Fiducial Inference for Mixed-Effects Models: A Frequentist Advancement for Small-Sample Problems

This dissertation develops a comprehensive fiducial inference framework for mixed-effects models, offering a robust frequentist alternative for statistical inference in small-sample settings. Traditional methods such as maximum likelihood estimation (MLE), Wald-type intervals, and bootstrap techniques often fail to provide accurate interval estimates when sample sizes are limited—a common scenario in epidemiologic studies, rare disease trials, and environmental monitoring. To address these limitations, this work builds on Fisher’s original fiducial argument and its modern extensions, proposing a structural-equation-based approach that avoids reliance on large-sample approximations and prior distributions. The proposed methodology is applied across three key domains. First, in agreement analysis using generalized linear mixed models (GLMMs), fiducial confidence intervals are developed for the concordance correlation coefficient (CCC), a widely used measure of reliability. This is particularly relevant in evaluating the consistency of AI-assisted medical diagnostics with human experts. Second, in the context of environmental and public health, fiducial methods are applied to random-effects calibration models, enabling precise interval estimation of unknown chemical concentrations across multiple laboratories. This approach accounts for both additive and multiplicative sources of measurement error, improving analytical precision in regulatory and exposure assessment settings. Third, an exact fiducial method is introduced for small area estimation (SAE) using linear mixed-effects models, allowing for accurate prediction intervals for domain-specific parameters without relying on asymptotic theory or computationally intensive resampling. Extensive simulation studies demonstrate that the proposed fiducial methods consistently achieve nominal coverage with narrower confidence intervals compared to classical approaches. The methods are shown to be robust across balanced and unbalanced designs, Gaussian and non-Gaussian responses, and both linear and non-linear mixed-effects models. Real-world applications—including interlaboratory studies of cadmium and copper concentrations—further validate the practical utility and computational efficiency of the fiducial framework. This research contributes a unified, scalable fiducial inference methodology that enhances analytical rigor in small-sample scenarios. It offers a valuable toolkit for statisticians and applied researchers working in biostatistics, environmental science, and public health. Future directions include extending the framework to high-dimensional data, deeper hierarchical models, and multiple testing problems.