Conference Agenda

A multistate cure model is a statistical framework used to analyze and represent the transitions individuals undergo between different states over time, accounting for the possibility of being cured by initial treatment. This model is particularly useful in pediatric oncology where a proportion of the patient population achieves cure through treatment and therefore will never experience certain events. Traditional multistate models do not account for this population heterogeneity. A multistate cure model can provide a more comprehensive understanding of the disease progression or remission patterns and aid in personalized treatment decisions and prognosis prediction. Despite its importance, no universal consensus exists on the structure of multistate cure models. Our study provides a novel framework for defining such models through a set of non-cure states. We develop a generalized algorithm based on the extended long data format, an extension of the traditional long data format, where a transition can be divided into two rows, each with a weight assigned reflecting the posterior probability of its cure status. The multistate cure model is built upon the current framework of multistate model and mixture cure model. The proposed algorithm makes use of the Expectation-Maximization (EM) algorithm and weighted likelihood representation such that it is highly flexible in model specification and easy to implement with standard packages. Additionally, it facilitates dynamic prediction. The state occupancy probabilities can be easily obtained within our multistate cure model framework, incorporating baseline covariates or even post-baseline information. The algorithm is applied on data from the European Society for Blood and Marrow Transplantation (EBMT). Standard errors of the estimated parameters in the EM algorithm are obtained via a non-parametric bootstrap procedure, while the method involving the calculation of the second-derivative matrix of the observed log-likelihood is also presented.

Background: Calibration is assessed on cause-specific absolute risks to determine agreement between predicted risks from the model and observed risks. For competing risks data, correct specification of more than one model may be required to ensure well-calibrated predicted risks for the event of interest. Furthermore, interest may be in the predicted risks of the event of interest, competing events and all-causes. Therefore, calibration must be assessed simultaneously using various measures.
Objectives: Evaluate calibration of prediction models for external validation using the cause-specific hazards (CSH) approach
Methods: We propose assessing miscalibration for CSH models using the complement of the cause-specific survival alongside assessment of calibration of the cause-specific absolute risks. We simulated various scenarios to illustrate how to identify which model(s) is mis-specified in an external validation setting. Calibration plots and calibration statistics (Calibration Slope, calibration-in-the-large) are presented alongside performance measures such as the Brier Score and Index of Prediction Accuracy. We propose using pseudo-values to calculate observed risks and we generate a smooth calibration curve with restricted cubic splines. We fitted flexible parametric survival models to the simulated data to flexibly estimate baseline CSH for prediction of individual cause-specific absolute risk. Methods will also be illustrated using a clinical dataset.
Results: Our simulations illustrate that miscalibration due to changes in the baseline CSH in external validation data are better identified using components from each cause-specific model. A mis-calibrated model on one cause, could lead to poor calibration on predicted absolute risks for each cause of interest, including the all-cause absolute risk. This is because prediction of a single cause-specific absolute risk is impacted by effects of variables on the cause of interest and competing events.
Conclusions: If accurate predictions for both all-cause and each cause-specific absolute risks are of interest, this is best achieved by developing and validating models via the CSH approach. For each cause-specific model, researchers should evaluate calibration plots separately using probabilities obtained using each respective cause-specific model components to reveal the cause of any miscalibration. Pseudo-values are also proposed to obtain observed individual cause-specific net or absolute risk and smoothed calibration curves.

Introduction:
Multi-state models describe processes where individuals transition between states over time. An important modelling choice is the time-scale, with two common choices available: (i) "clock-forward", measuring time from initial state entry, or (ii) "clock-reset", resetting time at each state entry.

Transition probabilities often depend on time of state entry. The “clock-forward” method incorporates this through delayed entry but struggles when some transitions are restricted to specific time windows after state entry. Such restricted transitions, where events occur only within a specific time-window after state entry, are common in medical settings, such as referrals at consultations, treatment initiation shortly after diagnosis, or early medication side effects. The “clock-reset” approach captures how the transition intensities of restricted transitions are equal to zero after the restricted time period has passed, but necessitates incorporating time of state entry as time-dependent covariate, complicating prediction.

Methods:

We consider a multi-state model where time since initial entry affects transition probabilities, and some transitions occur only within restricted time windows. Our motivating example is the referral of patients with musculoskeletal complains from general practitioners (GP) to specialists. Patients may have up to five consultations for a complaint, with referrals possible only during consultations. Between consultations, referral probability is zero.

We compared two time-scale approaches:

1. Clock-Forward: Time is measured from the initial consultation.
2. Clock-Reset: Time resets at each consultation, with time since first consultation included as time-dependent covariate. This approach captures how the baseline hazard of time-restricted transitions (referral) is equal to zero post-consultation.

Both models included baseline and time-dependent covariates, with the GP consultations as intermediate states and referral as absorbing state. Event probabilities were estimated by generating paths through the model. Model performance was assessed using discrimination (c-index) and calibration (O/E ratio & calibration plot).

