Range Ratio (RR)

Range Ratio (RR) is virtually identical to Range Difference (RD), but is calculated by dividing \(\mu_{max}\) by \(\mu_{min}\) rather than subtracting: 

$${RR}=\frac{\widehat{\mu}_{max}}{\widehat{\mu}_{min}},$$

where \(\mu_{max}\) is the best group average and \(\mu_{min}\) is the worst group average for a given outcome.

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

A survey design consistent estimator is

$$\widehat{RR}=\frac{y_{max}}{y_{min}}$$

where

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

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

Assume the least advantaged group and the most advantaged group are fixed and independent.  The variance for \(\widehat {RR}\) is: 

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

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

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

with

$$\left(\frac{{\partial}\widehat{RR}}{{\partial}\widehat{\mu}_{max}},\frac{{\partial}\widehat{RR}}{{\partial}\widehat{\mu}_{min}}\right)=\left(\frac{1}{\widehat{\mu}_{min}},-\frac{\widehat{\mu}_{max}}{\left(\widehat{\mu}_{min}\right)^{2}}\right)$$

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

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

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

Note that this variance estimator did not account for the uncertainties of identifying the least advantaged group and 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).