Exact one-sided inference for random-effects quantiles in negative binomial regression for ecological count data

Ecological count data often exhibit substantial heterogeneity beyond measured covariates and variable sampling effort. We study negative binomial regression with an additive latent effect on the log-mean and exposure offsets, leaving the random-effects distribution unspecified. Rather than estimating that distribution directly, we develop finite-sample valid one-sided inference for interpretable summaries of latent heterogeneity, with primary focus on heterogeneity quantiles that correspond to multiplicative departures from a covariate-adjusted baseline intensity. Our approach uses sample splitting to separate nuisance estimation from inference, constructs a randomized exceedance statistic on a held-out fold, and calibrates composite null tests through least-favorable mixing configurations. Exact p-values are obtained from Poisson–binomial tail probabilities, and test inversion yields a one-sided lower confidence bound with conditional finite-sample coverage. We also distinguish a sharper bounded calibration, when a scientifically credible upper support bound is available, from a fully distribution-free calibration that remains valid but is more conservative. Simulation experiments under NEON-like designs show conservative coverage and stable behavior across distinct heterogeneity regimes. An application to NEON small mammal live-trapping data illustrates how the proposed lower bound provides a transparent finite-sample statement about residual plot-night heterogeneity after adjusting for seasonality, habitat, site effects, and effort.