Statistics 512: Applied Linear Models
Topic 9
Topic Overview
This topic will cover
•
Random vs. Fixed Effects
•
Using
E
(
MS
) to obtain appropriate tests in a Random or Mixed Effects Model.
Chapter 25: Oneway Random Effects Design
Fixed Effects vs Random Effects
•
Up to this point we have been considering “fixed effects models”, in which the levels of
each factor were fixed in advance of the experiment and we were interested in differences
in response among those specific levels.
•
Now we will consider “random effects models”, in which the factor levels are meant
to be representative of a general population of possible levels.
We are interested in
whether that factor has a significant effect in explaining the response, but only in a
general way. For example, we’re not interested in a detailed comparison of level 2 vs.
level 3, say.
•
When we have both fixed and random effects, we call it a “mixed effects model”. The
main SAS procedure we will use is called “
proc mixed
” which allows for fixed and
random effects, but we can also use
glm
with a
random
statement.
We’ll start first
with a single random effect.
•
In some situations it is clear from the experiment whether an effect is fixed or random.
However there are also situations in which calling an effect fixed or random depends
on your point of view, and on your interpretation and understanding. So sometimes it
is a personal choice. This should become more clear with some examples.
Data for oneway design
•
Y
, the response variable
•
Factor with levels
i
= 1 to
r
•
Y
i,j
is the
j
th observation in cell
i
,
j
= 1 to
n
i
•
A balanced design has
n
=
n
i
1
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KNNL Example
•
KNNL page 1036 (
knnl1036.sas
)
•
Y
is the rating of a job applicant
•
Factor
A
represents five different personnel interviewers (officers),
r
= 5 levels
•
n
= 4
different
applicants were randomly chosen and interviewed by each interviewer
(i.e. 20 applicants) (applicant is
not
a factor since no applicant was interviewed more
than once)
•
The interviewers were selected at random
from the pool of interviewers and the appli
cants were randomly assigned to interviewers.
•
Here we are not so interested in the differences between the five interviewers that
happened to be picked (i.e. does Joe give higher ratings than Fred, is there a difference
between Ethel and Bob). Rather we are interested in quantifying and accounting for
the effect of “interviewer” in general. There are other interviewers in the “population”
(at the company) and we want to make inference about them too.
•
Another way to say this is that with fixed effects we were primarily interested in the
means
of the factor levels (and the differences between them). With random effects,
we are primarily interested in their
variances
.
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 Fall '08
 Staff
 Statistics, Variance, random effects, M SE, M SA

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