Time-dependent Covariate Effects Research Articles

A Symmetric Kernel Smoothing Estimation of the Time-Varying Coefficient for Medical Costs

In longitudinal studies, subjects are repeatedly observed at a set of distinct time points until the terminal event time. The time-varying coefficient model extends the parametric method and captures the dynamic trajectories of time-dependent covariate effects, thus enabling it to describe the potential relationship between the longitudinal variable and the observed time points. In this study, we propose a novel approach to the estimation of medical costs using a symmetric kernel smoothing method in the time-varying coefficient joint model. A smooth function of medical costs is derived by weighting the values of longitudinal data at all distinct observed time points via the combination of the kernel method and the inverse probability weighting method. For the simulation study, we first set up the true functions of time-varying coefficients; we then generated random samples for covariates and censored survival times. Subsequently, the longitudinal data of response variables could be produced. Further, numerical simulation experiments were conducted by using the proposed method and applying R code to the generated data. The estimated results for the parameters and non-parametric functions were compared with different settings. The numerical results illustrate that as the sample size increases, the bias and model-based standard errors decrease, and the performance improves with larger sample sizes. The estimates of functions in the model almost coincide with the true functions, as shown in the figures of the simulation study. Furthermore, the consistency of the obtained estimator is demonstrated via theoretical analysis, and a numerical simulation is performed to illustrate the performance of the proposed estimators. The proposed model is applied to a real-world data set acquired from a multicenter automatic defibrillator implantation trial (MADIT).

Bivariate INAR(1) model under negative binomial innovations with non-homogeneous over-dispersed indices and application

This paper introduces a new bivariate integer-valued autoregressive of order (1) (BINAR(1)) model with negative binomial (NB) innovations under non-stationary moments. The purpose of this time series process is mainly to model series that are affected by time-dependent covariate effects and that, in particular, exhibit different levels of over-dispersion which is a phenomenon commonly noticed in many real-life series applications. In this proposed model, the cross-correlation is induced locally by allowing the current counting series observation to relate with the previous-lagged observation of the other series or vice versa while the pair of NB innovations are assumed uncorrelated. The estimation of the regression, over-dispersion and dependence parameters is conducted using a generalized quasi-likelihood (GQL) approach since the specification of the likelihood function, under non-stationarity, is rather difficult to specify in the above situation. Monte-Carlo simulation experiments are executed to assess the quality of the GQL estimators. The model is also applied and compared with other bivariate time series models to some real-life series in Mauritius.

Partially linear Bayesian modeling of longitudinal rank and time-to-event data using accelerated failure time model with application to brain tumor data.

Joint modeling of longitudinal rank and time-to-event data with random effects model using a Bayesian approach is presented. Accelerated failure time (AFT) models can be used for the analysis of time-to-event data to estimate the effects of covariates on acceleration/deceleration of the survival time. The parametric AFT models require determining the event time distribution. So, we suppose that the time variable is modeled with Weibull AFT distribution. In many real-life applications, it is difficult to determine the appropriate distribution. To avoid this restriction, several semiparametric AFT models were proposed, containing spline-based model. So, we propose a flexible extension of the accelerated failure time model. Furthermore, the usual joint linear model, a joint partially linear model, is also considered containing the nonlinear effect of time on the longitudinal rank responses and nonlinear and time-dependent effects of covariates on the hazard. Also, a Bayesian approach that yields Bayesian estimates of the model's parameters is used. Some simulation studies are conducted to estimate parameters of the considered models. The model is applied to a real brain tumor patient's data set that underwent surgery. The results of analyzing data are presented to represent the method.

Flexible extension of the accelerated failure time model to account for nonlinear and time-dependent effects of covariates on the hazard.

The accelerated failure time model is an alternative to the Cox proportional hazards model in survival analysis. However, conclusions regarding the associations of prognostic factors with event times are valid only if the underlying modeling assumptions are met. In contrast to several flexible methods for relaxing the proportional hazards and linearity assumptions in the Cox model, formal investigation of the constant-over-time time ratio and linearity assumptions in the accelerated failure time model has been limited. Yet, in practice, prognostic factors may have time-dependent and/or nonlinear effects. Furthermore, parametric accelerated failure time models require correct specification of the baseline hazard function, which is treated as a nuisance parameter in the Cox proportional hazards model, and is rarely known in practice. To address these challenges, we propose a flexible extension of the accelerated failure time model where unpenalized regression B-splines are used to model (i) the baseline hazard function of arbitrary shape, (ii) the time-dependent covariate effects on the hazard, and (iii) nonlinear effects for continuous covariates. Simulations evaluate the accuracy of the time-dependent and/or nonlinear estimates, and of the resulting survival functions, in multivariable settings. The proposed flexible extension of the accelerated failure time model is applied to re-assess the effects of prognostic factors on mortality after septic shock.

