Lecture 22:
Random effects models
BMTRY 701
Biostatistical Methods II
Independence Assumption
 All of the regression assumptions we’ve
discussed thus far assume independence
 That is, patients (or other ‘units’) have outcomes
that are unrelated
 But what if they are?
•
•
•
•
the same person is measured multiple times
people from the same house are studied
people treated in the same hospital are studied
different tumors within the same patient are evaluated
 In all of those examples, the independence
assumption ‘falls apart’
How to deal with it?
 Two main approaches:
 Random effects model:
• include a ‘random intercept’ to account for correlation
• individuals who are ‘linked’ (i.e., from same house,
hospital, etc.) receive the same intercept
 Generalized estimating equations (GEE)
• model the correlation as part of the regression
• two part modeling:
 mean model
 covariance model
Nurse staffing in ICU example
 Hospitals in MD from 1994-1996, discharge data
 All patients with abdominal aortic surgery (AAS)
 Goal: evaluate the association between the
nurse-to-patient ratio in the ICU for risk of
medical and surgical complications after AAS.
 Data:
• patient outcomes (complications)
• nurse:patient ratio
 Issue: patients treated within the same hospital
are likely to have correlated outcomes
Random effects modeling
Standard logistic
model
Random effects
logistic model
logit( yi )   0  1 Nursei
logit( yij )   0  b j  1 Nurseij
b j ~ N (0,  )
2
Adding in the random effect
 Conditional on the random effect, the
observations within a hospital are independence
 Hence, independence is restored!
 Even so, random effects are considered
‘nuisance parameters’
• we generally don’t care about them
• they are necessary, but not interesting
 Our primary interest is still in β1
2
4
y
6
8
What does this look like? Linear Regression
0
2
4
6
x
8
10
Fitting Random Effects Models in R
library(nlme)
re.reg <- lme(y ~ x, random=~1|hospid)
o.reg <- lm(y~x)
bi <- re.reg$coefficients$random$hospid
b0 <- re.reg$coefficients$fixed[1]
b1 <- re.reg$coefficients$fixed[2]
par(mfrow=c(1,1))
plot(x,y)
abline(o.reg)
for(i in 1:20) {
lines(0:10, b0+b1*(0:10) + bi[i], col=2)
}
abline(o.reg, lwd=2)
Random Effects?
3
2
1
0
Frequency
4
5
Histogram of bi
-1.5
-1.0
-0.5
0.0
0.5
bi
1.0
1.5
2.0
Interpretation
 Recall “nuisance” parameters
 In most cases, we do not care about random
intercepts
 “Fixed” effects are interpreted in the same way
as in a standard regression model
Stata
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


xtreg: random effects linear regression
xtlogit: random effects logistic regression
xtpoisson: random effects poisson regression
stcox, ...shared(id): random effects Cox
regression
 Also, ‘cluster’ option in many regression
commands in Stata
Applied example
 http://www.acponline.org/clinical_information/jou
rnals_publications/ecp/sepoct01/pronovost.htm