Question

Please do all parts and show all calculations and R code if used.

3.3 Tumor counts: A cancer laboratory is estimating the rate of tumorigenesis in two strains of mice, A and B. They have tumor count data for 10 mice in strain A and 13 mice in strain B. Type A mice have been well studied, and information from other laboratories suggests that type A mice have tumor counts that are approximately Poisson-distributed with a mean of 12. Tumor count rates for type B mice are unknown, but type B mice are related to type A mice. The observed tumor counts for the two populations are (12.9. 12. 14. 13. 13. 15, 8. 15. 6); (11. 11. 10. 9. 9. 8. 7. 10, 6, 8. 8. 9. 7) УА ув a) Find the posterior distributions, means, variances and 95% quant ile- based confidence intervals for θΑ and θΒ, assuming a Poisson sampling distribution for each group and the following prior distribution: θΑ ~ gamma( 120. 10), θΒ ~ gamma( 12.1) , p@a. θ) = p@a ) × p(θΒ ) b) Compute and plot the posterior expectation of θΒ under the prior dis- tribution θв ~ gamma( 12x no, no) for each value of n0 E { 1.2, . . . . 50} Describe what sort of prior beliefs about θв would be necessary in or- der for the posterior expectation of θΒ to be close to that of θΑ c) Should knowledge about population A tell us anything about popu- lation B? Discuss whether or not it makes sense to have p(eA, 0B) - p@A) × p@B)

Answer 1