Prior work has shown that combining bootstrap imputation with tree-based machine learning variable selection methods can provide good performances achievable on fully observed data when covariate and outcome data are missing at random (MAR). This approach however is computationally expensive, especially on large-scale datasets. We propose an inference-based method, called RR-BART, which leverages the likelihood-based Bayesian machine learning technique, Bayesian additive regression trees, and uses Rubin's rule to combine the estimates and variances of the variable importance measures on multiply imputed datasets for variable selection in the presence of MAR data. We conduct a representative simulation study to investigate the practical operating characteristics of RR-BART, and compare it with the bootstrap imputation based methods. We further demonstrate the methods via a case study of risk factors for 3-year incidence of metabolic syndrome among middle-aged women using data from the Study of Women's Health Across the Nation (SWAN). The simulation study suggests that even in complex conditions of nonlinearity and nonadditivity with a large percentage of missingness, RR-BART can reasonably recover both prediction and variable selection performances, achievable on the fully observed data. RR-BART provides the best performance that the bootstrap imputation based methods can achieve with the optimal selection threshold value. In addition, RR-BART demonstrates a substantially stronger ability of detecting discrete predictors. Furthermore, RR-BART offers substantial computational savings. When implemented on the SWAN data, RR-BART adds to the literature by selecting a set of predictors that had been less commonly identified as risk factors but had substantial biological justifications.
We propose Robust Lasso-Zero, an extension of the Lasso-Zero methodology, initially introduced for sparse linear models, to the sparse corruptions problem. We give theoretical guarantees on the sign recovery of the parameters for a slightly simplified version of the estimator, called Thresholded Justice Pursuit. The use of Robust Lasso-Zero is showcased for variable selection with missing values in the covariates. In addition to not requiring the specification of a model for the covariates, nor estimating their covariance matrix or the noise variance, the method has the great advantage of hand…
The missing data issue is ubiquitous in health studies. Variable selection in the presence of both missing covariates and outcomes is an important statistical research topic but has been less studied. Existing literature focuses on parametric regression techniques that provide direct parameter estimates of the regression model. Flexible nonparametric machine learning methods considerably mitigate the reliance on the parametric assumptions, but do not provide as naturally defined variable importance measure as the covariate effect native to parametric models. We investigate a general variable s…
This paper provides an alternative to penalized estimators for estimation and vari- able selection in high dimensional linear regression models with measurement error or missing covariates. We propose estimation via bias corrected least squares after model selection. We show that by separating model selection and estimation, it is possible to achieve an improved rate of convergence of the L2 estimation error compared to the rate sqrt{s log p/n} achieved by simultaneous estimation and variable selection methods such as L1 penalized corrected least squares. If the correct model is selected with …
Two extensions of generalized linear models are considered. In the first one, response variables depend on multiple linear combinations of covariates. In the second one, only response variables are observed while the linear covariates are missing. We derive stochastic Lipschitz continuity results for the loss functions involved in the regression problems and apply them to get bounds on estimation error for Lasso. Multivariate comparison results on Rademacher complexity are obtained as tools to establish the stochastic Lipschitz continuity results.
Multi-task learning is a type of transfer learning that trains multiple tasks simultaneously and leverages the shared information between related tasks to improve the generalization performance. However, missing features in the input matrix is a much more difficult problem which needs to be carefully addressed. Removing records with missing values can significantly reduce the sample size, which is impractical for datasets with large percentage of missing values. Popular imputation methods often distort the covariance structure of the data, which causes inaccurate inference. In this paper we pr…
In the analysis of time-to-event data with multiple causes using a competing risks Cox model, often the cause of failure is unknown for some of the cases. The probability of a missing cause is typically assumed to be independent of the cause given the time of the event and covariates measured before the event occurred. In practice, however, the underlying missing-at-random assumption does not necessarily hold. Motivated by colorectal cancer subtype analysis, we develop semiparametric methods to conduct valid analysis, first when additional auxiliary variables are available for cases only. We c…
Healthcare cost prediction is a challenging task due to the high-dimensionality and high correlation among covariates. Additionally, the skewed, heavy-tailed, and often multi-modal nature of cost data can complicate matters further due to unobserved heterogeneity. In this study, we propose a novel framework for finite mixture regression models that incorporates covariate clustering methods to better account for the effects of clustered covariates on subgroups of the outcome, which enables a more accurate characterization of the complex distribution of the data. The proposed framework can be fo…
Missing values in covariates due to censoring by signal interference or lack of sensitivity in the measuring devices are common in industrial problems. We propose a full Bayesian solution to the prediction problem with an efficient Markov Chain Monte Carlo (MCMC) algorithm that updates all the censored covariate values jointly in a random scan Gibbs sampler. We show that the joint updating of missing covariate values can be at least two orders of magnitude more efficient than univariate updating. This increased efficiency is shown to be crucial for quickly learning the missing covariate values…
