Testing for Joint Significance Using Renewable Estimating Functions

Abstract

Testing for joint statistical significance arises from many applications such as mediation analysis, replicability analysis, and construction of causal knowledge graphs. When a null hypothesis involves multiple parameters, such a task of hypothesis testing is analytically of great difficulty.  Classical procedures, including the Sobel's test and the MaxP test, are known to be conservative in the type I error control, and a unified methodology to address this issue is an important open problem in the statistical literature. In this paper, we propose a new solution using renewable estimating functions (REFs) to construct a pivotal test statistic that is invariant to regions of the null parameters.  Consequently, the proposed method can overcome conservatism and achieve proper control of type I error and moreover improve statistical power. The key analytic behind our methodology resembles stochastic gradient descent, which incrementally decouples null cases to translate a test for complex composite nulls into that for a simple single null. We establish a full theoretical framework for this new methodology, including key large-sample properties. Through extensive simulation studies and real-world data applications, we demonstrate that the proposed method achieves superior finite-sample performance compared to existing alternatives.