Extended Relative Concentration Index (eRCI)

The extended Relative Concentration Index (eRCI) (6,7) is the relative version of eACI.  Similar to eACI, eRCI is extended from the standard Relative Concentration Index (RCI) (13) by incorporating an inequality aversion parameter \(\nu\).  It measures the extent to which health or illness is concentrated among particular socioeconomic groups (SEGs) in relative the population average.  By modifying the value of \(\nu\), researchers can explore different value judgements towards inequality aversion in the assessment of health disparities.

eRCI as defined by Wagstaff (20) has the following form:

$$\mathrm{{eRCI}} = \nu {\sum}_{j=1}^{J}p_j(1-R_j)^{\nu-1}-\frac{\nu}{\mu}{\sum}_{j=1}^{J}p_j\mu_j(1-R_j)^{\nu-1},$$

where \(\nu>0\) is the aversion parameter, \(\mu_{j}\) is the health status of SEG \(j;p_{j}\) is the population share of SEG \(j;\)\(\mu = \mathrm{\sum}_{j=1}^Jp_jy_j\) is the average health status in the population; \(R_j =\mathrm{\sum}_{\gamma = 1}^{j-1}p_\gamma - 0.5p_{j}\) is the relative rank of SEG \(j\) and it indicates the cumulative share of the population up to the midpoint of each SEG interval.

Point Estimator of \(\widehat{eRCI}\)

A survey design consistent estimator is

$$\mathrm{\widehat{eRCI}} = \nu {\sum}_{j=1}^{J}\widehat{p}_j(1-\widehat{R}_j)^{\nu-1}-\frac{\nu}{\widehat{\mu}}{\sum}_{j=1}^{J}\widehat{p}_j\widehat{\mu}_j(1-\widehat{R}_j)^{\nu-1},$$

where

$$\widehat{\mu}_{j}=\frac{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}\delta_{h\alpha{i}}^{j}w_{h\alpha{i}}y_{h\alpha{i}}}{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}\delta_{h\alpha{i}}^{j}w_{h\alpha{i}}}$$

is the estimated average health in SEG \(j\),

$$\widehat{p}_{j}=\frac{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}\delta_{h\alpha{i}}^{j}w_{h\alpha{i}}}{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}w_{h\alpha{i}}}$$

is the estimated population share of SEG \(j,\) and \(\widehat{R}_{j}\) is the estimated relative rank of SEG \(j.\)

Variance and Confidence Intervals of \(\widehat{eRCI}\)  

The variance of \(e\widehat{RCI}\) is: 

$$\mathrm{var}(\widehat{eRCI}) \cong {\sum}_{h=1}^H{\sum}_{\alpha=1}^{t_{h}}\left({Z}_{h\alpha}-\overline{Z}_{h}\right)\left({Z}_{h\alpha}-\overline{Z}_{h}\right)^{T},$$

where

$$Z_{h\alpha}={\sum}_{i=1}^{n_{h\alpha}}w_{h\alpha{i}}z_{h\alpha{i}},$$

$$\overline{Z}_{h}=\frac{{1}}{t_{h}}{\sum}_{\alpha=1}^{t_{h}}Z_{h\alpha},$$

and

$${z}_{h\alpha{i}}=\frac{{\partial}\widehat{eRCI}}{{\partial}{w}_{h\alpha{i}}}={\sum}_{j=1}^{J}\left(\frac{{\partial}\widehat{eRCI}}{\partial\widehat{p}_{j}}\times\frac{{\partial}\widehat{p}_{j}}{\partial{w}_{h\alpha{i}}}+\frac{{\partial}\widehat{eRCI}}{\partial\widehat{\mu}_{j}}\times\frac{{\partial}\widehat{\mu}_{j}}{\partial{w}_{h\alpha{i}}}\right),$$

with

$$\frac{{\partial}\widehat{eRCI}}{\partial\widehat{\mu}_{j}}=\frac{\nu\widehat{p}_{j}}{\widehat{\mu}}\left[\frac{{\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{\mu}_{j}\left(1-\widehat{R}_{j}\right)^{\nu-1}}{\widehat{\mu}}-\left(1-\widehat{R}_{j}\right)^{\nu-1}\right],$$

