Semi-supervised estimation of covariance with application to phenome-wide association studies with electronic medical records data

Electronic medical records data are valuable resources for discovery research. They contain detailed phenotypic information on individual patients, opening opportunities for simultaneously studying multiple phenotypes. A useful tool for such simultaneous assessment is the phenome-wide association study, which relates a genomic or biological marker of interest to a wide spectrum of disease phenotypes, typically defined by the diagnostic billing codes. One challenge arises when the biomarker of interest is expensive to measure on the entire electronic medical record cohort. Performing phenome-wide association study based on supervised estimation using only subjects who have marker measurements may yield limited power. In this paper, we focus on the setting where the marker is measured on a small fraction of the patients while a few surrogate markers such as historical measurements of the biomarker are available on a large number of patients. We propose an efficient semi-supervised estimation procedure to estimate the covariance between the biomarker and the billing code, leveraging the surrogate marker information. We employ surrogate marker values to impute the missing outcome via a two-step semi-non-parametric approach and demonstrate that our proposed estimator is always more efficient than the supervised counterpart without requiring the imputation model to be correct. We illustrate the proposed procedure by assessing the association between the C-reactive protein and some inflammatory diseases with an electronic medical record study of inflammatory bowel disease performed with the Partners HealthCare electronic medical record database where C-reactive protein was only measured for a small fraction of the patients due to budget constraints.