MacKenzie 2002 - Occupancy Model and Likelihood

Summary

Sites are closed during $T$ visits; a site is occupied with probability $\psi$ and, if occupied, the species is detected on visit $t$ with probability $p_t$ (never falsely detected). The all-zero history has a two-part probability, “occupied but missed” plus “absent”, and this mixture separates $\psi$ from $p$.

Overview

Notation: ${\psi }_i$ presence probability at site $i$; $p_{it}$ detection probability at site $i$, occasion $t$, given presence; $N$ surveyed sites; $T$ occasions; $n_t$ sites with a detection at $t$; $n_{\cdot }$ sites with at least one detection.

Sampling assumptions

(1) Sites are closed: none becomes occupied or abandoned during the survey period. (2) The species is never falsely detected when absent. (3) Detection at a site is independent of detection at other sites. Data per site are a 0/1 vector of detection histories.

Main Content

Analogy to closed-population capture-recapture

Sites play the role of individuals, except that the number of “never captured” sites is observed (the total $N$ is known); what is unknown is the share truly occupied. Applying a classical closed-population model (Concept - Closed-Population Capture-Recapture and Otis Models) to the detected sites only would estimate the number of occupied sites, but the occupancy formulation also lets $\psi$ depend on covariates.

Site-level likelihood

For a site with at least one detection (e.g. history 01010):

$$L_i={\psi }_i(1-p_{i1})p_{i2}(1-p_{i3})p_{i4}(1-p_{i5}).$$

For a never-detected site (00000):

$$L_k={\psi }_k\prod _{t=1}^5(1-p_{kt})+(1-{\psi }_k).$$

The model likelihood is the product over independent sites.

With fully site-specific ${\psi }_i,p_{it}$ the likelihood is over-parameterised; the probabilities must be constant or modelled through covariates.

Constant-parameter likelihood (Eq. 1)

With $\psi$ and $p_t$ constant across sites,

$$L(\psi ,\mathbf{p})={\psi }^{n_{\cdot }}\left[\prod _{t=1}^T{p}_t^{n_t}(1-p_t{)}^{n_{\cdot }-n_t}\right]{\left[\psi \prod _{t=1}^T(1-p_t)+(1-\psi )\right]}^{N-n_{\cdot }}.$$

Only $(n_1,\dots ,n_T,n_{\cdot })$ and $N$ are needed, so it can be maximised in a spreadsheet.

Why the factorisation holds. The $n_{\cdot }$ sites with a detection must be occupied; each contributes $\psi$ times a Bernoulli product over its visits, and collecting visit-wise terms gives ${\prod }_t{p}_t^{n_t}(1-p_t{)}^{n_{\cdot }-n_t}$. The $N-n_{\cdot }$ all-zero sites each contribute the mixture term.

Standard errors

The authors recommend the nonparametric bootstrap (resample $N$ sites with replacement, refit, take the SD of estimates; Buckland and Garthwaite 1991) over asymptotic second-derivative variances, because the asymptotic value is a lower bound and may be too small in small samples.

Covariates (Eq. 2-3)

Logistic covariate models

For a parameter $\theta \in \{\psi ,p\}$ with covariate matrix $X$ and coefficient vector $B$:

$$\theta =\frac{\exp (XB)}{1+\exp (XB)}.$$

With covariates on $\psi$, the average presence probability is $\hat{\psi }={\sum }_{i=1}^N{\hat{\psi }}_i/N$ (Eq. 3). Covariates for $\psi$ must be site-specific and time-constant (closure); covariates for $p$ may be time-varying and site-specific (e.g. temperature).

Contrast with mark-recapture: there, a time-varying individual covariate is observed only when the individual is captured. Here, site covariates can be recorded on every visit whether or not the species is detected. Time-varying covariates that cannot be measured independently of the detection process are not allowed.

Missing observations

If site $i$ is not sampled on occasion $t$, that occasion contributes no factor. Example: history 10-11 (no sampling at time 3) has likelihood $\psi p_1(1-p_2)p_4{p}_5$. This works only with the per-site likelihood, not the pooled Eq. 1.

Examples

Worked example: how much does non-detection tell us?

Setup. $\psi =0.5$, $p=0.3$, $T=5$ constant (illustrative, not from the paper). Solution. $P(00000)=0.5\cdot 0.7^5+0.5=0.5840$. By Bayes, $P(\text{occupied}\mid 00000)=0.5\cdot 0.7^5/0.5840=0.144$. Interpretation. After five empty visits a site still has a 14% chance of being occupied; the naive estimator treats this as 0%, which is its source of negative bias.

import numpy as np
from scipy.optimize import minimize
 
def nll(theta, n_t, n_dot, N):
    psi = 1/(1+np.exp(-theta[0])); p = 1/(1+np.exp(-theta[1:]))
    ll = n_dot*np.log(psi) + np.sum(n_t*np.log(p) + (n_dot-n_t)*np.log1p(-p))
    ll += (N-n_dot)*np.log(psi*np.prod(1-p) + 1-psi)
    return -ll
 
rng = np.random.default_rng(1)
N, T, psi, p = 200, 5, 0.6, 0.3
z = rng.random(N) < psi
y = (rng.random((N, T)) < p) & z[:, None]
n_t, n_dot = y.sum(0).astype(float), y.any(1).sum()
fit = minimize(nll, np.zeros(1+T), args=(n_t, n_dot, N))
print("psi_hat =", 1/(1+np.exp(-fit.x[0])), " naive =", n_dot/N)

Connections

• The all-zero mixture term is the zero-inflation device in Monsters and Mixtures; marginalising the binary latent state is the simplest case of Concept - Marginalising Latent Discrete Counts for HMC.
• Site effects with partial pooling extend the model: Hierarchical Models.

See Also

• MacKenzie 2002 - Site Occupancy with Imperfect Detection - Overview
• MacKenzie 2002 - Simulation Study and Maryland Anuran Results