Estimation and Inference for High Dimensional Generalized Linear Models: A Splitting and Smoothing Approach

In modern biomedical studies, focus has been shifted to estimate and explain the joint effects of high dimensional predictors (for example, molecular biomarkers) on a disease outcome (for example, onset of cancer). Quantifying the uncertainty of these estimates may aid in prevention strategies or treatment decisions for both patients and physicians. High dimensional inference, in terms of confidence intervals and hypothesis testing, has sparked interest, and much work has been done for linear regression. As there is lack of literature on inference for high dimensional generalized linear models, we propose a novel method with efficient algorithms to address this area, which accommodate a variety of outcomes, including normal, Binomial, and Poisson outcomes. We use multi-sample splitting, along with partial regression, to reduce the high dimensional problem to a sequence of low dimensional modeling. Splitting samples to two parts, we perform variable selection using one part, and conduct partial regression to estimate parameters with the rest of samples. Averaging the estimates over multiple splitting, we obtain the smoothed estimates, which are numerically stable and are consistent and asymptotically normal. We derive asymptotic results, which enable us to construct confidence intervals with proper coverage probabilities for all predictors. We conduct simulations to examine the finite sample performance of our proposal, apply our method to analyze a lung cancer cohort study, and have obtained some biologically meaningful results.