Theory of conjoint measurement
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 (Luce & Tukey 1964).
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., 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).
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[edit]
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).
Measurement and quantification[edit]
The classical / standard definition of measurement[edit]
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 1/100th 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:
Applications of conjoint measurement[edit]
Empirical applications of the theory of conjoint measurement have been sparse (Cliff 1992; Michell 2009).
Several empirical evaluations of the double cancellation have been conducted. Among these, Levelt, Riemersma & Bunt 1972 evaluated the axiom to the psychophysics of binaural loudness. They found the double cancellation axiom was rejected. Gigerenzer & Strube 1983 conducted a similar investigation and replicated Levelt, et al.' (1972) findings. 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. Luce & Steingrimsson 2011 found conjoint commutativity supported for binaural loudness and brightness.
Michell 1990 applied the theory to L. L. Thurstone's (1927) theory of paired comparisons, multidimensional scaling and Coombs' (1964) theory of unidimensional unfolding. He found support of the cancellation axioms only with Coombs' (1964) theory. 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) tested the cancellation axioms of upon the interstimulus midpoint orders obtained by the use of Coombs' (1964) theory of unidimensional unfolding. Coombs' theory in all three studies was applied to a set of six statements. These authors found that the axioms were satisfied, however, these were applications biased towards a positive result. With six stimuli, the probability of an interstimulus midpoint order satisfying the double cancellation axioms at random is .5874 (Michell, 1994). This is not an unlikely event. Kyngdon & Richards (2007) employed eight statements and found the interstimulus midpoint orders rejected the double cancellation condition.
Perline, Wright & Wainer 1979 applied conjoint measurement to item response data to a convict parole questionnaire and to intelligence test data gathered from Danish troops. They found considerable violation of the cancellation axioms in the parole questionnaire data, but not in the intelligence test data. Moreover, they recorded the supposed "no test" instances of double cancellation. 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 columns of their conjoint arrays (X) were defined by the demand placed upon working memory capacity through increasing numbers of working memory place keepers in letter series completion tasks. The rows were defined by levels of motivation (A), which consisted in different number of times available for completing the test. Their data (P) consisted of completion times and average number of series correct. They found support for the cancellation axioms, however, their study was biased by the small size of the conjoint arrays (3 × 3 is size) and by statistical techniques that did not take into consideration the ordinal restrictions imposed by the cancellation axioms.
Kyngdon (2011) used Karabatsos's (2001) order-restricted inference framework to test a conjoint matrix of reading item response proportions (P) where the examinee reading ability comprised the rows of the conjoint array (A) and the difficulty of the reading items formed the columns of the array (X). The levels of reading ability were identified via raw total test score and the levels of reading item difficulty were identified by the Lexile Framework for Reading (Stenner et al. 2006). 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. Kyngdon also tested simulated ability test response data using polynomial conjoint measurement. The data were generated using Humphry's extended frame of reference Rasch model (Humphry & Andrich 2008). He found support of distributive, single and double cancellation consistent with a distributive polynomial conjoint structure in three variables (Krantz & Tversky 1971).