THE COMMUTATIVE RULE AS NEW TEST FOR ADDITIVE CONJOINT MEASURMENT: THEORY AND DATA
Empirical evaluation of the key axiom underlying additive conjoint representation first studied the double cancellation axiom. That was shown to have considerable redundancy that made the statistical problems formidable. The special case called the Thomsen condition was shown to suffice and not to be redundant. However, it has the undesirable feature, for empirical purposes, of a statistical asymmetry in estimation. This led us to seek a symmetric replacement, which we have found in the commutative rule proposed by Falmagne (1976). In the presence of the usual assumptions, we show that the commutative rule is equivalent to the Thomsen condition, a result that appears to have been overlooked in the literature. We subject this property to empirical evaluation in both loudness and brightness. Current data show support for the commutative rule in both domains and thus for additivity.