Joint model with latent state for longitudinal and multistate data

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.

Joint modeling of longitudinal and competing-risk data using cumulative incidence functions for the failure submodels accounting for potential failure cause misclassification through double sampling

Most of the literature on joint modeling of longitudinal and competing-risk data is based on cause-specific hazards, although modeling of the cumulative incidence function (CIF) is an easier and more direct approach to evaluate the prognosis of an event. We propose a flexible class of shared parameter models to jointly model a normally distributed marker over time and multiple causes of failure using CIFs for the survival submodels, with CIFs depending on the "true" marker value over time (i.e., removing the measurement error). The generalized odds rate transformation is applied, thus a proportional subdistribution hazards model is a special case. The requirement that the all-cause CIF should be bounded by 1 is formally considered. The proposed models are extended to account for potential failure cause misclassification, where the true failure causes are available in a small random sample of individuals. We also provide a multistate representation of the whole population by defining mutually exclusive states based on the marker values and the competing risks. Based solely on the assumed joint model, we derive fully Bayesian posterior samples for state occupation and transition probabilities. The proposed approach is evaluated in a simulation study and, as an illustration, it is fitted to real data from people with HIV.

Modelling time-varying covariates effect on survival via functional data analysis: application to the MRC BO06 trial in osteosarcoma

Time-varying covariates are of great interest in clinical research since they represent dynamic patterns which reflect disease progression. In cancer studies biomarkers values change as functions of time and chemotherapy treatment is modified by delaying a course or reducing the dose intensity, according to patient’s toxicity levels. In this work, a Functional covariate Cox Model (FunCM) to study the association between time-varying processes and a time-to-event outcome is proposed. FunCM first exploits functional data analysis techniques to represent time-varying processes in terms of functional data. Then, information related to the evolution of the functions over time is incorporated into functional regression models for survival data through functional principal component analysis. FunCM is compared to a standard time-varying covariate Cox model, commonly used despite its limiting assumptions that covariate values are constant in time and measured without errors. Data from MRC BO06/EORTC 80931 randomised controlled trial for treatment of osteosarcoma are analysed. Time-varying covariates related to alkaline phosphatase levels, white blood cell counts and chemotherapy dose during treatment are investigated. The proposed method allows to detect differences between patients with different biomarkers and treatment evolutions, and to include this information in the survival model. These aspects are seldom addressed in the literature and could provide new insights into the clinical research.

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

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

Critical examination of current response shift methods and proposal for advancing new methods

Purpose

This work is part of an international, interdisciplinary initiative to synthesize research on response shift in results of patient-reported outcome measures. The objective is to critically examine current response shift methods. We additionally propose advancing new methods that address the limitations of extant methods.

Methods

Based on literature reviews, this critical examination comprises design-based, qualitative, individualized, and preference-based methods, latent variable models, and other statistical methods. We critically appraised their definition, operationalization, the type of response shift they can detect, whether they can adjust for and explain response shift, their assumptions, and alternative explanations. Overall limitations requiring new methods were identified.

Results

We examined 11 methods that aim to operationalize response shift, by assessing change in the meaning of one’s self-evaluation. Six of these methods distinguish between change in observed measurements (observed change) and change in the construct that was intended to be measured (target change). The methods use either (sub)group-based or individual-level analysis, or a combination. All methods have underlying assumptions to be met and alternative explanations for the inferred response shift effects. We highlighted the need to address the interpretation of the results as response shift and proposed advancing new methods handling individual variation in change over time and multiple time points.

Conclusion

No single response shift method is optimal; each method has strengths and limitations. Additionally, extra steps need to be taken to correctly interpret the results. Advancing new methods and conducting computer simulation studies that compare methods are recommended to move response shift research forward.

Supplementary Information

The online version of this article (10.1007/s11136-020-02755-4) contains supplementary material, which is available to authorized users.

Modelling of viral load dynamics and CD4 cell count progression in an antiretroviral naive cohort: using a joint linear mixed and multistate Markov model

Background

Patients infected with HIV may experience a succession of clinical stages before the disease diagnosis and their health status may be followed-up by tracking disease biomarkers. In this study, we present a joint multistate model for predicting the clinical progression of HIV infection which takes into account the viral load and CD4 count biomarkers.

Methods

The data is from an ongoing prospective cohort study conducted among antiretroviral treatment (ART) naïve HIV-infected women in the province of KwaZulu-Natal, South Africa. We presented a joint model that consists of two related submodels: a Markov multistate model for CD4 cell count transitions and a linear mixed effect model for longitudinal viral load dynamics.

