Assessing model fit in joint models of longitudinal and survival data with applications to cancer clinical trials

Joint modeling of longitudinal and competing risks for assessing blood oxygen saturation and its association with survival outcomes in COVID-19 patients

BACKGROUND

The objective of the present study is to evaluate the association between longitudinal and survival outcomes in the presence of competing risk events. To illustrate the application of joint modeling in clinical research, we assessed the blood oxygen saturation (SPO2) and its association with survival outcomes in coronavirus disease (COVID-19).

MATERIALS AND METHODS

In this prospective cohort study, we followed 300 COVID-19 patients, who were diagnosed with severe COVID-19 in the Rohani Hospital in Babol, the north of Iran from October 22, 2020 to March 5, 2021, where death was the event of interest, surviving was the competing risk event and SPO2 was the longitudinal outcome. Joint modeling analyses were compared to separate analyses for these data.

RESULT

The estimation of the association parameter in the joint modeling verified the association between longitudinal outcome SPO2 with survival outcome of death (Hazard Ratio (HR) = 0.33, P = 0.001) and the competing risk outcome of surviving (HR = 4.18, P < 0.001). Based on the joint modeling, longitudinal outcome (SPO2) decreased in hypertension patients (β = -0.28, P = 0.581) and increased in those with a high level of SPO2 on admission (β = 0.75, P = 0.03). Also, in the survival submodel in the joint model, the risk of death survival outcome increased in patients with diabetes comorbidity (HR = 4.38, P = 0.026).

CONCLUSION

The association between longitudinal measurements of SPO2 and survival outcomes of COVID-19 confirms that SPO2 is an important indicator in this disease. Thus, the application of this joint model can provide useful clinical evidence in the different areas of medical sciences.

Joint modeling of longitudinal changes of pulse rate and body temperature with time to recovery of pneumonia patients under treatment: a prospective cohort study

BACKGROUND

Pneumonia is the leading infectious cause of mortality worldwide and one of the most common lower respiratory tract infections that is contributing significantly to the burden of antibiotic consumption. The study aims to identify the determinants of the progress of pulse rate, body temperature and time to recovery of pneumonia patients.

METHOD

A prospective cohort study design was used from Felege Hiwot referral hospital on 214 sampled pneumonia patients from March 01, 2022 up to May 31, 2022. The Kaplan-Meier survival estimate and Log-Rank test was used to compare the survival time. Joint model of bivariate longitudinal and time to event model was used to identify factors of longitudinal change of pulse rate and body temperature with time to recovery jointly.

RESULT

As the follow up time of pneumonia patient's increase by one hour the average longitudinal change of pulse rate and body temperature were decreased by 0.4236 bpm and 0.0119 [Formula: see text]. The average longitudinal change of pulse rate and body temperature of patients who lived in rural was 1.4602 bpm and 0.1550 [Formula: see text] times less as compared to urban residence. Patients who had dangerous signs are significantly increased the average longitudinal change of pulse rate and body temperature by 2.042 bpm and 0.6031 [Formula: see text] as compared to patients who had no dangerous signs. A patient from rural residence was 1.1336 times more likely to experience the event of recovery as compared to urban residence. The estimated values of the association parameter for pulse rate and body temperature were -0.4236 bpm and -0.0119 respectively, which means pulse rate and body temperature were negatively related with patients recovery time.

CONCLUSION

Pulse rate and body temperature significantly affect the time to the first recovery of pneumonia patients who are receiving treatment. Age, residence, danger sign, comorbidity, baseline symptom and visiting time were the joint determinant factors for the longitudinal change of pulse rate, body temperature and time to recovery of pneumonia patients. The joint model approach provides precise dynamic predictions, widespread information about the disease transitions, and better knowledge of disease etiology.

