A joint normal-ordinal (probit) model for ordinal and continuous longitudinal data.

In biomedical studies, continuous and ordinal longitudinal variables are frequently encountered. In many of these studies it is of interest to estimate the effect of one of these longitudinal variables on the other. Time-dependent covariates have, however, several limitations; they can, for example, not be included when the data is not collected at fixed intervals. The issues can be circumvented by implementing joint models, where two or more longitudinal variables are treated as a response and modeled with a correlated random effect. Next, by conditioning on these response(s), we can study the effect of one or more longitudinal variables on another. We propose a normal-ordinal(probit) joint model. First, we derive closed-form formulas to estimate the model-based correlations between the responses on their original scale. In addition, we derive the marginal model, where the interpretation is no longer conditional on the random effects. As a consequence, we can make predictions for a subvector of one response conditional on the other response and potentially a subvector of the history of the response. Next, we extend the approach to a high-dimensional case with more than two ordinal and/or continuous longitudinal variables. The methodology is applied to a case study where, among others, a longitudinal ordinal response is predicted with a longitudinal continuous variable.

Bayesian joint modeling of multivariate longitudinal and survival outcomes using Gaussian copulas.

There is an increasing interest in the use of joint models for the analysis of longitudinal and survival data. While random effects models have been extensively studied, these models can be hard to implement and the fixed effect regression parameters must be interpreted conditional on the random effects. Copulas provide a useful alternative framework for joint modeling. One advantage of using copulas is that practitioners can directly specify marginal models for the outcomes of interest. We develop a joint model using a Gaussian copula to characterize the association between multivariate longitudinal and survival outcomes. Rather than using an unstructured correlation matrix in the copula model to characterize dependence structure as is common, we propose a novel decomposition that allows practitioners to impose structure (e.g., auto-regressive) which provides efficiency gains in small to moderate sample sizes and reduces computational complexity. We develop a Markov chain Monte Carlo model fitting procedure for estimation. We illustrate the method's value using a simulation study and present a real data analysis of longitudinal quality of life and disease-free survival data from an International Breast Cancer Study Group trial.

Choice of baseline hazards in joint modeling of longitudinal and time-to-event cancer survival data.

Longitudinal time-to-event analysis is a statistical method to analyze data where covariates are measured repeatedly. In survival studies, the risk for an event is estimated using Cox-proportional hazard model or extended Cox-model for exogenous time-dependent covariates. However, these models are inappropriate for endogenous time-dependent covariates like longitudinally measured biomarkers, Carcinoembryonic Antigen (CEA). Joint models that can simultaneously model the longitudinal covariates and time-to-event data have been proposed as an alternative. The present study highlights the importance of choosing the baseline hazards to get more accurate risk estimation. The study used colon cancer patient data to illustrate and compare four different joint models which differs based on the choice of baseline hazards [piecewise-constant Gauss-Hermite (GH), piecewise-constant pseudo-adaptive GH, Weibull Accelerated Failure time model with GH & B-spline GH]. We conducted simulation study to assess the model consistency with varying sample size (N = 100, 250, 500) and censoring (20 %, 50 %, 70 %) proportions. In colon cancer patient data, based on Akaike information criteria (AIC) and Bayesian information criteria (BIC), piecewise-constant pseudo-adaptive GH was found to be the best fitted model. Despite differences in model fit, the hazards obtained from the four models were similar. The study identified composite stage as a prognostic factor for time-to-event and the longitudinal outcome, CEA as a dynamic predictor for overall survival in colon cancer patients. Based on the simulation study Piecewise-PH-aGH was found to be the best model with least AIC and BIC values, and highest coverage probability(CP). While the Bias, and RMSE for all the models showed a competitive performance. However, Piecewise-PH-aGH has shown least bias and RMSE in most of the combinations and has taken the shortest computation time, which shows its computational efficiency. This study is the first of its kind to discuss on the choice of baseline hazards.

Anti-Spike IgG antibodies as correlates of protection against SARS-CoV-2 infection in the pre-Omicron and Omicron era.