Results

Viral load dynamics significantly affect the transition intensities of HIV/AIDS disease progression. The analysis also showed that patients with relatively high educational levels (β = − 0.004; 95% confidence interval [CI]:-0.207, − 0.064), high RBC indices scores (β = − 0.01; 95%CI:-0.017, − 0.002) and high physical health scores (β = − 0.001; 95%CI:-0.026, − 0.003) were significantly were associated with a lower rate of viral load increase over time. Patients with TB co-infection (β = 0.002; 95%CI:0.001, 0.004), having many sex partners (β = 0.007; 95%CI:0.003, 0.011), being younger age (β = 0.008; 95%CI:0.003, 0.012) and high liver abnormality scores (β = 0.004; 95%CI:0.001, 0.01) were associated with a higher rate of viral load increase over time. Moreover, patients with many sex partners (β = − 0.61; 95%CI:-0.94, − 0.28) and with a high liver abnormality score (β = − 0.17; 95%CI:-0.30, − 0.05) showed significantly reduced intensities of immunological recovery transitions. Furthermore, a high weight, high education levels, high QoL scores, high RBC parameters and being of middle age significantly increased the intensities of immunological recovery transitions.

Conclusion

Overall, from a clinical perspective, QoL measurement items, being of a younger age, clinical attributes, marital status, and educational status are associated with the current state of the patient, and are an important contributing factor to extend survival of the patients and guide clinical interventions. From a methodological perspective, it can be concluded that a joint multistate model approach provides wide-ranging information about the progression and assists to provide specific dynamic predictions and increasingly precise knowledge of diseases.

Estimation in the Cox survival regression model with covariate measurement error and a changepoint

The Cox regression model is a popular model for analyzing the relationship between a covariate vector and a survival endpoint. The standard Cox model assumes a constant covariate effect across the entire covariate domain. However, in many epidemiological and other applications, the covariate of main interest is subject to a threshold effect: a change in the slope at a certain point within the covariate domain. Often, the covariate of interest is subject to some degree of measurement error. In this paper, we study measurement error correction in the case where the threshold is known. Several bias correction methods are examined: two versions of regression calibration (RC1 and RC2, the latter of which is new), two methods based on the induced relative risk under a rare event assumption (RR1 and RR2, the latter of which is new), a maximum pseudo-partial likelihood estimator (MPPLE), and simulation-extrapolation (SIMEX). We develop the theory, present simulations comparing the methods, and illustrate their use on data concerning the relationship between chronic air pollution exposure to particulate matter PM₁₀ and fatal myocardial infarction (Nurses Health Study (NHS)), and on data concerning the effect of a subject's long-term underlying systolic blood pressure level on the risk of cardiovascular disease death (Framingham Heart Study (FHS)). The simulations indicate that the best methods are RR2 and MPPLE.

A curvilinear bivariate random changepoint model to assess temporal order of markers

In biomedical research, various longitudinal markers measuring different quantities are often collected over time. For example, repeated measures of psychometric scores are very informative about the degradation process toward dementia. These trajectories are generally nonlinear with an acceleration of the decline a few years before the diagnosis and a large heterogeneity between psychometric tests depending on the underlying cognitive function to be evaluated and the metrological properties of the test. Comparing the times of acceleration of the decline before diagnosis between cognitive tests is useful to better understand the natural history of the disease. Our objective is to propose a bivariate random changepoint model that allows for the comparison of the mean time of change between two markers. A frequentist approach is proposed that gives validated statistical tests to assess the temporal order of the changepoints. Using a spline transformation function, the model is designed to handle non-Gaussian data, that are common for cognitive scores which frequently exhibit a strong ceiling effect. The procedure is assessed through a simulation study and applied to a French cohort of elderly to identify the order of the decline of several cognitive scores. The whole methodology has been implemented in a R package freely available.

Using joint models to disentangle intervention effect types and baseline confounding: an application within an intervention study in prodromal Alzheimer's disease with Fortasyn Connect

BACKGROUND

Many prodromal Alzheimer's disease trials collect two types of data: the time until clinical diagnosis of dementia and longitudinal patient information. These data are often analysed separately, although they are strongly associated. By combining the longitudinal and survival data into a single statistical model, joint models can account for the dependencies between the two types of data.

