A Better Alternative to Non-parametric Approaches for Adjusting for Covariate Measurement Errors in Logistic Regression

In this article, we propose a flexible parametric (FP) approach for adjusting for covariate measurement errors in regression that can accommodate replicated measurements on the surrogate (mismeasured) version of the unobserved true covariate on all the study subjects or on a sub-sample of the study subjects as error assessment data. We utilize the general framework of the FP approach proposed by Hossain and Gustafson in 2009 for adjusting for covariate measurement errors in regression. The FP approach is then compared with the existing non-parametric approaches when error assessment data are available on the entire sample of the study subjects (complete error assessment data) considering covariate measurement error in a multiple logistic regression model. We also developed the FP approach when error assessment data are available on a sub-sample of the study subjects (partial error assessment data) and investigated its performance using both simulated and real life data. Simulation results reveal that, in comparable situations, the FP approach performs as good as or better than the competing non-parametric approaches in eliminating the bias that arises in the estimated regression parameters due to covariate measurement errors. Also, it results in better efficiency of the estimated parameters. Finally, the FP approach is found to perform adequately well in terms of bias correction, confidence coverage, and in achieving appropriate statistical power under partial error assessment data.     

Pseudo-empirical Bayes estimation of small area means under a nested error linear regression model with functional measurement errors

Small area estimation is studied under a nested error linear regression model with area level covariate subject to measurement error. Ghosh and Sinha (2007) obtained a pseudo-Bayes (PB) predictor of a small area mean and a corresponding pseudo-empirical Bayes (PEB) predictor, using the sample means of the observed covariate values to estimate the true covariate values. In this paper, we first derive an efficient PB predictor by using all the available data to estimate true covariate values. We then obtain a corresponding PEB predictor and show that it is asymptotically “optimal”. In addition, we employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator. Our results show that the proposed PEB predictor can lead to significant gain in efficiency over the previously proposed PEB predictor. Area level models are also studied.     

Joint Modeling of Event Time and Nonignorable Missing Longitudinal Data

Survival studies usually collect on each participant, both duration until some terminal event and repeated measures of a time-dependent covariate. Such a covariate is referred to as an internal time-dependent covariate. Usually, some subjects drop out of the study before occurence of the terminal event of interest. One may then wish to evaluate the relationship between time to dropout and the internal covariate. The Cox model is a standard framework for that purpose. Here, we address this problem in situations where the value of the covariate at dropout is unobserved. We suggest a joint model which combines a first-order Markov model for the longitudinaly measured covariate with a time-dependent Cox model for the dropout process. We consider maximum likelihood estimation in this model and show how estimation can be carried out via the EM-algorithm. We state that the suggested joint model may have applications in the context of longitudinal data with nonignorable dropout. Indeed, it can be viewed as generalizing Diggle and Kenward's model (1994) to situations where dropout may occur at any point in time and may be censored. Hence we apply both models and compare their results on a data set concerning longitudinal measurements among patients in a cancer clinical trial.     

On the clustering term in ecological analysis: how do different prior specifications affect results?

We study how different prior assumptions on the spatially structured heterogeneity term of the convolution hierarchical Bayesian model for spatial disease data could affect the results of an ecological analysis when response and exposure exhibit a strong spatial pattern. We show that in this case the estimate of the regression parameter could be strongly biased, both by analyzing the association between lung cancer mortality and education level on a real dataset and by a simulation experiment. The analysis is based on a hierarchical Bayesian model with a time dependent covariate in which we allow for a latency period between exposure and mortality, with time and space random terms and misaligned exposure-disease data.     

基于贝叶斯推断的HIV非线性混合效应联合模型研究

 在纵向数据研究中,混合效应模型的随机误差通常采用正态性假设。而诸如病毒载量和CD4细胞数目等病毒性数据通常呈现偏斜性,因此正态性假设可能影响模型结果甚至导致错误的结论。在HIV动力学研究中,病毒响应值往往与协变量相关,且协变量的测量值通常存在误差,为此论文中联立协变量过程建立具有偏正态分布的非线性混合效应联合模型,并用贝叶斯推断方法估计模型的参数。由于协变量能够解释个体内的部分变化,因此协变量过程的模型选择对病毒载量的拟合效果有重要的影响。该文提出了一次移动平均模型作为协变量过程的改进模型,比较后发现当协变量采用移动平均模型时,病毒载量模型的拟合效果更好。该结果对协变量模型的研究具有重要的指导意义。     

