Abstract
In this study, we consider the nonparametric quantile regression model with the covariates Missing at Random (MAR). Multiple imputation is becoming an increasingly popular approach for analyzing missing data, which combined with quantile regression is not well-developed. We propose an effective and accurate two-stage multiple imputation method for the model based on the quantile regression, which consists of initial imputation in the first stage and multiple imputation in the second stage. The estimation procedure makes full use of the entire dataset to achieve increased efficiency and we show the proposed two-stage multiple imputation estimator to be asymptotically normal. In simulation study, we compare the performance of the proposed imputation estimator with Complete Case (CC) estimator and other imputation estimators, e.g., the regression imputation estimator and k-Nearest-Neighbor imputation estimator. We conclude that the proposed estimator is robust to the initial imputation and illustrates more desirable performance than other comparative methods. We also apply the proposed multiple imputation method to an AIDS clinical trial data set to show its practical application.
Highlights
Where: m(·) = The unknown real function and Quantile regression has been widely used in analyzing the relationship between response and∈ = The error term covariates since its first introduction in (Koenker andBased on the above model, we consider the followingBassett, 1978)
We propose an effective and accurate two-stage multiple imputation method for the model based on the quantile regression, which consists of initial imputation in the first stage and multiple imputation in the second stage
Via the comparison of Average Mean Square Error (AMSE) values for the 7 estimators under the same sample size, the same missing function and the same quantile level, we conclude that the estimation performance of our proposed estimators TSMI1, TSMI2 and TSMI3 is uniformly better than that of the Complete Case (CC) estimator and the initial imputation estimators
Summary
Nonparametric regression does not assume that the relationship between response and covariates to be linear or satisfy some specified form, which might be more reasonable for most of data set and more flexible than parametric models. We pay more attention to the nonparametric quantile regression model (1.1) with the covariates missing at random, which has the following form Equation (1.2): Qτ (x,z) ≜ Qτ (Y | X = x,Z = z) = cτ + m(x,z). Based on the existing research and methods, we propose a two-stage multiple imputation method for nonparametric quantile regression with missing covariates, which greatly enriches the methods to cope with missing data in quantile regression.
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