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Dynamic Predictions with Time-Dependent Covariates in Survival Analysis using Joint Modeling and Landmarking
Dimitris Rizopoulos, Magdalena Murawska, Eleni-Rosalina Andrinopoulou, Geert Molenberghs, Johanna J. M. Takkenberg, Emmanuel Lesaffre
TL;DR
Accurate risk assessment requires using regularly collected tests and biomarkers to support prognosis and treatment decisions. The paper compares landmarking and joint modeling for dynamically updated survival predictions, finding a general gain from joint modeling while noting important scope and modeling limitations.
Problem
Accurate risk assessment requires methods that use regularly collected tests and biomarkers to support prognosis and treatment decisions.
Method
The paper contrasts landmarking and joint modeling to obtain dynamically updated survival probabilities, extending beyond formulations that use only the current biomarker value.
Results
The analysis generally finds a gain from using the joint modeling approach instead of landmarking.
Takeaways & Limitations
Comparing landmarking and joint modeling provides a basis for producing dynamically updated survival probabilities from longitudinal information.
Takeaways & Limitations
The developments focus on a single continuous longitudinal biomarker, while joint modeling makes strong assumptions about the path of time-dependent covariates.
Abstract
from arXiv · showhide
A key question in clinical practice is accurate prediction of patient prognosis. To this end, nowadays, physicians have at their disposal a variety of tests and biomarkers to aid them in optimizing medical care. These tests are often performed on a regular basis in order to closely follow the progression of the disease. In this setting it is of medical interest to optimally utilize the recorded information and provide medically-relevant summary measures, such as survival probabilities, that will aid in decision making. In this work we present and compare two statistical techniques that provide dynamically-updated estimates of survival probabilities, namely landmark analysis and joint models for longitudinal and time-to-event data. Special attention is given to the functional form linking the longitudinal and event time processes, and to measures of discrimination and calibration in the context of dynamic prediction.
1 Department of Biostatistics, Erasmus Medical Center, the Netherlands
The passage identifies a Department of Cardiothoracic Surgery at Erasmus Medical Center in the Netherlands.
- The Department of Cardiothoracic Surgery is affiliated with Erasmus Medical Center in the Netherlands.
3 Interuniversity Institute for Biostatistics and statistical Bioinformatics,
The passage lists an affiliation and the paper's main methodological keywords.
- The listed affiliation is Katholieke Universiteit Leuven and Universiteit Hasselt, Belgium.
- The keywords cover calibration, discrimination, prognostic modeling, risk prediction, and random effects.
1 Introduction
The introduction motivates dynamic survival prediction from repeated biomarkers and presents landmarking and joint modeling as approaches for comparing such predictions.
- Motivation: Many prognostic models use only a small fraction of available biomarker information, often relying on the last available measurement.
- Motivation: Repeated biomarker measurements may improve understanding of disease progression and prediction of event risk compared with a single measurement.
- Dynamic prediction: The statistical challenge is updating survival-probability estimates for a new patient as additional longitudinal information is recorded.
- Approaches: The paper contrasts landmarking with joint models for deriving dynamically updated survival probabilities.
- Approaches: It examines functional relationships between longitudinal and event-time processes and their effects on predictions.
- Evaluation: Discrimination and calibration measures are adapted to evaluate dynamic predictions in the longitudinal-biomarker setting.
2 Dynamic Individualized Predictions
The paper defines dynamic individualized survival prediction and develops landmarking and joint-modeling frameworks that update predictions using longitudinal information.
- Prediction target: Dynamic prediction estimates the probability that a new subject survives to a future time, conditional on survival and information observed up to a landmark time.
- Prediction target: Predictions can be updated when new information is recorded at a later time.
- Landmarking: Landmarking fits a Cox model to subjects at risk at the landmark time, using baseline covariates and the last available longitudinal response.
- Joint modeling: Joint models specify the longitudinal and event-time processes jointly, linking event risk to the true, unobserved marker trajectory.
- Longitudinal model: The paper focuses on linear mixed-effects models because the Aortic Valve marker, aortic gradient, is continuous.
- Joint modeling: Joint-model estimation uses conditional-independence assumptions and can proceed by maximum likelihood or Bayesian methods.
- Comparison: Landmarking is easier to implement, whereas joint modeling uses more information but requires specialized software and stronger modeling assumptions.
3 Functional Form
The section examines how different functional forms link longitudinal trajectories to event risk, because clinically relevant predictive features may include marker changes or history rather than only current levels. It presents alternative association structures for joint models and analogous landmarking summaries, while noting a landmarking limitation.
- Motivation: Functional-form choices determine which features of a longitudinal profile enter the event-risk model.Candidate features include current levels, rates of change, and summaries of the trajectory history.
- Joint modeling: Joint models have primarily used either subject-specific mean trajectories or random effects in the relative-risk predictor.The section motivates alternatives because increases or decreases in biomarker levels and whole-trajectory summaries may be more predictive.
- Joint modeling: The proposed association structures include marker level, slope, cumulative trajectory area, weighted history, and random effects.These parameterizations use different association parameters and therefore have different interpretations.
- Joint modeling: The slope formulation makes risk depend on both the marker level and its rate of change at time t.This distinguishes patients with equal marker levels but increasing versus decreasing trajectories.
- Joint modeling: History-based formulations relate risk to the accumulated or differentially weighted marker trajectory before t.A weighting example emphasizes marker levels in (t −3, t), with values closer to t receiving greater weight.