Modeling and maximum likelihood based inference of interval-censored data with unknown upper limits and time-dependent covariates

Due to the nature of study design or other reasons, the upper limits of the interval-censored data with multiple visits are unknown. A naïve approach is to treat the last observed time as the exact event time, which may induce biased estimators of the model parameters. In this paper, we first develop a Cox model with time-dependent covariates for the event time and a proportional hazards model with frailty for the gap time. We then construct the upper limits using the latent gap times to resolve the issue of interval-censored event time data with unknown upper limits. A data-augmentation technique and a Monte Carlo EM (MCEM) algorithm are developed to facilitate computation. Theoretical properties of the computational algorithm are also investigated. Additionally, new model comparison criteria are developed to assess the fit of the gap time data as well as the fit of the event time data conditional on the gap time data. Our proposed method compares favorably with competing methods in both simulation study and real data analysis.

Joint modeling of longitudinal changes of blood pressure and time to remission of hypertensive patients receiving treatment: Bayesian approach

INTRODUCTION

Hypertension is a widespread condition when the blood's force on the artery walls is extremely high to develop adverse health effects. This paper aimed to jointly model the longitudinal change of blood pressures (systolic and diastolic) and time to the first remission of hypertensive outpatients receiving treatment.

METHODS

A retrospective study design was used to collect appropriate data on longitudinal changes in blood pressure and time-to-event from the medical charts of 301 hypertensive outpatients under follow-up at Felege Hiwot referral hospital, Ethiopia. The data exploration was done using summary statistics measures, individual profile plots, Kaplan-Meier plots, and log-rank tests. To get wide-ranging information about the progression, joint multivariate models were employed.

RESULTS

A total of 301 hypertensive patients who take treatment was taken from Felege Hiwot referral hospital recorded between Sep. 2018 to Feb. 2021. Of this 153 (50.8%) were male, and 124 (49.2%) were residents from rural areas. About 83(27.6%), 58 (19.3%), 82 (27.2%), and 25 (8.3%) have a history of diabetes mellitus, cardiovascular disease, stroke, and HIV respectively. The median time of hypertensive patients to have first remission time was 11 months. The hazard of the patient's first remission time for males was 0.63 times less likely than the hazard for females. The time to attain the first remission for patients who had a history of diabetes mellitus was 46% lower than for those who had no history of diabetes mellitus.

CONCLUSION

Blood pressure dynamics significantly affect the time to the first remission of hypertensive outpatients receiving treatment. The patients who had a good follow-up, lower BUN, lower serum calcium, lower serum sodium, lower hemoglobin, and take the treatment enalapril showed an opportunity in decreasing their blood pressure. This compels patients to experience the first remission early. Besides, age, patient's history of diabetes, patient's history of cardiovascular disease, and treatment type were the joint determinant factors for the longitudinal change of BP and the first remission time. The Bayesian joint model approach provides specific dynamic predictions, wide-ranging information about the disease transitions, and better knowledge of disease etiology.

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 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.

Relationship of ejection fraction and natriuretic peptide trajectories in heart failure with baseline reduced and mid-range ejection fraction

BACKGROUND

The prognostic importance of trajectories of neurohormones relative to left ventricular function over time in heart failure with reduced and mid-range EF (HFrEF and HFmrEF) is poorly defined.

OBJECTIVE

To evaluate left ventricular ejection fraction (LVEF) and B-type natriuretic peptide (BNP) trajectories in HFrEF and HFmrEF.

METHODS

Analyses of LVEF and BNP trajectories after incident HF admissions presenting with abnormal LV systolic function were performed using 3 methods: a Cox proportional hazards model with time-varying covariates, a dual longitudinal-survival model with shared random effects, and an unsupervised analysis to capture 3 discrete trajectories for each parameter.

RESULTS

Among 1,158 patients (68.9 ± 13.0 years, 53.3% female), both time-varying LVEF measurements (P=.001) and log-transformed BNP measurements (p-values=2 × 10^-16) were independently associated with survival during 6 years after covariate adjustment. In the dual longitudinal/survival model, both LVEF and BNP trajectories again were independently associated with survival (P<.0001 in each model); however, LVEF was more dynamic than BNP (P <.0001 for time covariate in LVEF longitudinal model versus P=.88 for the time covariate in BNP longitudinal model). In the unsupervised analysis, 3 discrete LVEF trajectories (dividing the cohort into approximately thirds) and 3 discrete BNP trajectories were identified. Discrete LVEF and BNP trajectories had independent prognostic value in Kaplan-Meier analyses (P<.0001), and substantial membership variability across BNP and LVEF trajectories was noted.