Expected estimating equations to accommodate covariate measurement error

Estimating equations which are not necessarily likelihood-based score equations are becoming increasingly popular for estimating regression model parameters. This paper is concerned with estimation based on general estimating equations when true covariate data are missing for all the study subjects, but surrogate or mismeasured covariates are available instead. The method is motivated by the covariate measurement error problem in marginal or partly conditional regression of longitudinal data. We propose to base estimation on the expectation of the complete data estimating equation conditioned on available data. The regression parameters and other nuisance parameters are estimated simultaneously by solving the resulting estimating equations. The expected estimating equation (EEE) estimator is equal to the maximum likelihood estimator if the complete data scores are likelihood scores and conditioning is with respect to all the available data. A pseudo-EEE estimator, which requires less computation, is also investigated. Asymptotic distribution theory is derived. Small sample simulations are conducted when the error process is an order 1 autoregressive model. Regression calibration is extended to this setting and compared with the EEE approach. We demonstrate the methods on data from a longitudinal study of the relationship between childhood growth and adult obesity.     

Linear mode regression with covariate measurement error

We consider estimating the mode of a response given an error‐prone covariate. It is shown that ignoring measurement error typically leads to inconsistent inference for the conditional mode of the response given the true covariate, as well as misleading inference for regression coefficients in the conditional mode model. To account for measurement error, we first employ the Monte Carlo corrected score method (Novick & Stefanski, 2002) to obtain an unbiased score function based on which the regression coefficients can be estimated consistently. To relax the normality assumption on measurement error this method requires, we propose another method where deconvoluting kernels are used to construct an objective function that is maximized to obtain consistent estimators of the regression coefficients. Besides rigorous investigation on asymptotic properties of the new estimators, we study their finite sample performance via extensive simulation experiments, and find that the proposed methods substantially outperform a naive inference method that ignores measurement error. The Canadian Journal of Statistics 47: 262–280; 2019 © 2019 Statistical Society of Canada     

Bayesian inference for rare errors in populations with unequal unit sizes

We describe a Bayesian model for a scenario in which the population of errors contains many 0s and there is a known covariate. This kind of structure typically occurs in auditing, and we use auditing as the driving application of the method. Our model is based on a categorization of the error population together with a Bayesian nonparametric method of modelling errors within some of the categories. Inference is through simulation. We conclude with an example based on a data set provided by the UK's National Audit Office.     

Cox regression for mixed case interval-censored data with covariate errors

Covariate measurement error problems have been extensively studied in the context of right-censored data but less so for interval-censored data. Motivated by the AIDS Clinical Trial Group 175 study, where the occurrence time of AIDS was examined only at intermittent clinic visits and the baseline covariate CD4 count was measured with error, we describe a semiparametric maximum likelihood method for analyzing mixed case interval-censored data with mismeasured covariates under the proportional hazards model. We show that the estimator of the regression coefficient is asymptotically normal and efficient and provide a very stable and efficient algorithm for computing the estimators. We evaluate the method through simulation studies and illustrate it with AIDS data.     

Accelerated hazards mixture cure model

We propose a new cure model for survival data with a surviving or cure fraction. The new model is a mixture cure model where the covariate effects on the proportion of cure and the distribution of the failure time of uncured patients are separately modeled. Unlike the existing mixture cure models, the new model allows covariate effects on the failure time distribution of uncured patients to be negligible at time zero and to increase as time goes by. Such a model is particularly useful in some cancer treatments when the treat effect increases gradually from zero, and the existing models usually cannot handle this situation properly. We develop a rank based semiparametric estimation method to obtain the maximum likelihood estimates of the parameters in the model. We compare it with existing models and methods via a simulation study, and apply the model to a breast cancer data set. The numerical studies show that the new model provides a useful addition to the cure model literature.     

A Proportional Hazards Regression Model for the Subdistribution with Covariates‐adjusted Censoring Weight for Competing Risks Data

With competing risks data, one often needs to assess the treatment and covariate effects on the cumulative incidence function. Fine and Gray proposed a proportional hazards regression model for the subdistribution of a competing risk with the assumption that the censoring distribution and the covariates are independent. Covariate‐dependent censoring sometimes occurs in medical studies. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with proper adjustments for covariate‐dependent censoring. We consider a covariate‐adjusted weight function by fitting the Cox model for the censoring distribution and using the predictive probability for each individual. Our simulation study shows that the covariate‐adjusted weight estimator is basically unbiased when the censoring time depends on the covariates, and the covariate‐adjusted weight approach works well for the variance estimator as well. We illustrate our methods with bone marrow transplant data from the Center for International Blood and Marrow Transplant Research. Here, cancer relapse and death in complete remission are two competing risks.     

