Occupancy Estimaion Royle/Nichols

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Occupancy Estimaion Royle/Nichols

Royle and Nichols (2003) proposed a model to estimate the occupancy rate or the proportion of area occupied when heterogeneity in detection probability exists as a result of variation in abundance of the organism under study. The key feature of such problems that they exploit is that variation in abundance induces variation in detection probability. Thus, heterogeneity in abundance can be modeled as heterogeneity in detection probability. Moreover, this linkage between heterogeneity in abundance and heterogeneity in detection probability allows one to exploit a heterogeneous detection probability model to estimate the underlying distribution of abundances. Therefore, their method allows estimation of abundance from repeated observations’ of the presence or absence of animals without having to uniquely mark individuals in the population.

The model featured in Royle and Nichols (2003) uses the Poisson distribution as the statistical distribution to model abundance of the observed plots. The 2 parameters of this model are then r and lambda.  Assume that there are N animals on a plot. Then, the probability of observing 1 or more animals on the plot (and thus demonstrating that the plot is occupied) is p =  1 – (1 – r)^N.  The distribution of N across plots is assumed to follow a Poisson distribution with mean lambda.  For the Poisson distribution, the mean equals the variance.  The estimate of psi = probability that a plot is occupied is then the probability estimated from the Poisson distribution that 1 or more animals occur on the plot.

Therefore, the real parameters of the Royle/Nichols Poisson model are r and lambda, and average p [labeled expected value of p-hat, or E(p-hat)] and psi are computed as derived parameters.  To allow model averaging across the various occupancy models, psi is included as a derived parameter for the other occupancy data types as well.

A logical extension of the Royle/Nichols Poisson model is to allow a statistical distribution for N that does not require the mean to equal the variance.  The negative binomial distribution is often used for this purpose.  So, the Royle/Nichols negative binomial model has been included in MARK.  Besides r and lambda, one additional real parameter is required, labeled VarAdd in MARK.  VarAdd is the amount of additional variance added to the mean (lambda) over and above a Poisson distribution.  The advantage of coding the negative binomial in this fashion is that VarAdd = 0 approaches the Poisson distribution, so if VarAdd is fixed to a small value like 0.0001, the resulting set of estimates should be very similar to the Royle/Nichols Poisson model.  Note that fixing VarAdd to zero will usually result in a numerical error, so use a small value instead.  As with the Royle/Nichols Poisson data type, the estimates of psi and E(p-hat) are generated as derived parameters.

Although one might think that the estimated value of lambda is a mean density of animals for equal-sized plots, care must be taken in making this leap of faith.  If the animal populations are geographically and demographically closed as this model assumes, then lambda may be a reasonable estimate of animal density.  However, emigration/immigration between surveys will violate this assumption, and thus preclude the use of lambda as a density estimate.  There are 3 critical assumptions for these models to be useful.  First, animal detections must be independent, as assumed by the formula p =  1 – (1 – r)^N.   Second, N (the number of animals on a plot) is assumed constant across surveys of a plot.  In other words, there is population closure.  Finally,note that r is not a time-varying parameter, so detection probability of a single animal is assumed to be constant across time. Violation of any or all of these assumptions will produce questionable inferences.

An extension of the Royle/Nichols models that incorporate the actual counts are the Royle Count Models. However,even though bot sets of models are parameterized the same, the likelihoods are not comparable because different data are used.