Theil Index Health

Theil Index Health and the Decomposition of Health Inequality

Applied to health, the Theil index compares how a health outcome is shared out with how the population is shared out, and adds the logarithmic gaps into one figure of relative inequality. A value of zero means every person or group holds a share of the outcome equal to its share of the population, and the figure rises as health becomes concentrated in some people or groups. Analysts use it to ask where inequality sits, between regions or ethnic groups or among people inside them. It comes in two versions, Theil's T and the mean log deviation, which weight groups differently and can move in opposite directions on the same data. Like other measures of health inequality, it describes a distribution and leaves the judgement about fairness to a separate step.

Information theory behind the index

Henri Theil set out the measure in Economics and Information Theory in 1967, and Harper and Lynch note that economists often favour this general entropy class. Conceição and Ferreira note that Theil's first presentation rested on statistical information theory, and explain the index more intuitively in terms of groups. The index sums, across groups, the weighted logarithm of the ratio between each group's share of the resource and its share of the population, so a group whose two shares match contributes nothing.

Harper and Lynch apply the index to body mass index across social groups and to colorectal cancer mortality across racial groups in the United States, and the WHO handbook on health inequality monitoring applies it to service coverage across regions. With individual data the index is written as follows.

$$T = \frac{1}{N} \sum_{i=1}^{N} \frac{y_i}{\mu} \ln\left(\frac{y_i}{\mu}\right)$$

where $T$ is Theil's T, $N$ is the number of people, $y_i$ is the health outcome of person $i$ on a non-negative ratio scale with a true zero, $\mu$ is the population mean of the outcome (which must be positive) and $\ln$ is the natural logarithm. A person with a value of zero contributes zero, because $y \ln y$ tends to zero as $y$ falls to zero.

With grouped data, the WHO handbooks and the Health Equity Assessment Toolkit (HEAT) technical notes use population shares and ratios to the overall mean.

$$T_B = \sum_{j=1}^{J} p_j \, r_j \ln r_j, \qquad r_j = \frac{\mu_j}{\mu}$$

where $T_B$ is the Theil index across groups, $J$ is the number of groups, $p_j$ is the population share of group $j$, $\mu_j$ is the mean outcome in group $j$ and $r_j$ is its ratio to the overall mean $\mu$. WHO tools multiply the result by 1000 to make it easier to read.

Groups below the mean have negative terms, yet the WHO 2013 handbook notes that the index itself is never negative, and it equals zero only when every ratio is one. The handbook gives the index no maximum, and with $N$ equally weighted people its largest value is $\ln N$, reached when one person holds all the measured health.

Theil's T and the mean log deviation

Haughton and Khandker, in a World Bank handbook, place both versions in the generalised entropy family GE($\alpha$). GE(1) is Theil's T, and GE(0) is Theil's L, also called the mean log deviation (MLD). They note that for lower values of $\alpha$ the measure is more sensitive to changes in the lower tail of the distribution, and for higher values to changes in the upper tail.

$$L = \frac{1}{N} \sum_{i=1}^{N} \ln\left(\frac{\mu}{y_i}\right), \qquad L_B = \sum_{j=1}^{J} p_j \ln\left(\frac{1}{r_j}\right)$$

where $L$ is the mean log deviation for individuals, $L_B$ is its grouped form and the other symbols are as above. Because $\ln(\mu / y_i)$ has no finite value when $y_i$ is zero, the MLD cannot be computed for a distribution in which anyone has a value of zero.

Harper and Lynch note that Theil's T gives relatively more weight to the upper end of the health distribution. In their colorectal cancer example T is more affected by high mortality ratios and the MLD by large population shares, so T responds more to mortality change in the black population and the MLD to change in the much larger white population. Harper and colleagues add that, in the mean log deviation, the logarithm makes the contribution of a change of a given size depend on where in the distribution it occurs, a value judgement built into the measure.

The two versions can disagree in direction. In a WHO 2024 example from Indonesia, the MLD for the prevalence of children with zero doses of DTP vaccine across subnational regions fell slightly (389.2 in 2007 and 376.2 in 2017, multiplied by 1000), while the Theil index rose sharply (197.3 and 248.5).

Splitting inequality between and within groups

The property that sets the Theil family apart from the Gini and Atkinson measures is exact additive decomposition. Harper and Lynch state that when people are arranged into $J$ groups, Theil showed that the index is the exact sum of between-group disparity and a weighted average of within-group disparity.

$$T = \sum_{j=1}^{J} p_j \, r_j \ln r_j + \sum_{j=1}^{J} s_j T_j, \qquad s_j = p_j \, r_j$$

where the first sum is the between-group term $T_B$, $T_j$ is Theil's T calculated within group $j$ around that group's own mean, $s_j$ is group $j$'s share of the total health outcome and the other symbols are as above. For the mean log deviation the within-group weights are population shares instead, so $L = L_B + \sum_{j} p_j L_j$, where $L_j$ is the MLD within group $j$, as Conceição and Ferreira show for the general family.

