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 F-tests 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,
F-tests 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: Fixed-Effect 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 |
|
|
|
| R-Square |
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.
