Shape-Based Partially Linear Single-Index Cox Model for Alzheimer's Disease Conversion

Alzheimer's disease (AD) is the major cause of dementia in the elderly, and characterizing the time to conversion to AD is crucial in preventative treatment. While existing statistical methods have proven effective in modeling AD conversion involving various clinical, genetic, and neuroimaging predictors, limited research has explored scenarios where these predictors are shapes derived from the shape space, a nonlinear Hilbert space. In addition, the linear relationship assumption in existing methods may be violated, leading to substantial efficiency losses in real-world applications. To address these challenges, we propose a shape-based partially linear single-index Cox (SPLS-Cox) model that accommodates both scalar and shape predictors. This new development is motivated by establishing the likelihood of conversion to AD in 372 patients with mild cognitive impairment (MCI) enrolled in the Alzheimer's Disease Neuroimaging Initiative, leveraging the early shape-based markers of conversion extracted from the brain white matter region, corpus callosum (CC). These 372 MCI patients were followed over 48 months, during which 161 progressed to AD. Our SPLS-Cox model establishes both the estimation procedure and the pointwise confidence band. Simulation studies are conducted to evaluate the finite-sample performance of our SPLS-Cox. The real application reveals that the CC contour shape is a significant predictor for AD conversion.