MacKenzie 2002 - Site Occupancy with Imperfect Detection - Overview

Summary

MacKenzie, Nichols, Lachman, Droege, Royle and Langtimm (2002, Ecology 83(8): 2248-2255) show how to estimate the proportion of sites occupied by a species when it can be present yet go undetected. Repeated visits give detection histories; a two-part likelihood (occupancy $\psi$, detection $p$) recovers $\psi$. Key takeaway: the naive fraction of sites where the species was seen is biased low, and the bias can be corrected only with replicated surveys.

Overview

Research question. Non-detection at a site does not imply absence unless detection probability is 1. How can the fraction of occupied sites be estimated, with covariates and missing visits, from presence/absence surveys?

Contribution. A likelihood that parallels a closed-population mark-recapture model with an extra parameter for presence ($\psi$; the PDF text layer renders it as “c”). It is estimable from summary statistics alone, accommodates site covariates for $\psi$ and time-varying covariates for $p$, handles missing visits, and uses a nonparametric bootstrap for standard errors.

Key results.

• Simulation (243 scenarios, 500 data sets each): $\hat{\psi }$ is reasonably unbiased when $p\ge 0.3$ and $T\ge 5$; poor with $T=2$ unless $p\ge 0.5$; with low $p$, $\hat{\psi }$ tends to 1 and bootstrap SEs become spuriously small.
• Maryland Frogwatch data (29 sites after truncation): American toad $\hat{\psi }\approx 0.49$ vs naive 0.34 (a 44% increase); spring peeper $\hat{\psi }\approx 0.85$ vs naive 0.83.

Why it matters here. This is the base model behind PyMC example E28 and it shares its latent-state structure with N-mixture models (Barker 2018 - Reliability of N-Mixture Models - Overview) and closed-population capture-recapture (Concept - Closed-Population Capture-Recapture and Otis Models).

Motivating PyMC examples

From https://redam94.github.io/pymc-challenges/examples.html:

• E28 - “Imperfect detection: occupancy, N-mixture, capture-recapture” (Swiss breeding-bird survey crossbill and mallard; snowshoe hare capture histories; latent presence / abundance summed out in pm.CustomDist, finite-sample occupancy, naive vs corrected trends, visits x prior confounding).

Source Map

Paper section	Pages	Covered in
Abstract, Introduction (monitoring, metapopulations)	2248-2249	this note; MacKenzie 2002 - Extensions, Design Guidance and PyMC Sketch
Methods: Notation, Basic sampling situation (closure assumptions)	2249	MacKenzie 2002 - Occupancy Model and Likelihood
General likelihood, Eq. 1, bootstrap SE	2249-2250	MacKenzie 2002 - Occupancy Model and Likelihood
Extensions: covariates (Eq. 2-3), missing observations	2250	MacKenzie 2002 - Occupancy Model and Likelihood
Simulation study, Fig. 1	2251-2252	MacKenzie 2002 - Simulation Study and Maryland Anuran Results
Field study: Maryland anurans, Table 1	2252-2253	MacKenzie 2002 - Simulation Study and Maryland Anuran Results
Discussion: caution rules, design, dynamics, co-occurrence, heterogeneity	2253-2254	MacKenzie 2002 - Extensions, Design Guidance and PyMC Sketch
Literature cited, Appendix and Supplement (software, full simulation tables in ESA archive)	2254-2255	folder index (Sources section)

Connections

• Hierarchical view of the latent state: Hierarchical Models.
• Discrete latent states summed out: Concept - Marginalising Latent Discrete Counts for HMC.
• Zero-inflation analogy: Monsters and Mixtures.

See Also

• MacKenzie 2002 - Occupancy Model and Likelihood
• Barker 2018 - Reliability of N-Mixture Models - Overview
• Silverman 2020 - Multiple Systems Analysis for Modern Slavery - Overview