Bayesian Model Assessment in Joint Modeling of Longitudinal and Survival Data with Applications to Cancer Clinical Trials

New C-indices for assessing importance of longitudinal biomarkers in fitting competing risks survival data in the presence of partially masked causes

Competing risks survival data in the presence of partially masked causes are frequently encountered in medical research or clinical trials. When longitudinal biomarkers are also available, it is of great clinical importance to examine associations between the longitudinal biomarkers and the cause-specific survival outcomes. In this article, we propose a cause-specific C-index for joint models of longitudinal and competing risks survival data accounting for masked causes. We also develop a posterior predictive algorithm for computing the out-of-sample cause-specific C-index using Markov chain Monte Carlo samples from the joint posterior of the in-sample longitudinal and competing risks survival data. We further construct the Δ $$ \Delta $$ C-index to quantify the strength of association between the longitudinal and cause-specific survival data, or between the out-of-sample longitudinal and survival data. Empirical performance of the proposed assessment criteria is examined through an extensive simulation study. An in-depth analysis of the real data from large cancer prevention trials is carried out to demonstrate the usefulness of the proposed methodology.

Bayesian Model Assessment for Jointly Modeling Multidimensional Response Data with Application to Computerized Testing

Computerized assessment provides rich multidimensional data including trial-by-trial accuracy and response time (RT) measures. A key question in modeling this type of data is how to incorporate RT data, for example, in aid of ability estimation in item response theory (IRT) models. To address this, we propose a joint model consisting of a two-parameter IRT model for the dichotomous item response data, a log-normal model for the continuous RT data, and a normal model for corresponding paper-and-pencil scores. Then, we reformulate and reparameterize the model to capture the relationship between the model parameters, to facilitate the prior specification, and to make the Bayesian computation more efficient. Further, we propose several new model assessment criteria based on the decomposition of deviance information criterion (DIC) the logarithm of the pseudo-marginal likelihood (LPML). The proposed criteria can quantify the improvement in the fit of one part of the multidimensional data given the other parts. Finally, we have conducted several simulation studies to examine the empirical performance of the proposed model assessment criteria and have illustrated the application of these criteria using a real dataset from a computerized educational assessment program.

Bayesian criterion-based assessments of recurrent event models with applications to commercial truck driver behavior studies

Multitype recurrent events are commonly observed in transportation studies, since commercial truck drivers may encounter different types of safety critical events (SCEs) and take different lengths of on-duty breaks in a driving shift. Bayesian nonhomogeneous Poisson process models are a flexible approach to jointly model the intensity functions of the multitype recurrent events. For evaluating and comparing these models, the deviance information criterion (DIC) and the logarithm of the pseudo-marginal likelihood (LPML) are studied and Monte Carlo methods are developed for computing these model assessment measures. We also propose a set of new concordance indices (C-indices) to evaluate various discrimination abilities of a Bayesian multitype recurrent event model. Specifically, the within-event C-index quantifies adequacy of a given model in fitting the recurrent event data for each type, the between-event C-index provides an assessment of the model fit between two types of recurrent events, and the overall C-index measures the model's discrimination ability among multiple types of recurrent events simultaneously. Moreover, we jointly model the incidence of SCEs and on-duty breaks with driving behaviors using a Bayesian Poisson process model with time-varying coefficients and time-dependent covariates. An in-depth analysis of a real dataset from the commercial truck driver naturalistic driving study is carried out to demonstrate the usefulness and applicability of the proposed methodology.

Bayesian Design of Superiority Trials: Methods and Applications

In this paper, we lay out the basic elements of Bayesian sample size determination (SSD) for the Bayesian design of a two-arm superiority clinical trial. We develop a flowchart of the Bayesian SSD that highlights the critical components of a Bayesian design and provides a practically useful roadmap for designing a Bayesian clinical trial in real world applications. We empirically examine the amount of borrowing, the choice of noninformative priors, and the impact of model misspecification on the Bayesian type I error and power. A formal and statistically rigorous formulation of conditional borrowing within the decision rule framework is developed. Moreover, by extending the partial borrowing power priors, a new borrowing-by-parts power prior for incorporating historical data is proposed. Computational algorithms are also developed to calculate the Bayesian type I error and power. Extensive simulation studies are carried out to explore the operating characteristics of the proposed Bayesian design of a superiority trial.

