On corrected score approach for proportional hazards model with covariate measurement error

In the presence of covariate measurement error with the proportional hazards model, several functional modeling methods have been proposed. These include the conditional score estimator (Tsiatis and Davidian, 2001, Biometrika 88, 447-458), the parametric correction estimator (Nakamura, 1992, Biometrics 48, 829-838), and the nonparametric correction estimator (Huang and Wang, 2000, Journal of the American Statistical Association 95, 1209-1219) in the order of weaker assumptions on the error. Although they are all consistent, each suffers from potential difficulties with small samples and substantial measurement error. In this article, upon noting that the conditional score and parametric correction estimators are asymptotically equivalent in the case of normal error, we investigate their relative finite sample performance and discover that the former is superior. This finding motivates a general refinement approach to parametric and nonparametric correction methods. The refined correction estimators are asymptotically equivalent to their standard counterparts, but have improved numerical properties and perform better when the standard estimates do not exist or are outliers. Simulation results and application to an HIV clinical trial are presented.     

On latent-variable model misspecification in structural measurement error models for binary response

Summary .  We consider structural measurement error models for a binary response. We show that likelihood-based estimators obtained from fitting structural measurement error models with pooled binary responses can be far more robust to covariate measurement error in the presence of latent-variable model misspecification than the corresponding estimators from individual responses. Furthermore, despite the loss in information, pooling can provide improved parameter estimators in terms of mean-squared error. Based on these and other findings, we create a new diagnostic method to detect latent-variable model misspecification in structural measurement error models with individual binary response. We use simulation and data from the Framingham Heart Study to illustrate our methods.     

Combining complete multivariate outcomes with incomplete covariate information: a latent class approach

This work was motivated by the need to combine outcome information from a reference population with risk factor information from a screened subpopulation in a setting where the analytic goal was to study the association between risk factors and multiple binary outcomes. To achieve such an analytic goal, this article proposes a two-stage latent class procedure that first summarizes the commonalities among outcomes using a reference population sample, then analyzes the association between outcomes and risk factors. It develops a pseudo-maximum likelihood approach to estimating model parameters. The performance of the proposed method is evaluated in a simulation study and in an illustrative analysis of data from the Women's Health and Aging Study, a recent investigation of the causes and course of disability in older women. Combining information in the proposed way is found to improve both accuracy and precision in summarizing multiple categorical outcomes, which effectively diminishes ambiguity and bias in making risk factor inferences.     

Varying coefficients model with measurement error

Summary .   We propose a semiparametric partially varying coefficient model to study the relationship between serum creatinine concentration and the glomerular filtration rate (GFR) among kidney donors and patients with chronic kidney disease. A regression model is used to relate serum creatinine to GFR and demographic factors in which coefficient of GFR is expressed as a function of age to allow its effect to be age dependent. GFR measurements obtained from the clearance of a radioactively labeled isotope are assumed to be a surrogate for the true GFR, with the relationship between measured and true GFR expressed using an additive error model. We use locally corrected score equations to estimate parameters and coefficient functions, and propose an expected generalized cross-validation (EGCV) method to select the kernel bandwidth. The performance of the proposed methods, which avoid distributional assumptions on the true GFR and residuals, is investigated by simulation. Accounting for measurement error using the proposed model reduced apparent inconsistencies in the relationship between serum creatinine and GFR among different clinical data sets derived from kidney donor and chronic kidney disease source populations.     

Estimation in the Cox survival regression model with covariate measurement error and a changepoint

The Cox regression model is a popular model for analyzing the relationship between a covariate vector and a survival endpoint. The standard Cox model assumes a constant covariate effect across the entire covariate domain. However, in many epidemiological and other applications, the covariate of main interest is subject to a threshold effect: a change in the slope at a certain point within the covariate domain. Often, the covariate of interest is subject to some degree of measurement error. In this paper, we study measurement error correction in the case where the threshold is known. Several bias correction methods are examined: two versions of regression calibration (RC1 and RC2, the latter of which is new), two methods based on the induced relative risk under a rare event assumption (RR1 and RR2, the latter of which is new), a maximum pseudo-partial likelihood estimator (MPPLE), and simulation-extrapolation (SIMEX). We develop the theory, present simulations comparing the methods, and illustrate their use on data concerning the relationship between chronic air pollution exposure to particulate matter PM₁₀ and fatal myocardial infarction (Nurses Health Study (NHS)), and on data concerning the effect of a subject's long-term underlying systolic blood pressure level on the risk of cardiovascular disease death (Framingham Heart Study (FHS)). The simulations indicate that the best methods are RR2 and MPPLE.     

