Topic 30: Random Effects
Outline
• One-way random effects model
– Data
– Model
– Inference
Data for one-way random
effects model
• Y, the response variable
• Factor with levels i = 1 to r
• Yij is the jth observation in cell i, j = 1
to ni
• Almost identical model structure to
earlier one-way ANOVA.
• Difference in level of inference
Level of Inference
• In one-way ANOVA, interest was in
comparing the factor level means
• In random effects scenario, interest is
in the pop of factor level means, not
just the means of the r study levels
• Need to make assumptions about
population distribution
• Will take “random” draw from pop of
factor levels for use in study
KNNL Example
• KNNL p 1036
• Y is the rating of a job applicant
• Factor A represents five different
personnel interviewers (officers), r=5
• n=4 different applicants were
interviewed by each officer
• The interviewers were selected at
random and the applicants were
randomly assigned to interviewers
Read and check the data
data a1;
infile 'c:\...\CH25TA01.DAT';
input rating officer;
proc print data=a1;
run;
The data
Obs
1
2
3
4
5
6
7
8
9
10
rating
76
65
85
74
59
75
81
67
49
63
officer
1
1
1
1
2
2
2
2
3
3
The data
Obs
11
12
13
14
15
16
17
18
19
20
rating
61
46
74
71
85
89
66
84
80
79
officer
3
3
4
4
4
4
5
5
5
5
Plot the data
title1 'Plot of the data';
symbol1 v=circle i=none c=black;
proc gplot data=a1;
plot rating*officer/frame;
run;
Find and plot the means
proc means data=a1;
output out=a2 mean=avrate;
var rating;
by officer;
title1 'Plot of the means';
symbol1 v=circle i=join c=black;
proc gplot data=a2;
plot avrate*officer/frame;
run;
Random effects model
• Yij = μi + εij
Key
– the μi are iid N(μ, σμ2)
difference
– the εij are iid N(0, σ2)
– μi and εij are independent
• Yij ~ N(μ, σμ2 + σ2)
• Two sources of variation
• Observations with the same i are not
independent, covariance is σμ2
Random effects model
• This model is also called
– Model II ANOVA
– A variance components model
• The components of variance are
σμ2 and σ2
• The models that we previously
studied are called fixed effects
models
Random factor effects
model
• Yij = μ + i + εij
• i ~ N(0, σμ2) *****
• εij ~ N(0, σ2)
• Yij ~ N(μ, σμ2 + σ2)
Parameters
• There are three parameters in these
models
–μ
– σ μ2
– σ2
• The cell means (or factor levels) are
random variables, not parameters
• Inference focuses on these variances
Primary Hypothesis
• Want to know if H0: σμ2 = 0
• This implies all mi in model are equal
but also all mi not selected for
analysis are also equal.
• Thus scope is broader than fixed
effects case
• Need the factor levels of the study to
be “representative” of the population
Alternative Hypothesis
• We are sometimes interested in
estimating σμ2 /(σμ2 + σ2)
• This is the same as σμ2 /σY2
• In some applications it is called the
intraclass correlation coefficient
• It is the correlation between two
observations with the same I
• Also percent of total variation of Y
ANOVA table
• The terms and layout of the anova
table are the same as what we used
for the fixed effects model
• The expected mean squares (EMS)
are different because of the
additional random effects but F test
statistics are the same
• Be wary that hypotheses being
tested are different
EMS and parameter
estimates
•
•
•
•
•
E(MSA) = σ2 + nσμ2
E(MSE) = σ2
We use MSE to estimate σ2
Can use (MSA – MSE)/n to estimate σμ2
Question: Why might it we want an
alternative estimate for σμ2?
Main Hypotheses
• H 0: σ μ 2 = 0
• H 1: σ μ 2 ≠ 0
• Test statistic is F = MSA/MSE with r-1
and r(n-1) degrees of freedom, reject
when F is large, report the P-value
Run proc glm
proc glm data=a1;
class officer;
model rating=officer;
random officer/test;
run;
Model and error output
Source DF
Model
4
Error 15
Total 19
MS
394
73
F
P
5.39 0.0068
Random statement output
Source
Type III Expected MS
officer Var(Error) + 4 Var(officer)
Proc varcomp
proc varcomp data=a1;
class officer;
model rating=officer;
run;
Output
MIVQUE(0) Estimates
Variance Component
Var(officer)
Var(Error)
rating
80.41042
73.28333
Other methods are available
for estimation, minque is
the default
Proc mixed
proc mixed data=a1 cl;
class officer;
model rating=;
random officer/vcorr;
run;
Output
Covariance Parameter Estimates
Cov Parm
officer
Residual
Est Lower
80.4
73.2
24.4
39.9
Upper
1498
175
80.4104/(80.4104+73.2833)=.5232
Output from vcorr
Row Col1
1 1.0000
2 0.5232
3 0.5232
4 0.5232
Col2
0.5232
1.0000
0.5232
0.5232
Col3
0.5232
0.5232
1.0000
0.5232
Col4
0.5232
0.5232
0.5232
1.0000
Other topics
• Estimate and CI for μ, p1038
– Standard error involves a
combination of two variances
– Use MSA instead of MSE → r-1 df
• Estimate and CI for σμ2 /(σμ2 + σ2), p1040
• CIs for σμ2 and σ2, p1041-1047
– Available using Proc Mixed
Applications
• In the KNNL example we would like
σμ2 /(σμ2 + σ2) to be small, indicating
that the variance due to interviewer
is small relative to the variance due
to applicants
• In many other examples we would
like this quantity to be large,
– e.g., Are partners more likely to be
similar in sociability?
Last slide
• Start reading KNNL Chapter 25
• We used program topic30.sas to
generate the output for today