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LANCASTER UNIVERSITY
2010 EXAMINATIONS
PART II (Third year)
MATHEMATICS & STATISTICS 1 12 hours
Math 353: Survival and Reliability
In this module you should answer All Section A questions and 1 Section B question In section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40.
SECTION A
A1. A lifetime random variable T has a distribution function F (t) = P(T ≤ t). (a) Give a real-life example of a lifetime random variable T and state two features of the variable that should be taken into account in a subsequent analysis. [4] (b) Define the survivor function S(t) and show that S′(t) = −f (t), where f is the probability density function of T. [4] (c) Define the hazard function h(t) and the integrated hazard function H(t) of T. [4] (d) Show that the survivor function S(t) and the integrated hazard function H(t) are related by S(t) = exp{−H(t)}. (1) [6] (e) Using the relation in (1), show that the random variable H(T ) follows an Exponential distribution. [4] A2. In business mortality studies, it is believed that the early years of the business are the most difficult with most types of business. The longer a business survives, generally, other things being equal, the smaller becomes the probability of failure.
(a) Write down the instantaneous failure rate at time t conditional on the survival to time t in terms of the derivative of a conditional probability and show that the instantaneous failure rate is equal to the hazard function h(t). [5] (b) Show that h(t) is a constant, λ, if and only if T ∼ Exp(λ). [6]
(c) Which of the hazard functions in Figure 1 might be appropriate to describe the phe- nomenon of business mortality? Explain your reasoning. [3]
0 2 4 6 8 10
(a)
0 2 4 6 8 10
(b)
0 2 4 6 8 10
(c)
0 2 4 6 8 10
(d)
Figure 1: examples of hazard function (d) In comparison to your choice of hazard function, explain what other types of business mortality could be decribed by the remaining hazard functions. [6] (e) Suppose that h(t) = (^) t+1^1 , t ≥ 0. Find the corresponding density function f (t). [4]
A3. State assumptions of the Cox proportional hazard model. [4]
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SECTION B continued
B2. (a) The survival time of patients receiving either of two treatments for ovarian cancer are recorded. The covariates are age (in days/365.25) and treat that states which of two treatment categories each patient was allocated (1 = treatment, 0 = control). Below is the summary of the results of fitting various models.
Model Error dist. Covariates intercept age treat age*treat scale(1/α) 1 Exp. none 7. 2 Weibull none 7.111 0. 3 Exp. age 13.934 -0. 4 Exp. age, treat 12.123 -0.105 0. 5 Exp. age, treat, age∗treat 3.202 0.0445 8.917 -0. 6 Weibull age 12.397 -0.096 0. Null model Alternative model Likelihood ratio statistic 1 2 0. 1 3 12. 3 4 1. 3 5 2. 3 6 3. (i) From these results alone, state which model you would consider as appropriate for this data, explaining how you carry out likelihood ratio tests to guide your choice. [6] (ii) Write down the fitted model of your choice and interpret the results. [4] (iii) Suggest a diagnostic method and explain how you would apply your diagnostics. [4] (b) (i) Show that the Cox proportional hazard model can be written in terms of cumulative hazard function as H(t|x) = H 0 (t)g(x) where H 0 is the baseline cumulative hazard function. [4] (ii) The lifetimes Ti ∼ Exp(λi), i = 1,... , n where λi = exp(β′xi) for covariates x. Let m(·) be a monotone increasing function with m(0) = 0 defined on t > 0. (1) Show that m(Yi) follows a Cox proportional hazard model with baseline cumu- lative hazard function m−^1 (t). Write down the Cox proportional hazard model when m(t) = t^1 /^2 , t > 0 and identify baseline hazard function. [6]
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SECTION B continued
Question B2 continued
(2) Assume that the only covariate is the group indicator variable where xi = 1 if the subject is in treatment group and xi = 0 if the subject is in control group. Using the above result, assuming no censoring, suggest a method to simulate lifetime variables that follow a Cox-proportional hazard model with baseline hazard function h 0 (t) = 2t. [6]
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Gamma distribution:
lifetime T ∼ Gamma(α, λ) λ > 0 , α > 0 , t > 0
pdf f (t) = λαtα−^1 exp(−λt)/Γ(α)
moments E[T ] = α/λ, var [T ] = α/λ^2 ,
sums X 1 ,... , Xn iid Exp(λ) ⇒ T = ∑ni=1 Xi ∼ Gamma(n, λ).
