Predictive comparison of joint longitudinal-survival modeling: a case study illustrating competing approaches

The joint modeling of longitudinal and survival data has received extraordinary attention in the statistics literature recently, with models and methods becoming increasingly more complex. Most of these approaches pair a proportional hazards survival with longitudinal trajectory modeling through parametric or nonparametric specifications. In this paper we closely examine one data set previously analyzed using a two parameter parametric model for Mediterranean fruit fly (medfly) egg-laying trajectories paired with accelerated failure time and proportional hazards survival models. We consider parametric and nonparametric versions of these two models, as well as a proportional odds rate model paired with a wide variety of longitudinal trajectory assumptions reflecting the types of analyses seen in the literature. In addition to developing novel nonparametric Bayesian methods for joint models, we emphasize the importance of model selection from among joint and non joint models. The default in the literature is to omit at the outset non joint models from consideration. For the medfly data, a predictive diagnostic criterion suggests that both the choice of survival model and longitudinal assumptions can grossly affect model adequacy and prediction. Specifically for these data, the simple joint model used in by Tseng et al. (Biometrika 92:587–603, 2005) and models with much more flexibility in their longitudinal components are predictively outperformed by simpler analyses. This case study underscores the need for data analysts to compare on the basis of predictive performance different joint models and to include non joint models in the pool of candidates under consideration.     

Identification of Longitudinal Biomarkers in Survival Analysis for Competing Risks Data

In recent years, joint analysis of longitudinal measurements and survival data has received much attention. However, previous work has primarily focused on a single failure type for the event time. In this article, we consider joint modeling of repeated measurements and competing risks failure time data to allow for more than one distinct failure type in the survival endpoint so we fit a cause-specific hazards sub-model to allow for competing risks, with a separate latent association between longitudinal measurements and each cause of failure. Besides, previous work does not focus on the hypothesis to test a separate latent association between longitudinal measurements and each cause of failure. In this article, we derive a score test to identify longitudinal biomarkers or surrogates for a time to event outcome in competing risks data. With a carefully chosen definition of complete data, the maximum likelihood estimation of the cause-specific hazard functions is performed via an EM algorithm. We extend this work and allow random effects to be present in both the longitudinal biomarker and underlying survival function. The random effects in the biomarker are introduced via an explicit term while the random effect in the underlying survival function is introduced by the inclusion of frailty into the model.

We use simulations to explore how the number of individuals, the number of time points per individual and the functional form of the random effects from the longitudinal biomarkers considering heterogeneous baseline hazards in individuals influence the power to detect the association of a longitudinal biomarker and the survival time.     

Bayesian hierarchical models for food frequency assessment

The aim of this study is to assess the biases of a Food Frequency Questionnaire (FFQ) by comparing total energy intake (TEI) with total energy expenditure (TEE) obtained from doubly labelled water(DLW) biomarker after adjusting measurement errors in DLW. We develop several Bayesian hierarchical measurement error models of DLW with different distributional assumptions on TEI to obtain precise bias estimates of TEI. Inference is carried out by using MCMC simulation techniques in a fully Bayesian framework, and model comparisons are done via the mean square predictive error. Our results showed that the joint model with random effects under the Gamma distribution is the best fit model in terms of the MSPE and residual diagnostics, in which bias in TEI is not significant based on the 95% credible interval. The Canadian Journal of Statistics 38: 506–516; 2010 © 2010 Statistical Society of Canada     

A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects

This article studies a general joint model for longitudinal measurements and competing risks survival data. The model consists of a linear mixed effects sub-model for the longitudinal outcome, a proportional cause-specific hazards frailty sub-model for the competing risks survival data, and a regression sub-model for the variance–covariance matrix of the multivariate latent random effects based on a modified Cholesky decomposition. The model provides a useful approach to adjust for non-ignorable missing data due to dropout for the longitudinal outcome, enables analysis of the survival outcome with informative censoring and intermittently measured time-dependent covariates, as well as joint analysis of the longitudinal and survival outcomes. Unlike previously studied joint models, our model allows for heterogeneous random covariance matrices. It also offers a framework to assess the homogeneous covariance assumption of existing joint models. A Bayesian MCMC procedure is developed for parameter estimation and inference. Its performances and frequentist properties are investigated using simulations. A real data example is used to illustrate the usefulness of the approach.     

Application of trajectories from growth curve in identification of longitudinal biomarker for the multivariate survival data

In clinical studies, the researchers measure the patients' response longitudinally. In recent studies, Mixed models are used to determine effects in the individual level. In the other hand, Henderson et al. [3,4] developed a joint likelihood function which combines likelihood functions of longitudinal biomarkers and survival times. They put random effects in the longitudinal component to determine if a longitudinal biomarker is associated with time to an event. In this paper, we deal with a longitudinal biomarker as a growth curve and extend Henderson's method to determine if a longitudinal biomarker is associated with time to an event for the multivariate survival data.     

