Motivated by, but not limited to, association analyses of multiple genetic variants, we propose here a summary statistics‐based regression framework. The proposed method requires only variant‐specific summary statistics, and it unifies earlier methods based on individual‐level data as special cases. The resulting score test statistic, derived from a linear mixed‐effect regression model, inherently transforms the variant‐specific statistics using the precision matrix to improve power for detecting sparse alternatives. Furthermore, the proposed method can incorporate additional variant‐specific information with ease, facilitating omic‐data integration. We study the asymptotic properties of the proposed tests under the null and alternatives, and we investigate efficient P‐value calculation in finite samples. Finally, we provide supporting empirical evidence from extensive simulation studies and two applications.

Motivation: Mendelian randomization (MR) serves as a valuable tool for investigating causal relationships between exposures and disease outcomes in observational studies. However, MR methods, operating under classical assumptions, may yield biased estimates and inflated false-positive causal relationships when faced with realistic and complex correlated horizontal pleiotropy (CHP). While numerous MR methods have emerged to address CHP effects, limited methods can effectively handle relatively large direct effects, commonly known as idiosyncratic pleiotropy. Results: To address this gap, we propose an efficient and robust MR method to account for idiosyncratic and correlated pleiotropy, named RMR-ICP. Furthermore, Our method incorporates linkage disequilibrium structure using paralleled Gibbs sampling to enhance statistical power. The robustness and efficiency of our method are demonstrated through extensive simulation studies and applications. RMR-ICP is first used to analyze the effects of plasma proteins on stroke, followed by its application to conventional stroke risk factors. Our analysis reveals that Selectin E (SELE) exhibits a positive causal effect on the occurrence of any stroke. Only those specifically designed to account for idiosyncratic and CHP identified a significant positive causal effect of myeloperoxidase on ischemic stroke, with RMR-ICP providing stronger statistical evidence. Elevated Natriuretic Peptide B (BNP) levels specifically increase the risk of cardioembolic stroke (CES), though not with other stroke subtypes. This finding is consistent with previous studies suggesting that plasma BNP levels may help distinguish CES from other stroke types. Higher Waist-hip ratio (WHR) levels raise the risk across all stroke types. These findings provide new insights into identifying stroke-related risk factors. Availability and implementation: RMR-ICP is publicly available at https://github.com/QingCheng0218/RMR.ICP.

A model specification test is a statistical procedure used to assess whether a given statistical model accurately represents the underlying data-generating process. The smoothing-based nonparametric specification test is widely used due to its efficiency against “singular” local alternatives. However, large modern datasets create various computational problems when implementing the nonparametric specification test. The divide-and-conquer algorithm is highly effective for handling large datasets, as it can break down a large dataset into more manageable datasets. By applying divide-and-conquer, the nonparametric specification test can handle the computational problems induced by the massive size of the modern datasets, leading to improved scalability and efficiency and reduced processing time. However, the selection of smoothing parameters for optimal power of the distributed algorithm is an important problem. The rate of the smoothing parameter that ensures rate optimality of the test in the context of testing the specification of a nonlinear parametric regression function is studied in the literature. In this paper, we verified the uniqueness of the rate of the smoothing parameter that ensures the rate optimality of divide-and-conquer-based tests. By employing a penalty method to select the smoothing parameter, we obtain a test with an asymptotic normal null distribution and adaptiveness properties. The performance of this test is further illustrated through numerical simulations.