Question

An oncologist believes that 90% of cancer patients will respond to a new chemotherapy treatment and that it is unlikely that this proportion will be below 80%.

(a) Determine a beta prior on the response proportion which models the oncologist’s beliefs. (Hint: Find the beta which has mean µ = 0.9 and µ−2σ = 0.8, where σ is the standard deviation of the desired beta. Relate these conditions to the parameters defining a beta distribution.) During a trial, 30 patients were treated and 22 responded.

(b) Find the posterior distribution on the proportion.

(c) Give the MAP and Bayes estimators of the proportion.

(d) Find and interpret the 95% equi-tailed credible set for p and the 95% HDI.

(e) Find and interpret the posterior odds ratio and Bayes factor for the hypotheses

H0 : p ≥ 4/5 and H1 : p < 4/5 .

Answer 1