Mexhaz: An R Package for Fitting Flexible Hazard-Based Regression Models for Overall and Excess Mortality with a Random Effect

We present mexhaz, an R package for fitting flexible hazard-based regression models with the possibility to add time-dependent effects of covariates and to account for a twolevel hierarchical structure in the data through the inclusion of a normally distributed random intercept (i.e., a log-normally distributed shared frailty). Moreover, mexhazbased models can be fitted within the excess hazard setting by allowing the specification of an expected hazard in the model. These models are of common use in the context of the analysis of population-based cancer registry data. Follow-up time can be entered in the right-censored or counting process input style, the latter allowing models with delayed entries. The logarithm of the baseline hazard can be flexibly modeled with B-splines or restricted cubic splines of time. Parameters estimation is based on likelihood maximization: in deriving the contribution of each observation to the cluster-specific conditional likelihood, Gauss-Legendre quadrature is used to calculate the cumulative hazard; the cluster-specific marginal likelihoods are then obtained by integrating over the random effects distribution, using adaptive Gauss-Hermite quadrature. Functions to compute and plot the predicted (excess) hazard and (net) survival (possibly with cluster-specific predictions in the case of random effect models) are provided. We illustrate the use of the different options of the mexhaz package and compare the results obtained with those of other available R packages.

HIV incidence declines in a rural South African population: a G-imputation approach for inference

BackgroundAd hoc assumptions about the unobserved infection event, which is known only to occur between the latest-negative and earliest-positive test dates, can lead to biased HIV incidence rate estimates. Using a G-imputation approach, we infer the infection dates from covariate data to estimate the HIV incidence rate in a hyper-endemic South African setting.MethodsA large demographic surveillance system has annually tested a cohort of HIV-uninfected participants living in the KwaZulu-Natal province. Using this data, we estimated a cumulative baseline hazard function and the effects of time-dependent covariates on the interval censored infection dates. For each HIV-positive participant in the cohort, we derived a cumulative distribution function and sampled multiple infection dates conditional on the unique covariate values. We right censored the data at the imputed dates, calculated the annual HIV incidence rate per 100 person-years, and used Rubin’s rules to obtain the 95% confidence intervals.ResultsA total of 20,011 uninfected individuals with a repeat HIV test participated in the incidence cohort between 2005 and 2018. We observed 2,603 infections per 58,769 person-years of follow-up among women and 845 infections per 41,178 person-years of follow-up among men. Conditional on age and circumcision status (men only), the female HIV incidence rate declined by 25%, from 5.0 to 3.7 infections per 100 person-years between 2014 and 2018. During this period, the HIV incidence rate among men declined from 2.1 to 1.1 infections per 100 person-years—a reduction of 49%. We observed similar reductions in male and female HIV incidence conditional on condom-use, marital status, urban residential status, migration history, and the HIV prevalence in the surrounding community.ConclusionWe have followed participants in one of the world’s largest and longest running HIV cohorts to estimate long-term trends in the population-wide incidence of infection. Using a G-imputation approach, we present further evidence for HIV incidence rate declines in this hyper-endemic South African setting.

General regression model for the subdistribution of a competing risk under left-truncation and right-censoring.

Left-truncation poses extra challenges for the analysis of complex time-to-event data. We propose a general semiparametric regression model for left-truncated and right-censored competing risks data that is based on a novel weighted conditional likelihood function. Targeting the subdistribution hazard, our parameter estimates are directly interpretable with regard to the cumulative incidence function. We compare different weights from recent literature and develop a heuristic interpretation from a cure model perspective that is based on pseudo risk sets. Our approach accommodates external time-dependent covariate effects on the subdistribution hazard. We establish consistency and asymptotic normality of the estimators and propose a sandwich estimator of the variance. In comprehensive simulation studies we demonstrate solid performance of the proposed method. Comparing the sandwich estimator with the inverse Fisher information matrix, we observe a bias for the inverse Fisher information matrix and diminished coverage probabilities in settings with a higher percentage of left-truncation. To illustrate the practical utility of the proposed method, we study its application to a large HIV vaccine efficacy trial dataset.

