"Why functional measurement is (still) better than conjoint measurement: Judgement of numerosity by children and adolescents"

Supporters of Conjoint Measurement (CM) continue to cite an argument from Krantz, Luce, Suppes, and Tversky (1971) from a reanalysis of the city-occupation study by Sidowski and Anderson (1967). Anderson (1982) refuted this argument and demonstrated the superiority of Functional Measurement (FM); one of the ar- guments made by Anderson is that CM lacks power to reveal substantive interac- tions. Still, Coxon (2006, p. 9) recently concluded, “the interaction (in Sidowski & Anderson) is an artifact of the assumption that the rating scale is interval level...(because) an order preserving [monotone] additive representation is possiFunctional measurement vs conjoint measurement 11 ble” (p. 7). In the present study, first grade children, fourth grade children, and adolescents assessed judgments of numerosity based on size (area) and density. FM analyses revealed that adolescents followed the normative multiplicative rule, while children were generally additive. CM analyses, in contrast, were insensitive to the different integration rules followed by children and adolescents, i.e., an ad- ditive representation existed for all ages. Thus, an important developmental trend that is apparent both graphically and statistically in FM analyses disappeared in CM analyses. This not only supports Anderson’s arguments about the superiority of FM techniques, it provides a relevant counter-example to the CM arguments based on the Sidowski and Anderson study.