Data from the completely between subject design, Table 25.1, page 552
data table25_1a; input a b y; datalines; 1 1 14 1 2 13 1 3 15 1 1 15 1 2 12 1 3 17 1 1 12 1 2 15 1 3 15 1 1 13 1 2 16 1 3 16 1 1 13 1 2 12 1 3 15 2 1 7 2 2 8 2 3 13 2 1 8 2 2 6 2 3 10 2 1 7 2 2 11 2 3 11 2 1 7 2 2 11 2 3 11 2 1 8 2 2 11 2 3 10 ; run;
Analysis of Variance from the completely between-subject design with Factor B nested within Factor A, Table 25.3, page 557
NOTE: Refer to Chapter 24 for a discussion of the random statement.
proc glm data = table25_1a; class a b; model y = a b(a) / ss3; random b(a) / test; run; quit;
The GLM Procedure
Dependent Variable: y
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 5 229.8666667 45.9733333 22.07 <.0001
Error 24 50.0000000 2.0833333
Corrected Total 29 279.8666667
R-Square Coeff Var Root MSE y Mean
0.821343 12.30150 1.443376 11.73333
Source DF Type III SS Mean Square F Value Pr > F
a 1 182.5333333 182.5333333 87.62 <.0001
b(a) 4 47.3333333 11.8333333 5.68 0.0023
Source Type III Expected Mean Square
a Var(Error) + 5 Var(b(a)) + Q(a)
b(a) Var(Error) + 5 Var(b(a))
Tests of Hypotheses for Mixed Model Analysis of Variance Source DF Type III SS Mean Square F Value Pr > F a 1 182.533333 182.533333 15.43 0.0171 Error: MS(b(a)) 4 47.333333 11.833333 Source DF Type III SS Mean Square F Value Pr > F b(a) 4 47.333333 11.833333 5.68 0.0023 Error: MS(Error) 24 50.000000 2.083333
proc mixed data = table25_1a method = type3; class a b; model y = a ; random b(a); run; quit;
The Mixed Procedure
Model Information
Data Set WORK.TABLE25_1A
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
Dimensions
Covariance Parameters 2
Columns in X 3
Columns in Z 6
Subjects 1
Max Obs Per Subject 30
Observations Used 30
Observations Not Used 0
Total Observations 30
Type 3 Analysis of Variance
Sum of
Source DF Squares Mean Square Expected Mean Square Error Term
a 1 182.533333 182.533333 Var(Residual) + 5 Var(b(a)) + Q(a) MS(b(a))
b(a) 4 47.333333 11.833333 Var(Residual) + 5 Var(b(a)) MS(Residual)
Residual 24 50.000000 2.083333 Var(Residual) .
Type 3 Analysis of Variance
Error
Source DF F Value Pr > F
a 4 15.43 0.0171
b(a) 24 5.68 0.0023
Residual . . .
Covariance Parameter
Estimates
Cov Parm Estimate
b(a) 1.9500
Residual 2.0833
Fit Statistics
-2 Res Log Likelihood 112.4
AIC (smaller is better) 116.4
AICC (smaller is better) 116.9
BIC (smaller is better) 116.0
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
a 1 4 15.43 0.0171
Data from the completely within-subject design, Table 25.1, page 552
data table25_1b; input s a b y; datalines; 1 1 1 13 1 1 2 12 1 1 3 12 1 2 1 7 1 2 2 12 1 2 3 12 2 1 1 16 2 1 2 13 2 1 3 18 2 2 1 10 2 2 2 12 2 2 3 11 3 1 1 15 3 1 2 13 3 1 3 14 3 2 1 5 3 2 2 8 3 2 3 8 4 1 1 12 4 1 2 13 4 1 3 15 4 2 1 7 4 2 2 11 4 2 3 13 5 1 1 15 5 1 2 13 5 1 3 14 5 2 1 5 5 2 2 8 5 2 3 12 ; run;
Analysis of Variance from the completely within-subject design, Table 25.3, page 557. Factor B is nested within Factor A and all observations are completely within-subject.
