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The Mean Log Deviation (MLD) measures general disproportionality. It is a summary of the difference between the shares of health and shares of population after taking natural logarithm transformation respectively. The MLD can be estimated as follows (12):
$$\widehat{MLD} = -{\sum}_{j=1}^{J}p_j \ln (\frac{y_{j}}{\mu}) = \ln (\mu) - {\sum}_{j=1}^J p_j \ln(y_j),$$
where \(y_{j}\) is the health status of group j; pj is the population share of jth group; μ is the average health status in the population \(\mu = {\sum}_{j=1}^Jp_{j}y_{j}\).
The measure is population-weighted and is more sensitive to health differences further from the average rate (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).
Variance of \(\widehat{MLD}\)
The variance of \(\widehat{MLD}\) based on Taylor Series Linearization method is calculated as:
$$\mathrm{var}_{TL}(\widehat{MLD}) = {\sum}_{j=1}^{J}\hat{\sigma}_{j}^{2}p_{j}^{2} {(\frac{1}{\mu}-{\frac{1}{y_{j}}})^{2}.}$$
The standard error of \(\widehat{MLD}\) based on the Taylor Series linearization method is: \( \sqrt{\mathrm{var}_{TL}(\widehat{MDL})}\). See Ahn et al 2018 (4) for details on how \(\mathrm{var}_{TL}(\widehat{MLD})\) was derived.
Variance of \(\widehat{MLD}\) based on the Monte Carlo Simulation Based 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 social group. Calculate \(MLD^{(m)}\) using \(y_{j}^{(m)}\), and the variance of \(MLD\) is: \(\mathrm {var}_{MCS}(\widehat{MLD})=(M-1)^{-1}\mathrm {\sum}_{m=1}^{M}(\widehat{MLD}^{(m)}-\overline{\widehat{MLD}})^{2},\)
where \(\overline{\widehat{MLD}}=M^{-1}\mathrm{\sum}_{m=1}^{M}\widehat{MLD}^{(m)}.\) \(M=1,000\) is used in HD*Calc. The standard error of \(\widehat{MLD}\) based on the MCS method is: \( \sqrt{\mathrm{var}_{MCS}(\widehat{MDL})}\). Gamma distribution instead of truncated normal distribution was recommended to simulate age-adjusted rates (1).
95% Confidence Interval of \(\widehat{MLD}\)
The 95% confidence interval of \(\widehat{MLD}\) based on the Taylor Series Linearization method are:
$$\widehat{MLD} \pm 1.96 \times \sqrt{\mathrm{var}_{TL}(\widehat{MLD})}.$$
The lower and upper bounds of the 95% confidence interval of \(\widehat{MLD}\) based on the MCS method are the 2.5th percentile and 97.5th percentile of the \(1,000\widehat{MLD}^{(m)}\) values.
