Week 4 Lecture Slides (1)

Maximum is our estimate for t k exp zj l

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Unformatted text preview: exp β ZiT i R (tj ) = exp β Z(T) j exp β ZiT i R (tj ) Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β Multiplying the likelihood over all deaths gives the partial likelihood, whose maximum is our estimate for β : T k exp β Z(j ) ˆ = β L (β ) = is maximal β exp β ZiT j =1 i R (tj ) 16/45 This is a partial likelihood since it considers only likelihood of the deaths (censored observations contribute to the denominator) and does not depend on the times of death (just the order) (this is different to the Kaplan Meier approach). The numerator depends on information for the individual who experiences death; The denominator utilises information about all information about all lives who have not yet experienced death. ˆ β are asymptotically normal Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β Example A group of seven lives was observed over a period of time as part of a mortality investigation. Each of the lives was under observation at all ages from age 45 until death or policy expiry: Life 1 2 3 4 5 6 7 Sex F M F M M M F Age at Exit 47 50 52 55 64 65 65 Reason L D D L D L L where “L” means “Lapse” and “D” denotes “Death”. 17/45 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β The following proportional hazards model has been proposed: λ(t ; Z ) = λ0 (t ) e β Z where Z = 1 for males and 0 for females. Write down the partial likelihood. 18/45 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β Case 2: partial likelihood in present of ties in the data We will consider Breslow’s approximation: exp β sjT k L (β ) = j =1 dj exp β ZiT i R (tj ) where sj is the sum of the covariate vectors Z of the dj lives observed to die at time tj . This approximation works well when the number of ties are relatively small There are other methods like Efron’s method (see K&M, Section 8.4) - Breslow is an option in R (coxph) and SAS (proc phreg) 19/45 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β Example An investigation was carried out into the survival times (measured in months) of patients in hospital following liver transplants. For patient i , the covariates are zi 1 = 0 for placebo, 1 for treatment X, and zi 2 = weight of the patient (measured in kg). The observed lifetimes (with weights in brackets) were as follows: Placebo Treatment X 3 (83) 6∗ (58) 9 (68) 11 (73) 14 (75) 14 (68) 16 (86) 14∗ (49) Observations with an asterisk represent censored observations. Using Breslow’s approximation, what contribution to the partial likelihood is made by deaths at time 14? 20/45 Actuarial Statistics – Week 4: The Cox Regression Model Estimation of the regression parameters β Solution Denote β = (β1 , β2 ). Just prior to time 14, 4 lives are at risk and their covariates are (0, 75), (1, 68), (1, 49) and (0, 86). At time 14, there are two deaths with covariates (0, 75), (1, 68). Hence, the contribution to the partial likelihood made by deaths at time 14 is approximated by exp β (0 + 1, 75 + 68)T 2 [exp (β (0, 75...
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This document was uploaded on 04/03/2014.

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