# MACFIV: a novel framework for nonlinear causal inference in the body mass index–hypertension relationship with many weak and pleiotropic genetic instruments

## TL;DR

This paper introduces a new method to better understand the nonlinear relationship between body mass index and hypertension using genetic data.

## Contribution

The novel framework MACFIV improves causal inference by handling weak and pleiotropic genetic instruments with a two-stage model-averaged control function approach.

## Key findings

- MACFIV effectively estimates nonlinear causal relationships using weak and pleiotropic genetic instruments.
- Application to real data shows a U-shaped relationship between body mass index and hypertension.
- The method demonstrates robust performance in simulations and real-world datasets.

## Abstract

Causal inference is an essential approach for understanding biological processes. Traditional causal inference methods assume a linear relationship between different biological traits, whereas their true causal relationship may be nonlinear, such as U-shaped. Moreover, when the instrument set includes weak and pleiotropic genetic instruments, accurately capturing the shape of these relationships becomes challenging. To address these issues, we propose model-averaged control function-based instrumental variable regression, a two-stage framework based on a model-averaged control function approach to estimate the marginal effect function, which represents the derivative of the causal relationship. In the first stage, a model averaging technique is employed to estimate the control function, thereby reducing weak genetic instrument bias. In the second stage, B-spline approximation is applied to estimate the marginal effect function, while SCAD penalization is used to minimize pleiotropic instrument bias. We establish the asymptotic properties of the proposed estimator and demonstrate its robust performance through simulations. Application to the Atherosclerosis Risk in Communities dataset highlights a nonlinear causal relationship between body mass index and hypertension, with the proposed method effectively estimating the specific shape and trend of the relationship.