MODIFYING MIXED MODEL NEURAL NETWORKS FOR UNIT-LEVEL PROPORTION SMALL AREA ESTIMATION

Abstract

This paper develops a Small Area Estimation Mixed Model Neural Network (SAE-MNN) for unit-level estimation of area-level proportions under clustered binary responses. The method is motivated by a limitation of conventional generalized linear mixed model (GLMM)-based small area estimation, namely the linear specification of the fixed-effects component on the latent scale, which may be inadequate when auxiliary variables exhibit nonlinear effects and higher-order interactions. The proposed model replaces the linear predictor with a neural-network-based nonlinear function while retaining area-level random effects within a likelihood-based mixed-model framework. Thus, SAE-MNN combines nonlinear function approximation with the borrowing-strength mechanism required for coherent small area inference. The method is evaluated through a controlled simulation study and a real-data application. The simulation study considers four scenarios generated by combining two predictor structures (linear and interaction-based nonlinear) with two levels of between-area heterogeneity (small and large), and compares SAE-MNN with a conventional GLMM and a standard deep neural network (DNN). SAE-MNN performs robustly across all scenarios and is especially effective when nonlinear interaction and substantial area-level heterogeneity coexist. In the most complex scenario, SAE-MNN attains the lowest median RRMSE (8.389%), compared with 11.474% for GLMM and 18.826% for DNN, while maintaining empirical coverage between 0.89 and 0.94. In the real-data application, using KSA segment-level harvest proportions to estimate subdistrict-level monthly harvest proportions, SAE-MNN shows the closest overall agreement with the direct estimator, reproduces the main temporal pattern more faithfully than GLMM, and yields moderate shrinkage without excessive smoothing. Overall, the results indicate that SAE-MNN provides a principled compromise between the rigidity of GLMM and the purely predictive character of DNN, thereby extending hybrid small area estimation methodology to binary and proportional outcomes in data-rich settings with nonlinear auxiliary information and area-level heterogeneity.