This function fits the latent abundance mixture model described in
Royle and Nichols (2003).
The number of animals available for detection at site \(i\) is modelled as Poisson:
$$N_i \sim Poisson(\lambda_i)$$
We assume that all individuals at site \(i\) during sample \(j\) have identical
detection probabilities, \(r_{ij}\), and that detections are independent. The species
will be recorded if at least one individual is detected. Thus, the detection
probability for the species is linked to the
detection probability for an individual by
$$p_{ij} = 1 - (1 - r_{ij}) ^ {N_i}$$
Note that if \(N_i = 0\), then \(p_{ij} = 0\), and increasing values of \(N_i\) lead to higher values of \(p_{ij}\) The equation for the detection history is then:
$$y_{ij} \sim Bernoulli(p_{ij})$$
Covariates of \(\lambda_i\) are modelled with the log link
and covariates of \(r_{ij}\) are modelled with the logit link.