[R-sig-ME] reference on unidentifiability of observation-level variance in Bernoulli variables?

[Kent Holsinger]

Ben,

Here's how I think about it. (I know this isn't printed anywhere, but 
maybe it will be good enough for your purposes.)

Suppose we put a beta prior on p, Be(alpha*pi, alpha*(1-pi)), where 
alpha = (1-theta)/theta. Then the mean of p is pi and its variance is 
theta*pi*(1-pi). If we have only one observation, we can't estimate both 
pi and theta. The same argument would hold for *any* continuous 
distribution that has more than one parameter, i.e., any non-degenerate 
distribution.

Kent

On 11/15/11 3:53 PM, Ben Bolker wrote:
>    Hi folks,
>
>    It has often been discussed on this list and other R-help lists that
> overdispersion is (broadly speaking) unidentifiable for an ungrouped
> Bernoulli response variable (if some kind of grouping can be imposed,
> then it becomes identifiable). For example:
>
>
> https://stat.ethz.ch/pipermail/r-help/2008-February/154058.html
> http://finzi.psych.upenn.edu/R/Rhelp02a/archive/91242.html
>
> Peter Dalgaard:
>
>> The point being that you  cannot have a distribution on {0, 1} where>
> the variance is anything but  p(1-p) where p is the mean; if you put>  a
> distribution on p and integrate  it out, you still end up with the>
> same variance.
>
>     I am curious if anyone has a *printed* (book or peer-reviewed
> article) for this, or even can point to notes that actually go to the
> trouble of doing the integration and proving the statement.  I have
> looked in some of the usual places (MASS; McCullough, Searle, and
> Neuhaus; Zuur et al.) and haven't come across anything.
>
>    Something that showed the computation of the intraclass correlation
> and worked out the mean of the logistic-normal-binomial distribution as
> a function of the mean and variance of the underlying normal
> distribution would be nice too, although I'm guessing it doesn't have a
> straightforward analytical solution ...
>
>    I'm hoping I haven't missed anything obvious -- on the other hand, if
> I have it will be easy for someone to answer (and please don't be
> offended if it was in your book, which was on my shelf all the time and
> I forgot to look there ...)
>
>   thanks
>    Ben Bolker
>
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>


-- 
Kent E. Holsinger                kent at darwin.eeb.uconn.edu
                                  http://darwin.eeb.uconn.edu
-- Department of Ecology&  Evolutionary Biology
-- University of Connecticut, U-3043
-- Storrs, CT   06269-3043