[R] Want to fit random intercept in logistic regression (testing lmer and glmmML)

[Paul Johnson]

Greetings.  Here is sample code, with some comments. It shows how I
can simulate data and estimate glm with binomial family when there is
no individual level random error, but when I add random error into the
linear predictor, I have a difficult time getting reasonable estimates
of the model parameters or the variance component.

There are no clusters here, just individual level responses, so
perhaps I am misunderstanding the translation from my simple mixed
model to the syntax of glmmML and lmer.  I get roughly the same
(inaccurate) fixed effect parameter estimates from glmmML and lmer,
but the information they give on the variance components is quite
different.

Thanks in advance.

Now I paste in the example code

### Paul Johnson <pauljohn at ku.edu>
### 2006-03-08

N <- 1000
A <- -1
B <- 0.3


x <- 1 + 10 * rnorm(N)
eta <- A + B * x

pi <- exp(eta)/(1+exp(eta))

myunif <- runif(N)

y <- ifelse(myunif < pi, 1, 0)

plot(x,y, main=bquote( eta[i] == .(A) +   .(B) * x[i] ))

text ( 0.5*max(x), 0.5, expression( Prob( y[i] == 1) == frac( 1 , 1 +
exp(-eta[i] ))))


myglm1 <- glm ( y ~ x, family=binomial(link="logit") )
summary(myglm1)

## Just for fun....
myglm2 <- glm(y~x, family=quasibinomial)
summary(myglm2)



### Mixed model: random intercept with large variance

eta <- A + B * x + 5 * rnorm(N)
pi <- exp(eta)/(1+exp(eta))
myunif <- runif(N)
y <- ifelse(myunif < pi, 1, 0)

plot(x,y, main=bquote( eta[i] == .(A) +   .(B) * x[i] + u[i]))

text ( 0.5*max(x), 0.5, expression( Prob( y[i] == 1) == frac( 1 , 1 +
exp(-eta[i] ))))

### Parameter estimates go to hell, as expected
myglm3 <- glm ( y ~ x, family=binomial(link="logit") )
summary(myglm3)

### Why doesn't quasibinomial show more evidence of the random intercept?
myglm4 <- glm(y~x, family=quasibinomial)
summary(myglm4)



# Can I estimate with lmer?


library(lme4)

### With no repeated observations, what does lmer want?
### Clusters of size 1 ?
### In lme, I'd need random= ~ 1

idnumber <- 1: length(y)

mylmer1 <- lmer( y ~ x + ( 1 | idnumber ), family=binomial, method="Laplace" )
summary(mylmer1)


### Glmm wants clusters, and I don't have any clusters with more than
1 observation
###

library(glmmML)


myglmm1 <- glmmML(y~x, family=binomial, cluster = idnumber )

summary(myglmm1)

--
Paul E. Johnson
Professor, Political Science
1541 Lilac Lane, Room 504
University of Kansas