METHODS

We illustrate the major steps in a joint modelling approach, motivated by data from a prodromal Alzheimer's disease study: the LipiDiDiet trial.

RESULTS

By using joint models we are able to disentangle baseline confounding from the intervention effect and moreover, to investigate the association between longitudinal patient information and the time until clinical dementia diagnosis.

CONCLUSIONS

Joint models provide a valuable tool in the statistical analysis of clinical studies with longitudinal and survival data, such as in prodromal Alzheimer's disease trials, and have several added values compared to separate analyses.

Dealing with death when studying disease or physiological marker: the stochastic system approach to causality

The stochastic system approach to causality is applied to situations where the risk of death is not negligible. This approach grounds causality on physical laws, distinguishes system and observation and represents the system by multivariate stochastic processes. The particular role of death is highlighted, and it is shown that local influences must be defined on the random horizon of time of death. We particularly study the problem of estimating the effect of a factor V on a process of interest Y, taking death into account. We unify the cases where Y is a counting process (describing an event) and the case where Y is quantitative; we examine the case of observations in continuous and discrete time and we study the issue of whether the mechanism leading to incomplete data can be ignored. Finally, we give an example of a situation where we are interested in estimating the effect of a factor (blood pressure) on cognitive ability in elderly.

Hidden three-state survival model for bivariate longitudinal count data

A model is presented that describes bivariate longitudinal count data by conditioning on a progressive illness-death process where the two living states are latent. The illness-death process is modelled in continuous time, and the count data are described by a bivariate extension of the binomial distribution. The bivariate distributions for the count data approach include the correlation between two responses even after conditioning on the state. An illustrative data analysis is discussed, where the bivariate data consist of scores on two cognitive tests, and the latent states represent two stages of underlying cognitive function. By including a death state, possible association between cognitive function and the risk of death is accounted for.

A hypothesis testing procedure for random changepoint mixed models

In biomedical research, random changepoint mixed models are used to take into account an individual breakpoint in a biomarker trajectory. This may be observed in the cognitive decline measured by psychometric tests in the prediagnosis phase of Alzheimer's disease. The existence, intensity and duration of this accelerated decline can depend on individual characteristics. The main objective of our work is to propose inferential methods to assess the existence of this phase of accelerated decline, ie, the existence of a random changepoint. To do so, we use a mixed model with two linear phases and test the nullity of the parameter measuring the difference of slopes between the two phases. Because we face the issue of nuisance parameters being unidentifiable under the null hypothesis, the supremum of the classic score test statistic on these parameters is used. The asymptotic distribution of the supremum under the null is approached with a perturbation method based on the multiplier bootstrap. The performance of our testing procedure is assessed via simulations and the test is applied to the French cohort PAQUID of elderly subjects to study the shape of the prediagnosis decline according to educational level. The test is significant for both educational levels and the estimated trajectories confirmed that educational level is a good marker for cognitive reserve.

Shared parameter models for joint analysis of longitudinal and survival data with left truncation due to delayed entry - Applications to cystic fibrosis

Many longitudinal studies observe time to occurrence of a clinical event such as death, while also collecting serial measurements of one or more biomarkers that are predictive of the event, or are surrogate outcomes of interest. Joint modeling can be used to examine the relationship between the biomarker and the event, and also as a way of adjusting analyses of the biomarker for non-ignorable dropout. In settings such as registry studies, an additional complexity is caused when follow-up of subjects is delayed, referred to as left-truncation of follow-up in the survival analysis setting. If not adjusted for, this can cause bias in estimation of parameters of the survival distribution for the clinical event and in parameters of the longitudinal outcome such as the profile or rate of change over time because subjects may die or have the clinical event before follow-up starts. This paper illustrates how a broad class of shared parameter models can be used to jointly model a time to event outcome along with a longitudinal marker using available nonlinear mixed modeling software, when follow-up times are left truncated. Methods are applied to jointly model survival and decline in lung function in cystic fibrosis patients.

Joint Models of Longitudinal and Time-to-Event Data with More Than One Event Time Outcome: A Review

Methodological development and clinical application of joint models of longitudinal and time-to-event outcomes have grown substantially over the past two decades. However, much of this research has concentrated on a single longitudinal outcome and a single event time outcome. In clinical and public health research, patients who are followed up over time may often experience multiple, recurrent, or a succession of clinical events. Models that utilise such multivariate event time outcomes are quite valuable in clinical decision-making. We comprehensively review the literature for implementation of joint models involving more than a single event time per subject. We consider the distributional and modelling assumptions, including the association structure, estimation approaches, software implementations, and clinical applications. Research into this area is proving highly promising, but to-date remains in its infancy.

