Ecological Archives E084-015-A1

J. Andrew Royle and James D. Nichols. 2003. Estimating abundance from repeated presence–absence data or point counts. Ecology 84:777–790.

Appendix A. Simulation study design.

Although the avian point-count data analyzed for the example involved only 50 spatial samples, the fact that 11 replicate samples in time were available provides considerable information about the model parameters. For most problems, we feel that so many temporal replicates is unreasonable and believe that higher degrees of spatial replication are both more realistic and more attainable.

We chose values of $ \lambda$ to provide a broad range of values of occupancy rate, $ \psi$. Under the Poisson model we have that $ \psi$ = - log(1 - $ \lambda$ ). To achieve values of $ \psi$ = (0.99, 0.9, 0.8, 0.7, 0.6, 0.5, 0.4) we considered values of $ \lambda$ = (4.61, 2.30, 1.61, 1.20, 0.92, 0.69, 0.51). For each value of $ \lambda$, we chose r = 0.1 and r = 0.2 which dictates values of $ \bar{p}_{c}^{}$ which we feel are typical, or relatively low, for many capture-recapture and occupancy studies (Table 1, Appendix B).

We considered a broad range of values for T, the number of sampling occasions, and R, the number of sampling locations. Here we summarize the results only for T = 5 and for values of R = 500 (Table 1, Appendix B) and R = 200 (Table 2, Appendix B) because we feel that such values represent a compromise between achievable sampling designs in most monitoring and field study situations, and sufficient sample sizes so as to permit effective estimation.

In addition to the defining characteristics of each simulation (i.e., r, $ \lambda$, $ \psi$), Tables 1 and 2 (Appendix B) contain the implied values of $ \sigma_{c}^{}$ and $ \bar{p}_{c}^{}$. For the estimators of $ \lambda$ and $ \psi$, we present the mean and median of estimates from 2000 simulated data sets. In order to gage the impact of heterogeneity induced by varying abundance, we also estimated the occupancy rate under the ``constant p'' model described by MacKenzie et al. (2002) for each of the 2000 simulated data sets. The mean value of these estimates also is given in Tables 1 and 2 (Appendix B).

For the negative binomial simulations, we summarize results for data generated (Table 3, Appendix B) using $ \mu$ = (1.61, 0.69), only for r = 0.2, T = 5 and R = 200. For each data set generated under the negative binomial model, we fit both the Poisson and negative binomial models in order to assess the cost of neglecting the additional over-dispersion, in terms of estimating mean abundance ($ \mu$ under the negative binomial and $ \lambda$ under the Poisson model) and occupancy rate, $ \psi$. The over-dispersion parameter, a, was selected to suggest mean/variance ratios of 1.2, 2, and 4, so that the variance is 20% , 200%, and 400% larger than the Poisson case. We'll call these levels low, moderate, and extreme over-dispersion. Thus, for each value of $ \mu$, we used the following values of the negative binomial parameter a:

  low moderate extreme
$ \mu$ = 0.69 3.45 0.69 0.23
$ \mu$ = 1.61 8.05 1.61 0.54

Literature cited

MacKenzie, D. I., J. D. Nichols, G. B. Lachman, S. Droege, J. A. Royle, and C. A. Langtimm. 2002. Estimating site occupancy rates when detection probabilities are less than one. Ecology 83:2248–2255.



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