Artiﬁcial neural network for the joint modelling
of discrete cause-speciﬁc hazards
Elia M. Biganzoli
, Patrizia Boracchi
, Federico Ambrogi
di Statistica Medica e Biometria, Istituto Nazionale Tumori, Milano, Via Venezian 1,
20133 Milano, Italy
Istituto di Statistica Medica e Biometria, Universita
degli Studi di Milano, Italy
Received 9 May 2005; received in revised form 30 December 2005; accepted 11 January 2006
Artiﬁcial Intelligence in Medicine (2006) 37, 119—130
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 estima-
tion 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
* Corresponding author. Tel.: +39 02 23902065; fax: +39 02 50320866.
E-mail address: firstname.lastname@example.org (F. Ambrogi).
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