Population size estimation when multiple samples carrying the risk of misidentification are taken within the same capture occasion from the same individual

This paper introduces a new Poisson-extended latent multinomial model that accurately estimates population size in capture-recapture studies by simultaneously accounting for misidentification risks and repeated sampling of individuals within the same occasion, a scenario where existing models fail.

The Gist

Imagine you are trying to count how many unique guests are at a massive, chaotic party. You can't see the guests directly, so instead, you are looking for their footprints on the dance floor. This is similar to how scientists count wild animals (like otters) using non-invasive DNA sampling: they collect hair, scat, or saliva left behind rather than catching the animals.

However, there are two big problems with this "footprint counting" method:

1. The "Fake Footprint" Problem (Misidentification): Sometimes, the mud is muddy, or the DNA is a bit blurry. You might think a footprint belongs to "Guest A," but it actually belongs to "Guest B." Or, you might think two different footprints are from two different people, when they are actually from the same person. If you ignore this, you end up thinking there are more guests at the party than there actually are.
2. The "Multiple Footprints" Problem: In the past, scientists assumed that if they found a footprint, it was the only one that person left during that specific hour. But in reality, a single guest might leave a trail of five footprints in one hour. If your counting method assumes one person = one footprint, but that person actually left five, your math gets messed up.

The Old Way vs. The New Way

The Old Way:
Previous models were like a strict bouncer who said, "If I see a footprint, I count one person. If I see another footprint from the same person, I ignore it or assume it's a new person." This worked okay if people only left one footprint, but it failed miserably when guests left trails. It led to either over-counting (thinking there are more people because of the confusion) or under-counting (missing people entirely).

The New Way (This Paper):
The authors of this paper built a smarter calculator. They realized that when a guest (an animal) is at the party, they might leave multiple footprints (samples) at once.

They created a new mathematical tool (a "Poisson distribution" model) that asks a different question: "How likely is it that one person left 1, 2, 3, or more footprints in this hour?"

Think of it like this:

• Old Model: "I see 10 footprints. That must be 10 different people." (Wrong, if one person walked back and forth).
• New Model: "I see 10 footprints. Based on how messy the dance floor is, I calculate that this is likely 6 people, where some of them left extra footprints, and I'm also accounting for the fact that I might have mistaken a muddy smudge for a real footprint."

What Did They Find?

The researchers tested their new calculator with computer simulations (virtual parties) and real data from Eurasian otters (who leave scat in rivers, just like footprints in mud).

Here is the "Goldilocks" rule they discovered:

• Too Few Clues: If the animals are very shy and leave very few samples (like only 10% of the guests leaving a footprint), the new calculator still gets confused and underestimates the crowd. It thinks there are fewer people than there really are.
• Just Right: But, if the animals are active enough to leave a decent number of samples (about 23% to 36% of them leaving traces), the new model works perfectly. It correctly identifies that multiple footprints belong to the same person and filters out the "fake" ones.

The Bottom Line

This paper is a game-changer for wildlife conservation. It tells scientists: "Don't just count the samples you find; count how many samples each animal is likely to leave."

By using this new method, scientists can finally get an accurate headcount of animal populations even when the DNA samples are messy and when animals leave multiple clues at once. It's the difference between guessing the size of a crowd by counting blurry shadows versus actually knowing exactly how many people are dancing.

Technical Summary

Technical Summary: Population Size Estimation with Misidentification and Repeated Sampling

1. Problem Statement

Non-invasive capture-recapture (CR) monitoring, particularly methods utilizing DNA from sources like faeces, is a standard tool for estimating wildlife population sizes. However, these methods face two critical statistical challenges when occurring simultaneously:

• Misidentification: Genotyping errors or sample contamination can lead to the false classification of a single individual as multiple distinct individuals. Ignoring this risk typically results in a significant overestimation of population size.
• Repeated Sampling per Occasion: Existing models designed to correct for misidentification (e.g., Link et al., 2010) operate under the restrictive assumption that only one sample can be collected per individual per capture occasion. In reality, monitoring programs often collect multiple samples from the same individual during a single event (e.g., multiple faecal deposits).
• The Consequence: When models ignoring repeated observations are applied to datasets where multiple samples per individual exist, the estimates become biased. The current literature lacks a robust framework to handle both misidentification and multiple samples per individual within the same capture occasion.

2. Methodology

The authors propose a novel statistical framework to address these limitations by extending existing latent variable models.

• Model Extension: The study extends the Latent Multinomial Model (LMM) originally proposed by Link et al. (2010). While the LMM handles misidentification, it assumes a single observation per individual per occasion.
• Poisson Integration: To account for multiple samples, the authors introduce a Poisson distribution to model the number of samples collected from the same individual during a specific capture occasion. This allows the model to treat the count of samples as a stochastic variable rather than a fixed binary detection.
• Simulation Study: A series of simulations were conducted to evaluate the performance of the new Poisson-LMM under various conditions. Key variables included:
  • The expected number of samples per individual ($\lambda$).
  • The number of capture occasions ($T$).
  • The presence of misidentification errors.
• Empirical Application: The model was applied to a real-world dataset consisting of Eurasian otter (Lutra lutra) faecal samples (previously analyzed by Lampa et al., 2015) to demonstrate practical utility.

3. Key Contributions

• Theoretical Advancement: Development of a unified model that simultaneously corrects for genotyping misidentification and accommodates repeated sampling events within a single capture occasion.
• Relaxation of Assumptions: The model removes the unrealistic constraint of "one sample per individual per occasion," making it applicable to a broader range of non-invasive DNA monitoring programs.
• Threshold Identification: The study identifies specific statistical thresholds required for the model to yield unbiased results, providing practical guidelines for study design.

4. Results

The simulation and empirical results yielded the following insights:

• Performance Conditions for Unbiased Estimates:
  • The model produces unbiased population size estimates when the expected number of samples per individual ($\lambda$) is sufficiently high relative to the number of capture occasions.
  • Scenario A: With 5 capture occasions, unbiased estimates require $\lambda \geq 0.36$.
  • Scenario B: With 7 or more capture occasions, unbiased estimates require $\lambda \geq 0.23$.
• Performance Failure at Low Sampling Rates:
  • When $\lambda = 0.11$ (representing low detection probability where only ~42–62% of individuals are detected across 5, 7, and 9 occasions respectively), the model consistently underestimates the population size. This suggests that a minimum density of repeated samples is necessary for the Poisson component to effectively distinguish between true recaptures and misidentifications.
• Empirical Validation:
  • Application to the Eurasian otter dataset confirmed the presence of misidentifications, aligning with the original authors' expectations.
  • The new model successfully integrated these errors and the repeated sampling nature of the data to provide a more accurate population estimate than previous methods.

5. Significance

This research is pivotal for the field of wildlife ecology and population genetics for several reasons:

• Accuracy in Conservation: By correcting for the bias introduced by ignoring repeated samples, conservationists can obtain more accurate population sizes, which are critical for assessing extinction risk and managing protected areas.
• Optimization of Monitoring Effort: The identification of specific $\lambda$ thresholds helps researchers design more efficient monitoring programs. It indicates that simply increasing the number of capture occasions is not always sufficient; the intensity of sampling (number of samples per individual) must also meet a minimum threshold to ensure statistical validity.
• Robustness of Non-Invasive Methods: The study validates that non-invasive DNA sampling can remain a reliable tool for population estimation even when data is "messy" (containing both genotyping errors and multiple samples), provided the appropriate statistical model is used.