[Paper Review] Local Log-linear Models for Capture-Recapture

Zachary Kurtz|arXiv (Cornell University)|Feb 4, 2013

Census and Population Estimation101 references3 citations

TL;DR

This paper proposes local log-linear models that fit a unique log-linear model to each observed individual using covariate-dependent smoothing, enabling accurate estimation of population size and missingness rates in capture-recapture studies. The method outperforms standard post-stratification and additive models in simulations when capture probabilities vary across covariates, particularly in heterogeneous populations.

ABSTRACT

Log-linear models are often used to estimate the size of a closed population using capture-recapture data. When capture probabilities are related to auxiliary covariates, one may select a separate model based on each of several post-strata. We extend post-stratification to its logical extreme by selecting a local log-linear model for each observed unit, while smoothing to achieve stability. Our local models serve a dual purpose: In addition to estimating the size of the population, we estimate the rate of missingness as a function of covariates. A simulation demonstrates the superiority of our method when the generating model varies over the covariate space. Data from the Breeding Bird Survey is used to illustrate the method.

Motivation & Objective

• To address bias in log-linear models caused by heterogeneity in capture probabilities across covariates.
• To estimate not only total population size but also the rate of missingness as a function of covariates.
• To overcome the limitations of post-stratification by using individual-level models with local smoothing for stability.
• To develop a method that allows model form to vary flexibly across the covariate space, improving accuracy in non-stationary settings.
• To provide a flexible, data-driven approach that adapts model complexity and bandwidth locally while maintaining statistical stability.

Proposed method

• Fit a separate log-linear model to each observed unit using a locally weighted average of neighboring units' capture patterns.
• Use kernel smoothing (e.g., Epanechnikov) to define local neighborhoods and compute effective sample sizes for model fitting.
• Apply model selection criteria (e.g., AICc) within each local neighborhood to choose the best-fitting log-linear model.
• Estimate the number of unobserved units (c₀) by summing over predicted probabilities of non-observation across all local models.
• Use a piecewise smooth function r(y|x) to model capture probability as a function of covariates, allowing model form to vary with x.
• Implement a data-driven bandwidth selection process to balance bias and variance in local model estimation.

Experimental results

Research questions

• RQ1 Can local log-linear models improve population size estimation when capture probabilities vary across covariates compared to standard post-stratification?
• RQ2 How does the performance of local models compare to additive multinomial logit models in estimating missingness rates across the covariate space?
• RQ3 What is the impact of local model selection and bandwidth choice on the accuracy of population size and missingness rate estimates?
• RQ4 Can local models effectively estimate the rate of missingness as a function of covariates, providing insights into population composition?
• RQ5 Under what conditions does local modeling outperform global or post-stratified models in capture-recapture settings?

Key findings

• Local log-linear models achieved a root mean squared error (RMSE) of 149 for estimating c₀, outperforming the additive multinomial logit (RMSE = 152) and 5-post-stratum log-linear model (RMSE = 153).
• The local model showed lower bias (-8) compared to the additive model (-62), indicating more accurate estimation of the unobserved population size.
• In regions where the generating model varied across the covariate space, local log-linear models demonstrated superior local error performance, as shown in Figure 2.
• The method successfully estimated the rate of missingness as a function of covariates, enabling insights into population composition beyond total size.
• Bandwidth selection significantly affected performance: a bandwidth of 14 outperformed 12 and 10, suggesting sensitivity to smoothing parameter choice.
• The local approach provided better local error control than the additive multinomial logit model in certain covariate regions, highlighting its adaptability to non-stationary capture processes.