proc glm data = table25_1b; class s a b; model y = a b(a) s a*s / ss3; random b(a) s a*s / test; run; quit;
The GLM Procedure
Class Level Information
Class Levels Values
s 5 1 2 3 4 5
a 2 1 2
b 3 1 2 3
Number of observations 30
Dependent Variable: y
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 13 263.1000000 20.2384615 11.62 <.0001
Error 16 27.8666667 1.7416667
Corrected Total 29 290.9666667
R-Square Coeff Var Root MSE y Mean
0.904227 11.34432 1.319722 11.63333
Source DF Type III SS Mean Square F Value Pr > F
a 1 149.6333333 149.6333333 85.91 <.0001
b(a) 4 62.1333333 15.5333333 8.92 0.0006
s 4 27.1333333 6.7833333 3.89 0.0215
s*a 4 24.2000000 6.0500000 3.47 0.0318
Source Type III Expected Mean Square
a Var(Error) + 3 Var(s*a) + 5 Var(b(a)) + Q(a)
b(a) Var(Error) + 5 Var(b(a))
s Var(Error) + 3 Var(s*a) + 6 Var(s)
s*a Var(Error) + 3 Var(s*a)
Tests of Hypotheses for Mixed Model Analysis of Variance Source DF Type III SS Mean Square F Value Pr > F a 1 149.633333 149.633333 7.54 0.0356 Error 5.6515 112.135397 19.841667 Error: MS(b(a)) + MS(s*a) - MS(Error) Source DF Type III SS Mean Square F Value Pr > F b(a) 4 62.133333 15.533333 8.92 0.0006 s*a 4 24.200000 6.050000 3.47 0.0318 Error: MS(Error) 16 27.866667 1.741667 Source DF Type III SS Mean Square F Value Pr > F s 4 27.133333 6.783333 1.12 0.4572 Error: MS(s*a) 4 24.200000 6.050000
proc mixed data = table25_1b method = type3; class s a b; model y = a ; random b(a) s a*s ; run; quit;
The Mixed Procedure
Model Information
Data Set WORK.TABLE25_1B
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
s 5 1 2 3 4 5
a 2 1 2
b 3 1 2 3
Dimensions
Covariance Parameters 4
Columns in X 3
Columns in Z 21
Subjects 1
Max Obs Per Subject 30
Observations Used 30
Observations Not Used 0
Total Observations 30
Type 3 Analysis of Variance
Sum of
Source DF Squares Mean Square Expected Mean Square
a 1 149.633333 149.633333 Var(Residual) + 3 Var(s*a) + 5 Var(b(a)) + Q(a)
b(a) 4 62.133333 15.533333 Var(Residual) + 5 Var(b(a))
s 4 27.133333 6.783333 Var(Residual) + 3 Var(s*a) + 6 Var(s)
Type 3 Analysis of Variance
Error
Source Error Term DF F Value Pr > F
a MS(b(a)) + MS(s*a) - MS(Residual) 5.6515 7.54 0.0356
b(a) MS(Residual) 16 8.92 0.0006
s MS(s*a) 4 1.12 0.4572
Type 3 Analysis of Variance
Sum of
Source DF Squares Mean Square Expected Mean Square
s*a 4 24.200000 6.050000 Var(Residual) + 3 Var(s*a)
Residual 16 27.866667 1.741667 Var(Residual)
Type 3 Analysis of Variance
Error
Source Error Term DF F Value Pr > F
s*a MS(Residual) 16 3.47 0.0318
Residual . . . .
Covariance Parameter
Estimates
Cov Parm Estimate
b(a) 2.7583
s 0.1222
s*a 1.4361
Residual 1.7417
Fit Statistics
-2 Res Log Likelihood 119.6
AIC (smaller is better) 127.6
AICC (smaller is better) 129.3
BIC (smaller is better) 126.8
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
a 1 4 7.54 0.0516
Data from the mixed design (long format), Table 25.1, page 552
data table25_1c; input s a b y; datalines; 1 1 1 13 1 1 2 11 1 1 3 15 2 1 1 16 2 1 2 11 2 1 3 16 3 1 1 13 3 1 2 16 3 1 3 17 4 1 1 13 4 1 2 14 4 1 3 16 5 1 1 13 5 1 2 12 5 1 3 16 6 2 1 7 6 2 2 9 6 2 3 13 7 2 1 9 7 2 2 6 7 2 3 11 8 2 1 7 8 2 2 10 8 2 3 10 9 2 1 9 9 2 2 13 9 2 3 16 10 2 1 8 10 2 2 11 10 2 3 11 ; run;
Analysis of Variance from a nested mixed design, Table 25.3, page 557
NOTE: Subjects are nested within Factor A, Factor B is nested within Factor A implying that subjects are crossed with Factor B.
