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Unformatted text preview: Unobserved Heterogeneity Germ´ an Rodr´ ıguez [email protected] Spring, 2001; revised Spring 2005 This unit considers survival models with a random effect represent ing unobserved heterogeneity of frailty , a term first introduced by Vau pel et al. (1979). We consider models without covariates and then move on to the more general case. These notes are intended to complement Rodr´ ıguez (1995). 1 The Statistics of Heterogeneity Standard survival models assume homogeneity: all individuals are subject to the same risks embodied in the hazard λ ( t ) or the survivor functions S ( t ). Models with covariates relax this assumption by introducing observed sources of heterogeneity. Here we consider unobserved sources of hetero geneity that are not readily captured by covariates. 1.1 Conditional Hazard and Survival A popular approach to modelling such sources is the multiplicative frailty model, where the hazard for individual i at time t is λ i ( t ) = λ ( t  θ i ) = θ i λ ( t ) , the product of an individualspecific random effect θ i representing the indi vidual’s frailty , and a baseline hazard λ ( t ). Note that this is essentially a proportional hazards model. The individual hazard λ i ( t ) is interpreted as a conditional hazard given θ i . Associated with it we have a conditional survival function S i ( t ) = S ( t  θ i ) = S ( t ) θ i , representing the probability of being alive at t given the random effect θ i . 1 The twist is that the random effect θ i is not observed (perhaps not observable), but is assumed to have some a distribution with density g ( θ ). 1.2 Unconditional Hazard and Survival To obtain the unconditional survival function we need to “integrate out” the unobserved random effect: S ( t ) = Z ∞ S ( t  θ ) g ( θ ) dθ. We integrate from 0 to ∞ because frailty is nonnegative. If frailty was dis crete, taking values θ 1 ,...,θ k with probabilities π 1 ,...,π k then the integral would be replaced by a sum S ( t ) = X i S ( t  θ i ) π i . In both cases S ( t ) is the average S i ( t ). In a demographic context S ( t ) is often referred to as the population survivor function, while S i ( t ) is the individual survivor function. To obtain the unconditional hazard we start from the unconditional sur vival and take negative logs to obtain the cumulative hazard Λ( t ) = log S ( t ) = log Z ∞ S ( t  θ ) g ( θ ) dθ = log Z ∞ S ( t ) θ g ( θ ) dθ. The next step is to take derivatives w.r.t. t . Assuming that we can take the derivative operator inside the integral we find the unconditional hazard to be λ ( t ) = R ∞ d dt S ( t ) θ g ( θ ) dθ R ∞ S ( t ) θ g ( θ ) dθ = R ∞ θλ ( t ) S ( t ) θ g ( θ ) dθ R ∞ S ( t ) θ g ( θ ) dθ , where we used the fact that S ( t ) θ = e θ Λ ( t ) , so that d dt S ( t ) = e θ Λ ( t ) θλ ( t ) = θλ ( t ) e θ Λ ( t ) , and the last exponential can be recognized as S ( t ) θ ....
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This note was uploaded on 02/12/2008 for the course ECON 572 taught by Professor Rodriguez during the Spring '06 term at Princeton.
 Spring '06
 Rodriguez

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