Bayesian joint modelling of longitudinal and time to event data: a methodological review

Bridging the gap between two-stage and joint models: The case of tumor growth inhibition and overall survival models

Many clinical trials generate both longitudinal biomarker and time-to-event data. We might be interested in their relationship, as in the case of tumor size and overall survival in oncology drug development. Many well-established methods exist for analyzing such data either sequentially (two-stage models) or simultaneously (joint models). Two-stage modeling (2stgM) has been challenged (i) for not acknowledging that biomarkers are endogenous covariable to the survival submodel and (ii) for not propagating the uncertainty of the longitudinal biomarker submodel to the survival submodel. On the other hand, joint modeling (JM), which properly circumvents both problems, has been criticized for being time-consuming, and difficult to use in practice. In this paper, we explore a third approach, referred to as a novel two-stage modeling (N2stgM). This strategy reduces the model complexity without compromising the parameter estimate accuracy. The three approaches (2stgM, JM, and N2stgM) are formulated, and a Bayesian framework is considered for their implementation. Both real and simulated data were used to analyze the performance of such approaches. In all scenarios, our proposal estimated the parameters approximately as JM but without being computationally expensive, while 2stgM produced biased results.

Joint modeling of longitudinal CD4 count data and time to first occurrence of composite outcome

In this study, we jointly modeled longitudinal CD4 count data and survival outcome (time-to-first occurrence of composite outcome of death, cardiac tamponade or constriction) in other to investigate the effects of Mycobacterium indicus pranii immunotherapy and the CD4 count measurements on the hazard of the composite outcome among patients with HIV and tuberculous (TB) pericarditis. In this joint modeling framework, the models for longitudinal and the survival data are linked by an association structure. The association structure represents the hazard of the event for 1-unit increase in the longitudinal measurement. Models fitting and parameter estimation were carried out using R version 4.2.3. The association structure that represents the strength of the association between the hazard for an event at time point j and the area under the longitudinal trajectory up to the same time j provides the best fit. We found that 1-unit increase in CD4 count results in 2 % significant reduction in the hazard of the composite outcome. Among HIV and TB pericarditis individuals, the hazard of the composite outcome does not differ between of M.indicus pranii versus placebo. Application of joint models to investigate the effect of M.indicus pranii on the hazard of the composite outcome is limited. Hence, this study provides information on the effect of M.indicus pranii on the hazard of the composite outcome among HIV and TB pericarditis patients.

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.

Personalised follow-up and management schema for patients with screen-detected pulmonary nodules: A dynamic modelling study

BACKGROUND

Selecting the time target for follow-up testing in lung cancer screening is challenging. We aim to devise dynamic, personalized lung cancer screening schema for patients with pulmonary nodules detected through low-dose computed tomography.

METHODS

We developed and validated dynamic models using data of pulmonary nodule patients (aged 55-74 years) from the National Lung Screening Trial. We predicted patient-specific risk profiles at baseline (R0) and updated the risk evaluation results in repeated screening rounds (R1 and R2). We used risk cutoffs to optimize time-dependent sensitivity at an early decision point (3 months) and time-dependent specificity at a late decision point (1 year).

RESULTS

In validation, area under receiver operating characteristic curve for predicting 12-month lung cancer onset was 0.867 (95 % confidence interval: 0.827-0.894) and 0.807 (0.765-0.948) at R0 and R1-R2, respectively. The personalized schema, compared with National Comprehensive Cancer Network (NCCN) guideline and Lung-RADS, yielded lower rates of delayed diagnosis (1.7% vs. 1.7% vs. 6.9 %) and over-testing (4.9% vs. 5.6% vs. 5.6 %) at R0, and lower rates of delayed diagnosis (0.0% vs. 18.2% vs. 18.2 %) and over-testing (2.6% vs. 8.3% vs. 7.3 %) at R2. Earlier test recommendation among cancer patients was more frequent using the personalized schema (vs. NCCN: 29.8% vs. 20.9 %, p = 0.0065; vs. Lung-RADS: 33.2% vs. 22.8 %, p = 0.0025), especially for women, patients aged ≥65 years, and part-solid or non-solid nodules.

