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Topic 13 Handout:
Random Effects
Learning Goals:
(1) Understand the differences between fixed effects and random effects
models; (2) Be able to identify effects as either fixed or random; (3) Be able to utilize EMS
(expected mean squares) to determine (a) appropriate tests and (b) estimates for variances; (4) Be
able to interpret and draw appropriate conclusions from ANOVA models where random effects
are involved.
13.1
Three scenarios are given below.
For each of these, suggest at least one research
question of interest.
Then identify the response variable and all factors involved in the
study.
As you identify factors, determine the number of levels for each factor and
whether each factor should be treated as a fixed or random effect.
a.
Auto manufacturer wants to study the effects of differences between drivers (A)
and differences between cars (B) on gasoline consumption.
Four drivers were
selected at random, and additionally five cars of the same model with manual
transmissions were randomly taken from the assembly line.
Each driver drove
each car twice over a 40mile test course and the MPG were recorded.
b.
A researcher studied the sodium content of six brands of U.S. beers sold in a
metropolitan area.
For each beer, both the regular and light versions were
examined.
c.
Twelve job applicants were rated by each of the three personnel officers for a
company.
Each applicant was rated by each officer.
We want to explore whether
there are differences among the personnel officers.
Scenario I
:
(From KNNL Applied Linear Statistical Models
)
An automobile manufacturer
wishes to study the difference between drivers (factor A, 4 levels) and between cars (factor B, 5
levels) on gasoline consumption.
The data are given in the associated SAS file.
NOTE:
This scenario will simply be used for an inclass example.
All output has been provided
below on pages 13.
You may wish to refer to it as we talk about the example in class.
To treat effects as random, a “random” statement is required in the GLM coding:
proc
glm
;
class
driver car;
model
gas=drivercar;
random
driver car driver*car /
test
;
run
;
Note that the random statement must include interactions that are random as well.
The code
above creates the following output (somewhat edited for convenience):
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2
Dependent Variable: gas
Sum of
Source
DF
Squares
Mean Square
F Value
Pr > F
Model
19
377.4447500
19.8655132
113.03
<.0001
Error
20
3.5150000
0.1757500
Corrected Total
39
380.9597500
RSquare
Coeff Var
Root MSE
gas Mean
0.990773
1.395209
0.419225
30.04750
Source
DF
Type I SS
Mean Square
F Value
Pr > F
driver
3
280.2847500
93.4282500
531.60
<.0001
car
4
94.7135000
23.6783750
134.73
<.0001
driver*car
12
2.4465000
0.2038750
1.16
0.3715
Source
DF
Type III SS
Mean Square
F Value
Pr > F
driver
3
280.2847500
93.4282500
531.60
<.0001
car
4
94.7135000
23.6783750
134.73
<.0001
driver*car
12
2.4465000
0.2038750
1.16
0.3715
Source
Type III Expected Mean Square
driver
Var(Error) + 2 Var(driver*car) + 10 Var(driver)
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 Fall '08
 Staff

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