Interpretation of lmer's sigma for binomial data

Over-dispersion does not occur with a binary response variable so you  
don't need to test for it.

This does not mean that between-datum heterogeneity in the probability  
of success is absent, only that it cannot be observed. For example,  
take 1000 random draws from a binomial distribution with constant  
probability (0.5):

table(rbinom(1000, 1, 0.5))

and compare the frequency of outcomes with a 1000 draws from 1000  
binomial distributions with different probabilities of success (but  
with mean = 0.5)

table(rbinom(1000, 1, runif(1000)))

The data look the same, and so the between-datum heterogeneity  
(residual variance if you like) although it may exist cannot be  
estimated from the data.

Dear colleagues,

First of all, I apologize if this is not the appropriate forum for my
question.

I am trying to understand the meaning of the sigma slot in the
summary.mer class of the lme4 package (version 0.999375-34).  From what I
read in this mailing list's archive and elsewhere, sigma gives an
indication of under- or over-dispersed data.

So, I ran the following code:

   # Generate 20 repetitions of binary responses for size of 40
   # and a theoretical probability of 0.5
   repetitions <- 20
   size <- 40
   set.seed (1)
   success <- rbinom (repetitions, size, 0.5)

   # Compute the failure rate
   failure <- size - success

   # Add a dummy random effect variable, otherwise lmer barks
   block <- c (rep ("A", 10), rep ("B", 10))

   # Fit a binomial model using the actual counts
   library (lme4)
   m.binom <- lmer (cbind (success, failure) ~ 1 + (1 | block), family = binomial)

   # Fit a quasibinomial model using the proportions
   m.quasi <- lmer (success / (success + failure) ~ 1 + (1 | block), family = quasibinomial)

As expected, the value of summary(m.binom)@sigma is 1.  Since the data is
generated with rbinom and no under- or over-dispersion has been
explicitly introduced in it, I would expect that sigma for m.quasi is
also close to 1.  However, I see:

summary(m.quasi)@sigma

   [1] 0.07716662

I am sure I am terribly failing to understand something here.  What is
it?

Thanks in advance for your help.

Best regards,

Rafael Laboissiere