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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.