A general semiparametric approach to inference with marker-dependent hazard rate models

We examine a new general class of hazard rate models for duration data, containing a parametric and a nonparametric component. Both can be a mix of a time effect and possibly time-dependent covariate effects. A number of well-known models are special cases. In a counting process framework, a general profile likelihood estimator is developed and the parametric component of the model is shown to be asymptotically normal and efficient. Finite sample properties are investigated in simulations. The estimator is applied to investigate the long-run relationship between birth weight and later-life mortality.

Nonlinear and time-dependent effects of sparsely measured continuous time-varying covariates in time-to-event analysis.

Many flexible extensions of the Cox proportional hazards model incorporate time-dependent (TD) and/or nonlinear (NL) effects of time-invariant covariates. In contrast, little attention has been given to the assessment of such effects for continuous time-varying covariates (TVCs). We propose a flexible regression B-spline-based model for TD and NL effects of a TVC. To account for sparse TVC measurements, we added to this model the effect of time elapsed since last observation (TEL), which acts as an effect modifier. TD, NL, and TEL effects are estimated with the iterative alternative conditional estimation algorithm. Furthermore, a simulation extrapolation (SIMEX)-like procedure was adapted to correct the estimated effects for random measurement errors in the observed TVC values. In simulations, TD and NL estimates were unbiased if the TVC was measured with a high frequency. With sparse measurements, the strength of the effects was underestimated but the TEL estimate helped reduce the bias, whereas SIMEX helped further to correct for bias toward the null due to "white noise" measurement errors. We reassessed the effects of systolic blood pressure (SBP) and total cholesterol, measured at two-year intervals, on cardiovascular risks in women participating in the Framingham Heart Study. Accounting for TD effects of SBP, cholesterol and age, the NL effect of cholesterol, and the TEL effect of SBP improved substantially the model's fit to data. Flexible estimates yielded clinically important insights regarding the role of these risk factors. These results illustrate the advantages of flexible modeling of TVC effects.

State transition modeling of complex monitored health data

This article considers the analysis of complex monitored health data, where often one or several signals are reflecting the current health status that can be represented by a finite number of states, in addition to a set of covariates. In particular, we consider a novel application of a non-parametric state intensity regression method in order to study time-dependent effects of covariates on the state transition intensities. The method can handle baseline, time varying as well as dynamic covariates. Because of the non-parametric nature, the method can handle different data types and challenges under minimal assumptions. If the signal that is reflecting the current health status is of continuous nature, we propose the application of a weighted median and a hysteresis filter as data pre-processing steps in order to facilitate robust analysis. In intensity regression, covariates can be aggregated by a suitable functional form over a time history window. We propose to study the estimated cumulative regression parameters for different choices of the time history window in order to investigate short- and long-term effects of the given covariates. The proposed framework is discussed and applied to resuscitation data of newborns collected in Tanzania.

Time-varying feature selection for longitudinal analysis.

We propose time-varying coefficient model selection and estimation based on the spline approach, which is capable of capturing time-dependent covariate effects. The new penalty function utilizes local-region information for varying-coefficient estimation, in contrast to the traditional model selection approach focusing on the entire region. The proposed method is extremely useful when the signals associated with relevant predictors are time-dependent, and detecting relevant covariate effects in the local region is more scientifically relevant than those of the entire region. Our simulation studies indicate that the proposed model selection incorporating local features outperforms the global feature model selection approaches. The proposed method is also illustrated through a longitudinal growth and health study from National Heart, Lung, and Blood Institute.

Nonproportional hazards and unobserved heterogeneity in clustered survival data: When can we tell the difference?

