Semiparametric modeling of longitudinal measurements and time-to-event data--a two-stage regression calibration approach

Fast and flexible inference for joint models of multivariate longitudinal and survival data using integrated nested Laplace approximations

Modeling longitudinal and survival data jointly offers many advantages such as addressing measurement error and missing data in the longitudinal processes, understanding and quantifying the association between the longitudinal markers and the survival events, and predicting the risk of events based on the longitudinal markers. A joint model involves multiple submodels (one for each longitudinal/survival outcome) usually linked together through correlated or shared random effects. Their estimation is computationally expensive (particularly due to a multidimensional integration of the likelihood over the random effects distribution) so that inference methods become rapidly intractable, and restricts applications of joint models to a small number of longitudinal markers and/or random effects. We introduce a Bayesian approximation based on the integrated nested Laplace approximation algorithm implemented in the R package R-INLA to alleviate the computational burden and allow the estimation of multivariate joint models with fewer restrictions. Our simulation studies show that R-INLA substantially reduces the computation time and the variability of the parameter estimates compared with alternative estimation strategies. We further apply the methodology to analyze five longitudinal markers (3 continuous, 1 count, 1 binary, and 16 random effects) and competing risks of death and transplantation in a clinical trial on primary biliary cholangitis. R-INLA provides a fast and reliable inference technique for applying joint models to the complex multivariate data encountered in health research.

Modeling biomarker variability in joint analysis of longitudinal and time-to-event data

The role of visit-to-visit variability of a biomarker in predicting related disease has been recognized in medical science. Existing measures of biological variability are criticized for being entangled with random variability resulted from measurement error or being unreliable due to limited measurements per individual. In this article, we propose a new measure to quantify the biological variability of a biomarker by evaluating the fluctuation of each individual-specific trajectory behind longitudinal measurements. Given a mixed-effects model for longitudinal data with the mean function over time specified by cubic splines, our proposed variability measure can be mathematically expressed as a quadratic form of random effects. A Cox model is assumed for time-to-event data by incorporating the defined variability as well as the current level of the underlying longitudinal trajectory as covariates, which, together with the longitudinal model, constitutes the joint modeling framework in this article. Asymptotic properties of maximum likelihood estimators are established for the present joint model. Estimation is implemented via an Expectation-Maximization (EM) algorithm with fully exponential Laplace approximation used in E-step to reduce the computation burden due to the increase of the random effects dimension. Simulation studies are conducted to reveal the advantage of the proposed method over the two-stage method, as well as a simpler joint modeling approach which does not take into account biomarker variability. Finally, we apply our model to investigate the effect of systolic blood pressure variability on cardiovascular events in the Medical Research Council elderly trial, which is also the motivating example for this article.

Random survival forests with multivariate longitudinal endogenous covariates

Predicting the individual risk of clinical events using the complete patient history is a major challenge in personalized medicine. Analytical methods have to account for a possibly large number of time-dependent predictors, which are often characterized by irregular and error-prone measurements, and are truncated early by the event. In this work, we extended the competing-risk random survival forests to handle such endogenous longitudinal predictors when predicting event probabilities. The method, implemented in the R package DynForest, internally transforms the time-dependent predictors at each node of each tree into time-fixed features (using mixed models) that can then be used as splitting candidates. The final individual event probability is computed as the average of leaf-specific Aalen-Johansen estimators over the trees. Using simulations, we compared the performances of DynForest to accurately predict an event with (i) a joint modeling alternative when considering two longitudinal predictors only, and with (ii) a regression calibration method that ignores the informative truncation by the event when dealing with a large number of longitudinal predictors. Through an application in dementia research, we also illustrated how DynForest can be used to develop a dynamic prediction tool for dementia from multimodal repeated markers, and quantify the importance of each marker.

Characterization of direct and/or indirect genetic associations for multiple traits in longitudinal studies of disease progression

When quantitative longitudinal traits are risk factors for disease progression and subject to random biological variation, joint model analysis of time-to-event and longitudinal traits can effectively identify direct and/or indirect genetic association of single nucleotide polymorphisms (SNPs) with time-to-event. We present a joint model that integrates: (1) a multivariate linear mixed model describing trajectories of multiple longitudinal traits as a function of time, SNP effects, and subject-specific random effects and (2) a frailty Cox survival model that depends on SNPs, longitudinal trajectory effects, and subject-specific frailty accounting for dependence among multiple time-to-event traits. Motivated by complex genetic architecture of type 1 diabetes complications (T1DC) observed in the Diabetes Control and Complications Trial (DCCT), we implement a 2-stage approach to inference with bootstrap joint covariance estimation and develop a hypothesis testing procedure to classify direct and/or indirect SNP association with each time-to-event trait. By realistic simulation study, we show that joint modeling of 2 time-to-T1DC (retinopathy and nephropathy) and 2 longitudinal risk factors (HbA1c and systolic blood pressure) reduces estimation bias in genetic effects and improves classification accuracy of direct and/or indirect SNP associations, compared to methods that ignore within-subject risk factor variability and dependence among longitudinal and time-to-event traits. Through DCCT data analysis, we demonstrate feasibility for candidate SNP modeling and quantify effects of sample size and Winner's curse bias on classification for 2 SNPs identified as having indirect associations with time-to-T1DC traits. Joint analysis of multiple longitudinal and multiple time-to-event traits provides insight into complex traits architecture.

Absolute and relative risk estimation in the presence of outcome ascertainment gaps and competing risks

Incomplete coverage by cancer registries can lead to an underreporting of cancers and a resulting bias in risk estimates. When registries are defined by geographic region, gaps in observation can arise for individuals who reside outside of or migrate from the total registry catchment area. Moreover, the exact periods of non-observation for an individual may be unknown due to intermittent reporting of residential histories. The motivating example for this work is the U.S. Radiologic Technologist (USRT) study which ascertained cancer outcomes for a national cohort through 43 state/regional registries; similar gaps in outcome ascertainment can appear in other registry or electronic health record- based cohort studies. We propose a two-step procedure for estimating relative and absolute risk in these settings. First, using a mover stayer model fitted to individuals' known residential history, we obtain individual posterior probabilities of residing outside the registry catchment area each year. Second, we incorporate these probabilities in the survival data likelihood for competing risks to account for unobserved events. We assess the performance of the proposed method in extensive simulation studies. Compared to several simple alternative approaches, the proposed method reduces bias and improves efficiency. Finally, we apply the proposed method to a study of first primary lung cancers in the USRT cohort.