Haughton and Khandker state that the generalised entropy class can be decomposed across groups in this additive way and that the Gini index cannot. Harper and Lynch add the Atkinson measure to those that do not decompose perfectly. The between-group term needs only each group's population share and its ratio to the overall mean, so it can be computed from published group rates without individual records. Harper and Lynch point out that the within-group term requires continuous outcomes measured in individuals, which leaves its use for event-based outcomes such as incidence, mortality or screening unclear.

Harper and Lynch see the main advantage in tracking whether the share of total disparity lying between groups is rising or falling, since between-group disparity can rise because total disparity is rising.

Worked example with two areas and eight people

All figures are illustrative. Eight people in two areas are followed for 25 years, and the outcome is years lived in full health. Area 1 holds people with 4, 8, 12 and 16 years and Area 2 people with 10, 14, 16 and 20 years.

1. Overall mean. The total is 100 years, so $\mu = 100 / 8 = 12.5$ years.

2. Theil's T for everyone. The ratios $y_i / \mu$ are 0.32, 0.64, 0.96, 1.28, 0.80, 1.12, 1.28 and 1.60. The products $(y_i/\mu)\ln(y_i/\mu)$ are minus 0.3646, minus 0.2856, minus 0.0392, 0.3160, minus 0.1785, 0.1269, 0.3160 and 0.7520, which sum to 0.6430. Dividing by 8 gives $T = 0.0804$.

3. Between-area term. The area means are 10 and 15 years, so $r_1 = 0.8$ and $r_2 = 1.2$, each with a population share of 0.5. The term is $T_B = 0.5 \times 0.8 \times \ln 0.8 + 0.5 \times 1.2 \times \ln 1.2 = -0.0893 + 0.1094 = 0.0201$.

4. Within-area term. Around its own mean of 10, Area 1's ratios are 0.4, 0.8, 1.2 and 1.6, the products sum to 0.4258 and $T_1 = 0.4258 / 4 = 0.1064$. Around a mean of 15, Area 2's products sum to 0.1177, so $T_2 = 0.0294$. The areas hold 40 and 60 of the 100 years, so the weights are 0.4 and 0.6, and the within term is 0.4 × 0.1064 + 0.6 × 0.0294 = 0.0602, the sum of 0.0426 and 0.0177.

5. Check and share. The parts add to 0.0201 + 0.0602 = 0.0803, which matches the total of 0.0804 once rounding is removed (0.02014 + 0.06023 = 0.08037). About 25% of measured inequality lies between the areas and 75% within them.

6. Mean log deviation. The between term is $L_B = 0.5 \ln(12.5/10) + 0.5 \ln(12.5/15) = 0.1116 + (-0.0912) = 0.0204$. The within-area MLDs are 0.1218 and 0.0306, weighted by population shares of 0.5, giving 0.0762, and the total is $L = 0.0966$. On this measure only 21% lies between the areas, because the MLD gives more weight to the low values in Area 1.

7. Health shortfall and a uniform gain. Measured as shortfall from 25 years, the same people have 21, 17, 13, 9, 15, 11, 9 and 5 years, and T becomes 0.0747. Adding 2 healthy years to everyone lowers T to 0.0583, while multiplying every value by 1.2 leaves it at 0.0804.

A policy aimed only at the gap between the two area averages addresses the between-area share, about a quarter of measured inequality on T and about a fifth on the MLD.

Reading the index in monitoring and evaluation

The WHO 2013 handbook presents the Theil index as a useful measure of relative inequality across subgroups with no natural order, such as regions, where the slope index of inequality and the concentration index do not apply. It also warns that values may be hard for non-technical audiences to interpret. In its Egyptian example, regional inequality in antenatal care with at least four visits gave a Theil index of 176.78 (multiplied by 1000) in 1995, and the value roughly halved at each later survey.

Harper and Lynch met the same difficulty. Racial disparity in colorectal cancer mortality among people aged 45 to 64 gave a Theil index of 0.0124 in 1990 and 0.0198 in 2001, while the between-group variance, an absolute measure, stayed roughly constant. They note that, because entropy measures had been little used in health disparities research, whether such values are large or small was hard to judge.

