Survival Mixture Density Networks.

Survival analysis, the art of time-to-event modeling, plays an important role in clinical treatment decisions. Recently, continuous time models built from neural ODEs have been proposed for survival analysis. However, the training of neural ODEs is slow due to the high computational complexity of neural ODE solvers. Here, we propose an efficient alternative for flexible continuous time models, called Survival Mixture Density Networks (Survival MDNs). Survival MDN applies an invertible positive function to the output of Mixture Density Networks (MDNs). While MDNs produce flexible real-valued distributions, the invertible positive function maps the model into the time-domain while preserving a tractable density. Using four datasets, we show that Survival MDN performs better than, or similarly to continuous and discrete time baselines on concordance, integrated Brier score and integrated binomial log-likelihood. Meanwhile, Survival MDNs are also faster than ODE-based models and circumvent binning issues in discrete models.

Inverse-Weighted Survival Games.

Deep models trained through maximum likelihood have achieved state-of-the-art results for survival analysis. Despite this training scheme, practitioners evaluate models under other criteria, such as binary classification losses at a chosen set of time horizons, e.g. Brier score (BS) and Bernoulli log likelihood (BLL). Models trained with maximum likelihood may have poor BS or BLL since maximum likelihood does not directly optimize these criteria. Directly optimizing criteria like BS requires inverse-weighting by the censoring distribution. However, estimating the censoring model under these metrics requires inverse-weighting by the failure distribution. The objective for each model requires the other, but neither are known. To resolve this dilemma, we introduce Inverse-Weighted Survival Games. In these games, objectives for each model are built from re-weighted estimates featuring the other model, where the latter is held fixed during training. When the loss is proper, we show that the games always have the true failure and censoring distributions as a stationary point. This means models in the game do not leave the correct distributions once reached. We construct one case where this stationary point is unique. We show that these games optimize BS on simulations and then apply these principles on real world cancer and critically-ill patient data.

X-CAL: Explicit Calibration for Survival Analysis.

Survival analysis models the distribution of time until an event of interest, such as discharge from the hospital or admission to the ICU. When a model's predicted number of events within any time interval is similar to the observed number, it is called well-calibrated. A survival model's calibration can be measured using, for instance, distributional calibration (D-CALIBRATION) [Haider et al., 2020] which computes the squared difference between the observed and predicted number of events within different time intervals. Classically, calibration is addressed in post-training analysis. We develop explicit calibration (X-CAL), which turns D-CALIBRATION into a differentiable objective that can be used in survival modeling alongside maximum likelihood estimation and other objectives. X-CAL allows practitioners to directly optimize calibration and strike a desired balance between predictive power and calibration. In our experiments, we fit a variety of shallow and deep models on simulated data, a survival dataset based on MNIST, on length-of-stay prediction using MIMIC-III data, and on brain cancer data from The Cancer Genome Atlas. We show that the models we study can be miscalibrated. We give experimental evidence on these datasets that X-CAL improves D-CALIBRATION without a large decrease in concordance or likelihood.

Deep learning models for electrocardiograms are susceptible to adversarial attack.

Electrocardiogram (ECG) acquisition is increasingly widespread in medical and commercial devices, necessitating the development of automated interpretation strategies. Recently, deep neural networks have been used to automatically analyze ECG tracings and outperform physicians in detecting certain rhythm irregularities1. However, deep learning classifiers are susceptible to adversarial examples, which are created from raw data to fool the classifier such that it assigns the example to the wrong class, but which are undetectable to the human eye2,3. Adversarial examples have also been created for medical-related tasks4,5. However, traditional attack methods to create adversarial examples do not extend directly to ECG signals, as such methods introduce square-wave artefacts that are not physiologically plausible. Here we develop a method to construct smoothed adversarial examples for ECG tracings that are invisible to human expert evaluation and show that a deep learning model for arrhythmia detection from single-lead ECG6 is vulnerable to this type of attack. Moreover, we provide a general technique for collating and perturbing known adversarial examples to create multiple new ones. The susceptibility of deep learning ECG algorithms to adversarial misclassification implies that care should be taken when evaluating these models on ECGs that may have been altered, particularly when incentives for causing misclassification exist.