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Slope Index of Inequality (SII) captures the difference in the average health status between a person in the highest socioeconomic group and a person in the lowest socioeconomic group. SII was introduced by Preston, Haines, and Pamuk (10), and it is derived from the following simple linear regression model:
$$y_j = \beta_0 + \beta_1R_j$$
where \(\beta_0\) is the estimated health status of a hypothetical person at the bottom of the socioeconomic group hierarchy (i.e., a person whose relative rank \(R_j\) in the socioeconomic group distribution is zero), and \(\beta_1\), i.e., \(SII\), is the difference in average health status between the hypothetical person at the bottom of the socioeconomic group distribution and the hypothetical person at the top assuming linearity. \(SII \) is estimated using weighted least square using the population share \(p_j\) as the weight:
$$\widehat{SII}=\widehat{\beta}_1=\frac{{\sum}_{j=1}^Jp_jR_j(y_j-\mu)}{{\sum}_{j=1}^Jp_jR_j^2-({\sum}_{j=1}^Jp_jR_j)^2},$$
where \(y_j\) is the health status of group \(j\); \(p_j\) is the population share of the jth group; μ is the average level of health status in the population, \(\mu = {\sum}_{j=1}^Jp_jy_j\); \(R_j\) is the relative rank of the \(j\)th socioeconomic group, \({R}_{j}={\sum}_{\gamma=1}^{j-1}({p}_{j}+0.5{p}_{j})\).
Variance of \(\widehat{SII}\)
The variance of based on weighted least square method can be calculated as:
$$\mathrm{var}_{TL}(\widehat{SII})=\frac{{\sum}_{j=1}^Jp_j^2R^2_j\hat{\sigma}^2_j + \left[{\sum}_{j=1}^Jp_jR_j\right]^2{\sum}_{j=1}^Jp_j^2\hat{\sigma}^2_j-2{\sum}_{j=1}^J(p_jR_j){\sum}_{j=1}^J(p^2_jR_j\hat{\sigma}^2_j)}{\left[{\sum}_{j=1}^Jp_jR_j^2-({\sum}_{j=1}^Jp_jR_j)^2\right]^2}.$$
The standard error of \(\widehat{SII}\) based on weighted least square method is: \(\sqrt{\mathrm{var}_{TL}(\widehat{SII})}\). See Ahn et al 2018 (4) for details on how \(\mathrm{var}_{TL}(\widehat{SII})\) was derived. We use notation \(\mathrm{var}_{TL}\) for consistency purpose.
Variance of \(\widehat{SII}\) based on Monte Carlo Simulation 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 socio-economic group. Calculate \(\widehat{SII}\) using \(y_j^{(m)}\). Then:
$$\mathrm{var}_{MCS}(\widehat{SII})=(M-1)^{-1}{\sum}_{m=1}^M(\widehat{SII}^{(m)}-\overline{\widehat{SII}})^2,$$
where \(\overline{\widehat{SII}}=M^{-1}{\sum}_{m=1}^M\widehat{SII}^{(m)}\). \(M = 1,000\) are used in HD*Calc. Randomly generate M age-adjusted rates \(y_{j}^{(m)},m=1,...,M,\) using the distribution: \(y_{j}^{(m)}\sim Gamma(mean=y_{j},var=\widehat{\sigma}_{j}^{2})\) for each social group. The standard error of \(\widehat{SII}\) based on the MCS method is: \(\sqrt{\mathrm{var}_{MCS}(\widehat{SII})}\). Gamma distribution instead of truncated normal distribution was recommended to simulate age-adjusted rates (1).
95% Confidence Interval of \(\widehat{SII}\)
The 95% confidence interval of \(\widehat{SII}\) based on the weighted least square method is:
$$\widehat{SII} \pm 1.96 \times \sqrt{\mathrm{var}_{TL}(\widehat{SII})}.$$
The two bounds of the 95% confidence interval of \(\widehat{SII}\) based on the MCS approach are the 2.5th percentile and 97.5th percentile of the \(1,000\widehat{SII}^{(m)}\) values.
