Math and Medicine: Homework Assignment 4
Due on February 5
1. The Welch paper more or less assumes that mortality is constant (and thus survival is
exponential) over relatively short periods of time. Suppose this is reasonable, but that an
unmeasured covariate creates different values of m for two groups of people of the same age.
a. Pick two different but reasonable values of m and a reasonable value of md (mortality
due to the disease) and find the treatment benefit for each group.
b. If we didn’t know about the groups, we would only know the average survival time
for the whole population. Supposing that the two groups are equal in size, find the
expected lifespan for the population, and the resulting value of m.
c. What is the treatment benefit? Does the unknown heterogeneity make much of a
difference?
d. Discuss how you would include one additional factor that deviates from this idealized
scenario.
2. The program welch.R on the web site simulates the process in the paper using the Gompertz
survival curve, for which the hazard as a function of age is
h(a) = h0 eba .
h0 is a baseline hazard, and b describes how fast hazard changes with age.
a. Set b = 0. Find the value of h0 for which the mean lifespan is close to that with the
basic parameters (mean lifespan = 63.7). How do the curves change? What does this
mean? Do you have any idea why the expected remaining lifespan curve isn’t constant
like it should be?
b. Now make b larger and find a value of h0 that again produces a mean lifespan of around
63.7. How does this differ from the basic case? Does anything go wrong?
c. Finally, throw caution to the winds and make b negative (but not too big), and try to
find a value of h0 that produces the mean lifespan of 63.7. How does this differ from
the basic case? Does anything go wrong?