Multivariate survival data are frequently encountered in biomedical applications in the form of clustered failures (or recurrent events data). A popular way of analyzing such data is by using shared frailty models, which assume that the proportional hazards assumption holds conditional on an unobserved cluster‐specific random effect. Such models are often incorporated in more complicated joint models in survival analysis.If the random effect distribution has finite expectation, then the conditional proportional hazards assumption does not carry over to the marginal models. It has been shown that, for univariate data, this makes it impossible to distinguish between the presence of unobserved heterogeneity (eg, due to missing covariates) and marginal nonproportional hazards. We show that time‐dependent covariate effects may falsely appear as evidence in favor of a frailty model also in the case of clustered failures or recurrent events data, when the cluster size or number of recurrent events is small. When true unobserved heterogeneity is present, the presence of nonproportional hazards leads to overestimating the frailty effect. We show that this phenomenon is somewhat mitigated as the cluster size grows.We carry out a simulation study to assess the behavior of test statistics and estimators for frailty models in such contexts. The gamma, inverse Gaussian, and positive stable shared frailty models are contrasted using a novel software implementation for estimating semiparametric shared frailty models. Two main questions are addressed in the contexts of clustered failures and recurrent events: whether covariates with a time‐dependent effect may appear as indication of unobserved heterogeneity and whether the additional presence of unobserved heterogeneity can be detected in this case. Finally, the practical implications are illustrated in a real‐world data analysis example.

Instrumental variable estimation in semi-parametric additive hazards models.

SummaryInstrumental variable methods allow unbiased estimation in the presence of unmeasured confounders when an appropriate instrumental variable is available. Two‐stage least‐squares and residual inclusion methods have recently been adapted to additive hazard models for censored survival data. The semi‐parametric additive hazard model which can include time‐independent and time‐dependent covariate effects is particularly suited for the two‐stage residual inclusion method, since it allows direct estimation of time‐independent covariate effects without restricting the effect of the residual on the hazard. In this article, we prove asymptotic normality of two‐stage residual inclusion estimators of regression coefficients in a semi‐parametric additive hazard model with time‐independent and time‐dependent covariate effects. We consider the cases of continuous and binary exposure. Estimation of the conditional survival function given observed covariates is discussed and a resampling scheme is proposed to obtain simultaneous confidence bands. The new methods are compared to existing ones in a simulation study and are applied to a real data set. The proposed methods perform favorably especially in cases with exposure‐dependent censoring.

Weighted NPMLE for the Subdistribution of a Competing Risk

ABSTRACTDirect regression modeling of the subdistribution has become popular for analyzing data with multiple, competing event types. All general approaches so far are based on nonlikelihood-based procedures and target covariate effects on the subdistribution. We introduce a novel weighted likelihood function that allows for a direct extension of the Fine–Gray model to a broad class of semiparametric regression models. The model accommodates time-dependent covariate effects on the subdistribution hazard. To motivate the proposed likelihood method, we derive standard nonparametric estimators and discuss a new interpretation based on pseudo risk sets. We establish consistency and asymptotic normality of the estimators and propose a sandwich estimator of the variance. In comprehensive simulation studies, we demonstrate the solid performance of the weighted nonparametric maximum likelihood estimation in the presence of independent right censoring. We provide an application to a very large bone marrow transplant dataset, thereby illustrating its practical utility. Supplementary materials for this article are available online.

Structural identification and variable selection in high-dimensional varying-coefficient models

ABSTRACTVarying-coefficient models have been widely used to investigate the possible time-dependent effects of covariates when the response variable comes from normal distribution. Much progress has been made for inference and variable selection in the framework of such models. However, the identification of model structure, that is how to identify which covariates have time-varying effects and which have fixed effects, remains a challenging and unsolved problem especially when the dimension of covariates is much larger than the sample size. In this article, we consider the structural identification and variable selection problems in varying-coefficient models for high-dimensional data. Using a modified basis expansion approach and group variable selection methods, we propose a unified procedure to simultaneously identify the model structure, select important variables and estimate the coefficient curves. The unique feature of the proposed approach is that we do not have to specify the model structure in advance, therefore, it is more realistic and appropriate for real data analysis. Asymptotic properties of the proposed estimators have been derived under regular conditions. Furthermore, we evaluate the finite sample performance of the proposed methods with Monte Carlo simulation studies and a real data analysis.

Bayes linear kinematics in a dynamic survival model

Bayes linear kinematics and Bayes linear Bayes graphical models provide an extension of Bayes linear methods so that full conditional updates may be combined with Bayes linear belief adjustment. In this paper we investigate the application of this approach to survival analysis with time-dependent covariate effects, a more complicated problem than previous applications. We use a piecewise-constant hazard function with a prior in which covariate effects are correlated over time. The need for computationally intensive methods is avoided and the relatively simple structure facilitates interpretation. Our approach eliminates the problem of non-commutativity which was observed in earlier work by Gamerman. We apply the technique to data on survival times for leukemia patients.

