Range Difference (RD)

Range difference (RD) compares the health of the least advantaged social group to the most advantaged social group and is calculated as

$${RD} = \mu_{max} -\mu_{min},$$ 

Point Estimator of \(\widehat{RD}\)

A survey design consistent estimator is

$$\widehat{RD} = \widehat{\mu}_{max} -\widehat{\mu}_{min},$$

where

$$\widehat\mu_{j}=\frac{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\delta}^{j}_{h\alpha{i}}{w}_{h\alpha{i}}{y}_{h\alpha{i}}}{{\sum}_{h=1}^{H}{\sum}_{h=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\delta}^{j}_{h\alpha{i}}{w}_{h\alpha{i}}}\epsilon(max, min)$$

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

Assume the least advantaged group and the most advantaged group are fixed and independent.  The variance for is

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

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

$${z}_{h\alpha{i}}=\frac{{\partial}\widehat{RD}}{{\partial}{w}_{h\alpha{i}}}={\sum}_{j\epsilon(max, min)}^{J}\frac{{\partial}\widehat{RD}}{\partial\widehat{\mu}_{j}}\times\frac{{\partial}\widehat{\mu}_{j}}{\partial{w}_{h\alpha{i}}},$$

with

$$\left(\frac{{\partial}\widehat{RD}}{{\partial}\widehat{\mu}_{max}},\frac{{\partial}\widehat{RD}}{{\partial}\widehat{\mu}_{min}}\right)=\left(1,-1\right),$$

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

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

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

Note that this variance estimator did not account for the uncertainties of identifying the least advantaged group and the most advanced group, and the negative correlation between these two groups. Further research on this topic is needed. For details about the derivations, see Li et al (18).