Estimation of Adaptive Truncated Spline Nonparametric Regression Models Based on Linear Mixed Models

Abstract

Spline nonparametric regression is a flexible approach for modeling data that do not follow standard curve patterns. However, a major challenge arises when data exhibit spatial heterogeneity specifically, varying curvature across different locations as seen in drug pharmacokinetics data. Standard Spline approaches with a single global smoothing parameter often fail to capture these characteristics, leading to over smoothing at peak concentrations or undersmoothing during the elimination phase. This study aims to construct an adaptive truncated spline regression estimator using the Linear Mixed Model (LMM) approach. Within this framework, the Spline function is represented as a combination of fixed and random effects, where the variance of the random effects is allowed to vary locally. Parameter estimation is conducted using the Restricted Maximum Likelihood (REML) method. Application to Theophylline concentration data shows that the adaptive model provides the best balance between accuracy and efficiency. This model yielded an AIC value of 447,83 and a GCV of 2,4080, which are comparable to the standard Spline, but with significantly lower complexity (Effective Degrees of Freedom / EDF = 4,46) compared to the standard Spline (EDF = 9.26). These results demonstrate that the adaptive method is capable of producing a parsimonious and biologically representative model.