Functional approach of flexibly modelling generalized longitudinal data and survival time

We propose a flexible functional approach for modelling generalized longitudinal data and survival time using principal components. In the proposed model the longitudinal observations can be continuous or categorical data, such as Gaussian, binomial or Poisson outcomes. We generalize the traditional joint models that treat categorical data as continuous data by using some transformations, such as CD4 counts. The proposed model is data-adaptive, which does not require pre-specified functional forms for longitudinal trajectories and automatically detects characteristic patterns. The longitudinal trajectories observed with measurement error or random error are represented by flexible basis functions through a possibly nonlinear link function, combining dimension reduction techniques resulting from functional principal component (FPC) analysis. The relationship between the longitudinal process and event history is assessed using a Cox regression model. Although the proposed model inherits the flexibility of non-parametric methods, the estimation procedure based on the EM algorithm is still parametric in computation, and thus simple and easy to implement. The computation is simplified by dimension reduction for random coefficients or FPC scores. An iterative selection procedure based on Akaike information criterion (AIC) is proposed to choose the tuning parameters, such as the knots of spline basis and the number of FPCs, so that appropriate degree of smoothness and fluctuation can be addressed. The effectiveness of the proposed approach is illustrated through a simulation study, followed by an application to longitudinal CD4 counts and survival data which were collected in a recent clinical trial to compare the efficiency and safety of two antiretroviral drugs.     

The proportional hazards model with covariate measurement error

The proportional hazards regression model is commonly used to evaluate the relationship between survival and covariates. Covariates are frequently measured with error. Substituting mismeasured values for the true covariates leads to biased estimation. Hu et al. (Biometrics 88 (1998) 447) have proposed to base estimation in the proportional hazards model with covariate measurement error on a joint likelihood for survival and the covariate variable. Nonparametric maximum likelihood estimation (NPMLE) was used and simulations were conducted to assess the asymptotic validity of this approach. In this paper, we derive a rigorous proof of asymptotic normality of the NPML estimators.     

The estimation of gross flows in the presence of measurement error using auxiliary variables

Classification error can lead to substantial biases in the estimation of gross flows from longitudinal data. We propose a method to adjust flow estimates for bias, based on fitting separate multinomial logistic models to the classification error probabilities and the true state transition probabilities using values of auxiliary variables. Our approach has the advantages that it does not require external information on misclassification rates, it permits the identification of factors that are related to misclassification and true transitions and it does not assume independence between classification errors at successive points in time. Constraining the prediction of the stocks to agree with the observed stocks protects against model misspecification. We apply the approach to data on women from the Panel Study of Income Dynamics with three categories of labour force status. The model fitted is shown to have interpretable coefficient estimates and to provide a good fit. Simulation results indicate good performance of the model in predicting the true flows and robustness against departures from the model postulated.     

Two-step and likelihood methods for joint models of longitudinal and survival data

We compare the commonly used two-step methods and joint likelihood method for joint models of longitudinal and survival data via extensive simulations. The longitudinal models include LME, GLMM, and NLME models, and the survival models include Cox models and AFT models. We find that the full likelihood method outperforms the two-step methods for various joint models, but it can be computationally challenging when the dimension of the random effects in the longitudinal model is not small. We thus propose an approximate joint likelihood method which is computationally efficient. We find that the proposed approximation method performs well in the joint model context, and it performs better for more “continuous” longitudinal data. Finally, a real AIDS data example shows that patients with higher initial viral load or lower initial CD4 are more likely to drop out earlier during an anti-HIV treatment.     

Comparing crossing hazard rate functions by joint modelling of survival and longitudinal data

Abstract

It is one of the important issues in survival analysis to compare two hazard rate functions to evaluate treatment effect. It is quite common that the two hazard rate functions cross each other at one or more unknown time points, representing temporal changes of the treatment effect. In certain applications, besides survival data, we also have related longitudinal data available regarding some time-dependent covariates. In such cases, a joint model that accommodates both types of data can allow us to infer the association between the survival and longitudinal data and to assess the treatment effect better. In this paper, we propose a modelling approach for comparing two crossing hazard rate functions by joint modelling survival and longitudinal data. Maximum likelihood estimation is used in estimating the parameters of the proposed joint model using the EM algorithm. Asymptotic properties of the maximum likelihood estimators are studied. To illustrate the virtues of the proposed method, we compare the performance of the proposed method with several existing methods in a simulation study. Our proposed method is also demonstrated using a real dataset obtained from an HIV clinical trial.     