Joint models for longitudinal and time-to-event data: a review of reporting quality with a view to meta-analysis

BACKGROUND

Joint models for longitudinal and time-to-event data are commonly used to simultaneously analyse correlated data in single study cases. Synthesis of evidence from multiple studies using meta-analysis is a natural next step but its feasibility depends heavily on the standard of reporting of joint models in the medical literature. During this review we aim to assess the current standard of reporting of joint models applied in the literature, and to determine whether current reporting standards would allow or hinder future aggregate data meta-analyses of model results.

METHODS

We undertook a literature review of non-methodological studies that involved joint modelling of longitudinal and time-to-event medical data. Study characteristics were extracted and an assessment of whether separate meta-analyses for longitudinal, time-to-event and association parameters were possible was made.

RESULTS

The 65 studies identified used a wide range of joint modelling methods in a selection of software. Identified studies concerned a variety of disease areas. The majority of studies reported adequate information to conduct a meta-analysis (67.7% for longitudinal parameter aggregate data meta-analysis, 69.2% for time-to-event parameter aggregate data meta-analysis, 76.9% for association parameter aggregate data meta-analysis). In some cases model structure was difficult to ascertain from the published reports.

CONCLUSIONS

Whilst extraction of sufficient information to permit meta-analyses was possible in a majority of cases, the standard of reporting of joint models should be maintained and improved. Recommendations for future practice include clear statement of model structure, of values of estimated parameters, of software used and of statistical methods applied.

Joint modelling of longitudinal and multi-state processes: application to clinical progressions in prostate cancer

Joint modelling of longitudinal and survival data is increasingly used in clinical trials on cancer. In prostate cancer for example, these models permit to account for the link between longitudinal measures of prostate-specific antigen (PSA) and time of clinical recurrence when studying the risk of relapse. In practice, multiple types of relapse may occur successively. Distinguishing these transitions between health states would allow to evaluate, for example, how PSA trajectory and classical covariates impact the risk of dying after a distant recurrence post-radiotherapy, or to predict the risk of one specific type of clinical recurrence post-radiotherapy, from the PSA history. In this context, we present a joint model for a longitudinal process and a multi-state process, which is divided into two sub-models: a linear mixed sub-model for longitudinal data and a multi-state sub-model with proportional hazards for transition times, both linked by a function of shared random effects. Parameters of this joint multi-state model are estimated within the maximum likelihood framework using an EM algorithm coupled with a quasi-Newton algorithm in case of slow convergence. It is implemented under R, by combining and extending mstate and JM packages. The estimation program is validated by simulations and applied on pooled data from two cohorts of men with localized prostate cancer. Thanks to the classical covariates available at baseline and the repeated PSA measurements, we are able to assess the biomarker's trajectory, define the risks of transitions between health states and quantify the impact of the PSA dynamics on each transition intensity. Copyright © 2016 John Wiley & Sons, Ltd.

Joint latent class model for longitudinal data and interval-censored semi-competing events: Application to dementia

Joint models are used in ageing studies to investigate the association between longitudinal markers and a time-to-event, and have been extended to multiple markers and/or competing risks. The competing risk of death must be considered in the elderly because death and dementia have common risk factors. Moreover, in cohort studies, time-to-dementia is interval-censored since dementia is assessed intermittently. So subjects can develop dementia and die between two visits without being diagnosed. To study predementia cognitive decline, we propose a joint latent class model combining a (possibly multivariate) mixed model and an illness-death model handling both interval censoring (by accounting for a possible unobserved transition to dementia) and semi-competing risks. Parameters are estimated by maximum-likelihood handling interval censoring. The correlation between the marker and the times-to-events is captured by latent classes, homogeneous sub-groups with specific risks of death, dementia, and profiles of cognitive decline. We propose Markovian and semi-Markovian versions. Both approaches are compared to a joint latent-class model for competing risks through a simulation study, and applied in a prospective cohort study of cerebral and functional ageing to distinguish different profiles of cognitive decline associated with risks of dementia and death. The comparison highlights that among subjects with dementia, mortality depends more on age than on duration of dementia. This model distinguishes the so-called terminal predeath decline (among healthy subjects) from the predementia decline.

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.