1 3 6 2 2 3 8 2 3 3 6 1 1 4 11 1 2 4 6 1 3 4 9 2 1 4 10 2 2 4 11 2 3 4 11 1 1 4 9 1 2 4 8 1 3 4 11 2 1 4 8 2 2 4 9 2 3 4 11 1 1 5 9 1 2 5 7 1 3 5 8 2 1 5 9 2 2 5 11 2 3 5 10 1 1 5 11 1 2 5 5 1 3 5 8 2 1 5 10 2 2 5 8 2 3 5 8 ; run;
Table 24.7, page 540. Three-factorial analysis of variance treating factor A as fixed and factors B and C as random.
NOTE: The specification of random factors is defined through the random statement and the test option performs hypothesis tests for each effect specified in the model. It is pivotal to know which to factors to name and evaluate as random. See Table 24.3, page 535 for discussion of assigning error terms in designs with random factors.
proc glm data= table24_1; class a b c ; model y = a|b|c / ss3; random a*b a*c /test; test h = b c e = b*c; test h= a*b a*c e= a*b*c; run; quit;
Class Level Information
Class Levels Values
a 2 1 2
b 3 1 2 3
c 5 1 2 3 4 5
Number of observations 60
Dependent Variable: y
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 29 218.0833333 7.5201149 3.86 0.0002
Error 30 58.5000000 1.9500000
Corrected Total 59 276.5833333
R-Square Coeff Var Root MSE y Mean
0.788491 14.57138 1.396424 9.583333
Source DF Type III SS Mean Square F Value Pr > F
a 1 14.01666667 14.01666667 7.19 0.0118
b 2 32.43333333 16.21666667 8.32 0.0013
a*b 2 40.03333333 20.01666667 10.26 0.0004
c 4 62.66666667 15.66666667 8.03 0.0002
a*c 4 54.40000000 13.60000000 6.97 0.0004
b*c 8 13.73333333 1.71666667 0.88 0.5440
a*b*c 8 0.80000000 0.10000000 0.05 0.9999
Source Type III Expected Mean Square
a Var(Error) + 6 Var(a*c) + 10 Var(a*b) + Q(a,a*b*c)
b Var(Error) + 10 Var(a*b) + Q(b,b*c,a*b*c)
a*b Var(Error) + 10 Var(a*b) + Q(a*b*c)
c Var(Error) + 6 Var(a*c) + Q(c,b*c,a*b*c)
a*c Var(Error) + 6 Var(a*c) + Q(a*b*c)
b*c Var(Error) + Q(b*c,a*b*c)
a*b*c Var(Error) + Q(a*b*c)
Tests of Hypotheses for Mixed Model Analysis of Variance
Dependent Variable: y
Source DF Type III SS Mean Square F Value Pr > F
* a 1 14.016667 14.016667 0.44 0.5417
Error 4.0648 128.717475 31.666667
Error: MS(a*b) + MS(a*c) - MS(Error)
* This test assumes one or more other fixed effects are zero.
Source DF Type III SS Mean Square F Value Pr > F
* b 2 32.433333 16.216667 0.81 0.5524
Error: MS(a*b) 2 40.033333 20.016667
* This test assumes one or more other fixed effects are zero.
Source DF Type III SS Mean Square F Value Pr > F
a*b 2 40.033333 20.016667 10.26 0.0004
a*c 4 54.400000 13.600000 6.97 0.0004
* b*c 8 13.733333 1.716667 0.88 0.5440
a*b*c 8 0.800000 0.100000 0.05 0.9999
Error: MS(Error) 30 58.500000 1.950000
* This test assumes one or more other fixed effects are zero.
Source DF Type III SS Mean Square F Value Pr > F
* c 4 62.666667 15.666667 1.15 0.4471
Error: MS(a*c) 4 54.400000 13.600000
* This test assumes one or more other fixed effects are zero.