Longitudinal uric acid has nonlinear association with kidney failure and mortality in chronic kidney disease

We investigated the shape of the relationship between longitudinal uric acid (UA) and the hazard of kidney failure and death in chronic kidney disease (CKD) patients, and attempted to identify thresholds associated with increased hazards. We included CKD stage 3-5 patients from the CKD-REIN cohort with one serum UA measurement at cohort entry. We used cause-specific multivariate Cox models including a spline function of current values of UA (cUA), estimated from a separate linear mixed model. We followed 2781 patients (66% men, median age, 69 years) for a median of 3.2 years with a median of five longitudinal UA measures per patient. The hazard of kidney failure increased with increasing cUA, with a plateau between 6 and 10 mg/dl and a sharp increase above 11 mg/dl. The hazard of death had a U-shape relationship with cUA, with a hazard twice higher for 3 or 11 mg/dl, compared to 5 mg/dl. In CKD patients, our results indicate that UA above 10 mg/dl is a strong risk marker for kidney failure and death and that low UA levels below 5 mg/dl are associated with death before kidney failure.

Using joint models to study the association between CD4 count and the risk of death in TB/HIV data

Background

The association structure linking the longitudinal and survival sub-models is of fundamental importance in the joint modeling framework and the choice of this structure should be made based on the clinical background of the study. However, this information may not always be accessible and rationale for selecting this association structure has received relatively little attention in the literature. To this end, we aim to explore four alternative functional forms of the association structure between the CD4 count and the risk of death and provide rationale for selecting the optimal association structure for our data. We also aim to compare the results obtained from the joint model to those obtained from the time-varying Cox model.

Methods

We used data from the Centre for the AIDS Programme of Research in South Africa (CAPRISA) AIDS Treatment programme, the Starting Antiretroviral Therapy at Three Points in Tuberculosis (SAPiT) study, an open-label, three armed randomised, controlled trial between June 2005 and July 2010 (N=642). In our analysis, we combined the early and late integrated arms and compared results to the sequential arm. We utilized the Deviance Information Criterion (DIC) to select the final model with the best structure, with smaller values indicating better model adjustments to the data.

Results

Patient characteristics were similar across the study arms. Combined integrated therapy arms had a reduction of 55% in mortality (HR:0.45, 95% CI:0.28-0.72) compared to the sequential therapy arm. The joint model with a cumulative effects functional form was chosen as the best association structure. In particular, our joint model found that the area under the longitudinal profile of CD4 count was strongly associated with a 21% reduction in mortality (HR:0.79, 95% CI:0.72-0.86). Where as results from the time-varying Cox model showed a 19% reduction in mortality (HR:0.81, 95% CI:0.77-0.84).

Conclusions

In this paper we have shown that the “current value” association structure is not always the best structure that expresses the correct relationship between the outcomes in all settings, which is why it is crucial to explore alternative clinically meaningful association structures that links the longitudinal and survival processes.

Individual dynamic prediction of clinical endpoint from large dimensional longitudinal biomarker history: a landmark approach

Background

The individual data collected throughout patient follow-up constitute crucial information for assessing the risk of a clinical event, and eventually for adapting a therapeutic strategy. Joint models and landmark models have been proposed to compute individual dynamic predictions from repeated measures to one or two markers. However, they hardly extend to the case where the patient history includes much more repeated markers. Our objective was thus to propose a solution for the dynamic prediction of a health event that may exploit repeated measures of a possibly large number of markers.

Methods

We combined a landmark approach extended to endogenous markers history with machine learning methods adapted to survival data. Each marker trajectory is modeled using the information collected up to the landmark time, and summary variables that best capture the individual trajectories are derived. These summaries and additional covariates are then included in different prediction methods adapted to survival data, namely regularized regressions and random survival forests, to predict the event from the landmark time. We also show how predictive tools can be combined into a superlearner. The performances are evaluated by cross-validation using estimators of Brier Score and the area under the Receiver Operating Characteristic curve adapted to censored data.

Results

We demonstrate in a simulation study the benefits of machine learning survival methods over standard survival models, especially in the case of numerous and/or nonlinear relationships between the predictors and the event. We then applied the methodology in two prediction contexts: a clinical context with the prediction of death in primary biliary cholangitis, and a public health context with age-specific prediction of death in the general elderly population.

Conclusions

Our methodology, implemented in R, enables the prediction of an event using the entire longitudinal patient history, even when the number of repeated markers is large. Although introduced with mixed models for the repeated markers and methods for a single right censored time-to-event, the technique can be used with any other appropriate modeling technique for the markers and can be easily extended to competing risks setting.

Supplementary Information

The online version contains supplementary material available at (10.1186/s12874-022-01660-3).

Joint modelling of longitudinal measurements and survival times via a multivariate copula approach

Joint modelling of longitudinal and time-to-event data is usually described by a joint model which uses shared or correlated latent effects to capture associations between the two processes. Under this framework, the joint distribution of the two processes can be derived straightforwardly by assuming conditional independence given the random effects. Alternative approaches to induce interdependency into sub-models have also been considered in the literature and one such approach is using copulas to introduce non-linear correlation between the marginal distributions of the longitudinal and time-to-event processes. The multivariate Gaussian copula joint model has been proposed in the literature to fit joint data by applying a Monte Carlo expectation-maximisation algorithm. In this paper, we propose an exact likelihood estimation approach to replace the more computationally expensive Monte Carlo expectation-maximisation algorithm and we consider results based on using both the multivariate Gaussian and t copula functions. We also provide a straightforward way to compute dynamic predictions of survival probabilities, showing that our proposed model is comparable in prediction performance to the shared random effects joint model.

Development of clinical prediction models for renal and cardiovascular outcomes and mortality in patients with type 2 diabetes and chronic kidney disease using time-varying predictors

AIMS

To develop a set of prediction models for end-stage kidney disease (ESKD), cardiovascular outcomes, and mortality in patients with type 2 diabetes (T2D) and chronic kidney disease (CKD) using commonly measured clinical variables.

