The comparison of fitted non-linear exposure-response relationships in Cox models using smoothing methods through simulations
by
Usha Govindarajulu
Brigham and Women's Hospital, Harvard Medical School
Coauthors: Elizabeth Malloy (American University); Bhaswati Ganguli (University of Calcutta); Donna Speigelman (Harvard School of Public Health); Ellen Eisen, (Harvard School of Public Health)

Background: We examined the behavior of three alternative splines and fractional polynomial for modeling environmental epidemiology data. Model fit can only be examined when the true exposure-response curve is known. Therefore, we examined performance of p-splines, restricted cubic splines, natural splines, and fractional polynomials in simulation studies.

Methods: Survival data were generated under six plausible exposure-response scenarios: null, linear, sine, threshold, logarithmic and quadratic, with a right skewed exposure distribution. Cox models with each spline or fractional polynomial were fit to 1000 datasets, of size 2000, for each scenario. Model fit was measured by RMSE. Relative bias was defined by the area difference between the fitted and true curves.

Results: For a true null, linear, or quadratic function, fractional polynomial had a smaller RMSE compared to the splines (psplines, NS, RCS) and overall smallest IQR. The p-splines (either selected by AIC or df = 4) had the second smallest RMSE and performed the best for log function.. Under a threshold model, RCS fit best. Fewer poorly fitting P-splines were selected by 4 degrees of freedom (df ) criterion than by AIC. Under the null, p-splines selected by 4 df rejected the null hypothesis slightly more than natural splines or RCS, but close to 5% of the time. When the true curve was nonlinear, the power to reject linearity was better for p-splines than for NS or RCS. There was not much difference in magnitude of the bias across different splines fit to the same true curve, although the bias did vary by the nature of the curve, from 10% for linear or threshold, to 20% for quadratic, to 30% for a logarithmic curve.

Conclusions: All methods performed well for the typical exposure-response scenarios examined, but p-splines (4 df) fit the data best overall. The model fit was best (lowest median RMSE and IQR) for all true positive scenarios except the threshold model. When non-linearity exists, the model selected by AIC rejected linearity more often than the others. Despite some advantages, AIC selected more poor fitting models than the fixed degrees of freedom default (df=4) in the scenarios considered here.

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Date received: July 19, 2007