Models for predicting the time of a future event are crucial for risk assessment, across a diverse range of applications. Existing time-to-event (survival) models have focused primarily on preserving pairwise ordering of estimated event times, or relative risk. Model calibration is relatively under explored, despite its critical importance in time-to-event applications. We present a survival function estimator for probabilistic predictions in time-to-event models, based on a neural network model for draws from the distribution of event times, without explicit assumptions on the form of the distribution. This is done like in adversarial learning, but we achieve learning without a discriminator or adversarial objective. The proposed estimator can be used in practice as a means of estimating and comparing conditional survival distributions, while accounting for the predictive uncertainty of probabilistic models. Extensive experiments show that the proposed model outperforms existing approaches, trained both with and without adversarial learning, in terms of both calibration and concentration of time-to-event distributions.
Existing calibration literature in predictive models has primarily focused on recalibration techniques for predictions from classification or regression models . For classification tasks, the Brier score is a commonly used proper score metric, quantifying the accuracy of probabilistic predictions, and thus it is often used to assess calibration. The Brier score has also been used to asses calibration in time-to-event models @cite_0 @cite_14 @cite_2 @cite_11 , however, this score has to be evaluated at pre-specified (thresholded) time horizons. Alternatively, S-CRPS @cite_12 considers the integral of the Brier score evaluated at all possible thresholds @cite_13 , which is a more principled and comprehensive approach than calibration at pre-specifying time horizon thresholds.
@cite_14 @cite_0 @cite_2 @cite_13 @cite_12 @cite_11
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Censoring is a common phenomenon that arises in many longitudinal studies where an event of interest could not be recorded within the given time frame. Censoring causes missing time-to-event labels, and this effect is compounded when dealing with datasets which have high amounts of censored instances. In addition, dependent censoring in the data, where censoring is dependent on the covariates in the data leads to bias in standard survival estimators. This motivates us to develop an approach for pre-processing censored data which calibrates the right censored (RC) times in an attempt to reduce the bias in the survival estimators. This calibration is done using an imputation method which estimates the sparse inverse covariance matrix over the dataset in an iterative convergence framework. During estimation, we apply row and column-based regularization to account for both row and column-wise correlations between different instances while imputing them. This is followed by comparing these imputed censored times with the original RC times to obtain the final calibrated RC times. These calibrated RC times can now be used in the survival dataset in place of the original RC times for more effective prediction. One of the major benefits of our calibration approach is that it is a pre-processing method for censored data which can be used in conjunction with any survival prediction algorithm and improve its performance. We evaluate the goodness of our approach using a wide array of survival prediction algorithms which are applied over crowdfunding data, electronic health records (EHRs), and synthetic censored datasets. Experimental results indicate that our calibration method improves the AUC values of survival prediction algorithms, compared to applying them directly on the original survival data. We propose Lomax delegate racing (LDR) to explicitly model the mechanism of survival under competing risks and to interpret how the covariates accelerate or decelerate the time to event. LDR explains non-monotonic covariate effects by racing a potentially infinite number of sub-risks, and consequently relaxes the ubiquitous proportional-hazards assumption which may be too restrictive. Moreover, LDR is naturally able to model censoring and missing event times or types. For inference, we develop a Gibbs sampler under data augmentation for moderately sized data, along with a stochastic gradient descent maximum a posteriori inference algorithm for big data applications. Illustrative experiments are provided on both synthetic and real datasets, and comparison with various benchmark algorithms for survival analysis with competing risks demonstrates distinguished performance of LDR. Scoring rules assess the quality of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and on the event or value that materializes. A scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionF if he or she issues the probabilistic forecast F, rather than G ≠ F. It is strictly proper if the maximum is unique. In prediction problems, proper scoring rules encourage the forecaster to make careful assessments and to be honest. In estimation problems, strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand. This article reviews and develops the theory of proper scoring rules on general probability spaces, and proposes and discusses examples thereof. Proper scoring rules derive from convex functions and relate to information measures, entropy functions, and Bregman divergences. In the case of categorical variables, we prove a rigorous version of the ... Personalized probabilistic forecasts of time to event (such as mortality) can be crucial in decision making, especially in the clinical setting. Inspired by ideas from the meteorology literature, we approach this problem through the paradigm of maximizing sharpness of prediction distributions, subject to calibration. In regression problems, it has been shown that optimizing the continuous ranked probability score (CRPS) instead of maximum likelihood leads to sharper prediction distributions while maintaining calibration. We introduce the Survival-CRPS, a generalization of the CRPS to the time to event setting, and present right-censored and interval-censored variants. To holistically evaluate the quality of predicted distributions over time to event, we present the Survival-AUPRC evaluation metric, an analog to area under the precision-recall curve. We apply these ideas by building a recurrent neural network for mortality prediction, using an Electronic Health Record dataset covering millions of patients. We demonstrate significant benefits in models trained by the Survival-CRPS objective instead of maximum likelihood. The importance of survival analysis in many disciplines (especially in medicine) has led to the development of a variety of approaches to modeling the survival function. Models constructed via various approaches o er di erent strengths and weaknesses in terms of discriminative performance and calibration, but no one model is best across all datasets or even across all time horizons within a single dataset. Because we require both good calibration and good discriminative performance over di erent time horizons, conventional model selection and ensemble approaches are not applicable. This paper develops a novel approach that combines the collective intelligence of different underlying survival models to produce a valid survival function that is well-calibrated and o ers superior discriminative performance at di erent time horizons. Empirical results show that our approach provides signi cant gains over the benchmarks on a variety of real-world datasets.