[Paper Review] Two-stage approaches to the analysis of occupancy data I: The homogeneous case

Natalie Karavarsamis, Richard Huggins|arXiv (Cornell University)|Mar 30, 2018

Spatial and Panel Data AnalysisEconomics, Econometrics and Finance10 references4 citations

TL;DR

This paper proposes a two-stage inference approach for homogeneous occupancy models using orthogonal parameterization and partial likelihood to simplify maximum likelihood estimation. By transforming parameters to ensure orthogonality between detection and occupancy probabilities, it enables separate estimation via conditional and partial likelihoods, yielding efficient, analytically tractable estimators with minimal computational burden and only a small loss in efficiency compared to full likelihood methods.

ABSTRACT

Occupancy models are used in statistical ecology to estimate species dispersion. The two components of an occupancy model are the detection and occupancy probabilities, with the main interest being in the occupancy probabilities. We show that for the homogeneous occupancy model there is an orthogonal transformation of the parameters that gives a natural two-stage inference procedure based on a conditional likelihood. We then extend this to a partial likelihood that gives explicit estimators of the model parameters. By allowing the separate modelling of the detection and occupancy probabilities, the extension of the two-stage approach to more general models has the potential to simplify the computational routines used there.

Motivation & Objective

• To simplify the complex full likelihood estimation in occupancy models, which involves joint modeling of detection and occupancy probabilities.
• To reduce computational burden in models with many covariates, where full likelihood becomes infeasible due to the exponential growth in possible models.
• To develop a two-stage inference procedure that allows separate modeling of detection and occupancy probabilities, enhancing computational stability and efficiency.
• To explore the use of orthogonal parameterization and partial likelihood to derive explicit, analytically tractable estimators for occupancy and detection probabilities.

Proposed method

• Apply an orthogonal transformation of parameters, defining η = ψθ (probability of detecting an occupied site) and retaining p (detection probability per visit), which renders the parameters asymptotically independent.
• Use a conditional likelihood for p based on the number of detections at occupied sites, yielding a binomial likelihood for detection probabilities.
• Derive a partial likelihood that decomposes into two independent binomial likelihoods for η and p, enabling explicit analytic estimators.
• Implement the two-stage approach: first estimate p via conditional likelihood, then estimate ψ via the transformed parameter η, avoiding joint maximization.
• Validate the method through simulations and real data application to frog detection data, comparing efficiency and bias to full likelihood.
• Use the R package VGAM to implement the conditional likelihood approach and compare results with full likelihood estimation.

Experimental results

Research questions

• RQ1 Can orthogonal parameterization in homogeneous occupancy models lead to a natural two-stage inference procedure that simplifies maximum likelihood estimation?
• RQ2 How does the partial likelihood approach compare to full likelihood in terms of efficiency and computational feasibility?
• RQ3 What is the impact of small detection probabilities and low numbers of survey occasions on estimator bias and standard error accuracy?
• RQ4 Can the two-stage method maintain high efficiency while reducing computational complexity in models with many covariates?

Key findings

• The orthogonal transformation of parameters η = ψθ and p results in asymptotically independent maximum likelihood estimators, simplifying inference and reducing computational complexity.
• The detection probability estimator p derived from conditional likelihood is equivalent to a standard binomial estimator, enabling straightforward implementation.
• The partial likelihood approach yields explicit analytic estimators for both ψ and p, with the estimator for ψ achieving over 90% relative efficiency compared to full likelihood.
• Simulations show no significant bias in either method, and estimated standard errors are reliable, though slight overestimation occurs for small p and τ.
• In the real data application to 27 sites with τ=4, the partial likelihood gave p̃=0.889 and ψ̃=0.556, while full likelihood gave p̂=0.780 and ψ̂=0.557, indicating a notable difference in detection probability estimates.
• The discrepancy in p estimates suggests potential heterogeneity in detection probabilities across survey occasions, as the partial likelihood ignores first-visit information used in the full likelihood.