Nonparametric covariate adjustment in estimating hazard ratios

In randomized clinical trials with time‐to‐event outcomes, the hazard ratio is commonly used to quantify the treatment effect relative to a control. The Cox regression model is commonly used to adjust for relevant covariates to obtain more accurate estimates of the hazard ratio between treatment groups. However, it is well known that the treatment hazard ratio based on a covariate‐adjusted Cox regression model is conditional on the specific covariates and differs from the unconditional hazard ratio that is an average across the population. Therefore, covariate‐adjusted Cox models cannot be used when the unconditional inference is desired. In addition, the covariate‐adjusted Cox model requires the relatively strong assumption of proportional hazards for each covariate. To overcome these challenges, a nonparametric randomization‐based analysis of covariance method was proposed to estimate the covariate‐adjusted hazard ratios for multivariate time‐to‐event outcomes. However, empirical evaluations of the performance (power and type I error rate) of the method have not been studied. Although the method is derived for multivariate situations, for most registration trials, the primary endpoint is a univariate outcome. Therefore, this approach is applied to univariate outcomes, and performance is evaluated through a simulation study in this paper. Stratified analysis is also investigated. As an illustration of the method, we also apply the covariate‐adjusted and unadjusted analyses to an oncology trial. Copyright © 2015 John Wiley & Sons, Ltd.     

Overcoming biases and misconceptions in ecological studies

The aggregate data study design provides an alternative group level analysis to ecological studies in the estimation of individual level health risks. An aggregate model is derived by aggregating a plausible individual level relative rate model within groups, such that population-based disease rates are modelled as functions of individual level covariate data. We apply an aggregate data method to a series of fictitious examples from a review paper by Greenland and Robins which illustrated the problems that can arise when using the results of ecological studies to make inference about individual health risks. We use simulated data based on their examples to demonstrate that the aggregate data approach can address many of the sources of bias that are inherent in typical ecological analyses, even though the limited between-region covariate variation in these examples reduces the efficiency of the aggregate study. The aggregate method has the potential to estimate exposure effects of interest in the presence of non-linearity, confounding at individual and group levels, effect modification, classical measurement error in the exposure and non-differential misclassification in the confounder.     

SMALL AREA ESTIMATION USING SURVEY WEIGHTS WITH FUNCTIONAL MEASUREMENT ERROR IN THE COVARIATE

Nested error linear regression models using survey weights have been studied in small area estimation to obtain efficient model‐based and design‐consistent estimators of small area means. The covariates in these nested error linear regression models are not subject to measurement errors. In practical applications, however, there are many situations in which the covariates are subject to measurement errors. In this paper, we develop a nested error linear regression model with an area‐level covariate subject to functional measurement error. In particular, we propose a pseudo‐empirical Bayes (PEB) predictor to estimate small area means. This predictor borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. We also employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator.     

An Additive–Multiplicative Restricted Mean Residual Life Model

The mean residual life measures the expected remaining life of a subject who has survived up to a particular time. When survival time distribution is highly skewed or heavy tailed, the restricted mean residual life must be considered. In this paper, we propose an additive–multiplicative restricted mean residual life model to study the association between the restricted mean residual life function and potential regression covariates in the presence of right censoring. This model extends the proportional mean residual life model using an additive model as its covariate dependent baseline. For the suggested model, some covariate effects are allowed to be time‐varying. To estimate the model parameters, martingale estimating equations are developed, and the large sample properties of the resulting estimators are established. In addition, to assess the adequacy of the model, we investigate a goodness of fit test that is asymptotically justified. The proposed methodology is evaluated via simulation studies and further applied to a kidney cancer data set collected from a clinical trial.     

Latent class based multiple imputation approach for missing categorical data

In this paper we propose a latent class based multiple imputation approach for analyzing missing categorical covariate data in a highly stratified data model. In this approach, we impute the missing data assuming a latent class imputation model and we use likelihood methods to analyze the imputed data. Via extensive simulations, we study its statistical properties and make comparisons with complete case analysis, multiple imputation, saturated log-linear multiple imputation and the Expectation–Maximization approach under seven missing data mechanisms (including missing completely at random, missing at random and not missing at random). These methods are compared with respect to bias, asymptotic standard error, type I error, and 95% coverage probabilities of parameter estimates. Simulations show that, under many missingness scenarios, latent class multiple imputation performs favorably when jointly considering these criteria. A data example from a matched case–control study of the association between multiple myeloma and polymorphisms of the Inter-Leukin 6 genes is considered.     