Tests of Hypotheses Using the Type III MS for b*c as an Error Term
Source DF Type III SS Mean Square F Value Pr > F
b 2 32.43333333 16.21666667 9.45 0.0078
c 4 62.66666667 15.66666667 9.13 0.0045
Tests of Hypotheses Using the Type III MS for a*b*c as an Error Term
Source DF Type III SS Mean Square F Value Pr > F
a*b 2 40.03333333 20.01666667 200.17 <.0001
a*c 4 54.40000000 13.60000000 136.00 <.0001
Data from Table 2.1, page 16.
data table2_1; input fact a s y @@; datalines; 1 1 1 0 2 1 1 1 3 1 1 1 4 1 1 1 5 1 1 1 6 1 1 1 7 1 1 1 8 1 1 1 9 1 1 1 10 1 1 1 11 1 1 1 12 1 1 1 13 1 1 1 14 1 1 0 15 1 1 1 16 1 1 1 17 1 1 0 18 1 1 1 19 1 1 0 20 1 1 1 1 1 2 0 2 1 2 0 3 1 2 1 4 1 2 1 5 1 2 1 6 1 2 0 7 1 2 0 8 1 2 1 9 1 2 1 10 1 2 0 11 1 2 1 12 1 2 1 13 1 2 1 14 1 2 1 15 1 2 1 16 1 2 0 17 1 2 1 18 1 2 0 19 1 2 0 20 1 2 1 1 1 3 1 2 1 3 1 3 1 3 0 4 1 3 0 5 1 3 0 6 1 3 1 7 1 3 1 8 1 3 0 9 1 3 1 10 1 3 1 11 1 3 1 12 1 3 0 13 1 3 1 14 1 3 1 15 1 3 0 16 1 3 0 17 1 3 0 18 1 3 1 19 1 3 1 20 1 3 0 1 1 4 1 2 1 4 1 3 1 4 1 4 1 4 1 5 1 4 0 6 1 4 1 7 1 4 0 8 1 4 1 9 1 4 1 10 1 4 1 11 1 4 0 12 1 4 0 13 1 4 1 14 1 4 1 15 1 4 1 16 1 4 1 17 1 4 1 18 1 4 0 19 1 4 1 20 1 4 1 1 1 5 1 2 1 5 0 3 1 5 1 4 1 5 0 5 1 5 0 6 1 5 0 7 1 5 0 8 1 5 1 9 1 5 0 10 1 5 1 11 1 5 1 12 1 5 0 13 1 5 1 14 1 5 0 15 1 5 0 16 1 5 0 17 1 5 1 18 1 5 0 19 1 5 1 20 1 5 1 1 1 6 0 2 1 6 0 3 1 6 0 4 1 6 1 5 1 6 0 6 1 6 0 7 1 6 1 8 1 6 0 9 1 6 1 10 1 6 0 11 1 6 0 12 1 6 1 13 1 6 1 14 1 6 1 15 1 6 0 16 1 6 0 17 1 6 0 18 1 6 0 19 1 6 0 20 1 6 0 1 1 7 1 2 1 7 1 3 1 7 0 4 1 7 1 5 1 7 0 6 1 7 1 7 1 7 0 8 1 7 1 9 1 7 0 10 1 7 1 11 1 7 1 12 1 7 1 13 1 7 1 14 1 7 0 15 1 7 1 16 1 7 1 17 1 7 1 18 1 7 0 19 1 7 0 20 1 7 0 1 1 8 1 2 1 8 0 3 1 8 1 4 1 8 0 5 1 8 1 6 1 8 1 7 1 8 1 8 1 8 1 9 1 8 1 10 1 8 0 11 1 8 0 12 1 8 1 13 1 8 0 14 1 8 0 15 1 8 1 16 1 8 0 17 1 8 1 18 1 8 1 19 1 8 0 20 1 8 1 1 1 9 1 2 1 9 0 3 1 9 0 4 1 9 1 5 1 9 1 6 1 9 1 7 1 9 1 8 1 9 0 9 1 9 0 10 1 9 1 11 1 9 1 12 1 9 1 13 1 9 1 14 1 9 1 15 1 9 0 16 1 9 0 17 1 9 1 18 1 9 0 19 1 9 1 20 1 9 0 1 1 10 1 2 1 10 1 3 1 10 1 4 1 10 0 5 1 10 0 6 1 10 1 7 1 10 0 8 1 10 1 9 1 10 0 10 1 10 0 11 1 10 1 12 1 10 0 13 1 10 1 14 1 10 0 15 1 10 1 16 1 