1 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.

2 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.

3 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)

4 PH Relative survival model of Esteve

5  For the k-th time-segment, k = 1 … r PIECEWISE PH MODEL Test of the PH:  j2 =  j3 = ……  jr = 0

6 PARAMETRIC TIME-BY- COVARIATE INTERACTION  For the k-th time-segment

7 CUBIC SPLINE FUNCTIONS FOR MODELLING TIME-BY-COVARIATE INTERACTIONS

8 RESTRICTED CUBIC SPLINE FUNCTIONS FOR MODELLING TIME-BY- COVARIATE INTERACTION

9 TESTS  Any type of dependence with time  j 1 =  j 2 = 0  Non linear dependence  j 2 = 0  Effect of covariate Zj  j 0 =  j 1 =  j 2 = 0

10 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.  5 th, 50 th and 95 th quantiles

11 APPLICATION: PATIENTS 2075 cases of colon cancer diagnosed between 76 and 90 (Burgundy Registry of Digestive Cancers) end of follow-up: December 31, deaths at 5 years Median survival time of 12 months

12  Prognostic factors: * gender * age (< 65, 65-74,  75) * periods of diagnosis (76-78, 79-81, 82-84, 85-87, 88-90) * cancer TNM stage

13 RESULTS

14 Comparison of crude (Cox model) and relative survival (Esteve model) Proportional Hazard model in multivariate analyses Click for larger picture

15 Testing the Proportional Hazard assumption in multivariate Relative Survival analysis Click for larger picture

16 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

17 Age

18 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

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21 1,50 -1,50 0,00

22 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