Semiparametric approaches for joint modeling of longitudinal and survival data with time-varying coefficients

Summary .   We study joint modeling of survival and longitudinal data. There are two regression models of interest. The primary model is for survival outcomes, which are assumed to follow a time-varying coefficient proportional hazards model. The second model is for longitudinal data, which are assumed to follow a random effects model. Based on the trajectory of a subject's longitudinal data, some covariates in the survival model are functions of the unobserved random effects. Estimated random effects are generally different from the unobserved random effects and hence this leads to covariate measurement error. To deal with covariate measurement error, we propose a local corrected score estimator and a local conditional score estimator. Both approaches are semiparametric methods in the sense that there is no distributional assumption needed for the underlying true covariates. The estimators are shown to be consistent and asymptotically normal. However, simulation studies indicate that the conditional score estimator outperforms the corrected score estimator for finite samples, especially in the case of relatively large measurement error. The approaches are demonstrated by an application to data from an HIV clinical trial.     

A penalized spline approach to functional mixed effects model analysis

In this article, we propose penalized spline (P-spline)-based methods for functional mixed effects models with varying coefficients. We decompose longitudinal outcomes as a sum of several terms: a population mean function, covariates with time-varying coefficients, functional subject-specific random effects, and residual measurement error processes. Using P-splines, we propose nonparametric estimation of the population mean function, varying coefficient, random subject-specific curves, and the associated covariance function that represents between-subject variation and the variance function of the residual measurement errors which represents within-subject variation. Proposed methods offer flexible estimation of both the population- and subject-level curves. In addition, decomposing variability of the outcomes as a between- and within-subject source is useful in identifying the dominant variance component therefore optimally model a covariance function. We use a likelihood-based method to select multiple smoothing parameters. Furthermore, we study the asymptotics of the baseline P-spline estimator with longitudinal data. We conduct simulation studies to investigate performance of the proposed methods. The benefit of the between- and within-subject covariance decomposition is illustrated through an analysis of Berkeley growth data, where we identified clearly distinct patterns of the between- and within-subject covariance functions of children's heights. We also apply the proposed methods to estimate the effect of antihypertensive treatment from the Framingham Heart Study data.     

Estimating marginal and incremental effects on health outcomes using flexible link and variance function models

We propose an extension to the estimating equations in generalized linear models to estimate parameters in the link function and variance structure simultaneously with regression coefficients. Rather than focusing on the regression coefficients, the purpose of these models is inference about the mean of the outcome as a function of a set of covariates, and various functionals of the mean function used to measure the effects of the covariates. A commonly used functional in econometrics, referred to as the marginal effect, is the partial derivative of the mean function with respect to any covariate, averaged over the empirical distribution of covariates in the model. We define an analogous parameter for discrete covariates. The proposed estimation method not only helps to identify an appropriate link function and to suggest an underlying distribution for a specific application but also serves as a robust estimator when no specific distribution for the outcome measure can be identified. Using Monte Carlo simulations, we show that the resulting parameter estimators are consistent. The method is illustrated with an analysis of inpatient expenditure data from a study of hospitalists.     

Multiple Imputation Methods for Estimating Regression Coefficients in the Competing Risks Model with Missing Cause of Failure

We propose a method to estimate the regression coefficients in a competing risks model where the cause-specific hazard for the cause of interest is related to covariates through a proportional hazards relationship and when cause of failure is missing for some individuals. We use multiple imputation procedures to impute missing cause of failure, where the probability that a missing cause is the cause of interest may depend on auxiliary covariates, and combine the maximum partial likelihood estimators computed from several imputed data sets into an estimator that is consistent and asymptotically normal. A consistent estimator for the asymptotic variance is also derived. Simulation results suggest the relevance of the theory in finite samples. Results are also illustrated with data from a breast cancer study.     

Efficacy of repeated measures in regression models with measurement error.