CONCLUSIONS

The personalized schema is easy-to-implement and more accurate compared with rule-based protocols. The results highlight value of personalized approaches in realizing efficient nodule management.

Achieving Precision Healthcare through Nanomedicine and Enhanced Model Systems

The ability to customize medical choices according to an individual's genetic makeup and biomarker patterns marks a significant advancement toward overall improved healthcare for both individuals and society at large. By transitioning from the conventional one-size-fits-all approach to tailored treatments that can account for predispositions of different patient populations, nanomedicines can be customized to target the specific molecular underpinnings of a patient's disease, thus mitigating the risk of collateral damage. However, for these systems to reach their full potential, our understanding of how nano-based therapeutics behave within the intricate human body is necessary. Effective drug administration to the targeted organ or pathological niche is dictated by properties such as nanocarrier (NC) size, shape, and targeting abilities, where understanding how NCs change their properties when they encounter biomolecules and phenomena such as shear stress in flow remains a major challenge. This Review specifically focuses on vessel-on-a-chip technology that can provide increased understanding of NC behavior in blood and summarizes the specialized environment of the joint to showcase advanced tissue models as approaches to address translational challenges. Compared to conventional cell studies or animal models, these advanced models can integrate patient material for full customization. Combining such models with nanomedicine can contribute to making personalized medicine achievable.

Modeling multiple correlated end-organ disease trajectories: A tutorial for multistate and joint models with applications in diabetes complications

State-of-the-art biostatistics methods allow for the simultaneous modeling of several correlated non-fatal disease processes over time, but there is no clear guidance on the optimal analysis in most settings. An example occurs in diabetes, where it is not known with certainty how microvascular complications of the eyes, kidneys, and nerves co-develop over time. In this article, we propose and contrast two general model frameworks for studying complications (sequential state and parallel trajectory frameworks) and review multivariate methods for their analysis, focusing on multistate and joint modeling. We illustrate these methods in a tutorial format using the long-term follow-up from the Diabetes Control and Complications Trial and Epidemiology of Diabetes Interventions and Complications study public data repository. A formal comparison of prediction error and discrimination is included. Multistate models are particularly advantageous for determining the order and timing of complications, but require discretization of the longitudinal outcomes and possibly a very complex state space process. Intermittent observation of the states must be accounted for, and discretization is a probable disadvantage in this setting. In contrast, joint models can account for variations of continuous biomarkers over time and are particularly designed for modeling complex association structures between the complications and for performing dynamic predictions of an outcome of interest to inform clinical decisions (eg, a late-stage complication). We found that both models have helpful features that can better-inform our understanding of the complex trajectories that complications may take and can therefore help with decision making for patients presenting with diabetes complications.

Diagnostic performance and longitudinal analysis of fungal biomarkers in COVID-19 associated pulmonary aspergillosis

Objectives

Galactomannan lateral flow assay (GM-LFA) is a reliable test for COVID-19 associated pulmonary aspergillosis (CAPA) diagnosis. We aimed to assess the diagnostic performance of GM-LFA with different case definitions, the association between the longitudinal measurements of serum GM-ELISA, GM-LFA, and the risk of death.

Methods

Serum and nondirected bronchial lavage (NBL) samples were periodically collected. The sensitivity and specificity analysis for GM-LFA was done in different time periods. Longitudinal analysis was done with the joint model framework.

Results

A total of 207 patients were evaluated. On the day of CAPA diagnosis, serum GM-LFA had a sensitivity of 42 % (95 % CI: 23-63) and specificity of 82 % (95 % CI: 78-84), while NBL GM-LFA had a sensitivity of 73 % (95 % CI: 45-92), specificity of 85 % (95 % CI: 76-91) for CAPA. Sensitivity decreased through the following days in both samples. Univariate joint model analysis showed that increasing GM-LFA and GM-ELISA levels were associated with increased mortality, and that effect remained same with serum GM-ELISA in multivariate joint model analysis.

