A weighted estimating equation for linear regression with missing covariate data

Linear regression is one of the most popular statistical techniques. In linear regression analysis, missing covariate data occur often. A recent approach to analyse such data is a weighted estimating equation. With weighted estimating equations, the contribution to the estimating equation from a complete observation is weighted by the inverse 'probability of being observed'. In this paper, we propose a weighted estimating equation in which we wrongly assume that the missing covariates are multivariate normal, but still produces consistent estimates as long as the probability of being observed is correctly modelled. In simulations, these weighted estimating equations appear to be highly efficient when compared to the most efficient weighted estimating equation as proposed by Robins et al. and Lipsitz et al. However, these weighted estimating equations, in which we wrongly assume that the missing covariates are multivariate normal, are much less computationally intensive than the weighted estimating equations given by Lipsitz et al. We compare the weighted estimating equations proposed in this paper to the efficient weighted estimating equations via an example and a simulation study. We only consider missing data which are missing at random; non-ignorably missing data are not addressed in this paper.     

Multinomial logistic regression with missing outcome data: An application to cancer subtypes

Many diseases such as cancer and heart diseases are heterogeneous and it is of great interest to study the disease risk specific to the subtypes in relation to genetic and environmental risk factors. However, due to logistic and cost reasons, the subtype information for the disease is missing for some subjects. In this article, we investigate methods for multinomial logistic regression with missing outcome data, including a bootstrap hot deck multiple imputation (BHMI), simple inverse probability weighted (SIPW), augmented inverse probability weighted (AIPW), and expected estimating equation (EEE) estimators. These methods are important approaches for missing data regression. The BHMI modifies the standard hot deck multiple imputation method such that it can provide valid confidence interval estimation. Under the situation when the covariates are discrete, the SIPW, AIPW, and EEE estimators are numerically identical. When the covariates are continuous, nonparametric smoothers can be applied to estimate the selection probabilities and the estimating scores. These methods perform similarly. Extensive simulations show that all of these methods yield unbiased estimators while the complete-case (CC) analysis can be biased if the missingness depends on the observed data. Our simulations also demonstrate that these methods can gain substantial efficiency compared with the CC analysis. The methods are applied to a colorectal cancer study in which cancer subtype data are missing among some study individuals.     

Identification of the optimal treatment regimen in the presence of missing covariates

Covariates associated with treatment-effect heterogeneity can potentially be used to make personalized treatment recommendations towards best clinical outcomes. Methods for treatment-selection rule development that directly maximize treatment-selection benefits have attracted much interest in recent years, due to the robustness of these methods to outcome modeling. In practice, the task of treatment-selection rule development can be further complicated by missingness in data. Here, we consider the identification of optimal treatment-selection rules for a binary disease outcome when measurements of an important covariate from study participants are partly missing. Under the missing at random assumption, we develop a robust estimator of treatment-selection rules under the direct-optimization paradigm. This estimator targets the maximum selection benefits to the population under correct specification of at least one mechanism from each of the two sets—missing data or conditional covariate distribution, and treatment assignment or disease outcome model. We evaluate and compare performance of the proposed estimator with alternative direct-optimization estimators through extensive simulation studies. We demonstrate the application of the proposed method through a real data example from an Alzheimer's disease study for developing covariate combinations to guide the treatment of Alzheimer's disease.     

Population-calibrated multiple imputation for a binary/categorical covariate in categorical regression models

Multiple imputation (MI) has become popular for analyses with missing data in medical research. The standard implementation of MI is based on the assumption of data being missing at random (MAR). However, for missing data generated by missing not at random mechanisms, MI performed assuming MAR might not be satisfactory. For an incomplete variable in a given data set, its corresponding population marginal distribution might also be available in an external data source. We show how this information can be readily utilised in the imputation model to calibrate inference to the population by incorporating an appropriately calculated offset termed the “calibrated-δ adjustment.” We describe the derivation of this offset from the population distribution of the incomplete variable and show how, in applications, it can be used to closely (and often exactly) match the post-imputation distribution to the population level. Through analytic and simulation studies, we show that our proposed calibrated-δ adjustment MI method can give the same inference as standard MI when data are MAR, and can produce more accurate inference under two general missing not at random missingness mechanisms. The method is used to impute missing ethnicity data in a type 2 diabetes prevalence case study using UK primary care electronic health records, where it results in scientifically relevant changes in inference for non-White ethnic groups compared with standard MI. Calibrated-δ adjustment MI represents a pragmatic approach for utilising available population-level information in a sensitivity analysis to explore potential departures from the MAR assumption.     

