The invariance properties of the mutual information index of multigroup segregation

In the context of educational segregation by ethnic group, it has been argued that rigorous pairwise segregation comparisons over time or across space should be invariant in two situations: when the ethnic composition of the population changes while the distribution of each ethnic group over the schools remains constant (invariance 1), or when the size distribution of schools changes while the ethnic composition of each school remains constant (invariance 2). This paper makes two contributions to the segregation literature. First, it argues by means of the Mutual Information or M index, which is neither invariant 1 nor 2, that both properties have strong implications, and it provides reasons to defend that the overall segregation index need not satisfy either one. Second, nevertheless, it is shown that in pairwise comparisons this index admits two decompositions into three terms. In the first decomposition, a term is invariant 1 and also satisfies a weak version of invariance 2. In the second decomposition, a term is invariant 2 and also satisfies a weak version of invariance 1. It is shown that these decompositions can be used to reach the analogous ones obtained in Deutsch et al. (2006).