MacKenzie 2002 - Extensions, Design Guidance and PyMC Sketch

Summary

The discussion gives red flags for untrustworthy $\hat{\psi }$, minimum-effort guidance for visit numbers, and four extensions (robust design, colonisation/extinction, species co-occurrence, heterogeneity). A PyMC sketch shows how the latent presence indicator is marginalised so NUTS can be used.

Overview

The paper positions occupancy as a cheaper state variable than abundance for large-scale monitoring and metapopulation work. Its discussion is the source of widely cited design rules of thumb.

Main Content

Caution rule (Discussion)

Doubt an estimate of $\psi$ very close to 1 when per-visit detection is low ($<0.15$), especially with few visits ($<7$): the data cannot distinguish a genuinely absent species from one merely not detected.

Design guidance.

• With only two visits per site, the true occupancy should exceed 0.7 and per-visit detection should exceed 0.3, and even then precision may be low.
• More visits improve precision and (at low $p$) accuracy. Multiple searches in a single trip (e.g. two 3-minute calling surveys in one night) can substitute when travel is costly.
• If many patches must be surveyed, repeat-visit a subset to estimate $p$ (possibly as a function of site covariates) and apply it to single-visit sites.
• Think about the likely analysis when designing the survey.

Extensions the authors foresee.

1. Dynamics: combine with Pollock’s robust design. Within each closed period estimate $\psi$; across periods model colonisation and extinction probabilities, analogous to open-population birth and death. This removes the need for $p=1$ or stationarity assumed by earlier metapopulation work (Hanski 1992, 1994; Erwin et al. 1998).
2. Species co-occurrence: parameterise ${\psi }_{AB}$ in addition to ${\psi }_A,{\psi }_B$. Parameters grow exponentially: three species need four extra (${\psi }_{AB},{\psi }_{AC},{\psi }_{BC},{\psi }_{ABC}$); six species need 57 extra.
3. Heterogeneity: unmodelled site heterogeneity in $\psi$ or $p$ could be handled with the finite-mixture approach of Pledger (2000), jackknife (Burnham and Overton 1978) or coverage estimators (Chao et al. 1992).
4. Continuous detection processes: Poisson-family models (Boyce et al. 2001).

Software and the Maryland data were provided in ESA’s Electronic Data Archive supplement (E083-041-S1).

Examples

PyMC sketch (untested, PyMC 5 style)

Marginalise the binary presence $z_i$ so that gradient-based sampling works. For each site, the log-likelihood is log(psi) + sum(logp_det) if detected at least once, else logaddexp(log(psi) + sum(log1m(p)), log1m(psi)).

import pymc as pm, pytensor.tensor as pt, numpy as np
# y: (N, T) 0/1 array, X_psi: (N, k), W_p: (N, T, m)
with pm.Model() as occ:
    b_psi = pm.Normal("b_psi", 0, 1.5, shape=X_psi.shape[1])
    b_p = pm.Normal("b_p", 0, 1.5, shape=W_p.shape[2])
    psi = pm.math.sigmoid(pt.dot(X_psi, b_psi))            # (N,)
    p = pm.math.sigmoid(pt.dot(W_p, b_p))                  # (N, T)
    ever = y.max(1) > 0
    ll_det = pt.log(psi) + pt.sum(y*pt.log(p) + (1-y)*pt.log1p(-p), axis=1)
    ll_nodet = pt.logaddexp(pt.log(psi) + pt.sum(pt.log1p(-p), axis=1),
                            pt.log1p(-psi))
    pm.Potential("lik", pt.sum(pt.where(ever, ll_det, ll_nodet)))

For a missing visit, mask its term out of both sums. This is the same trick as pymc_extras.marginalize mentioned for E28; see Concept - Marginalising Latent Discrete Counts for HMC.

Connections

• Visit-number and low-detection warnings anticipate the weak-identification results of N-mixture models: Barker 2018 - Reliability of N-Mixture Models - Overview.
• Heterogeneity mixtures lead to latent class MSE: Concept - Latent Class and Dirichlet Process Models for Multiple Systems Estimation.
• Prior choices on logit-scale coefficients: Prior Predictive Checking.

See Also

• MacKenzie 2002 - Site Occupancy with Imperfect Detection - Overview
• Concept - Closed-Population Capture-Recapture and Otis Models