Comparing smoothing techniques in Cox models for exposure–response relationships

To allow for non-linear exposure-response relationships, we applied flexible non-parametric smoothing techniques to models of time to lung cancer mortality in two occupational cohorts with skewed exposure distributions. We focused on three different smoothing techniques in Cox models: penalized splines, restricted cubic splines, and fractional polynomials. We compared standard software implementations of these three methods based on their visual representation and criterion for model selection. We propose a measure of the difference between a pair of curves based on the area between them, standardized by the average of the areas under the pair of curves. To capture the variation in the difference over the range of exposure, the area between curves was also calculated at percentiles of exposure and expressed as a percentage of the total difference. The dose-response curves from the three methods were similar in both studies over the denser portion of the exposure range, with the difference between curves up to the 50th percentile less than 1 per cent of the total difference. A comparison of inverse variance weighted areas applied to the data set with a more skewed exposure distribution allowed us to estimate area differences with more precision by reducing the proportion attributed to the upper 1 per cent tail region. Overall, the penalized spline and the restricted cubic spline were closer to each other than either was to the fractional polynomial.