Shape Mediation Analysis in Alzheimer's Disease Studies

As a crucial tool in neuroscience, mediation analysis has been developed and widely adopted to elucidate the role of intermediary variables derived from neuroimaging data. Typically, structural equation models (SEMs) are employed to investigate the influences of exposures on outcomes, with model coefficients being interpreted as causal effects. While existing SEMs have proven to be effective tools for mediation analysis involving various neuroimaging-related mediators, limited research has explored scenarios where these mediators are derived from the shape space. In addition, the linear relationship assumption adopted in existing SEMs may lead to substantial efficiency losses and decreased predictive accuracy in real-world applications. To address these challenges, we introduce a novel framework for shape mediation analysis, designed to explore the causal relationships between genetic exposures and clinical outcomes, whether mediated or unmediated by shape-related factors while accounting for potential confounding variables. Within our framework, we apply the square-root velocity function to extract elastic shape representations, which reside within the linear Hilbert space of square-integrable functions. Subsequently, we introduce a two-layer shape regression model to characterize the relationships among neurocognitive outcomes, elastic shape mediators, genetic exposures, and clinical confounders. Both estimation and inference procedures are established for unknown parameters along with the corresponding causal estimands. The asymptotic properties of estimated quantities are investigated as well. Both simulated studies and real-data analyses demonstrate the superior performance of our proposed method in terms of estimation accuracy and robustness when compared to existing approaches for estimating causal estimands.