Validation of the alternating conditional estimation algorithm for estimation of flexible extensions of Cox's proportional hazards model with nonlinear constraints on the parameters.

Standard optimization algorithms for maximizing likelihood may not be applicable to the estimation of those flexible multivariable models that are nonlinear in their parameters. For applications where the model's structure permits separating estimation of mutually exclusive subsets of parameters into distinct steps, we propose the alternating conditional estimation (ACE) algorithm. We validate the algorithm, in simulations, for estimation of two flexible extensions of Cox's proportional hazards model where the standard maximum partial likelihood estimation does not apply, with simultaneous modeling of (1) nonlinear and time-dependent effects of continuous covariates on the hazard, and (2) nonlinear interaction and main effects of the same variable. We also apply the algorithm in real-life analyses to estimate nonlinear and time-dependent effects of prognostic factors for mortality in colon cancer. Analyses of both simulated and real-life data illustrate good statistical properties of the ACE algorithm and its ability to yield new potentially useful insights about the data structure.

Power and sample size calculation for the additive hazard model

ABSTRACTSample size determination is essential during the planning phases of a study. When the study endpoint is the time to an event, Cox proportional hazard model is the traditional technique to analyze the effects of covariates on survival time. In contrast to the proportional hazard model, the additive hazard (AH) model specifies that the effect of covariates additively increases or decreases the hazard function. The popularity of this model is that it gives a more intuitive interpretation without the proportional hazard assumption. Because there is no literature estimating the required sample size based on the AH model, we provide a flexible formula for calculating the required sample size. The proposed formula incorporates time-independent and time-dependent effects of covariates without complicated mathematical calculations. The performance of the method was evaluated by extensive simulations. Finally, some pilot studies are shown to illustrate the applications.

Generalized Quasi-Likelihood Ratio Tests for Semiparametric Analysis of Covariance Models in Longitudinal Data

ABSTRACTWe model generalized longitudinal data from multiple treatment groups by a class of semiparametric analysis of covariance models, which take into account the parametric effects of time dependent covariates and the nonparametric time effects. In these models, the treatment effects are represented by nonparametric functions of time and we propose a generalized quasi-likelihood ratio test procedure to test if these functions are identical. Our estimation procedure is based on profile estimating equations combined with local linear smoothers. We find that the much celebrated Wilks phenomenon which is well established for independent data still holds for longitudinal data if a working independence correlation structure is assumed in the test statistic. However, this property does not hold in general, especially when the working variance function is misspecified. Our empirical study also shows that incorporating correlation into the test statistic does not necessarily improve the power of the test. The proposed methods are illustrated with simulation studies and a real application from opioid dependence treatments. Supplementary materials for this article are available online.

A multilevel excess hazard model to estimate net survival on hierarchical data allowing for non-linear and non-proportional effects of covariates.

The excess hazard regression model is an approach developed for the analysis of cancer registry data to estimate net survival, that is, the survival of cancer patients that would be observed if cancer was the only cause of death. Cancer registry data typically possess a hierarchical structure: individuals from the same geographical unit share common characteristics such as proximity to a large hospital that may influence access to and quality of health care, so that their survival times might be correlated. As a consequence, correct statistical inference regarding the estimation of net survival and the effect of covariates should take this hierarchical structure into account. It becomes particularly important as many studies in cancer epidemiology aim at studying the effect on the excess mortality hazard of variables, such as deprivation indexes, often available only at the ecological level rather than at the individual level. We developed here an approach to fit a flexible excess hazard model including a random effect to describe the unobserved heterogeneity existing between different clusters of individuals, and with the possibility to estimate non-linear and time-dependent effects of covariates. We demonstrated the overall good performance of the proposed approach in a simulation study that assessed the impact on parameter estimates of the number of clusters, their size and their level of unbalance. We then used this multilevel model to describe the effect of a deprivation index defined at the geographical level on the excess mortality hazard of patients diagnosed with cancer of the oral cavity. Copyright © 2016 John Wiley & Sons, Ltd.