Kunst Mackenbach Relative Index (KMI)

Kunst and Mackenbach (17) modified this definition of the RII slightly by dividing the estimated health of the hypothetical person at the bottom of the socioeconomic group (SEG) distribution: 

$${KMI} = \frac{\beta_{0}}{\beta_{0} + {\beta}_{1}} = \frac{{\beta}_0}{{\beta}_0 + \mathrm{SII}}=\frac{4}{RII+2}-1=\frac{\left(\frac{\mu}{SII}-\frac{1}{2}\right)\frac{ACI}{2}}{\left(\frac{\mu}{SII}+\frac{1}{2}\right)\frac{ACI}{2}}$$

$$=1-2\frac{RCI}{4{\sum}_{j=1}^{J}p_{j}R_{j}^{2}+RCI-1},$$

where \({\beta}_0 = {\mu}_j - {\beta}_1R_j,\) and \(R_j\) is the relative rank of SEG \(j.\)

Thus, the Kunst-Mackenbach Relative Index (KMI) is more like a traditional relative risk measure in that it compares the health of the extremes of the social distribution by computing the ratio of health status of the most disadvantaged to the most advantaged.  Thus, if the index is 1.5 then the average health status of the most disadvantaged is 1.5 times as high as that of the most advantaged.  Note that the values of \(R_j=0\) or 1 do not correspond to the lowest and highest categories but to the extremes of these categories.  They therefore represent extreme, possibly hypothetical subgroups.  As noted above, the use of the SII, RII, ACI,  RCI and KMI indices depends on having a socioeconomic group classification scheme with an inherent ordering.

Point Estimator of \(\widehat{KMI}\)

A survey design consistent estimator is:

$$\widehat{KMI}=1-2\frac{\widehat{RCI}}{4{\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{R}_{j}^{2}+\widehat{RCI}-1},$$

where \(\widehat{RCI}=\frac{2}{\widehat{\mu}}\left({\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{\mu}_{j}\widehat{R}_{j}\right)-1\) is the estimated RCI,

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

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

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

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

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

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

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

with

$$\frac{\partial\widehat{KMI}}{\partial\widehat{\mu}_{j}}=\left(\frac{\widehat{KMI}^{2}-1}{2\widehat{RCI}}\right)\frac{\partial\widehat{RCI}}{\partial\widehat{\mu}_{j}}=\frac{\left(1+\widehat{KMI}^{2}\right)}{-4}\frac{\partial\widehat{RII}}{\partial\widehat{\mu}_{j}}$$

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

$$\frac{\partial\widehat{RII}}{\partial\widehat{p}_{j}}=\left(\frac{\widehat{KMI}^{2}-1}{2\widehat{RCI}}\right)\frac{\partial\widehat{RCI}}{\partial\widehat{p}_{j}}+\frac{2\left(1-\widehat{KMI}\right)^{2}}{\widehat{RCI}}\frac{\partial}{\partial\widehat{p}_{j}}\left({\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{R}_{j}^{2}\right)=\frac{\left(1+\widehat{KMI}^{2}\right)}{-4}\frac{\partial\widehat{RII}}{\partial\widehat{\mu}_{j}}$$

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

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

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

For details about the derivations of \(\widehat{KMI}\), see Li et al. (18).