Barker 2018 - N-Mixture Model and the Capture-Recapture View

Summary

The N-mixture model is a closed-population capture-recapture model whose recapture information has been thrown away. Decomposing the Mt/M0 likelihood into $L_1\times L_2\times L_3$ shows that the part carrying most information about $N$ ($L_1$, hypergeometric recaptures) is missing, leaving only the weak terms $L_2{L}_3$.

Overview

Strategies for abundance under imperfect detection: (i) capture-recapture with auxiliary data to estimate $p$ directly; (ii) N-mixture models, which assume constant $p$ (after covariates) plus a hierarchical model for $N$; (iii) index models for relative abundance assuming constant $p$.

N-mixture model

For sites $i=1,\dots ,I$ and visits $j=1,\dots ,J$,

$$n_{ij}\mid N_i\stackrel{iid}{\sim }\text{Binomial}(N_i,p),N_i\stackrel{iid}{\sim }\text{Poisson}(\nu )$$

(or negative binomial, zero-inflated Poisson). Assumptions: (a) a closed population of $N_i$ animals at site $i$ across visits; (b) no animal counted twice within a visit; (c) detection $p$ constant after covariate adjustment. With a single site and $J>1$, $n_j\sim \text{Binomial}(N,p)$ lets both $N$ and $p$ be estimated, but only through second-moment information (Carroll and Lombard 1985).

Limitations listed by the authors: (i) a defined closed population is a strong assumption and an ill-defined sampling area induces individual heterogeneity in $p$; (ii) Poisson counts conditional on $N$ may be preferable if individuals may be counted repeatedly; (iii) uncontrolled variation in $p$ among visits breaks the model; (iv) estimating a binomial index $N$ is difficult and these difficulties flow through (Dennis et al. 2015: infinite or very large $\hat{N}$ when $p$ and $J$ are small).

Main Content

Counts as incomplete capture-recapture data

Let $\omega =({\omega }_1,\dots ,{\omega }_J)$ be a capture history and $x_{\omega }$ the number of individuals with history $\omega$ (so $x_{000}$ is unseen). With three occasions the unmarked counts are

$$n_1=x_{111}+x_{101}+x_{110}+x_{100},n_2=x_{111}+x_{110}+x_{011}+x_{010},n_3=x_{111}+x_{101}+x_{011}+x_{001}.$$

The $x_{\omega }$ are now latent (a latent multinomial; Link et al. 2010), and $(n_1,\dots ,n_J)$ are not sufficient statistics for any model in the Otis series M0, …, Mtbh, even assuming constant $p$.

Likelihood decomposition under model Mt (Eq. 1)

$$[\{x_{\omega }\}\mid N,\mathbf{p}]\propto \underset{L_1=[\{m_j\}\mid \{n_j\},N]}{\underbrace{\prod _{j=2}^J\left(\genfrac{}{}{0ex}{}{M_j}{m_j}\right)\left(\genfrac{}{}{0ex}{}{N-M_j}{n_j-m_j}\right)/\left(\genfrac{}{}{0ex}{}{N}{n_j}\right)}}\times \underset{L_2=[\{n_j\}\mid N,\mathbf{p}]}{\underbrace{\prod _{j=1}^J\left(\genfrac{}{}{0ex}{}{N}{n_j}\right)p_j^{n_j}(1-p_j{)}^{N-n_j}}}$$

where $m_j$ is the number of marked animals caught in sample $j$ and $M_j$ the number of marked animals in the population just before sample $j$. $L_1$ extends the two-sample hypergeometric model conditional on the captures $\{n_j\}$ and can be treated as a partial likelihood that contains virtually all the information about $N$ (Schofield and Barker 2016).

Decomposition under M0 (Eq. 2) and what N-mixtures keep

Setting $p_1=\cdots =p_J=p$ and writing $t={\sum }_j{n}_j$,

$$[\{x_{\omega }\}\mid N,p]\propto L_1\times \underset{L_2=[\{n_j\}\mid N,t]}{\underbrace{\frac{{\prod }_j\left(\genfrac{}{}{0ex}{}{N}{n_j}\right)}{\left(\genfrac{}{}{0ex}{}{JN}{t}\right)}}}\times \underset{L_3=[t\mid N,p]}{\underbrace{\left(\genfrac{}{}{0ex}{}{JN}{t}\right)p^t(1-p{)}^{JN-t}}}.$$

(i) $L_2$ carries some information about $N$ given $t$; this is what N-mixtures exploit. (ii) $L_2{L}_3$ is the likelihood for $n_j\mid N\stackrel{iid}{\sim }\text{Binomial}(N,p)$. Relative to $L_1$, the residual information in $L_2{L}_3$ is small (Fig. 1).

With $m_j$ missing, summing out $L_1$ in Eq. 1 leaves $L_2$ only, ${\prod }_j\left(\genfrac{}{}{0ex}{}{N}{n_j}\right)p_j^{n_j}(1-p_j{)}^{N-n_j}$, so distinct $p_j$ with unmarked counts is over-specified: reliable inference is impossible. Identifiability under M0 depends on equal $p_j$ (in capture-recapture it does not, owing to the dominance of $L_1$) and on binomial variation (see Barker 2018 - Poisson Alternative and Model Discrimination).

Figure 1 plots the contribution of $L_1,L_2,L_3$ for a simulated $k=5$ study with closed population of 20 and $p=0.25$ per occasion: $L_1$ dominates the shape of the likelihood in $N$.

N-mixtures and the multivariate Poisson (Sec. 2.1)

With $N_i\sim \text{Poisson}(\nu )$ and $\pi =\{{\pi }_{\omega }\}$ simple functions of the capture histories and $p$, the complete-data likelihood factorises into independent Poissons:

Complete-data likelihood (Eq. 5)

$$[\{x_{i\omega }\},\{N_i\}\mid p]=\prod _i\text{Multinomial}(\{x_{i\omega }\};N_i,\pi )\text{Poisson}(N_i;\nu )=\prod _i\prod _{\omega }\frac{e^{-\nu {\pi }_{\omega }}(\nu {\pi }_{\omega }{)}^{x_{i\omega }}}{x_{i\omega }!}.$$

Summing out the latent quantities with $\{n_{ij}\}$ held fixed gives a multivariate Poisson marginal for $\{n_{ij}\}$ (Dennis et al. 2015), with marginal mean equal to marginal variance.

Examples

Poisson thinning check

With $\nu =30$, $p=0.1$: $P(n=2)={\sum }_N\text{Poisson}(N;30)\text{Binomial}(2;N,0.1)=0.2240$, equal to $\text{Poisson}(2;3)=0.2240$. The marginal count is Poisson with mean $\nu p$, so the single-visit marginal cannot separate $\nu$ from $p$; only the covariance across visits within a site ($\nu p^2$) can.

import numpy as np
from scipy.stats import poisson, binom
nu, p = 30, 0.1
lhs = sum(poisson.pmf(N, nu)*binom.pmf(2, N, p) for N in range(200))
print(lhs, poisson.pmf(2, nu*p))   # identical

Connections

• The Mt/M0 series and Lincoln-Petersen estimator: Concept - Closed-Population Capture-Recapture and Otis Models.
• Marginalising $N_i$ in practice (truncate at $N_{\max }$): Concept - Marginalising Latent Discrete Counts for HMC.
• Latent multinomial / incomplete table view reappears in MSE: Silverman 2020 - Poisson Log-Linear Model for Multiple Lists.

See Also

• Barker 2018 - Reliability of N-Mixture Models - Overview
• Barker 2018 - Poisson Alternative and Model Discrimination