The tree-structured varying coefficient (TSVC) model is a flexible approach for generalized regression, where the linear effects of the covariates are allowed to vary with the values of effect modifiers. Relevant effect modifiers and interactions are identified using recursive partitioning. In TSVC models, analogously to other semi- and nonparametric regression approaches, one needs to account for the cost of data-driven model building when deriving the model degrees of freedom (DoF). To address this issue, we develop an easy-to-apply formula to approximate the DoF of a TSVC model. This formula is employed for model selection based on the Bayesian information criterion (BIC) and compared to the naive solution, setting the DoF to the number of free model parameters, in a simulation study. To illustrate the proposed DoF method, TSVC models using BIC-based selection were fitted to data from the Survey of Health, Ageing, and Retirement in Europe. Results indicated that calculation of the DoF using the proposed formula resulted in more accurate selection results with improved predictive ability.

翻译：树结构变系数（Tree-Structured Varying Coefficient, TSVC）模型是广义回归的一种灵活方法，其中协变量的线性效应允许随效应修饰变量的取值而变化。通过递归分割识别相关的效应修饰变量及其交互作用。在TSVC模型中，与其他半参数和非参数回归方法类似，在推导模型自由度（Degrees of Freedom, DoF）时需考虑数据驱动模型构建的成本。为解决此问题，我们开发了一个易于应用的公式来近似TSVC模型的自由度。该公式用于基于贝叶斯信息准则（BIC）的模型选择，并在模拟研究中与将自由度设置为自由模型参数数量的朴素解进行比较。为说明所提出的自由度方法，我们将基于BIC选择的TSVC模型拟合至欧洲健康、老龄化和退休调查数据。结果表明，使用所提出的公式计算自由度能够实现更准确的选择结果，并提升预测能力。