Theil Index (T)

The Theil Index (T), developed by economist Henri Theil (15), measures general disproportionality.  The formula is as follows:

$${T} = {\sum}_{j=1}^{J}p_{j}r_{j}\ln(r_j),$$

where \(p_{j}\) is the population share of group \(j,r_{j}=\frac{\mu_{j}}{\mu}\) is the ratio of the health status of group \(j\) relative to the average health status in the population \(\mu={\sum}_{j=1}^{J}p_{j}\mu_{j}.\)

T is population-weighted and is more sensitive to health differences further from the population average (by using the logarithm) and may be used for both ordered socioeconomic groups (e.g., education) and unordered groups (e.g., gender, race).

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

A survey design consistent estimator is

$$\widehat{T} = {\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{r}_{j}\ln(\widehat{r}_{j}),$$

where

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

is the estimated population share of group \(j,\widehat{r}_{j}=\frac{\widehat{\mu}_{j}}{\widehat{\mu}}\) is the estimated ratio of the health status in group

$$j,\widehat\mu_{j}=\frac{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\delta}^{j}_{h\alpha{i}}{w}_{h\alpha{i}}{y}_{h\alpha{i}}}{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_{h}}{\sum}_{i=1}^{n_{h\alpha}}{\delta}^{j}_{h\alpha{i}}{w}_{h\alpha{i}}},$$

relative to the estimated average health in the population \(\widehat{\mu}={\sum}_{j=1}^{J}\widehat{p}_{j}\widehat{\mu}_{j}\).

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

Variance of \(\widehat{T} \) is:

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

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

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

with

$$\frac{\partial\widehat{ID}_{isp}}{\partial\widehat{\mu}_{j}}=\frac{100\left(-1\right)^{1=I\{\widehat{\mu}_{j}>\widehat{\mu}_{ref}\}}}{\left(J-1\right)\widehat{\mu}_{ref}}forj\neq{ref},$$

$$\frac{\partial\widehat{ID}_{isp}}{\partial\widehat{\mu}_{j}}=\frac{100}{\left(J-1\right)\widehat{\mu}_{ref}^{2}}{\sum}_{j=1}^{J-1}\widehat{\mu}_{j}\left(-1\right)^{I\{\widehat{\mu}_{j}>\widehat{\mu}_{ref}\}} forj=ref$$

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

The 95% confidence intervals of \(\widehat{ID}_{isp}\) are:

$$\widehat{ID}_{isp}\pm1.96\times\sqrt{\mathrm{var}(\widehat{ID}_{isp})}.$$