Computational Statistics & Data Analysis 42 (2003) 243–262 www.elsevier.com/locate/csda Joint modelling of cause-specic hazard functions with cubic splines: an application to a large series of breast cancer patients Patrizia Boracchi a , Elia Biganzoli b; ∗ , Ettore Marubini a; b a Istituto di Statistica Medica e Biometria, Universit a degli Studi di Milano, Italy b Unit a di Statistica Medica e Biometria, Istituto Nazionale per lo Studio e la Cura dei Tumori, Via G.Venezian 1, 20133, Milano, Italy Received 1 November 2000; received in revised form 1 February 2002 Abstract The time of appearance of several kinds of relapses after a therapeutic intervention is of increasing interest in oncology. Typically, in breast cancer patients, events of clinical interest are intra-breast tumor recurrences and distant metastases, which act in a competitive way when considered as rst failure. The evaluation of dierential eects of clinical and biological vari- ables on each event can improve the knowledge on the course of the disease and the targeting of future therapy. A simple tool for the joint smoothed estimation of cause-specic hazards func- tions and continuous covariate eects has been developed. Within the framework of generalized linear models with Poisson error, an extension of the piecewise exponential model is proposed, based on grouping follow-up times and continuous covariates. Interpolation of cause-specic hazards is obtained by resorting to cubic splines, which are piecewise polynomials of simple implementation with standard statistical software; their exibility and smoothness are easily con- trolled by the number of knots and constraints on polynomial derivatives. The approach was applied to a data set of 2233 breast cancer patients treated with conservative surgery. It al- lowed modelling time-dependent and cause-specic eects of covariates on the hazard functions. c 2002 Published by Elsevier Science B.V. Keywords: Competing risks; Hazard regression; GLMs; Splines; Breast cancer * Corresponding author. Tel.: +39-0223903203; fax: +39-022362930. E-mail address: biganzoli@institutotumori.mi.it (E. Biganzoli). 0167-9473/02/$-see front matter c 2002 Published by Elsevier Science B.V. PII:S0167-9473(02)00122-6