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Extended Relative Concentration Index (eRCI)

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 $$\widehat{RCI}$$.