Anti-Spike IgG antibodies against SARS-CoV-2, which are elicited by vaccination and infection, are correlates of protection against infection with pre-Omicron variants. Whether this association can be generalized to infections with Omicron variants is unclear. We conducted a retrospective cohort study with 8457 blood donors in Tyrol, Austria, analyzing 15,340 anti-Spike IgG antibody measurements from March 2021 to December 2022 assessed by Abbott SARS-CoV-2 IgG II chemiluminescent microparticle immunoassay. Using a Bayesian joint model, we estimated antibody trajectories and adjusted hazard ratios for incident SARS-CoV-2 infection ascertained by self-report or seroconversion of anti-Nucleocapsid antibodies. At the time of their earliest available anti-Spike IgG antibody measurement (median November 23, 2021), participants had a median age of 46.0 years (IQR 32.8-55.2), with 45.3% being female, 41.3% having a prior SARS-CoV-2 infection, and 75.5% having received at least one dose of a COVID-19 vaccine. Among 6159 participants with endpoint data, 3700 incident SARS-CoV-2 infections with predominantly Omicron sublineages were recorded over a median of 8.8 months (IQR 5.7-12.4). The age- and sex-adjusted hazard ratio for SARS-CoV-2 associated with having twice the anti-Spike IgG antibody titer was 0.875 (95% credible interval 0.868-0.881) overall, 0.842 (0.827-0.856) during 2021, and 0.884 (0.877-0.891) during 2022 (all p < 0.001). The associations were similar in females and males (Pinteraction = 0.673) and across age (Pinteraction = 0.590). Higher anti-Spike IgG antibody titers were associated with reduced risk of incident SARS-CoV-2 infection across the entire observation period. While the magnitude of association was slightly weakened in the Omicron era, anti-Spike IgG antibody continues to be a suitable correlate of protection against newer SARS-CoV-2 variants.

Factor-augmented transformation models for interval-censored failure time data.

Interval-censored failure time data frequently arise in various scientific studies where each subject experiences periodical examinations for the occurrence of the failure event of interest, and the failure time is only known to lie in a specific time interval. In addition, collected data may include multiple observed variables with a certain degree of correlation, leading to severe multicollinearity issues. This work proposes a factor-augmented transformation model to analyze interval-censored failure time data while reducing model dimensionality and avoiding multicollinearity elicited by multiple correlated covariates. We provide a joint modeling framework by comprising a factor analysis model to group multiple observed variables into a few latent factors and a class of semiparametric transformation models with the augmented factors to examine their and other covariate effects on the failure event. Furthermore, we propose a nonparametric maximum likelihood estimation approach and develop a computationally stable and reliable expectation-maximization algorithm for its implementation. We establish the asymptotic properties of the proposed estimators and conduct simulation studies to assess the empirical performance of the proposed method. An application to the Alzheimer's Disease Neuroimaging Initiative (ADNI) study is provided. An R package ICTransCFA is also available for practitioners. Data used in preparation of this article were obtained from the ADNI database.

A latent variable approach to jointly modeling longitudinal and cumulative event data using a weighted two-stage method.

Ecological momentary assessment (EMA), a data collection method commonly employed in mHealth studies, allows for repeated real-time sampling of individuals' psychological, behavioral, and contextual states. Due to the frequent measurements, data collected using EMA are useful for understanding both the temporal dynamics in individuals' states and how these states relate to adverse health events. Motivated by data from a smoking cessation study, we propose a joint model for analyzing longitudinal EMA data to determine whether certain latent psychological states are associated with repeated cigarette use. Our method consists of a longitudinal submodel-a dynamic factor model-that models changes in the time-varying latent states and a cumulative risk submodel-a Poisson regression model-that connects the latent states with the total number of events. In the motivating data, both the predictors-the underlying psychological states-and the event outcome-the number of cigarettes smoked-are partially unobservable; we account for this incomplete information in our proposed model and estimation method. We take a two-stage approach to estimation that leverages existing software and uses importance sampling-based weights to reduce potential bias. We demonstrate that these weights are effective at reducing bias in the cumulative risk submodel parameters via simulation. We apply our method to a subset of data from a smoking cessation study to assess the association between psychological state and cigarette smoking. The analysis shows that above-average intensities of negative mood are associated with increased cigarette use.

Novel non-linear models for clinical trial analysis with longitudinal data: A tutorial using SAS for both frequentist and Bayesian methods.