CONCLUSION

Although LVEF trajectories have greater temporal variation, BNP trajectories provide additive prognostication and an even stronger association with survival times in heart failure patients with abnormal LV systolic function.

Diagnostics for a two-stage joint survival model

A two-stage joint survival model is used to analyse time to event outcomes that could be associated with biomakers that are repeatedly collected over time. A Two-stage joint survival model has limited model checking tools and is usually assessed using standard diagnostic tools for survival models. The diagnostic tools can be improved and implemented. Time-varying covariates in a two-stage joint survival model might contain outlying observations or subjects. In this study we used the variance shift outlier model (VSOM) to detect and down-weight outliers in the first stage of the two-stage joint survival model. This entails fitting a VSOM at the observation level and a VSOM at the subject level, and then fitting a combined VSOM for the identified outliers. The fitted values were then extracted from the combined VSOM which were then used as time-varying covariate in the extended Cox model. We illustrate this methodology on a dataset from a multi-centre randomised clinical trial. A multi-centre trial showed that a combined VSOM fits the data better than an extended Cox model. We noted that implementing a combined VSOM, when desired, has a better fit based on the fact that outliers are down-weighted.

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.

Longitudinal Tumor Size and Neutrophil-to-Lymphocyte Ratio Are Prognostic Biomarkers for Overall Survival in Patients With Advanced Non-Small Cell Lung Cancer Treated With Durvalumab

Therapy optimization remains an important challenge in the treatment of advanced non-small cell lung cancer (NSCLC). We investigated tumor size (sum of the longest diameters (SLD) of target lesions) and neutrophil-to-lymphocyte ratio (NLR) as longitudinal biomarkers for survival prediction. Data sets from 335 patients with NSCLC from study NCT02087423 and 202 patients with NSCLC from study NCT01693562 of durvalumab were used for model qualification and validation, respectively. Nonlinear Bayesian joint models were designed to assess the impact of longitudinal measurements of SLD and NLR on patient subgrouping (by Response Evaluation Criteria in Solid Tumors 1.1 criteria at 3 months after therapy start), long-term survival, and precision of survival predictions. Various validation scenarios were investigated. We determined a more distinct patient subgrouping and a substantial increase in the precision of survival estimates after the incorporation of longitudinal measurements. The highest performance was achieved using a multivariate SLD and NLR model, which enabled predictions of NSCLC clinical outcomes.

Childhood growth prior to screen-detected celiac disease: prospective follow-up of an at-risk birth cohort

OBJECTIVES

To determine the association between childhood growth prior to the development of celiac disease (CD) and CD autoimmunity (CDA) identified by periodic serological screening.

STUDY DESIGN

The Diabetes Autoimmunity Study in the Young cohort includes 1979 genetically at-risk children from Denver, Colorado, with annual growth measurements from age nine months until ten years. Between 1993 and February 2019, 120 children developed CDA defined by persistent positive tissue transglutaminase autoantibodies (TGA); among these, 71 met our criteria for CD based on histopathological findings or high TGA levels. Age- and sex-specific z-scores of weight, body mass index (BMI), and height prior to seroconversion were derived using US reference charts as standards. Joint modeling of serial growth measurements was used to estimate adjusted hazard ratios (aHRs) accounting for celiac-associated human leukocyte antigens, early-life feeding practices, and socio-demographics.

RESULTS

In the first 10 years of life, there were no significant associations between the child's current weight, BMI and height and the risk of screening-detected CDA or CD, neither was the weight nor BMI velocity associated with CDA or CD as identified by screening (all aHRs approximated 1). Increased height velocity was associated with later CD, but not CDA, development (aHR per 0.01-z score/year, 1.28; 95% confidence interval [CI] 1.18-1.38 and 1.03; 0.97-1.09, respectively).

CONCLUSIONS

In the first 10 years of life, from prospectively collected serial growth measurements, we found no evidence of impaired childhood growth before CD and CDA development as identified through early and periodic screening.

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.

