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# Relative Index of Inequality (RII)

The Slope Index of Inequality (SII) is a measure of absolute disparity.  Dividing this estimated slope by the average population health, however, provides a relative disparity measure, the Relative Index of Inequality RII (16) which is estimated as:

$$\widehat{RII} = \frac{SII}{\mu}=\frac{\widehat{\beta}_1}{\mu}=\frac{1}{{\sum}_{j=1}^Jp_jR_j^2-({\sum}_{j=1}^Jp_jR_j)^2}\left[\frac{{\sum}_{j=1}^Jp_jR_jy_j}{{\sum}_{j=1}^Jp_jy_j}-{\sum}_{j=1}^Jp_jR_j\right],$$

where μ is the average health status of the population, $$\mu={\sum}_{j=1}^{J}p_{j}y_{j}$$; yj is the health status of group j; pj is the population share of the jth group; Rj is the relative rank of the jth socioeconomic group, $$R_{j}={\sum}_{\gamma=1}^{j-1}p_{\gamma}-0.5p_{j}$$.

The interpretation of $$RII$$ is similar to the $$SII$$, but it now measures the proportionate (in regard to the average population level) rather than the absolute increase or decrease in health between the highest and lowest socioeconomic groups.

## Variance of $$\widehat{RII}$$

Variance of $$\widehat{RII}$$ based on Taylor Series linearization method can be estimated as:

$$\mathrm{var}_{TL}(\widehat{RII}) = \frac{{\sum}_{j=1}^J\left[p_jR_j\mu-p_j{\sum}_{j=1}^Jp_jR_jy_j\right]^2\hat{\sigma}^2_j}{\mu^4\left[{\sum}_{j=1}^Jp_jR_j^2-\left({\sum}_{j=1}^Jp_jR_j\right)^2\right]^2}$$

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

## Variance of $$\widehat{RII}$$ based on Monte Carlo Simulation Method

Randomly generate M age-adjusted rates $$y_{j}^{(m)}=1,...,M$$ using the distribution:  $$y_j^{(m)} \sim Gamma(mean=y_j, var=\hat{\sigma}^2_j)$$ for each socioeconomic group.  Calculate $$\widehat{RII}^{(m)}$$ using $$y_{j}^{(m)}$$.  Then:

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

where $$\overline{\widehat{RII}} = M^{-1}{\sum}_{m=1}^M(\widehat{RII}^{(m)} - \overline{\widehat{RII}})^2$$.  $$M=1,000$$ is used in HD*Calc.  The standard error of $$\widehat{RII}$$ based on the MCS method is:  $$\sqrt{\mathrm{var}_{MCS}(\widehat{RII})}.$$

Gamma distribution instead of truncated normal distribution was recommended to simulate age-adjusted rates (1).

## 95% Confidence Interval of $$\widehat{RII}$$

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

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

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