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Yu et al 2018 (5) adapts the extended Absolute Concentration Index (eACI) with aversion parameters for assessment of health disparities using age-adjusted rates estimated from population-based data. It may only be used with socio-economic groups that have a natural ordering, such as income or education groups. The extended can be calculated as:
$$\widehat{eACI}=\mu (\widehat{eRCI})=\nu\mu {\sum}_{j=1}^{J}p_j(1-R_{j})^{\nu-1}-\nu{\sum}_{j=1}^{J}p_{j}y_{j}(1-R_{j})^{\nu-1},$$
where \(\nu>0\) is the aversion parameter, \(y_{j}\) 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, \(\mu={\sum}_{j=1}^{J}p_{j}y_{j};R_{j}\) is the relative rank of the jth socio-economic group, \(R_{j}={\sum}_{\gamma=1}^{j-1}p_{\gamma}-0.5p_{j}\). Rj essentially indicates the cumulative share of the population up to the midpoint of each group interval.
Variance of \(\widehat{eACI}\)
The variance of \(\widehat{eACI}\) based on Taylor Series linearization method can be calculated as:
$$var_{TL}(\widehat{eACI})=\nu^{2}{\sum}_{j=1}^{J} \widehat{\sigma}_{j}^{2}p_{j}^{{2}}\left[{\sum}_{j=1}^{J}p_{j}(1-R_{j})^{\nu-1}-(1-R_{j})^{\nu-1}\right]^{2}.$$
The standard error of \(eACI\) based on the Taylor Series linearization method is: \(\sqrt{var_{TL}(\widehat{eACI})}\). See Yu et al 2018 (5) for details on how \(var_{TL}(\widehat{eACI})\) was derived.
Variance of \(\widehat{eACI}\) based on Monte Carlo Simulation (MCS) 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 socio-economic group. Then calculate \(\widehat{eACI}^{(m)}\)using \(y_{j}^{(m)}\). Thus:
$$var_{MCS}(\widehat{eACI})=(M-1)^{-1}{\sum}_{m=1}^{M}(\widehat{eACI}^{(m)}-\overline{\widehat{eACI}})^{2},$$
where \(\overline{\widehat{eACI}} = M^{-1}{\sum}_{m=1}^{M}\widehat{eACI}^{(m)}\). \(M=1,000\) is 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 \(\widehat{eACI}\) based on the MCS method is: \(\sqrt{var_{MCS}(\widehat{eACI})}\). Gamma distribution instead of truncated normal distribution was recommended to simulate age-adjusted rates (1).
95% Confidence Interval of \(\widehat{eACI}\)
The 95% confidence interval of \(\widehat{eACI}\) based on Taylor Series Linearization variance estimation method is: \(\widehat{eACI}\pm1.96\times\sqrt{var_{TL}(\widehat{eACI})}.\)
The lower and upper bounds of the 95% confidence interval of \(\widehat{eACI}\) based on the MCS method are the 2.5th percentile and 97.5th percentile of the 1,000 \(\widehat{eACI}^{(m)}\) values.
When the aversion parameter \(\nu=2\), eACI reduces to the Absolute Concentration Index (ACI) (8,9), which measures the extent to which health or illness is concentrated among particular socio-economic groups on the absolute scale. ACI is a measure of the covariance between social rank and health, and it is derived by plotting the cumulative share of the population, ranked by socio-economic status, against the cumulative amount of ill health (i.e., the cumulative contribution of each subgroup to the mean level of health in the population). To be specific, estimate of ACI is written as:
$$\widehat{ACI}={\sum}_{j=1}^{J}p_{j}(2R_{j}-1)y_{j}.$$
The variance of \(\widehat{ACI}\) based on Taylor Series linearization method reduces to:
$$var_{TL}\widehat{ACI}={\sum}_{j=1}^{J}p_{j}^{2}\widehat{\sigma}_{j}^{2}(2R_{j}-1)^{2}.$$
The standard error of \(\widehat{ACI}\) based on Taylor Series linearization method is: \(\sqrt{var_{TL}(\widehat{ACI})}\). See Ahn et al 2018 (4) for more details on ACI.
The variance of \(\widehat{ACI}\) based on the MCS method and the 95% confidence intervals based on both analytic and MCS methods have the same format as those for \(\widehat{eACI}\) when \(\nu=2{.}\)
