Population Density Estimators for Right-Censored Distance Sampling

1. Problem Statement

Context: The Point-Centered Quarter Method (PCQM) is a standard, efficient technique for estimating population density in ecology (particularly for trees) without requiring full census plots. It relies on measuring distances from a sampling point to the nearest individuals in defined sectors.
The Challenge:

• Right-Censoring: In practical field surveys, a maximum search radius ($C$) is often imposed due to time constraints, safety, or visibility limits. If no individual is found within $C$, the observation is "right-censored" ($R > C$). Ignoring these censored sectors leads to significant positive bias in density estimates.
• Spatial Aggregation: Classical PCQM estimators assume Complete Spatial Randomness (CSR), modeled by a homogeneous Poisson process. However, natural populations often exhibit spatial aggregation (clustering) due to limited dispersal or environmental heterogeneity. Under aggregation, Poisson-based estimators systematically underestimate true density.
• The Gap: While methods exist to correct for right-censoring under CSR, and methods exist to handle aggregation under uncensored conditions, no systematic framework currently integrates Negative Binomial Distribution (NBD) modeling with right-censored distance sampling. Existing censored methods fail when applied to aggregated populations.

2. Methodology

The authors propose a unified framework developing both moment-based and maximum likelihood (MLE) estimators under two core spatial models: the Poisson model (for CSR) and the Negative Binomial Distribution (NBD) model (for aggregation).

A. Poisson Model (CSR Case)

• Moment-Based Estimators: The authors extend classical estimators (Cottam-type and Pollard-type) by replacing empirical moments with censoring-adjusted moments.
  • They derive a correction factor using the lower incomplete gamma function to account for the probability of censoring.
  • For the special case of the nearest neighbor ($\ell=1$), their adjusted Cottam-type estimator reduces exactly to the Warde-Petran estimator.
• Maximum Likelihood Estimator (MLE): They derive the likelihood function for censored data, incorporating the probability of observing a distance $> C$.
  • For $\ell=1$, a closed-form solution is derived.
  • For $\ell > 1$, the MLE is obtained via numerical optimization.

B. Negative Binomial Model (Aggregation Case)

This is the novel contribution of the paper.

• Moment-Based Estimators: To handle censoring under aggregation, the authors propose approximating the conditional expectation of censored distances.
  • Key Approximation: They calculate the conditional expectation of censored distances assuming CSR ($k \to \infty$) as an initial step. This approximation is justified as having negligible bias under moderate censoring and vanishing as aggregation increases.
  • Using these adjusted moments, they extend the Shen et al. (2020) NBD estimator and Morisita's first estimator to the censored setting.
• Maximum Likelihood Estimator (MLE):
  • They construct a likelihood function combining the NBD probability density for observed distances and the cumulative distribution function (involving the regularized incomplete beta function) for censored sectors.
  • Parameters (density $\lambda$ and aggregation $k$) are estimated via numerical maximization of the log-likelihood.

3. Key Contributions

1. First Unified Framework: This study provides the first comprehensive set of estimators that simultaneously address right-censoring and spatial aggregation in distance sampling.
2. Theoretical Extension: It successfully extends classical moment-based and likelihood-based theory from uncensored data to right-censored data for both Poisson and NBD models.
3. Practical Approximation: The development of a computationally tractable method to estimate moments for censored aggregated data using a CSR-based approximation for the conditional expectation.
4. Open-Source Toolkit: The authors provide an R package (TruncatedPCQM) implementing all proposed estimators.

4. Results

The methods were evaluated through extensive simulations (Thomas cluster processes) and applications to fully censused forest data (Barro Colorado Island and Harvard Forest).

• CSR Populations:
  • All Poisson-based censored estimators (Cottam, Pollard, MLE) performed well with low bias.
  • The Warde-Petran estimator (existing method) showed substantial positive bias at low densities where censoring was severe.
• Aggregated Populations:
  • NBD-based MLE ($\hat{\lambda}^{(c)}_{n,MLE}$) consistently achieved the highest accuracy and robustness.
  • It maintained a median relative bias below 20% across a wide range of ecological scenarios, including strong clustering and varying search radii.
  • Other methods (including existing CSR-based corrections and moment-based NBD estimators) failed to consistently guarantee this level of accuracy, often exhibiting higher bias or variance.
• Trade-offs ($C$ vs. $\ell$):
  • Increasing the nearest-neighbor order ($\ell$) generally improves estimation in uncensored data but increases the probability of censoring under a fixed radius $C$.
  • Under severe truncation, lower-order designs ($\ell=1$ or $2$) offer a better balance between information content and censoring-induced bias.

5. Significance

• Ecological Relevance: The study addresses a pervasive real-world constraint (truncated search radii) that has previously forced ecologists to either discard data or use biased estimators when studying aggregated populations.
• Improved Accuracy: By validating the NBD-based MLE, the paper provides a rigor tested tool that significantly reduces estimation error in forest surveys where spatial clustering is the norm.
• Decision Support: The findings offer clear guidance for field ecologists: if spatial aggregation is suspected (common in forests), the NBD-based MLE should be preferred over traditional Poisson-based methods, even when data is censored.
• Foundation for Future Work: This framework bridges the gap between theoretical spatial statistics and practical field constraints, enabling more reliable biodiversity assessments and conservation planning.