Imputations for High Missing Rate Data in Covariates Via Semi-supervised Learning Approach

Abstract

Advancements in data collection techniques and the heterogeneity of data resources can yield high percentages of missing observations on variables, such as block-wise missing data. Under missing-data scenarios, traditional methods such as the simple average, k-nearest neighbor, multiple, and regression imputations may lead to results that are unstable or unable be computed. Motivated by the concept of semi-supervised learning, we propose a novel approach with which to fill in missing values in covariates that have high missing rates. Specifically, we consider the missing and nonmissing subjects in any covariate as the unlabeled and labeled target outputs, respectively, and treat their corresponding responses as the unlabeled and labeled inputs. This innovative setting allows us to impute a large number of missing data without imposing any model assumptions. In addition, the resulting imputation has a closed form for continuous covariates, and it can be calculated efficiently. An analogous procedure is applicable for discrete covariates. We further employ the nonparametric techniques to show the theoretical properties of imputed covariates. Simulation studies and an online consumer finance example are presented to illustrate the usefulness of the proposed method.

Keywords:

• Block-wise missing
• Cross-validation
• High missing rate data
• Interchangeable imputation
• Semi-supervised imputation

Supplementary Material

The online supplementary material includes four components. Appendix A presents six technical conditions, Appendix B provides the proof of Theorem 1, Appendix C provides additional simulations to assess the robustness of SSI against data non-normality, tuning parameter selection and different missing mechanisms, and Appendix D presents simulation results for sequentially semi-supervised imputation, SSSI. Note that the conditions are used only for the theoretical proofs, and not for the practical imputations of SSI.

Acknowledgments

The authors are grateful to the editor, associate editor, and anonymous referees for their insightful comments and constructive suggestions.

Additional information

Funding

Wei Lan’s research was supported by the National Natural Science Foundation of China (NSFC,71991472, 12171395, 11931014, 71532001), the Joint Lab of Data Science and Business Intelligence at Southwestern University of Finance and Economics, and the Fundamental Research Funds for the Central Universities (JBK1806002). Xuerong Chen’s research was supported by the National Natural Science Foundation of China (NSFC,11871402,11931014) and the Fundamental Research Funds for the Central Universities (JBK1806002). Tao Zou’s research was supported by ANU College of Business and Economics Early Career Researcher Grant, the RSFAS Cross Disciplinary Grant.