Correlation between variables subject to an order restriction, with application to scientometric indices

Variables subject to an order restriction, for instance Y ≤ X, have a bivariate distribution over a non-rectangular joint domain that entails a non-null and potentially large structural relation even if the variables show no association (in the sense that particular ranges of values of X do not co-occur with particular ranges of values of Y). Order restrictions affect a number of scientometric indices (including the h index and its variants) that are routinely subjected to correlational analyses to assess whether they provide redundant information, but these correlations are contaminated by the structural relation. This paper proposes an alternative definition of association between variables subject to an order restriction that eliminates their structural relation and reverts to the conventional definition when applied to variables that are not subject to order restrictions. This alternative definition is illustrated in a number of theoretical cases and it is also applied to empirical data involving scientometric indices subject to an order restriction. A test statistic is also derived which allows testing for the significance of an association between variables subject to an order restriction.