Volume 31 - Article 22 | Pages 659-686

Unobserved population heterogeneity: A review of formal relationships

By James W. Vaupel, Trifon I. Missov

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Date received:15 Nov 2013
Date published:17 Sep 2014
Word count:2907
Keywords:gamma-distributed frailty, Gompertz-Makeham force of mortality, relative-risk models, unobserved heterogeneity


Background: Survival models accounting for unobserved heterogeneity (frailty models) play an important role in mortality research, yet there is no article that concisely summarizes useful relationships.

Objective: We present a list of important mathematical relationships that govern populations in which individuals differ from each other in unobserved ways. For some relationships we present proofs that, albeit formal, tend to be simple and intuitive.

Methods: We organize the article in a progression, starting with general relationships and then turning to models with stronger and stronger assumptions.

Results: We start with the general case, in which we do not assume any structure of the underlying baseline hazard, the frailty distribution, or their link to one another. Then we sequentially assume, first, a relative-risk model; second, a gamma distribution for frailty; and, finally, a Gompertz and Gompertz-Makeham specification for baseline mortality.

Comments: The article might serve as a handy overall reference to frailty models, especially for mortality research.

Author's Affiliation

James W. Vaupel - Max-Planck-Institut für Demografische Forschung, Germany [Email]
Trifon I. Missov - Max-Planck-Institut für Demografische Forschung, Germany [Email]

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