Weighted quantile regression for analyzing health care cost data with missing covariates

Analysis of health care cost data is often complicated by a high level of skewness, heteroscedastic variances and the presence of missing data. Most of the existing literature on cost data analysis have been focused on modeling the conditional mean. In this paper, we study a weighted quantile regression approach for estimating the conditional quantiles health care cost data with missing covariates. The weighted quantile regression estimator is consistent, unlike the naive estimator, and asymptotically normal. Furthermore, we propose a modified BIC for variable selection in quantile regression when the covariates are missing at random. The quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost and is naturally adapted to the skewness and heterogeneity of the cost data. The method is semiparametric in the sense that it does not require to specify the likelihood function for the random error or the covariates. We investigate the weighted quantile regression procedure and the modified BIC via extensive simulations. We illustrate the application by analyzing a real data set from a health care cost study. Copyright © 2013 John Wiley & Sons, Ltd.     

Direct estimation of the area under the receiver operating characteristic curve in the presence of verification bias

The area under a receiver operating characteristic (ROC) curve (AUC) is a commonly used index for summarizing the ability of a continuous diagnostic test to discriminate between healthy and diseased subjects. If all subjects have their true disease status verified, one can directly estimate the AUC nonparametrically using the Wilcoxon statistic. In some studies, verification of the true disease status is performed only for a subset of subjects, possibly depending on the result of the diagnostic test and other characteristics of the subjects. Because estimators of the AUC based only on verified subjects are typically biased, it is common to estimate the AUC from a bias-corrected ROC curve. The variance of the estimator, however, does not have a closed-form expression and thus resampling techniques are used to obtain an estimate. In this paper, we develop a new method for directly estimating the AUC in the setting of verification bias based on U-statistics and inverse probability weighting (IPW). Closed-form expressions for the estimator and its variance are derived. We also show that the new estimator is equivalent to the empirical AUC derived from the bias-corrected ROC curve arising from the IPW approach.     

Power analyses for longitudinal study designs with missing data

Existing methods for power analysis for longitudinal study designs are limited in that they do not adequately address random missing data patterns. Although the pattern of missing data can be assessed during data analysis, it is unknown during the design phase of a study. The random nature of the missing data pattern adds another layer of complexity in addressing missing data for power analysis. In this paper, we model the occurrence of missing data with a two-state, first-order Markov process and integrate the modelling information into the power function to account for random missing data patterns. The Markov model is easily specified to accommodate different anticipated missing data processes. We develop this approach for the two most popular longitudinal models: the generalized estimating equations (GEE) and the linear mixed-effects model under the missing completely at random (MCAR) assumption. For GEE, we also limit our consideration to the working independence correlation model. The proposed methodology is illustrated with numerous examples that are motivated by real study designs.     

Semiparametric regression models for repeated measures of mortal cohorts with non‐monotone missing outcomes and time‐dependent covariates

We propose a semiparametric marginal modeling approach for longitudinal analysis of cohorts with data missing due to death and non‐response to estimate regression parameters interpreted as conditioned on being alive. Our proposed method accommodates outcomes and time‐dependent covariates that are missing not at random with non‐monotone missingness patterns via inverse‐probability weighting. Missing covariates are replaced by consistent estimates derived from a simultaneously solved inverse‐probability‐weighted estimating equation. Thus, we utilize data points with the observed outcomes and missing covariates beyond the estimated weights while avoiding numerical methods to integrate over missing covariates. The approach is applied to a cohort of elderly female hip fracture patients to estimate the prevalence of walking disability over time as a function of body composition, inflammation, and age. Copyright © 2010 John Wiley & Sons, Ltd.     

