To fit this model by classical methods, we reduce the number of parameters to be estimated by forming an integrated likelihood function. This is accomplished by integrating the joint probability of yi = (yi1, yi2,..., yi1o) and Ni over the admissible values of Ni to obtain the marginal probability of the counts yi given only the metapopulation-level parameters A and p:
In this equation f denotes the pmf of the binomial distribution and g denotes the pmf of the Poisson distribution. The marginal probability q(yi|A,p) cannot be expressed in closed form; therefore, in practice this probability is computed numerically by substituting a sufficiently large number for the upper index of summation in Eq. (8.3.1).
To compute maximum likelihood estimates of A and p, we assume independence among sample locations and find the value of these parameters that maximizes the likelihood function L(A,p) = Hn=i q(yi|A,p). Panel 8.1 contains R code for computing the logarithm of this likelihood function. When unconstrained optimizers, such as R's built-in functions nlm or optim, are used, a 1-to-1 transformation of the model's parameters may be needed to ensure that the estimates are confined to their admissible range of values. For example, one normally would compute the MLE of log(A) and then obtain the MLE of A by inverting the transformation. In this way, the MLE of A is guaranteed to be strictly positive (as required in the model), whereas the MLE of log(A) may be positive or negative. Transformation is also useful when estimating detection probabilities. For example, one normally would compute the MLE of logit(p) to ensure that an estimate of p is confined to the (0,1) interval. Such transformations were used in the analysis of ovenbird point counts; this yielded the following MLEs (and 95 percent confidence limits): A = 2.12 (1.66, 2.69), p = 0.32 (0.27, 0.37). Based on these estimates, we conclude that in any given sampling occasion only 1 ovenbird is detected for every 3 that are present. In addition, we estimate that the average abundance of ovenbirds is 2.12 birds per sample location.
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