Longitudinal data from clinical trials are commonly analyzed using mixed models for repeated measures (MMRM) when the time variable is categorical or linear mixed-effects models (ie, random effects model) when the time variable is continuous. In these models, statistical inference is typically based on the absolute difference in the adjusted mean change (for categorical time) or the rate of change (for continuous time). Previously, we proposed a novel approach: modeling the percentage reduction in disease progression associated with the treatment relative to the placebo decline using proportional models. This concept of proportionality provides an innovative and flexible method for simultaneously modeling different cohorts, multivariate endpoints, and jointly modeling continuous and survival endpoints. Through simulated data, we demonstrate the implementation of these models using SAS procedures in both frequentist and Bayesian approaches. Additionally, we introduce a novel method for implementing MMRM models (ie, analysis of response profile) using the nlmixed procedure.

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.

Improved mortality prediction for pediatric acute liver failure using dynamic prediction strategy.

OBJECTIVES: To develop and validate a prediction tool for pediatric acute liver failure (PALF) mortality risks that captures the rapid and heterogeneous clinical course for accurate and updated prediction. METHODS: Data included 1144 participants with PALF enrolled during three phases of the PALF registry study over 15 years. Using joint modeling, we built a dynamic prediction tool for mortality by combining longitudinal trajectories of multiple laboratory and clinical variables. The predictive performance for 7-day and 21-day mortality was assessed using the area under curve (AUC) through cross-validation and split-by-time validation. RESULTS: We constructed a prognostic joint model that combines the temporal trajectories of international normalized ratio, total bilirubin, hepatic encephalopathy, platelet count, and serum creatinine. Dynamic prediction using updated information improved predictive performance over static prediction using the information at enrollment (Day 0) only. In cross-validation, AUC increased from 0.784 to 0.887 when measurements obtained between Days 1 and 2 were incorporated. AUC remained similar when we used the earlier subset of the sample for training and the later subset for testing. CONCLUSIONS: Serial measurements of five variables in the first few days of PALF capture the dynamic clinical course of the disease and improve risk prediction for mortality. Continuous disease monitoring and updating risk prognosis are beneficial for timely and judicious medical decisions.

Relationship between prostate-specific antigen, alkaline phosphatase levels, and time-to-tumor shrinkage: understanding the progression of prostate cancer in a longitudinal study.

BACKGROUND: This study delves into the complex interplay among prostate-specific antigen, alkaline phosphatase, and the temporal dynamics of tumor shrinkage in prostate cancer. By investigating the longitudinal trajectories and time-to-prostate cancer tumor shrinkage, we aim to untangle the intricate patterns of these biomarkers. This understanding is pivotal for gaining profound insights into the multifaceted aspects of prostate cancer progression. The joint model approach serves as a comprehensive framework, facilitating the elucidation of intricate interactions among these pivotal elements within the context of prostate cancer . METHODS: A new joint model under a shared parameters strategy is proposed for mixed bivariate longitudinal biomarkers and event time data, for obtaining accurate estimates in the presence of missing covariate data. The primary innovation of our model resides in its effective management of covariates with missing observations. Built upon established frameworks, our joint model extends its capabilities by integrating mixed longitudinal responses and accounting for missingness in covariates, thus confronting this particular challenge. We posit that these enhancements bolster the model's utility and dependability in real-world contexts characterized by prevalent missing data. The main objective of this research is to provide a model-based approach to get full information from prostate cancer data collected with patients' baseline characteristics ( Age , body mass index ( BMI ), GleasonScore , Grade , and Drug ) and two longitudinal endogenous covariates ( Platelets and Bilirubin ). RESULTS: The results reveal a clear association between prostate-specific antigen and alkaline phosphatase biomarkers in the context of time-to-prostate cancer tumor shrinkage. This underscores the interconnected dynamics of these key indicators in gauging disease progression. CONCLUSIONS: The analysis of the prostate cancer dataset, incorporating a joint evaluation of mixed longitudinal prostate-specific antigen and alkaline phosphatase biomarkers alongside tumor status, has provided valuable insights into disease progression. The results demonstrate the effectiveness of the proposed joint model, as evidenced by accurate estimates. The shared variables associated with both longitudinal biomarkers and event times consistently deviate from zero, highlighting the robustness and reliability of the model in capturing the complex dynamics of prostate cancer progression. This approach holds promise for enhancing our understanding and predictive capabilities in the clinical assessment of prostate cancer.