Ignoring measurement error may cause bias in the estimation of regression parameters. When the true covariates are unobservable, multiple imprecise measurements can be used in the analysis to correct for the associated bias. We suggest a simple estimating procedure that gives consistent estimates of regression parameters by using the repeated measurements with error. The relative Pitman efficiency of our estimator based on models with and without measurement error has been found to be a simple function of the number of replicates and the ratio of intra- to inter-variance of the true covariate. The procedure thus provides a guide for deciding the number of repeated measurements in the design stage. An example from a survey study is presented.     

Regularized estimation in the accelerated failure time model with high-dimensional covariates

We consider two regularization approaches, the LASSO and the threshold-gradient-directed regularization, for estimation and variable selection in the accelerated failure time model with multiple covariates based on Stute's weighted least squares method. The Stute estimator uses Kaplan-Meier weights to account for censoring in the least squares criterion. The weighted least squares objective function makes the adaptation of this approach to multiple covariate settings computationally feasible. We use V-fold cross-validation and a modified Akaike's Information Criterion for tuning parameter selection, and a bootstrap approach for variance estimation. The proposed method is evaluated using simulations and demonstrated on a real data example.     

Correcting for measurement error in individual-level covariates in nonlinear mixed effects models

The nonlinear mixed effects model is used to represent data in pharmacokinetics, viral dynamics, and other areas where an objective is to elucidate associations among individual-specific model parameters and covariates; however, covariates may be measured with error. For additive measurement error, we show substitution of mismeasured covariates for true covariates may lead to biased estimators for fixed effects and random effects covariance parameters, while regression calibration may eliminate bias in fixed effects but fail to correct that in covariance parameters. We develop methods to take account of measurement error that correct this bias and may be implemented with standard software, and we demonstrate their utility via simulation and application to data from a study of HIV dynamics.     

Robust best linear estimation for regression analysis using surrogate and instrumental variables

We investigate methods for regression analysis when covariates are measured with errors. In a subset of the whole cohort, a surrogate variable is available for the true unobserved exposure variable. The surrogate variable satisfies the classical measurement error model, but it may not have repeated measurements. In addition to the surrogate variables that are available among the subjects in the calibration sample, we assume that there is an instrumental variable (IV) that is available for all study subjects. An IV is correlated with the unobserved true exposure variable and hence can be useful in the estimation of the regression coefficients. We propose a robust best linear estimator that uses all the available data, which is the most efficient among a class of consistent estimators. The proposed estimator is shown to be consistent and asymptotically normal under very weak distributional assumptions. For Poisson or linear regression, the proposed estimator is consistent even if the measurement error from the surrogate or IV is heteroscedastic. Finite-sample performance of the proposed estimator is examined and compared with other estimators via intensive simulation studies. The proposed method and other methods are applied to a bladder cancer case-control study.     

A smoothed corrected score approach for proportional hazards model with misclassified discretized covariates induced by error-contaminated continuous time-dependent exposure

We consider the proportional hazards model in which the covariates include the discretized categories of a continuous time-dependent exposure variable measured with error. Naively ignoring the measurement error in the analysis may cause biased estimation and erroneous inference. Although various approaches have been proposed to deal with measurement error when the hazard depends linearly on the time-dependent variable, it has not yet been investigated how to correct when the hazard depends on the discretized categories of the time-dependent variable. To fill this gap in the literature, we propose a smoothed corrected score approach based on approximation of the discretized categories after smoothing the indicator function. The consistency and asymptotic normality of the proposed estimator are established. The observation times of the time-dependent variable are allowed to be informative. For comparison, we also extend to this setting two approximate approaches, the regression calibration and the risk-set regression calibration. The methods are assessed by simulation studies and by application to data from an HIV clinical trial.     

