I like your idea of doing a back-of-the-envelope calculation assuming that there are just two risks. Suppose that risk #1 has survival probability s1 and risk #2 has survival probability s2. Assume s1 < s2. Let’s compare two alternative interventions:
Intervention 1: Increase the survival probability for risk #1 by some small amount x.
Intervention 2: Increase the survival probability for risk #2 by that same amount x.
Now we have:
Intervention 1 will increase the overall survival probability from s1*s2 to (s1+x)*s2. That is an increase of x*s2.
Intervention 2 will increase the overall survival probability from s1*s2 to s1*(s2+x). That is an increase of x*s1.
So the increase in overall survival probability produced by intervention #1 is 100 * (s2/s1 − 1) percent greater than the increase in overall survival probability produced by intervention #2.
Plugging in some toy numbers:
Suppose that risk #1 has survival probability 90% and risk #2 has survival probability 99%. Then an intervention that increases s1 by a small amount will produce 10% more increase in overall survival probability than an intervention that increases s2 by that same small amount.
If risk #1 had survival probability 80% instead, then intervention 1 would produce approximately 24% more increase in overall survival probability than intervention 2.
So a risk’s base survival probability does make some difference to how much one increases overall survival probability by mitigating that risk. But overall I agree with you that for risks with survival probability greater than 90%, the difference is modest.
I like your idea of doing a back-of-the-envelope calculation assuming that there are just two risks. Suppose that risk #1 has survival probability s1 and risk #2 has survival probability s2. Assume s1 < s2. Let’s compare two alternative interventions:
Intervention 1: Increase the survival probability for risk #1 by some small amount x.
Intervention 2: Increase the survival probability for risk #2 by that same amount x.
Now we have:
Intervention 1 will increase the overall survival probability from s1*s2 to (s1+x)*s2. That is an increase of x*s2.
Intervention 2 will increase the overall survival probability from s1*s2 to s1*(s2+x). That is an increase of x*s1.
So the increase in overall survival probability produced by intervention #1 is 100 * (s2/s1 − 1) percent greater than the increase in overall survival probability produced by intervention #2.
Plugging in some toy numbers:
Suppose that risk #1 has survival probability 90% and risk #2 has survival probability 99%. Then an intervention that increases s1 by a small amount will produce 10% more increase in overall survival probability than an intervention that increases s2 by that same small amount.
If risk #1 had survival probability 80% instead, then intervention 1 would produce approximately 24% more increase in overall survival probability than intervention 2.
So a risk’s base survival probability does make some difference to how much one increases overall survival probability by mitigating that risk. But overall I agree with you that for risks with survival probability greater than 90%, the difference is modest.
Thanks for the elaboration, Adam! That makes sense.