Example 41.8 Mixed Model Analysis of Variance with 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 41.8.1 through Output 41.8.4.
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 tests based on person and machine*person being treated as random effects. As you can see in Output 41.8.4, this requires that a linear combination of mean squares be constructed to test both the machine and person hypotheses; thus, tests that use Satterthwaite approximations are needed.
Output 41.8.1
Summary Information about Groups
Output 41.8.2
FixedEffect Model Analysis of Variance
The GLM Procedure
Dependent Variable: rating
17 
3061.743333 
180.102549 
206.41 
<.0001 
26 
22.686667 
0.872564 


43 
3084.430000 



0.992645 
1.560754 
0.934111 
59.85000 
2 
1648.664722 
824.332361 
944.72 
<.0001 
5 
1008.763583 
201.752717 
231.22 
<.0001 
10 
404.315028 
40.431503 
46.34 
<.0001 
2 
1238.197626 
619.098813 
709.52 
<.0001 
5 
1011.053834 
202.210767 
231.74 
<.0001 
10 
404.315028 
40.431503 
46.34 
<.0001 
Output 41.8.3
Expected Values of Type III Mean Squares
Var(Error) + 2.137 Var(machine*person) + Q(machine) 
Var(Error) + 2.2408 Var(machine*person) + 6.7224 Var(person) 
Var(Error) + 2.3162 Var(machine*person) 
Output 41.8.4
Mixed Model Analysis of Variance
The GLM Procedure
Tests of Hypotheses for Mixed Model Analysis of Variance
Dependent Variable: rating
2 
1238.197626 
619.098813 
16.57 
0.0007 
10.036 
375.057436 
37.370384 


5 
1011.053834 
202.210767 
5.17 
0.0133 
10.015 
392.005726 
39.143708 


10 
404.315028 
40.431503 
46.34 
<.0001 
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 41.8.5.
proc mixed data=machine method=type3;
class machine person;
model rating = machine;
random person machine*person;
run;
Output 41.8.5
PROC MIXED Mixed Model Analysis of Variance (Partial Output)
The Mixed Procedure
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 
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 
10 
404.315028 
40.431503 
Var(Residual) + 2.3162 Var(machine*person) 
MS(Residual) 
26 
46.34 
<.0001 
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 58,
The MIXED Procedure,
for more details.