The WHO 2024 manual notes that the Theil index and the MLD were first developed to measure economic inequality, where the resource in question, wealth, is subject to redistribution.

In economic evaluation the measure has a narrower role. The distributional cost-effectiveness analysis tutorial by Asaria, Griffin and Cookson compares screening strategies using gap measures, the slope and relative indices of inequality, the Gini coefficient and Atkinson and Kolm indices with stated inequality aversion parameters, and no Theil index. Conceição and Ferreira, following Sen, class the Theil index among objective measures that contain no notion of social welfare, so it can show where inequality lies without saying how much health a decision maker should give up to reduce it.

Theil index beside the Gini, Atkinson and concentration indices

Several measures summarise the same distribution and answer different questions. The Gini coefficient has a direct reading from the Lorenz curve but no exact split between and within groups. The Atkinson index carries an explicit inequality aversion parameter, which the Theil index lacks.

The concentration index and the slope index of inequality need groups with a natural order, such as wealth or education, because they rely on cumulative fractional rank. The Theil index ignores order, so it suits regions, ethnic groups or combinations of factors. Harper and Lynch note that decomposable measures also let cross-classified groups, such as gender and education, be examined jointly.

Choices that change the result and common misreadings

Most errors with the Theil index come from treating one number as if it were free of the analyst's choices. Two analysts with the same survey can report different, equally correct values.

1. Achievement or shortfall. The index is relative, so healthy years and years of shortfall give different values, as do survival and mortality. The analysis should state which direction is measured and keep it fixed across comparisons.
2. Uniform gains. Haughton and Khandker list mean independence among the properties of a good measure, and doubling every value leaves the index unchanged. Adding the same gain to everyone reduces it, as step 7 shows, so a fall in the Theil index can coexist with unchanged absolute gaps.
3. Zeros and negative values. Theil's T accepts zeros but the MLD does not, and neither accepts negative values, so raw change scores need transforming first.
4. Grouping. Grouped data capture only between-group inequality. Conceição and Ferreira stress that total inequality among individuals is almost never observed directly, and Harper and Lynch leave open how the number of groups compared affects the level.
5. Missing groups and uncertainty. The HEAT notes treat the index as missing if any subgroup estimate or population share is missing, and give a variance formula based on subgroup standard errors.
6. Description and fairness. A high Theil index shows concentration of the outcome. Whether that concentration is unjust is a question of health inequity, which needs a normative judgement the index does not supply.

Sources

• Asaria M, Griffin S, Cookson R. Distributional cost-effectiveness analysis: a tutorial. Medical Decision Making. 2016;36(1):8-19. https://doi.org/10.1177/0272989X15583266
• Conceição P, Ferreira P. The Young Person's Guide to the Theil Index: Suggesting Intuitive Interpretations and Exploring Analytical Applications. UTIP Working Paper 14. Austin, TX: University of Texas Inequality Project; 2000. https://utip.lbj.utexas.edu/papers/utip_14.pdf
• Harper S, King NB, Meersman SC, Reichman ME, Breen N, Lynch J. Implicit value judgments in the measurement of health inequalities. Milbank Quarterly. 2010;88(1):4-29. https://doi.org/10.1111/j.1468-0009.2010.00587.x
• Harper S, Lynch J. Methods for Measuring Cancer Disparities: Using Data Relevant to Healthy People 2010 Cancer-Related Objectives. NCI Cancer Surveillance Monograph Series, Number 6. NIH Publication No. 05-5777. Bethesda, MD: National Cancer Institute; 2005. https://drum.lib.umd.edu/items/4f9ca2ad-c0fa-4aa2-a4ff-acc1d27b5502
• Haughton J, Khandker SR. Handbook on Poverty and Inequality. Washington, DC: World Bank; 2009. Chapter 6, Inequality measures. https://documents1.worldbank.org/curated/en/488081468157174849/pdf/483380PUB0Pove101OFFICIAL0USE0ONLY1.pdf
• World Health Organization. Handbook on health inequality monitoring: with a special focus on low- and middle-income countries. Geneva: World Health Organization; 2013. https://iris.who.int/handle/10665/85345
• World Health Organization. Health inequality monitoring: harnessing data to advance health equity. Geneva: World Health Organization; 2024. https://iris.who.int/server/api/core/bitstreams/7ed039ab-4a7f-4f66-9b4b-264008d98f81/content
• World Health Organization. Health Equity Assessment Toolkit (HEAT and HEAT Plus): technical notes. Version 7.1. Geneva: World Health Organization; 2026. https://srhdpeuwpubsa.blob.core.windows.net/whdh/HIDR/HEAT_HEATPlus_technical_notes.pdf