METHODS

We studied 1432 participants with T2D and CKD enrolled in the Chronic Renal Insufficiency Cohort, followed for a median period of 7 years. We used Cox proportional-hazards models to model the six outcomes (ESKD, stroke, myocardial infarction (MI), congestive heart failure (CHF), death before ESKD, and all-cause mortality). We internally evaluated these models using concordance and calibration measures.

RESULTS

The newly developed six prediction models included 15 predictors: age at diabetes diagnosis, sex, blood pressure, body mass index, hemoglobin A1c, high density lipoprotein cholesterol, urine protein-to-creatinine ratio, estimated glomerular filtration rate, smoking status, and history of stroke, MI, CHF, ESKD, and amputation. The resulting models demonstrated good/strong discrimination (cross-validation C-index range: 0.70 to 0.90) and calibration.

CONCLUSIONS

This study provided an internally validated and useful tool for predicting individual adverse outcomes and mortality in patients with T2D and CKD. These models may inform optimal use of targeted health interventions.

Penalized regression calibration: A method for the prediction of survival outcomes using complex longitudinal and high-dimensional data

Longitudinal and high‐dimensional measurements have become increasingly common in biomedical research. However, methods to predict survival outcomes using covariates that are both longitudinal and high‐dimensional are currently missing. In this article, we propose penalized regression calibration (PRC), a method that can be employed to predict survival in such situations. PRC comprises three modeling steps: First, the trajectories described by the longitudinal predictors are flexibly modeled through the specification of multivariate mixed effects models. Second, subject‐specific summaries of the longitudinal trajectories are derived from the fitted mixed models. Third, the time to event outcome is predicted using the subject‐specific summaries as covariates in a penalized Cox model. To ensure a proper internal validation of the fitted PRC models, we furthermore develop a cluster bootstrap optimism correction procedure that allows to correct for the optimistic bias of apparent measures of predictiveness. PRC and the CBOCP are implemented in the R package pencal, available from CRAN. After studying the behavior of PRC via simulations, we conclude by illustrating an application of PRC to data from an observational study that involved patients affected by Duchenne muscular dystrophy, where the goal is predict time to loss of ambulation using longitudinal blood biomarkers.

A two-stage approach for joint modeling of longitudinal measurements and competing risks data

Joint modeling of longitudinal measurements and time-to-event data is used in many practical studies of medical sciences. Most of the time, particularly in clinical studies and health inquiry, there are more than one event and they compete for failing an individual. In this situation, assessing the competing risk failure time is important. In most cases, implementation of joint modeling involves complex calculations. Therefore, we propose a two-stage method for joint modeling of longitudinal measurements and competing risks (JMLC) data based on the full likelihood approach via the conditional EM (CEM) algorithm. In the first stage, a linear mixed effect model is used to estimate the parameters of the longitudinal sub-model. In the second stage, we consider a cause-specific sub-model to construct competing risks data and describe an approximation for the joint log-likelihood that uses the estimated parameters of the first stage. We express the results of a simulation study and perform this method on the "standard and new anti-epileptic drugs" trial to check the effect of drug assaying on the treatment effects of lamotrigine and carbamazepine through treatment failure.

Two-Stage Joint Model for Multivariate Longitudinal and Multistate Processes, with Application to Renal Transplantation Data

In longitudinal studies, clinicians usually collect longitudinal biomarkers’ measurements over time until an event such as recovery, disease relapse, or death occurs. Joint modeling approaches are increasingly used to study the association between one longitudinal and one survival outcome. However, in practice, a patient may experience multiple disease progression events successively. So instead of modeling of a single event, progression of the disease as a multistate process should be modeled. On the other hand, in such studies, multivariate longitudinal outcomes may be collected and their association with the survival process is of interest. In the present study, we applied a joint model of various longitudinal biomarkers and transitions between different health statuses in patients who underwent renal transplantation. The full joint likelihood approaches are faced with the complexities in computation of the likelihood. So, here, we have proposed two-stage modeling of multivariate longitudinal outcomes and multistate conditions to avoid these complexities. The proposed model showed reliable results compared to the joint model in case of joint modeling of univariate longitudinal biomarker and the multistate process.

Methods to Account for Uncertainty in Latent Class Assignments When Using Latent Classes as Predictors in Regression Models, with Application to Acculturation Strategy Measures

Latent class models have become a popular means of summarizing survey questionnaires and other large sets of categorical variables. Often these classes are of primary interest to better understand complex patterns in data. Increasingly, these latent classes are reified into predictors of other outcomes of interests, treating the most likely class as the true class to which an individual belongs even though there is uncertainty in class membership. This uncertainty can be viewed as a form of measurement error in predictors, leading to bias in the estimates of the regression parameters associated with the latent classes. Despite this fact, there is very limited literature treating latent class predictors as measurement error models. Most applications ignore this issue and fit a two-stage model that treats the modal class prediction as truth. Here, we develop two approaches-one likelihood-based, the other Bayesian-to implement a joint model for latent class analysis and outcome prediction. We apply these methods to an analysis of how acculturation behaviors predict depression in South Asian immigrants to the United States. A simulation study gives guidance for when a two-stage model can be safely implemented and when the joint model may be required.

Revisiting methods for modeling longitudinal and survival data: Framingham Heart Study

Background

Statistical methods for modeling longitudinal and time-to-event data has received much attention in medical research and is becoming increasingly useful. In clinical studies, such as cancer and AIDS, longitudinal biomarkers are used to monitor disease progression and to predict survival. These longitudinal measures are often missing at failure times and may be prone to measurement errors. More importantly, time-dependent survival models that include the raw longitudinal measurements may lead to biased results. In previous studies these two types of data are frequently analyzed separately where a mixed effects model is used for the longitudinal data and a survival model is applied to the event outcome.

Methods

In this paper we compare joint maximum likelihood methods, a two-step approach and a time dependent covariate method that link longitudinal data to survival data with emphasis on using longitudinal measures to predict survival. We apply a Bayesian semi-parametric joint method and maximum likelihood joint method that maximizes the joint likelihood of the time-to-event and longitudinal measures. We also implement the Two-Step approach, which estimates random effects separately, and a classic Time Dependent Covariate Model. We use simulation studies to assess bias, accuracy, and coverage probabilities for the estimates of the link parameter that connects the longitudinal measures to survival times.