- Landmarking: Landmarking can incorporate slopes, areas, and weighted observed histories through Cox models fitted at landmark time t, but lacks an analogue of the random-effects formulation.The slope uses the last two measurements, while the area is based on observed measurements up to t.
4 Measuring Predictive Performance
The section adapts discrimination and calibration measures to dynamic survival prediction using information available up to a landmark time. It defines dynamic concordance through weighted time-specific AUCs and evaluates prediction error over future time points or intervals.
- Overview: Dynamic discrimination and calibration measures are adapted to prediction settings with time-dependent longitudinal information.The measures require an estimate of πj(u | t) and therefore apply to both landmarking and joint modeling.
- Discrimination: Dynamic sensitivity and specificity classify subjects using πj(t + ∆t | t) relative to a threshold c.Subjects with predicted survival probability at or below c are cases, while those above c are controls.
- Discrimination: The dynamic AUC measures whether subjects experiencing an event receive lower predicted survival than subjects who do not.It is evaluated among comparable subject pairs with measurements available through t.
- Discrimination: The dynamic concordance index summarizes time-specific concordance probabilities over follow-up using a weighted average of AUCs.Weights account for the changing number of subjects contributing comparisons at different time points.
- Calibration: Prediction error is measured at future time u and can also be averaged over an interval, with censoring accounted for in the estimator.These error measures can compare predictive accuracy between nested models and quantify explained variation.
5 Analysis of the Aortic Valve Dataset
The Aortic Valve analysis uses repeated aortic-gradient measurements and survival outcomes to fit flexible joint models and compare them with landmarking for dynamic prediction. Joint models generally provide better predictive accuracy and discrimination than landmarking in this dataset, with M4 offering the best overall balance.
- Data and outcomes: Re-operation-free survival differed minimally between sub-coronary implantation and root replacement, with a slight late advantage for sub-coronary implantation.
- Joint-model specification: The longitudinal submodel uses square-root-transformed aortic gradients and natural cubic splines to represent subject-specific trajectories by intervention group.The transformation addresses right skewness, while spline basis functions provide a flexible time specification.
- Joint-model specification: Four survival submodels impose different association structures between the longitudinal and event processes, including time-dependent and time-independent parameterizations.Bayesian estimation used MCMC with one chain of 115,000 iterations and a 15,000-iteration burn-in.
- Dynamic prediction: Joint models performed better than landmarking in both accuracy and discrimination, while M4 had the best accuracy and respectable discrimination relative to M1 and M2.M4 had the smallest prediction error at year 9.5 and over the full interval, whereas M1 and M2 provided the strongest discrimination.
6 Simulations
Simulations compare landmarking with joint modeling under four association scenarios using repeated synthetic longitudinal and survival datasets. Joint modeling generally yields more accurate survival predictions, although both approaches perform similarly in Scenario III.
- Simulation design: Each scenario used a different functional form for the association between the longitudinal and survival processes.The simulations followed a joint-modeling framework without assuming constant biomarker levels between visits.
- Evaluation: Predictions from seven models were evaluated against gold-standard survival probabilities using root mean squared prediction errors.Ten censored subjects were excluded from fitting and predicted at ten time points after their last longitudinal measurement.
- Results: Joint modeling appeared more accurate than landmarking, with the clearest differences in Scenarios I, II, and IV.
- Results: Both approaches produced similarly accurate results in Scenario III.
7 Discussion
The paper contrasts landmarking and joint modeling for dynamically updating survival probabilities, finding generally greater gains from joint modeling while noting its heavier modeling and computational demands. Landmarking remains attractive for practical implementation and multiple biomarkers, whereas joint models offer greater flexibility but face scalability and model-selection challenges.
- The work contrasts landmarking and joint modeling for producing dynamically updated survival-probability predictions.Both approaches are compared within the paper’s dynamic-prediction framework.
- Joint modeling accommodates greater flexibility in time-dependent covariate processes but requires more modeling assumptions and is generally more computationally intensive.Its fitting burden can increase substantially as the random-effects dimension grows.
- Joint modeling generally provides a gain over landmarking in the simulation study and Aortic Valve dataset analysis.The reported result is general rather than tied to a single numerical performance estimate.
- Landmarking is straightforward for multiple biomarkers because additional markers can enter the Cox-model linear predictor as baseline covariates.This advantage applies when several continuous or categorical markers are recorded.
- With multiple longitudinal outcomes or competing risks, choosing the functional parameterization for each outcome and event becomes a demanding model-selection exercise.Both extensions increase the number of possible models.
- Landmarking is readily implemented in standard Cox-model software, while joint-model prediction and evaluation are available through the JM and JMbayes R packages.The packages implement joint models, dynamic predictions, and calibration and discrimination measures.
A Simulation Settings
The simulation study specifies parameter values for longitudinal and survival submodels across four scenarios. These settings include fixed effects, a diagonal random-effects covariance matrix, and scenario-specific survival parameters.
- The longitudinal submodels specify fixed effects β1 = 0.93, β2 = −0.6, β3 = 0.63, β4 = 0.42, β5 = 1.1, and β6 = 0.54.
- The random-effects diagonal covariance matrix uses D11 = 0.49, D22 = 4.52, and D33 = 2.33.
- The simulations use four scenarios with parameter values for the survival submodels reported in Table 5.The table is identified as containing the survival-submodel parameter settings.