We see that the parameter 0 contributes to the intercept in this formulation, and therefore trades off with a function of r. While this precise result does not hold with the logit link function, it does suggest that we should be concerned with the identifiability of the additional parameter. Fortunately, consideration of the complementary log-log (cloglog) link suggests a parameterization that should yield identifiability of the extra-binomial parameter. Namely, in order to have 0 be multiplicative with log(Ni) in Eq. (4.4.2), we require
We refer to this model as the quasi-binomial detection model (Royle, 2008b). The likelihood specification in R is shown in Panel 4.1. We will fit this model to data shortly. Other models for detection might be reasonable but we have not explored or cataloged the possibilities. Royle and Link (2006) describe a general form of dependence, and we discuss the case of 'no dependence' in Section 4.5.
We have so far only considered the case where local abundance is assumed to be Poisson. However, many choices for g are possible. One that is usually used to accommodate overdispersion relative to the Poisson is the negative binomial. A convenient parameterization is that based on the mean,
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