Results

Simulation results demonstrate that the Two-Step approach performed best at estimating the link parameter when variability in the longitudinal measure is low but is somewhat biased downwards when the variability is high. Bayesian semi-parametric and maximum likelihood joint methods yield higher link parameter estimates with low and high variability in the longitudinal measure. The Time Dependent Covariate method resulted in consistent underestimation of the link parameter. We illustrate these methods using data from the Framingham Heart Study in which lipid measurements and Myocardial Infarction data were collected over a period of 26 years.

Conclusions

Traditional methods for modeling longitudinal and survival data, such as the time dependent covariate method, that use the observed longitudinal data, tend to provide downwardly biased estimates. The two-step approach and joint models provide better estimates, although a comparison of these methods may depend on the underlying residual variance.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12874-021-01207-y.

Flexible multivariate joint model of longitudinal intensity and binary process for medical monitoring of frequently collected data

A frequent problem in longitudinal studies is that data may be assessed at subject-selected, irregularly spaced time-points, resulting in highly unbalanced outcome data, inducing bias, especially if availability of data is directly related to outcome. Our aim was to develop a multivariate joint model in a mixed outcomes framework to minimize irregular sampling bias. We demonstrate using blood glucose monitoring throughout pregnancy and risk of preterm birth among women with type 1 diabetes mellitus. Blood glucose measurements were unequally spaced and intensity of sampling varied between and within individuals over time. Multivariate linear mixed effects submodel for the longitudinal outcome (blood glucose), Poisson model for the intensity of glucose sampling, and logistic regression model for binary process (preterm birth) were specified. Association between models is captured through shared random effects. Markov chain Monte Carlo methods were used to fit the model. The multivariate joint model provided better prediction, compared with a joint model with a multivariate linear mixed effects submodel (ignoring intensity of glucose sampling) and a two-stage model. Most association parameters were significant in the preterm birth outcome model, signifying improvement of predictive ability of the binary endpoint by sharing random effects between glucose monitoring and preterm birth. A simulation study is presented to illustrate the effectiveness of the multivariate joint modeling approach.

A Two-Stage Approach for Bayesian Joint Models of Longitudinal and Survival Data: Correcting Bias with Informative Prior

Joint models of longitudinal and survival outcomes have gained much popularity in recent years, both in applications and in methodological development. This type of modelling is usually characterised by two submodels, one longitudinal (e.g., mixed-effects model) and one survival (e.g., Cox model), which are connected by some common term. Naturally, sharing information makes the inferential process highly time-consuming. In particular, the Bayesian framework requires even more time for Markov chains to reach stationarity. Hence, in order to reduce the modelling complexity while maintaining the accuracy of the estimates, we propose a two-stage strategy that first fits the longitudinal submodel and then plug the shared information into the survival submodel. Unlike a standard two-stage approach, we apply a correction by incorporating an individual and multiplicative fixed-effect with informative prior into the survival submodel. Based on simulation studies and sensitivity analyses, we empirically compare our proposal with joint specification and standard two-stage approaches. The results show that our methodology is very promising, since it reduces the estimation bias compared to the other two-stage method and requires less processing time than the joint specification approach.

A novel calibration framework for survival analysis when a binary covariate is measured at sparse time points

The goals in clinical and cohort studies often include evaluation of the association of a time-dependent binary treatment or exposure with a survival outcome. Recently, several impactful studies targeted the association between initiation of aspirin and survival following colorectal cancer (CRC) diagnosis. The value of this exposure is zero at baseline and may change its value to one at some time point. Estimating this association is complicated by having only intermittent measurements on aspirin-taking. Commonly used methods can lead to substantial bias. We present a class of calibration models for the distribution of the time of status change of the binary covariate. Estimates obtained from these models are then incorporated into the proportional hazard partial likelihood in a natural way. We develop non-parametric, semiparametric, and parametric calibration models, and derive asymptotic theory for the methods that we implement in the aspirin and CRC study. We further develop a risk-set calibration approach that is more useful in settings in which the association between the binary covariate and survival is strong.

Assessing Importance of Biomarkers: a Bayesian Joint Modeling Approach of Longitudinal and Survival Data with Semicompeting Risks

Longitudinal biomarkers such as patient-reported outcomes (PROs) and quality of life (QOL) are routinely collected in cancer clinical trials or other studies. Joint modeling of PRO/QOL and survival data can provide a comparative assessment of patient-reported changes in specific symptoms or global measures that correspond to changes in survival. Motivated by a head and neck cancer clinical trial, we develop a class of trajectory-based models for longitudinal and survival data with disease progression. Specifically, we propose a class of mixed effects regression models for longitudinal measures, a cure rate model for the disease progression time (T _P ), and a Cox proportional hazards model with time-varying covariates for the overall survival time (T _D ) to account for T _P and treatment switching. Under the semi-competing risks framework, the disease progression is the nonterminal event, the occurrence of which is subject to a terminal event of death. The properties of the proposed models are examined in detail. Within the Bayesian paradigm, we derive the decompositions of the deviance information criterion (DIC) and the logarithm of the pseudo marginal likelihood (LPML) to assess the fit of the longitudinal component of the model and the fit of each survival component, separately. We further develop ΔDIC as well as ΔLPML to determine the importance and contribution of the longitudinal data to the model fit of the T _P and T _D data.

Use of a pooled cohort to impute cardiovascular disease risk factors across the adult life course

BACKGROUND

In designing prevention strategies, it may be useful to understand how early and midlife cardiovascular disease risk factor (CVDRF) exposures affect outcomes that primarily occur in mid to late life. Few single US cohorts have followed participants from early adulthood to late life.

METHODS

We pooled four prospective cohorts that represent segments of the adult life course, and studied 15 001 White and Black adults aged 18 to 95 years at enrollment. We imputed early and midlife exposure to body mass index (BMI), glucose, lipids and blood pressure (BP). CVDRF trajectories were estimated using linear mixed models. Using the best linear unbiased predictions, we obtained person-specific estimates of CVDRF trajectories beginning at age 20 until each participant's end of follow-up. We then calculated for each CVDRF, summary measures of early and midlife exposure as time-weighted averages (TWAs).