Results:

We analyzed 2,358,750 primary care episodes, with 1,842,533 transitions and 23,884 referrals. The choice of time-scale significantly influenced estimated covariate effects for time-restricted transitions and predicted event probabilities. A simulation study will be presented that further investigates strategies for incorporating time-restricted transitions in multi-state models.

Conclusion:

We present insights into how the time-scale selection (“clock-forward” or “clock-reset”) impacts multi-state models, particularly for transitions with restricted time windows. In our application, the two approaches yield different results in terms of estimated covariate effects and predicted probabilities. The simulation study will give more insight into modeling strategies for time-restricted transitions in multi-state model.

Introduction

Modern survival analysis relies on dynamic survival prediction models that adapt to changing conditions using time-dependent covariates. Landmarking systematically assesses individuals at risk over time, while mixed-model landmarking improves accuracy by incorporating longitudinal trajectories. However, competing risks, where multiple events influence outcomes, challenge traditional methods, leading to biased estimates and flawed clinical decisions. Integrating machine learning with mixed-model landmarking addresses these limitations, enabling more precise, data-driven predictions for real-world applications.

Methods

This study integrates landmarking and competing risks using various machine learning techniques. We assess the predictive performance of Random Survival Forest (RSF), LightGBM, XGBoost, and Bayesian Additive Regression Trees (BART) through simulation studies and real-world data from a multicenter Phase III breast cancer trial. The proposed mixed-model landmarking approaches are rigorously evaluated using several performance metrics including Area Under the Curve (AUC), C-index, and Brier score.

Results

Results show that in terms of calibration, discrimination, and prediction accuracy, mixed-model landmarking routinely performs better than standard landmarking. Simulation studies highlight the necessity for specialized models by showing that ignoring competing hazards significantly overestimates occurrence probabilities. Mixed-model landmarking outperforms RSF in terms of prediction, achieving lower Brier scores (as low as 0.021) and higher AUC values (up to 0.883). These results are supported by real-world data, which demonstrates how well longitudinal data and machine learning can be combined for dynamic prediction. RSF stands out as the most reliable machine learning technique, outperforming gradient-boosting models in terms of calibration and discrimination.

Conclusion

This study illustrates the benefits of mixed-model landmarking within competing risk frameworks, filling a significant gap in survival analysis. Predictive accuracy is improved by integrating cutting-edge machine learning techniques, providing reliable solutions for a range of clinical applications. Besides, dynamic modeling frameworks driven by machine learning are useful tools for enhancing survival prediction in medical research because they can handle high-dimensional data and nonlinear interactions with ease.

Introduction The “illness-death model” describes a simple multistate process where subjects can move from the initial state to the final state (death) possibly transiting to an intermediate state (illness). This framework applies also when the intermediate state consists in a therapeutic intervention administered after waiting some time since the initial event, e.g. cardiopathic patients waiting for a possible heart transplant.

Methods When the aim is to estimate and compare the survival of patients on the waiting list vs after being transplanted, it is important to check the validity of the Markov assumption in order to select the most appropriate time scale (clock-forward or clock-reset) guiding the mortality after transplant and to assess the role of the waiting time [1].

We analyzed data from a cohort of nearly 1000 patients affected by severe cardiomyopathy included in a waiting list for heart transplant but with low priority. The process is non-markovian because the mortality rate since transplant tends to increase along with a longer waiting time.

Results We extended a non-parametric method to estimate survival in the presence of a time-dependent intervention [2], accounting for the impact of waiting time, to answer the question: what is the survival of a patient that will never be transplanted compared to a patient transplanted immediately after entry in list? Using a landmark approach, we also answer questions like: what is the survival of a patient alive at a certain time after entry on list and that will never be transplanted compared to a patient alive and transplanted at that time?

Finally, we used predictions from a double-scale Cox model [3] to estimate profile-specific survival without vs after transplant, adjusting for baseline (i.e. at entry on list) covariates.

Conclusion We show an original approach for survival prediction in the presence of a time-dependent treatment covariate accounting for the waiting time until treatment switch in a non-markovian process.

References

[1] Putter H, Fiocco M, Geskus RB. Tutorial in biostatistics: competing risks and multi‐state models. Statistics in Medicine 2007; 26:2389-2430.

[2] Bernasconi DP, Rebora P, Iacobelli S, Valsecchi MG, Antolini L. Survival probabilities with time dependent treatment indicator: quantities and non-parametric estimators. Statistics in Medicine 2015; 35(7):1032-48.

[3] Tassistro E, Bernasconi DP, Rebora P, Valsecchi MG, Antolini L. Modelling the hazard of transition into the absorbing state in the illness-death model. Biometrical Journal 2019; 62(3):836-851.