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

[Rolf Turner]

I would like to second the request for enlightenment on this issue.
In particular, how does this fit with the recent posting from Jarrod
Hadfield:

> 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.


Hadfield's posting makes perfectly good sense to me.  I have done
simulation experiments similar to that which he suggested, and have
found what he said I would find.  Also I did some simple theoretical
manipulations which further confirmed his assertions.

So what is the quasibinomial family ***doing***, then?

	cheers,

		Rolf Turner

On 2/08/2010, at 10:27 AM, Rafael Laboissiere wrote:

> 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
> 
> _______________________________________________
> R-sig-mixed-models at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models

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