Annals of Applied Statistics

A reference-invariant health disparity index based on Rényi divergence

Makram Talih

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One of four overarching goals of Healthy People 2020 (HP2020) is to achieve health equity, eliminate disparities, and improve the health of all groups. In health disparity indices (HDIs) such as the mean log deviation (MLD) and Theil index (TI), disparities are relative to the population average, whereas in the index of disparity (IDisp) the reference is the group with the least adverse health outcome. Although the latter may be preferable, identification of a reference group can be affected by statistical reliability. To address this issue, we propose a new HDI, the Rényi index (RI), which is reference-invariant. When standardized, the RI extends the Atkinson index, where a disparity aversion parameter can incorporate societal values associated with health equity. In addition, both the MLD and TI are limiting cases of the RI. Also, a symmetrized Rényi index (SRI) can be constructed, resulting in a symmetric measure in the two distributions whose relative entropy is being evaluated. We discuss alternative symmetric and reference-invariant HDIs derived from the generalized entropy (GE) class and the Bregman divergence, and argue that the SRI is more robust than its GE-based counterpart to small changes in the distribution of the adverse health outcome. We evaluate the design-based standard errors and bootstrapped sampling distributions for the SRI, and illustrate the proposed methodology using data from the National Health and Nutrition Examination Survey (NHANES) on the 2001–04 prevalence of moderate or severe periodontitis among adults aged 45–74, which track Oral Health objective OH-5 in HP2020. Such data, which use a binary individual-level outcome variable, are typical of HP2020 data.

Article information

Ann. Appl. Stat., Volume 7, Number 2 (2013), 1217-1243.

First available in Project Euclid: 27 June 2013

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Epidemiological methods health inequalities alpha–gamma divergence survey data Taylor series linearization rescaled bootstrap


Talih, Makram. A reference-invariant health disparity index based on Rényi divergence. Ann. Appl. Stat. 7 (2013), no. 2, 1217--1243. doi:10.1214/12-AOAS621.

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Supplemental materials

  • Supplementary material A: Technical appendix: Decomposability. Expressions and variance calculations for the total or aggregate RI and SRI and their within-group components when individual-level data are continuous.
  • Supplementary material B: Additional case study from NHANES. Disparities in mean total blood cholesterol levels ($\mu\mathrm{g/dL}$) in U.S. adults aged 20 and over, 2005–08.
  • Supplementary material C: R syntax and output files. Syntax and output from case studies comparing the equally-weighted and population-weighted RI and SRI; their group-specific, between-, and within-group components; and their design-based standard errors and sampling distributions, obtained via Taylor series linearization, balanced repeated replication, and rescaled bootstrap. Syntax is reverse-compatible with that in Borrell and Talih (2011, 2012).