Quantile regression and empirical likelihood for the analysis of longitudinal data with monotone missing responses due to dropout,with applications to quality of life measurements from clinical trials

The analysis of quality of life (QoL) data can be challenging due to the skewness of responses and the presence of missing data. In this paper, we propose a new weighted quantile regression method for estimating the conditional quantiles of QoL data with responses missing at random. The proposed method makes use of the correlation information within the same subject from an auxiliary mean regression model to enhance the estimation efficiency and takes into account of missing data mechanism. The asymptotic properties of the proposed estimator have been studied and simulations are also conducted to evaluate the performance of the proposed estimator. The proposed method has also been applied to the analysis of the QoL data from a clinical trial on early breast cancer, which motivated this study.     

Analysis of non-ignorable missing and left-censored longitudinal data using a weighted random effects tobit model

In a longitudinal study with response data collected during a hospital stay, observations may be missing because of the subject's discharge from the hospital prior to completion of the study or the death of the subject, resulting in non-ignorable missing data. In addition to non-ignorable missingness, there is left-censoring in the response measurements because of the inherent limit of detection. For analyzing non-ignorable missing and left-censored longitudinal data, we have proposed to extend the theory of random effects tobit regression model to weighted random effects tobit regression model. The weights are computed on the basis of inverse probability weighted augmented methodology. An extensive simulation study was performed to compare the performance of the proposed model with a number of competitive models. The simulation study shows that the estimates are consistent and that the root mean square errors of the estimates are minimal for the use of augmented inverse probability weights in the random effects tobit model. The proposed method is also applied to the non-ignorable missing and left-censored interleukin-6 biomarker data obtained from the Genetic and Inflammatory Markers of Sepsis study.     

A comparison of multiple imputation and fully augmented weighted estimators for Cox regression with missing covariates

Several approaches exist for handling missing covariates in the Cox proportional hazards model. The multiple imputation (MI) is relatively easy to implement with various software available and results in consistent estimates if the imputation model is correct. On the other hand, the fully augmented weighted estimators (FAWEs) recover a substantial proportion of the efficiency and have the doubly robust property. In this paper, we compare the FAWEs and the MI through a comprehensive simulation study. For the MI, we consider the multiple imputation by chained equation and focus on two imputation methods: Bayesian linear regression imputation and predictive mean matching. Simulation results show that the imputation methods can be rather sensitive to model misspecification and may have large bias when the censoring time depends on the missing covariates. In contrast, the FAWEs allow the censoring time to depend on the missing covariates and are remarkably robust as long as getting either the conditional expectations or the selection probability correct due to the doubly robust property. The comparison suggests that the FAWEs show the potential for being a competitive and attractive tool for tackling the analysis of survival data with missing covariates. Copyright © 2010 John Wiley & Sons, Ltd.     

A multiple imputation-based sensitivity analysis approach for data subject to missing not at random

Missingness mechanism is in theory unverifiable based only on observed data. If there is a suspicion of missing not at random, researchers often perform a sensitivity analysis to evaluate the impact of various missingness mechanisms. In general, sensitivity analysis approaches require a full specification of the relationship between missing values and missingness probabilities. Such relationship can be specified based on a selection model, a pattern-mixture model or a shared parameter model. Under the selection modeling framework, we propose a sensitivity analysis approach using a nonparametric multiple imputation strategy. The proposed approach only requires specifying the correlation coefficient between missing values and selection (response) probabilities under a selection model. The correlation coefficient is a standardized measure and can be used as a natural sensitivity analysis parameter. The sensitivity analysis involves multiple imputations of missing values, yet the sensitivity parameter is only used to select imputing/donor sets. Hence, the proposed approach might be more robust against misspecifications of the sensitivity parameter. For illustration, the proposed approach is applied to incomplete measurements of level of preoperative Hemoglobin A1c, for patients who had high-grade carotid artery stenosisa and were scheduled for surgery. A simulation study is conducted to evaluate the performance of the proposed approach.     

