Abstract
Neurodegenerative diseases such as Alzheimer's and Parkinson's often exhibit complex, multivariate longitudinal outcomes that require advanced statistical methods to comprehensively evaluate treatment efficacy. The Longitudinal Rank Sum Test (LRST) offers a nonparametric framework to assess global treatment effects across multiple longitudinal endpoints without requiring multiplicity corrections. This study develops a robust methodology for power and sample size estimation specific to the LRST, integrating theoretical derivations, asymptotic properties, and practical estimation techniques under large sample conditions. Validation through numerical simulations demonstrates the accuracy of the proposed methods, while real-world applications to clinical trials in Alzheimer's disease (AD) and Parkinson's disease (PD) highlight their practical significance. This framework facilitates the design of efficient, well-powered trials, advancing the evaluation of treatments for complex diseases with multivariate longitudinal outcomes.