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# Kunst Mackenbach Relative Index (KMI)

Kunst and Mackenbach (17) modified this definition of the RII slightly by dividing the estimated health of the hypothetical person at the bottom of the socio-economic group distribution by the estimated health of the hypothetical person at the top of the socio-economic group distribution which is denoted as KMI.  Measure KMI can be estimated as:

$$\widehat{KMI} = \frac{\widehat{\beta}_0}{\widehat{\beta}_0 + \widehat{\beta}_1} = \frac{\widehat{\beta}_0}{\widehat{\beta}_0 + \mathrm{SII}}$$

where $$\hat{\beta}_0 = y_j - \widehat{\beta}_1R_j$$, $$R_j$$ is the relative rank of the jth socioeconomic group.

Thus, the Kunst-Mackenbach Relative Index (KMI) is more like a traditional relative risk measure in that it compares the health of the extremes of the socio-economic distribution, but it is estimated using the data on all socioeconomic groups and is weighted to account for socioeconomic-group size.  As noted above, the use of the $$SII$$ and $$RII$$ indices depends on having a socioeconomic group classification scheme with an inherent ordering.

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

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

$$\mathrm{var}_{TL}(\widehat{KMI})={\sum}_{j=1}^J\left[\frac{p_j\widehat{\beta}_1}{(\widehat{\beta}_0+\widehat{\beta}_1)^2}-\frac{p_j\left[R_j-{\sum}_{j=1}^J\left(p_jR_j\right)\right]\mu}{(\widehat{\beta}_0+\widehat{\beta}_1)^2\left[{\sum}_{j=1}^Jp_jR_j^2-\left({\sum}_{j=1}^Jp_jR_j\right)^2\right]^2}\right]^2\hat{\sigma}_j^2,$$

where μ is the average health status of the population, $$\mu={\sum}_{j=1}^Jp_jy_j$$; yj is the health status of group j; pj is the population share of the jth socio-economic group; Rj is the relative rank of the jth socio-economic group, $$R_j = {\sum}_{\gamma=1}^{j-1}p_\gamma - 0.5p_j$$.

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

Randomly generate M age-adjusted rates $$y_j^{(m)}, m = 1, \ldots, M$$ using the distribution:  $$y_j^{(m)} \sim N(y_j, \hat{\sigma}^2_j)$$for each socio-economic group.  Calculate $$\mathrm{KMI}^{(m)}$$ using $$y_j^{(m)}$$.  Then:

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

where $$\overline{\widehat{KMI}} = M^{-1}{\sum}_{m=1}^M \widehat{KMI}^{(m)}$$.  M=1,000 are used in HD*Calc. The standard error of $$\widehat{KMI}$$ based on the MCS method is:  $$\sqrt{\mathrm{var}_{MCS}(\widehat{KMI})}$$.  Gamma distribution instead of truncated normal distribution was recommended to simulate age-adjusted rates (1).  See Ahn et al 2018 (4) for details on how $$\mathrm{var}_{TL}(\widehat{KMI})$$ was derived.

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

The 95% confidence interval of $$\widehat{KMI}$$ based on Taylor Series linearization variance are:

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

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