A Review of Statistical Methods for Handling Nonignorable Missing Data using Instrument Approach

Nonignorable missing data, where the probability of missingness depends on unobserved values, presents a significant challenge in statistical analysis. Traditional methods often rely on strong parametric assumptions that are difficult to verify and may lead to biased estimates if misspecified. Recent advances have introduced the concept of a nonresponse instrument or shadow variable as a powerful tool to enhance model identifiability without requiring full parametric specification. This paper provides a comprehensive review of statistical methods that leverage instrumental variables to address nonignorable missingness, focusing on two predominant semiparametric frameworks: one with a parametric data model and a nonparametric propensity model, and the other with a parametric propensity model and a nonparametric data model. We discuss key developments, methodological insights, and remaining challenges in this rapidly evolving field.