A Stochastic Regression Model for General Trend Analysis of Longitudinal Continuous Data

A predictive continuous time model is developed for continuous panel data to assess the effect of time‐varying covariates on the general direction of the movement of a continuous response that fluctuates over time. This is accomplished by reparameterizing the infinitesimal mean of an Ornstein–Uhlenbeck processes in terms of its equilibrium mean and a drift parameter, which assesses the rate that the process reverts to its equilibrium mean. The equilibrium mean is modeled as a linear predictor of covariates. This model can be viewed as a continuous time first‐order autoregressive regression model with time‐varying lag effects of covariates and the response, which is more appropriate for unequally spaced panel data than its discrete time analog. Both maximum likelihood and quasi‐likelihood approaches are considered for estimating the model parameters and their performances are compared through simulation studies. The simpler quasi‐likelihood approach is suggested because it yields an estimator that is of high efficiency relative to the maximum likelihood estimator and it yields a variance estimator that is robust to the diffusion assumption of the model. To illustrate the proposed model, an application to diastolic blood pressure data from a follow‐up study on cardiovascular diseases is presented. Missing observations are handled naturally with this model.     

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.     

Maximum likelihood estimation in the additive hazards model

The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As the non-parametric model, additive hazards offer a very flexible way of modeling time-varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper, we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non-negative for all covariate values in their domain. We show that the maximum likelihood estimator may be obtained by separately maximizing the log-likelihood contribution of each event time point, and we show that the maximizing problem is equivalent to fitting a series of Poisson regression models with an identity link under non-negativity constraints. We derive an analytic solution to the maximum likelihood estimator. We contrast the maximum likelihood estimator with the ordinary least-squares estimator in a simulation study and show that the maximum likelihood estimator has smaller mean squared error than the ordinary least-squares estimator. An illustration with data on patients with carcinoma of the oropharynx is provided.     

On Method of Moments Estimation in Linear Mixed Effects Models with Measurement Error on Covariates and Response with Application to a Longitudinal Study of Gene-Environment Interaction

We study a linear mixed effects model for longitudinal data, where the response variable and covariates with fixed effects are subject to measurement error. We propose a method of moment estimation that does not require any assumption on the functional forms of the distributions of random effects and other random errors in the model. For a classical measurement error model we apply the instrumental variable approach to ensure identifiability of the parameters. Our methodology, without instrumental variables, can be applied to Berkson measurement errors. Using simulation studies, we investigate the finite sample performances of the estimators and show the impact of measurement error on the covariates and the response on the estimation procedure. The results show that our method performs quite satisfactory, especially for the fixed effects with measurement error (even under misspecification of measurement error model). This method is applied to a real data example of a large birth and child cohort study.     

Laplace approximation in measurement error models

Likelihood analysis for regression models with measurement errors in explanatory variables typically involves integrals that do not have a closed-form solution. In this case, numerical methods such as Gaussian quadrature are generally employed. However, when the dimension of the integral is large, these methods become computationally demanding or even unfeasible. This paper proposes the use of the Laplace approximation to deal with measurement error problems when the likelihood function involves high-dimensional integrals. The cases considered are generalized linear models with multiple covariates measured with error and generalized linear mixed models with measurement error in the covariates. The asymptotic order of the approximation and the asymptotic properties of the Laplace-based estimator for these models are derived. The method is illustrated using simulations and real-data analysis.     

Analysis of covariance with incomplete data via semiparametric model transformations

We propose a method for fitting semiparametric models such as the proportional hazards (PH), additive risks (AR), and proportional odds (PO) models. Each of these semiparametric models implies that some transformation of the conditional cumulative hazard function (at each t) depends linearly on the covariates. The proposed method is based on nonparametric estimation of the conditional cumulative hazard function, forming a weighted average over a range of t-values, and subsequent use of least squares to estimate the parameters suggested by each model. An approximation to the optimal weight function is given. This allows semiparametric models to be fitted even in incomplete data cases where the partial likelihood fails (e.g., left censoring, right truncation). However, the main advantage of this method rests in the fact that neither the interpretation of the parameters nor the validity of the analysis depend on the appropriateness of the PH or any of the other semiparametric models. In fact, we propose an integrated method for data analysis where the role of the various semiparametric models is to suggest the best fitting transformation. A single continuous covariate and several categorical covariates (factors) are allowed. Simulation studies indicate that the test statistics and confidence intervals have good small-sample performance. A real data set is analyzed.     

