Restricted cubic splines for modelling periodic data

In regression modelling the non-linear relationships between explanatory variables and outcome are often effectively modelled using restricted cubic splines (RCS). We focus on situations where the values of the outcome change periodically over time and we define an extension of RCS that considers periodicity by introducing numerical constraints. Practical examples include the estimation of seasonal variations, a common aim in virological research, or the study of hormonal fluctuations within menstrual cycle. Using real and simulated data with binary outcomes we show that periodic RCS can perform better than other methods proposed for periodic data. They greatly reduce the variability of the estimates obtained at the extremes of the period compared to cubic spline methods and require the estimation of fewer parameters; cosinor models perform similarly to the best cubic spline model and their estimates are generally less variable, but only if an appropriate number of harmonics is used. Periodic RCS provide a useful extension of RCS for periodic data when the assumption of equality of the outcome at the beginning and end of the period is scientifically sensible. The implementation of periodic RCS is freely available in peRiodiCS R package and the paper presents examples of their usage for the modelling of the seasonal occurrence of the viruses.