Slope Index of Inequality (SII)

Slope Index of Inequality (SII) captures the difference in the average health status between a person in the highest socioeconomic group (SEG) and a person in the lowest SEG. SII was introduced by Preston, Haines, and Pamuk (10), and it is derived from the following simple linear regression model:

 $$\mu_{j} = \beta_{0} + \beta_{1}R_{j}forj=1-J$$

where \(\beta_{0}\) is the estimated health status of a hypothetical person at the bottom of the SEG hierarchy (i.e., a person whose relative rank \(R_{j}\) in the SEG distribution is zero), and\(\beta_{1},\) i.e. \(SII\), is the difference in average health status between the hypothetical person at the bottom of the SEG distribution and the hypothetical person at the top assuming linearity (i.e. \(R_{j}=0\) vs. .  Because the relative rank is based on the cumulative proportions of the population (from 0 to 1), a “one-unit” change in relative rank is equivalent to moving from the bottom to the top of the SEG distribution.  Because the regression is run on grouped data, it is often estimated via weighted least squares, with weights equal to the population share of group j.

$${SII}={\beta}_1=\frac{{\sum}_{j=1}^Jp_jR_j(\mu_j-\mu)}{{\sum}_{j=1}^Jp_jR_j^2-\left({\sum}_{j=1}^Jp_jR_j\right)^2},$$

where \(\mu_{j}\) is the average health status of SEG  \(j; {p}_{j}\) is the population share of SEG \(j;{R}_{j}={\sum}_{\gamma}^{j-1}{p}_{\gamma}-0.5{p}_{j}\) is the relative rank of SEG\(j\), and  \(\mu={\sum}_{j=1}^{J}{p}_{j}{\mu}_{j}\) is the average health status in the population.

Point Estimator of SII

A design consistent estimator is

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

where

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

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

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

is the estimated population share of SEG \(j\), 

$$\widehat{R}_{j}={\sum}_{\gamma}^{j-1}\widehat{p}_{\gamma}-0.5\widehat{p}_{j}$$

is the estimated relative rank of SEG \(j\), and \(\widehat{\mu}={\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{\mu}_{j}\) is the estimated population average of health status.

\(\widehat{SII}\)is closely related to the standard\(ACI\)(denoted also as \(\widehat{eACI}_{2}\) ) in that 

$$\widehat{SII}=\frac{{1}}{{2}}\widehat{eACI}_{\nu=2}/\left({\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{R}_{j}^{2}-\widehat{R}^{2}\right).$$

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

The variance for\(\widehat{SII}\)is:  

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

where

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

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

and

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

with

$$\frac{\partial\widehat{SII}}{\partial\widehat{p}_{j}}=\frac{\widehat{SII}}{\widehat{eACI}_{\nu=2}}\frac{\partial\widehat{eACI}_{\nu=2}}{\partial\widehat{p}_{j}}-\frac{2\widehat{SII}^{2}}{\widehat{eACI}_{\nu=2}}\frac{\partial}{\partial{p}_{j}}\left({\sum}_{j=1}^{J}{p}_{j}{R}_{j}^{2}\right),$$

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

$$\frac{\partial\widehat{SII}}{\partial\widehat{\mu}_{j}}=\frac{\widehat{SII}}{\widehat{eACI}}\frac{\partial\widehat{eACI}}{\partial\widehat{\mu}_{j}},$$

and

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

95% Confidence Interval of\(\widehat{SII}\)are:

The 95% confidence interval of  based on the weighted least square method is: 

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

For details about the derivations, see Li et al. (18).