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# Index of Disparity (IDisp)

The Index of Disparity $$(ID_{isp})$$ summarizes the absolute difference in the average health status between several social group rates and a reference rate, and expresses the summed absolute differences as a proportion of the reference rate.  This measure was formally introduced by Pearcy and Keppel (11) and is estimated as:

$$\widehat{ID_{isp}} = 100 \times \frac{1}{J-1}\mathrm{\sum}^J_{j=1, j \neq \mathrm{ref}}\frac{\vert y_j-y_{\mathrm{ref}}\vert}{y_{\mathrm{ref}}},$$

where $$y_{\mathrm{ref}}$$ is the health status indicator in the reference group.  While in principle, any reference group may be chosen, the authors recommend the best group rate as the comparison since that represents the rate desirable for all groups to achieve.  In this case it is not necessary to take the absolute value of the rate differences since they will all be positive. Hence, we can rewrite $$ID_{isp}$$ as:

$$\widehat{ID_{isp}} = 100 \times \frac{1}{J-1}{\sum}^{J}_{j=1, j \neq \mathrm{ref}}\left(\frac{y_j}{y_\mathrm{ref}}-1\right).$$

The variance of $$\widehat{ID_{isp}}$$ based on Taylor Series linearization method can be calculated as:

$$\mathrm{var}_{TL}(\widehat{ID_{\mathrm{ISP}}})=\frac{1}{y^2_{\mathrm{ref}}}\left[\frac{1}{(J-1)^2}{\sum}_{j=1,j\neq\mathrm{ref}}^J \hat{\sigma}^2_j+(\mathrm{ID}_\mathrm{isp}+100)^2\hat{\sigma}^2_{\mathrm{ref}}\right].$$

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

## Variance of $$\widehat{{ID}_{isp}}$$ Based on the Monte Carlo Simulation Based 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 $$ID_{isp}^{(m)}$$ using $$y_{j}^{(m)}$$, and the variance of $$\widehat{ID_{isp}}$$ is:

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

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

## 95% Confidence Interval of $$\widehat{ID_{isp}}$$

The 95% confidence interval of $$\widehat{ID_{isp}}$$ based on Taylor Series Linearization variance is:

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

The lower and upper boundaries of the 95% confidence interval of $$\widehat{ID_{isp}}$$ based on the MCS approach are the 2.5th percentile and 97.5th percentile of the $$1,000\widehat{ID}_{isp}^{(m)}$$ values.