Artificial neural network for the joint modelling of discrete cause-specific hazards

Objective: Artificial neural network (ANN) based regression methods have been introduced for modelling censored survival data to account for complex prognostic patterns. In the framework of ANN extensions of generalized linear models for survival data, PLANN is a partial logistic ANN, suitable for smoothed discrete hazard estimation as a function of time and covariates. An extension of PLANN for competing risks analysis (PLANNCR) is now proposed for discrete or grouped survival times, resorting to the multinomial likelihood. Methods and materials: PLANNCR is built by assigning input nodes to the explanatory variables with the time interval treated as an ordinal variable. The logistic function is used as activation for the hidden nodes of the network, whereas the softmax, which corresponds to the canonical link of generalized linear models for polytomous regression, is adopted for multiple output nodes, to provide a smoothed estimation of discrete conditional event probabilities for each event. The Kullback-Leibler distance is used as error function for the target vectors, amounting to half of the deviance of a multinomial logistic regression model. PLANNCR can jointly model non-linear, non-proportional and non-additive effects on cause-specific hazards (CSHs). The degree of smoothing is modulated by the number of hidden nodes and penalization of the error function (weight decay). Model optimisation is achieved by quasi-Newton algorithms, while non-linear cross-validation (NCV) and the Network Information Criterion (NIC) were adopted for model selection. PLANNCR was applied to data on 1793 women with primary invasive breast cancer, histologically N-, who underwent surgery at the Milan Cancer Institute between 1981 and 1986. Results: Differential effects of covariates and time on the shape of the CSH for the three main failure causes, namely intra-breast tumor recurrences, distant metastases and contralateral breast cancer, have been enlightened. Conclusions: PLANNCR can be suitably adopted in an exploratory framework for a thorough evaluation of the disease dynamics in the presence of competing risks.