Joint analysis of longitudinal measurements and spatially clustered competing risks HIV/AIDS data

The joint modeling of repeated measurements and time-to-event provides a general framework to describe better the link between the progression of disease through longitudinal measurements and time-to-event outcome. In the survival data, a sample of individuals is frequently grouped into clusters. In some applications, these clusters could be arranged spatially, for example, based on geographical regions. There are two benefits of considering spatial variation in these data, enhancing the efficiency and accuracy of the parameters estimations, and investigating the survivorship spatial pattern. On the other hand, in survival data, there is a situation that subjects are supposed to experience more than one type of event potentially, but the occurrence of one type of event prevents the occurrence of the others. In this article, we considered a Bayesian joint model of longitudinal and competing risks outcomes for spatially clustered HIV/AIDS data. The data were from a registry-based study carried in Hamadan Province, Iran, from December 1997 to June 2020. In this joint model, a linear mixed effects model was used for the longitudinal submodel and a cause-specific hazard model with spatial and spatial-risk random effects was used for the survival submodel. Also, a latent structure was defined by random effects to link both event times and longitudinal processes. We used a univariate intrinsic conditional autoregressive (ICAR) distribution and a multivariate ICAR distribution for modeling the areal spatial and spatial-risk random effects, respectively. The performance of our proposed model using simulation studies and analysis of HIV/AIDS data were assessed.

BRIDGING RANDOMIZED CONTROLLED TRIALS AND SINGLE-ARM TRIALS USING COMMENSURATE PRIORS IN ARM-BASED NETWORK META-ANALYSIS

Network meta-analysis (NMA) is a powerful tool to compare multiple treatments directly and indirectly by combining and contrasting multiple independent clinical trials. Because many NMAs collect only a few eligible randomized controlled trials (RCTs), there is an urgent need to synthesize different sources of information, e.g., from both RCTs and single-arm trials. However, single-arm trials and RCTs may have different populations and quality, so that assuming they are exchangeable may be inappropriate. This article presents a novel method using a commensurate prior on variance (CPV) to borrow variance (rather than mean) information from single-arm trials in an arm-based (AB) Bayesian NMA. We illustrate the advantages of this CPV method by reanalyzing an NMA of immune checkpoint inhibitors in cancer patients. Comprehensive simulations investigate the impact on statistical inference of including single-arm trials. The simulation results show that the CPV method provides efficient and robust estimation even when the two sources of information are moderately inconsistent.

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.

Joint modelling of longitudinal and survival data in the presence of competing risks with applications to prostate cancer data

This research is motivated from the data from a large Selenium and Vitamin E Cancer Prevention Trial (SELECT). The prostate specific antigens (PSAs) were collected longitudinally, and the survival endpoint was the time to low-grade cancer or the time to high-grade cancer (competing risks). In this article, the goal is to model the longitudinal PSA data and the time-to-prostate cancer (PC) due to low- or high-grade. We consider the low-grade and high-grade as two competing causes of developing PC. A joint model for simultaneously analysing longitudinal and time-to-event data in the presence of multiple causes of failure (or competing risk) is proposed within the Bayesian framework. The proposed model allows for handling the missing causes of failure in the SELECT data and implementing an efficient Markov chain Monte Carlo sampling algorithm to sample from the posterior distribution via a novel reparameterization technique. Bayesian criteria, ΔDICSurv, and ΔWAICSurv, are introduced to quantify the gain in fit in the survival sub-model due to the inclusion of longitudinal data. A simulation study is conducted to examine the empirical performance of the posterior estimates as well as ΔDICSurv and ΔWAICSurv and a detailed analysis of the SELECT data is also carried out to further demonstrate the proposed methodology.

Risk factor identification in cystic fibrosis by flexible hierarchical joint models

Cystic fibrosis (CF) is a lethal autosomal disease hallmarked by respiratory failure. Maintaining lung function and minimizing frequency of acute respiratory events known as pulmonary exacerbations are essential to survival. Jointly modeling longitudinal lung function and exacerbation occurrences may provide better inference. We propose a shared-parameter joint hierarchical Gaussian process model with flexible link function to investigate the impacts of both demographic and time-varying clinical risk factors on lung function decline and to examine the associations between lung function and occurrence of pulmonary exacerbation. A two-level Gaussian process is used to capture the nonlinear longitudinal trajectory, and a flexible link function is introduced to the joint model in order to analyze binary process. Bayesian model assessment criteria are provided in examining the overall performance in joint models and marginal fitting in each submodel. We conduct simulation studies and apply the proposed model in a local CF center cohort. In the CF application, a nonlinear structure is supported in modeling both the longitudinal continuous and binary processes. A negative association is detected between lung function and pulmonary exacerbation by the joint model. The importance of risk factors, including gender, diagnostic status, insurance status, and BMI, is examined in joint models.

