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Modelling time dependent hazard ratios in relative survival: application to colon cancer.

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## Modelling time dependent hazard ratios in relative survival: application to colon cancer.

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**Modelling time dependent hazard ratios in relative**survival: application to colon cancer. BOLARD P, QUANTIN C, ABRAHAMOWICZ M, ESTEVE J, GIORGI R, CHADHA-BOREHAM H, BINQUET C, FAIVRE J.**INTRODUCTION** Previous results of the Flexible generalisation of the Cox model • - The PH hypothesis does not hold for most prognostic factors for all-causes mortality in colon cancer • Some of these effects may reflect the inability of the method to separate cancer-related mortality from all-causes mortality • - Analyses of our BRDC colon cancer data require simultaneous modelling of both relative survival and possibly non-proportional hazards.**METHODS** PH Relative survival model of Esteve et al. Non PH Relative survival models piecewise PH model parametric time-by-covariate interaction non-parametric time-by-covariate (spline)**PIECEWISE PH MODEL** For the k-th time-segment, k = 1 … r Test of the PH: j2 = j3 = …… jr = 0**PARAMETRIC TIME-BY-COVARIATE INTERACTION** For the k-th time-segment**CUBIC SPLINE FUNCTIONS FOR MODELLING**TIME-BY-COVARIATE INTERACTIONS**RESTRICTED CUBIC SPLINE FUNCTIONS FOR MODELLING**TIME-BY-COVARIATE INTERACTION**TESTS** Any type of dependence with time j1 = j2 = 0 Non linear dependence j2 = 0 Effect of covariate Zj j0 = j1 = j2 = 0**NUMBER OF KNOTS AND THEIR LOCATION** Number: can be restricted between 3 and 5 knots in most cases [Stone ] 3 knots. Location: * both - quantiles of the distribution function of deaths. - percentiles of the distribution function of the follow-up times. * In our restricted cubic spline model, we cannot fix the knots too near the extremes because of the linearity constraints. 5th,50th and 95th quantiles**APPLICATION: PATIENTS**2075 cases of colon cancer diagnosed between 76 and 90 (Burgundy Registry of Digestive Cancers) end of follow-up: December 31, 1994. 1334 deaths at 5 years Median survival time of 12 months** Prognostic factors:*** gender * age (< 65, 65-74, 75) * periods of diagnosis (76-78, 79-81, 82-84, 85-87, 88-90) * cancer TNM stage**Comparison of crude (Cox model) and relative survival**(Esteve model) Proportional Hazard model in multivariate analyses Click for larger picture**Testing the Proportional Hazard assumption in multivariate**Relative Survival analysis Click for larger picture**Change of the Hazard Ratio associated to age (reference**category: < 65 years) using piecewise Proportional Hazard models in crude and relative survival Click for larger picture**Test of proportional hazard assumption obtained with model 3**using restricted cubic spline functions for modelling different time-by-covariate interactions. Click for larger picture**1,50**0,00 -1,50**CONCLUSION** Both flexible modelling of non-proportional hazards and the relative survival approach are important: differences between relative survival and the conventional Cox model. Restricted cubic spline model * better fit than a linear time-by-covariate interaction * more parsimonious than a piecewise PH relative survival model * allows to represent both simple and complex patterns of changes Number and the location of knots