Exact Likelihoods for N-mixture models with Time-to-Detection Data

TL;DR

This paper develops explicit likelihood formulas for N-mixture models incorporating time-to-detection data, showing how such data can improve abundance estimates under certain conditions, and provides practical tools for implementation.

Contribution

The paper introduces closed-form likelihood expressions for time-to-detection data in N-mixture models, enhancing estimation accuracy and providing an R package for practical application.

Findings

Recording times-to-detection can improve estimation precision when no double counting occurs.

Times-to-detection data do not add information in the presence of double counting with exponential arrival times.

New likelihood formulas are derived from order statistics theory for these models.

Abstract

This paper is concerned with the formulation of $N$-mixture models for estimating the abundance and probability of detection of a species from binary response, count and time-to-detection data. A modelling framework, which encompasses time-to-first-detection within the context of detection/non-detection and time-to-each-detection and time-to-first-detection within the context of count data, is introduced. Two observation processes which depend on whether or not double counting is assumed to occur are also considered. The main focus of the paper is on the derivation of explicit forms for the likelihoods associated with each of the proposed models. Closed-form expressions for the likelihoods associated with time-to-detection data are new and are developed from the theory of order statistics. A key finding of the study is that, based on the assumption of no double counting, the likelihoods associated with times-to-detection together with count data are the product of the likelihood for the counts alone and a term which depends on the detection probability parameter. This result demonstrates that, in this case, recording times-to-detection could well improve precision in estimation over recording counts alone. In contrast, for the double counting protocol with exponential arrival times, no information was found to be gained by recording times-to-detection in addition to the count data. An R package and an accompanying vignette are also introduced in order to complement the algebraic results and to demonstrate the use of the models in practice.