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Yu et al 2018 (5) adapts the extended Relative Concentration Index (eRCI) (6, 7) with aversion parameters \(\nu\) for assessment of health disparities using age-adjusted rates estimated from population-based data. The eRCI is the relative version of the eACI. The eRCI may only be used with socio-economic groups that have an inherent ranking, such as income or education groups. The eRCI can be calculated as:
$$\mathrm{\widehat{eRCI}} = \nu {\sum}_{j=1}^{J}p_j(1-R_j)^{\nu-1}-\frac{\nu}{\mu}{\sum}_{j=1}^{J}p_jy_j(1-R_j)^{\nu-1},$$
where \(\nu>0\) is the aversion parameter, \(y_{j}\) is the health status of group j; pj is the population share of the \(j^{th}\) group; μ is the average level of health status in the population, \(\mu = \mathrm{\sum}_{j=1}^Jp_jy_j;R_j\) is the relative rank of the jth socio-economic group, \(R_j =\mathrm{\sum}_{\gamma = 1}^{j-1}p_\gamma - 0.5p_{j}.\) Rj essentially indicates the cumulative share of the population up to the midpoint of each group interval.
Variance of \(\widehat{eRCI}\)
The variance of \(\widehat{eRCI}\) based on Taylor Series linearization method can be estimated as:
$$\mathrm{var}_{TL}(\widehat{eRCI}) = \frac{1}{\mu^2}{\sum}_{j=1}^J\hat{\sigma}^2_jp^2_j\left[\frac{\nu{\sum}_{j=1}^Jp_j(1-R_j)^{\nu-1}y_j}{\mu} - \nu(1-R_j)^{\nu-1}\right]^2$$
The standard error of \(eRCI\) based on the Taylor Series linearization method is: \( \sqrt{\mathrm{var}_{TL}(\widehat{eRCI})}\). See Yu et al 2018 (5) for details on how \({\mathrm{var}_{TL}(\widehat{eRCI})}\) was derived.
Variance of \(\widehat{eRCI}\) based on the Monte Carlo Simulation-based (MCS) method
Randomly generate M age-adjusted rates \(y_{j}^{(m)},m=1,...,M,\) using the distribution: \(y_j^{(m)} \sim Gamma(mean=y_j,var= \hat{\sigma}^2_j)\)for each socio-economic group. Then calculate \(eRCI^{(m)}\) using \(y_{j}^{(m)}\). Thus:
$$\mathrm {var}_{MCS}(\widehat{eRCI})=(M-1)^{-1}\mathrm{\sum}_{m=1}^{M}(\widehat{eRCI}^{(m)}-\overline{\widehat{eRCI}})^{2},$$
where \(\overline{\widehat{eRCI}}=M^{-1}\mathrm{\sum}_{m=1}^{M}\widehat{eRCI}^{(m)}\). \(M=1,000\) is used in HD*Calc. The standard error of \(\widehat{eRCI}\) based on the MCS method is: \( \sqrt{\mathrm{var}_{MCS}(\widehat{eRCI})}\). Gamma distribution instead of truncated normal distribution was recommended to simulate age-adjusted rates (1).
95% Confidence Interval of \(\widehat{eRCI}\)
The 95% confidence interval of \(\widehat {eRCI}\) based on Taylor Series Linearization variance estimate is:
$$\widehat{eRCI} \pm 1.96 \times \sqrt{\mathrm{var}_{TL}(\widehat{eRCI})}.$$
The lower and upper boundaries of the 95% confidence interval of \(\widehat {eRCI}\) based on the MCS approach are the 2.5th percentile and 97.5th percentile of the \(1000\widehat{eRCI}^{(m)}\) values.
When the aversion parameter \(\nu=2\), eRCI reduces to the Relative Concentration Index (RCI) (13), which measures the extent to which health or illness is concentrated among particular socio-economic groups on the absolute scale. To be specific, estimate of RCI is written as:
$$\widehat{RCI} = \frac{2}{\mu}\left({\sum}_{j=1}^Jp_jy_jR_j\right)-1=\frac{{\sum}_{j=1}^Jp_j(2R_j-1)y_j}{\mu}$$
The variance of \(\widehat{RCI}\) based on Taylor Series linearization method reduces to:
$${\mathrm{var}_{TL}(\widehat{RCI})}= \frac{1}{{\mu}^{2}}\mathrm{\sum}_{j=1}^J{\widehat\sigma}_{j}^{2}{p}_{j}^{2}({2R}_{j} -1-RCI)^{2}.$$
The standard error of \(\widehat{RCI}\) based on Taylor Series linearization method is: \( \sqrt{\mathrm{var}_{TL}(\widehat{RCI})}\). See Kakwani et al (13) and Ahn et al 2018 (4) for more details on RCI.
The variance of \(\widehat{RCI}\) based on the MCS method and the 95% confidence intervals based on both analytic and MCS methods are the same as those for \(\widehat{eRCI}\) with \(\nu = 2\).
One of the reasons the RCI is favored by some is that “it reflects the socio-economic dimension to inequalities in health” (14, p.548). That is, a downward health gradient (such that health worsens with socio-economic group rank) results in a positive \(\widehat{RCI},\) whereas an upward health gradient results in a negative \(\widehat{RCI}\).
