Single-Index Measurement Error Jump Regression Model in Alzheimer's Disease Studies

Alzheimer's disease (AD) is the major cause of dementia in the elderly, and investigations on the impact of risk factors on neurocognitive performance are crucial in preventative treatment. While existing statistical regression models, such as single-index models, have proven effective tools for uncovering the relationship between the neurocognitive scores and covariates of interest such as demographic information, clinical variables, and neuroimaging features, limited research has explored scenarios where jump discontinuities exist in the regression patterns and the covariates are unobservable but measured with errors, which are common in real applications. To address these challenges, we propose a single-index measurement error jump regression model (SMEJRM) that can handle both jump discontinuities and measurement errors in image covariates introduced by different image processing software. This development is motivated by data from 168 patients in the Alzheimer's Disease Neuroimaging Initiative. We establish both the estimation procedure and the corresponding asymptotic results. Simulation studies are conducted to evaluate the finite sample performance of our SMEJRM and the estimation procedure. The real application reveals that jump discontinuities do exist in the relationship between neurocognitive scores and some covariates of interest in this study.