Why cubic splines

That said, there maybe situations in which categorisation of a continuous variable could be helpful. Categorisation also lends itself to visual presentation of the data that are more natural and familiar to most readers, or an outcome could be based on a particular variable falling within a prescribed window.

In any event, we caution the reader to carefully consider the ramifications of categorisation of a continuous variable if, indeed, such categorisation occurs. In conclusion, restricted cubic splines are a flexible tool to model complex, non-linear relationships between a continuous variable and an outcome. In general, categorising continuous variables will lead to loss of information and poor predictions particularly if splitting into only two groups , and this approach should be avoided in most settings [ 7 , 8 ], or at minimum used with caution.

When faced with continuous data, we recommend the exploration of non-linear associations between the continuous variable being modelled and outcome and we argue that a useful method for such exploration is the use of restricted cubic splines. This approach can be implemented with many statistical software programmes currently available e.

This app models various non-linear relationships and compares predictions between a conventional logistic regression model and a model using a restricted cubic spline.

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Correspondence to J. Reprints and Permissions. Gauthier, J. Cubic splines to model relationships between continuous variables and outcomes: a guide for clinicians.

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Download PDF. Subjects Prognosis Translational research. Full size image. Modelling non-linear shapes: example of the surface of an egg. Example 1, Simulated data. Example 2, Real data. References 1. Article PubMed Google Scholar 8. A cubic spline is a spline constructed of piecewise third-order polynomials which pass through a set of control points.

The second derivative of each polynomial is commonly set to zero at the endpoints, since this provides a boundary condition that completes the system of equations. This produces a so-called "natural" cubic spline and leads to a simple tridiagonal system which can be solved easily to give the coefficients of the polynomials. However, this choice is not the only one possible, and other boundary conditions can be used instead. Consider 1-dimensional spline for a set of points. Following Bartels et al.

Taking the derivative of in each interval then gives. Solving 2 - 5 for , , , and then gives. This gives a total of equations for the unknowns. To obtain two more conditions, require that the second derivatives at the endpoints be zero, so. Rearranging all these equations Bartels et al. Bartels, R. This method gives an interpolating polynomial that is smoother and has smaller error than some other interpolating polynomials such as Lagrange polynomial and Newton polynomial.

The spline functions S x satisfying this type of boundary condition are called periodic splines. There are several methods that can be used to find the spline function S x according to its corresponding conditions. Since there are 4n coefficients to determine with 4n conditions, we can easily plug the values we know into the 4n conditions and then solve the system of equations.