BOLT-SSI: A Statistical Approach to Screening Interaction Effects for Ultra-High Dimensional Data

Abstract

Detecting interaction effects among predictors on the response variable is a crucial step in various applications. In this paper, we first propose a simple method for sure screening interactions (SSI). Although its computation complexity is $O(p^2n)$, SSI works well for problems of moderate dimensionality (e.g., $p=10^3\sim10^4$), without the heredity assumption. To ultra-high dimensional problems (e.g., $p = 10^6$), motivated by discretization associated Boolean representation and operations and the contingency table for discrete variables, we propose a fast algorithm, named BOLT-SSI. The statistical theory has been established for SSI and BOLT-SSI, guaranteeing their sure screening property. The performance of SSI and BOLT-SSI are evaluated by comprehensive simulation and real case studies. Numerical results demonstrate that SSI and BOLT-SSI can often outperform their competitors in terms of computational efficiency and statistical accuracy. The proposed method can be applied for fully detecting interactions with more than 300,000 predictors. Based on this study, we believe that there is a great need to rethink the relationship between statistical accuracy and computational efficiency. We have shown that the computational performance of a statistical method can often be greatly improved by exploring the advantages of computational architecture with a tolerable loss of statistical accuracy.