Range Ratio (RR)

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

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

While in the context of social group comparisons the is typically based on comparing, for example, the least advantaged group (e.g., the lowest social group) to the most advantaged group, in the context of comparing it to summary measures of health disparity we calculate it as a range measure.  That is, at each time point it measures the relative difference in the rates of the least advantaged and most advantaged group (i.e., the relative range), regardless of their social group status. 

Variance of \(\widehat {RR}\)

Assume the least advantaged group and the highest advantaged group are fixed and independent. The variance of \(\widehat {RR}\) based on Taylor Series linearization method can be calculated as:

$$\mathrm{var}_{TL}(\widehat{RR})=\frac{1}{y^2_{\mathit{min}}}(\hat{\sigma}^2_\mathit{max}+\widehat{RR}^2\hat{\sigma}^2_\mathit{min})$$

The standard error of \(\widehat {RR}\) based on the Taylor Series linearization method is:  \( \sqrt{\mathrm{var}_{TL}(\widehat{RR})}\).  See Ahn et al 2018 (4) for details on how \(\mathrm{var}_{TL}(\widehat{RR})\) was derived.

Further research is needed to account for the variability that the least advantaged group and the highest advantaged group are unknown and are correlated.

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

Randomly generate M age-adjusted rates \(y_{j}^{(m)},m=1,...M,\) using the distribution \(y_j^{(m)}\sim Gamma(mean=y_j,var=\hat{\sigma}^2_j)\) for each social group.  Calculate \(\widehat {RR}^{(m)}\) using \(y_{B}^{(m)}\) and \(y_{w}^{(m)}\), and the associated variance is: 

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

where \(\overline{\widehat{RR}}=M {\sum}_{m=1}^M\widehat{RR}^{(m)}\).  \(M=1,000\) is used in HD*Calc.  The standard error of \(\widehat {RR}\) based on the MCS method is:  \( \sqrt{\mathrm{var}_{MCS}(\widehat{RR})}\).  Gamma distribution instead of truncated normal distribution was recommended to simulate age-adjusted rates (1).  

95% Confidence Level of \(\widehat{RR}\)

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

$$\mathrm{var}_{TL}(\log (\widehat{RR})) = \frac{\hat{\sigma}^2_\mathit{max}}{y^2_\mathit{max}} + \frac{\hat{\sigma}^2_\mathit{min}}{y^2_\mathit{min}}$$. 

Thus the 95% confidence interval of \(\mathrm{log}\widehat{(RR)}\) is:  \(\mathrm{log}(\widehat{RR}) \pm 1.96 \times \sqrt{\mathrm{var}_{TL}(\mathrm{log}(\widehat{RR})})\).  Therefore the 95% confidence interval of \(\widehat{RR}\) is:   

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

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