Multiple imputation in the presence of an incomplete binary variable created from an underlying continuous variable

Multiple imputation (MI) is used to handle missing at random (MAR) data. Despite warnings from statisticians, continuous variables are often recoded into binary variables. With MI it is important that the imputation and analysis models are compatible; variables should be imputed in the same form they appear in the analysis model. With an encoded binary variable more accurate imputations may be obtained by imputing the underlying continuous variable. We conducted a simulation study to explore how best to impute a binary variable that was created from an underlying continuous variable. We generated a completely observed continuous outcome associated with an incomplete binary covariate that is a categorized version of an underlying continuous covariate, and an auxiliary variable associated with the underlying continuous covariate. We simulated data with several sample sizes, and set 25% and 50% of data in the covariate to MAR dependent on the outcome and the auxiliary variable. We compared the performance of five different imputation methods: (a) Imputation of the binary variable using logistic regression; (b) imputation of the continuous variable using linear regression, then categorizing into the binary variable; (c, d) imputation of both the continuous and binary variables using fully conditional specification (FCS) and multivariate normal imputation; (e) substantive-model compatible (SMC) FCS. Bias and standard errors were large when the continuous variable only was imputed. The other methods performed adequately. Imputation of both the binary and continuous variables using FCS often encountered mathematical difficulties. We recommend the SMC-FCS method as it performed best in our simulation studies.     

Semiparametric inference for a 2-stage outcome-auxiliary-dependent sampling design with continuous outcome

Two-stage design has long been recognized to be a cost-effective way for conducting biomedical studies. In many trials, auxiliary covariate information may also be available, and it is of interest to exploit these auxiliary data to improve the efficiency of inferences. In this paper, we propose a 2-stage design with continuous outcome where the second-stage data is sampled with an "outcome-auxiliary-dependent sampling" (OADS) scheme. We propose an estimator which is the maximizer for an estimated likelihood function. We show that the proposed estimator is consistent and asymptotically normally distributed. The simulation study indicates that greater study efficiency gains can be achieved under the proposed 2-stage OADS design by utilizing the auxiliary covariate information when compared with other alternative sampling schemes. We illustrate the proposed method by analyzing a data set from an environmental epidemiologic study.     

Random effects logistic regression analysis with auxiliary covariates

We study a semiparametric estimation method for the random effects logistic regression when there is auxiliary covariate information about the main exposure variable. We extend the semiparametric estimator of Pepe and Fleming (1991, Journal of the American Statistical Association 86, 108-113) to the random effects model using the best linear unbiased prediction approach of Henderson (1975, Biometrics 31, 423-448). The method can be used to handle the missing covariate or mismeasured covariate data problems in a variety of real applications. Simulation study results show that the proposed method outperforms the existing methods. We analyzed a data set from the Collaborative Perinatal Project using the proposed method and found that the use of DDT increases the risk of preterm births among U.S. children.     

Estimated Pseudopartial‐Likelihood Method for Correlated Failure Time Data with Auxiliary Covariates

Summary As biological studies become more expensive to conduct, statistical methods that take advantage of existing auxiliary information about an expensive exposure variable are desirable in practice. Such methods should improve the study efficiency and increase the statistical power for a given number of assays. In this article, we consider an inference procedure for multivariate failure time with auxiliary covariate information. We propose an estimated pseudopartial likelihood estimator under the marginal hazard model framework and develop the asymptotic properties for the proposed estimator. We conduct simulation studies to evaluate the performance of the proposed method in practical situations and demonstrate the proposed method with a data set from the studies of left ventricular dysfunction ( SOLVD Investigators, 1991 , New England Journal of Medicine 325 , 293–302).     

