Example 30.8: Mixed Model Analysis of Variance Using the RANDOM Statement
Milliken and Johnson (1984) present an
example of an unbalanced mixed model.
Three machines, which are considered as a fixed effect, and six
employees, which are considered a random effect, are studied.
Each employee operates each machine for
either one, two, or three different times.
The dependent variable is an overall rating, which takes
into account the number and quality of components produced.
The following statements form the data set and
perform a mixed model analysis of variance by
requesting the TEST option in the RANDOM statement.
Note that the machine*person interaction is declared as
a random effect; in general, when an interaction involves
a random effect, it too should be declared as random.
The results of the analysis are shown in Output 30.8.1
through Output 30.8.4.
data machine;
input machine person rating @@;
datalines;
1 1 52.0 1 2 51.8 1 2 52.8 1 3 60.0 1 4 51.1 1 4 52.3
1 5 50.9 1 5 51.8 1 5 51.4 1 6 46.4 1 6 44.8 1 6 49.2
2 1 64.0 2 2 59.7 2 2 60.0 2 2 59.0 2 3 68.6 2 3 65.8
2 4 63.2 2 4 62.8 2 4 62.2 2 5 64.8 2 5 65.0 2 6 43.7
2 6 44.2 2 6 43.0 3 1 67.5 3 1 67.2 3 1 66.9 3 2 61.5
3 2 61.7 3 2 62.3 3 3 70.8 3 3 70.6 3 3 71.0 3 4 64.1
3 4 66.2 3 4 64.0 3 5 72.1 3 5 72.0 3 5 71.1 3 6 62.0
3 6 61.4 3 6 60.5
;
proc glm data=machine;
class machine person;
model rating=machine person machine*person;
random person machine*person / test;
run;
The TEST option in the RANDOM statement requests that
PROC GLM determine the appropriate Ftests based on person
and machine*person being treated as random effects.
As you can see in Output 30.8.4, this requires that a
linear combination of mean squares be constructed to
test both the machine and person hypotheses; thus,
Ftests using Satterthwaite approximations are used.
Output 30.8.1: Summary Information on Groups
Class Level Information 
Class 
Levels 
Values 
machine 
3 
1 2 3 
person 
6 
1 2 3 4 5 6 
Number of observations 
44 

Output 30.8.2: FixedEffect Model Analysis of Variance
The GLM Procedure 
Dependent Variable: rating 
Source 
DF 
Sum of Squares 
Mean Square 
F Value 
Pr > F 
Model 
17 
3061.743333 
180.102549 
206.41 
<.0001 
Error 
26 
22.686667 
0.872564 


Corrected Total 
43 
3084.430000 



RSquare 
Coeff Var 
Root MSE 
rating Mean 
0.992645 
1.560754 
0.934111 
59.85000 
Source 
DF 
Type I SS 
Mean Square 
F Value 
Pr > F 
machine 
2 
1648.664722 
824.332361 
944.72 
<.0001 
person 
5 
1008.763583 
201.752717 
231.22 
<.0001 
machine*person 
10 
404.315028 
40.431503 
46.34 
<.0001 
Source 
DF 
Type III SS 
Mean Square 
F Value 
Pr > F 
machine 
2 
1238.197626 
619.098813 
709.52 
<.0001 
person 
5 
1011.053834 
202.210767 
231.74 
<.0001 
machine*person 
10 
404.315028 
40.431503 
46.34 
<.0001 

Output 30.8.3: Expected Values of Type III Mean Squares
Source 
Type III Expected Mean Square 
machine 
Var(Error) + 2.137 Var(machine*person) + Q(machine) 
person 
Var(Error) + 2.2408 Var(machine*person) + 6.7224 Var(person) 
machine*person 
Var(Error) + 2.3162 Var(machine*person) 

Output 30.8.4: Mixed Model Analysis of Variance
The GLM Procedure 
Tests of Hypotheses for Mixed Model Analysis of Variance 
Dependent Variable: rating 
Source 
DF 
Type III SS 
Mean Square 
F Value 
Pr > F 
machine 
2 
1238.197626 
619.098813 
16.57 
0.0007 
Error 
10.036 
375.057436 
37.370384 


Error: 0.9226*MS(machine*person) + 0.0774*MS(Error) 
Source 
DF 
Type III SS 
Mean Square 
F Value 
Pr > F 
person 
5 
1011.053834 
202.210767 
5.17 
0.0133 
Error 
10.015 
392.005726 
39.143708 


Error: 0.9674*MS(machine*person) + 0.0326*MS(Error) 
Source 
DF 
Type III SS 
Mean Square 
F Value 
Pr > F 
machine*person 
10 
404.315028 
40.431503 
46.34 
<.0001 
Error: MS(Error) 
26 
22.686667 
0.872564 



Note that you can also use the MIXED procedure to analyze mixed
models. The following statements use PROC MIXED to reproduce the mixed
model analysis of variance; the relevant part of the PROC MIXED results is
shown in Output 30.8.5
proc mixed data=machine method=type3;
class machine person;
model rating = machine;
random person machine*person;
run;
Output 30.8.5: PROC MIXED Mixed Model Analysis of Variance (Partial Output)
Type 3 Analysis of Variance 
Source 
DF 
Sum of Squares 
Mean Square 
Expected Mean Square 
Error Term 
Error DF 
F Value 
Pr > F 
machine 
2 
1238.197626 
619.098813 
Var(Residual) + 2.137 Var(machine*person) + Q(machine) 
0.9226 MS(machine*person) + 0.0774 MS(Residual) 
10.036 
16.57 
0.0007 
person 
5 
1011.053834 
202.210767 
Var(Residual) + 2.2408 Var(machine*person) + 6.7224 Var(person) 
0.9674 MS(machine*person) + 0.0326 MS(Residual) 
10.015 
5.17 
0.0133 
machine*person 
10 
404.315028 
40.431503 
Var(Residual) + 2.3162 Var(machine*person) 
MS(Residual) 
26 
46.34 
<.0001 
Residual 
26 
22.686667 
0.872564 
Var(Residual) 
. 
. 
. 
. 

The advantage of PROC MIXED is that it offers more versatility for
mixed models; the disadvantage is that it can be less computationally
efficient for large data sets. See Chapter 41, "The MIXED Procedure,"
for more details.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.