In: Statistics and Probability
Patients with high blood pressure were randomly allocated to receive one of two treatments. All patients had been treated with cognitive behaviour therapy (CBT) and were then assigned (at random) to recieve no further treatment (NFT) or beta-blockers (BB). Six weeks later it was noted whether or not each of the patients showed a decrease in their blood pressure. For the NFT group 15 out of 50 patients showed a decrease after 6 weeks, whereas this happened for 26 out of 50 of the BB patients. Assume a Binomial distribution for each group, with individual sample sizes ???? and individuals parameters ???? so that the following model applies: ??1 ∼ Bi(50, ??1) ??2 ∼ Bi(50, ??2) where ??1 and ??2 denote the numbers of patient experiencing a decrease in the NFT and BB groups respectively.
1. Formulate the joint likelihood and joint log-likelihood functions.
2. Find the maximum likelihood estimates for ??1 and ??2.
3. Check that you have found maximum likelihood estimates.
4. Calculate a 95% CI for the difference in proportions (i.e. ??1 − ??2 ) of patients likely to have a decrease between the treatments.
5. What can we say about the probabilities of blood pressure decrease due to the BB treatment in question?