A semiparametric transition model with latent traits for longitudinal multistate data

SUMMARY: We propose a general multistate transition model. The model is developed for the analysis of repeated episodes of multiple states representing different health status. Transitions among multiple states are modeled jointly using multivariate latent traits with factor loadings. Different types of state transition are described by flexible transition-specific nonparametric baseline intensities. A state-specific latent trait is used to capture individual tendency of the sojourn in the state that cannot be explained by covariates and to account for correlation among repeated sojourns in the same state within an individual. Correlation among sojourns across different states within an individual is accounted for by the correlation between the different latent traits. The factor loadings for a latent trait accommodate the dependence of the transitions to different competing states from a same state. We obtain the semiparametric maximum likelihood estimates through an expectation-maximization (EM) algorithm. The method is illustrated by studying repeated transitions between independence and disability states of activities of daily living (ADL) with death as an absorbing state in a longitudinal aging study. The performance of the estimation procedure is assessed by simulation studies.     

An estimator for the proportional hazards model with multiple longitudinal covariates measured with error

In many longitudinal studies, it is of interest to characterize the relationship between a time-to-event (e.g. survival) and several time-dependent and time-independent covariates. Time-dependent covariates are generally observed intermittently and with error. For a single time-dependent covariate, a popular approach is to assume a joint longitudinal data-survival model, where the time-dependent covariate follows a linear mixed effects model and the hazard of failure depends on random effects and time-independent covariates via a proportional hazards relationship. Regression calibration and likelihood or Bayesian methods have been advocated for implementation; however, generalization to more than one time-dependent covariate may become prohibitive. For a single time-dependent covariate, Tsiatis and Davidian (2001) have proposed an approach that is easily implemented and does not require an assumption on the distribution of the random effects. This technique may be generalized to multiple, possibly correlated, time-dependent covariates, as we demonstrate. We illustrate the approach via simulation and by application to data from an HIV clinical trial.     

A zero‐augmented generalized gamma regression calibration to adjust for covariate measurement error: A case of an episodically consumed dietary intake

Measurement error in exposure variables is a serious impediment in epidemiological studies that relate exposures to health outcomes. In nutritional studies, interest could be in the association between long‐term dietary intake and disease occurrence. Long‐term intake is usually assessed with food frequency questionnaire (FFQ), which is prone to recall bias. Measurement error in FFQ‐reported intakes leads to bias in parameter estimate that quantifies the association. To adjust for bias in the association, a calibration study is required to obtain unbiased intake measurements using a short‐term instrument such as 24‐hour recall (24HR). The 24HR intakes are used as response in regression calibration to adjust for bias in the association. For foods not consumed daily, 24HR‐reported intakes are usually characterized by excess zeroes, right skewness, and heteroscedasticity posing serious challenge in regression calibration modeling. We proposed a zero‐augmented calibration model to adjust for measurement error in reported intake, while handling excess zeroes, skewness, and heteroscedasticity simultaneously without transforming 24HR intake values. We compared the proposed calibration method with the standard method and with methods that ignore measurement error by estimating long‐term intake with 24HR and FFQ‐reported intakes. The comparison was done in real and simulated datasets. With the 24HR, the mean increase in mercury level per ounce fish intake was about 0.4; with the FFQ intake, the increase was about 1.2. With both calibration methods, the mean increase was about 2.0. Similar trend was observed in the simulation study. In conclusion, the proposed calibration method performs at least as good as the standard method.     

Nonparametric correction for covariate measurement error in a stratified Cox model

Stratified Cox regression models with large number of strata and small stratum size are useful in many settings, including matched case-control family studies. In the presence of measurement error in covariates and a large number of strata, we show that extensions of existing methods fail either to reduce the bias or to correct the bias under nonsymmetric distributions of the true covariate or the error term. We propose a nonparametric correction method for the estimation of regression coefficients, and show that the estimators are asymptotically consistent for the true parameters. Small sample properties are evaluated in a simulation study. The method is illustrated with an analysis of Framingham data.     

Joint models for a primary endpoint and multiple longitudinal covariate processes

Summary .   Joint models are formulated to investigate the association between a primary endpoint and features of multiple longitudinal processes. In particular, the subject-specific random effects in a multivariate linear random-effects model for multiple longitudinal processes are predictors in a generalized linear model for primary endpoints. Li, Zhang, and Davidian (2004, Biometrics 60 , 1–7) proposed an estimation procedure that makes no distributional assumption on the random effects but assumes independent within-subject measurement errors in the longitudinal covariate process. Based on an asymptotic bias analysis, we found that their estimators can be biased when random effects do not fully explain the within-subject correlations among longitudinal covariate measurements. Specifically, the existing procedure is fairly sensitive to the independent measurement error assumption. To overcome this limitation, we propose new estimation procedures that require neither a distributional or covariance structural assumption on covariate random effects nor an independence assumption on within-subject measurement errors. These new procedures are more flexible, readily cover scenarios that have multivariate longitudinal covariate processes, and can be implemented using available software. Through simulations and an analysis of data from a hypertension study, we evaluate and illustrate the numerical performances of the new estimators.     