We study problems with multiple missing covariates and partially observed responses. We develop a new framework to handle complex missing covariate scenarios via inverse probability weighting, regression adjustment, and a multiply-robust procedure. We apply our framework to three classical problems: the Cox model from survival analysis, missing response, and binary treatment from causal inference. We also discuss how to handle missing covariates in these scenarios, and develop associated identifying theories and asymptotic theories. We apply our procedure to simulations and an Alzheimer's dise…
Attrition is a common occurrence in cluster randomised trials (CRTs) which leads to missing outcome data. Two approaches for analysing such trials are cluster-level analysis and individual-level analysis. This paper compares the performance of unadjusted cluster-level analysis, baseline covariate adjusted cluster-level analysis and linear mixed model (LMM) analysis, under baseline covariate dependent missingness (CDM) in continuous outcomes, in terms of bias, average estimated standard error and coverage probability. The methods of complete case analysis (CCA) and multiple imputation (MI) are …
Missing outcomes are a commonly occurring problem for cluster randomised trials, which can lead to biased and inefficient inference if ignored or handled inappropriately. Two approaches for analysing such trials are cluster-level analysis and individual-level analysis. In this study, we assessed the performance of unadjusted cluster-level analysis, baseline covariate adjusted cluster-level analysis, random effects logistic regression (RELR) and generalised estimating equations (GEE) when binary outcomes are missing under a baseline covariate dependent missingness mechanism. Missing outcomes we…
In this paper, we propose an empirical likelihood-based weighted estimator of regression parameter in quantile regression model with nonignorable missing covariates. The proposed estimator is computationally simple and achieves semiparametric efficiency if the probability of missingness on the fully observed variables is correctly specified. The efficiency gain of the proposed estimator over the complete-case-analysis estimator is quantified theoretically and illustrated via simulation and a real data application.
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 (see, e.g., Zhu and Goldberg, 2009 and Chapelle et al., 2010), we propose a novel approach with which to fill in missing values in covariates that have high missing rat…
We consider the situation of estimating Cox regression in which some covariates are subject to missing, and there exists additional information (including observed event time, censoring indicator and fully observed covariates) which may be predictive of the missing covariates. We propose to use two working regression models: one for predicting the missing covariates and the other for predicting the missing probabilities. For each missing covariate observation, these two working models are used to define a nearest neighbor imputing set. This set is then used to nonparametrically impute covariat…
Researchers regularly perform conditional prediction using imputed values of missing data. However, applications of imputation often lack a firm foundation in statistical theory. This paper originated when we were unable to find analysis substantiating claims that imputation of missing data has good frequentist properties when data are missing at random (MAR). We focused on the use of observed covariates to impute missing covariates when estimating conditional means of the form E(y|x, w). Here y is an outcome whose realizations are always observed, x is a covariate whose realizations are alway…
Complete randomization allows for consistent estimation of the average treatment effect based on the difference in means of the outcomes without strong modeling assumptions on the outcome-generating process. Appropriate use of the pretreatment covariates can further improve the estimation efficiency. However, missingness in covariates is common in experiments and raises an important question: should we adjust for covariates subject to missingness, and if so, how? The unadjusted difference in means is always unbiased. The complete-covariate analysis adjusts for all completely observed covariate…
In this paper, we present three estimators of the ROC curve when missing observations arise among the biomarkers. Two of the procedures assume that we have covariates that allow to estimate the propensity and the estimators are obtained using an inverse probability weighting method or a smoothed version of it. The other one assumes that the covariates are related to the biomarkers through a regression model which enables us to construct convolution--based estimators of the distribution and quantile functions. Consistency results are obtained under mild conditions. Through a numerical study we …
In recent years, spatial and spatio-temporal modeling have become an important area of research in many fields (epidemiology, environmental studies, disease mapping). In this work we propose different spatial models to study hospital recruitment, including some potentially explicative variables. Interest is on the distribution per geographical unit of the ratio between the number of patients living in this geographical unit and the population in the same unit. Models considered are within the framework of Bayesian Latent Gaussian models. Our response variable is assumed to follow a binomial di…
Missing covariates in regression or classification problems can prohibit the direct use of advanced tools for further analysis. Recent research has realized an increasing trend towards the usage of modern Machine Learning algorithms for imputation. It originates from their capability of showing favourable prediction accuracy in different learning problems. In this work, we analyze through simulation the interaction between imputation accuracy and prediction accuracy in regression learning problems with missing covariates when Machine Learning based methods for both, imputation and prediction a…