A Bayesian joint model for multivariate longitudinal and time-to-event data with application to ALL maintenance studies.

The most common type of cancer diagnosed among children is the Acute Lymphocytic Leukemia (ALL). A study was conducted by Tata Translational Cancer Research Center (TTCRC) Kolkata, in which 236 children (diagnosed as ALL patients) were treated for the first two years (approximately) with two standard drugs (6MP and MTx) and were then followed nearly for the next 3 years. The goal is to identify the longitudinal biomarkers that are associated with time-to-relapse, and also to assess the effectiveness of the drugs. We develop a Bayesian joint model in which a linear mixed model is used to jointly model three biomarkers (i.e. white blood cell count, neutrophil count, and platelet count) and a semi-parametric proportional hazards model is used to model the time-to-relapse. Our proposed joint model can assess the effects of different covariates on the progression of the biomarkers, and the effects of the biomarkers (and the covariates) on time-to-relapse. In addition, the proposed joint model can impute the missing longitudinal biomarkers efficiently. Our analysis shows that the white blood cell (WBC) count is not associated with time-to-relapse, but the neutrophil count and the platelet count are significantly associated with it. We also infer that a lower dose of 6MP and a higher dose of MTx jointly result in a lower relapse probability in the follow-up period. Interestingly, we find that relapse probability is the lowest for the patients classified into the "high-risk" group at presentation. The effectiveness of the proposed joint model is assessed through the extensive simulation studies.

The effect of obesity phenotype changes on cardiovascular outcomes in adults older than 40 years in the prospective cohort of the Tehran lipids and glucose study (TLGS): joint model of longitudinal and time-to-event data.

BACKGROUND: Obesity is a worldwide health concern with serious clinical effects, including myocardial infarction (MI), stroke, cardiovascular diseases (CVDs), and all-cause mortality. The present study aimed to assess the association of obesity phenotypes and different CVDs and mortality in males and females by simultaneously considering the longitudinal and survival time data. METHODS: In the Tehran Lipid and Glucose Study (TLGS), participants older than three years were selected by a multi-stage random cluster sampling method and followed for about 19 years. In the current study, individuals aged over 40 years without a medical history of CVD, stroke, MI, and coronary heart disease were included. Exclusions comprised those undergoing treatment for CVD and those with more than 30% missing information or incomplete data. Joint modeling of longitudinal binary outcome and survival time data was applied to assess the dependency and the association between the changes in obesity phenotypes and time to occurrence of CVD, MI, stroke, and CVD mortality. To account for any potential sex-related confounding effect on the association between the obesity phenotypes and CVD outcomes, sex-specific analysis was carried out. The analysis was performed using packages (JMbayes2) of R software (version 4.2.1). RESULTS: Overall, 6350 adults above 40 years were included. In the joint modeling of CVD outcome among males, literates and participants with a family history of diabetes were at lower risk of CVD compared to illiterates and those with no family history of diabetes in the Bayesian Cox model. Current smokers were at higher risk of CVD compared to non-smokers. In a logistic mixed effects model, odds of obesity phenotype was higher among participants with low physical activity, family history of diabetes and older age compared to males with high physical activity, no family history of diabetes and younger age. In females, based on the results of the Bayesian Cox model, participants with family history of diabetes, family history of CVD, abnormal obesity phenotype and past smokers had a higher risk of CVD compared to those with no history of diabetes, CVD and nonsmokers. In the obesity varying model, odds of obesity phenotype was higher among females with history of diabetes and older age compared to those with no history of diabetes and who were younger. There was no significant variable associated with MI among males in the Bayesian Cox model. Odds of obesity phenotype was higher in males with low physical activity compared to those with high physical activity in the obesity varying model, whereas current smokers were at lower odds of obesity phenotype than nonsmokers. In females, risk of MI was higher among those with family history of diabetes compared to those with no history of diabetes in the Bayesian Cox model. In the logistic mixed effects model, a direct and significant association was found between age and obesity phenotype. In males, participants with history of diabetes, abnormal obesity phenotype and older age were at higher risk of stroke in the Bayesian Cox model compared to males with no history of diabetes, normal obesity phenotype and younger persons. In the obesity varying model, odds of obesity phenotype was higher in males with low physical activity, family history of diabetes and older age compared to those with high physical activity, no family history of diabetes and who were younger. Smokers had a lower odds of obesity phenotype than nonsmokers. In females, past smokers and those with family history of diabetes were at higher risk of stroke compared to nonsmokers and females with no history of diabetes in the Bayesian Cox model. In the obesity varying model, females with family history of diabetes and older ages had a higher odds of obesity phenotype compared to those with no family history of diabetes and who were younger. Among males, risk of CVD mortality was lower in past smokers compared to nonsmokers in the survival model. A direct and significant association was found between age and CVD mortality. Odds of obesity phenotype was higher in males with a history of diabetes than in those with no family history of diabetes in the logistic mixed effects model. CONCLUSIONS: It seems that modifications to metabolic disorders may have an impact on the heightened incidence of CVDs. Based on this, males with obesity and any type of metabolic disorder had a higher risk of CVD, stroke and CVD mortality (excluding MI) compared to those with a normal body mass index (BMI) and no metabolic disorders. Females with obesity and any type of metabolic disorder were at higher risk of CVD(, MI and stroke compared to those with a normal BMI and no metabolic disorders suggesting that obesity and metabolic disorders are related. Due to its synergistic effect on high blood pressure, metabolic disorders raise the risk of CVD.

