[R-sig-ME] Interpretation of lmer's sigma for binomial data

[Rafael Laboissiere]

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