Allowing for missing outcome data and incomplete uptake of randomised interventions, with application to an Internet-based alcohol trial

Missing outcome data and incomplete uptake of randomised interventions are common problems, which complicate the analysis and interpretation of randomised controlled trials, and are rarely addressed well in practice. To promote the implementation of recent methodological developments, we describe sequences of randomisation-based analyses that can be used to explore both issues. We illustrate these in an Internet-based trial evaluating the use of a new interactive website for those seeking help to reduce their alcohol consumption, in which the primary outcome was available for less than half of the participants and uptake of the intervention was limited. For missing outcome data, we first employ data on intermediate outcomes and intervention use to make a missing at random assumption more plausible, with analyses based on general estimating equations, mixed models and multiple imputation. We then use data on the ease of obtaining outcome data and sensitivity analyses to explore departures from the missing at random assumption. For incomplete uptake of randomised interventions, we estimate structural mean models by using instrumental variable methods. In the alcohol trial, there is no evidence of benefit unless rather extreme assumptions are made about the missing data nor an important benefit in more extensive users of the intervention. These findings considerably aid the interpretation of the trial's results. More generally, the analyses proposed are applicable to many trials with missing outcome data or incomplete intervention uptake. To facilitate use by others, Stata code is provided for all methods.     

Copula selection models for non-Gaussian outcomes that are missing not at random

Missing not at random (MNAR) data pose key challenges for statistical inference because the substantive model of interest is typically not identifiable without imposing further (eg, distributional) assumptions. Selection models have been routinely used for handling MNAR by jointly modeling the outcome and selection variables and typically assuming that these follow a bivariate normal distribution. Recent studies have advocated parametric selection approaches, for example, estimated by multiple imputation and maximum likelihood, that are more robust to departures from the normality assumption compared with those assuming that nonresponse and outcome are jointly normally distributed. However, the proposed methods have been mostly restricted to a specific joint distribution (eg, bivariate t-distribution). This paper discusses a flexible copula-based selection approach (which accommodates a wide range of non-Gaussian outcome distributions and offers great flexibility in the choice of functional form specifications for both the outcome and selection equations) and proposes a flexible imputation procedure that generates plausible imputed values from the copula selection model. A simulation study characterizes the relative performance of the copula model compared with the most commonly used selection models for estimating average treatment effects with MNAR data. We illustrate the methods in the REFLUX study, which evaluates the effect of laparoscopic surgery on long-term quality of life in patients with reflux disease. We provide software code for implementing the proposed copula framework using the R package GJRM .     

Incorporating missingness for estimation of marginal regression models with multiple source predictors

Multiple informant data refers to information obtained from different individuals or sources used to measure the same construct; for example, researchers might collect information regarding child psychopathology from the child's teacher and the child's parent. Frequently, studies with multiple informants have incomplete observations; in some cases the missingness of informants is substantial. We introduce a Maximum Likelihood (ML) technique to fit models with multiple informants as predictors that permits missingness in the predictors as well as the response. We provide closed form solutions when possible and analytically compare the ML technique to the existing Generalized Estimating Equations (GEE) approach. We demonstrate that the ML approach can be used to compare the effect of the informants on response without standardizing the data. Simulations incorporating missingness show that ML is more efficient than the existing GEE method. In the presence of MCAR missing data, we find through a simulation study that the ML approach is robust to a relatively extreme departure from the normality assumption. We implement both methods in a study investigating the association between physical activity and obesity with activity measured using multiple informants (children and their mothers).     

Analysis of longitudinal studies with death and drop-out: a case study

The analysis of longitudinal data has recently been an active area of biostatistical research. Two main approaches to analysis have emerged, the first concentrating on modelling evolution of marginal distributions of the main response variable of interest and the other on subject-specific trajectories. In epidemiology the analysis is usually complicated by missing data and by death of study participants. Motivated by a study of cognitive decline in the elderly, this paper argues that these two types of incomplete follow-up may need to be treated differently in the analysis, and proposes an extension to the marginal modelling approach. The problem of informative drop-out is also discussed. The methods are implemented in the 'Stata' statistical package.     