A marginalized two-part joint model for a longitudinal biomarker and a terminal event with application to advanced head and neck cancers.

The sum of the longest diameter (SLD) of the target lesions is a longitudinal biomarker used to assess tumor response in cancer clinical trials, which can inform about early treatment effect. This biomarker is semicontinuous, often characterized by an excess of zeros and right skewness. Conditional two-part joint models were introduced to account for the excess of zeros in the longitudinal biomarker distribution and link it to a time-to-event outcome. A limitation of the conditional two-part model is that it only provides an effect of covariates, such as treatment, on the conditional mean of positive biomarker values, and not an overall effect on the biomarker, which is often of clinical relevance. As an alternative, we propose in this article, a marginalized two-part joint model (M-TPJM) for the repeated measurements of the SLD and a terminal event, where the covariates affect the overall mean of the biomarker. Our simulation studies assessed the good performance of the marginalized model in terms of estimation and coverage rates. Our application of the M-TPJM to a randomized clinical trial of advanced head and neck cancer shows that the combination of panitumumab in addition with chemotherapy increases the odds of observing a disappearance of all target lesions compared to chemotherapy alone, leading to a possible indirect effect of the combined treatment on time to death.

Multivariate joint model under competing risks to predict death of hospitalized patients for SARS-CoV-2 infection.

During the coronavirus disease 2019 (COVID-19) pandemic, several clinical prognostic scores have been proposed and evaluated in hospitalized patients, relying on variables available at admission. However, capturing data collected from the longitudinal follow-up of patients during hospitalization may improve prediction accuracy of a clinical outcome. To answer this question, 327 patients diagnosed with COVID-19 and hospitalized in an academic French hospital between January and July 2020 are included in the analysis. Up to 59 biomarkers were measured from the patient admission to the time to death or discharge from hospital. We consider a joint model with multiple linear or nonlinear mixed-effects models for biomarkers evolution, and a competing risks model involving subdistribution hazard functions for the risks of death and discharge. The links are modeled by shared random effects, and the selection of the biomarkers is mainly based on the significance of the link between the longitudinal and survival parts. Three biomarkers are retained: the blood neutrophil counts, the arterial pH, and the C-reactive protein. The predictive performances of the model are evaluated with the time-dependent area under the curve (AUC) for different landmark and horizon times, and compared with those obtained from a baseline model that considers only information available at admission. The joint modeling approach helps to improve predictions when sufficient information is available. For landmark 6 days and horizon of 30 days, we obtain AUC [95% CI] 0.73 [0.65, 0.81] and 0.81 [0.73, 0.89] for the baseline and joint model, respectively (p = 0.04). Statistical inference is validated through a simulation study.

