Nonlinear Models

So we see these four forms of models have the following primary characteristics. The actual differences will depend on how well each accounts for the actual relationship. For example, the linear model will be best for representing a linear relationship, but terrible for representing a U shaped curve.

• Linear: Interpretability. Efficient, easy to communicate, and supported by almost all modeling packages. Accuracy may be problematic for some relationships. This is the continuous model seen most (almost exclusively) in the medical literature (but see categories).
  
• Categories: Interpretability. Efficient, easy to communicate and supported by almost all modeling packages. This can be an effective way to handle nonlinear responses and is frequently used in the medical literature (e.g. blood pressure ranges). Requires selection of the cut points and error may be large near the boundaries).
  
• Spline: Potentially a very accurate (or over) fit. Useful for checking fine grained behavior with respect to a variable (e.g. to evaluate accuracy of FP, linear, or categories models). Available in coxph.
  
• Fractional Polynomials: Can give a more accurate fit than linear while limiting model complexity. The coefficients are not interpretable, but the hazard ratio plot above is. Availability is limited (I know of support in the mfp, rms, and coxphw R packages. mfp was used separately to determine the FPs used in this document.)

Fractional polynomials is a somewhat obscure technique so here is a reference:
Multivariable regression model building by using fractional polynomials: Description of SAS, STATA and R programs [@Sauerbrei_2006]

Here is some discussion of splines versus fractional polynomials:
Polynomials and Splines: Fractional polynomials example data set

For comparisons of Penalized Splines (used above), Fractional Polynomials, and Restricted Cubic Splines see:
Comparing smoothing techniques in Cox models for exposure-response relationships [@Govindarajulu_2007]
Application of Smoothing Methods for Determining of the Effcting Factors on the Survival Rate of Gastric Cancer Patients [@Noorkojuri_2013]

Based on the hazard ratio plots and AIC results above, for the Age variable I would use the linear model for maximum interpretability, the spline model for the most accurate fit, or the fractional polynomials model for the best combination of accuracy and model simplicity/efficency.