A joint latent class changepoint model to improve the prediction of time to graft failure

Summary.  The reciprocal of serum creatinine concentration, RC, is often used as a biomarker to monitor renal function. It has been observed that RC trajectories remain relatively stable after transplantation until a certain moment, when an irreversible decrease in the RC levels occurs. This decreasing trend commonly precedes failure of a graft. Two subsets of individuals can be distinguished according to their RC trajectories: a subset of individuals having stable RC levels and a subset of individuals who present an irrevocable decrease in their RC levels. To describe such data, the paper proposes a joint latent class model for longitudinal and survival data with two latent classes. RC trajectories within latent class one are modelled by an intercept-only random-effects model and RC trajectories within latent class two are modelled by a segmented random changepoint model. A Bayesian approach is used to fit this joint model to data from patients who had their first kidney transplantation in the Leiden University Medical Center between 1983 and 2002. The resulting model describes the kidney transplantation data very well and provides better predictions of the time to failure than other joint and survival models.     

Jointly modeling skew longitudinal survival data with missingness and mismeasured covariates

Jointly modeling longitudinal and survival data has been an active research area. Most researches focus on improving the estimating efficiency but ignore many data features frequently encountered in practice. In the current study, we develop the joint models that concurrently accounting for longitudinal and survival data with multiple features. Specifically, the proposed model handles skewness, missingness and measurement errors in covariates which are typically observed in the collection of longitudinal survival data from many studies. We employ a Bayesian inferential method to make inference on the proposed model. We applied the proposed model to an real data study. A few alternative models under different conditions are compared. We conduct extensive simulations in order to evaluate how the method works.     

Jointly modeling longitudinal proportional data and survival times with an application to the quality of life data in a breast cancer trial

Motivated by the joint analysis of longitudinal quality of life data and recurrence free survival times from a cancer clinical trial, we present in this paper two approaches to jointly model the longitudinal proportional measurements, which are confined in a finite interval, and survival data. Both approaches assume a proportional hazards model for the survival times. For the longitudinal component, the first approach applies the classical linear mixed model to logit transformed responses, while the second approach directly models the responses using a simplex distribution. A semiparametric method based on a penalized joint likelihood generated by the Laplace approximation is derived to fit the joint model defined by the second approach. The proposed procedures are evaluated in a simulation study and applied to the analysis of breast cancer data motivated this research.     

Joint models with multiple longitudinal outcomes and a time-to-event outcome: a corrected two-stage approach

Joint models for longitudinal and survival data have gained a lot of attention in recent years, with the development of myriad extensions to the basic model, including those which allow for multivariate longitudinal data, competing risks and recurrent events. Several software packages are now also available for their implementation. Although mathematically straightforward, the inclusion of multiple longitudinal outcomes in the joint model remains computationally difficult due to the large number of random effects required, which hampers the practical application of this extension. We present a novel approach that enables the fitting of such models with more realistic computational times. The idea behind the approach is to split the estimation of the joint model in two steps: estimating a multivariate mixed model for the longitudinal outcomes and then using the output from this model to fit the survival submodel. So-called two-stage approaches have previously been proposed and shown to be biased. Our approach differs from the standard version, in that we additionally propose the application of a correction factor, adjusting the estimates obtained such that they more closely resemble those we would expect to find with the multivariate joint model. This correction is based on importance sampling ideas. Simulation studies show that this corrected two-stage approach works satisfactorily, eliminating the bias while maintaining substantial improvement in computational time, even in more difficult settings.

     

Joint modeling of censored longitudinal and event time data

Censoring of a longitudinal outcome often occurs when data are collected in a biomedical study and where the interest is in the survival and or longitudinal experiences of a study population. In the setting considered herein, we encountered upper and lower censored data as the result of restrictions imposed on measurements from a kinetic model producing “biologically implausible” kidney clearances. The goal of this paper is to outline the use of a joint model to determine the association between a censored longitudinal outcome and a time to event endpoint. This paper extends Guo and Carlin's [6] paper to accommodate censored longitudinal data, in a commercially available software platform, by linking a mixed effects Tobit model to a suitable parametric survival distribution. Our simulation results showed that our joint Tobit model outperforms a joint model made up of the more naïve or “fill-in” method for the longitudinal component. In this case, the upper and/or lower limits of censoring are replaced by the limit of detection. We illustrated the use of this approach with example data from the hemodialysis (HEMO) study [3] and examined the association between doubly censored kidney clearance values and survival.     