RESULTS

In the pooled cohort, 33.7% were Black and 54.8% were female. CVDRF summary measures worsened in midlife compared with early life and varied by sex and race. In particular, systolic and diastolic BP were consistently higher over the adult life course among men, and BMI was higher among Blacks, particularly Black women. Simulation studies suggested acceptable imputation accuracy, especially for the younger cohorts. Correlations of true and imputed CVDRF summary measures ranged from 0.53 to 0.99, and agreement ranged from 67% to 99%.

CONCLUSIONS

These results suggest that imputed CVDRFs may be accurate enough to be useful in assessing the effects of early and midlife exposures on later life outcomes.

A Birnbaum-Saunders Model for Joint Survival and Longitudinal Analysis of Congestive Heart Failure Data

We consider a parametric joint modelling of longitudinal measurements and survival times, motivated by a study conducted at the Heart Institute (Incor), São Paulo, Brazil, with the objective of evaluating the impact of B-type Natriuretic Peptide (BNP) collected at different instants on the survival of patients with Congestive Heart Failure (CHF). We employ a linear mixed model for the longitudinal response and a Birnbaum-Saunders model for the survival times, allowing the inclusion of subjects without longitudinal observations. We derive maximum likelihood estimators of the joint model parameters and conduct a simulation study to compare the true survival probabilities with dynamic predictions obtained from the fit of the proposed joint model and to evaluate the performance of the method for estimating the model parameters.The proposed joint model is applied to the cohort of 1609 patients with CHF, of which 1080 have no BNP measurements. The parameter estimates and their standard errors obtained via: i) the traditional approach, where only individuals with at least one measurement of the longitudinal response are included and ii) the proposed approach, which includes survival information from all individuals, are compared with those obtained via marginal (longitudinal and survival) models.

Assessing the Relationship between Gestational Glycemic Control and Risk of Preterm Birth in Women with Type 1 Diabetes: A Joint Modeling Approach

BACKGROUND

Characterizing maternal glucose sampling over the course of the entire pregnancy is an important step toward improvement in prediction of adverse birth outcome, such as preterm birth, for women with type 1 diabetes mellitus (T1DM).

OBJECTIVES

To characterize the relationship between the gestational glycemic profile and risk of preterm birth using a joint modeling approach.

METHODS

A joint model was developed to simultaneously characterize the relationship between a longitudinal outcome (daily blood glucose sampling) and an event process (preterm birth). A linear mixed effects model using natural cubic splines was fitted to predict the longitudinal submodel. Covariates included mother's age at last menstrual period, age at diabetes onset, body mass index, hypertension, retinopathy, and nephropathy. Various association structures (value, value plus slope, and area under the curve) were examined before selecting the final joint model. We compared the joint modeling approach to the time-dependent Cox model (TDCM).

RESULTS

A total of 16,480 glucose readings over gestation (range: 50-260 days) with 32 women (28%) having preterm birth was included in the study. Mother's age at last menstrual period and age at diabetes onset were statistically significant (beta = 1.29, 95% CI 1.10, 1.72; beta = 0.84, 95% CI 0.62, 0.98) for the longitudinal submodel, reflecting that older women tended to have higher mean blood glucose and those with later diabetes onset tended to have a lower mean blood glucose level. The presence of nephropathy was statistically significant in the event submodel (beta = 2.29, 95% CI 1.05, 4.48). Cumulative association parameterization provided the best joint model fit. The joint model provided better fit compared to the time-dependent Cox model (DIC (JM) = 19,895; DIC (TDCM) = 19,932).

CONCLUSION

The joint model approach was able to simultaneously characterize the glycemic profile and assess the risk of preterm birth and provided additional insights and a better model fit compared to the time-dependent Cox model.

Imputing missing time-dependent covariate values for the discrete time Cox model

We describe a procedure for imputing missing values of time-dependent covariates in a discrete time Cox model using the chained equations method. The procedure multiply imputes the missing values for each time-period in a time-sequential manner, using covariates from the current and previous time-periods as well as the survival outcome. The form of the outcome variable used in the imputation model depends on the functional form of the time-dependent covariate(s) and differs from the case of Cox regression with only baseline covariates. This time-sequential approach provides an approximation to a fully conditional approach. We illustrate the procedure with data on diabetics, evaluating the association of their glucose control with the risk of selected cancers. Using simulations we show that the suggested estimator performed well (in terms of bias and coverage) for completely missing at random, missing at random and moderate non-missing-at-random patterns. However, for very strong non-missing-at-random patterns, the estimator was seriously biased and the coverage was too low. The procedure can be implemented using multiple imputation with the Fully conditional Specification (FCS) method (MI procedure in SAS with FCS statement or similar packages in other software, e.g. MICE in R). For use with event times on a continuous scale, the events would need to be grouped into time-intervals.

Survival analysis with time-dependent covariates subject to missing data or measurement error: Multiple Imputation for Joint Modeling (MIJM)

Modern epidemiological studies collect data on time-varying individual-specific characteristics, such as body mass index and blood pressure. Incorporation of such time-dependent covariates in time-to-event models is of great interest, but raises some challenges. Of specific concern are measurement error, and the non-synchronous updating of covariates across individuals, due for example to missing data. It is well known that in the presence of either of these issues the last observation carried forward (LOCF) approach traditionally used leads to bias. Joint models of longitudinal and time-to-event outcomes, developed recently, address these complexities by specifying a model for the joint distribution of all processes and are commonly fitted by maximum likelihood or Bayesian approaches. However, the adequate specification of the full joint distribution can be a challenging modeling task, especially with multiple longitudinal markers. In fact, most available software packages are unable to handle more than one marker and offer a restricted choice of survival models. We propose a two-stage approach, Multiple Imputation for Joint Modeling (MIJM), to incorporate multiple time-dependent continuous covariates in the semi-parametric Cox and additive hazard models. Assuming a primary focus on the time-to-event model, the MIJM approach handles the joint distribution of the markers using multiple imputation by chained equations, a computationally convenient procedure that is widely available in mainstream statistical software. We developed an R package "survtd" that allows MIJM and other approaches in this manuscript to be applied easily, with just one call to its main function. A simulation study showed that MIJM performs well across a wide range of scenarios in terms of bias and coverage probability, particularly compared with LOCF, simpler two-stage approaches, and a Bayesian joint model. The Framingham Heart Study is used to illustrate the approach.

