Handling missing data by deleting completely observed records

When data are missing, analyzing records that are completely observed may cause bias or inefficiency. Existing approaches in handling missing data include likelihood, imputation and inverse probability weighting. In this paper, we propose three estimators inspired by deleting some completely observed data in the regression setting. First, we generate artificial observation indicators that are independent of outcome given the observed data and draw inferences conditioning on the artificial observation indicators. Second, we propose a closely related weighting method. The proposed weighting method has more stable weights than those of the inverse probability weighting method (Zhao, L., Lipsitz, S., 1992. Designs and analysis of two-stage studies. Statistics in Medicine 11, 769–782). Third, we improve the efficiency of the proposed weighting estimator by subtracting the projection of the estimating function onto the nuisance tangent space. When data are missing completely at random, we show that the proposed estimators have asymptotic variances smaller than or equal to the variance of the estimator obtained from using completely observed records only. Asymptotic relative efficiency computation and simulation studies indicate that the proposed weighting estimators are more efficient than the inverse probability weighting estimators under wide range of practical situations especially when the missingness proportion is large.     

A comparison study of nonparametric imputation methods

Consider estimation of a population mean of a response variable when the observations are missing at random with respect to the covariate. Two common approaches to imputing the missing values are the nonparametric regression weighting method and the Horvitz-Thompson (HT) inverse weighting approach. The regression approach includes the kernel regression imputation and the nearest neighbor imputation. The HT approach, employing inverse kernel-estimated weights, includes the basic estimator, the ratio estimator and the estimator using inverse kernel-weighted residuals. Asymptotic normality of the nearest neighbor imputation estimators is derived and compared to kernel regression imputation estimator under standard regularity conditions of the regression function and the missing pattern function. A comprehensive simulation study shows that the basic HT estimator is most sensitive to discontinuity in the missing data patterns, and the nearest neighbors estimators can be insensitive to missing data patterns unbalanced with respect to the distribution of the covariate. Empirical studies show that the nearest neighbor imputation method is most effective among these imputation methods for estimating a finite population mean and for classifying the species of the iris flower data.     

Proportional hazards model for competing risks data with missing cause of failure

We consider the semiparametric proportional hazards model for the cause-specific hazard function in analysis of competing risks data with missing cause of failure. The inverse probability weighted equation and augmented inverse probability weighted equation are proposed for estimating the regression parameters in the model, and their theoretical properties are established for inference. Simulation studies demonstrate that the augmented inverse probability weighted estimator is doubly robust and the proposed method is appropriate for practical use. The simulations also compare the proposed estimators with the multiple imputation estimator of Lu and Tsiatis (2001). The application of the proposed method is illustrated using data from a bone marrow transplant study.     

Linear regression analysis of survival data with missing censoring indicators

Linear regression analysis has been studied extensively in a random censorship setting, but typically all of the censoring indicators are assumed to be observed. In this paper, we develop synthetic data methods for estimating regression parameters in a linear model when some censoring indicators are missing. We define estimators based on regression calibration, imputation, and inverse probability weighting techniques, and we prove all three estimators are asymptotically normal. The finite-sample performance of each estimator is evaluated via simulation. We illustrate our methods by assessing the effects of sex and age on the time to non-ambulatory progression for patients in a brain cancer clinical trial.     

Augmented inverse probability weighted fractional imputation in quantile regression

By employing all the observed information and the optimal augmentation term, we propose an augmented inverse probability weighted fractional imputation method (AFI) to handle covariates missing at random in quantile regression. Compared with the existing completely case analysis, inverse probability weighting, multiple imputation and fractional imputation based on quantile regression model with missing covarites, we carry out simulation study to investigate its performance in estimation accuracy and efficiency, computational efficiency and estimation robustness. We also talk about the influence of imputation replicates in our AFI. Finally, we apply our methodology to part of the National Health and Nutrition Examination Survey data.     

Estimation of average treatment effects based on parametric propensity score model

In this paper, the estimation of average treatment effects is examined given that the propensity score is of a parametric form with some unknown parameters. Under the assumption that the treatment is ignorable given some observed characteristics, the MLEs for those unknown parameters in the probability assignment model have been achieved firstly and then three estimators have been defined by the inverse probability weighted, regression and imputation methods, respectively. All the estimators are shown asymptotically normal and more importantly, the substantial efficiency gains of the first two estimates have been obtained theoretically compared with the existing estimators in Hahn (1998) and Hirano et al. (2003), i.e., the inverse weighted probability estimator and the regression estimator have smaller asymptotic variances. Our simulation analysis verifies the theoretical results in terms of biases, SEs and MSEs.     

