How to calculate per year risk given risk over 6 years

Question

Risk of death in group A over 6 years is 2%. Risk of death in group B over 6 years is 6%. How do I calculate the risk of death per year in each group?

I am modeling the time to event in a cancer treatment group. I am using SAS.

Answer 1

You have to assume a constant hazard of death within each group to get a single value for "the risk of death per year in each group." That means that you are assuming an exponential model for survival. You might want to look at this page for discussion of when and whether that might be a reasonable assumption. More typically, the risk of death changes over time.

If such a model is nevertheless OK, the cumulative probability of an event over time (restricted to be $t \ge0$) is characterized by a single parameter $\lambda$:

$$F(t;\lambda) = 1- \exp(-\lambda t).$$

You know that at $t$ = 6 years, group A has a fractional cumulative risk of $F_A(6)=0.02$, while group B has a fractional cumulative risk of $F_B(6)=0.06$. Plug those values into the above formula to get corresponding values of $\lambda$, in units of per year. I get 0.00337 for group A and 0.01031 for group B. Those are the rate coefficients in units of risk per year, a standard way to present such results. (Those values aren't exactly what you would get for the risk after exactly 1 year if you plugged those coefficients back into the formula above, although they are very close in this case.)

Note that your survival curves seem to have very few events, so your estimates of 6-year survival are probably very inexact.