Theory of conjoint measurement
It has been argued that this is due to the high level of formal mathematics involved (e.g., Cliff 1992) and that the theory cannot account for the "noisy" data typically discovered in psychological research (e.g., Perline, Wright & Wainer 1979).
It has been argued that the Rasch model is a stochastic variant of the theory of conjoint measurement (e.g., Brogden 1977; Embretson & Reise 2000; Fischer 1995; Keats 1967; Kline 1998; Scheiblechner 1999), however, this has been disputed (e.g., Karabatsos, 2001; Kyngdon, 2008).
Order restricted methods for conducting probabilistic tests of the cancellation axioms of conjoint measurement have been developed in the past decade (e.g., Karabatsos, 2001; Davis-Stober, 2009).
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.
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) harv error: no target: CITEREFMichell1999 (help).
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.
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).
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).
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.
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.
In the familiar everyday instances, temperature is measured using instruments calibrated in either the Fahrenheit or Celsius scales.
The greater the number of levels in both A and X, the less probable it is that the cancellation axioms are satisfied at random (Arbuckle & Larimer 1976; McClelland 1977) and the more stringent test of quantity the application of conjoint measurement becomes.
As they involve infinitistic concepts, the solvability and Archimedean axioms are not amenable to direct testing in any finite empirical situation.
It is most commonly interpreted as the latter, given that most behavioural scientists consider that their tests and surveys "measure" attributes on so-called "interval scales" (Kline 1998).
van der Ven 1980 proposed a scaling method for conjoint structures but he also did not discuss the unit.
Hence it would seem that application of conjoint measurement requires some prior descriptive theory of the relevant natural system.
Gigerenzer & Strube 1983 observed that the evaluation of double cancellation involves considerable redundancy that complicates its empirical testing.
Therefore, Steingrimsson & Luce 2005 evaluated instead the equivalent Thomsen condition axiom, which avoids this redundancy, and found the property supported in binaural loudness.
Luce & Steingrimsson 2011, summarized the literature to that date, including the observation that the evaluation of the Thomsen Condition also involves an empirical challenge that they find remedied by the conjoint commutativity axiom, which they show to be equivalent to the Thomsen Condition.
However, the statistical techniques employed by Michell (1990) in testing Thurstone's theory and multidimensional scaling did not take into consideration the ordinal constraints imposed by the cancellation axioms (van der Linden 1994).
(Johnson 2001), Kyngdon (2006), Michell (1994) and (Sherman 1993) harv error: no target: CITEREFSherman1993 (help) tested the cancellation axioms of upon the interstimulus midpoint orders obtained by the use of Coombs' (1964) theory of unidimensional unfolding.
With six stimuli, the probability of an interstimulus midpoint order satisfying the double cancellation axioms at random is .5874 (Michell, 1994).
Kyngdon & Richards (2007) employed eight statements and found the interstimulus midpoint orders rejected the double cancellation condition.
Interpreting these correctly as instances in support of double cancellation (Michell, 1988), the results of Perline, Wright & Wainer 1979 are better than what they believed.
Stankov & Cregan 1993 applied conjoint measurement to performance on sequence completion tasks.
The rows were defined by levels of motivation (A), which consisted in different number of times available for completing the test.
Kyngdon found that satisfaction of the cancellation axioms was obtained only through permutation of the matrix in a manner inconsistent with the putative Lexile measures of item difficulty.