A Bayesian quantile joint modeling of multivariate longitudinal and time-to-event data.

Linear mixed models are traditionally used for jointly modeling (multivariate) longitudinal outcomes and event-time(s). However, when the outcomes are non-Gaussian a quantile regression model is more appropriate. In addition, in the presence of some time-varying covariates, it might be of interest to see how the effects of different covariates vary from one quantile level (of outcomes) to the other, and consequently how the event-time changes across different quantiles. For such analyses linear quantile mixed models can be used, and an efficient computational algorithm can be developed. We analyze a dataset from the Acute Lymphocytic Leukemia (ALL) maintenance study conducted by Tata Medical Center, Kolkata. In this study, the patients suffering from ALL were treated with two standard drugs (6MP and MTx) for the first two years, and three biomarkers (e.g. lymphocyte count, neutrophil count and platelet count) were longitudinally measured. After treatment the patients were followed nearly for the next three years, and the relapse-time (if any) for each patient was recorded. For this dataset we develop a Bayesian quantile joint model for the three longitudinal biomarkers and time-to-relapse. We consider an Asymmetric Laplace Distribution (ALD) for each outcome, and exploit the mixture representation of the ALD for developing a Gibbs sampler algorithm to estimate the regression coefficients. Our proposed model allows different quantile levels for different biomarkers, but still simultaneously estimates the regression coefficients corresponding to a particular quantile combination. We infer that a higher lymphocyte count accelerates the chance of a relapse while a higher neutrophil count and a higher platelet count (jointly) reduce it. Also, we infer that across (almost) all quantiles 6MP reduces the lymphocyte count, while MTx increases the neutrophil count. Simulation studies are performed to assess the effectiveness of the proposed approach.

Studying the association between longitudinal mammographic density measurements and breast cancer risk: a joint modelling approach.

BACKGROUND: Researchers have suggested that longitudinal trajectories of mammographic breast density (MD) can be used to understand changes in breast cancer (BC) risk over a woman's lifetime. Some have suggested, based on biological arguments, that the cumulative trajectory of MD encapsulates the risk of BC across time. Others have tried to connect changes in MD to the risk of BC. METHODS: To summarize the MD-BC association, we jointly model longitudinal trajectories of MD and time to diagnosis using data from a large ([Formula: see text]) mammography cohort of Swedish women aged 40-80 years. Five hundred eighteen women were diagnosed with BC during follow-up. We fitted three joint models (JMs) with different association structures; Cumulative, current value and slope, and current value association structures. RESULTS: All models showed evidence of an association between MD trajectory and BC risk ([Formula: see text] for current value of MD, [Formula: see text] and [Formula: see text] for current value and slope of MD respectively, and [Formula: see text] for cumulative value of MD). Models with cumulative association structure and with current value and slope association structure had better goodness of fit than a model based only on current value. The JM with current value and slope structure suggested that a decrease in MD may be associated with an increased (instantaneous) BC risk. It is possible that this is because of increased screening sensitivity rather than being related to biology. CONCLUSION: We argue that a JM with a cumulative association structure may be the most appropriate/biologically relevant model in this context.

Two-part joint model for a longitudinal semicontinuous marker and a terminal event with application to metastatic colorectal cancer data.

Joint models for a longitudinal biomarker and a terminal event have gained interests for evaluating cancer clinical trials because the tumor evolution reflects directly the state of the disease. A biomarker characterizing the tumor size evolution over time can be highly informative for assessing treatment options and could be taken into account in addition to the survival time. The biomarker often has a semicontinuous distribution, i.e., it is zero inflated and right skewed. An appropriate model is needed for the longitudinal biomarker as well as an association structure with the survival outcome. In this article, we propose a joint model for a longitudinal semicontinuous biomarker and a survival time. The semicontinuous nature of the longitudinal biomarker is specified by a two-part model, which splits its distribution into a binary outcome (first part) represented by the positive versus zero values and a continuous outcome (second part) with the positive values only. Survival times are modeled with a proportional hazards model for which we propose three association structures with the biomarker. Our simulation studies show some bias can arise in the parameter estimates when the semicontinuous nature of the biomarker is ignored, assuming the true model is a two-part model. An application to advanced metastatic colorectal cancer data from the GERCOR study is performed where our two-part model is compared to one-part joint models. Our results show that treatment arm B (FOLFOX6/FOLFIRI) is associated to higher SLD values over time and its positive association with the terminal event leads to an increased risk of death compared to treatment arm A (FOLFIRI/FOLFOX6).

