The theory of conjoint measurement (also known as conjoint measurement or additive conjoint measurement) is a general, formal theory of continuous
quantity
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...
. 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
In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenat ...
. 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
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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
The Rasch model, named after Georg Rasch, is a psychometric model for analyzing categorical data, such as answers to questions on a reading assessment or questionnaire responses, as a function of the trade-off between the respondent's abilities, ...
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
Conjoint analysis is a survey-based statistical technique used in market research that helps determine how people value different attributes (feature, function, benefits) that make up an individual product or service.
The objective of conjoint an ...
, which is a statistical-experiments methodology employed in
marketing
Marketing is the process of exploring, creating, and delivering value to meet the needs of a target market in terms of goods and services; potentially including selection of a target audience; selection of certain attributes or themes to emph ...
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
Norman Robert Campbell (1880–1949) was an English physicist and philosopher of science.
Early life
Norman Robert Campbell was born in 1880. He was the son of William Middleton Campbell, Governor of the Bank of England, and his wife Edith Ag ...
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
Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.
Early life and education
Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Chris ...
(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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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
Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, Ca ...
(1964), who proposed a hierarchy of cancellation conditions for the indirect testing of the solvability and Archimedean
axioms
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
, 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
Patrick Colonel Suppes (; March 17, 1922 – November 17, 2014) was an American philosopher who made significant contributions to philosophy of science, the theory of measurement, the foundations of quantum mechanics, decision theory, psychology ...
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
Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister.
Bayesian () refers either to a range of concepts and approaches that relate to statistical methods based on Bayes' theorem, or a followe ...
Markov chain Monte Carlo
In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain ...
methodology for
psychometric
Psychometrics is a field of study within psychology concerned with the theory and technique of measurement. Psychometrics generally refers to specialized fields within psychology and education devoted to testing, measurement, assessment, and ...
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
Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, thought, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characte ...
. 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
Prospect theory is a theory of behavioral economics and behavioral finance that was developed by Daniel Kahneman and Amos Tversky in 1979. The theory was cited in the decision to award Kahneman the 2002 Nobel Memorial Prize in Economics.
Based ...
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
The Allais paradox is a choice problem designed by to show an inconsistency of actual observed choices with the predictions of expected utility theory.
Statement of the problem
The Allais paradox arises when comparing participants' choices in two ...
. 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
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
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
Anders Celsius (; 27 November 170125 April 1744) was a Swedish astronomer, physicist and mathematician. He was professor of astronomy at Uppsala University from 1730 to 1744, but traveled from 1732 to 1735 visiting notable observatories in Germ ...
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:
: