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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 individual-specific 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 non-negative. 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