Flexible Bayesian additive joint models with an application to type 1 diabetes research

Flexible joint model for time-to-event and non-Gaussian longitudinal outcomes

In medical studies, repeated measurements of biomarkers and time-to-event data are often collected during the follow-up period. To assess the association between these two outcomes, joint models are frequently considered. The most common approach uses a linear mixed model for the longitudinal part and a proportional hazard model for the survival part. The latter assumes a linear relationship between the survival covariates and the log hazard. In this work, we propose an extension allowing the inclusion of nonlinear covariate effects in the survival model using Bayesian penalized B-splines. Our model is valid for non-Gaussian longitudinal responses since we use a generalized linear mixed model for the longitudinal process. A simulation study shows that our method gives good statistical performance and highlights the importance of taking into account the possible nonlinear effects of certain survival covariates. Data from patients with a first progression of glioblastoma are analysed to illustrate the method.

Automatic search intervals for the smoothing parameter in penalized splines

The selection of smoothing parameter is central to the estimation of penalized splines. The best value of the smoothing parameter is often the one that optimizes a smoothness selection criterion, such as generalized cross-validation error (GCV) and restricted likelihood (REML). To correctly identify the global optimum rather than being trapped in an undesired local optimum, grid search is recommended for optimization. Unfortunately, the grid search method requires a pre-specified search interval that contains the unknown global optimum, yet no guideline is available for providing this interval. As a result, practitioners have to find it by trial and error. To overcome such difficulty, we develop novel algorithms to automatically find this interval. Our automatic search interval has four advantages. (i) It specifies a smoothing parameter range where the associated penalized least squares problem is numerically solvable. (ii) It is criterion-independent so that different criteria, such as GCV and REML, can be explored on the same parameter range. (iii) It is sufficiently wide to contain the global optimum of any criterion, so that for example, the global minimum of GCV and the global maximum of REML can both be identified. (iv) It is computationally cheap compared with the grid search itself, carrying no extra computational burden in practice. Our method is ready to use through our recently developed R package gps ( ≥  version 1.1). It may be embedded in more advanced statistical modeling methods that rely on penalized splines.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11222-022-10178-z.

Joint Modelling Approaches to Survival Analysis via Likelihood-Based Boosting Techniques

Joint models are a powerful class of statistical models which apply to any data where event times are recorded alongside a longitudinal outcome by connecting longitudinal and time-to-event data within a joint likelihood allowing for quantification of the association between the two outcomes without possible bias. In order to make joint models feasible for regularization and variable selection, a statistical boosting algorithm has been proposed, which fits joint models using component-wise gradient boosting techniques. However, these methods have well-known limitations, i.e., they provide no balanced updating procedure for random effects in longitudinal analysis and tend to return biased effect estimation for time-dependent covariates in survival analysis. In this manuscript, we adapt likelihood-based boosting techniques to the framework of joint models and propose a novel algorithm in order to improve inference where gradient boosting has said limitations. The algorithm represents a novel boosting approach allowing for time-dependent covariates in survival analysis and in addition offers variable selection for joint models, which is evaluated via simulations and real world application modelling CD4 cell counts of patients infected with human immunodeficiency virus (HIV). Overall, the method stands out with respect to variable selection properties and represents an accessible way to boosting for time-dependent covariates in survival analysis, which lays a foundation for all kinds of possible extensions.

Frailty modeling under a selective sampling protocol: an application to type 1 diabetes related autoantibodies

In studies following selective sampling protocols for secondary outcomes, conventional analyses regarding their appearance could provide misguided information. In the large type 1 diabetes prevention and prediction (DIPP) cohort study monitoring type 1 diabetes-associated autoantibodies, we propose to model their appearance via a multivariate frailty model, which incorporates a correlation component that is important for unbiased estimation of the baseline hazards under the selective sampling mechanism. As further advantages, the frailty model allows for systematic evaluation of the association and the differences in regression parameters among the autoantibodies. We demonstrate the properties of the model by a simulation study and the analysis of the autoantibodies and their association with background factors in the DIPP study, in which we found that high genetic risk is associated with the appearance of all the autoantibodies, whereas the association with sex and urban municipality was evident for IA-2A and IAA autoantibodies.

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

BACKGROUND

In clinical research, there is an increasing interest in joint modelling of longitudinal and time-to-event data, since it reduces bias in parameter estimation and increases the efficiency of statistical inference. Inference and prediction from frequentist approaches of joint models have been extensively reviewed, and due to the recent popularity of data-driven Bayesian approaches, a review on current Bayesian estimation of joint model is useful to draw recommendations for future researches.

METHODS

We have undertaken a comprehensive review on Bayesian univariate and multivariate joint models. We focused on type of outcomes, model assumptions, association structure, estimation algorithm, dynamic prediction and software implementation.

RESULTS

A total of 89 articles have been identified, consisting of 75 methodological and 14 applied articles. The most common approach to model the longitudinal and time-to-event outcomes jointly included linear mixed effect models with proportional hazards. A random effect association structure was generally used for linking the two sub-models. Markov Chain Monte Carlo (MCMC) algorithms were commonly used (93% articles) to estimate the model parameters. Only six articles were primarily focused on dynamic predictions for longitudinal or event-time outcomes.

CONCLUSION

Methodologies for a wide variety of data types have been proposed; however the research is limited if the association between the two outcomes changes over time, and there is also lack of methods to determine the association structure in the absence of clinical background knowledge. Joint modelling has been proved to be beneficial in producing more accurate dynamic prediction; however, there is a lack of sufficient tools to validate the prediction.