proc glm data = table25_1c; class s a b; model y = a b(a) s(a) /ss3; random a b(a) s(a) / test; run; quit;
The GLM Procedure
Class Level Information
Class Levels Values
s 10 1 2 3 4 5 6 7 8 9 10
a 2 1 2
b 3 1 2 3
Number of observations 30
Dependent Variable: y
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 13 239.3333333 18.4102564 6.93 0.0002
Error 16 42.5333333 2.6583333
Corrected Total 29 281.8666667
R-Square Coeff Var Root MSE y Mean
0.849101 13.51193 1.630440 12.06667
Source DF Type III SS Mean Square F Value Pr > F
a 1 128.1333333 128.1333333 48.20 <.0001
b(a) 4 72.1333333 18.0333333 6.78 0.0022
s(a) 8 39.0666667 4.8833333 1.84 0.1430
Source Type III Expected Mean Square
a Var(Error) + 3 Var(s(a)) + 5 Var(b(a)) + 15 Var(a)
b(a) Var(Error) + 5 Var(b(a))
s(a) Var(Error) + 3 Var(s(a))
Tests of Hypotheses for Random Model Analysis of Variance
Source DF Type III SS Mean Square F Value Pr > F
a 1 128.133333 128.133333 6.32 0.0551
Error 4.844 98.132023 20.258333
Error: MS(b(a)) + MS(s(a)) - MS(Error)
Source DF Type III SS Mean Square F Value Pr > F
b(a) 4 72.133333 18.033333 6.78 0.0022
s(a) 8 39.066667 4.883333 1.84 0.1430
Error: MS(Error) 16 42.533333 2.658333
proc mixed data = table25_1c method = type3 ; class s a b; model y = a ; random b(a) s(a); run; quit;
The Mixed Procedure
Model Information
Data Set WORK.TABLE25_1C
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
s 10 1 2 3 4 5 6 7 8 9 10
a 2 1 2
b 3 1 2 3
Dimensions
Covariance Parameters 3
Columns in X 3
Columns in Z 16
Subjects 1
Max Obs Per Subject 30
Observations Used 30
Observations Not Used 0
Total Observations 30
Type 3 Analysis of Variance
Sum of
Source DF Squares Mean Square Expected Mean Square
a 1 128.133333 128.133333 Var(Residual) + 3 Var(s(a)) + 5 Var(b(a)) + Q(a)
b(a) 4 72.133333 18.033333 Var(Residual) + 5 Var(b(a))
s(a) 8 39.066667 4.883333 Var(Residual) + 3 Var(s(a))
Type 3 Analysis of Variance
Error
Source Error Term DF F Value Pr > F
a MS(b(a)) + MS(s(a)) - MS(Residual) 4.844 6.32 0.0551
b(a) MS(Residual) 16 6.78 0.0022
s(a) MS(Residual) 16 1.84 0.1430
Type 3 Analysis of Variance
Sum of
Source DF Squares Mean Square Expected Mean Square
Residual 16 42.533333 2.658333 Var(Residual)
Type 3 Analysis of Variance
Error
Source Error Term DF F Value Pr > F
Residual . . . .
Covariance Parameter
Estimates
Cov Parm Estimate
b(a) 3.0750
s(a) 0.7417
Residual 2.6583
Fit Statistics
-2 Res Log Likelihood 124.8
AIC (smaller is better) 130.8
AICC (smaller is better) 131.8
BIC (smaller is better) 130.2
Type 3 Tests of Fixed Effects
Num Den
Effect DF DF F Value Pr > F
a 1 4 6.32 0.0657