Joint analysis of nonlinear heterogeneous longitudinal data and binary outcome: an application to AIDS clinical studies

Finite mixture models are currently used to analyze heterogeneous longitudinal data. By releasing the homogeneity restriction of nonlinear mixed-effects (NLME) models, finite mixture models not only can estimate model parameters but also cluster individuals into one of the pre-specified classes with class membership probabilities. This clustering may have clinical significance, which might be associated with a clinically important binary outcome. This article develops a joint modeling of a finite mixture of NLME models for longitudinal data in the presence of covariate measurement errors and a logistic regression for a binary outcome, linked by individual latent class indicators, under a Bayesian framework. Simulation studies are conducted to assess the performance of the proposed joint model and a naive two-step model, in which finite mixture model and logistic regression are fitted separately, followed by an application to a real data set from an AIDS clinical trial, in which the viral dynamics and dichotomized time to the first decline of CD4/CD8 ratio are analyzed jointly.     

A Bayesian test of homogeneity of association parameter using transition modelling of longitudinal mixed responses

In this paper, a Bayesian framework using a joint transition model for analysing longitudinal mixed ordinal and continuous responses is considered. The joint model considers a multivariate mixed model for the responses in which a transitive cumulative logistic regression model and an autoregressive regression model are used to model ordinal and continuous responses, respectively. Also, to take into account the association between longitudinal ordinal and continuous responses, a dynamic association parameter is used. A test is conducted to see whether this parameter is time-invariant and another test is presented to see whether this parameter is equal to zero or significantly far from zero. Our approach is applied to longitudinal PIAT (Peabody Individual Achievement Test) data where the Bayesian estimates of parameters are obtained.     

Penalized spline joint models for longitudinal and time-to-event data

The joint models for longitudinal data and time-to-event data have recently received numerous attention in clinical and epidemiologic studies. Our interest is in modeling the relationship between event time outcomes and internal time-dependent covariates. In practice, the longitudinal responses often show non linear and fluctuated curves. Therefore, the main aim of this paper is to use penalized splines with a truncated polynomial basis to parameterize the non linear longitudinal process. Then, the linear mixed-effects model is applied to subject-specific curves and to control the smoothing. The association between the dropout process and longitudinal outcomes is modeled through a proportional hazard model. Two types of baseline risk functions are considered, namely a Gompertz distribution and a piecewise constant model. The resulting models are referred to as penalized spline joint models; an extension of the standard joint models. The expectation conditional maximization (ECM) algorithm is applied to estimate the parameters in the proposed models. To validate the proposed algorithm, extensive simulation studies were implemented followed by a case study. In summary, the penalized spline joint models provide a new approach for joint models that have improved the existing standard joint models.     

A joint marginalized multilevel model for longitudinal outcomes

The shared-parameter model and its so-called hierarchical or random-effects extension are widely used joint modeling approaches for a combination of longitudinal continuous, binary, count, missing, and survival outcomes that naturally occurs in many clinical and other studies. A random effect is introduced and shared or allowed to differ between two or more repeated measures or longitudinal outcomes, thereby acting as a vehicle to capture association between the outcomes in these joint models. It is generally known that parameter estimates in a linear mixed model (LMM) for continuous repeated measures or longitudinal outcomes allow for a marginal interpretation, even though a hierarchical formulation is employed. This is not the case for the generalized linear mixed model (GLMM), that is, for non-Gaussian outcomes. The aforementioned joint models formulated for continuous and binary or two longitudinal binomial outcomes, using the LMM and GLMM, will naturally have marginal interpretation for parameters associated with the continuous outcome but a subject-specific interpretation for the fixed effects parameters relating covariates to binary outcomes. To derive marginally meaningful parameters for the binary models in a joint model, we adopt the marginal multilevel model (MMM) due to Heagerty [13] and Heagerty and Zeger [14] and formulate a joint MMM for two longitudinal responses. This enables to (1) capture association between the two responses and (2) obtain parameter estimates that have a population-averaged interpretation for both outcomes. The model is applied to two sets of data. The results are compared with those obtained from the existing approaches such as generalized estimating equations, GLMM, and the model of Heagerty [13]. Estimates were found to be very close to those from single analysis per outcome but the joint model yields higher precision and allows for quantifying the association between outcomes. Parameters were estimated using maximum likelihood. The model is easy to fit using available tools such as the SAS NLMIXED procedure.     

The Method to Identify a Biomarker and to Evaluate Its Efficiency for Survival by Using the Joint Model of the Accelerate Failure Time and Longitudinal Data

In this article, we discuss how to identify longitudinal biomarkers in survival analysis under the accelerated failure time model and also discuss the effectiveness of biomarkers under the accelerated failure time model. Two methods proposed by Shcemper et al. are deployed to measure the efficacy of biomarkers. We use simulations to explore how the factors can influence the power of a score test to detect the association of a longitudinal biomarker and the survival time. These factors include the functional form of the random effects from the longitudinal biomarkers, in the different number of individuals, and time points per individual. The simulations are used to explore how the number of individuals, the number of time points per individual influence the effectiveness of the biomarker to predict survival at the given endpoint under the accelerated failure time model. We illustrate our methods using a prothrombin index as a predictor of survival in liver cirrhosis patients.