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

Abstract

Multi-state models (MSMs) are very useful for describing complicated event history data. These models may be considered 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 multi-state models 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 modeled linearly. In practice, however, the effect of a given continuous predictor can be unknown, and its form may be different in all permitted 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 that each continuous covariate has on the outcome 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, and the results were compared against those obtained through the traditional Cox regression model. This application revealed hitherto unreported effects: whereas DNA index is only an important nonlinear predictor of recurrence, the percentage of cells in phase S is a significant predictor of both recurrence and mortality. All analyses were performed using software written by the authors.