Sufficient Dimension Reduction With Missing Predictors

• Published in: Journal of the American Statistical Association Vol. 103; no. 482; pp. 822 - 831
• Main Authors: Li, Lexin, Lu, Wenbin
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.06.2008; American Statistical Association; Taylor & Francis Ltd
• Subjects: Analytical estimating; Applications; Covariance; Data analysis; Data lines; Dimensional analysis; Dimensionality reduction; Double robustness; Estimate reliability; Estimation; Estimation methods; Estimators; Exact sciences and technology; General topics; Linear inference, regression; Linear regression; Mathematics; Missing at random; Missing covariates; Missing data; Modeling; Multivariate analysis; Parametric models; Probability and statistics; Reduction; Regression analysis; Sciences and techniques of general use; Sliced inverse regression; Statistical analysis; Statistical inference; Statistical methods; Statistical models; Statistics; Sufficient dimension reduction; Theory and Methods; United States > US; Variance estimation; Missing at random; Probability distribution; Parametric model; Asymptotic convergence; Variance; Parametric method; Sufficient dimension reduction; Missing covariates; Data analysis; Covariance analysis; Average; Probability; Variance analysis; Mean estimation; Data reduction; Statistical method; Statistical regression; Dimension reduction; Reduction method; Simulation; Double robustness; Observation data; Application; Sliced inverse regression
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/016214508000000283

In high-dimensional data analysis, sufficient dimension reduction (SDR) methods are effective in reducing the predictor dimension, while retaining full regression information and imposing no parametric models. However, it is common in high-dimensional data that a subset of predictors may have missing observations. Existing SDR methods resort to the complete-case analysis by removing all the subjects with missingness in any of the predictors under inquiry. Such an approach does not make effective use of the data and is valid only when missingness is independent of both observed and unobserved quantities. In this article, we propose a new class of SDR estimators under a more general missingness mechanism that allows missingness to depend on the observed data. We focus on a widely used SDR method, sliced inverse regression, and propose an augmented inverse probability weighted sliced inverse regression estimator (AIPW-SIR). We show that AIPW-SIR is doubly robust and asymptotically consistent and demonstrate that AIPW-SIR is more effective than the complete-case analysis through both simulations and real data analysis. We also outline the extension of the AIPW strategy to other SDR methods, including sliced average variance estimation and principal Hessian directions.