$$\frac{{\partial}\widehat{\mu}_{j}}{{\partial}{w}_{h\alpha{i}}}=\frac{{\partial}_{h\alpha{i}}^{j}\left({y}_{h\alpha{1}}-\widehat{\mu}_{j}\right)}{{\sum}_{h=1}^{H}{\sum}_{a=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}\delta_{h\alpha{i}}{w_{h\alpha{i}}}},$$

$$\frac{\partial\widehat{MLD}}{\partial\widehat{\mu}_{j}}=\widehat{p}_{j}\left(\frac{1}{\widehat{\mu}}-\frac{1}{\widehat{\mu}_{j}}\right),$$

$$\frac{\partial\widehat{eRCI}}{\partial\widehat{p}_{j}}=\nu\left[\left(1-\frac{\mu_{j}}{\mu}\right)\left(1-R_{j}\right)^{\nu-1}+\frac{\nu-1}{2}p_{j}\left(1-\frac{\mu_{j}}{\mu}\right)\left(1-R_{j}\right)^{\nu-2}-\left(\nu-1\right){\sum}_{j}^{J}p_{j}\left(1-\frac{\mu_{j}}{\mu}\right)\left(1-R_{j}\right)^{\nu-2}\right]+\nu\frac{\mu_{j}}{\widehat{\mu}^{2}}{\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{\mu}_{j}\left(1-\widehat{R}_{j}\right)^{\nu-1}$$

$$\frac{{\partial}\widehat{p}_{j}}{{\partial}{w}_{h\alpha{i}}}=\frac{{\delta}_{h\alpha{i}}^{j}-\widehat{p}_{j}}{{\sum}_{h=1}^{H}{\sum}_{a=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{w_{h\alpha{i}}}},$$

The 95% confidence intervals of \(\widehat{eRCI}\) are:

$$\widehat{eRCI}\pm1.96\times\sqrt{\mathrm{var}(\widehat{eRCI})}.$$

In Relation to RCI

When \(\nu=2,\)eRCI reduces to RCI (13,14), thus:

$$\widehat{eRCI}_{\nu=2}=\widehat{RCI}=\frac{2}{\mu}\left({\sum}-{J=1}^{J}\widehat{p}_{j}\widehat{\mu}_{j}\widehat{R}_{j}\right)-1,$$

with partial derivatives

$$\frac{\partial\widehat{RCI}}{\partial\widehat{\mu}_{j}}=\frac{\nu\widehat{p}_{j}}{\widehat{\mu}}\left[\frac{{\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{\mu}_{j}\widehat{R}_{j}\left(1-\widehat{R}_{j}\right)^{\nu-1}}{\widehat{\mu}}\right],$$

$$\frac{\partial\widehat{RCI}}{\partial\widehat{p}_{j}}={\nu}\left[\left(1-\frac{\widehat{\mu}_{j}}{\widehat{\mu}}\right)\left(1-\widehat{R}_{j}\right)^{\nu-1}+\frac{\nu-1}{2}\widehat{p}_{j}\left(1-\frac{\widehat{\mu}_{j}}{\widehat{\mu}}\right)\left(1-\widehat{R}_{j}\right)^{\nu-2}-\left(\nu-1\right){\sum}_{j=1}^{J}\widehat{p}_{j}\left(1-\frac{\widehat{\mu}_{j}}{\widehat{\mu}}\right)\left(1-\widehat{R}_{j}\right)^{\nu-2}\right]+\nu\frac{\widehat{\mu}}{\widehat{\mu}}{\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{\mu}_{j}\left(1-\widehat{R}_{j}\right)^{\nu-1},$$

The remaining two partial derivatives needed for calculating Taylor deviate \(z_{h\alpha{i}},\) i.e. \(\frac{\partial\widehat{p}_{j}}{\partial{w}_{h\alpha{i}}}\) and \(\frac{\partial\widehat{\mu}_{j}}{\partial{w}_{h\alpha{i}}}\), are unchanged.  For details about the derivations of RCI, see Li et al (18) and Yu et al (21).