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.     

A new nonparametric approach for baseline covariate adjustment for two-group comparative studies

SUMMARY: We consider two-armed clinical trials in which the response and/or the covariates are observed on either a binary, ordinal, or continuous scale. A new general nonparametric (NP) approach for covariate adjustment is presented using the notion of a relative effect to describe treatment effects. The relative effect is defined by the probability of observing a higher response in the experimental than in the control arm. The notion is invariant under monotone transformations of the data and is therefore especially suitable for ordinal data. For a normal or binary distributed response the relative effect is the transformed effect size or the difference of response probability, respectively. An unbiased and consistent NP estimator for the relative effect is presented. Further, we suggest a NP procedure for correcting the relative effect for covariate imbalance and random covariate imbalance, yielding a consistent estimator for the adjusted relative effect. Asymptotic theory has been developed to derive test statistics and confidence intervals. The test statistic is based on the joint behavior of the estimated relative effect for the response and the covariates. It is shown that the test statistic can be used to evaluate the treatment effect in the presence of (random) covariate imbalance. Approximations for small sample sizes are considered as well. The sampling behavior of the estimator of the adjusted relative effect is examined. We also compare the probability of a type I error and the power of our approach to standard covariate adjustment methods by means of a simulation study. Finally, our approach is illustrated on three studies involving ordinal responses and covariates.     

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.     

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.     

Maximum likelihood methods for nonignorable missing responses and covariates in random effects models

This article analyzes quality of life (QOL) data from an Eastern Cooperative Oncology Group (ECOG) melanoma trial that compared treatment with ganglioside vaccination to treatment with high-dose interferon. The analysis of this data set is challenging due to several difficulties, namely, nonignorable missing longitudinal responses and baseline covariates. Hence, we propose a selection model for estimating parameters in the normal random effects model with nonignorable missing responses and covariates. Parameters are estimated via maximum likelihood using the Gibbs sampler and a Monte Carlo expectation maximization (EM) algorithm. Standard errors are calculated using the bootstrap. The method allows for nonmonotone patterns of missing data in both the response variable and the covariates. We model the missing data mechanism and the missing covariate distribution via a sequence of one-dimensional conditional distributions, allowing the missing covariates to be either categorical or continuous, as well as time-varying. We apply the proposed approach to the ECOG quality-of-life data and conduct a small simulation study evaluating the performance of the maximum likelihood estimates. Our results indicate that a patient treated with the vaccine has a higher QOL score on average at a given time point than a patient treated with high-dose interferon.     

Regression Analysis with a Misclassified Covariate from a Current Status Observation Scheme

Summary Naive use of misclassified covariates leads to inconsistent estimators of covariate effects in regression models. A variety of methods have been proposed to address this problem including likelihood, pseudo‐likelihood, estimating equation methods, and Bayesian methods, with all of these methods typically requiring either internal or external validation samples or replication studies. We consider a problem arising from a series of orthopedic studies in which interest lies in examining the effect of a short‐term serological response and other covariates on the risk of developing a longer term thrombotic condition called deep vein thrombosis. The serological response is an indicator of whether the patient developed antibodies following exposure to an antithrombotic drug, but the seroconversion status of patients is only available at the time of a blood sample taken upon the discharge from hospital. The seroconversion time is therefore subject to a current status observation scheme, or Case I interval censoring, and subjects tested before seroconversion are misclassified as nonseroconverters. We develop a likelihood‐based approach for fitting regression models that accounts for misclassification of the seroconversion status due to early testing using parametric and nonparametric estimates of the seroconversion time distribution. The method is shown to reduce the bias resulting from naive analyses in simulation studies and an application to the data from the orthopedic studies provides further illustration.     