10 0 17 1 10 0 18 1 10 1 19 1 10 1 20 1 10 0 1 1 11 1 2 1 11 1 3 1 11 1 4 1 11 0 5 1 11 0 6 1 11 1 7 1 11 0 8 1 11 0 9 1 11 0 10 1 11 1 11 1 11 0 12 1 11 1 13 1 11 0 14 1 11 1 15 1 11 0 16 1 11 1 17 1 11 0 18 1 11 1 19 1 11 1 20 1 11 1 1 1 12 1 2 1 12 1 3 1 12 1 4 1 12 0 5 1 12 0 6 1 12 1 7 1 12 0 8 1 12 0 9 1 12 1 10 1 12 0 11 1 12 0 12 1 12 1 13 1 12 1 14 1 12 0 15 1 12 0 16 1 12 1 17 1 12 0 18 1 12 0 19 1 12 1 20 1 12 1 1 1 13 1 2 1 13 1 3 1 13 1 4 1 13 1 5 1 13 1 6 1 13 0 7 1 13 1 8 1 13 0 9 1 13 1 10 1 13 1 11 1 13 1 12 1 13 0 13 1 13 1 14 1 13 1 15 1 13 1 16 1 13 0 17 1 13 0 18 1 13 0 19 1 13 0 20 1 13 1 1 1 14 0 2 1 14 0 3 1 14 0 4 1 14 0 5 1 14 0 6 1 14 1 7 1 14 0 8 1 14 1 9 1 14 1 10 1 14 1 11 1 14 1 12 1 14 1 13 1 14 0 14 1 14 1 15 1 14 0 16 1 14 0 17 1 14 1 18 1 14 1 19 1 14 0 20 1 14 0 1 1 15 1 2 1 15 0 3 1 15 0 4 1 15 1 5 1 15 0 6 1 15 1 7 1 15 0 8 1 15 1 9 1 15 1 10 1 15 0 11 1 15 0 12 1 15 0 13 1 15 1 14 1 15 0 15 1 15 1 16 1 15 1 17 1 15 1 18 1 15 1 19 1 15 1 20 1 15 0 1 1 16 0 2 1 16 1 3 1 16 1 4 1 16 1 5 1 16 1 6 1 16 1 7 1 16 1 8 1 16 0 9 1 16 1 10 1 16 1 11 1 16 1 12 1 16 0 13 1 16 0 14 1 16 1 15 1 16 0 16 1 16 1 17 1 16 0 18 1 16 1 19 1 16 1 20 1 16 1 1 2 17 1 2 2 17 1 3 2 17 1 4 2 17 1 5 2 17 0 6 2 17 1 7 2 17 1 8 2 17 0 9 2 17 0 10 2 17 0 11 2 17 0 12 2 17 1 13 2 17 1 14 2 17 1 15 2 17 1 16 2 17 0 17 2 17 0 18 2 17 1 19 2 17 0 20 2 17 1 1 2 18 1 2 2 18 1 3 2 18 0 4 2 18 1 5 2 18 1 6 2 18 1 7 2 18 0 8 2 18 0 9 2 18 1 10 2 18 0 11 2 18 1 12 2 18 0 13 2 18 1 14 2 18 1 15 2 18 0 16 2 18 0 17 2 18 0 18 2 18 1 19 2 18 1 20 2 18 1 1 2 19 1 2 2 19 1 3 2 19 0 4 2 19 0 5 2 19 1 6 2 19 1 7 2 19 1 8 2 19 0 9 2 19 0 10 2 19 0 11 2 19 0 12 2 19 0 13 2 19 0 14 2 19 0 15 2 19 1 16 2 19 1 17 2 19 1 18 2 19 1 19 2 19 0 20 2 19 1 1 2 20 0 2 2 20 1 3 2 20 0 4 2 20 1 5 2 20 1 6 2 20 1 7 2 20 1 8 2 20 1 9 2 20 0 10 2 20 0 11 2 20 0 12 2 20 1 13 2 20 1 14 2 20 0 15 2 20 0 16 2 20 0 17 2 20 1 18 2 20 0 19 2 20 0 20 2 20 0 1 2 21 1 2 2 21 1 3 2 21 1 4 2 21 0 5 2 21 1 6 2 21 1 7 2 21 1 8 2 21 1 9 2 21 1 10 2 21 1 11 2 21 1 12 2 21 1 13 2 21 1 14 2 21 1 15 2 21 1 16 2 21 0 17 2 21 1 18 2 21 1 19 2 21 0 20 2 21 0 1 2 22 1 2 2 22 1 