title: Estimating negative binomial parameters from occurrence data with detection times.
abstract: The negative binomial distribution is a common model for the analysis of count data in biology and ecology. In many applications, we may not observe the complete frequency count in a quadrat but only that a species occurred in the quadrat. If only occurrence data are available then the two parameters of the negative binomial distribution, the aggregation index and the mean, are not identifiable. This can be overcome by data augmentation or through modeling the dependence between quadrat occupancies. Here, we propose to record the (first) detection time while collecting occurrence data in a quadrat. We show that under what we call proportionate sampling, where the time to survey a region is proportional to the area of the region, that both negative binomial parameters are estimable. When the mean parameter is larger than two, our proposed approach is more efficient than the data augmentation method developed by Solow and Smith (, Am. Nat. 176, 96-98), and in general is cheaper to conduct. We also investigate the effect of misidentification when collecting negative binomially distributed data, and conclude that, in general, the effect can be simply adjusted for provided that the mean and variance of misidentification probabilities are known. The results are demonstrated in a simulation study and illustrated in several real examples.
authors.affiliation: Institute of Statistics, National Chung Hsing University, Taiwan. wenhan@nchu.edu.tw.
authors.forename: Wen-Han
authors.initials: WH
authors.lastname: Hwang
authors.affiliation: Department of Mathematics and Statistics, The University of Melbourne, Australia.
authors.forename: Richard
authors.initials: R
authors.lastname: Huggins
authors.affiliation: School of Mathematics and Statistics Evolution & Ecology Research Centre, The University of New South Wales, Australia.
authors.forename: Jakub
authors.initials: J
authors.lastname: Stoklosa
mesh_terms: D016010:Binomial Distribution; D001699:Biometry / Q000379:methods*; D003198:Computer Simulation; D006801:Humans; D015233:Models, Statistical; D011336:Probability; D015983:Selection Bias; D013997:Time Factors
keywords: Aggregation index; Cost analysis; Misidentification; Negative binomial distribution; Presence-absence data