Summarizing quantile estimates of weight at length to compare condition: Geographic and temporal variation in Arctic Grayling
Authors: Cade Brian S. , Gilham Andrew T. 
Published: April 2024

Abstract

Objective
We developed a novel, simple approach to summarize variation in quantile regression estimates of weight at length across multiple levels of grouping factors (locations or years) and demonstrate its utility for comparing fish condition using Arctic Grayling Thymallus arcticus data. We compared condition among 28 geographic locations of Arctic Grayling populations and among 12 and 25 years for two populations.

Methods
We summarized variation among locations and among years within locations using percentiles (maximum, 75th, 50th, 25th, and minimum) of predicted quantiles of weight at length, allowing us to simplify the distributions of predicted quantiles of weight into highest to lowest fourths (quartiles) across locations or years. One part of this summary information, 75th percentiles of median (0.50 quantile) predicted weights at length across populations, is identical to summary statistics used to develop standard weight equations for computing relative weights. However, our approach avoids biases inherent in fitting a single standard weight equation and can accommodate among-fish variation in weight at length via multiple quantiles when developing reference standards for comparative or diagnostic purposes. A selected new weight at length can be compared with the distribution of quantile predictions among locations in terms of weights or as proportionate weights similar to relative weight.

Result
Arctic Grayling in 28 geographic locations, from the southern limit of their distribution in Montana and Wyoming north through Canada and Alaska to the Arctic coast, had considerable variation in allometric exponents in weight–length relationships and in predicted weights at length. Predicted weights of Arctic Grayling for heavier fish (0.90 quantiles) in populations with less (lower quartiles) to greater (upper quartiles) weights were 127–194 g and 336–510 g at total lengths = 250 and 350 mm, respectively. Lighter fish (0.10 quantiles) in populations with less to greater weights were 100–133 g and 259–364 g, at the same total lengths, respectively. We provide statistical code and tabled values that can be used as reference standards as a replacement for standard weight equations for comparing new samples of fishes. We also quantified annual variation in condition within two intensively sampled populations, the Kuparuk River in Alaska (25 years, 1985–2012) and Red Rock Creek, Montana (12 years, 1994–2016). Annual distributions of weights at length were greater for Arctic Grayling in Red Rock Creek than those in the Kuparuk River. Variation in predicted distributions of weights at length among years was also greater for Red Rock Creek, although fish from both locations exhibited substantial among-year variation that exceeded half the variation estimated among geographic locations.

Conclusion
Our estimates indicated that there was substantial spatial and temporal variation in condition of Arctic Grayling that might be investigated further to help understand constraints on populations to aid conservation and management of the species. Our summary approach can be easily extended to other investigations of fish weight at length relationships where multiple quantile estimates are made across many levels of grouping factors.