Relative Index of Inequality (RII)

The Relative Index of Inequality (RII) (16) is the slope index inequality (SII) relative to the average health in the population, and RII is also related to the extend Relative Concentration Index (eRCI) when the aversion parameter \(\nu=2,\) i.e. the standard Relative Concentration Index (RCI):

$$RII=\frac{SII}{\mu}=\frac{RCI}{2\left({\sum}_{j=1}^{J}{p}_{j}{R}_{j}^{2}-\overline{R}^{2}\right)}$$

where \(\mu={\sum}_{j=1}^{J}p_{j}\mu_{j}\) is the average health status in the population; \(\mu_{j}\) is the health status of socioeconomic group (SEG) \(j;p_j\mu_j\) is the population share of SEG \(j;\) and R\(R_{j}={\sum}_{\gamma=1}^{j-1}p_\gamma-0.5p_{j}\) is the relative rank of SEG  and it indicates the cumulative share of the population up to the midpoint of each SEG interval.

The interpretation of RII is similar to SII, but it now measures the proportionate (in regard to the average population level) rather than the absolute increase or decrease in health between the highest and lowest SEGs.

Point Estimator of \(\widehat{RII}\)

A survey design consistent estimator is:

$$\widehat{RII}=\frac{\widehat{RII}}{2\left({\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{R}_{j}^{2}-\overline{\widehat{R}}^{2}\right)},$$

where

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

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

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

The variance for \(\widehat{RII}\) is

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

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

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

with

$$\frac{\partial\widehat{RII}}{\partial\widehat{\mu}}=\frac{2\widehat{p}_{j}\widehat{R}_{j}-\left(1+\widehat{RCI}\right)\widehat{p}_{j}}{\widehat{\mu}\left(2{\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{R}_{j}-.5\right)},$$

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

$$\frac{\partial\widehat{RII}}{\partial\widehat{p}_{j}}=\frac{1}{2}\frac{\frac{\partial\widehat{RCI}}{\partial\widehat{p}_{j}}\left({\sum}_{j=1}^{J}p_{j}{R}_{j}^{2}-\overline{R}^{2}\right)-\widehat{RCI}\frac{\partial}{\partial{p}_{j}}\left({\sum}_{j=1}^{J}{p}_{j}{R}_{j}^{2}\right)}{\left({\sum}_{j=1}^{J}p_{j}R_{j}^{2}-\overline{R}^{2}\right)^{2}},$$

where

$$\frac{\partial}{\partial{p}_{j}}\left({\sum}_{j=1}^{J}{p}_{j}{R}_{j}^{2}\right)=R_{j}\left({R}_{j}-{p}_{j}\right)=2{\sum}_{j}^{J}{p}_r{R}_{r}$$

and

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

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

$$\widehat{RII} \pm 1.96 \times \sqrt{\mathrm{var}_{TL}(\widehat{RII})}.$$