# Accurately Estimating Correlations Between Demographic Parameters: A Response to Riecke Et al. (2024)

## TL;DR

The paper compares different statistical methods for estimating correlations between survival and recovery rates in animal populations, finding that results are reliable only with large sample sizes.

## Contribution

The study evaluates the performance of Gamma(1,1) and Uniform(0,5) prior distributions in estimating demographic correlations with varying sample sizes.

## Key findings

- Large sample sizes yield reliable correlation estimates with either prior distribution.
- Small sample sizes lead to uncertain and ambiguous correlation estimates.
- Annual survival is more uncertain than annual recovery in small samples.

## Abstract

Correlations between annual recovery and survival probabilities estimated from tag‐recovery data have been used to quantify the demographic response of exploited populations to harvest. Deane et al. (2023) evaluated the bias and certainty of correlation parameters between recovery and survival probabilities estimated as random effects drawn from bivariate normal distributions relative to different prior distributions and sample size combinations. Riecke et al. (2024) observed that we incorrectly parameterized a precision matrix with Gamma priors and suggested using a Gamma(1,1) prior distribution for the standard deviations as an alternative. Riecke et al. (2024) provided results from tag‐recovery models that estimate mortality hazard rates after fitting these models to tag‐recovery datasets with large sample sizes. Here, we fit tag‐recovery models to the data we previously simulated (Deane et al. 2023) while using Gamma(1,1) as the prior distribution for standard deviations while parameterizing these models to estimate recovery and survival in discrete time or to estimate cause‐specific mortality as hazard rates. We compare our new results to previous results obtained while using Uniform(0,5) prior distribution for the standard deviations. When sample sizes were large, correlation estimates obtained with either prior distribution provided similarly reliable parameter recovery and inference, replicating results of Riecke et al. (2024). With smaller sample sizes similar to those available for most duck populations in North America, correlations estimated with either prior distribution were uncertain and ambiguous. With decreasing sample sizes, annual survival was estimated with increasing uncertainty when compared to annual recovery, likely contributing to the poor ability to estimate correlation. Consistent with the original interpretation of Deane et al. (2023) and previous literature, we found correlations were often estimated with high uncertainty such that the sign (+ or –) may be the only attribute of these parameters that can be reliably interpreted.

Correlations between annual recovery and survival probabilities estimated as random effects drawn from multivariate normal distributions have been used to quantify the demographic response of exploited populations to harvest. We fit tag‐recovery models to simulated datasets using Uniform(0,5) and Gamma(1,1) as prior distributions for standard deviations. When sample sizes were large, correlation estimates obtained with either prior distribution provided similarly reliable parameter recovery, and when sample sizes were more modest, correlation estimates using either prior distribution provided mostly unreliable inference.