Accounting for random observation time in risk prediction with longitudinal markers: An imputation approach

Longitudinally measured biomarkers are useful to predict the risk of clinical endpoints, since subject-specific marker trajectory contains additional information on pathology and critical windows. The work is motivated by the Scandinavian Fetal Growth Study, aiming at predicting pregnancy outcomes with repeated ultrasound measurements during pregnancy. While the observation time of markers often varies across individuals, it is not well understood how the variations affect risk prediction. Existing methods of longitudinal risk prediction, such as shared random effects model and pattern mixture model, construct a prediction implicitly as a function of the biomarkers and their observation time. Methods that ignore the longitudinal structure, such as sufficient dimension reduction and logistic regression, have better interpretability regarding how a biomarker measured at specific time window contributes to the disease risk, but often have reduced accuracy because of ignoring the observation time information. We propose a novel imputation approach to handle the random observation time, while preserving the direct interpretation. Through extensive simulation studies and analyses of the Scandinavian Fetal Growth Study data, we systematically compared the discrimination and calibration performance of different risk prediction methods, and found that the imputation method has comparable performance to longitudinal methods with an advantage of better interpretability.

Bayesian Semiparametric Joint Regression Analysis of Recurrent Adverse Events and Survival in Esophageal Cancer Patients

We propose a Bayesian semiparametric joint regression model for a recurrent event process and survival time. Assuming independent latent subject frailties, we define marginal models for the recurrent event process intensity and survival distribution as functions of the subject's frailty and baseline covariates. A robust Bayesian model, called Joint-DP, is obtained by assuming a Dirichlet process for the frailty distribution. We present a simulation study that compares posterior estimates under the Joint-DP model to a Bayesian joint model with lognormal frailties, a frequentist joint model, and marginal models for either the recurrent event process or survival time. The simulations show that the Joint-DP model does a good job of correcting for treatment assignment bias, and has favorable estimation reliability and accuracy compared with the alternative models. The Joint-DP model is applied to analyze an observational dataset from esophageal cancer patients treated with chemo-radiation, including the times of recurrent effusions of fluid to the heart or lungs, survival time, prognostic covariates, and radiation therapy modality.

Impact of Steroid Therapy on Early Growth in Infants with Biliary Atresia: The Multicenter Steroids in Biliary Atresia Randomized Trial

OBJECTIVE

To investigate the impact of corticosteroid therapy on the growth of participants in the Steroids in Biliary Atresia Randomized Trial (START) conducted through the Childhood Liver Disease Research Network. The primary analysis in START indicated that steroids did not have a beneficial effect on drainage in a cohort of infants with biliary atresia. We hypothesized that steroids would have a detrimental effect on growth in these infants.

STUDY DESIGN

A total of 140 infants were enrolled in START, with 70 randomized to each treatment arm: steroid and placebo. Length, weight, and head circumference were obtained at baseline and follow-up visits to 24 months of age.

RESULTS

Patients treated with steroids had significantly lower length and head circumference z scores during the first 3 months post-hepatoportoenterostomy (HPE), and significantly lower weight until 12 months. Growth trajectories in the steroid and placebo arms differed significantly for length (P < .0001), weight (P = .009), and head circumference (P < .0001) with the largest impact noted for those with successful HPE. Growth trajectory for head circumference was significantly lower in patients treated with steroids irrespective of HPE status, but recovered during the second 6 months of life.

CONCLUSIONS

Steroid therapy following HPE in patients with biliary atresia is associated with impaired length, weight, and head circumference growth trajectories for at least 6 months post-HPE, especially impacting infants with successful bile drainage.

TRIAL REGISTRATION

ClinicalTrials.gov: NCT00294684.

An approximate joint model for multiple paired longitudinal outcomes and time-to-event data

Joint modeling of multivariate paired longitudinal data and time-to-event data presents computational challenges that supersede full likelihood estimation due to the large dimensional random effects vector needed to capture correlation due to clustering with respect to pairs, subjects, and outcomes. We propose an alternative, computationally simpler approach to estimation of complex shared parameter models where missing data is imputed based on the Posterior Predictive Distribution from a Conditional Linear Model (CLM) approximation. Existing methods for complete data are then implemented to obtain estimates of the event time model parameters. Our method is applied to examine the effects of discordant growth in anthropometric measures of longitudinal fetal growth in twin fetuses and the timing of birth. Simulation results are presented to show that our method performs relatively well with moderate measurement errors under certain CLM approximations.

Heterogeneous growth bent-cable models for time-to-event and longitudinal data: application to AIDS studies

The major limitations of growth curve mixture models for HIV/AIDS data are the usual assumptions of normality and monophasic curves within latent classes. This article addresses these limitations by using non-normal skewed distributions and multiphasic patterns for outcomes of prospective studies. For such outcomes, new skew-t (ST) distributions are proposed for modeling heterogeneous growth trajectories, which exhibit not abrupt but gradual multiphasic changes from a declining trend to an increasing trend over time. We assess these clinically important features of longitudinal HIV/AIDS data using the bent-cable framework within a context of a joint modeling of time-to-event process and response process. A real dataset from an AIDS clinical study is used to illustrate the proposed methods.

Two-stage model for multivariate longitudinal and survival data with application to nephrology research

In many follow-up studies different types of outcomes are collected including longitudinal measurements and time-to-event outcomes. Commonly, it is of interest to study the association between them. Joint modeling approaches of a single longitudinal outcome and survival process have recently gained increasing attention from both frequentist and Bayesian perspective. However, in many studies several longitudinal biomarkers are of interest and instead of selecting one single biomarker, the relationships between all these outcomes and their association with survival needs to be investigated. Our motivating study comes from Peritoneal Dialysis Programme in Nephrology research from Nephrology Unit, CHP (Hospital de Santo António), Porto, Portugal in which the interest relies on the possible association between various biomarkers (calcium, phosphate, parathormone, and creatinine) and the patients' survival. To this aim, we propose a two-stage model-based approach for multivariate longitudinal and survival data that allowed us to study such complex association structure. The multivariate model suggested in this paper provided new insights in the area of nephrology research showing valid results in comparison with those models studying each longitudinal biomarker with survival separately.