Association of Gluten Intake During the First 5 Years of Life With Incidence of Celiac Disease Autoimmunity and Celiac Disease Among Children at Increased Risk

IMPORTANCE

High gluten intake during childhood may confer risk of celiac disease.

OBJECTIVES

To investigate if the amount of gluten intake is associated with celiac disease autoimmunity and celiac disease in genetically at-risk children.

DESIGN, SETTING, AND PARTICIPANTS

The participants in The Environmental Determinants of Diabetes in the Young (TEDDY), a prospective observational birth cohort study designed to identify environmental triggers of type 1 diabetes and celiac disease, were followed up at 6 clinical centers in Finland, Germany, Sweden, and the United States. Between 2004 and 2010, 8676 newborns carrying HLA antigen genotypes associated with type 1 diabetes and celiac disease were enrolled. Screening for celiac disease with tissue transglutaminase autoantibodies was performed annually in 6757 children from the age of 2 years. Data on gluten intake were available in 6605 children (98%) by September 30, 2017.

EXPOSURES

Gluten intake was estimated from 3-day food records collected at ages 6, 9, and 12 months and biannually thereafter until the age of 5 years.

MAIN OUTCOMES AND MEASURES

The primary outcome was celiac disease autoimmunity, defined as positive tissue transglutaminase autoantibodies found in 2 consecutive serum samples. The secondary outcome was celiac disease confirmed by intestinal biopsy or persistently high tissue transglutaminase autoantibody levels.

RESULTS

Of the 6605 children (49% females; median follow-up: 9.0 years [interquartile range, 8.0-10.0 years]), 1216 (18%) developed celiac disease autoimmunity and 447 (7%) developed celiac disease. The incidence for both outcomes peaked at the age of 2 to 3 years. Daily gluten intake was associated with higher risk of celiac disease autoimmunity for every 1-g/d increase in gluten consumption (hazard ratio [HR], 1.30 [95% CI, 1.22-1.38]; absolute risk by the age of 3 years if the reference amount of gluten was consumed, 28.1%; absolute risk if gluten intake was 1-g/d higher than the reference amount, 34.2%; absolute risk difference, 6.1% [95% CI, 4.5%-7.7%]). Daily gluten intake was associated with higher risk of celiac disease for every 1-g/d increase in gluten consumption (HR, 1.50 [95% CI, 1.35-1.66]; absolute risk by age of 3 years if the reference amount of gluten was consumed, 20.7%; absolute risk if gluten intake was 1-g/d higher than the reference amount, 27.9%; absolute risk difference, 7.2% [95% CI, 6.1%-8.3%]).

CONCLUSIONS AND RELEVANCE

Higher gluten intake during the first 5 years of life was associated with increased risk of celiac disease autoimmunity and celiac disease among genetically predisposed children.

A joint model for multiple dynamic processes and clinical endpoints: Application to Alzheimer's disease

As other neurodegenerative diseases, Alzheimer's disease, the most frequent dementia in the elderly, is characterized by multiple progressive impairments in the brain structure and in clinical functions such as cognitive functioning and functional disability. Until recently, these components were mostly studied independently because no joint model for multivariate longitudinal data and time to event was available in the statistical community. Yet, these components are fundamentally interrelated in the degradation process toward dementia and should be analyzed together. We thus propose a joint model to simultaneously describe the dynamics of multiple correlated components. Each component, defined as a latent process, is measured by one or several continuous markers (not necessarily Gaussian). Rather than considering the associated time to diagnosis as in standard joint models, we assume diagnosis corresponds to the passing above a covariate-specific threshold (to be estimated) of a pathological process that is modeled as a combination of the component-specific latent processes. This definition captures the clinical complexity of diagnoses such as dementia diagnosis but also benefits from simplifications for the computation of maximum likelihood estimates. We show that the model and estimation procedure can also handle competing clinical endpoints. The estimation procedure, implemented in an R package, is validated by simulations and the method is illustrated on a large French population-based cohort of cerebral aging in which we focused on the dynamics of three clinical manifestations and the associated risk of dementia and death before dementia.