Inverse Probability Weighting with Missing Predictors of Treatment Assignment or Missingness

Inverse probability weighting (IPW) can deal with confounding in non randomized studies. The inverse weights are probabilities of treatment assignment (propensity scores), estimated by regressing assignment on predictors. Problems arise if predictors can be missing. Solutions previously proposed include assuming assignment depends only on observed predictors and multiple imputation (MI) of missing predictors. For the MI approach, it was recommended that missingness indicators be used with the other predictors. We determine when the two MI approaches, (with/without missingness indicators) yield consistent estimators and compare their efficiencies.We find that, although including indicators can reduce bias when predictors are missing not at random, it can induce bias when they are missing at random. We propose a consistent variance estimator and investigate performance of the simpler Rubin’s Rules variance estimator. In simulations we find both estimators perform well. IPW is also used to correct bias when an analysis model is fitted to incomplete data by restricting to complete cases. Here, weights are inverse probabilities of being a complete case. We explain how the same MI methods can be used in this situation to deal with missing predictors in the weight model, and illustrate this approach using data from the National Child Development Survey.     

Estimating treatment effects on the marginal recurrent event mean in the presence of a terminating event

In biomedical studies where the event of interest is recurrent (e.g., hospitalization), it is often the case that the recurrent event sequence is subject to being stopped by a terminating event (e.g., death). In comparing treatment options, the marginal recurrent event mean is frequently of interest. One major complication in the recurrent/terminal event setting is that censoring times are not known for subjects observed to die, which renders standard risk set based methods of estimation inapplicable. We propose two semiparametric methods for estimating the difference or ratio of treatment-specific marginal mean numbers of events. The first method involves imputing unobserved censoring times, while the second methods uses inverse probability of censoring weighting. In each case, imbalances in the treatment-specific covariate distributions are adjusted out through inverse probability of treatment weighting. After the imputation and/or weighting, the treatment-specific means (then their difference or ratio) are estimated nonparametrically. Large-sample properties are derived for each of the proposed estimators, with finite sample properties assessed through simulation. The proposed methods are applied to kidney transplant data.     

Nonparametric estimation in the illness-death model using prevalent data

We study nonparametric estimation of the illness-death model using left-truncated and right-censored data. The general aim is to estimate the multivariate distribution of a progressive multi-state process. Maximum likelihood estimation under censoring suffers from problems of uniqueness and consistency, so instead we review and extend methods that are based on inverse probability weighting. For univariate left-truncated and right-censored data, nonparametric maximum likelihood estimation can be considerably improved when exploiting knowledge on the truncation distribution. We aim to examine the gain in using such knowledge for inverse probability weighting estimators in the illness-death framework. Additionally, we compare the weights that use truncation variables with the weights that integrate them out, showing, by simulation, that the latter performs more stably and efficiently. We apply the methods to intensive care units data collected in a cross-sectional design, and discuss how the estimators can be easily modified to more general multi-state models.     

Proportional hazards regression with interval-censored and left-truncated data

This paper considers the estimation of the regression coefficients in the Cox proportional hazards model with left-truncated and interval-censored data. Using the approaches of Pan [A multiple imputation approach to Cox regression with interval-censored data, Biometrics 56 (2000), pp. 199–203] and Heller [Proportional hazards regression with interval censored data using an inverse probability weight, Lifetime Data Anal. 17 (2011), pp. 373–385], we propose two estimates of the regression coefficients. The first estimate is based on a multiple imputation methodology. The second estimate uses an inverse probability weight to select event time pairs where the ordering is unambiguous. A simulation study is conducted to investigate the performance of the proposed estimators. The proposed methods are illustrated using the Centers for Disease Control and Prevention (CDC) acquired immunodeficiency syndrome (AIDS) Blood Transfusion Data.     

Model selection and model averaging for semiparametric partially linear models with missing data

We study model selection and model averaging in semiparametric partially linear models with missing responses. An imputation method is used to estimate the linear regression coefficients and the nonparametric function. We show that the corresponding estimators of the linear regression coefficients are asymptotically normal. Then a focused information criterion and frequentist model average estimators are proposed and their theoretical properties are established. Simulation studies are performed to demonstrate the superiority of the proposed methods over the existing strategies in terms of mean squared error and coverage probability. Finally, the approach is applied to a real data case.     

Dimension reduction in estimating equations with covariates missing at random

To estimate parameters defined by estimating equations with covariates missing at random, we consider three bias-corrected nonparametric approaches based on inverse probability weighting, regression and augmented inverse probability weighting. However, when the dimension of covariates is not low, the estimation efficiency will be affected due to the curse of dimensionality. To address this issue, we propose a two-stage estimation procedure by using the dimension-reduced kernel estimation in conjunction with bias-corrected estimating equations. We show that the resulting three estimators are asymptotically equivalent and achieve the desirable properties. The impact of dimension reduction in nonparametric estimation of parameters is also investigated. The finite-sample performance of the proposed estimators is studied through simulation, and an application to an automobile data set is also presented.     

Ensemble Approaches to Estimating the Population Mean with Missing Response

We propose new ensemble approaches to estimate the population mean for missing response data with fully observed auxiliary variables. We first compress the working models according to their categories through a weighted average, where the weights are proportional to the square of the least‐squares coefficients of model refitting. Based on the compressed values, we develop two ensemble frameworks, under which one is to adjust weights in the inverse probability weighting procedure and the other is built upon an additive structure by reformulating the augmented inverse probability weighting function. The asymptotic normality property is established for the proposed estimators through the theory of estimating functions with plugged‐in nuisance parameter estimates. Simulation studies show that the new proposals have substantial advantages over existing ones for small sample sizes, and an acquired immune deficiency syndrome data example is used for illustration.     