Dynamic thresholds and a summary ROC curve: Assessing prognostic accuracy of longitudinal markers

Cancer patients, chronic kidney disease patients, and subjects infected with HIV are routinely monitored over time using biomarkers that represent key health status indicators. Furthermore, biomarkers are frequently used to guide initiation of new treatments or to inform changes in intervention strategies. Since key medical decisions can be made on the basis of a longitudinal biomarker, it is important to evaluate the potential accuracy associated with longitudinal monitoring. To characterize the overall accuracy of a time-dependent marker, we introduce a summary ROC curve that displays the overall sensitivity associated with a time-dependent threshold that controls time-varying specificity. The proposed statistical methods are similar to concepts considered in disease screening, yet our methods are novel in choosing a potentially time-dependent threshold to define a positive test, and our methods allow time-specific control of the false-positive rate. The proposed summary ROC curve is a natural averaging of time-dependent incident/dynamic ROC curves and therefore provides a single summary of net error rates that can be achieved in the longitudinal setting.

A Bayesian two-stage regression approach of analysing longitudinal outcomes with endogeneity and incompleteness

Abstract:

Two-stage regression methods are typically used for handling endogeneity in the simultaneous equations models in economics and other social sciences. However, the problem is challenging in the presence of incomplete response and/or incomplete endogenous covariate(s). We propose a Bayesian approach for the joint modelling of incomplete longitudinal continuous response and an incomplete count endogenous covariate, where the incompleteness is caused by the censorship through a selection mechanism. We define latent continuous variables which are left-censored at zero and develop a Gibbs sampling algorithm for the simultaneous estimation of the model parameters. We consider partially varying coefficients regression models containing covariates with fixed and time-varying effects on the response. Our work is motivated by a sample dataset from the Health and Retirement Study (HRS) for modelling the out-of-pocket medical cost, where the number of hospital admissions is considered as an endogenous covariate. Our analysis addresses some of the previously unanswered questions on the physical and financial health of the older population based on HRS data. Simulation studies are performed for assessing the usefulness of the proposed method compared to its competitors.

Joint modeling of survival time and longitudinal outcomes with flexible random effects

Joint models with shared Gaussian random effects have been conventionally used in analysis of longitudinal outcome and survival endpoint in biomedical or public health research. However, misspecifying the normality assumption of random effects can lead to serious bias in parameter estimation and future prediction. In this paper, we study joint models of general longitudinal outcomes and survival endpoint but allow the underlying distribution of shared random effect to be completely unknown. For inference, we propose to use a mixture of Gaussian distributions as an approximation to this unknown distribution and adopt an Expectation-Maximization (EM) algorithm for computation. Either AIC and BIC criteria are adopted for selecting the number of mixtures. We demonstrate the proposed method via a number of simulation studies. We illustrate our approach with the data from the Carolina Head and Neck Cancer Study (CHANCE).

The use of repeated blood pressure measures for cardiovascular risk prediction: a comparison of statistical models in the ARIC study

Many prediction models have been developed for the risk assessment and the prevention of cardiovascular disease in primary care. Recent efforts have focused on improving the accuracy of these prediction models by adding novel biomarkers to a common set of baseline risk predictors. Few have considered incorporating repeated measures of the common risk predictors. Through application to the Atherosclerosis Risk in Communities study and simulations, we compare models that use simple summary measures of the repeat information on systolic blood pressure, such as (i) baseline only; (ii) last observation carried forward; and (iii) cumulative mean, against more complex methods that model the repeat information using (iv) ordinary regression calibration; (v) risk-set regression calibration; and (vi) joint longitudinal and survival models. In comparison with the baseline-only model, we observed modest improvements in discrimination and calibration using the cumulative mean of systolic blood pressure, but little further improvement from any of the complex methods. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

Robust Dynamic Risk Prediction with Longitudinal Studies

Providing accurate and dynamic age-specific risk prediction is a crucial step in precision medicine. In this manuscript, we introduce an approach for estimating the τ-year age-specific absolute risk directly via a flexible varying coefficient model. The approach facilitates the utilization of predictors varying over an individual's lifetime. By using a nonparametric inverse probability weighted kernel estimating equation, the age-specific effects of risk factors are estimated without requiring the specification of the functional form. The approach allows borrowing information across individuals of similar ages, and therefore provides a practical solution for situations where the longitudinal information is only measured sparsely. We evaluate the performance of the proposed estimation and inference procedures with numerical studies, and make comparisons with existing methods in the literature. We illustrate the performance of our proposed approach by developing a dynamic prediction model using data from the Framingham Study.

Joint two-part Tobit models for longitudinal and time-to-event data

In this article, we show how Tobit models can address problems of identifying characteristics of subjects having left-censored outcomes in the context of developing a method for jointly analyzing time-to-event and longitudinal data. There are some methods for handling these types of data separately, but they may not be appropriate when time to event is dependent on the longitudinal outcome, and a substantial portion of values are reported to be below the limits of detection. An alternative approach is to develop a joint model for the time-to-event outcome and a two-part longitudinal outcome, linking them through random effects. This proposed approach is implemented to assess the association between the risk of decline of CD4/CD8 ratio and rates of change in viral load, along with discriminating between patients who are potentially progressors to AIDS from patients who do not. We develop a fully Bayesian approach for fitting joint two-part Tobit models and illustrate the proposed methods on simulated and real data from an AIDS clinical study.

Flexible Bayesian additive joint models with an application to type 1 diabetes research

The joint modeling of longitudinal and time-to-event data is an important tool of growing popularity to gain insights into the association between a biomarker and an event process. We develop a general framework of flexible additive joint models that allows the specification of a variety of effects, such as smooth nonlinear, time-varying and random effects, in the longitudinal and survival parts of the models. Our extensions are motivated by the investigation of the relationship between fluctuating disease-specific markers, in this case autoantibodies, and the progression to the autoimmune disease type 1 diabetes. Using Bayesian P-splines, we are in particular able to capture highly nonlinear subject-specific marker trajectories as well as a time-varying association between the marker and event process allowing new insights into disease progression. The model is estimated within a Bayesian framework and implemented in the R-package bamlss.