Semiparametric models for missing covariate and response data in regression models

We consider a class of semiparametric models for the covariate distribution and missing data mechanism for missing covariate and/or response data for general classes of regression models including generalized linear models and generalized linear mixed models. Ignorable and nonignorable missing covariate and/or response data are considered. The proposed semiparametric model can be viewed as a sensitivity analysis for model misspecification of the missing covariate distribution and/or missing data mechanism. The semiparametric model consists of a generalized additive model (GAM) for the covariate distribution and/or missing data mechanism. Penalized regression splines are used to express the GAMs as a generalized linear mixed effects model, in which the variance of the corresponding random effects provides an intuitive index for choosing between the semiparametric and parametric model. Maximum likelihood estimates are then obtained via the EM algorithm. Simulations are given to demonstrate the methodology, and a real data set from a melanoma cancer clinical trial is analyzed using the proposed methods.     

Generalized case-control sampling under generalized linear models

A generalized case-control (GCC) study, like the standard case-control study, leverages outcome-dependent sampling (ODS) to extend to nonbinary responses. We develop a novel, unifying approach for analyzing GCC study data using the recently developed semiparametric extension of the generalized linear model (GLM), which is substantially more robust to model misspecification than existing approaches based on parametric GLMs. For valid estimation and inference, we use a conditional likelihood to account for the biased sampling design. We describe analysis procedures for estimation and inference for the semiparametric GLM under a conditional likelihood, and we discuss problems with estimation and inference under a conditional likelihood when the response distribution is misspecified. We demonstrate the flexibility of our approach over existing ones through extensive simulation studies, and we apply the methodology to an analysis of the Asset and Health Dynamics Among the Oldest Old study, which motives our research. The proposed approach yields a simple yet versatile solution for handling ODS in a wide variety of possible response distributions and sampling schemes encountered in practice.     

A semiparametric method for analyzing matched case-control family studies with a continuous outcome and proband sampling

We consider matched case-control familial studies which match a group of patients, called "case probands," with a group of disease-free subjects, called "control probands," using a set of family-level matching variables. Family members of each proband are then recruited into the study. Of interest here is the familial aggregation of the response variable and the effects of subject-specific covariates on the response. We propose an estimating equation approach to jointly estimate the main effects and intrafamilial correlations for matched family studies with a continuous outcome. Only knowledge of the first two joint moments of the response variable is required. The induced estimators for the main effects and intrafamilial correlations are consistent and asymptotically normally distributed. We apply the proposed method to sleep apnea data. A simulation study demonstrates the usefulness of our approach.     

A flexible approach to measurement error correction in case-control studies

SUMMARY: We investigate the use of prospective likelihood methods to analyze retrospective case-control data where some of the covariates are measured with error. We show that prospective methods can be applied and the case-control sampling scheme can be ignored if one adequately models the distribution of the error-prone covariates in the case-control sampling scheme. Indeed, subject to this, the prospective likelihood methods result in consistent estimates and information standard errors are asymptotically correct. However, the distribution of such covariates is not the same in the population and under case-control sampling, dictating the need to model the distribution flexibly. In this article, we illustrate the general principle by modeling the distribution of the continuous error-prone covariates using the skewnormal distribution. The performance of the method is evaluated through simulation studies, which show satisfactory results in terms of bias and coverage. Finally, the method is applied to the analysis of two data sets which refer, respectively, to a cholesterol study and a study on breast cancer.     

On Combining Family‐Based and Population‐Based Case–Control Data in Association Studies

Summary Combining data collected from different sources can potentially enhance statistical efficiency in estimating effects of environmental or genetic factors or gene–environment interactions. However, combining data across studies becomes complicated when data are collected under different study designs, such as family‐based and unrelated individual‐based case–control design. In this article, we describe likelihood‐based approaches that permit the joint estimation of covariate effects on disease risk under study designs that include cases, relatives of cases, and unrelated individuals. Our methods accommodate familial residual correlation and a variety of ascertainment schemes. Extensive simulation experiments demonstrate that the proposed methods for estimation and inference perform well in realistic settings. Efficiencies of different designs are contrasted in the simulation. We applied the methods to data from the Colorectal Cancer Family Registry.     

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.     

