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In the context of measuring health disparities, the Range Difference (RD) is often used to compare the health of less-advantaged social groups to more-advantaged. Here we use RD as a summary measure of the gap between the best \((max)\) and the worst \((min)\) social groups for a given outcome (i.e., the absolute range). Estimate of RD is calculated as:
$$\widehat{RD} = y_{max} -y_{min}$$
Variance of \(\widehat{RD}\)
Assume the lowest advantaged group and the most advantaged group are fixed and independent. The variance for \(\widehat{RD}\) is:
$$\mathrm{var}_{TL}(\widehat{RD}) = \hat{\sigma}^2_{\mathit{max}}+\hat{\sigma}^2_{\mathit{min}}$$
The standard error of \(\widehat{RD}\) is:
$$\sqrt{\mathrm{var}_{TL}(\widehat{RD})}$$
No Taylor series linearization approximation is needed to derive this variance since \(\widehat{RD}\) is already a linear function of \(Y=(y_{max},y_{min})\). We still use notation to be consistent with the variance notation for other health disparity measures. Further research is needed to account for the variability that the least advantaged group and the most advantaged group are unknown and are correlated.
Variance of \(\widehat{RD}\) based on Monte Carlo Simulation (MCS) Method
Randomly generate M age-adjusted rates \(y_j^{(m)},m = 1, \ldots,M\) using the distribution: \(y_{j}^{(m)}\sim Gamma(mean=y_{j},var=\widehat{\sigma}_{j}^{2})\) for each social group. Then calculate \(\widehat{RD}^{(m)}\) using \(y_j^{(m)}\). Thus:
$$\mathrm{var}_{MCS}(\widehat{RD}) = (M-1)^{-1}{\sum}_{m=1}^M(\widehat{RD}^{(m)}-\overline{\widehat{RD}})^2$$
where \(\overline{\widehat{RD}}=M^{-1}{\sum}_{m=1}^{M}\widehat{RD}^{(m)}\). \(M=1,000\) is used in HD*Calc. The standard error of \(\widehat{RD}\) based on MCS method is \(\sqrt{\mathrm{var}_{MCS}(\widehat{RD})}\). Gamma distribution instead of truncated normal distribution was recommended to simulate age-adjusted rates (1).
95% Confidence Interval of \(\widehat{RD}\)
The 95% confidence interval of \(\widehat{RD}\) based on the analytic method is:
$$\widehat{RD} \pm 1.96 \times \sqrt{\mathrm{var}_{TL}(\widehat{RD})}.$$
The lower and upper bound of the 95% confidence interval of \(\widehat{RD}\) based on the MCS method are the 2.5th percentile and 97.5th percentile of the \(1,000 \widehat{RD}^{(m)}\) values.