Adjustment for Missingness Using Auxiliary Information in Semiparametric Regression

Summary .  In this article, we study the estimation of mean response and regression coefficient in semiparametric regression problems when response variable is subject to nonrandom missingness. When the missingness is independent of the response conditional on high-dimensional auxiliary information, the parametric approach may misspecify the relationship between covariates and response while the nonparametric approach is infeasible because of the curse of dimensionality. To overcome this, we study a model-based approach to condense the auxiliary information and estimate the parameters of interest nonparametrically on the condensed covariate space. Our estimators possess the double robustness property, i.e., they are consistent whenever the model for the response given auxiliary covariates or the model for the missingness given auxiliary covariate is correct. We conduct a number of simulations to compare the numerical performance between our estimators and other existing estimators in the current missing data literature, including the propensity score approach and the inverse probability weighted estimating equation. A set of real data is used to illustrate our approach.     

Continuous Covariates in Mark‐Recapture‐Recovery Analysis: A Comparison of Methods

Summary Time varying, individual covariates are problematic in experiments with marked animals because the covariate can typically only be observed when each animal is captured. We examine three methods to incorporate time varying, individual covariates of the survival probabilities into the analysis of data from mark‐recapture‐recovery experiments: deterministic imputation, a Bayesian imputation approach based on modeling the joint distribution of the covariate and the capture history, and a conditional approach considering only the events for which the associated covariate data are completely observed (the trinomial model). After describing the three methods, we compare results from their application to the analysis of the effect of body mass on the survival of Soay sheep (Ovis aries) on the Isle of Hirta, Scotland. Simulations based on these results are then used to make further comparisons. We conclude that both the trinomial model and Bayesian imputation method perform best in different situations. If the capture and recovery probabilities are all high, then the trinomial model produces precise, unbiased estimators that do not depend on any assumptions regarding the distribution of the covariate. In contrast, the Bayesian imputation method performs substantially better when capture and recovery probabilities are low, provided that the specified model of the covariate is a good approximation to the true data‐generating mechanism.     

Weighted Likelihood Method for Grouped Survival Data in Case–Cohort Studies with Application to HIV Vaccine Trials

Summary Grouped failure time data arise often in HIV studies. In a recent preventive HIV vaccine efficacy trial, immune responses generated by the vaccine were measured from a case–cohort sample of vaccine recipients, who were subsequently evaluated for the study endpoint of HIV infection at prespecified follow‐up visits. Gilbert et al. (2005, Journal of Infectious Diseases 191 , 666–677) and Forthal et al. (2007, Journal of Immunology 178, 6596–6603) analyzed the association between the immune responses and HIV incidence with a Cox proportional hazards model, treating the HIV infection diagnosis time as a right‐censored random variable. The data, however, are of the form of grouped failure time data with case–cohort covariate sampling, and we propose an inverse selection probability‐weighted likelihood method for fitting the Cox model to these data. The method allows covariates to be time dependent, and uses multiple imputation to accommodate covariate data that are missing at random. We establish asymptotic properties of the proposed estimators, and present simulation results showing their good finite sample performance. We apply the method to the HIV vaccine trial data, showing that higher antibody levels are associated with a lower hazard of HIV infection.     

Design and Inference for Cancer Biomarker Study with an Outcome and Auxiliary‐Dependent Subsampling

Summary In cancer research, it is important to evaluate the performance of a biomarker (e.g., molecular, genetic, or imaging) that correlates patients' prognosis or predicts patients' response to treatment in a large prospective study. Due to overall budget constraint and high cost associated with bioassays, investigators often have to select a subset from all registered patients for biomarker assessment. To detect a potentially moderate association between the biomarker and the outcome, investigators need to decide how to select the subset of a fixed size such that the study efficiency can be enhanced. We show that, instead of drawing a simple random sample from the study cohort, greater efficiency can be achieved by allowing the selection probability to depend on the outcome and an auxiliary variable; we refer to such a sampling scheme as outcome and auxiliary‐dependent subsampling (OADS). This article is motivated by the need to analyze data from a lung cancer biomarker study that adopts the OADS design to assess epidermal growth factor receptor (EGFR) mutations as a predictive biomarker for whether a subject responds to a greater extent to EGFR inhibitor drugs. We propose an estimated maximum‐likelihood method that accommodates the OADS design and utilizes all observed information, especially those contained in the likelihood score of EGFR mutations (an auxiliary variable of EGFR mutations) that is available to all patients. We derive the asymptotic properties of the proposed estimator and evaluate its finite sample properties via simulation. We illustrate the proposed method with a data example.     

Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information

This article presents a new method for maximum likelihood estimation of logistic regression models with incomplete covariate data where auxiliary information is available. This auxiliary information is extraneous to the regression model of interest but predictive of the covariate with missing data. Ibrahim (1990, Journal of the American Statistical Association 85, 765-769) provides a general method for estimating generalized linear regression models with missing covariates using the EM algorithm that is easily implemented when there is no auxiliary data. Vach (1997, Statistics in Medicine 16, 57-72) describes how the method can be extended when the outcome and auxiliary data are conditionally independent given the covariates in the model. The method allows the incorporation of auxiliary data without making the conditional independence assumption. We suggest tests of conditional independence and compare the performance of several estimators in an example concerning mental health service utilization in children. Using an artificial dataset, we compare the performance of several estimators when auxiliary data are available.     

Semiparametric Bayesian Analysis of Nutritional Epidemiology Data in the Presence of Measurement Error

Summary : We propose a semiparametric Bayesian method for handling measurement error in nutritional epidemiological data. Our goal is to estimate nonparametrically the form of association between a disease and exposure variable while the true values of the exposure are never observed. Motivated by nutritional epidemiological data, we consider the setting where a surrogate covariate is recorded in the primary data, and a calibration data set contains information on the surrogate variable and repeated measurements of an unbiased instrumental variable of the true exposure. We develop a flexible Bayesian method where not only is the relationship between the disease and exposure variable treated semiparametrically, but also the relationship between the surrogate and the true exposure is modeled semiparametrically. The two nonparametric functions are modeled simultaneously via B‐splines. In addition, we model the distribution of the exposure variable as a Dirichlet process mixture of normal distributions, thus making its modeling essentially nonparametric and placing this work into the context of functional measurement error modeling. We apply our method to the NIH‐AARP Diet and Health Study and examine its performance in a simulation study.     

Statistical Inference for a Two‐Stage Outcome‐Dependent Sampling Design with a Continuous Outcome

Summary The two‐stage case–control design has been widely used in epidemiology studies for its cost‐effectiveness and improvement of the study efficiency ( White, 1982 , American Journal of Epidemiology 115, 119–128; Breslow and Cain, 1988 , Biometrika 75, 11–20). The evolution of modern biomedical studies has called for cost‐effective designs with a continuous outcome and exposure variables. In this article, we propose a new two‐stage outcome‐dependent sampling (ODS) scheme with a continuous outcome variable, where both the first‐stage data and the second‐stage data are from ODS schemes. We develop a semiparametric empirical likelihood estimation for inference about the regression parameters in the proposed design. Simulation studies were conducted to investigate the small‐sample behavior of the proposed estimator. We demonstrate that, for a given statistical power, the proposed design will require a substantially smaller sample size than the alternative designs. The proposed method is illustrated with an environmental health study conducted at National Institutes of Health.     

Estimation and interpretation of incidence rate difference for recurrent events when the estimation model is misspecified

Recurrent events data are common in experimental and observational studies. It is often of interest to estimate the effect of an intervention on the incidence rate of the recurrent events. The incidence rate difference is a useful measure of intervention effect. A weighted least squares estimator of the incidence rate difference for recurrent events was recently proposed for an additive rate model in which both the baseline incidence rate and the covariate effects were constant over time. In this article, we relax this model assumption and examine the properties of the estimator under the additive and multiplicative rate models assumption in which the baseline incidence rate and covariate effects may vary over time. We show analytically and numerically that the estimator gives an appropriate summary measure of the time‐varying covariate effects. In particular, when the underlying covariate effects are additive and time‐varying, the estimator consistently estimates the weighted average of the covariate effects over time. When the underlying covariate effects are multiplicative and time‐varying, and if there is only one binary covariate indicating the intervention status, the estimator consistently estimates the weighted average of the underlying incidence rate difference between the intervention and control groups over time. We illustrate the method with data from a randomized vaccine trial.     

A pseudo-bayesian shrinkage approach to regression with missing covariates

Summary We consider the linear regression of outcome Y on regressors W and Z with some values of W missing, when our main interest is the effect of Z on Y, controlling for W. Three common approaches to regression with missing covariates are (i) complete‐case analysis (CC), which discards the incomplete cases, and (ii) ignorable likelihood methods, which base inference on the likelihood based on the observed data, assuming the missing data are missing at random ( Rubin, 1976b ), and (iii) nonignorable modeling, which posits a joint distribution of the variables and missing data indicators. Another simple practical approach that has not received much theoretical attention is to drop the regressor variables containing missing values from the regression modeling (DV, for drop variables). DV does not lead to bias when either (i) the regression coefficient of W is zero or (ii) W and Z are uncorrelated. We propose a pseudo‐Bayesian approach for regression with missing covariates that compromises between the CC and DV estimates, exploiting information in the incomplete cases when the data support DV assumptions. We illustrate favorable properties of the method by simulation, and apply the proposed method to a liver cancer study. Extension of the method to more than one missing covariate is also discussed.     

Score test for linkage in generalized linear models

We derive a test for linkage in a Generalized Linear Mixed Model (GLMM) framework which provides a natural adjustment for marginal covariate effects. The method boils down to the score test of a quasi-likelihood derived from the GLMM, it is computationally inexpensive and can be applied to arbitrary pedigrees. In particular, for binary traits, relative pairs of different nature (affected and discordant) and individuals with different covariate values can be naturally combined in a single test. The model introduced could explain a number of situations usually described as gene by covariate interaction phenomena, and offers substantial gains in efficiency compared to methods classically used in those instances.     

On estimation and prediction for spatial generalized linear mixed models

We use spatial generalized linear mixed models (GLMM) to model non-Gaussian spatial variables that are observed at sampling locations in a continuous area. In many applications, prediction of random effects in a spatial GLMM is of great practical interest. We show that the minimum mean-squared error (MMSE) prediction can be done in a linear fashion in spatial GLMMs analogous to linear kriging. We develop a Monte Carlo version of the EM gradient algorithm for maximum likelihood estimation of model parameters. A by-product of this approach is that it also produces the MMSE estimates for the realized random effects at the sampled sites. This method is illustrated through a simulation study and is also applied to a real data set on plant root diseases to obtain a map of disease severity that can facilitate the practice of precision agriculture.     

Structural inference in transition measurement error models for longitudinal data

We propose a new class of models, transition measurement error models, to study the effects of covariates and the past responses on the current response in longitudinal studies when one of the covariates is measured with error. We show that the response variable conditional on the error-prone covariate follows a complex transition mixed effects model. The naive model obtained by ignoring the measurement error correctly specifies the transition part of the model, but misspecifies the covariate effect structure and ignores the random effects. We next study the asymptotic bias in naive estimator obtained by ignoring the measurement error for both continuous and discrete outcomes. We show that the naive estimator of the regression coefficient of the error-prone covariate is attenuated, while the naive estimators of the regression coefficients of the past responses are generally inflated. We then develop a structural modeling approach for parameter estimation using the maximum likelihood estimation method. In view of the multidimensional integration required by full maximum likelihood estimation, an EM algorithm is developed to calculate maximum likelihood estimators, in which Monte Carlo simulations are used to evaluate the conditional expectations in the E-step. We evaluate the performance of the proposed method through a simulation study and apply it to a longitudinal social support study for elderly women with heart disease. An additional simulation study shows that the Bayesian information criterion (BIC) performs well in choosing the correct transition orders of the models.     

Bias and Sensitivity Analysis When Estimating Treatment Effects from the Cox Model with Omitted Covariates

Summary

Omission of relevant covariates can lead to bias when estimating treatment or exposure effects from survival data in both randomized controlled trials and observational studies. This paper presents a general approach to assessing bias when covariates are omitted from the Cox model. The proposed method is applicable to both randomized and non‐randomized studies. We distinguish between the effects of three possible sources of bias: omission of a balanced covariate, data censoring and unmeasured confounding. Asymptotic formulae for determining the bias are derived from the large sample properties of the maximum likelihood estimator. A simulation study is used to demonstrate the validity of the bias formulae and to characterize the influence of the different sources of bias. It is shown that the bias converges to fixed limits as the effect of the omitted covariate increases, irrespective of the degree of confounding. The bias formulae are used as the basis for developing a new method of sensitivity analysis to assess the impact of omitted covariates on estimates of treatment or exposure effects. In simulation studies, the proposed method gave unbiased treatment estimates and confidence intervals with good coverage when the true sensitivity parameters were known. We describe application of the method to a randomized controlled trial and a non‐randomized study.     

