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Biostatistics Advance Access originally published online on September 28, 2005
Biostatistics 2006 7(2):213-224; doi:10.1093/biostatistics/kxj002
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© The Author 2005. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Relative index of inequality: definition, estimation, and inference

Jamie C. Sergeant*

Nuffield College, Oxford OX1 1NF, UK jamie.sergeant{at}nuf.ox.ac.uk

David Firth

Department of Statistics, University of Warwick, Coventry CV4 7AL, UK d.firth{at}warwick.ac.uk

* To whom correspondence should be addressed.

The relative index of inequality (RII) is a commonly used measure of the extent to which the occurrence of an outcome such as chronic illness or early death varies with socioeconomic status or some other background variable. The standard RII estimator applies only to linear variation in incidence rates. In this paper a general definition of the RII is introduced, alternative approaches to point estimation are considered, and a parametric bootstrap method is suggested for the construction of approximate confidence intervals. Estimation based on cubic splines fitted by maximum penalized likelihood is developed in some detail, and the proposed approach handles naturally the commonly needed adjustment for a ‘standardizing’ covariate such as age. Death rates in a large longitudinal study in England and Wales from 1996–2000 are analyzed in order to illustrate the various methods. A small simulation study explores the relative merits of different estimators. The approach based on cubic splines is found to reduce bias substantially, at the expense of some increase in variance, when variation in incidence rates is nonlinear.

Keywords: Bootstrap; Cross-validation; Health inequality; Interval-censored covariate; Penalized likelihood; Slope index; Smoothing spline


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