Questions 1,2. All parts.
ISE 505: Modeling for Health Policy and Medical Decision Making Homework 3 1. You would like to perform a probabilistic sensitivity analysis on your discrete time Markov model of health outcomes (a model like in homework 1) with annual transitions. There are three health states (healthy, sick, and very sick). You have the following transition matrix: Where p1, p2, and p3 are transition probabilities. The literature indicates that the mean of p1 is 0.4 and the variance of p1 should be 0.04. You’d like to model each of the parameters (p1, p2, and p3) as a beta distribution. a. What are the alpha and beta parameters associated with the beta distribution for p1? b. Why is the beta distribution commonly used when modeling probabilities? You do some more digging though the literature and find means and 95% confidence intervals for p2 and p3, and from these you calculate that in your PSA it would be appropriate to use p2 ~ beta(4,4) and p3 ~ beta(1.5,2). c. Draw 1000 samples from each of these distributions and plot a histogram of each. d. Your population of 100 people starts in health state 1 ([100 0 0]’). Find the total number of life years lived in after 10 years have passed for each of the 1000 parameter sets of (p1, p2, p3). For simplicity, assume transitions happen at the beginning of the year for all people, and do not consider the half-cycle adjustment or discounting – although these would be concerns in a real analysis. How many people are healthy, on average over your 1000 samples, at the end of the analysis period? e. The QALY weights for these health states are [ 1 0.8 0.2] and the costs are [1 1.4 3] (in thousands). What are the total QALYs and costs over the analysis period, averaged over the 1000 samples? f. Suppose there is an intervention that improves the quality of life for health state 3 so the QALY weights become [1, 0.8, 0.7] and additional costs are incurred: cost weights of [1, 1.4, 5]. Plot a scatterplot of the cost-QALY plane, showing the base case and the intervention using the output of each of the 1000 samples. g. Find the net monetary benefit of the base case and the intervention over all the samples and plot the cost-effectiveness acceptability curve. 2. A variety of vaccinations require follow up shots – for instance, Hepatitis A and B vaccinations (Twinrix) vaccinations require three doses, with the second dose roughly 1 month after the first dose and the third roughly 6 months after the first dose. Therefore vaccination clinics often have to schedule follow-up visits for patients getting follow-up doses as well as serving walk-in patients getting their first dose. A vaccination clinic in LA is considering how to schedule follow up visits and staff the clinic. Since their capacity is limited, they can divert patients requiring follow-up doses to the nearby Keck vaccination clinic, but they cannot turn away new patients. They can choose how many follow-up visits to schedule, but the average number of walk-in, new patients they serve every day is given below: Time Number of Walk-ins 8am 1 9am 4 10am 6 11am 3 12pm 2 1pm 5 2pm 6 3pm 7 4pm 3 5pm 0 There are five vaccination technicians available to be hired for $125 an hour. They have agreed to work a total of 35 person-hours a day between the five of them. There must have a minimum of two technicians at the clinic during operating hours (8am-5pm). Each technician can administer 2 vaccinations an hour. The city of LA public health department will reimburse the clinic $150 per new patient and $60 for each follow-up dose that a patient receives. The clinic charges follow-up patients an additional $10 between 11am – 1pm and $5 after 2pm since these are popular patient hours and they have historically had too many follow-up patients to handle. Despite this, they still have far too many patients and divert many patients to Keck every hour. a. Suppose the clinic wishes to maximize profits. What optimization problem would you solve? Explain what each constraint means. Hint: make the decision variables the number of follow-up patients to schedule at every hour and the number of technicians to schedule every hour. b. Relax the problem to assume that you can hire non-integer number of technicians every hour. How many technicians will they hire for each hour, on average? What profit will they make? c. How many patients would they see, on average? d. If they could convince the five technicians to work a maximum of 50 person-hours in total (instead of 35), how much profit would they make? e. How many more patients would they then serve? f. If their goal was to see as many patients as possible instead of maximizing profit, how much money would they lose (with 50 person-hours)?