Deep Neural Network With a Smooth Monotonic Output Layer for Dynamic Risk Prediction

Risk prediction is a key component of survival analysis across various fields, including medicine, public health, economics, engineering, and others. The fundamental concern of risk prediction lies in the joint distribution of risk factors and the time to event. The recent success of survival analysis has already been extended to dynamic risk prediction, which incorporates multiple longitudinal observations into predictive models. However, existing methods often rely on parametric model assumptions or discretely approximate survival functions, potentially introducing more bias in predictions. To address these limitations, we introduce a deep neural network featuring a novel output layer termed the Smooth Monotonic Output Layer (SMOL). This model avoids discretization as well as parametric model assumptions. At its core, SMOL takes a general vector as the input and constructs a monotonic, differentiable function via B-splines. Employing SMOL as the output layer allows for direct, nonparametric estimation of monotonic functions of interest, such as survival and cumulative distribution functions. We performed extensive experiments utilizing data from the Cardiovascular Disease Lifetime Risk Pooling Project (LRPP), which harmonized individual data from multiple longitudinal community-based cardiovascular disease (CVD) studies. Our results demonstrate that the proposed approach achieves state-of-the-art accuracy in predicting individual-level risk for atherosclerotic CVD.