Many psychological attributes — loudness, preference, utility — cannot be physically concatenated the way lengths or masses can. Conjoint measurement, introduced by R. Duncan Luce and John Tukey in 1964, solved this fundamental problem by showing that interval-scale measurement is possible whenever an attribute depends jointly on two or more independent factors whose effects combine additively.
The Additive Model
Consider stimuli that vary on two dimensions, A and P. A subject judges which of two stimuli (a, p) or (b, q) is greater on some attribute. Conjoint measurement asks: under what conditions can we assign numerical values φ₁ to levels of A and φ₂ to levels of P such that the overall ordering is captured by the sum φ₁(a) + φ₂(p)?
2. Independence: (a, p) ≽ (b, p) for all p ⟹ (a, q) ≽ (b, q) for all q
3. Thomsen condition: if (a, q) ~ (c, p) and (c, r) ~ (b, q), then (a, r) ~ (b, p)
4. Archimedean: no infinite sequences of equally spaced elements
Significance for Psychology
The independence axiom is the most psychologically substantive: it states that the ordering of levels on one factor does not depend on the level of the other factor. This is empirically testable. If independence fails, the simple additive model is rejected, and more complex polynomial or non-additive representations may be needed.
Krantz and Tversky extended conjoint measurement to polynomial models where the factors can interact multiplicatively. This generalization accommodates situations where the joint effect is not simply additive — for example, when the perceived quality of a product depends on the product of two factors rather than their sum.
Conjoint measurement demonstrated that rigorous quantitative measurement does not require a physical concatenation operation. This was a landmark result for psychology, where attributes like sensation, preference, and utility had long resisted the kind of measurement possible in physics.