Joint model robustness compared with the time-varying covariate Cox model to evaluate the association between a longitudinal marker and a time-to-event endpoint

BACKGROUND

The recent progress in medical research generates an increasing interest in the use of longitudinal biomarkers for characterizing the occurrence of an outcome. The present work is motivated by a study, where the objective was to explore the potential of the long pentraxin 3 (PTX3) as a prognostic marker of Acute Graft-versus-Host Disease (GvHD) after haematopoietic stem cell transplantation. Time-varying covariate Cox model was commonly used, despite its limiting assumptions that marker values are constant in time and measured without error. A joint model has been developed as a viable alternative; however, the approach is computationally intensive and requires additional strong assumptions, in which the impacts of their misspecification were not sufficiently studied.

METHODS

We conduct an extensive simulation to clarify relevant assumptions for the understanding of joint models and assessment of its robustness under key model misspecifications. Further, we characterize the extent of bias introduced by the limiting assumptions of the time-varying covariate Cox model and compare its performance with a joint model in various contexts. We then present results of the two approaches to evaluate the potential of PTX3 as a prognostic marker of GvHD after haematopoietic stem cell transplantation.

RESULTS

Overall, we illustrate that a joint model provides an unbiased estimate of the association between a longitudinal marker and the hazard of an event in the presence of measurement error, showing improvement over the time-varying Cox model. However, a joint model is severely biased when the baseline hazard or the shape of the longitudinal trajectories are misspecified. Both the Cox model and the joint model correctly specified indicated PTX3 as a potential prognostic marker of GvHD, with the joint model providing a higher hazard ratio estimate.

CONCLUSIONS

Joint models are beneficial to investigate the capability of the longitudinal marker to characterize time-to-event endpoint. However, the benefits are strictly linked to the correct specification of the longitudinal marker trajectory and the baseline hazard function, indicating a careful consideration of assumptions to avoid biased estimates.

A Joint Modeling Approach for Childhood Meat, Fish and Egg Consumption and the Risk of Advanced Islet Autoimmunity

Several dietary factors have been suspected to play a role in the development of advanced islet autoimmunity (IA) and/or type 1 diabetes (T1D), but the evidence is fragmentary. A prospective population-based cohort of 6081 Finnish newborn infants with HLA-DQB1-conferred susceptibility to T1D was followed up to 15 years of age. Diabetes-associated autoantibodies and diet were assessed at 3- to 12-month intervals. We aimed to study the association between consumption of selected foods and the development of advanced IA longitudinally with Cox regression models (CRM), basic joint models (JM) and joint latent class mixed models (JLCMM). The associations of these foods to T1D risk were also studied to investigate consistency between alternative endpoints. The JM showed a marginal association between meat consumption and advanced IA: the hazard ratio adjusted for selected confounding factors was 1.06 (95% CI: 1.00, 1.12). The JLCMM identified two classes in the consumption trajectories of fish and a marginal protective association for high consumers compared to low consumers: the adjusted hazard ratio was 0.68 (0.44, 1.05). Similar findings were obtained for T1D risk with adjusted hazard ratios of 1.13 (1.02, 1.24) for meat and 0.45 (0.23, 0.86) for fish consumption. Estimates from the CRMs were closer to unity and CIs were narrower compared to the JMs. Findings indicate that intake of meat might be directly and fish inversely associated with the development of advanced IA and T1D, and that disease hazards in longitudinal nutritional epidemiology are more appropriately modeled by joint models than with naive approaches.

Nonlinear association structures in flexible Bayesian additive joint models

Joint models of longitudinal and survival data have become an important tool for modeling associations between longitudinal biomarkers and event processes. The association between marker and log hazard is assumed to be linear in existing shared random effects models, with this assumption usually remaining unchecked. We present an extended framework of flexible additive joint models that allows the estimation of nonlinear covariate specific associations by making use of Bayesian P-splines. Our joint models are estimated in a Bayesian framework using structured additive predictors for all model components, allowing for great flexibility in the specification of smooth nonlinear, time-varying, and random effects terms for longitudinal submodel, survival submodel, and their association. The ability to capture truly linear and nonlinear associations is assessed in simulations and illustrated on the widely studied biomedical data on the rare fatal liver disease primary biliary cirrhosis. All methods are implemented in the R package bamlss to facilitate the application of this flexible joint model in practice.

Joint modeling of longitudinal autoantibody patterns and progression to type 1 diabetes: results from the TEDDY study

AIMS

The onset of clinical type 1 diabetes (T1D) is preceded by the occurrence of disease-specific autoantibodies. The level of autoantibody titers is known to be associated with progression time from the first emergence of autoantibodies to the onset of clinical symptoms, but detailed analyses of this complex relationship are lacking. We aimed to fill this gap by applying advanced statistical models.

METHODS

We investigated data of 613 children from the prospective TEDDY study who were persistent positive for IAA, GADA and/or IA2A autoantibodies. We used a novel approach of Bayesian joint modeling of longitudinal and survival data to assess the potentially time- and covariate-dependent association between the longitudinal autoantibody titers and progression time to T1D.

RESULTS

For all autoantibodies we observed a positive association between the titers and the T1D progression risk. This association was estimated as time-constant for IA2A, but decreased over time for IAA and GADA. For example the hazard ratio [95% credibility interval] for IAA (per transformed unit) was 3.38 [2.66, 4.38] at 6 months after seroconversion, and 2.02 [1.55, 2.68] at 36 months after seroconversion.

CONCLUSIONS

These findings indicate that T1D progression risk stratification based on autoantibody titers should focus on time points early after seroconversion. Joint modeling techniques allow for new insights into these associations.