9 (Atkinson, 2011) might be called for. Sklar’s Theorem, from the statistics literature, tells us that any joint distribution function can be decomposed into the marginal distribution functions of each dimension (corresponding to the dashboard approach), and a **copula function** , which captures the information of the degree of interdependency between the dimensions. Copulas, which are the third approach we want to highlight, have already been used to study the relationship between health and income (Decancq 2009, Quinn 2007). Formally, the copula function is the joint distribution function of _n_ vectors whose elements are the _relative position_ of every individual in each of the _n_ dimensions of well-being. As in the case of the margins, once the copula function is constructed, one could perform a stochastic dominance analysis to assess whether the degree of interdependency between the components has changed in an unambiguous way. In addition, and in the same way that information on unidimensional inequality can be summarized by an index (such as the Gini coefficient or the mean log deviation), measures of rank correlation allow us to order distributions unambiguously in terms of their degree of interdependency (Decancq, 2009).

As an example, we reproduce from Decancq (2009) a graph showing the evolution of the Spearman correlation coefficient, a commonly used measure of rank correlation, in Russia between 1995 and 2003 (Figure 3). The author considers three dimensions: standard of living, health, and schooling, represented respectively by household income, self-assessed health, and years of schooling as primary ranking variables (another three variables are used as secondary ranking variables in case of ties). By way of context, the Russian Human Development Index increased from the beginning to end of the period, albeit with a deterioration around the 1998 financial crises.

In this figure, the dashed line represents the 95% confidence interval obtained by Monte Carlo simulations, while the dotted line represents an alternative computation of the same confidence interval, by bootstrapping. The figure illustrates that the degree of dependence across these three dimensions of well-being has been increasing throughout most of the period: as the average well-being was rising, so was the degree of interdependency between the dimensions of well-being.

Although they differ in technical complexity, these three alternative techniques (Venndiagrammatic representations of the dependency structure; multivariate stochastic dominance and the analysis of copula functions) would seem to avoid the disadvantages associated with both the scalar indices and the dashboard approach. Like the dashboard approach (but unlike scalar indices), they do not require the use of generally arbitrary weights to aggregate across dimensions, with their unpalatable implications in terms of trade-offs. Like scalar indices (but unlike the dashboard approach) they incorporate information about how the deprivations are _jointly_ distributed and allow analysts to take into account different levels or changes in the extent of overlap or correlation between them. This is a menu of analytical approaches that