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 *j*th 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, \(\mu={\sum}_{j=1}^Jp_jy_j\); *y _{j} *is the health status of group

*j*;

*p*is the population share of the

_{j}*j*th socio-economic group;

*R*is the relative rank of the

_{j}*j*th 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 2.5^{th }percentile and the 97.5^{th} percentile of the \(1,000\mathrm{KMI}^{(m)} \) values.