Ascertainment bias in rate ratio estimation from case-sibling control studies of variable age-at-onset diseases

Motivated by a Finnish case-control study of early onset diabetes in which diabetic children are matched to sibling controls, we investigate ascertainment bias of the usual rate ratio estimator from case-control data under simplex complete ascertainment of families during a fixed interval of time. Analytic results indicate that the assumptions necessary for valid estimation are that the disease is rare and the factors under study are exchangeable--essentially that the covariate distribution does not depend on calendar time or birth order. Further, we found that the rare disease assumption could be dropped by restricting to cases that were diagnosed during the enrollment period of the study or including all cases but eliminating the proband as a control for non-enrollment-period cases. An important consequence of this work is that standard family-based case-control studies are subject to ascertainment bias if exchangeability of the covariates under investigation does not hold.     

A Bayesian Approach to Joint Mixed‐Effects Models with a Skew‐Normal Distribution and Measurement Errors in Covariates

Summary In recent years, nonlinear mixed‐effects (NLME) models have been proposed for modeling complex longitudinal data. Covariates are usually introduced in the models to partially explain intersubject variations. However, one often assumes that both model random error and random effects are normally distributed, which may not always give reliable results if the data exhibit skewness. Moreover, some covariates such as CD4 cell count may be often measured with substantial errors. In this article, we address these issues simultaneously by jointly modeling the response and covariate processes using a Bayesian approach to NLME models with covariate measurement errors and a skew‐normal distribution. A real data example is offered to illustrate the methodologies by comparing various potential models with different distribution specifications. It is showed that the models with skew‐normality assumption may provide more reasonable results if the data exhibit skewness and the results may be important for HIV/AIDS studies in providing quantitative guidance to better understand the virologic responses to antiretroviral treatment.     

Regression models for infant mortality data in Norwegian siblings, using a compound Poisson frailty distribution with random scale

The power variance function distributions, which include the gamma and compound Poisson (CP) distributions among others, are commonly used in frailty models for family data. In a previous paper, we presented a frailty model constructed by randomizing the scale parameter in a CP distribution. When combined with a parametric baseline hazard, this yields a model with heterogeneity on both the individual and the family level and a subgroup with zero frailty, corresponding to people not experiencing the event. In this paper, we discuss covariates in the model. Depending on where the covariates are inserted in the model, one may have proportional hazards at the individual level, the family level, and a larger group level (for covariates shared by many families, e.g. ethnic groups) or get accelerated failure times. Each of these alternatives gives a specific interpretation of the covariate effects. An application to data infant mortality in siblings from the Medical Birth Registry of Norway is included. We compare the results for some of the different covariate modeling options.     

Evaluating the use of friend or family controls in epidemiologic case-control studies

BackgroundTraditional methodologies for identifying and recruiting controls in epidemiologic case-control studies, such as random digit dialing or neighborhood walk, suffer from declining response rates. Here, we revisit the feasibility and comparability of using alternative sources of controls, specifically friend and family controls.MethodsWe recruited from a recently completed case-control study of non-Hodgkin lymphoma (NHL) among women in Los Angeles County where controls from the parent study were ascertained by neighborhood walk. We calculated participation rates and compared questionnaire responses between the friend/family controls and the original matched controls from the parent study.ResultsOf the 182 NHL case patients contacted, 111 (61%) agreed to participate in our feasibility study. 70 (63%) provided contact information for potential friend and/or family member controls. We were able to successfully contact and recruit a friend/family member for 92% of the case patients. This represented 46 friend controls and 54 family controls. Family controls significantly differed from original matched controls by sex and household income. Other characteristics were similar between friend controls and the original study’s neighborhood controls.ConclusionThe apparent comparability of neighborhood controls to friend and family controls among respondents in this study suggests that these alternative methods of control identification can serve as a complementary source of eligible controls in epidemiologic case-control studies.     