Quantifying the Race Stratified Impact of Socioeconomics on Graft Outcomes in Kidney Transplant Recipients

BACKGROUND

Socioeconomic status (SES) is a significant determinant of health outcomes and may be an important component of the causal chain surrounding racial disparities in kidney transplantation. The social adaptability index (SAI) is a validated and quantifiable measure of SES, with a lack of studies analyzing this measure longitudinally or between races.

METHODS

Longitudinal cohort study in adult kidney transplantation transplanted at a single-center between 2005 and 2012. The SAI score includes 5 domains (employment, education, marital status, substance abuse and income), each with a minimum of 0 and maximum of 3 for an aggregate of 0 to 15 (higher score → better SES).

RESULTS

One thousand one hundred seventy-one patients were included; 624 (53%) were African American (AA) and 547 were non-AA. African Americans had significantly lower mean baseline SAI scores (AAs 6.5 vs non-AAs 7.8; P < 0.001). Cox regression analysis demonstrated that there was no association between baseline SAI and acute rejection in non-AAs (hazard ratio [HR], 0.92; 95% confidence interval [95% CI], 0.81-1.05), whereas it was a significant predictor of acute rejection in AAs (HR, 0.89; 95% CI, 0.80-0.99). Similarly, a 2-stage approach to joint modelling of time to graft loss and longitudinal SAI did not predict graft loss in non-AAs (HR, 1.01; 95% CI, 0.28-3.62), whereas it was a significant predictor of graft loss in AAs (HR, 0.23; 95% CI, 0.06-0.93).

CONCLUSIONS

After controlling for confounders, SAI scores were associated with a lower risk of acute rejection and graft loss in AA kidney transplant recipients, whereas neither baseline nor follow-up SAI predicted outcomes in non-AA kidney transplant recipients.

Overall Graft Loss Versus Death-Censored Graft Loss: Unmasking the Magnitude of Racial Disparities in Outcomes Among US Kidney Transplant Recipients

BACKGROUND

Black kidney transplant recipients experience disproportionately high rates of graft loss. This disparity has persisted for 40 years, and improvements may be impeded based on the current public reporting of overall graft loss by US regulatory organizations for transplantation.

METHODS

Longitudinal cohort study of kidney transplant recipients using a data set created by linking Veterans Affairs and US Renal Data System information, including 4918 veterans transplanted between January 2001 and December 2007, with follow-up through December 2010. Multivariable analysis was conducted using 2-stage joint modeling of random and fixed effects of longitudinal data (linear mixed model) with time to event outcomes (Cox regression).

RESULTS

Three thousand three hundred six non-Hispanic whites (67%) were compared with 1612 non-Hispanic black (33%) recipients with 6.0 ± 2.2 years of follow-up. In the unadjusted analysis, black recipients were significantly more likely to have overall graft loss (hazard ratio [HR], 1.19; 95% confidence interval [95% CI], 1.07-1.33), death-censored graft loss (HR, 1.67; 95% CI, 1.45-1.92), and lower mortality (HR, 0.83; 95% CI, 0.72-0.96). In fully adjusted models, only death-censored graft loss remained significant (HR, 1.38; 95% CI, 1.12-1.71; overall graft loss [HR, 1.08; 95% CI, 0.91-1.28]; mortality [HR, 0.84; 95% CI, 0.67-1.06]). A composite definition of graft loss reduced the magnitude of disparities in blacks by 22%.

CONCLUSIONS

Non-Hispanic black kidney transplant recipients experience a substantial disparity in graft loss, but not mortality. This study of US data provides evidence to suggest that researchers should focus on using death-censored graft loss as the primary outcome of interest to facilitate a better understanding of racial disparities in kidney transplantation.

Performance of joint modelling of time-to-event data with time-dependent predictors: an assessment based on transition to psychosis data

Joint modelling has emerged to be a potential tool to analyse data with a time-to-event outcome and longitudinal measurements collected over a series of time points. Joint modelling involves the simultaneous modelling of the two components, namely the time-to-event component and the longitudinal component. The main challenges of joint modelling are the mathematical and computational complexity. Recent advances in joint modelling have seen the emergence of several software packages which have implemented some of the computational requirements to run joint models. These packages have opened the door for more routine use of joint modelling. Through simulations and real data based on transition to psychosis research, we compared joint model analysis of time-to-event outcome with the conventional Cox regression analysis. We also compared a number of packages for fitting joint models. Our results suggest that joint modelling do have advantages over conventional analysis despite its potential complexity. Our results also suggest that the results of analyses may depend on how the methodology is implemented.

A longitudinal, observational study with many repeated measures demonstrated improved precision of individual survival curves using Bayesian joint modeling of disability and survival

BACKGROUND/STUDY CONTEXT: It has not been previously demonstrated whether Bayesian joint modeling (BJM) of disability and survival can, under certain conditions, improve precision of individual survival curves.

METHODS

A longitudinal, observational study wherein 754 initially nondisabled community-dwelling adults in greater New Haven, Connecticut, were observed on a monthly basis for over 10 years.

RESULTS

In this study, BJM exploited many monthly observations to demonstrate, relative to a separate survival model with adjustment, improved precision of individual survival curves, permitting detection of significant differences between survival curves of two similar individuals. The gain in precision was lost when using only those observations from intervals of 6, 9, or 12 months.

CONCLUSION

When there are many repeated measures, BJM of longitudinal functional disability and interval-censored survival can potentially increase the precision of individual survival curves relative to those from a separate survival model. This may facilitate the identification of significant differences between individual survival curves, a useful result usually precluded by the large variability inherent to individual-level estimates from stand-alone survival models.

Joint modeling of survival and longitudinal non-survival data: current methods and issues. Report of the DIA Bayesian joint modeling working group

Explicitly modeling underlying relationships between a survival endpoint and processes that generate longitudinal measured or reported outcomes potentially could improve the efficiency of clinical trials and provide greater insight into the various dimensions of the clinical effect of interventions included in the trials. Various strategies have been proposed for using longitudinal findings to elucidate intervention effects on clinical outcomes such as survival. The application of specifically Bayesian approaches for constructing models that address longitudinal and survival outcomes explicitly has been recently addressed in the literature. We review currently available methods for carrying out joint analyses, including issues of implementation and interpretation, identify software tools that can be used to carry out the necessary calculations, and review applications of the methodology.