Statistical inference for partially shape-constrained function-on-scalar linear regression models

Functional linear regression models are widely used to link functional/longitudinal outcomes with multiple scalar predictors, identifying time-varying covariate effects through regression coefficient functions. Beyond assessing statistical significance, characterizing the shapes of coefficient functions is crucial for drawing interpretable scientific conclusions. Existing studies on shape-constrained analysis primarily focus on global shapes, which require strict prior knowledge of functional relationships across the entire domain. This often leads to misspecified regression models due to a lack of prior information, making them impractical for real-world applications. To address this, a flexible framework is introduced to identify partial shapes in regression coefficient functions. The proposed partial shape-constrained analysis enables researchers to validate functional shapes within a targeted sub-domain, avoiding the misspecification of shape constraints outside the sub-domain of interest. The method also allows for testing different sub-domains for individual covariates and multiple partial shape constraints across composite sub-domains. Our framework supports both kernel- and spline-based estimation approaches, ensuring robust performance with flexibility in computational preference. Finite-sample experiments across various scenarios demonstrate that the proposed framework significantly outperforms the application of global shape constraints to partial domains in both estimation and inference procedures. The inferential tool particularly maintains the type I error rate at the nominal significance level and exhibits increasing power with larger sample sizes, confirming the consistency of the test procedure. The practicality of partial shape-constrained inference is demonstrated through two applications: a clinical trial on NeuroBloc for type A-resistant cervical dystonia and the National Institute of Mental Health Schizophrenia Study.