Multiple imputation step-selection analysis: Improving estimation accuracy of travel distance accounting for route uncertainty

This paper introduces Multiple Imputation Step Selection Analysis (MiSSA), a novel method that improves the accuracy of travel distance estimation in animal movement studies by addressing the underestimation bias inherent in conventional Integrated Step Selection Analysis (iSSA) caused by linear interpolation assumptions.

The Gist

The Big Picture: Connecting the Dots in Animal Movements

Imagine you are trying to figure out how a squirrel gets from one tree to another. You have a camera that takes a photo of the squirrel every 10 minutes.

• Photo 1: The squirrel is at the base of an Oak tree.
• Photo 2 (10 mins later): The squirrel is at the base of a Pine tree.

The Old Way (iSSA):
The traditional method (called iSSA) assumes the squirrel walked in a perfectly straight line between those two photos. It draws a straight ruler line from the Oak to the Pine.

• The Problem: Squirrels don't walk in straight lines! They zigzag, climb over rocks, chase butterflies, and hide from hawks. By drawing a straight line, the old method underestimates how far the squirrel actually ran. It's like measuring the distance of a winding hiking trail by measuring the straight-line distance between the start and end points on a map. You miss all the fun (and the exercise) in the middle.

The New Way (MiSSA):
The authors of this paper, Shiori Takeshige and Yusaku Ohkubo, came up with a smarter way called MiSSA (Multiple Imputation Step-Selection Analysis).

Instead of guessing one straight line, MiSSA says: "We don't know exactly where the squirrel was in those 10 minutes, so let's imagine 1,000 different possible paths it could have taken."

The Analogy: The "Choose Your Own Adventure" Book

Think of the time between two photos as a blank page in a "Choose Your Own Adventure" book.

1. The Old Method: It picks one random path and says, "This is what happened." It's confident, but often wrong.
2. The New Method (MiSSA): It generates 1,000 different storylines.
   • Storyline A: The squirrel went straight up the hill.
   • Storyline B: The squirrel took a detour to a berry bush.
   • Storyline C: The squirrel ran in a wide circle to avoid a dog.

It calculates the distance for all 1,000 stories, then averages them out. This gives a much more realistic picture of the total distance traveled, even though the camera didn't record every single step.

Why Does This Matter?

1. Fixing the "Blurry Photo" Problem
Many animals (like small birds or mice) are tracked with devices that can't take photos very often because the batteries are small or the animals are too tiny for heavy cameras. This results in "blurry" data with big gaps.

• Analogy: If you watch a movie but only see frames 1 and 100, you might think the character teleported. MiSSA fills in frames 2 through 99 with plausible scenes so you can see the real action.

2. Saving Animals
Conservationists need to know exactly how far animals travel to protect them.

• The Highway Problem: If a road cuts through a forest, we need to know if animals are taking a long, dangerous detour around it or if they are crossing it directly.
• The Result: Because the old method underestimated the distance, it might have made us think animals were moving faster and more directly than they really were. With MiSSA, we get a truer picture. This helps us build better wildlife bridges or protect the right areas, ensuring animals don't get hit by cars or lose their homes.

The "Magic" Ingredient: Multiple Imputation

The paper uses a statistical trick called Multiple Imputation.

• Simple Explanation: Instead of trying to guess the one missing piece of a puzzle, you create many different versions of the missing piece based on what you know. Then, you look at the average of all those versions.
• In the Paper: They used math to generate thousands of "fake" but realistic paths between the real GPS points. By analyzing all these fake paths together, they cancelled out the errors and found the true average distance.

The Results: Did It Work?

The authors tested this in two ways:

1. Computer Simulations: They created a fake world with fake animals and knew the exact distance they traveled.
   • Result: The old method (iSSA) was consistently too low (it said the animals walked 84 meters when they actually walked 100). The new method (MiSSA) was almost spot on (104 meters).
2. Real World Test: They looked at real data from a Fisher (a type of wild cat-like animal).
   • Result: Again, the new method estimated a longer, more realistic travel distance than the old method.

The Takeaway

This paper is about filling in the blanks with better math.

When we track animals, we often miss the details of their journey. The old way of analyzing this data was like drawing a straight line between two dots. The new way (MiSSA) is like imagining all the possible winding roads the animal could have taken, calculating the distance for each, and finding the truth in the middle.

This helps scientists understand animal behavior better, which leads to better decisions on how to protect wildlife and their habitats. It turns a "best guess" into a "well-informed estimate."

Technical Summary

1. Problem Statement

The Core Issue: Integrated Step Selection Analysis (iSSA) is a standard method in movement ecology for estimating habitat selection and movement parameters (step length and turning angle). However, iSSA relies on linear interpolation between consecutive observed location points.

• The Flaw: When tracking data has low temporal resolution (long intervals between fixes), the actual animal path is rarely a straight line. By assuming a straight line, iSSA systematically underestimates the true travel distance (step length).
• Consequences: This bias leads to inaccurate estimation of movement kernels (e.g., Gamma distribution parameters for step length) and potentially flawed inferences regarding habitat selection, as animals often move tortuously in preferred habitats and linearly in unsuitable ones.
• Limitations of Current Solutions: While high-resolution tracking exists, it is often impractical for small species due to device weight constraints, battery life, or the need for recapture. Furthermore, high-frequency data can introduce noise (measurement error), and existing imputation attempts in ecology have sometimes failed due to simplistic strategies (e.g., single imputation).

