Joint Models simultaneously model longitudinal and survival outcomes, leveraging patterns in patients' longitudinal trajectory to improve the prediction of survival outcomes. The classical parametric joint models, however, rely on fixed parametric assumptions, making them susceptible to bias under model misspecification and smaller sample sizes. We propose a deep joint model, DeepAJM, that does not require any parametric assumptions, while retaining a partially interpretable, per-longitudinal-outcome association structure. The joint model uses an encoder-decoder (sequence-to-sequence) architecture to learn the latent structure in patients' time-varying covariate trajectories. The model links the longitudinal processes to the survival processes through a learned interpretable association structure, in which each longitudinal output from the decoder gets remodulated by baseline covariates before it contributes to the risk scores from the survival head of the architecture. The model was evaluated on three datasets ( a cardiovascular-disease EHR cohort, a primary biliary cirrhosis (PBC2) dataset, and a simulated dataset) against a classical parametric joint model, TransformerJM, DA-LSTM and a Cox-based survival-only model. All models were assessed using C-index, integrated brier score (IBS), time-dependent AUROC, and time-dependent AUPRC. Our model achieved the best discrimination in terms of the C-index, time-dependent AUROC, and AUPRC across all datasets.
Figures & tables
Figure 1: Construction of training and test sequences used across all evaluated methods.
Figure 2: DeepAJM Model Architecture
Dataset
DeepAJM
PJM
TransformerJM
DA-LSTM
DA-LSTM (t)
Surv-only
CVD
0.9114
0.5124
0.6609
0.6621
0.8474
0.6166
PBC2
0.8073
0.5031
0.5876
0.7188
0.6817
0.5543
Simulated
0.6410
0.6422
0.5897
0.5895
0.6316
0.4609
Table 1: Concordance index (C-index) across datasets for DeepAJM and benchmarked models.
Dataset
DeepAJM
PJM
TransformerJM
DA-LSTM
DA-LSTM (t)
Surv-only
CVD
0.1122
0.3070
0.2592
0.2138
0.1733
0.2203
PBC2
0.1012
0.1584
0.1322
0.5895
0.3860
0.1360
Simulated
0.1330
0.0743
0.1107
0.1342
0.1956
0.1066
Table 2: Integrated Brier Score (IBS) across datasets for DeepAJM and benchmarked models.
Dataset
DeepAJM
PJM
TransformerJM
DA-LSTM
DA-LSTM (t)
Surv-only
CVD
0.9374
0.5127
0.6884
0.7028
0.8898
0.6190
PBC2
0.8878
0.4497
0.6600
0.7758
0.7928
0.5623
Simulated
0.7483
0.5551
0.5034
0.5194
0.6876
0.4333
Table 3: Time-dependent AUROC across datasets for DeepAJM and benchmarked models.
Dataset
DeepAJM
PJM
TransformerJM
DA-LSTM
DA-LSTM (t)
Surv-only
CVD
0.8548
0.3635
0.5500
0.5777
0.7988
0.4479
PBC2
0.5270
0.1752
0.3206
0.4090
0.4490
0.1687
Simulated
0.2462
0.1046
0.1276
0.1527
0.2201
0.1261
Table 4: Time-dependent AUPRC across datasets for DeepAJM and benchmarked models.
Survival analysis aims to model how covariates and time jointly shape the time-to-event distribution under right censoring. Classical methods such as the Cox model and generalised additive models (GAMs) require interactions and time-varying effects to be manually specified, which is increasingly impractical on rich clinical datasets. We introduce KAPLAN-HR, a B-spline Kolmogorov-Arnold Network (KAN) for nonparametric estimation of the conditional hazard as a joint function of covariates and time. A single-layer KAPLAN-HR model recovers a GAM, while deeper architectures capture interactions and time-varying effects through composition. We establish a convergence rate for the nonparametric KAN hazard estimator that depends only on the smoothness of the underlying KAN representation and not on the covariate dimension, thereby mitigating the curse of dimensionality for KAN-representable targets. In evaluations over six clinical benchmark datasets, KAPLAN-HR matches or exceeds the predictive performance of established statistical and deep learning survival methods.
Stelios Boulitsakis Logothetis, Angela Wood, Pietro Liò
In randomized trials involving multiple treatments, bivariate survival outcomes present significant analytical challenges for making decisions. This paper addresses the problem of deriving optimal individualized treatment rules to maximize the joint survival probability beyond fixed time points (t1,t2) through deep neural networks, while accounting for right censoring. We propose a novel approach that models treatment rules via stochastic policies, coupling marginal accelerated failure time models via link function to capture bivariate dependence. To enhance robustness and effectiveness of decision making, we introduce an adaptive prediction-powered method that leverages auxiliary predictions from machine learning models.
Kun Ren, Yifan Cui, Wen Su
Department of Biostatistics, City University of Hong Kong · Center for Data Science, Zhejiang University
Time-to-event data is widespread across the life sciences and engineering, but it is typically encountered together with censoring, which complicates the application of standard machine learning methods. Deep Cox models have emerged as a popular method for analyzing time-to-event data because they gracefully handle censoring and can be used with unstructured data such as clinical text reports, genomic sequences, and pathology images. However, their predicted survival probabilities are often poorly calibrated, thus limiting their practical utility. In this paper, we propose a novel post hoc calibration method for Deep Cox models that uses isotonic regression to refine predicted survival probabilities without affecting discriminative power. We establish favorable theoretical guarantees, including a double-robustness property and asymptotic calibration. Experiments on synthetic and real-world clinical data demonstrate the empirical effectiveness of our method.