Randomized Controlled Trials (RCTs) are often considered the gold standard for estimating causal effect, but they may lack external validity when the population eligible to the RCT is substantially different from the target population. Having at hand a sample of the target population of interest allows us to generalize the causal effect. Identifying the treatment effect in the target population requires covariates to capture all treatment effect modifiers that are shifted between the two sets. Standard estimators then use either weighting (IPSW), outcome modeling (G-formula), or combine the tw…
In many applications, survey data are collected from different survey centers in different regions. It happens that in some circumstances, response variables are completely observed while the covariates have missing values. In this paper, we propose a joint spatial regression model for the response variable and missing covariates via a sequence of one-dimensional conditional spatial regression models. We further construct a joint spatial model for missing covariate data mechanisms. The properties of the proposed models are examined and a Markov chain Monte Carlo sampling algorithm is used to s…
Multiple imputation (MI) has become one of the main procedures used to treat missing data, but the guidelines from the methodological literature are not easily transferred to multilevel research. For models including random slopes, proper MI can be difficult, especially when the covariate values are partially missing. In the present article, we discuss applications of MI in multilevel random-coefficient models, theoretical challenges posed by slope variation, and the current limitations of standard MI software. Our findings from three simulation studies suggest that (a) MI is able to recover m…
In this paper we recast the problem of missing values in the covariates of a regression model as a latent Gaussian Markov random field (GMRF) model in a fully Bayesian framework. Our proposed approach is based on the definition of the covariate imputation sub-model as a latent effect with a GMRF structure. We show how this formulation works for continuous covariates and provide some insight on how this could be extended to categorical covariates. The resulting Bayesian hierarchical model naturally fits within the integrated nested Laplace approximation (INLA) framework, which we use for model …
In longitudinal data a response variable is measured over time, or under different conditions, for a cohort of individuals. In many situations all intended measurements are not available which results in missing values. If the missing value is never followed by an observed measurement, this leads to dropout pattern. The missing values could be in the response variable, the covariates or in both. The missingness mechanism is termed non-random when the probability of missingness depends on the missing value and may be on the observed values. In this case the missing values should be considered i…
Handling incomplete outcomes and covariates in cluster-randomized trials: doubly-robust estimation, efficiency considerations, and sensitivity analysis
In this paper, we consider a weighted local linear estimator based on the inverse selection probability for nonparametric regression with missing covariates at random. The asymptotic distribution of the maximal deviation between the estimator and the true regression function is derived and an asymptotically accurate simultaneous confidence band is constructed. The estimator for the regression function is shown to be oracally efficient in the sense that it is uniformly indistinguishable from that when the selection probabilities are known. Finite sample performance is examined via simulation st…
We focus on the problem of generalizing a causal effect estimated on a randomized controlled trial (RCT) to a target population described by a set of covariates from observational data. Available methods such as inverse propensity sampling weighting are not designed to handle missing values, which are however common in both data sources. In addition to coupling the assumptions for causal effect identifiability and for the missing values mechanism and to defining appropriate estimation strategies, one difficulty is to consider the specific structure of the data with two sources and treatment an…
The standard quantile regression model assumes a linear relationship at the quantile of interest and that all variables are observed. We relax these assumptions by considering a partial linear model while allowing for missing linear covariates. To handle the potential bias caused by missing data we propose a weighted objective function using inverse probability weighting. Our work examines estimators using parametric and nonparametric estimates of the missing probability. For variable selection of the linear terms in the presence of missing data we consider a penalized and weighted objective f…
Where performance comparison of healthcare providers is of interest, characteristics of both patients and the health condition of interest must be balanced across providers for a fair comparison. This is unlikely to be feasible within observational data, as patient population characteristics may vary geographically and patient care may vary by characteristics of the health condition. We simulated data for patients and providers, based on a previously utilized real-world dataset, and separately considered both binary and continuous covariate-effects at the upper level. Multilevel latent class (…
Missing data is a common problem in clinical data collection, which causes difficulty in the statistical analysis of such data. To overcome problems caused by incomplete data, we propose a new imputation method called projective resampling imputation mean estimation (PRIME), which can also address ``the curse of dimensionality" problem in imputation with less information loss. We use various sample sizes, missing-data rates, covariate correlations, and noise levels in simulation studies, and all results show that PRIME outperformes other methods such as iterative least-squares estimation (ILSE…