Review and Comparison of Computational Approaches for Joint Longitudinal and Time-to-Event Models

Joint models for longitudinal and time-to-event data are useful in situations where an association exists between a longitudinal marker and an event time. These models are typically complicated due to the presence of shared random effects and multiple submodels. As a consequence, software implementation is warranted that is not prohibitively time consuming. While methodological research in this area continues, several statistical software procedures exist to assist in the fitting of some joint models. We review the available implementation for frequentist and Bayesian models in the statistical programming languages R, SAS, and Stata. A description of each procedure is given including estimation techniques, input and data requirements, available options for customization, and some available extensions, such as competing risks models. The software implementations are compared and contrasted through extensive simulation, highlighting their strengths and weaknesses. Data from an ongoing trial on adrenal cancer patients is used to study different nuances of software fitting on a practical example.

Modelling cancer outcomes of bone metastatic patients: combining survival data with N-Telopeptide of type I collagen (NTX) dynamics through joint models

Background

Joint models (JM) have emerged as a promising statistical framework to concurrently analyse survival data and multiple longitudinal responses. This is particularly relevant in clinical studies where the goal is to estimate the association between time-to-event data and the biomarkers evolution. In the context of oncological data, JM can indeed provide interesting prognostic markers for the event under study and thus support clinical decisions and treatment choices. However, several problems arise when dealing with this type of data, such as the high-dimensionality of the covariates space, the lack of knowledge about the function structure of the time series and the presence of missing data, facts that may hamper the accurate estimation of the JM.

Methods

We propose to apply JM for the analysis of bone metastatic patients and infer the association of their survival with several covariates, in particular the N-Telopeptide of Type I Collagen (NTX) dynamics. This biomarker has been identified as a relevant prognostic factor in patients with metastatic cancer, but only using static information in some specific time points.

Results

We extended this analysis using the full NTX time series for a larger cohort of patients with bone metastasis, and compared the results obtained by the JM and the extended Cox regression model. Imputation based on fuzzy clustering was used to deal with missing values and several functions for NTX evolution were compared, such as rational, exponential and cubic splines.

Conclusions

The JM obtained confirm the association between NTX values and patients’ response, attesting the importance of this time series, and additionally provide a deep understanding of the key survival covariates.

Flexible longitudinal linear mixed models for multiple censored responses data

In biomedical studies and clinical trials, repeated measures are often subject to some upper and/or lower limits of detection. Hence, the responses are either left or right censored. A complication arises when more than one series of responses is repeatedly collected on each subject at irregular intervals over a period of time and the data exhibit tails heavier than the normal distribution. The multivariate censored linear mixed effect (MLMEC) model is a frequently used tool for a joint analysis of more than one series of longitudinal data. In this context, we develop a robust generalization of the MLMEC based on the scale mixtures of normal distributions. To take into account the autocorrelation existing among irregularly observed measures, a damped exponential correlation structure is considered. For this complex longitudinal structure, we propose an exact estimation procedure to obtain the maximum-likelihood estimates of the fixed effects and variance components using a stochastic approximation of the EM algorithm. This approach allows us to estimate the parameters of interest easily and quickly as well as to obtain the standard errors of the fixed effects, the predictions of unobservable values of the responses, and the log-likelihood function as a byproduct. The proposed method is applied to analyze a set of AIDS data and is examined via a simulation study.

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 Modeling and Inference for Nonignorably Missing Longitudinal Binary Response Data with Applications to HIV Prevention Trials

Missing data are frequently encountered in longitudinal clinical trials. To better monitor and understand the progress over time, one must handle the missing data appropriately and examine whether the missing data mechanism is ignorable or nonignorable. In this article, we develop a new probit model for longitudinal binary response data. It resolves a challenging issue for estimating the variance of the random effects, and substantially improves the convergence and mixing of the Gibbs sampling algorithm. We show that when improper uniform priors are specified for the regression coefficients of the joint multinomial model via a sequence of one-dimensional conditional distributions for the missing data indicators under nonignorable missingness, the joint posterior distribution is improper. A variation of Jeffreys prior is thus established as a remedy for the improper posterior distribution. In addition, an efficient Gibbs sampling algorithm is developed using a collapsing technique. Two model assessment criteria, the deviance information criterion (DIC) and the logarithm of the pseudomarginal likelihood (LPML), are used to guide the choices of prior specifications and to compare the models under different missing data mechanisms. We report on extensive simulations conducted to investigate the empirical performance of the proposed methods. The proposed methodology is further illustrated using data from an HIV prevention clinical trial.

