Erreygers and Van Ourti's (2011b) comment on my paper (Wagstaff, 2011) leaves me with the overall impression that a consensus is emerging on some key issues in this field and that the areas of disagreement are narrowing.
The main claim of my paper was that binary variables are amenable to both relative and absolute inequality analyses, which have the properties of a ratio-scale variable.
In their comment, Erreygers and Van Ourti said that they agree that a binary variable can be used in an inequality analysis. In their earlier paper (Erreygers and Van Ourti, 2011a), they could perhaps have been a little clearer on this point. There they said that inequality cannot be undertaken on nominal and ordinal variables, and in their Table 1, they classified binary variables as ordinal variables. One has to dig down to a footnote in their paper to see their suggestion that 'one might overcome the ordinal nature by assuming that it expresses the presence of a certain condition in percentage points'. My reading of the literature is that there is actually a general agreement that binary variables can safely be treated as ratio-scale variables; it is not a question of one needing to 'overcome' their ordinal nature. It is also a little surprising that their admission that binary variables are amenable to inequality analyses is buried in a footnote! Anyway, the key thing is that we agree.
There also seems to be a broad agreement that one can apply the concepts of relative and absolute inequalities to binary variables. We all agree that some of the transformation tests that one usually does cannot be performed in the case of binary variables-one cannot for example double everyone's immunization status. Erreygers and Van Ourti would like us to think in terms of 'quasi-relativity' and 'quasi-absoluteness', defining precisely the transformations that are possible. I am not particularly convinced: the term 'quasi' is rather misleading (we are still talking about the same concepts after all); and it is not clear what is being gained by adding this added layer of complexity to the analysis. But we should not be hung up on semantics. We all agree that some transformations cannot be carried out but that we can basically still talk of absolute and relative inequalities in the case of binary variables.
We are also in agreement that with binary (and other bounded) variables two challenges arise that do not arise with unbounded variables: (a) the range of the concentration index is linked to the mean of the variable, a point made originally by Wagstaff (2005); and (b) the ranking of countries, time periods, and so on can vary depending on whether the good attribute is coded 0 or 1, a point made by Clarke et al. (2002); Erreygers (2009a) referred to this latter issue as the mirror property. My normalization, W, gets around the first problem, and as Erreygers (2009a) noted, it gets around the second one too. Erreygers' normalization, E, also gets around both problems. Erreygers' (2009a) paper, the debate in the Journal of Health Economics that followed (Erreygers, 2009b;Wagstaff, 2009), and these two papers in Health Economics (Erreygers and Van Ourti, 2011b;