Regression analysis of mean quality-adjusted lifetime with censored data

In clinical trials of chronic diseases such as acquired immunodeficiency syndrome, cancer, or cardiovascular diseases, the concept of quality-adjusted lifetime (QAL) has received more and more attention. In this paper, we consider the problem of how the covariates affect the mean QAL when the data are subject to right censoring. We allow a very general form for the mean model as a function of covariates. Using the idea of inverse probability weighting, we first construct a simple weighted estimating equation for the parameters in our mean model. We then find the form of the most efficient estimating equation, which yields the most efficient estimator for the regression parameters. Since the most efficient estimator depends on the distribution of the health history processes, and thus cannot be estimated nonparametrically, we consider different approaches for improving the efficiency of the simple weighted estimating equation using observed data. The applicability of these methods is demonstrated by both simulation experiments and a data example from a breast cancer clinical trial study.     

Longitudinal Data Analysis with Event Time as a Covariate

We consider the estimation of a nonparametric smooth function of some event time in a semiparametric mixed effects model from repeatedly measured data when the event time is subject to right censoring. The within-subject correlation is captured by both cross-sectional and time-dependent random effects, where the latter is modeled by a nonhomogeneous Ornstein–Uhlenbeck stochastic process. When the censoring probability depends on other variables in the model, which often happens in practice, the event time data are not missing completely at random. Hence, the complete case analysis by eliminating all the censored observations may yield biased estimates of the regression parameters including the smooth function of the event time, and is less efficient. To remedy, we derive the likelihood function for the observed data by modeling the event time distribution given other covariates. We propose a two-stage pseudo-likelihood approach for the estimation of model parameters by first plugging an estimator of the conditional event time distribution into the likelihood and then maximizing the resulting pseudo-likelihood function. Empirical evaluation shows that the proposed method yields negligible biases while significantly reduces the estimation variability. This research is motivated by the project of hormone profile estimation around age at the final menstrual period for the cohort of women in the Michigan Bone Health and Metabolism Study.     

A Nonrecursive Estimator for the Cox Model

The Cox regression model is one of the most widely used models to incorporate covariates. The frequently used partial likelihood estimator of the regression parameter has to be computed iteratively. In this paper we propose a noniterative estimator for the regression parameter and show that under certain conditions it dominates another noniterative estimator derived by Kalbfleish and Prentice. The new estimator is demonstrated on lifetime data of rats having been subject to insult with a carcinogen.     

Approximate Bayesian evaluation of multiple treatment effects

We propose an approximate Bayesian method for comparing an experimental treatment to a control based on a randomized clinical trial with multivariate patient outcomes. Overall treatment effect is characterized by a vector of parameters corresponding to effects on the individual patient outcomes. We partition the parameter space into four sets where, respectively, the experimental treatment is superior to the control, the control is superior to the experimental, the two treatments are equivalent, and the treatment effects are discordant. We compute posterior probabilities of the parameter sets by treating an estimator of the parameter vector like a random variable in the Bayesian paradigm. The approximation may be used in any setting where a consistent, asymptotically normal estimator of the parameter vector is available. The method is illustrated by application to a breast cancer data set consisting of multiple time-to-event outcomes with covariates and to count data arising from a cross-classification of response, infection, and treatment in an acute leukemia trial.     

Cox Regression in Nested Case–Control Studies with Auxiliary Covariates

Summary Nested case–control (NCC) design is a popular sampling method in large epidemiological studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors. Thomas' maximum partial likelihood estimator is commonly used to estimate the regression parameters in Cox's model for NCC data. In this article, we consider a situation in which failure/censoring information and some crude covariates are available for the entire cohort in addition to NCC data and propose an improved estimator that is asymptotically more efficient than Thomas' estimator. We adopt a projection approach that, heretofore, has only been employed in situations of random validation sampling and show that it can be well adapted to NCC designs where the sampling scheme is a dynamic process and is not independent for controls. Under certain conditions, consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed. Furthermore, a simplified approximate estimator is proposed when the disease is rare. Extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with Thomas' estimator and other competing estimators. Moreover, sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated, and we find that the biases are reasonably small in realistic situations. We further demonstrate the proposed method with data from studies on Wilms' tumor.