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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?