Bayesian Meta-Regression Model Using Heavy-Tailed Random-effects with Missing Sample Sizes for Self-thinning Meta-data

Motivated by the self-thinning meta-data, a random-effects meta-analysis model with unknown precision parameters is proposed with a truncated Poisson regression model for missing sample sizes. The random effects are assumed to follow a heavy-tailed distribution to accommodate outlying aggregate values in the response variable. The logarithm of the pseudo-marginal likelihood (LPML) is used for model comparison. In addition, in order to determine which self-thinning law is more supported by the meta-data, a measure called "Plausibility Index (PI)" is developed. A simulation study is conducted to examine empirical performance of the proposed methodology. Finally, the proposed model and the PI measure are applied to analyze a self-thinning meta-data set in details.

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.

Flexible link functions in a joint hierarchical Gaussian process model

Many longitudinal studies often require jointly modeling a biomarker and an event outcome, in order to provide more accurate inference and dynamic prediction of disease progression. Cystic fibrosis (CF) studies have illustrated the benefits of these models, primarily examining the joint evolution of lung-function decline and survival. We propose a novel joint model within the shared-parameter framework that accommodates nonlinear lung-function trajectories, in order to provide more accurate inference on lung-function decline over time and to examine the association between evolution of lung function and risk of a pulmonary exacerbation (PE) event recurrence. Specifically, a two-level Gaussian process (GP) is used to estimate the nonlinear longitudinal trajectories and a flexible link function is introduced for a more accurate depiction of the binary process on the event outcome. Bayesian model assessment is used to evaluate each component of the joint model in simulation studies and an application to longitudinal data on patients receiving care from a CF center. A nonlinear structure is suggested by both longitudinal continuous and binary evaluations. Including a flexible link function improves model fit to these data. The proposed hierarchical GP model with a flexible power link function where Laplace distribution is the baseline (spep) has the best fit of all joint models considered, characterizing how accelerated lung-function decline corresponds to increased odds of experiencing another PE.

Bayesian flexible hierarchical skew heavy-tailed multivariate meta regression models for individual patient data with applications

A flexible class of multivariate meta-regression models are proposed for Individual Patient Data (IPD). The methodology is well motivated from 26 pivotal Merck clinical trials that compare statins (cholesterol lowering drugs) in combination with ezetimibe and statins alone on treatment-naïve patients and those continuing on statins at baseline. The research goal is to jointly analyze the multivariate outcomes, Low Density Lipoprotein Cholesterol (LDL-C), High Density Lipoprotein Cholesterol (HDL-C), and Triglycerides (TG). These three continuous outcome measures are correlated and shed much light on a subject's lipid status. The proposed multivariate meta-regression models allow for different skewness parameters and different degrees of freedom for the multivariate outcomes from different trials under a general class of skew t-distributions. The theoretical properties of the proposed models are examined and an efficient Markov chain Monte Carlo (MCMC) sampling algorithm is developed for carrying out Bayesian inference under the proposed multivariate meta-regression model. In addition, the Conditional Predictive Ordinates (CPOs) are computed via an efficient Monte Carlo method. Consequently, the logarithm of the pseudo marginal likelihood and Bayesian residuals are obtained for model comparison and assessment, respectively. A detailed analysis of the IPD meta data from the 26 Merck clinical trials is carried out to demonstrate the usefulness of the proposed methodology.

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.

A new Bayesian joint model for longitudinal count data with many zeros, intermittent missingness, and dropout with applications to HIV prevention trials

In longitudinal clinical trials, it is common that subjects may permanently withdraw from the study (dropout), or return to the study after missing one or more visits (intermittent missingness). It is also routinely encountered in HIV prevention clinical trials that there is a large proportion of zeros in count response data. In this paper, a sequential multinomial model is adopted for dropout and subsequently a conditional model is constructed for intermittent missingness. The new model captures the complex structure of missingness and incorporates dropout and intermittent missingness simultaneously. The model also allows us to easily compute the predictive probabilities of different missing data patterns. A zero-inflated Poisson mixed-effects regression model is assumed for the longitudinal count response data. We also propose an approach to assess the overall treatment effects under the zero-inflated Poisson model. We further show that the joint posterior distribution is improper if uniform priors are specified for the regression coefficients under the proposed model. Variations of the g-prior, Jeffreys prior, and maximally dispersed normal prior are thus established as remedies for the improper posterior distribution. An efficient Gibbs sampling algorithm is developed using a hierarchical centering technique. A modified logarithm of the pseudomarginal likelihood and a concordance based area under the curve criterion are used to compare the models under different missing data mechanisms. We then conduct an extensive simulation study to investigate the empirical performance of the proposed methods and further illustrate the methods using real data from an HIV prevention clinical trial.