Regression on Quantile Residual Life

Summary A time‐specific log‐linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time‐to‐event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance–covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long‐term follow‐up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.     

Robustness of Generalized Estimating Equation (GEE) Tests of Significance against Misspecification of the Error Structure Model

Generalized linear model analyses of repeated measurements typically rely on simplifying mathematical models of the error covariance structure for testing the significance of differences in patterns of change across time. The robustness of the tests of significance depends, not only on the degree of agreement between the specified mathematical model and the actual population data structure, but also on the precision and robustness of the computational criteria for fitting the specified covariance structure to the data. Generalized estimating equation (GEE) solutions utilizing the robust empirical sandwich estimator for modeling of the error structure were compared with general linear mixed model (GLMM) solutions that utilized the commonly employed restricted maximum likelihood (REML) procedure. Under the conditions considered, the GEE and GLMM procedures were identical in assuming that the data are normally distributed and that the variance‐covariance structure of the data is the one specified by the user. The question addressed in this article concerns relative sensitivity of tests of significance for treatment effects to varying degrees of misspecification of the error covariance structure model when fitted by the alternative procedures. Simulated data that were subjected to monte carlo evaluation of actual Type I error and power of tests of the equal slopes hypothesis conformed to assumptions of ordinary linear model ANOVA for repeated measures except for autoregressive covariance structures and missing data due to dropouts. The actual within‐groups correlation structures of the simulated repeated measurements ranged from AR(1) to compound symmetry in graded steps, whereas the GEE and GLMM formulations restricted the respective error structure models to be either AR(1), compound symmetry (CS), or unstructured (UN). The GEE‐based tests utilizing empirical sandwich estimator criteria were documented to be relatively insensitive to misspecification of the covariance structure models, whereas GLMM tests which relied on restricted maximum likelihood (REML) were highly sensitive to relatively modest misspecification of the error correlation structure even though normality, variance homogeneity, and linearity were not an issue in the simulated data.Goodness‐of‐fit statistics were of little utility in identifying cases in which relatively minor misspecification of the GLMM error structure model resulted in inadequate alpha protection for tests of the equal slopes hypothesis. Both GEE and GLMM formulations that relied on unstructured (UN) error model specification produced nonconservative results regardless of the actual correlation structure of the repeated measurements. A random coefficients model produced robust tests with competitive power across all conditions examined. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A simulation-based marginal method for longitudinal data with dropout and mismeasured covariates

Longitudinal data often contain missing observations and error-prone covariates. Extensive attention has been directed to analysis methods to adjust for the bias induced by missing observations. There is relatively little work on investigating the effects of covariate measurement error on estimation of the response parameters, especially on simultaneously accounting for the biases induced by both missing values and mismeasured covariates. It is not clear what the impact of ignoring measurement error is when analyzing longitudinal data with both missing observations and error-prone covariates. In this article, we study the effects of covariate measurement error on estimation of the response parameters for longitudinal studies. We develop an inference method that adjusts for the biases induced by measurement error as well as by missingness. The proposed method does not require the full specification of the distribution of the response vector but only requires modeling its mean and variance structures. Furthermore, the proposed method employs the so-called functional modeling strategy to handle the covariate process, with the distribution of covariates left unspecified. These features, plus the simplicity of implementation, make the proposed method very attractive. In this paper, we establish the asymptotic properties for the resulting estimators. With the proposed method, we conduct sensitivity analyses on a cohort data set arising from the Framingham Heart Study. Simulation studies are carried out to evaluate the impact of ignoring covariate measurement error and to assess the performance of the proposed method.