Nonlinear multilevel joint model for individual lesion kinetics and survival to characterize intra-individual heterogeneity in patients with advanced cancer.

In advanced cancer patients, tumor burden is calculated using the sum of the longest diameters (SLD) of the target lesions, a measure that lumps all lesions together and ignores intra-patient heterogeneity. Here, we used a rich dataset of 342 metastatic bladder cancer patients treated with a novel immunotherapy agent to develop a Bayesian multilevel joint model that can quantify heterogeneity in lesion dynamics and measure their impact on survival. Using a nonlinear model of tumor growth inhibition, we estimated that dynamics differed greatly among lesions, and inter-lesion variability accounted for 21% and 28% of the total variance in tumor shrinkage and treatment effect duration, respectively. Next, we investigated the impact of individual lesion dynamics on survival. Lesions located in the liver and in the bladder had twice as much impact on the instantaneous risk of death compared to those located in the lung or the lymph nodes. Finally, we evaluated the utility of individual lesion follow-up for dynamic predictions. Consistent with results at the population level, the individual lesion model outperformed a model relying only on SLD, especially at early landmark times and in patients with liver or bladder target lesions. Our results show that an individual lesion model can characterize the heterogeneity in tumor dynamics and its impact on survival in advanced cancer patients.

Multilevel joint model of longitudinal continuous and binary outcomes for hierarchically structured data.

Joint modeling has been a useful strategy for incorporating latent associations between different types of outcomes simultaneously, often focusing on a longitudinal continuous outcome characterized by an LME submodel and a terminal event subject to a Cox proportional hazard or parametric survival submodel. Applications to hierarchical longitudinal studies have been less frequent, particularly with respect to a binary process, which is commonly specified by a GLMM. Furthermore, many of the joint model developments have not allowed for investigations of nested effects, such as those arising from multicenter studies. To fill this gap, we propose a multilevel joint model that encompasses the LME submodel and GLMM through a Bayesian approach. Motivated by the need for timely detection of pulmonary exacerbation and characterization of irregularly observed lung function measurements in people living with cystic fibrosis (CF) receiving care across multiple centers, we apply the model to the data arising from US CF Foundation Patient Registry. In parallel, we examine the extent of bias induced by a non-hierarchical model. Our simulation study and application results show that incorporating the center effect along with individual stochastic variation over time within the LME submodel improves model estimation and prediction. Given that the center effect is evident in lung function observed in the CF population, accounting for center-specific power parameters by incorporating the symmetric power exponential power (spep) link function in the GLMM can facilitate more accurate conclusions in clinical studies.

A comparison of two approaches to dynamic prediction: Joint modeling and landmark modeling.

Joint modeling and landmark modeling are two mainstream approaches to dynamic prediction in longitudinal studies, that is, the prediction of a clinical event using longitudinally measured predictor variables available up to the time of prediction. It is an important research question to the methodological research field and also to practical users to understand which approach can produce more accurate prediction. There were few previous studies on this topic, and the majority of results seemed to favor joint modeling. However, these studies were conducted in scenarios where the data were simulated from the joint models, partly due to the widely recognized methodological difficulty on whether there exists a general joint distribution of longitudinal and survival data so that the landmark models, which consists of infinitely many working regression models for survival, hold simultaneously. As a result, the landmark models always worked under misspecification, which caused difficulty in interpreting the comparison. In this paper, we solve this problem by using a novel algorithm to generate longitudinal and survival data that satisfies the working assumptions of the landmark models. This innovation makes it possible for a "fair" comparison of joint modeling and landmark modeling in terms of model specification. Our simulation results demonstrate that the relative performance of these two modeling approaches depends on the data settings and one does not always dominate the other in terms of prediction accuracy. These findings stress the importance of methodological development for both approaches. The related methodology is illustrated with a kidney transplantation dataset.