Joint modeling of multivariate longitudinal data and survival data in several observational studies of Huntington's disease

BACKGROUND

Joint modeling is appropriate when one wants to predict the time to an event with covariates that are measured longitudinally and are related to the event. An underlying random effects structure links the survival and longitudinal submodels and allows for individual-specific predictions. Multiple time-varying and time-invariant covariates can be included to potentially increase prediction accuracy. The goal of this study was to estimate a multivariate joint model on several longitudinal observational studies of Huntington's disease, examine external validity performance, and compute individual-specific predictions for characterizing disease progression. Emphasis was on the survival submodel for predicting the hazard of motor diagnosis.

METHODS

Data from four observational studies was analyzed: Enroll-HD, PREDICT-HD, REGISTRY, and Track-HD. A Bayesian approach to estimation was adopted, and external validation was performed using a time-varying AUC measure. Individual-specific cumulative hazard predictions were computed based on a simulation approach. The cumulative hazard was used for computing predicted age of motor onset and also for a deviance residual indicating the discrepancy between observed diagnosis status and model-based status.

RESULTS

The joint model trained in a single study had very good performance in discriminating among diagnosed and pre-diagnosed participants in the remaining test studies, with the 5-year mean AUC = .83 (range .77-.90), and the 10-year mean AUC = .86 (range .82-.92). Graphical analysis of the predicted age of motor diagnosis showed an expected strong relationship with the trinucleotide expansion that causes Huntington's disease. Graphical analysis of the deviance-type residual revealed there were individuals who converted to a diagnosis despite having relatively low model-based risk, others who had not yet converted despite having relatively high risk, and the majority falling between the two extremes.

CONCLUSIONS

Joint modeling is an improvement over traditional survival modeling because it considers all the longitudinal observations of covariates that are predictive of an event. Predictions from joint models can have greater accuracy because they are tailored to account for individual variability. These predictions can provide relatively accurate characterizations of individual disease progression, which might be important in the timing of interventions, determining the qualification for appropriate clinical trials, and general genotypic analysis.

Joint hierarchical Gaussian process model with application to personalized prediction in medical monitoring

A two-level Gaussian process (GP) joint model is proposed to improve personalized prediction of medical monitoring data. The proposed model is applied to jointly analyze multiple longitudinal biomedical outcomes, including continuous measurements and binary outcomes, to achieve better prediction in disease progression. At the population level of the hierarchy, two independent GPs are used to capture the nonlinear trends in both the continuous biomedical marker and the binary outcome, respectively; at the individual level, a third GP, which is shared by the longitudinal measurement model and the longitudinal binary model, induces the correlation between these two model components and strengthens information borrowing across individuals. The proposed model is particularly advantageous in personalized prediction. It is applied to the motivating clinical data on cystic fibrosis disease progression, for which lung function measurements and onset of acute respiratory events are monitored jointly throughout each patient's clinical course. The results from both the simulation studies and the cystic fibrosis data application suggest that the inclusion of the shared individual-level GPs under the joint model framework leads to important improvements in personalized disease progression prediction.

Bayesian joint modeling of bivariate longitudinal and competing risks data: An application to study patient-ventilator asynchronies in critical care patients

Mechanical ventilation is a common procedure of life support in intensive care. Patient-ventilator asynchronies (PVAs) occur when the timing of the ventilator cycle is not simultaneous with the timing of the patient respiratory cycle. The association between severity markers and the events death or alive discharge has been acknowledged before, however, little is known about the addition of PVAs data to the analyses. We used an index of asynchronies (AI) to measure PVAs and the SOFA (sequential organ failure assessment) score to assess overall severity. To investigate the added value of including the AI, we propose a Bayesian joint model of bivariate longitudinal and competing risks data. The longitudinal process includes a mixed effects model for the SOFA score and a mixed effects beta regression model for the AI. The survival process is defined in terms of a cause-specific hazards model for the competing risks death or alive discharge. Our model indicates that the SOFA score is strongly related to vital status. PVAs are positively associated with alive discharge but there is not enough evidence that PVAs provide a more accurate indication of death prognosis than the SOFA score alone.