Consistency of modified kernel regression estimation for functional data

Abstract

Missing data are common in medical and social science studies and often pose a serious challenge in data analysis. Multiple imputation methods are popular and natural tools for handling missing data, replacing each missing value with a set of plausible values that represent the uncertainty about the underlying values. We consider a case of missing at random (MAR) and investigate the estimation of the marginal mean of an outcome variable in the presence of missing values when a set of fully observed covariates is available. We propose a new nonparametric multiple imputation (MI) approach that uses two working models to achieve dimension reduction and define the imputing sets for the missing observations. Compared with existing nonparametric imputation procedures, our approach can better handle covariates of high dimension, and is doubly robust in the sense that the resulting estimator remains consistent if either of the working models is correctly specified. Compared with existing doubly robust methods, our nonparametric MI approach is more robust to the misspecification of both working models; it also avoids the use of inverse-weighting and hence is less sensitive to missing probabilities that are close to 1. We propose a sensitivity analysis for evaluating the validity of the working models, allowing investigators to choose the optimal weights so that the resulting estimator relies either completely or more heavily on the working model that is likely to be correctly specified and achieves improved efficiency. We investigate the asymptotic properties of the proposed estimator, and perform simulation studies to show that the proposed method compares favorably with some existing methods in finite samples. The proposed method is further illustrated using data from a colorectal adenoma study.

A Nonparametric Multiple Imputation Approach for Data with Missing Covariate Values with Application to Colorectal Adenoma Data

“…It replaces each missing value with a set of plausible values that incorporates the uncertainty about the underlying value to be imputed. We previously proposed a multiple imputation approach to impute event times for censored observations in survival analysis (Hsu et al, 2006) and to impute outcomes for subjects with missing outcomes in estimation of population mean (Long et al, 2012). We proposed using two predictive scores to define a neighborhood to impute event times for each censored case and to impute outcomes for each missing outcome case.…”

A nonparametric multiple imputation approach for missing categorical data

“…Past literature [8, 11] has suggested that the use of two working models for predicting missing values and missingness probabilities in NNMI can induce a double robustness property when Y is continuous. We surmise that would hold as well when Y is categorical.…”

“…For the latter approach, the method using multinomial logistic regressions for the outcome model is denoted as NNMI MLR ( NN , ω 1 ,…; ω M ), and that using cumulative logistic regressions is denoted as NNMI CLR ( NN , ω 1 ,…; ω M ). The previous work on NNMI [8, 11] has demonstrated that bias increases while SD and SE decreases when NN increased. It is suggested that NN= 3 or 5 in general result in slightly lower MSEs.…”

“…Previous literature [8, 11] has suggested that a small, non-zero weight is often necessary for the score from the missingness probability model if Y is continuous. Because M =3 here in our setting, we consider several combinations for ( ω 1 , ω 2 ; ω 3 ) because these weights might reflect the applicant’s belief on the validity of the three working models.…”

“…Similar ideas have been proposed in [11] and [8]. In those contexts, the NNMI approach is applied to impute missing at random continuous variables and produces reasonable results under a variety of model misspecifications.…”