Sufficient dimension reduction for censored predictors

Motivated by a study conducted to evaluate the associations of 51 inflammatory markers and lung cancer risk, we propose several approaches of varying computational complexity for analyzing multiple correlated markers that are also censored due to lower and/or upper limits of detection, using likelihood-based sufficient dimension reduction (SDR) methods. We extend the theory and the likelihood-based SDR framework in two ways: (i) we accommodate censored predictors directly in the likelihood, and (ii) we incorporate variable selection. We find linear combinations that contain all the information that the correlated markers have on an outcome variable (i.e., are sufficient for modeling and prediction of the outcome) while accounting for censoring of the markers. These methods yield efficient estimators and can be applied to any type of outcome, including continuous and categorical. We illustrate and compare all methods using data from the motivating study and in simulations. We find that explicitly accounting for the censoring in the likelihood of the SDR methods can lead to appreciable gains in efficiency and prediction accuracy, and also outperformed multiple imputations combined with standard SDR.