Pair Ratio (PR)

The Pair Ratio (PR) is virtually identical to the pair difference (PD), but is calculated by dividing \(y_1\) by\(y_2\) rather than subtracting.  The equation is: 

$$\widehat{PR}=\frac{y_1}{y_2},$$

where, again, \(y_2\) 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.

Variance of \(\widehat{PR}\)

The variance of \(\widehat{PR}\) based on the Taylor Series linearization method can be estimated as: 

$$\mathrm{var}_{TL} \left( \widehat {PR} \right) = \frac{1}{y^2_2}(\sigma^2_2 + RR^{*2} \sigma_1^2).$$

The standard error of \(\widehat{PR}\) based on the Taylor Series linearization method is \(\sqrt{\mathrm{var}_{TL}(\widehat{PR})}\).

Variance of \(\widehat{PR}\) based on the Monte Carlo Simulation Method

Randomly generate M age-adjusted rates \(y_j^{(m)}, m = 1, \ldots, M, \) using the distribution:  \(y_j^{(m)} \sim N(y_j, \hat{\sigma}^2_j)\) for each of the two chosen social groups, \(j=1...,J\).  Then calculate \(\widehat{PR}^{(m)}\) using \(y_j^{(m)}.\)  Thus:

$$\mathrm{var}_{MCS}(\widehat{PR}) = (M-1)^{-1}{\sum}_{m=1}^M(\widehat{PR}^{(m)}-\overline{\widehat{PR}})^2,$$

where \(\overline{\widehat{PR}} = M^{-1}{\sum}_{m=1}^M\widehat{PR}^{(m)}.\)  M = 1,000 are used in HD*Calc.  The standard error of \(\widehat{PR}\) based on the MCS method is:  \(\sqrt{\mathrm{var}_{MCS}(\widehat{PR})}\).

95% Confidence Interval of \(\widehat{PR}\)

To compute the 95% confidence interval of \(\widehat{PR}\), we utilize the variance of \(\widehat{PR}\) after logarithm transformation. The variance of \(\log(\widehat{PR})\) using Taylor Series Linearization is: 

$$\mathrm{var}_{TL}\left(\log(\widehat{PR}\right)) = \frac{\widehat{\sigma}_1^2}{y_1^2} + \frac{\widehat{\sigma}_2^2}{y_2^2}.$$ 

Thus, the 95% confidence interval of \(\log(RR^*)\) is:

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

The 95% confidence interval of \(\widehat{PR}\) based on Taylor Series Linearization method is:

$$\left(\exp(\log(\widehat{PR})-1.96 \times \sqrt{\mathrm{var}_{TL}(\log(\widehat{PR})}\right), \exp\left(\log(\widehat{PR})+1.96 \times \sqrt{\mathrm{var}_{TL}(\log(\widehat{PR}))})\right).$$

The upper and lower bounds of the 95% confidence interval of \(\widehat{PR}\) based on the MCS method are the 2.5th percentile and the 97.5th percentile of the 1,000 \(\widehat{PR}^{(m)}\) values.