[ADMB Users] Variance of ZINB - and partitioning variance

dave fournier davef at otter-rsch.com
Thu Jan 31 11:49:11 PST 2013

I read in the Getting Started with glmmADMB guide that using
family="nbinom2" uses the
parameterization Var = mu(1+ (mu/k))                    (1)

I assume mu is a mean parameter, but what is k ? And how do these relate
to the estimates from the model output  ?:

You don't seem to have received an answer to this question.  Think of 
"i" as indexing the observations.
Let mu_i be the predicted mean for the i'th measurement.  Then the above 
formula is more
informatively written as

    Var_i = mu_i * (1+mu_i/k)

so there are a lot of mu_i's and only 1 k.  This k is simply a way of 
parameterizing the relationship
between the mu_i and the Var_i.  If k is smaller Var_i is larger. There 
are models for producing
negative binomial outcomes where k has a "physical" interpretation, but 
one may prefer to have it
simply defined by the relationship (1). It depends on your point of 
view. Taking this point of view
permits you to wonder about the possibility of more general 
relationships between var_i and mu_i
such as

          Var_i = mu_i*(tau + a*mu)   , tau>=1, a>=0   (2)

where a=1/k extends the possible values of k
or even

          Var_i = mu_i*(tau + a*mu^b)   , tau>=1, a>=0   (3)

As usual it would be interesting to test all of this stuff on a large 
number of data sets. Maybe one day we will
get this automated.  As for ICC I guess now that you know var_i it 
should be not too difficult to calculate ICC
(whatever it is).

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