Conditional estimation for generalized linear models when covariates are subject-specific parameters in a mixed model for longitudinal measurements

The relationship between a primary endpoint and features of longitudinal profiles of a continuous response is often of interest, and a relevant framework is that of a generalized linear model with covariates that are subject-specific random effects in a linear mixed model for the longitudinal measurements. Naive implementation by imputing subject-specific effects from individual regression fits yields biased inference, and several methods for reducing this bias have been proposed. These require a parametric (normality) assumption on the random effects, which may be unrealistic. Adapting a strategy of Stefanski and Carroll (1987, Biometrika74, 703-716), we propose estimators for the generalized linear model parameters that require no assumptions on the random effects and yield consistent inference regardless of the true distribution. The methods are illustrated via simulation and by application to a study of bone mineral density in women transitioning to menopause.     

A measurement error model for time-series studies of air pollution and mortality

One barrier to interpreting the observational evidence concerning the adverse health effects of air pollution for public policy purposes is the measurement error inherent in estimates of exposure based on ambient pollutant monitors. Exposure assessment studies have shown that data from monitors at central sites may not adequately represent personal exposure. Thus, the exposure error resulting from using centrally measured data as a surrogate for personal exposure can potentially lead to a bias in estimates of the health effects of air pollution. This paper develops a multi-stage Poisson regression model for evaluating the effects of exposure measurement error on estimates of effects of particulate air pollution on mortality in time-series studies. To implement the model, we have used five validation data sets on personal exposure to PM10. Our goal is to combine data on the associations between ambient concentrations of particulate matter and mortality for a specific location, with the validation data on the association between ambient and personal concentrations of particulate matter at the locations where data have been collected. We use these data in a model to estimate the relative risk of mortality associated with estimated personal-exposure concentrations and make a comparison with the risk of mortality estimated with measurements of ambient concentration alone. We apply this method to data comprising daily mortality counts, ambient concentrations of PM10measured at a central site, and temperature for Baltimore, Maryland from 1987 to 1994. We have selected our home city of Baltimore to illustrate the method; the measurement error correction model is general and can be applied to other appropriate locations.Our approach uses a combination of: (1) a generalized additive model with log link and Poisson error for the mortality-personal-exposure association; (2) a multi-stage linear model to estimate the variability across the five validation data sets in the personal-ambient-exposure association; (3) data augmentation methods to address the uncertainty resulting from the missing personal exposure time series in Baltimore. In the Poisson regression model, we account for smooth seasonal and annual trends in mortality using smoothing splines. Taking into account the heterogeneity across locations in the personal-ambient-exposure relationship, we quantify the degree to which the exposure measurement error biases the results toward the null hypothesis of no effect, and estimate the loss of precision in the estimated health effects due to indirectly estimating personal exposures from ambient measurements.     

Latent model for correlated binary data with diagnostic error

We propose a methodology for modeling correlated binary data measured with diagnostic error. A shared random effect is used to induce correlations in repeated true latent binary outcomes and in observed responses and to link the probability of a true positive outcome with the probability of having a diagnosis error. We evaluate the performance of our proposed approach through simulations and compare it with an ad hoc approach. The methodology is illustrated with data from a study that assessed the probability of corneal arcus in patients with familial hypercholesterolemia.     

Expected estimating equations for missing data, measurement error, and misclassification, with application to longitudinal nonignorable missing data

Summary .   Missing data, measurement error, and misclassification are three important problems in many research fields, such as epidemiological studies. It is well known that missing data and measurement error in covariates may lead to biased estimation. Misclassification may be considered as a special type of measurement error, for categorical data. Nevertheless, we treat misclassification as a different problem from measurement error because statistical models for them are different. Indeed, in the literature, methods for these three problems were generally proposed separately given that statistical modeling for them are very different. The problem is more challenging in a longitudinal study with nonignorable missing data. In this article, we consider estimation in generalized linear models under these three incomplete data models. We propose a general approach based on expected estimating equations (EEEs) to solve these three incomplete data problems in a unified fashion. This EEE approach can be easily implemented and its asymptotic covariance can be obtained by sandwich estimation. Intensive simulation studies are performed under various incomplete data settings. The proposed method is applied to a longitudinal study of oral bone density in relation to body bone density.