The weighted least square based estimators with censoring indicators missing at random

In this paper, we study linear regression analysis when some of the censoring indicators are missing at random. We define regression calibration estimate, imputation estimate and inverse probability weighted estimate for the regression coefficient vector based on the weighted least squared approach due to Stute (1993), and prove all the estimators are asymptotically normal. A simulation study was conducted to evaluate the finite properties of the proposed estimators, and a real data example is provided to illustrate our methods.     

Combining Inverse Probability Weighting and Multiple Imputation to Improve Robustness of Estimation

Inverse probability weighting (IPW) and multiple imputation are two widely adopted approaches dealing with missing data. The former models the selection probability, and the latter models data distribution. Consistent estimation requires correct specification of corresponding models. Although the augmented IPW method provides an extra layer of protection on consistency, it is usually not sufficient in practice as the true data‐generating process is unknown. This paper proposes a method combining the two approaches in the same spirit of calibration in sampling survey literature. Multiple models for both the selection probability and data distribution can be simultaneously accounted for, and the resulting estimator is consistent if any model is correctly specified. The proposed method is within the framework of estimating equations and is general enough to cover regression analysis with missing outcomes and/or missing covariates. Results on both theoretical and numerical investigation are provided.     

Mean response estimation with missing response in the presence of high-dimensional covariates

This paper studies the problem of mean response estimation where missingness occurs to the response but multiple-dimensional covariates are observable. Two main challenges occur in this situation: curse of dimensionality and model specification. The non parametric imputation method relieves model specification but suffers curse of dimensionality, while some model-based methods such as inverse probability weighting (IPW) and augmented inverse probability weighting (AIPW) methods are the opposite. We propose a unified non parametric method to overcome the two challenges with the aiding of sufficient dimension reduction. It imposes no parametric structure on propensity score or conditional mean response, and thus retains the non parametric flavor. Moreover, the estimator achieves the optimal efficiency that a double robust estimator can attain. Simulations were conducted and it demonstrates the excellent performances of our method in various situations.     

Using response probability and ratio imputation in the estimation under callbacks

Although the response model has been frequently applied to nonresponse weighting adjustment or imputation, the estimation under callbacks has been relatively underdeveloped in the response model. We propose an estimator under callbacks using both the response probability and the ratio imputation and a replication variance estimator of the estimator. We also study the estimation of the response probability. A simulation study illustrates our technique.     

Evaluating bias correction in weighted proportional hazards regression

Often in observational studies of time to an event, the study population is a biased (i.e., unrepresentative) sample of the target population. In the presence of biased samples, it is common to weight subjects by the inverse of their respective selection probabilities. Pan and Schaubel (Can J Stat 36:111–127, 2008) recently proposed inference procedures for an inverse selection probability weighted (ISPW) Cox model, applicable when selection probabilities are not treated as fixed but estimated empirically. The proposed weighting procedure requires auxiliary data to estimate the weights and is computationally more intense than unweighted estimation. The ignorability of sample selection process in terms of parameter estimators and predictions is often of interest, from several perspectives: e.g., to determine if weighting makes a significant difference to the analysis at hand, which would in turn address whether the collection of auxiliary data is required in future studies; to evaluate previous studies which did not correct for selection bias. In this article, we propose methods to quantify the degree of bias corrected by the weighting procedure in the partial likelihood and Breslow-Aalen estimators. Asymptotic properties of the proposed test statistics are derived. The finite-sample significance level and power are evaluated through simulation. The proposed methods are then applied to data from a national organ failure registry to evaluate the bias in a post-kidney transplant survival model.     

Semiparametric Analysis of Isotonic Errors-in-Variables Regression Models with Missing Response

This article is concerned with the estimation problem in the semiparametric isotonic regression model when the covariates are measured with additive errors and the response is missing at random. An inverse marginal probability weighted imputation approach is developed to estimate the regression parameters and a least-square approach under monotone constraint is employed to estimate the functional component. We show that the proposed estimator of the regression parameter is root-n consistent and asymptotically normal and the isotonic estimator of the functional component, at a fixed point, is cubic root-n consistent. A simulation study is conducted to examine the finite-sample properties of the proposed estimators. A data set is used to demonstrate the proposed approach.     

Effects of correlated covariates on the asymptotic efficiency of matching and inverse probability weighting estimators for causal inference

In observational studies, the overall aim when fitting a model for the propensity score is to reduce bias for an estimator of the causal effect. To make the assumption of an unconfounded treatment plausible researchers might include many, possibly correlated, covariates in the propensity score model. In this paper, we study how the asymptotic efficiency of matching and inverse probability weighting estimators for average causal effects change when the covariates are correlated. We investigate the case with multivariate normal covariates, a logistic model for the propensity score and linear models for the potential outcomes and show results under different model assumptions. We show that the correlation can both increase and decrease the large sample variances of the estimators, and that the correlation affects the asymptotic efficiency of the estimators differently, both with regard to direction and magnitude. Moreover, the strength of the confounding towards the outcome and the treatment plays an important role.