Proportional Hazards Models

BackgroundMedical practitioners use survival models to explore and understand the relationships between patients' covariates (e.g. clinical and genetic features) and the effectiveness of various treatment options. Standard survival models like the linear Cox proportional hazards model require extensive feature engineering or prior medical knowledge to model treatment interaction at an individual level. While nonlinear survival methods, such as neural networks and survival forests, can inherently model these high-level interaction terms, they have yet to be shown as effective treatment recommender systems.MethodsWe introduce DeepSurv, a Cox proportional hazards deep neural network and state-of-the-art survival method for modeling interactions between a patient's covariates and treatment effectiveness in order to provide personalized treatment recommendations.ResultsWe perform a number of experiments training DeepSurv on simulated and real survival data. We demonstrate that DeepSurv performs as well as or better than other state-of-the-art survival models and validate that DeepSurv successfully models increasingly complex relationships between a patient's covariates and their risk of failure. We then show how DeepSurv models the relationship between a patient's features and effectiveness of different treatment options to show how DeepSurv can be used to provide individual treatment recommendations. Finally, we train DeepSurv on real clinical studies to demonstrate how it's personalized treatment recommendations would increase the survival time of a set of patients.ConclusionsThe predictive and modeling capabilities of DeepSurv will enable medical researchers to use deep neural networks as a tool in their exploration, understanding, and prediction of the effects of a patient's characteristics on their risk of failure.

ObjectivesThe choice of an adequate sample size for a Cox regression analysis is generally based on the rule of thumb derived from simulation studies of a minimum of 10 events per variable (EPV). One simulation study suggested scenarios in which the 10 EPV rule can be relaxed. The effect of a range of binary predictors with varying prevalence, reflecting clinical practice, has not yet been fully investigated.Study design and settingWe conducted an extended resampling study using a large general-practice data set, comprising over 2 million anonymized patient records, to examine the EPV requirements for prediction models with low-prevalence binary predictors developed using Cox regression. The performance of the models was then evaluated using an independent external validation data set. We investigated both fully specified models and models derived using variable selection.ResultsOur results indicated that an EPV rule of thumb should be data driven and that EPV ≥ 20 ​ generally eliminates bias in regression coefficients when many low-prevalence predictors are included in a Cox model.ConclusionHigher EPV is needed when low-prevalence predictors are present in a model to eliminate bias in regression coefficients and improve predictive accuracy.

Artificial neural networks (ANN) are computing architectures with many interconnections of simple neural-inspired computing elements, and have been applied to biomedical fields such as imaging analysis and diagnosis. We have developed a new ANN framework called Cox-nnet to predict patient prognosis from high throughput transcriptomics data. In 10 TCGA RNA-Seq data sets, Cox-nnet achieves the same or better predictive accuracy compared to other methods, including Cox-proportional hazards regression (with LASSO, ridge, and mimimax concave penalty), Random Forests Survival and CoxBoost. Cox-nnet also reveals richer biological information, at both the pathway and gene levels. The outputs from the hidden layer node provide an alternative approach for survival-sensitive dimension reduction. In summary, we have developed a new method for accurate and efficient prognosis prediction on high throughput data, with functional biological insights. The source code is freely available at https://github.com/lanagarmire/cox-nnet.

IntroductionNitrate and nitrite are naturally occurring in both plant- and animal-sourced foods, are used as additives in the processing of meat, and are found in water. There is growing evidence that they exhibit a spectrum of health effects, depending on the dietary source. The aim of the study was to examine source-dependent associations between dietary intakes of nitrate/nitrite and both all-cause and cause-specific mortality.MethodsIn 52,247 participants of the Danish Diet, Cancer and Health Study, associations between source-dependent nitrate and nitrite intakes--calculated using comprehensive food composition and national drinking water quality monitoring databases--and all-cause, cardiovascular disease (CVD)-related, and cancer-related mortality over 27 years were examined using restricted cubic splines within Cox proportional hazards models adjusting for demographic, lifestyle, and dietary confounders. Analyses were stratified by factors hypothesised to influence the formation of carcinogenic N-nitroso compounds (namely, smoking and dietary intakes of vitamin C, vitamin E, folate, and polyphenols).ResultsPlant-sourced nitrate intake was inversely associated with all-cause mortality [HRQ5vsQ1: 0.83 (0.80, 0.87)] while higher risks of all-cause mortality were seen for higher intakes of naturally occurring animal-sourced nitrate [1.09 (1.04, 1.14)], additive permitted meat-sourced nitrate [1.19 (1.14, 1.25)], and tap water-sourced nitrate [1.19 (1.14, 1.25)]. Similar source-dependent associations were seen for nitrite and for CVD-related and cancer-related mortality except that naturally occurring animal-sourced nitrate and tap water-sourced nitrate were not associated with cancer-related mortality and additive permitted meat-sourced nitrate was not associated with CVD-related mortality. No clear patterns emerged in stratified analyses.ConclusionNitrate/nitrite from plant sources are inversely associated while those from naturally occurring animal-sources, additive-permitted meat sources, and tap water-sources are positively associated with mortality.