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Modelling heterogeneity in survival data

Published online by Cambridge University Press:  14 July 2016

Philip Hougaard
Philip Hougaard*
Affiliation:
Novo Research Institute
*
∗ Postal address: Biostatistical Department, Novo Research Institute, Novo Alle, 2880 Bagsværd, Denmark.
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Abstract

Ordinary survival models implicitly assume that all individuals in a group have the same risk of death. It may, however, be relevant to consider the group as heterogeneous, i.e. a mixture of individuals with different risks. For example, after an operation each individual may have constant hazard of death. If risk factors are not included, the group shows decreasing hazard. This offers two fundamentally different interpretations of the same data. For instance, Weibull distributions with shape parameter less than 1 can be generated as mixtures of constant individual hazards. In a proportional hazards model, neglect of a subset of the important covariates leads to biased estimates of the other regression coefficients. Different choices of distributions for the unobserved covariates are discussed, including binary, gamma, inverse Gaussian and positive stable distributions, which show both qualitative and quantitative differences. For instance, the heterogeneity distribution can be either identifiable or unidentifiable. Both mathematical and interpretational consequences of the choice of distribution are considered. Heterogeneity can be evaluated by the variance of the logarithm of the mixture distribution. Examples include occupational mortality, myocardial infarction and diabetes.

Keywords

PROPORTIONAL HAZARDSOMITTED COVARIATESMIXTURESINVERSE GAUSSIANGAMMA DISTRIBUTIONSSTABLE DISTRIBUTIONS
Type
Short Communications
Information
Journal of Applied Probability , Volume 28 , Issue 3 , September 1991 , pp. 695 - 701
Copyright
Copyright © Applied Probability Trust 1991 

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