notes unobserved Heterogeneity

notes unobserved Heterogeneity - Unobserved Heterogeneity...

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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 0 ( t ) , the product of an individual-specific random effect θ i representing the indi- vidual’s frailty , and a baseline hazard λ 0 ( 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 0 ( t ) θ i , representing the probability of being alive at t given the random effect θ i . 1
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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 ) = 0 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 ) = 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 0 S ( t | θ ) g ( θ ) = - log 0 S 0 ( 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 ) = - 0 d dt S 0 ( t ) θ g ( θ ) 0 S 0 ( t ) θ g ( θ ) = 0 θλ 0 ( t ) S 0 ( t ) θ g ( θ ) 0 S 0 ( t ) θ g ( θ ) , where we used the fact that S 0 ( t ) θ = e - θ Λ 0 ( t ) , so that d dt S 0 ( t ) = - e - θ Λ 0 ( t ) θλ 0 ( t ) = - θλ 0 ( t ) e - θ Λ 0 ( t ) , and the last exponential can be recognized as S 0 ( t ) θ . 2
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Note that the population hazard λ ( t ) is a weighted average of the individ- ual hazards λ i ( t ) with weights equal to the density of θ times the probability of surviving to t : S ( t | θ ) g ( θ ) = S 0 ( t ) θ g ( θ ) , Why can’t we calculate the population hazard as a simple average of the individual hazards, the way we calculated the population survivor function?
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  • Spring '06
  • Rodriguez
  • Probability theory, frailty, inverse gaussian frailty

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