On Data Enriched Logistic Regression
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Biomedical researchers usually study the effects of certain exposures on disease risks among a well-defined population. To achieve this goal, the gold standard is to design a trial with an appropriate sample from that population. Due to the high cost of such trials, usually the sample size collected is limited and is not enough to accurately estimate some exposures' effect. In this paper, we discuss how to leverage the information from external `big data' (data with much larger sample size) to improve the estimation accuracy at the risk of introducing small bias. We proposed a family of weighted estimators to balance the bias increase and variance reduction when including the big data. We connect our proposed estimator to the established penalized regression estimators. We derive the optimal weights using both second order and higher order asymptotic expansions. Using extensive simulation studies, we showed that the improvement in terms of mean square error (MSE) for the regression coefficient can be substantial even with finite sample sizes and our weighted method outperformed the existing methods such as penalized regression and James Stein's approach. Also we provide theoretical guarantee that the proposed estimators will never lead to asymptotic MSE larger than the maximum likelihood estimator using small data only in general. We applied our proposed methods to the Asia Cohort Consortium China cohort data to estimate the relationships between age, BMI, smoking, alcohol use and mortality.
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