From plummer@iarc.fr  Wed Sep 22 10:42:18 1999
From: plummer@iarc.fr (plummer@iarc.fr)
Date: Wed, 22 Sep 1999 11:42:18 +0200 (MET DST)
Subject: quasi-likelihood in glm (PR#284)
Message-ID: <199909220942.LAA16986@pubhealth.ku.dk>

I have been using quasi-likelihood to analyze some overdispersed lesion
count data which requires the variance function "mu^2" (specifically,
I am using glm with the family quasi(var="mu^2", link="log")). Some of
the counts are zero which led to problems fitting the model in R.

1) glm was unable to find starting values
2) after I supplied initial values from the fit of a Poisson
   model, glm could not proceed because the deviance was infinite.

Splus 5.0 had no problems fitting the model.  I was able to
reproduce the Splus results by editing the quasi() family
generator and using the unnormalized quasi-likelihood

- log(mu) - y/mu                  (*)

in place of the normalized version

log(y/mu) - (y - mu)/mu            (**)

Further investigation of Splus shows how it overcomes these
two problems
1) A fudge factor (in this case 0.167) is added to zero observations 
   so that they can be used as starting values. 
2) The contribution to the deviance is (*) for zero observations
   and (**) for the rest.

I suggest that R also uses this approach for quasi-likelihood
models.
Martyn

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quasi-likelihood in glm (PR#284)
plummer@iarc.fr