Although the dissimilarity index might therefore be seen as a measure of inequality of opportunity, the HOI itself clearly cannot.<sup>17</sup> It is intended – and defined – as a measure of average access, adjusted (or penalized) by inequality of opportunity. Unsurprisingly, therefore, it is closely correlated with other indicators of “level of development”. This association is already clear in Figure 7, which ranks the average HOI for all countries in Table 3, ranging from 9.6 in Niger, to 91.6 in Chile. There is almost no overlap in HOI between the African and the Latin American sub-samples, and the correlation between the HOI and GNI per capita for these countries is 0.89 (p-value: 0.0005).

Perhaps more striking is the correlation with the UNDP’s Human Development Index which is even higher (at 0.94) and highly statistically significant. Figure 8 presents the scatter plot. This is remarkable because the two indices are constructed on the basis of completely different data. Until 2010 (the year used in Figure 8), the Human Development Index was calculated as a simple average of three normalized indices in the dimensions of health, income and education.<sup>18</sup> The income index used GNP per capita, and the health index was based on life expectancy at birth, while the education index combined information on literacy and the gross school enrolment ratio. Of these four basic components, only one is close to the indicators used to construct the HOI, namely gross enrolment ratio, which is related to the “school attendance” data used in the first column of Table 3. The other four components of the HOI, listed above, do not enter directly into the computation of the HDI, and neither does the latter explicitly adjust for dissimilarity across types in any way. Conversely, life expectancy at birth, GDP per capita and literacy do not enter the HOI explicitly.

A correlation of 0.94 between these two indices, albeit calculated only over a non-representative sample of 39 countries in two of the world’s regions, suggests two things. First, it suggests that the average coverage rates of services like access to water, electricity, etc. are highly correlated with the constituent elements of the HDI. Second, it suggests that the HOI is determined, to a very large extent, by the first term in the product π�<sup>j</sup> �1-D<sup>j</sup> � . In fact, the correlations between average coverage and the component-specific HOI in this sample are extremely high: they are greater than 0.99 for school attendance; access to water; access to electricity; and having finished primary school on time. It is 0.987 for access to sanitation. This implies, of course, that the penalty for inequality of opportunity, �1-D<sup>j</sup> � , accounts for a much smaller share of the variance in the HOI than mean coverage.

A final international comparison issue our data can shed light on is the association between the dissimilarity index (the measure of inequality of opportunity contained within the HOI) and the index of inequality of economic opportunity (IEO-R). The dissimilarity index can be interpreted as the proportion

> 17 A possible caveat with viewing the dissimilarity index within the HOI as a measure of inequality of opportunity is that the index is typically calculated “for children”. This justifies the use of certain variables - like geographic location or education of the adults in the household - as circumstances, which are clearly in the realm of choices for the adults. The argument is that the index applies to children, and these are circumstances from their perspective.  But this then raises the issue of age of responsibility, and whether or not all inequalities in access to services for children below a certain age should not be considered inequality of opportunity. Under that view, unequal access to water or sanitation among five-year olds within the same type (i.e. sharing identical observed circumstances) should also be counted as inequality of opportunity.

> 18 The correlation with the inequality-adjusted Human Development Index introduced for the first time in 2011 is almost the same: 0.95.

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