Estimating Marginal Effects in Accelerated Failure Time Models for Serial Sojourn Times Among Repeated Events

Recurrent event data are commonly encountered in longitudinal studies when events occur repeatedly over time for each study subject. An accelerated failure time (AFT) model on the sojourn time between recurrent events is considered in this article. This model assumes that the covariate effect and the subject-specific frailty are additive on the logarithm of sojourn time, and the covariate effect maintains the same over distinct episodes, while the distributions of the frailty and the random error in the model are unspecified. With the ordinal nature of recurrent events, two scale transformations of the sojourn times are derived to construct semiparametric methods of log-rank type for estimating the marginal covariate effects in the model. The proposed estimation approaches/inference procedures also can be extended to the bivariate events, which alternate themselves over time. Examples and comparisons are presented to illustrate the performance of the proposed methods.     

To adjust or not to adjust for baseline when analyzing repeated binary responses? The case of complete data when treatment comparison at study end is of interest

The benefits of adjusting for baseline covariates are not as straightforward with repeated binary responses as with continuous response variables. Therefore, in this study, we compared different methods for analyzing repeated binary data through simulations when the outcome at the study endpoint is of interest. Methods compared included chi‐square, Fisher's exact test, covariate adjusted/unadjusted logistic regression (Adj.logit/Unadj.logit), covariate adjusted/unadjusted generalized estimating equations (Adj.GEE/Unadj.GEE), covariate adjusted/unadjusted generalized linear mixed model (Adj.GLMM/Unadj.GLMM). All these methods preserved the type I error close to the nominal level. Covariate adjusted methods improved power compared with the unadjusted methods because of the increased treatment effect estimates, especially when the correlation between the baseline and outcome was strong, even though there was an apparent increase in standard errors. Results of the Chi‐squared test were identical to those for the unadjusted logistic regression. Fisher's exact test was the most conservative test regarding the type I error rate and also with the lowest power. Without missing data, there was no gain in using a repeated measures approach over a simple logistic regression at the final time point. Analysis of results from five phase III diabetes trials of the same compound was consistent with the simulation findings. Therefore, covariate adjusted analysis is recommended for repeated binary data when the study endpoint is of interest. Copyright © 2015 John Wiley & Sons, Ltd.     

A generalized estimating equation approach to modelling incompatible data formats with covariate measurement error: application to human immunodeficiency virus immune markers

The integration of technological advances into research studies often raises an issue of incompatibility of data. This problem is common to longitudinal and multicentre studies, taking the form of changes in the definitions, acquisition of data or measuring instruments of some study variables. In our case of studying the relationship between a marker of immune response to human immunodeficiency virus and human immunodeficiency virus infection status, using data from the Multi-Center AIDS Cohort Study, changes in the manufactured tests used for both variables occurred throughout the study, resulting in data with different manufactured scales. In addition, the latent nature of the immune response of interest necessitated a further consideration of a measurement error component. We address the general issue of incompatibility of data, together with the issue of covariate measurement error, in a unified, generalized linear model setting with inferences based on the generalized estimating equation framework. General conditions are constructed to ensure consistent estimates and their variances for the primary model of interest, with the asymptotic behaviour of resulting estimates examined under a variety of modelling scenarios. The approach is illustrated by modelling a repeated ordinal response with incompatible formats, as a function of a covariate with incompatible formats and measurement error, based on the Multi-Center AIDS Cohort Study data.     

Unified Inference for Sparse and Dense Longitudinal Data in Time‐varying Coefficient Models

Time‐varying coefficient models are widely used in longitudinal data analysis. These models allow the effects of predictors on response to vary over time. In this article, we consider a mixed‐effects time‐varying coefficient model to account for the within subject correlation for longitudinal data. We show that when kernel smoothing is used to estimate the smooth functions in time‐varying coefficient models for sparse or dense longitudinal data, the asymptotic results of these two situations are essentially different. Therefore, a subjective choice between the sparse and dense cases might lead to erroneous conclusions for statistical inference. In order to solve this problem, we establish a unified self‐normalized central limit theorem, based on which a unified inference is proposed without deciding whether the data are sparse or dense. The effectiveness of the proposed unified inference is demonstrated through a simulation study and an analysis of Baltimore MACS data.