This paper deals with robust marginal estimation under a general regression model when missing data occur in the response and also in some of covariates. The target is a marginal location parameter which is given through an $M-$functional. To obtain robust Fisher--consistent estimators, properly defined marginal distribution function estimators are considered. These estimators avoid the bias due to missing values by assuming a missing at random condition. Three methods are considered to estimate the marginal distribution function which allows to obtain the $M-$location of interest: the well-kn…
Semi-parametric methods are often used for the estimation of intervention effects on correlated outcomes in cluster-randomized trials (CRTs). When outcomes are missing at random (MAR), Inverse Probability Weighted (IPW) methods incorporating baseline covariates can be used to deal with informative missingness. Also, augmented generalized estimating equations (AUG) correct for imbalance in baseline covariates but need to be extended for MAR outcomes. However, in the presence of interactions between treatment and baseline covariates, neither method alone produces consistent estimates for the mar…
This article focuses on measurement error in covariates in regression analyses in which the aim is to estimate the association between one or more covariates and an outcome, adjusting for confounding. Error in covariate measurements, if ignored, results in biased estimates of parameters representing the associations of interest. Studies with variables measured with error can be considered as studies in which the true variable is missing, for either some or all study participants. We make the link between measurement error and missing data and describe methods for correcting for bias due to cov…
We propose a general Bayesian nonparametric (BNP) approach to causal inference in the point treatment setting. The joint distribution of the observed data (outcome, treatment, and confounders) is modeled using an enriched Dirichlet process. The combination of the observed data model and causal assumptions allows us to identify any type of causal effect - differences, ratios, or quantile effects, either marginally or for subpopulations of interest. The proposed BNP model is well-suited for causal inference problems, as it does not require parametric assumptions about the distribution of confoun…
Missing data is a common problem in clinical data collection, which causes difficulty in the statistical analysis of such data. In this article, we consider the problem under a framework of a semiparametric partially linear model when observations are subject to missingness with complex patterns. If the correct model structure of the additive partially linear model is available, we propose to use a new imputation method called Partial Replacement IMputation Estimation (PRIME), which can overcome problems caused by incomplete data in the partially linear model. Also, we use PRIME in conjunction…
Doubly robust target inference for generalized linear regression with completely missing covariates
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Department of Statistics and Data Science, Tsinghua University（清华大学统计与数据科学系）
Augmented transfer regression learning for completely missing covariates
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Department of Statistics, Colorado State University（科罗拉多州立大学统计学系）
Logistic regression is a common classification method in supervised learning. Surprisingly, there are very few solutions for performing logistic regression with missing values in the covariates. We suggest a complete approach based on a stochastic approximation version of the EM algorithm to do statistical inference with missing values including the estimation of the parameters and their variance, derivation of confidence intervals and a model selection procedure. We also tackle the problem of prediction for new observations (on a test set) with missing covariate data. The methodology is compu…
Measurement error (ME) and missing values in covariates are often unavoidable in disciplines that deal with data, and both problems have separately received considerable attention during the past decades. However, while most researchers are familiar with methods for treating missing data, accounting for ME in covariates of regression models is less common. In addition, ME and missing data are typically treated as two separate problems, despite practical and theoretical similarities. Here, we exploit the fact that missing data in a continuous covariate is an extreme case of classical ME, allowi…
Marginal structural models (MSMs) are commonly used to estimate causal intervention effects in longitudinal non-randomised studies. A common issue when analysing data from observational studies is the presence of incomplete confounder data, which might lead to bias in the intervention effect estimates if they are not handled properly in the statistical analysis. However, there is currently no recommendation on how to address missing data on covariates in MSMs under a variety of missingness mechanisms encountered in practice. We reviewed existing methods to handling missing data in MSMs and per…
Integrative analysis of datasets generated by multiple cohorts is a widely-used approach for increasing sample size, precision of population estimators, and generalizability of analysis results in epidemiological studies. However, often each individual cohort dataset does not have all variables of interest for an integrative analysis collected as a part of an original study. Such cohort-level missingness poses methodological challenges to the integrative analysis since missing variables have traditionally: (1) been removed from the data for complete case analysis; or (2) been completed by miss…
In many applied fields incomplete covariate vectors are commonly encountered. It is well known that this can be problematic when making inference on model parameters, but its impact on prediction performance is less understood. We develop a method based on covariate dependent partition models that seamlessly handles missing covariates while completely avoiding any type of imputation. The method we develop allows in-sample predictions as well as out-of-sample prediction, even if the missing pattern in the new subjects' incomplete covariate vector was not seen in the training data. Any data type…
