Survival and Disability Models: Probability Analysis and Rates
School
University of New South Wales**We aren't endorsed by this school
Course
COMM 1140
Subject
Statistics
Date
Dec 12, 2024
Pages
1
Uploaded by CommodoreFalconPerson558
Week 9: Discussion QuestionsQ1Consider the basic survival model with two states: A (alive) and D (dead). In a certain country, theforce of mortality,µx, in the age range 90-105 years is given by:Age range (years)µx90≤x <950.1095≤x <1000.15100≤x <1050.20The head of state sends a congratulatory card on a citizen’s 100th birthday and again on reaching age 105.Find the probability that a person aged exactly 93 will receive a congratulatory card for reaching age 100 butNOT receive a second congratulatory card for reaching age 105.Q2Consider a disability model with three states: Healthy (H), Disabled (DI), and Dead (DE). The transi-tions between these states are governed by a non-homogeneous continuous-time Markov chain. The transitionrates are given as follows:The rate of transitioning from H to DI at timetisλH,DI(t) = 0.02 + 0.001t.The rate of transitioning from DI to DE at timetisλDI,DE(t) = 0.03 + 0.002t.The rate of transitioning from H to DE at timetisλH,DE(t) = 0.01.There are no transitions from DI to H or from DE to any other state.(a) Write down the generator matrixQ(t) for this non-homogeneous continuous-time Markov chain.(b) Given that an individual is Healthy at timet= 0, compute the probability that the individual will stayhealth over [0,10].(c) Compute the probability that an individual who is disabled at timet= 0 will not be alive at timet= 10.(d) Compute the probability that an individual who is healthy at timet= 0 will be disabled at timet= 10.(Hint: You can use R if necessary.)Q3Consider a large study of employment status over a period of time. The total time period individualswere employed was 120,000 years, and the total time period individuals were unemployed was 10,000 years.During the study, 12,000 individuals lost their jobs, and 11,000 unemployed individuals found new jobs. Thenumber of individuals who retired was 200 from the employed group and 100 from the unemployed group.Suppose that retirement is an absorbing state. The transition rate between stateiandjare denoted byλij,wherei, j= Employed(E),Unemployed(U), and Retired(R).(a) Write down the likelihood and derive the maximum likelihood estimates forλEU,λER,λUE, andλUR.(b) Estimate the transition ratesλEU,λER,λUE, andλURand find their standard errors.1