Conclusion

GM-LFA, particularly in NBL samples, seems to be a reliable method for CAPA diagnosis. For detecting patients with higher risk of mortality, longitudinal measurement of serum GM-ELISA can be useful.

Bayesian joint models for multi-regional clinical trials

In recent years, multi-regional clinical trials (MRCTs) have increased in popularity in the pharmaceutical industry due to their ability to accelerate the global drug development process. To address potential challenges with MRCTs, the International Council for Harmonisation released the E17 guidance document which suggests the use of statistical methods that utilize information borrowing across regions if regional sample sizes are small. We develop an approach that allows for information borrowing via Bayesian model averaging in the context of a joint analysis of survival and longitudinal data from MRCTs. In this novel application of joint models to MRCTs, we use Laplace's method to integrate over subject-specific random effects and to approximate posterior distributions for region-specific treatment effects on the time-to-event outcome. Through simulation studies, we demonstrate that the joint modeling approach can result in an increased rejection rate when testing the global treatment effect compared with methods that analyze survival data alone. We then apply the proposed approach to data from a cardiovascular outcomes MRCT.

Using a Bayesian model of the joint distribution of pain and time on medication to decide on pain medication for neuropathy

The PAIN-CONTRoLS trial compared four medications in treating Cryptogenic sensory polyneuropathy. The primary outcome was a utility function that combined two outcomes, patients' pain score reduction and patients' quit rate. However, additional analysis of the individual outcomes could also be leveraged to inform selecting an optimal medication for future patients. We demonstrate how joint modeling of longitudinal and time-to-event data from PAIN-CONTRoLS can be used to predict the effects of medication in a patient-specific manner and helps to make patient-focused decisions. A joint model was used to evaluate the two outcomes while accounting for the association between the longitudinal process and the time-to-event processes. Results suggested no significant association between the patients' pain scores and time to the medication quit in the PAIN-CONTRoLS study, but the joint model still provided robust estimates and a better model fit. Using the model estimates, given patients' baseline characteristics, a drug profile on both the pain reduction and medication time could be obtained for each drug, providing information on how likely they would quit and how much pain reduction they should expect. Our analysis suggested that drugs viable for one patient may not be beneficial for others.

Heart rate and QRS duration as biomarkers predict the immediate outcome from pulseless electrical activity

INTRODUCTION

Pulseless electrical activity (PEA) is commonly observed in in-hospital cardiac arrest (IHCA). Universally available ECG characteristics such as QRS duration (QRSd) and heart rate (HR) may develop differently in patients who obtain ROSC or not. The aim of this study was to assess prospectively how QRSd and HR as biomarkers predict the immediate outcome of patients with PEA.

METHOD

We investigated 327 episodes of IHCA in 298 patients at two US and one Norwegian hospital. We assessed the ECG in 559 segments of PEA nested within episodes, measuring QRSd and HR during pauses of compressions, and noted the clinical state that immediately followed PEA. We investigated the development of HR, QRSd, and transitions to ROSC or no-ROSC (VF/VT, asystole or death) in a joint longitudinal and competing risks statistical model.

RESULTS

Higher HR, and a rising HR, reflect a higher transition intensity ("hazard") to ROSC (p < 0.001), but HR was not associated with the transition intensity to no-ROSC. A lower QRSd and a shrinking QRSd reflect an increased transition intensity to ROSC (p = 0.023) and a reduced transition intensity to no-ROSC (p = 0.002).

CONCLUSION

HR and QRSd convey information of the immediateoutcome during resuscitation from PEA. These universally available and promising biomarkers may guide the emergency team in tailoring individual treatment.

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.

Approximate Bayesian inference for joint linear and partially linear modeling of longitudinal zero-inflated count and time to event data

Joint modeling of zero-inflated count and time-to-event data is usually performed by applying the shared random effect model. This kind of joint modeling can be considered as a latent Gaussian model. In this paper, the approach of integrated nested Laplace approximation (INLA) is used to perform approximate Bayesian approach for the joint modeling. We propose a zero-inflated hurdle model under Poisson or negative binomial distributional assumption as sub-model for count data. Also, a Weibull model is used as survival time sub-model. In addition to the usual joint linear model, a joint partially linear model is also considered to take into account the non-linear effect of time on the longitudinal count response. The performance of the method is investigated using some simulation studies and its achievement is compared with the usual approach via the Bayesian paradigm of Monte Carlo Markov Chain (MCMC). Also, we apply the proposed method to analyze two real data sets. The first one is the data about a longitudinal study of pregnancy and the second one is a data set obtained of a HIV study.

