On Data-Enriched Logistic Regression

Biomedical researchers typically investigate the effects of specific exposures on disease risks within a well-defined population. The gold standard for such studies is to design a trial with an appropriately sampled cohort. However, due to the high cost of such trials, the collected sample sizes are often limited, making it difficult to accurately estimate the effects of certain exposures. In this paper, we discuss how to leverage the information from external "big data" (datasets with significantly larger sample sizes) to improve the estimation accuracy at the risk of introducing a small amount of bias. We propose a family of weighted estimators to balance bias increase and variance reduction when incorporating the big data. We establish a connection between our proposed estimator and the well-known penalized regression estimators. We derive optimal weights using both second-order and higher-order asymptotic expansions. Through extensive simulation studies, we demonstrate that the improvement in mean square error (MSE) for the regression coefficient can be substantial even with finite sample sizes, and our weighted method outperformed existing approaches such as penalized regression and James-Stein estimator. Additionally, we provide a theoretical guarantee that the proposed estimators will never yield an asymptotic MSE larger than the maximum likelihood estimator using small data only in general. Finally, we apply our proposed methods to the Asia Cohort Consortium China cohort data to estimate the relationships between age, BMI, smoking, alcohol use, and mortality.