The theory of conjoint measurement (also known as conjoint measurement or additive conjoint measurement) is a general, formal theory of continuous quantity. It was independently discovered by the French economist Gérard Debreu (1960) and by the American mathematical psychologist R. Duncan Luce and statistician John Tukey .
The theory concerns the situation where at least two natural attributes, A and X, non-interactively relate to a third attribute, P. It is not required that A, X or P are known to be quantities. Via specific relations between the levels of P, it can be established that P, A and X are continuous quantities. Hence the theory of conjoint measurement can be used to quantify attributes in empirical circumstances where it is not possible to combine the levels of the attributes using a side-by-side operation or concatenation. The quantification of psychological attributes such as attitudes, cognitive abilities and utility is therefore logically plausible. This means that the scientific measurement of psychological attributes is possible. That is, like physical quantities, a magnitude of a psychological quantity may possibly be expressed as the product of a real number and a unit magnitude.
Application of the theory of conjoint measurement in psychology, however, has been limited. It has been argued that this is due to the high level of formal mathematics involved (e.g., ) and that the theory cannot account for the "noisy" data typically discovered in psychological research (e.g., ). It has been argued that the Rasch model is a stochastic variant of the theory of conjoint measurement (e.g., ; ; ; ; ; ), 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).
The theory of conjoint measurement is (different but) related to conjoint analysis, which is a statistical-experiments methodology employed in marketing to estimate the parameters of additive utility functions. Different multi-attribute stimuli are presented to respondents, and different methods are used to measure their preferences about the presented stimuli. The coefficients of the utility function are estimated using alternative regression-based tools.
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 .
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 . In the first article of the inaugural issue of the Journal of Mathematical Psychology, 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. and Amos Tversky (1967) developed what became known as polynomial conjoint measurement, with 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 .
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 (; ). Later enumeration studies focused on polynomial conjoint measurement (; ). 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 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 . George Karabatsos and his associates (Karabatsos, 2001; ) 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).
Measurement and quantification
The classical / standard definition of measurement
In physics and metrology, the standard definition of measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit magnitude of the same kind (de Boer, 1994/95; Emerson, 2008). For example, the statement "Peter's hallway is 4 m long" expresses a measurement of an hitherto unknown length magnitude (the hallway's length) as the ratio of the unit (the metre in this case) to the length of the hallway. The number 4 is a real number in the strict mathematical sense of this term.
For some other quantities, invariant are ratios between attribute differences. Consider temperature, for example. In the familiar everyday instances, temperature is measured using instruments calibrated in either the Fahrenheit or Celsius scales. What are really being measured with such instruments are the magnitudes of temperature differences. For example, Anders Celsius defined the unit of the Celsius scale to be 1/100 of the difference in temperature between the freezing and boiling points of water at sea level. A midday temperature measurement of 20 degrees Celsius is simply the difference of the midday temperature and the temperature of the freezing water divided by the difference of the Celsius unit and the temperature of the freezing water.
Formally expressed, a scientific measurement is:
where Q is the magnitude of the quantity, r is a real number and [Q] is a unit magnitude of the same kind.
Extensive and intensive quantity
Length is a quantity for which natural concatenation operations exist. That is, we can combine in a side-by-side fashion lengths of rigid steel rods, for example, such that the additive relations between lengths is readily observed. If we have four 1 m lengths of such rods, we can place them end to end to produce a length of 4 m. Quantities capable of concatenation are known as extensive quantities and include mass, time, electrical resistance and plane angle. These are known as base quantities in physics and metrology.
Temperature is a quantity for which there is an absence of concatenation operations. We cannot pour a volume of water of temperature 40 °C into another bucket of water at 20 °C and expect to have a volume of water with a temperature of 60 °C. Temperature is therefore an intensive quantity.
Psychological attributes, like temperature, are considered to be intensive as no way of concatenating such attributes has been found. But this is not to say that such attributes are not quantifiable. The theory of conjoint measurement provides a theoretical means of doing this.
Theory
Consider two natural attributes A, and X. It is not known that either A or X is a continuous quantity, or that both of them are. Let a, b, and c represent three independent, identifiable levels of A; and let x, y and z represent three independent, identifiable levels of X. A third attribute, P, consists of the nine ordered pairs of levels of A and X. That is, (a, x), (b, y),..., (c, z) (see Figure 1). The quantification of A, X and P depends upon the behaviour of the relation holding upon the levels of P. These relations are presented as axioms in the theory of conjoint measurement.
Single cancellation or independence axiom
The single cancellation axiom is as follows. The relation upon P satisfies single cancellation if and only if for all a and b in A, and x in X, (a, x) > (b, x) is implied for every w in X such that (a, w) > (b, w). Similarly, for all x and y in X and a in A, (a, x) > (a, y) is implied for every d in A such that (d, x) > (d, y). What this means is that if any two levels, a, b, are ordered, then this order holds irrespective of each and every level of X. The same holds for any two levels, x and y of X with respect to each and every level of A.