This paper presents an advanced approach to Mendelian Randomization (MR) by utilizing Functional Instrumental Variables (FIVs) to investigate causal relationships. Traditional MR, which primarily uses single nucleotide polymorphisms (SNPs) as instrumental variables (IVs), often faces challenges such as weak instruments and pleiotropy. To overcome these limitations, the paper proposes a novel Functional Mendelian Randomization (FMR) framework via FIVs. In this framework, SNPs are modeled as a function of their genomic locations within a gene, and this function serves as a FIV. The proposed approach effectively adjusts for the presence of functional direct effects (pleiotropy), enhancing the accuracy of causal inference. A ”smoothness” assumption, as the generalization of the InSIDE(Instrument Strength Independent of Direct Effect) assumption, is proposed to ensure the identifiability of the causality effect. Additionally, an Evalue type statistic is introduced to quantify the robustness of the assumption. The paper develops both theoretical foundations and simulation studies to validate the proposed methodology. The method is applied to the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database to examine the association between gene expressions and AD-related metrics, an area not previously studied due to the pleiotropic effects often exhibited by gene expression as exposure. The proposed method identifies the causal effect of MPHOSPH9 and APOE gene expression on cerebrospinal fluid β-Amyloid 42 (Aβ42). 

 

Keywords— Causal Inference, Functional Data Analysis, Instrumental Variables, Mendelian Randomization, Pleiotropy