Model averaging for estimation of optimum rate and critical values under model uncertainty

Abstract

Fertilizer recommendations rely on fitted yield response models to estimate optimum rates and critical values, but these estimates are sensitive to the assumed functional form. This study evaluated how model uncertainty affects estimates of agronomic optimum nitrogen (N) rate (AONR) and critical soil test values (CSTV) for phosphorus (P), and whether model averaging improves inference. Simulation experiments were conducted using linear plateau, quadratic plateau, quadratic, and Mitscherlich functions as data generating processes. Estimation approaches included single model fits, Akaike information criterion (AIC) weighted model averaging, bootstrap aggregation (bagging), and Bayesian model averaging (BMA). Performance was assessed using bias, variance, and mean squared error (MSE). Two field case studies were used to evaluate practical implications. Single model estimators performed well only under correct specification but showed large bias under misspecification. When the true response was linear plateau, the Mitscherlich model overestimated AONR by 289 kg N ha⁻¹ and produced a MSE of 173,423, compared to 352 for the true model. Model averaging reduced error across approaches, although no single method was uniformly superior. A reduction in (MSE to 4,765 and bias to 55 kg N ha⁻¹ was achieved using BMA, while AIC-weighted averaging and bagging also reduced bias and variance relative to single model estimates. In field data, AONR estimates ranged from 121 to 241 kg N ha⁻¹ across models, whereas model averaged estimates were more stable (163–188 kg N ha⁻¹). Similarly, CSTV estimates ranged from 12 to 27 mg P kg⁻¹ for individual models and were more consistent under model averaging (18–20 mg P kg⁻¹). Model uncertainty is a major source of variability in fertilizer recommendations. Model averaging reduces sensitivity to functional form and provides more stable and reliable estimates of optimal rates and critical values.