2. Methodology: Multiple Imputation Step-Selection Analysis (MiSSA)

The authors propose MiSSA, a novel framework that integrates Multiple Imputation (MI) techniques into the iSSA workflow to account for route uncertainty.

A. Core Concept

Instead of treating the path between two observed points ($X_t$ and $X_{t+\tau}$) as a single straight line, MiSSA treats the intermediate path as missing data. It generates multiple plausible "pseudo-complete" datasets representing different possible routes the animal could have taken.

B. Workflow Steps

1. Imputation of Used Steps:

   • For every observed step between $X_t$ and $X_{t+\tau}$, the method generates $M$ imputed datasets.
   • Geometric Construction: A perpendicular bisector is drawn between the start and end points. Candidate intermediate points are generated along this bisector (e.g., at 1-meter intervals).
   • Path Recovery: Paths are reconstructed connecting $X_t \to \text{Candidate Point} \to X_{t+\tau}$.
   • Filtering: Paths exceeding a biological plausibility threshold (e.g., $>2$ standard deviations of observed step lengths) are optionally rejected to reduce computational cost.
   • Sampling: One imputed path is randomly selected for each dataset based on the density of step length and turning angle distributions. This is repeated $M$ times (e.g., $M=100$ to $1000$) to create $M$ distinct datasets.

2. Conditional Logistic Regression (per Dataset):

   • For each of the $M$ imputed datasets, a standard conditional logistic regression is performed.
   • Response Variable: 1 for the "imputed used step," 0 for "available steps" (randomly generated steps).
   • Predictors: Habitat covariates, step length, log-transformed step length, and cosine of the turning angle.
   • This yields $M$ sets of regression coefficients ($\hat{\beta}_1, \hat{\beta}_2, ..., \hat{\beta}_M$).

3. Pooling Results (Rubin's Rule):

   • The $M$ sets of coefficients are combined using Rubin's Rules for multiple imputation.
   • Point Estimate: The final coefficient is the average of the $M$ estimates: $\bar{\beta} = \frac{1}{M}\sum \hat{\beta}_m$.
   • Variance Estimation: The total variance ($V_M$) integrates both the within-imputation variance (average variance of the $M$ models) and the between-imputation variance (variability of estimates across the $M$ models).
   • Parameter Update: The pooled coefficients are used to update the parameters of the movement kernels (Gamma distribution for step length, Von Mises for turning angle) to reflect the corrected travel distances.

3. Key Contributions

• Novel Framework: First application of rigorous multiple imputation statistics specifically to correct step-length bias in iSSA.
• Route Uncertainty Modeling: Moves beyond the "straight-line" assumption by explicitly modeling the probability distribution of unobserved paths.
• Statistical Rigor: Properly accounts for the uncertainty introduced by imputation through Rubin's rules, providing valid confidence intervals for movement parameters.
• Applicability to Low-Resolution Data: Enables the use of legacy or low-frequency tracking data (e.g., geolocators) without the severe underestimation bias inherent in traditional methods.

4. Results

The authors validated MiSSA through two approaches:

A. Simulation Study

• Setup: Simulated animal movements in virtual landscapes with varying spatial autocorrelation. Data was downsampled to simulate low-resolution tracking (interval $2\tau$).
• Findings:
  • Bias Reduction: Traditional iSSA consistently underestimated the expected step length (by ~15-25% in tested scenarios). MiSSA estimates were significantly closer to the true values.
  • Accuracy: Across all landscape types and parameter sets (Shape, Scale, Kappa), MiSSA yielded lower Mean Squared Error (MSE) than iSSA.
  • Robustness: While accuracy decreased slightly when true parameters were very large, MiSSA still outperformed iSSA in all tested scenarios.

B. Empirical Case Study (Fisher Pekania pennanti)

• Data: Used real tracking data from a Fisher.
• Findings:
  • The estimated expected step length (Gamma distribution mean) was 53.48 for iSSA vs. 64.81 for MiSSA.
  • This ~21% increase in estimated travel distance demonstrates that accounting for route uncertainty substantially alters the inferred movement behavior and habitat selection parameters.

5. Significance and Implications

• Conservation Management: Accurate travel distance estimation is critical for identifying functional movement corridors and barriers. Underestimating distance can lead to incorrect assessments of how animals navigate human-altered landscapes (e.g., crossing roads or avoiding urban areas).
• Data Integration: MiSSA facilitates the comparison of datasets with different temporal resolutions (e.g., combining high-resolution GPS data with low-resolution geolocator data), enabling more robust meta-analyses.
• Methodological Advancement: It bridges the gap between movement ecology and missing data statistics, offering a solution to a long-standing problem in the field without requiring expensive, high-frequency hardware upgrades.
• Future Directions: The authors note limitations, such as the need to validate habitat selection coefficients specifically and the sensitivity of the method to the range of candidate imputed paths, suggesting future work should focus on adaptive thresholding for different species and behaviors.