Mean Log Deviation (MLD)

Mean Log Deviation (MLD) (7) measures the general disproportionality as

$${MLD} = -{\sum}_{j=1}^{J}p_j \ln (\frac{\mu_{j}}{\mu}) = \ln (\mu)- {\sum}_{j=1}^J p_j \ln(\mu_j),$$

where \(\mu_{j}\) is the health status of group \(j;p_{j}\) is the population share of group \(j;\mu\) is the average health status in the population \(\mu = {\sum}_{j=1}^Jp_{j}\mu_{j}.\)  

The measure is population-weighted and is more sensitive to health differences further from the population average (by the use of the logarithm) and may be used for both ordered socioeconomic groups (e.g., education) and unordered groups (e.g., gender, race).  MLD is a special case of the generalized entropy index (Shorrocks 1980).

Point Estimator of \(\widehat{MLD}\)

A survey design consistent estimator is

$$\widehat{MHD}=ln\left(\widehat{\mu}\right)-{\sum}_{j=1}^{J}\widehat{p}_{j}ln\left(\widehat{\mu}_{j}\right),$$

where

$$\widehat{\mu}_{j}=\frac{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}\delta_{h\alpha{i}}^{j}w_{h\alpha{i}}y_{h\alpha{i}}}{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}\delta_{h\alpha{i}}^{j}w_{h\alpha{i}}}$$

is the estimated average health in group \(j\),

$$\widehat{p}_{j}=\frac{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}\delta_{h\alpha{i}}^{j}w_{h\alpha{i}}}{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}w_{h\alpha{i}}}$$

is the estimated population share of group \(j\), and \(\widehat{\mu}={\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{\mu}_{j}\) is the estimated average health in the population.

Variance and Confidence Intervals of \(\widehat{MLD}\)

The variance for \(\widehat{MLD}\) is

$$var\left(\widehat{MLD}\right)\cong{\sum}_{h=1}^{H}\frac{t_{h}}{t_{h}-1}{\sum}_{\alpha=1}^{t_{h}}\left(Z_{h\alpha}-\overline{Z}_{h}\right)\left(Z_{h\alpha}-\overline{Z}_{h}\right)^{T},$$

where

$$Z_{h\alpha}={\sum}_{i=1}^{n_{h\alpha}}w_{h\alpha{i}}z_{h\alpha{i}},$$

$$\overline{Z}_{h}=\frac{{1}}{t_{h}}{\sum}_{\alpha=1}^{t_{h}}Z_{h\alpha},$$

and

$${z}_{h\alpha{i}}=\frac{{\partial}\widehat{MLD}}{{\partial}{w}_{h\alpha{i}}}={\sum}_{j=1}^{J}\left(\frac{{\partial}\widehat{MLD}}{\partial\widehat{p}_{j}}\times\frac{{\partial}\widehat{p}_{j}}{\partial{w}_{h\alpha{i}}}+\frac{{\partial}\widehat{MLD}}{\partial\widehat{\mu}_{j}}\times\frac{{\partial}\widehat{\mu}_{j}}{\partial{w}_{h\alpha{i}}}\right),$$

with

$$\frac{\partial\widehat{MLD}}{\partial\widehat{\mu}_{j}}=\widehat{p}_{j}\left(\frac{1}{\widehat{\mu}}-\frac{1}{\widehat{\mu}_{j}}\right),$$

$$\frac{{\partial}\widehat{\mu}_{j}}{{\partial}{w}_{h\alpha{i}}}=\frac{{\partial}_{h\alpha{i}}^{j}\left({y}_{h\alpha{i}}-\widehat{\mu}_{j}\right)}{{\sum}_{h=1}^{H}{\sum}_{a=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\partial}_{h\alpha{i}}^{j}{w_{h\alpha{i}}}},$$

$$\frac{\partial\widehat{MLD}}{\partial\widehat{p}_{j}}=\frac{\widehat{\mu}_{j}}{\widehat{\mu}}-ln\left(\widehat{\mu}_{j}\right),$$

and

$$\frac{{\partial}\widehat{p}_{j}}{{\partial}{w}_{h\alpha{i}}}=\frac{{\partial}_{h\alpha{i}}^{j}-\widehat{p}_{j}}{{\sum}_{h=1}^{H}{\sum}_{a=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{w_{h\alpha{i}}}},$$

The 95% confidence intervals of \(\widehat{MLD}\) are:

$$\widehat{MLD}\pm1.96\times\sqrt{\mathrm{var}(\widehat{MLD})}.$$

For details about the derivations, see Li et al. (18).