Joint modeling of longitudinal and survival data with a covariate subject to a limit of detection

We develop and study an innovative method for jointly modeling longitudinal response and time-to-event data with a covariate subject to a limit of detection. The joint model assumes a latent process based on random effects to describe the association between longitudinal and time-to-event data. We study the role of the association parameter on the regression parameters estimators. We model the longitudinal and survival outcomes using linear mixed-effects and Weibull frailty models, respectively. Because of the limit of detection, missing covariate (explanatory variable, x) values may lead to the non-ignorable missing, resulting in biased parameter estimates with poor coverage probabilities of the confidence interval. We define and estimate the probability of missing due to the limit of detection. Then we develop a novel joint density and hence the likelihood function that incorporates the effect of left-censored covariate. Monte Carlo simulations show that the estimators of the proposed method are approximately unbiased and provide expected coverage probabilities for both longitudinal and survival submodels parameters. We also present an application of the proposed method using a large clinical dataset of pneumonia patients obtained from the Genetic and Inflammatory Markers of Sepsis study.

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

Joint models for longitudinal and survival data are routinely used in clinical trials or other studies to assess a treatment effect while accounting for longitudinal measures such as patient-reported outcomes (PROs). In the Bayesian framework, the deviance information criterion (DIC) and the logarithm of the pseudo marginal likelihood (LPML) are two well-known Bayesian criteria for comparing joint models. However, these criteria do not provide separate assessments of each component of the joint model. In this paper, we develop a novel decomposition of DIC and LPML to assess the fit of the longitudinal and survival components of the joint model, separately. Based on this decomposition, we then propose new Bayesian model assessment criteria, namely, ΔDIC and ΔLPML, to determine the importance and contribution of the longitudinal (survival) data to the model fit of the survival (longitudinal) data. Moreover, we develop an efficient Monte Carlo method for computing the Conditional Predictive Ordinate (CPO) statistics in the joint modeling setting. A simulation study is conducted to examine the empirical performance of the proposed criteria and the proposed methodology is further applied to a case study in mesothelioma.

JMFit: A SAS Macro for Joint Models of Longitudinal and Survival Data

Joint models for longitudinal and survival data now have a long history of being used in clinical trials or other studies in which the goal is to assess a treatment effect while accounting for a longitudinal biomarker such as patient-reported outcomes or immune responses. Although software has been developed for fitting the joint model, no software packages are currently available for simultaneously fitting the joint model and assessing the fit of the longitudinal component and the survival component of the model separately as well as the contribution of the longitudinal data to the fit of the survival model. To fulfill this need, we develop a SAS macro, called JMFit. JMFit implements a variety of popular joint models and provides several model assessment measures including the decomposition of AIC and BIC as well as ΔAIC and ΔBIC recently developed in Zhang et al. (2014). Examples with real and simulated data are provided to illustrate the use of JMFit.

Joint modelling of longitudinal and survival data: incorporating delayed entry and an assessment of model misspecification

A now common goal in medical research is to investigate the inter-relationships between a repeatedly measured biomarker, measured with error, and the time to an event of interest. This form of question can be tackled with a joint longitudinal-survival model, with the most common approach combining a longitudinal mixed effects model with a proportional hazards survival model, where the models are linked through shared random effects. In this article, we look at incorporating delayed entry (left truncation), which has received relatively little attention. The extension to delayed entry requires a second set of numerical integration, beyond that required in a standard joint model. We therefore implement two sets of fully adaptive Gauss-Hermite quadrature with nested Gauss-Kronrod quadrature (to allow time-dependent association structures), conducted simultaneously, to evaluate the likelihood. We evaluate fully adaptive quadrature compared with previously proposed non-adaptive quadrature through a simulation study, showing substantial improvements, both in terms of minimising bias and reducing computation time. We further investigate, through simulation, the consequences of misspecifying the longitudinal trajectory and its impact on estimates of association. Our scenarios showed the current value association structure to be very robust, compared with the rate of change that we found to be highly sensitive showing that assuming a simpler trend when the truth is more complex can lead to substantial bias. With emphasis on flexible parametric approaches, we generalise previous models by proposing the use of polynomials or splines to capture the longitudinal trend and restricted cubic splines to model the baseline log hazard function. The methods are illustrated on a dataset of breast cancer patients, modelling mammographic density jointly with survival, where we show how to incorporate density measurements prior to the at-risk period, to make use of all the available information. User-friendly Stata software is provided.

