Barker 2018 - Reliability of N-Mixture Models - Overview

Summary

Barker, Schofield, Link and Sauer (2018, Biometrics 74(1): 369-377) argue that N-mixture models, which estimate abundance $N$ and detection $p$ from repeated unmarked counts, rest on an identifiability that comes from the binomial and constant-$p$ assumptions, not from the data. Plausible alternatives in which $N$ is non-identifiable or does not exist fit practically as well. Reliable inference from counts alone is limited to relative abundance.

Overview

Why this paper stands in for Royle (2004). Royle’s original N-mixture paper (Biometrics 60: 108-115) had no legal open PDF reachable here; this paper restates the model, explains where its identification comes from, and is the most cited critique of it. It is therefore used as the source for the N-mixture topic. Royle’s paper remains a follow-on ingest.

The model. Replicate counts $n_{ij}\mid N_i\sim \text{Binomial}(N_i,p)$ at sites $i=\mathrm{1..}I$ and visits $j=\mathrm{1..}J$, with $N_i\sim \text{Poisson}(\nu )$ or a heavier-parameterised alternative.

Contributions.

1. A capture-recapture view: counts are incomplete data of a latent multinomial; not marking animals discards the dominant term of the likelihood.
2. A counterexample: Poisson sampling with replacement (or a double-Poisson-lognormal count model) has the same first two moments, makes $N$ non-identifiable, and is hard to tell apart by AIC.
3. A recommendation: fit Poisson regression (with covariates and random effects) for relative abundance; collect auxiliary data (recaptures) if absolute abundance is needed.

Key numbers. In the simulations the competing double Poisson-lognormal model was preferred or within 1 AIC unit in 82% / 59% / 41% of data sets simulated under the N-mixture model (scenarios A/B/C, $E[n]=3,9,15$).

Motivating PyMC examples

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

• E28 - “Imperfect detection: occupancy, N-mixture, capture-recapture” (Swiss breeding-bird mallard counts; N-mixture truncation check, Poisson vs NegBin). The example’s identifiability caveats are exactly this paper’s.

Source Map

Paper section	Covered in
Summary; 1 Introduction; 1.1 N-mixture model; 1.2 structure	this note; Barker 2018 - N-Mixture Model and the Capture-Recapture View
2 A capture-recapture perspective (Eq. 1-2, Fig. 1)	Barker 2018 - N-Mixture Model and the Capture-Recapture View
2.1 N-mixtures and the multivariate Poisson (Eq. 5-6)	Barker 2018 - N-Mixture Model and the Capture-Recapture View
3 Binomial vs Poisson models (Eq. 3-4)	Barker 2018 - Poisson Alternative and Model Discrimination
3.1 Can we discriminate among models? (Table 1, Fig. 2)	Barker 2018 - Poisson Alternative and Model Discrimination
4 Relative abundance: Poisson models (Eq. 7-10)	Barker 2018 - Relative Abundance, Alder Flycatcher Example and Discussion
4.1 Alder flycatcher example (Table 2, Fig. 3)	Barker 2018 - Relative Abundance, Alder Flycatcher Example and Discussion
5 Discussion; 6 Supplementary materials; References	Barker 2018 - Relative Abundance, Alder Flycatcher Example and Discussion

Connections

• Roots in the occupancy model: MacKenzie 2002 - Site Occupancy with Imperfect Detection - Overview.
• Capture-recapture models M0, Mt, Mtbh: Concept - Closed-Population Capture-Recapture and Otis Models.
• Overdispersion and zero-inflation alternatives: Monsters and Mixtures. Random-effect Poisson as partial pooling: Hierarchical Models.
• Identification-by-prior-structure problems also appear in Aleshin-Guendel 2020 - Identifiability of Latent Class MSE Models - Overview.

See Also

• Concept - Marginalising Latent Discrete Counts for HMC
• SBC Case Studies (a calibration check would flag the weak identification)