were derived by the non-parametric bootstrap method, with 5000 samples taken from the original data set by sampling with replacement. We then established the cumulative incidence curves of dementia and of death under different sets of $a$ and $a^*$. These curves reflect the cumulative failure rates over time for a particular cause (e.g. dementia), acounting for the presence of other competing events (e.g. death). To estimate the curves, we considered dementia and death as two terminal states of a multi-state model where the transition from dementia to death was treated as an absorbing state, i.e. the one that subjects never exist [@putter07]. For pedagogic purposes, we only use the results of the natural effect model (1) (i.e. without interaction between $a$ and $a^*$) to establish these curves.

   Model  Coefficient             Estimate        95%CI        p-value  
  ------- ---------------------- ---------- ----------------- --------- --
          **Dementia** ($j=1$)                                          
          $\alpha_{1j}$            0.599     (0.090; 1.001)     0.007   
          $\alpha_{2j}$            0.012     (-0.014; 0.051)    0.438   
          **Death** ($j=2$)                                             
          $\alpha_{1j}$            -0.126    (-0.483; 0.143)    0.462   
          $\alpha_{2j}$            0.001     (-0.013; 0.011)    0.913   
          **Dementia** ($j=1$)                                          
          $\alpha_{1j}$            0.591     (0.093; 1.003)     0.007   
          $\alpha_{2j}$            0.002     (-0.012; 0.017)    0.827   
          $\alpha_{3j}$            0.017     (-0.023; 0.079)    0.520   
          **Death** ($j=2$)                                             
          $\alpha_1$               -0.121    (-0.480; 0.148)    0.485   
          $\alpha_2$               0.005     (-0.011; 0.023)    0.584   
          $\alpha_3$               -0.009    (-0.049; 0.022)    0.632   

  : Data analysis: estimation 