# Simulation of woman's age of getting breast cancer (cumulative incidence rate)

I am writing a programme to simulate the age at which women will get breast cancer. I have data on the cumulative incidence rate for the whole population.

What I am doing right now is using Monte Carlo methods at every 5 year age step from 0 years old to the age of women have cancer. But this seems stupid and inefficient, because I am dealing with a huge number of women.

One suggestion is to directly use the cumulative incidence rate:

1. Generate a random number from 0 to 1.
2. Find the age which has the same value of probability as this random number, and apply that simulated age for that woman having cancer.

So I have two questions:

1. What are your suggestions to do this process?
2. Is the suggestion I have mentioned above correct? I'm concerned that cumulative incidence rate means the risk of having cancer BEFORE this age rather than AT this age?

## 1 Answer

The cumulative incidence is the probability of developing cancer at or before the given age. The probability of developing cancer at a given age is equal to the cumulative incidence at that age minus the cumulative incidence just before that age (because she has cancer now, but did not before). If you take the time interval small enough (and divide by the size of the time interval) then you are essentially taking a derivative. Finding probabilities for continuous variables involves doing an integral. In this case the derivative and integral cancel each other out so that the method 2 above works.