Bayesian joint modeling of longitudinal and spatial survival AIDS data

Joint analysis of longitudinal and survival data has received increasing attention in the recent years, especially for analyzing cancer and AIDS data. As both repeated measurements (longitudinal) and time-to-event (survival) outcomes are observed in an individual, a joint modeling is more appropriate because it takes into account the dependence between the two types of responses, which are often analyzed separately. We propose a Bayesian hierarchical model for jointly modeling longitudinal and survival data considering functional time and spatial frailty effects, respectively. That is, the proposed model deals with non-linear longitudinal effects and spatial survival effects accounting for the unobserved heterogeneity among individuals living in the same region. This joint approach is applied to a cohort study of patients with HIV/AIDS in Brazil during the years 2002-2006. Our Bayesian joint model presents considerable improvements in the estimation of survival times of the Brazilian HIV/AIDS patients when compared with those obtained through a separate survival model and shows that the spatial risk of death is the same across the different Brazilian states. Copyright © 2016 John Wiley & Sons, Ltd.

Joint modelling of repeated measurement and time-to-event data: an introductory tutorial

BACKGOUND

The term 'joint modelling' is used in the statistical literature to refer to methods for simultaneously analysing longitudinal measurement outcomes, also called repeated measurement data, and time-to-event outcomes, also called survival data. A typical example from nephrology is a study in which the data from each participant consist of repeated estimated glomerular filtration rate (eGFR) measurements and time to initiation of renal replacement therapy (RRT). Joint models typically combine linear mixed effects models for repeated measurements and Cox models for censored survival outcomes. Our aim in this paper is to present an introductory tutorial on joint modelling methods, with a case study in nephrology.

METHODS

We describe the development of the joint modelling framework and compare the results with those obtained by the more widely used approaches of conducting separate analyses of the repeated measurements and survival times based on a linear mixed effects model and a Cox model, respectively. Our case study concerns a data set from the Chronic Renal Insufficiency Standards Implementation Study (CRISIS). We also provide details of our open-source software implementation to allow others to replicate and/or modify our analysis.

RESULTS

The results for the conventional linear mixed effects model and the longitudinal component of the joint models were found to be similar. However, there were considerable differences between the results for the Cox model with time-varying covariate and the time-to-event component of the joint model. For example, the relationship between kidney function as measured by eGFR and the hazard for initiation of RRT was significantly underestimated by the Cox model that treats eGFR as a time-varying covariate, because the Cox model does not take measurement error in eGFR into account.

CONCLUSIONS

Joint models should be preferred for simultaneous analyses of repeated measurement and survival data, especially when the former is measured with error and the association between the underlying error-free measurement process and the hazard for survival is of scientific interest.

Joint modeling of longitudinal and survival data with missing and left-censored time-varying covariates

We propose a joint model for longitudinal and survival data with time-varying covariates subject to detection limits and intermittent missingness at random. The model is motivated by data from the Multicenter AIDS Cohort Study (MACS), in which HIV+ subjects have viral load and CD4 cell count measured at repeated visits along with survival data. We model the longitudinal component using a normal linear mixed model, modeling the trajectory of CD4 cell count by regressing on viral load, and other covariates. The viral load data are subject to both left censoring because of detection limits (17%) and intermittent missingness (27%). The survival component of the joint model is a Cox model with time-dependent covariates for death because of AIDS. The longitudinal and survival models are linked using the trajectory function of the linear mixed model. A Bayesian analysis is conducted on the MACS data using the proposed joint model. The proposed method is shown to improve the precision of estimates when compared with alternative methods.