Theory of Conjoint Measurement - Historical Overview

Historical Overview

In the 1930s, the British Association for the Advancement of Science established the Ferguson Committee to investigate the possibility of psychological attributes being measured scientifically. The British physicist and measurement theorist Norman Robert Campbell was an influential member of the committee. In its Final Report (Ferguson, et al., 1940), Campbell and the Committee concluded that because psychological attributes were not capable of sustaining concatenation operations, such attributes could not be continuous quantities. Therefore, they could not be measured scientifically. This had important ramifications for psychology, the most significant of these being the creation in 1946 of the operational theory of measurement by Harvard psychologist Stanley Smith Stevens. Stevens' non-scientific theory of measurement is widely held as definitive in psychology and the behavioural sciences generally (Michell 1999).

Whilst the German mathematician Otto Hölder (1901) anticipated features of the theory of conjoint measurement, it was not until the publication of Luce & Tukey's seminal 1964 paper that the theory received its first complete exposition. Luce & Tukey's presentation was algebraic and is therefore considered more general than Debreu's (1960) topological work, the latter being a special case of the former (Luce & Suppes 2002). In the first article of the inaugural issue of the Journal of Mathematical Psychology, Luce & Tukey 1964 proved that via the theory of conjoint measurement, attributes not capable of concatenation could be quantified. N.R. Campbell and the Ferguson Committee were thus proven wrong. That a given psychological attribute is a continuous quantity is a logically coherent and empirically testable hypothesis.

Appearing in the next issue of the same journal were important papers by Dana Scott (1964), who proposed a hierarchy of cancellation conditions for the indirect testing of the solvability and Archimedean axioms, and David Krantz (1964) who connected the Luce & Tukey work to that of Hölder (1901).

Work soon focused on extending the theory of conjoint measurement to involve more than just two attributes. Krantz 1968 and Amos Tversky (1967) developed what became known as polynomial conjoint measurement, with Krantz 1968 providing a schema with which to construct conjoint measurement structures of three or more attributes. Later, the theory of conjoint measurement (in its two variable, polynomial and n-component forms) received a thorough and highly technical treatment with the publication of the first volume of Foundations of Measurement, which Krantz, Luce, Tversky and philosopher Patrick Suppes cowrote (Krantz et al. 1971).

Shortly after the publication of Krantz, et al., (1971), work focused upon developing an "error theory" for the theory of conjoint measurement. Studies were conducted into the number of conjoint arrays that supported only single cancellation and both single and double cancellation (Arbuckle & Larimer 1976; McClelland 1977). Later enumeration studies focused on polynomial conjoint measurement (Karabatsos & Ullrich 2002; Ullrich & Wilson 1993). These studies found that it is highly unlikely that the axioms of the theory of conjoint measurement are satisfied at random, provided that more than three levels of at least one of the component attributes has been identified.

Joel Michell (1988) later identified that the "no test" class of tests of the double cancellation axiom was empty. Any instance of double cancellation is thus either an acceptance or a rejection of the axiom. Michell also wrote at this time a non-technical introduction to the theory of conjoint measurement (Michell 1990) which also contained a schema for deriving higher order cancellation conditions based upon Scott's (1964) work. Using Michell's schema, Ben Richards (Kyngdon & Richards, 2007) discovered that some instances of the triple cancellation axiom are "incoherent" as they contradict the single cancellation axiom. Moreover, he identified many instances of the triple cancellation which are trivially true if double cancellation is supported.

The axioms of the theory of conjoint measurement are not stochastic; and given the ordinal constraints placed on data by the cancellation axioms, order restricted inference methodology must be used (Iverson & Falmagne 1985). George Karabatsos and his associates (Karabatsos, 2001; Karabatsos & Sheu 2004) developed a Bayesian Markov chain Monte Carlo methodology for psychometric applications. Karabatsos & Ullrich 2002 demonstrated how this framework could be extended to polynomial conjoint structures. Karabatsos (2005) generalised this work with his multinomial Dirichlet framework, which enabled the probabilistic testing of many non-stochastic theories of mathematical psychology. More recently, Clintin Davis-Stober (2009) developed a frequentist framework for order restricted inference that can also be used to test the cancellation axioms.

Perhaps the most notable (Kyngdon, 2011) use of the theory of conjoint measurement was in the prospect theory proposed by the Israeli - American psychologists Daniel Kahneman and Amos Tversky (Kahneman & Tversky, 1979). Prospect theory was a theory of decision making under risk and uncertainty which accounted for choice behaviour such as the Allais Paradox. David Krantz wrote the formal proof to prospect theory using the theory of conjoint measurement. In 2002, Kahneman received the Nobel Memorial Prize in Economics for prospect theory (Birnbaum, 2008).