Index of Disparity (IDisp)

The Index of Disparity \((ID_{isp})\) summarizes the absolute difference in the average health status between several social groups and a reference social group.  It expresses the summed absolute differences as a proportion of the average health in the reference group.  This measure was formally introduced by Pearcy and Keppel (11) and is calculated as: 

$${ID_{isp}} = 100 \times \frac{1}{J-1}\mathrm{\sum}^J_{j=1, j \neq \mathrm{ref}}\frac{\vert \mu_j-\mu_{\mathrm{ref}}\vert}{\mu_{\mathrm{ref}}},$$

where \(\mu_{\mathrm{ref}}\) is the average health status in the reference group.  While in principle, any reference group may be chosen, it is often recommended to use the best-off group as the comparison since it represents the health status desirable for all groups to achieve.  In this case, it is not necessary to take the absolute values of the differences since they will all be positive.  Hence, \(ID_{isp}\) can be rewritten as:

$${ID_{isp}} = 100 \times \frac{1}{J-1}{\sum}^{J}_{j=1, j \neq \mathrm{ref}}\left(\frac{\mu_j}{\mu_\mathrm{ref}}-1\right).$$

Point Estimator of \(\widehat{{ID}_{isp}}\)

A survey design consistent estimator is

$$\widehat{ID}_{isp}= 100 \times \frac{1}{J-1}{\sum}^{J}_{j=1, j \neq \mathrm{ref}}\left(\frac{\widehat\mu_j}{\widehat\mu_\mathrm{ref}}-1\right),$$

where

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

Variance and Confidence Intervals of \(\widehat{{ID}_{isp}}\)

The variance of \(\widehat{ID_{isp}}\) is: 

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

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

$${z}_{h\alpha{i}}=\frac{{\partial}\widehat{ID}_{isp}}{{\partial}{w}_{h\alpha{i}}}={\sum}_{j=1}^{J}\frac{{\partial}\widehat{ID}_{isp}}{\partial\widehat{\mu}_{j}}\times\frac{{\partial}\widehat{\mu}_{j}}{\partial{w}_{h\alpha{i}}},$$

with

$$\frac{\partial\widehat{ID}_{isp}}{\partial\widehat{\mu}_{j}}=\frac{100\left(-1\right)^{1=I\{\widehat{\mu}_{j}>\widehat{\mu}_{ref}\}}}{\left(J-1\right)\widehat{\mu}_{ref}}forj\neq{ref},$$

$$\frac{\partial\widehat{ID}_{isp}}{\partial\widehat{\mu}_{j}}=\frac{100}{\left(J-1\right)\widehat{\mu}_{ref}^{2}}{\sum}_{j=1}^{J-1}\widehat{\mu}_{j}\left(-1\right)^{I\{\widehat{\mu}_{j}>\widehat{\mu}_{ref}\}} forj=ref$$

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

The 95% confidence intervals of \(\widehat{ID}_{isp}\) are:

$$\widehat{ID}_{isp}\pm1.96\times\sqrt{\mathrm{var}(\widehat{ID}_{isp})}.$$

Note that this variance estimator assumes the reference is identified a prior based on study objectives, thus this process does not contribute to the variance of \(\widehat{ID}_{isp}.\)  Further research on this topic is needed.  For details about the derivations, see Li et al. (18).