Aggregation properties of relative impact and other classical indicators: Convexity issues and the Yule-Simpson paradox

Among classical bibliometric indicators, direct and relative impact measures for countries or other players in science are appealing and standard. Yet, as shown in this article, they may exhibit undesirable statistical properties, or at least ones that pose questions of interpretation in evaluation and benchmarking contexts. In this article, we address two such properties namely sensitivity to the Yule-Simpson effect, and a problem related to convexity. The Yule-Simpson effect can occur for direct impacts and, in a variant form, for relative impact, causing an apparent incoherence between field values and the aggregate (all-fields) value. For relative impacts, it may result in a severe form of ‘out-range’ of aggregate values, where a player’s relative impact shifts from ‘good’ to ‘bad’, or conversely. Out-range and lack of convexity in general are typical of relative impact indicators. Using empirical data, we suggest that, for relative impact measures, ‘out-range’ due to lack of convexity is not exceptional. The Yule-Simpson effect is less frequent, and especially occurs for small players with particular specialisation profiles.