Pair Ratio (PR)

The Pair Ratio (PR) is virtually identical to the pair difference (PD), but is calculated by dividing \(\mu_k\) by \(\mu_l\) rather than subtracting.  The equation is: 

$${PR}=\frac{{\mu}_{k}}{{\mu}_{l}},$$

where, again, \(\mu_l\) is the health status in the reference group.  This is a relative disparity measure similar to the Range Ratio, but in this case the user selects the reference group and the comparison group.

$$\widehat{PR}=\frac{\widehat{\mu}_{k}}{\widehat{\mu}_{l}},$$

where

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

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

The variance for \(\widehat{PR}\) is:

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

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

$${z}_{h\alpha{i}}=\frac{{\partial}\widehat{RD}}{{\partial}{w}_{h\alpha{i}}}={\sum}_{j\epsilon(k,l)}\frac{{\partial}\widehat{RD}}{\partial\widehat{\mu}_{j}}\times\frac{{\partial}\widehat{\mu}_{j}}{\partial{w}_{h\alpha{i}}},$$

with

$$\left(\frac{{\partial}\widehat{PR}}{{\partial}\widehat{\mu}_{k}},\frac{{\partial}\widehat{PR}}{{\partial}\widehat{\mu}_{l}}\right)=\left(\frac{1}{\widehat{\mu}_{k}},-\frac{\widehat{\mu}_{l}}{\left(\widehat{\mu}_{k}\right)^{2}}\right)$$

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

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

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