3 2 22 0 4 2 22 1 5 2 22 1 6 2 22 1 7 2 22 1 8 2 22 0 9 2 22 1 10 2 22 0 11 2 22 1 12 2 22 1 13 2 22 1 14 2 22 1 15 2 22 1 16 2 22 0 17 2 22 1 18 2 22 1 19 2 22 1 20 2 22 1 1 2 23 1 2 2 23 1 3 2 23 0 4 2 23 1 5 2 23 0 6 2 23 1 7 2 23 1 8 2 23 1 9 2 23 1 10 2 23 1 11 2 23 1 12 2 23 1 13 2 23 0 14 2 23 1 15 2 23 1 16 2 23 1 17 2 23 1 18 2 23 0 19 2 23 1 20 2 23 1 1 2 24 1 2 2 24 1 3 2 24 0 4 2 24 1 5 2 24 0 6 2 24 1 7 2 24 1 8 2 24 1 9 2 24 1 10 2 24 1 11 2 24 1 12 2 24 1 13 2 24 1 14 2 24 0 15 2 24 1 16 2 24 1 17 2 24 1 18 2 24 1 19 2 24 0 20 2 24 1 1 2 25 1 2 2 25 1 3 2 25 1 4 2 25 0 5 2 25 1 6 2 25 1 7 2 25 1 8 2 25 0 9 2 25 1 10 2 25 0 11 2 25 0 12 2 25 0 13 2 25 0 14 2 25 0 15 2 25 0 16 2 25 0 17 2 25 0 18 2 25 1 19 2 25 1 20 2 25 1 1 2 26 1 2 2 26 0 3 2 26 1 4 2 26 1 5 2 26 1 6 2 26 1 7 2 26 0 8 2 26 0 9 2 26 1 10 2 26 0 11 2 26 1 12 2 26 1 13 2 26 1 14 2 26 1 15 2 26 1 16 2 26 1 17 2 26 1 18 2 26 1 19 2 26 1 20 2 26 1 1 2 27 1 2 2 27 0 3 2 27 0 4 2 27 0 5 2 27 1 6 2 27 1 7 2 27 0 8 2 27 0 9 2 27 0 10 2 27 1 11 2 27 1 12 2 27 1 13 2 27 0 14 2 27 1 15 2 27 0 16 2 27 1 17 2 27 0 18 2 27 1 19 2 27 1 20 2 27 1 1 2 28 1 2 2 28 1 3 2 28 1 4 2 28 0 5 2 28 1 6 2 28 1 7 2 28 1 8 2 28 1 9 2 28 1 10 2 28 1 11 2 28 1 12 2 28 1 13 2 28 0 14 2 28 0 15 2 28 1 16 2 28 1 17 2 28 0 18 2 28 1 19 2 28 1 20 2 28 1 1 2 29 1 2 2 29 0 3 2 29 1 4 2 29 1 5 2 29 0 6 2 29 1 7 2 29 1 8 2 29 1 9 2 29 1 10 2 29 1 11 2 29 1 12 2 29 1 13 2 29 1 14 2 29 0 15 2 29 1 16 2 29 1 17 2 29 1 18 2 29 0 19 2 29 0 20 2 29 1 1 2 30 1 2 2 30 1 3 2 30 1 4 2 30 1 5 2 30 1 6 2 30 1 7 2 30 1 8 2 30 0 9 2 30 1 10 2 30 0 11 2 30 0 12 2 30 1 13 2 30 0 14 2 30 1 15 2 30 0 16 2 30 0 17 2 30 1 18 2 30 1 19 2 30 0 20 2 30 1 1 2 31 1 2 2 31 1 3 2 31 1 4 2 31 1 5 2 31 0 6 2 31 1 7 2 31 0 8 2 31 1 9 2 31 0 10 2 31 1 11 2 31 0 12 2 31 0 13 2 31 0 14 2 31 1 15 2 31 1 16 2 31 0 17 2 31 0 18 2 31 1 19 2 31 1 20 2 31 1 1 2 32 1 2 2 32 1 3 2 32 1 4 2 32 1 5 2 32 1 6 2 32 1 7 2 32 1 8 2 32 1 9 2 32 1 10 2 32 1 11 2 32 1 12 2 32 1 13 2 32 1 14 2 32 0 15 2 32 1 16 2 32 1 17 2 32 1 18 2 32 1 19 2 32 1 20 2 32 1 ; run;
Treating facts as random and A and S(A) as fixed, Table 24.11, page 547.
