Flexible hazard ratio curves for continuous predictors in multi-state models: an application to breast cancer data

Multi-state models (MSMs) are very useful for describing complicated event history data. These models may be considered as a generalization of survival analysis where survival is the ultimate outcome of interest but where intermediate (transient) states are identified. One major goal in clinical applications of MSMs is to study the relationship between the different covariates and disease evolution. Usually, MSMs are assumed to be parametric, and the effects of continuous predictors on log-hazards are modelled linearly. In practice, however, the effect of a given continuous predictor can be unknown, and its form may be different in all transitions. In this paper, we propose a P-spline approach that allows for non-linear relationships between continuous predictors and survival in the multi-state framework. To better understand the effects at each transition, results are expressed in terms of hazard ratio curves, taking a specific covariate value as reference. Confidence bands for these curves are also derived. The proposed methodology was applied to a database on breast cancer, using a progressive three-state model. This application revealed hitherto unreported effects: whereas DNA index is only an important non-linear predictor of recurrence, the percentage of cells in phase S is a significant predictor of both recurrence and mortality.