Extended Absolute Concentration Index (eACI)

The extended Absolute Concentration Index (eACI) (6,7) was an extension from the standard Absolute Concentration Index (ACI) (8,9) by incorporating an inequality aversion parameter.  It measures the extent to which health or illness is concentrated among particular socioeconomic groups (SEGs) on the absolute scale.  By modifying the value of, researchers can explore different value judgements towards inequality aversion in the assessment of health disparities.

eACI expressed in its aggregated form is calculated as:

$${eACI}=\nu\mu {\sum}_{j=1}^{J}p_j(1-R_{j})^{\nu-1}-\nu {\sum}_{j=1}^{J}{p_j}{\mu_{j}}(1-R_{j})^{\nu-1},$$

where \(\nu>0\) is the aversion parameter, \(\mu{j}\) is the average health status in SEG \(j:p_{j}\) is the population share of SEG \(j:R_{j}={\sum}_{\gamma=1}^{j-1}{p_{\gamma}}-.05{p_{j}}\) is the relative risk of SEG j and \(\mu={\sum}_{j=1}^{J}{p_{j}}{\mu_{j}}\) is the average health status in the population.

When \(\nu=2\), eACI reduces to ACI calculated as:

$${ACI}=2\left[\left({\sum}_{j=1}^{J}p_{j}{\mu}_{j}{\sum}_{j=1}^{j-1}p_{r}\right)+\frac{{1}}{{2}}{\sum}_{j=1}^{J}{{\mu}_{j}{p}_{j}}^{2}\right]-{\mu}.$$

Point Estimator of eACI

A design consistent estimator is 

$$\widehat{eACI}=\nu\widehat{\mu}{\sum}_{j=1}^{J}\widehat{p}_{j}(1-R_{j})^{\nu-1}-\nu {\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{\mu}_{j}(1-\widehat{R}_{j})^{\nu-1},$$

where

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

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

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

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

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

The variance of \(\widehat{eACI}\) is:  

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

where \(Z_{h\alpha}={\sum}_{i=1}^{n_{h\alpha}}w_{h\alpha{i}}z_{h\alpha{i}},\) \(\overline{Z}=\frac{{1}}{{t_{h}}}{\sum}_{\alpha=1}^{t_{h}}{Z}_{h\alpha}\) and

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

with

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

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

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

and

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

The 95% confidence intervals of are:

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

In Relation to ACI

When \(\nu=2,\) eACI reduces to ACI, thus

$$\widehat{eACI}_{\nu=2}=\widehat{ACI}=2\left({\sum}_{j=1}^{j}\widehat{p}_{j}\widehat{\mu}_{j}\widehat{R}_{j}\right)-\widehat{\mu}_{j},$$

with partial derivatives

$$\frac{{\partial\widehat{ACI}}}{{\partial\widehat{p}_{j}}}=2\left[\widehat{\mu}_{j}{\sum}_{r=1}^{j=1}\widehat{p}_{r}+{\sum}_{r=j}^{J}\widehat{p}_{r}{\mu}_{r}\right]-\widehat{\mu}_{j}$$

$$\frac{{{\partial}\widehat{ACI}}}{{{\partial}\widehat{\mu}_{j}}}=\widehat{p}_{j}\left(2\widehat{R}_{j}-1\right)$$

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