Informative presence and observation in routine health data: A review of methodology for clinical risk prediction

Objective

Informative presence (IP) is the phenomenon whereby the presence or absence of patient data is potentially informative with respect to their health condition, with informative observation (IO) being the longitudinal equivalent. These phenomena predominantly exist within routinely collected healthcare data, in which data collection is driven by the clinical requirements of patients and clinicians. The extent to which IP and IO are considered when using such data to develop clinical prediction models (CPMs) is unknown, as is the existing methodology aiming at handling these issues. This review aims to synthesize such existing methodology, thereby helping identify an agenda for future methodological work.

Materials and Methods

A systematic literature search was conducted by 2 independent reviewers using prespecified keywords.

Results

Thirty-six articles were included. We categorized the methods presented within as derived predictors (including some representation of the measurement process as a predictor in the model), modeling under IP, and latent structures. Including missing indicators or summary measures as predictors is the most commonly presented approach amongst the included studies (24 of 36 articles).

Discussion

This is the first review to collate the literature in this area under a prediction framework. A considerable body relevant of literature exists, and we present ways in which the described methods could be developed further. Guidance is required for specifying the conditions under which each method should be used to enable applied prediction modelers to use these methods.

Conclusions

A growing recognition of IP and IO exists within the literature, and methodology is increasingly becoming available to leverage these phenomena for prediction purposes. IP and IO should be approached differently in a prediction context than when the primary goal is explanation. The work included in this review has demonstrated theoretical and empirical benefits of incorporating IP and IO, and therefore we recommend that applied health researchers consider incorporating these methods in their work.

Longitudinal FEV1 and Exacerbation Risk in COPD: Quantifying the Association Using Joint Modelling

BACKGROUND

Lung function, measured as forced expiratory volume in one second (FEV1), and exacerbations are two endpoints evaluated in chronic obstructive pulmonary disease (COPD) clinical trials. Joint analysis of these endpoints could potentially increase statistical power and enable assessment of efficacy in shorter and smaller clinical trials.

OBJECTIVE

To evaluate joint modelling as a tool for analyzing treatment effects in COPD clinical trials by quantifying the association between longitudinal improvements in FEV1 and exacerbation risk reduction.

METHODS

A joint model of longitudinal FEV1 and exacerbation risk was developed based on patient-level data from a Phase III clinical study in moderate-to-severe COPD (1740 patients), evaluating efficacy of fixed-dose combinations of a long-acting bronchodilator, formoterol, and an inhaled corticosteroid, budesonide. Two additional studies (1604 and 1042 patients) were used for external model validation and parameter re-estimation.

RESULTS

A significant (p<0.0001) association between FEV1 and exacerbation risk was estimated, with an approximate 10% reduction in exacerbation risk per 100 mL improvement in FEV1, consistent across trials and treatment arms. The risk reduction associated with improvements in FEV1 was relatively small compared to the overall exacerbation risk reduction for treatment arms including budesonide (10-15% per 160 µg budesonide). High baseline breathlessness score and previous history of exacerbations also influenced the risk of exacerbation.

CONCLUSION

Joint modelling can be used to co-analyze longitudinal FEV1 and exacerbation data in COPD clinical trials. The association between the endpoints was consistent and appeared unrelated to treatment mechanism, suggesting that improved lung function is indicative of an exacerbation risk reduction. The risk reduction associated with improved FEV1 was, however, generally small and no major impact on exacerbation trial design can be expected based on FEV1 alone. Further exploration with other longitudinal endpoints should be considered to further evaluate the use of joint modelling in analyzing COPD clinical trials.

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.