Occupancy Royle Counts

Occupancy Royle Count Models

Royle (2004) suggested the use of N-mixture models for estimating poulation size from spatially replicated counts. These models are a form of occupancy model in that the counts are of the animals observed, not the true number of animals existing on a plot. These models are quite similar to the Royle and Nicols (2004) occupancy models where both Poisson and Negative Binomial models are assumed. However, the count models include the actual count of animals for each occasion, so that this formulation should be more efficient than the Royle/Nichols models.

The parameters for the Poisson model are identical to the parameters of the Royle/Nicols model, consisting of the 2 parameters 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 Counts 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 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, and is described in Royle (2004). So, the Royle counts 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 counts 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 counts 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.

To enter the observed counts, the encounter histories are greatly modified. Instead of the usual LDLD... paper, the 2 characters for each occasion are used to enter the actual count, from 00 up to 99. This format limits the maximum count to 99 individuals. The following is an example of the MARK encounter histories file for simulated data using the Poisson model:

/* Occasions=7 r=0.5 lambda=1 */

00000000000000 1;

01010101000100 1;

00000000000000 1;

01000002020000 1;

00000000000000 1;

02000100020001 1;

00010101010201 1;

00000000000000 1;

00000001010001 1;

02010201000201 1;

00000102020101 1;

02030201010300 1;

01010202020102 1;

02000101020001 1;

01010101020100 1;

00010100010000 1;

01000102010202 1;

00000001000000 1;

02010201010001 1;

01010101000000 1;

01010000000000 1;

00000101000001 1;

02020200010101 1;

01010101010001 1;

01010101010100 1;

01000202010000 1;

00000000000000 1;

00000100010100 1;

02000302020202 1;

00000000000000 1;

01020201010102 1;

00000000000000 1;

00000000000100 1;

01010101000100 1;

02010101010101 1;

00000000000000 1;

02000100000000 1;

01000001000001 1;

01010000010001 1;

00000000000000 1;

00010001000000 1;

02040302040404 1;

01010000000101 1;

01010100000100 1;

02020102010100 1;

00010001010100 1;

00000000000000 1;

00000000010001 1;

00000001010001 1;

00000000000000 1;

02020102030001 1;

00000000000000 1;

00010001000000 1;

00000000000100 1;

00000001010000 1;

00000000000000 1;

01010001000001 1;

01010101000101 1;

01010101010200 1;

00030102020001 1;

00000101010101 1;

00010000010101 1;

01000102010302 1;

02000001010202 1;

00000000000000 1;

01000100000000 1;

00000101010101 1;

00000000000000 1;

01000001000101 1;

01000100010102 1;

01000000010101 1;

00010100000100 1;

00000100010001 1;

00000000000000 1;

02010002030201 1;

00000000000000 1;

01000101000000 1;

00000000000000 1;

01020102010100 1;

00000000000000 1;

01000001000000 1;

01010101010100 1;

00000000000000 1;

01010202010100 1;

01000101010100 1;

03010202010003 1;

00000000000000 1;

01010202000201 1;

01010002000101 1;

00010101010100 1;

00000000000000 1;

00010101000200 1;

01030101010000 1;

01010000010101 1;

The MARK files for this example are available at http://welcome.warnercnr.colostate.edu/~gwhite/mark/Examples/POISSON COUNTS.zip. A second example of simulated negative binomial data is available at http://welcome.warnercnr.colostate.edu/~gwhite/mark/Examples/NEGBIN COUNTS.zip.