Models for the propensity score that contemplate the positivity assumption and their application to missing data and causality

Generalized linear models are often assumed to fit propensity scores, which are used to compute inverse probability weighted (IPW) estimators. To derive the asymptotic properties of IPW estimators, the propensity score is supposed to be bounded away from zero. This condition is known in the literature as strict positivity (or positivity assumption), and, in practice, when it does not hold, IPW estimators are very unstable and have a large variability. Although strict positivity is often assumed, it is not upheld when some of the covariates are unbounded. In real data sets, a data‐generating process that violates the positivity assumption may lead to wrong inference because of the inaccuracy in the estimations. In this work, we attempt to conciliate between the strict positivity condition and the theory of generalized linear models by incorporating an extra parameter, which results in an explicit lower bound for the propensity score. An additional parameter is added to fulfil the overlap assumption in the causal framework.     

A flexible parametric approach for estimating continuous‐time inverse probability of treatment and censoring weights

Marginal structural Cox models are used for quantifying marginal treatment effects on outcome event hazard function. Such models are estimated using inverse probability of treatment and censoring (IPTC) weighting, which properly accounts for the impact of time‐dependent confounders, avoiding conditioning on factors on the causal pathway. To estimate the IPTC weights, the treatment assignment mechanism is conventionally modeled in discrete time. While this is natural in situations where treatment information is recorded at scheduled follow‐up visits, in other contexts, the events specifying the treatment history can be modeled in continuous time using the tools of event history analysis. This is particularly the case for treatment procedures, such as surgeries. In this paper, we propose a novel approach for flexible parametric estimation of continuous‐time IPTC weights and illustrate it in assessing the relationship between metastasectomy and mortality in metastatic renal cell carcinoma patients. Copyright © 2016 John Wiley & Sons, Ltd.     

Multivariate test power approximations for balanced linear mixed models in studies with missing data

Multilevel and longitudinal studies are frequently subject to missing data. For example, biomarker studies for oral cancer may involve multiple assays for each participant. Assays may fail, resulting in missing data values that can be assumed to be missing completely at random. Catellier and Muller proposed a data analytic technique to account for data missing at random in multilevel and longitudinal studies. They suggested modifying the degrees of freedom for both the Hotelling–Lawley trace F statistic and its null case reference distribution. We propose parallel adjustments to approximate power for this multivariate test in studies with missing data. The power approximations use a modified non‐central F statistic, which is a function of (i) the expected number of complete cases, (ii) the expected number of non‐missing pairs of responses, or (iii) the trimmed sample size, which is the planned sample size reduced by the anticipated proportion of missing data. The accuracy of the method is assessed by comparing the theoretical results to the Monte Carlo simulated power for the Catellier and Muller multivariate test. Over all experimental conditions, the closest approximation to the empirical power of the Catellier and Muller multivariate test is obtained by adjusting power calculations with the expected number of complete cases. The utility of the method is demonstrated with a multivariate power analysis for a hypothetical oral cancer biomarkers study. We describe how to implement the method using standard, commercially available software products and give example code. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal auxiliary‐covariate‐based two‐phase sampling design for semiparametric efficient estimation of a mean or mean difference,with application to clinical trials

To address the objective in a clinical trial to estimate the mean or mean difference of an expensive endpoint Y, one approach employs a two‐phase sampling design, wherein inexpensive auxiliary variables W predictive of Y are measured in everyone, Y is measured in a random sample, and the semiparametric efficient estimator is applied. This approach is made efficient by specifying the phase two selection probabilities as optimal functions of the auxiliary variables and measurement costs. While this approach is familiar to survey samplers, it apparently has seldom been used in clinical trials, and several novel results practicable for clinical trials are developed. We perform simulations to identify settings where the optimal approach significantly improves efficiency compared to approaches in current practice. We provide proofs and R code. The optimality results are developed to design an HIV vaccine trial, with objective to compare the mean ‘importance‐weighted’ breadth (Y) of the T‐cell response between randomized vaccine groups. The trial collects an auxiliary response (W) highly predictive of Y and measures Y in the optimal subset. We show that the optimal design‐estimation approach can confer anywhere between absent and large efficiency gain (up to 24 % in the examples) compared to the approach with the same efficient estimator but simple random sampling, where greater variability in the cost‐standardized conditional variance of Y given W yields greater efficiency gains. Accurate estimation of E[Y | W] is important for realizing the efficiency gain, which is aided by an ample phase two sample and by using a robust fitting method. Copyright © 2013 John Wiley & Sons, Ltd.