proc glm data = table2_1; class a s fact; model y = a|s(a)|fact @2 / ss3; test h=a e=s(a); random s(a) a*fact/test; run; quit;
Class Level Information
Class Levels Values
a 2 1 2
s 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
fact 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of observations 640
Dependent Variable: y
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 69 22.4875000 0.3259058 1.46 0.0119
Error 570 127.0062500 0.2228180
Corrected Total 639 149.4937500
R-Square Coeff Var Root MSE y Mean
0.150424 75.15001 0.472036 0.628125
Source DF Type III SS Mean Square F Value Pr > F
a 1 2.02500000 2.02500000 9.09 0.0027
s(a) 30 10.86875000 0.36229167 1.63 0.0201
fact 19 5.68125000 0.29901316 1.34 0.1505
a*fact 19 3.91250000 0.20592105 0.92 0.5526
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 1 2.02500000 2.02500000 5.59 0.0247
Source Type III Expected Mean Square
a Var(Error) + 16 Var(a*fact) + 20 Var(s(a)) + Q(a)
s(a) Var(Error) + 20 Var(s(a))
fact Var(Error) + 16 Var(a*fact) + Q(fact)
a*fact Var(Error) + 16 Var(a*fact)
Tests of Hypotheses for Mixed Model Analysis of Variance
Dependent Variable: y
Source DF Type III SS Mean Square F Value Pr > F
a 1 2.025000 2.025000 5.86 0.0264
Error 17.821 6.155438 0.345395
Error: MS(s(a)) + MS(a*fact) - MS(Error)
Source DF Type III SS Mean Square F Value Pr > F
s(a) 30 10.868750 0.362292 1.63 0.0201
a*fact 19 3.912500 0.205921 0.92 0.5526
Error: MS(Error) 570 127.006250 0.222818
Source DF Type III SS Mean Square F Value Pr > F
fact 19 5.681250 0.299013 1.45 0.2118
Error: MS(a*fact) 19 3.912500 0.205921
Treating Factor A and Factor B as fixed and Factor C as random in a three factor between subject design, Table 24.12, page 549.
proc mixed data= table24_1 method = type3; class a b c ; model y = a|b ; random c a*c b*c a*b*c ; run; quit;
Model Information
Data Set WORK.TABLE24_1
Dependent Variable y
Covariance Structure Variance Components
Estimation Method Type 3
Residual Variance Method Factor
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Containment
Class Level Information
Class Levels Values
a 2 1 2
b 3 1 2 3
c 5 1 2 3 4 5
Dimensions
Covariance Parameters 5
Columns in X 12
Columns in Z 60
Subjects 1
Max Obs Per Subject 60
Observations Used 60
Observations Not Used 0
Total Observations 60
Type 3 Analysis of Variance
Sum of
Source DF Squares Mean Square Expected Mean Square
a 1 14.016667 14.016667 Var(Residual) + 2 Var(a*b*c) + 6 Var(a*c) + Q(a,a*b)
b 2 32.433333 16.216667 Var(Residual) + 2 Var(a*b*c) + 4 Var(b*c) + Q(b,a*b)
a*b 2 40.033333 20.016667 Var(Residual) + 2 Var(a*b*c) + Q(a*b)
c 4 62.666667 15.666667 Var(Residual) + 2 Var(a*b*c) + 4 Var(b*c) + 6 Var(a*c) + 12 Var(c)
a*c 4 54.400000 13.600000 Var(Residual) + 2 Var(a*b*c) + 6 Var(a*c)
b*c 8 13.733333 1.716667 Var(Residual) + 2 Var(a*b*c) + 4 Var(b*c)
a*b*c 8 0.800000 0.100000 Var(Residual) + 2 Var(a*b*c)
Residual 30 58.500000 1.950000 Var(Residual)
Type 3 Analysis of Variance
Error
Source Error Term DF F Value Pr > F
a MS(a*c) 4 1.03 0.3674
b MS(b*c) 8 9.45 0.0078
a*b MS(a*b*c) 8 200.17 <.0001
c MS(a*c) + MS(b*c) - MS(a*b*c) 4.9678 1.03 0.4745
a*c MS(a*b*c) 8 136.00 <.0001
b*c MS(a*b*c) 8 17.17 0.0003
a*b*c MS(Residual) 30 0.05 0.9999
Residual . . . .
Covariance Parameter
Estimates
Cov Parm Estimate
c 0.03750
a*c 2.2500
b*c 0.4042
a*b*c -0.9250
Residual 1.9500
Fit Statistics
-2 Res Log Likelihood 194.4
AIC (smaller is better) 204.4
AICC (smaller is better) 205.7
BIC (smaller is better) 202.5
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
a 1 4 1.03 0.3674
b 2 8 9.45 0.0078
a*b 2 8 200.17 <.0001