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The absolute disparity between two health status indicators is the simple arithmetic difference. It is estimated as:
$$\widehat{PD} = y_1 - y_2$$
where \(y_1\) and \(y_2\) are health status in two chosen social groups. In this case \(y_2\) serves as the reference group health status and \(\widehat{PD}\) is expressed in the same units as \(y_1\) and \(y_2\), a typical disparity measure that uses the absolute difference between two rates. This is similar to the Range Difference but in this case the user selects the reference group and the comparison group.
Variance of \(\widehat{PD}\)
The Variance for \(\widehat{PD}\) is:
$$\mathrm{var}_{TL}(\widehat{PD})=\hat{\sigma}_1^2+\hat{\sigma}_2^2.$$
The standard error of \(\widehat{PD}\) is: \(\sqrt{\mathrm{var}_{TL}(\widehat{PD})}\). No Taylor series approximation is needed to derive this variance since \(\widehat{PD}\) is already a linear function of \(y_1\) and \(y_2\). We still use \(\mathrm{var}_{TL}\) to denote the variance for consistency purpose.
Variance of \(\widehat{PD}\) based on Monte Carlo Simulation
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 group, j = 1,2. Then calculate \(\widehat{PD}^{(m)}\) using \(y_j^{(m)}, j = 1, 2.\) Thus:
$$\mathrm{var}_{MCS}(\widehat{PD}) = (M-1)^{-1}{\sum}_{m=1}^M(\widehat{PD}^{(m)}-\overline{\widehat{PD}})^2,$$
where \(\overline{\widehat{PD}} = M^{-1}{\sum}_{m=1}^M\widehat{PD}^{(m)}.\) M = 1000 are used in HD*Calc. The standard error of \(\widehat{PD}\) based on the MCS method is: \(\sqrt{\mathrm{var}_{MCS}(\widehat{PD})}.\)
95% Confidence Interval of \(\widehat{PD}\)
The 95% confidence interval of \(\widehat{PD}\) is:
$$\widehat{PD} \pm 1.96 \times \sqrt{\mathrm{var}_{TL}(\widehat{PD})}.$$
The lower and upper boundaries of the 95% confidence interval of \(\widehat{PD}\) based on the MCS approach is the 2.5th percentile and the 97.5th percentile of the 1,000 \(\widehat{PD}^{(m)}\) values.