A general method for dealing with misclassification in regression: the misclassification SIMEX

We have developed a new general approach for handling misclassification in discrete covariates or responses in regression models. The simulation and extrapolation (SIMEX) method, which was originally designed for handling additive covariate measurement error, is applied to the case of misclassification. The statistical model for characterizing misclassification is given by the transition matrix Pi from the true to the observed variable. We exploit the relationship between the size of misclassification and bias in estimating the parameters of interest. Assuming that Pi is known or can be estimated from validation data, we simulate data with higher misclassification and extrapolate back to the case of no misclassification. We show that our method is quite general and applicable to models with misclassified response and/or misclassified discrete regressors. In the case of a binary response with misclassification, we compare our method to the approach of Neuhaus, and to the matrix method of Morrissey and Spiegelman in the case of a misclassified binary regressor. We apply our method to a study on caries with a misclassified longitudinal response.     

Likelihood methods for incomplete longitudinal binary responses with incomplete categorical covariates

We consider longitudinal studies in which the outcome observed over time is binary and the covariates of interest are categorical. With no missing responses or covariates, one specifies a multinomial model for the responses given the covariates and uses maximum likelihood to estimate the parameters. Unfortunately, incomplete data in the responses and covariates are a common occurrence in longitudinal studies. Here we assume the missing data are missing at random (Rubin, 1976, Biometrika 63, 581-592). Since all of the missing data (responses and covariates) are categorical, a useful technique for obtaining maximum likelihood parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). In using the EM algorithm with missing responses and covariates, one specifies the joint distribution of the responses and covariates. Here we consider the parameters of the covariate distribution as a nuisance. In data sets where the percentage of missing data is high, the estimates of the nuisance parameters can lead to highly unstable estimates of the parameters of interest. We propose a conditional model for the covariate distribution that has several modeling advantages for the EM algorithm and provides a reduction in the number of nuisance parameters, thus providing more stable estimates in finite samples.     

Nonparametric tests for continuous covariate effects with multistate survival data

SUMMARY: In clinical trials and observational studies, it is often of scientific interest to evaluate the effects of covariates on complex multistate event probabilities. With discrete covariates, nonparametric tests may be constructed using estimates of the relevant quantities. With continuous covariates, a common approach is to arbitrarily discretize the covariates, which may lead to substantial information loss. Another strategy is to formulate the covariate effects in a regression model. Model-based tests may have either low power or be biased under misspecification. We propose nonparametric tests not requiring arbitrary discretization. The tests involve integrals of estimates continuously indexed by dichotomizations of the covariates. General asymptotic results are derived under null and alternative hypotheses, and verified using empirical process theory in several special cases. The tests are consistent under stochastic ordering, which arises naturally with multistate data. A novel nonparametric measure of covariate effect is studied as a natural byproduct of the testing procedure. Simulation studies and two real data analyses demonstrate the gains of the new testing procedure over those based either on categorization or on regression models.     

Resampling-based multiple testing methods with covariate adjustment: application to investigation of antiretroviral drug susceptibility

Summary .   Identifying genetic mutations that cause clinical resistance to antiretroviral drugs requires adjustment for potential confounders, such as the number of active drugs in a HIV-infected patient's regimen other than the one of interest. Motivated by this problem, we investigated resampling-based methods to test equal mean response across multiple groups defined by HIV genotype, after adjustment for covariates. We consider construction of test statistics and their null distributions under two types of model: parametric and semiparametric. The covariate function is explicitly specified in the parametric but not in the semiparametric approach. The parametric approach is more precise when models are correctly specified, but suffer from bias when they are not; the semiparametric approach is more robust to model misspecification, but may be less efficient. To help preserve type I error while also improving power in both approaches, we propose resampling approaches based on matching of observations with similar covariate values. Matching reduces the impact of model misspecification as well as imprecision in estimation. These methods are evaluated via simulation studies and applied to a data set that combines results from a variety of clinical studies of salvage regimens. Our focus is on relating HIV genotype to viral susceptibility to abacavir after adjustment for the number of active antiretroviral drugs (excluding abacavir) in the patient's regimen.