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In the theory of conjoint measurement, the quantitative structure of natural attributes can be discovered in the absence of natural concatenation operations. It is the most general theory of measurement known to science. Fundamental and derived measurement in physics (Campbell, 1920), often thought to be the most general paths to quantification, are actually instances of the theory of conjoint measurement (Michell, 1990).

Conjoint measurement was independently discovered by the economist Debreu (1960) and by the mathematical psychologist R. Duncan Luce and statistician John Tukey (Luce & Tukey, 1964). Luce & Tukey's exposition was algebraic and is therefore considered more general than Debreu's topological work, the latter being a special case of the former (Luce & Suppes, 2002). The significance of the theory of conjoint measurement lies in the fact that non-geometric or "intensive" properties and systems can be quantified. Hence the quantification of such things as psychological attributes (e.g. attitudes, cognitive abilities and utility) is a logical possibility.

[edit] Background

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. In the first article of the inaugural issue of the Journal of Mathematical Psychology, Luce & Tukey (1964) proved that two attributes could be simultaneously quantified if and only if certain ordinal relations held upon a third attribute. 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, Luce, Suppes & Tversky, 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 "two way tables" 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.

Later, Joel Michell (1988) 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) recently 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). Recently, 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.

[edit] The theory of conjoint measurement

[edit] Axioms

An additive conjoint system is simply defined as follows:

$Q = f\left(A, X\right)$

where $A = \big\{a, b, c,\ldots \big\}$ and $X = \big\{x, y, z,\ldots \big\}$ are non-empty (and possibly infinite) sets that are disjoint (that is, $A \cap X = \varnothing$ ) and f is a non-interactive, additive function (i.e. is a monotonic function or a function preserved under monontone transformation). Let " $\succsim$ " be a simple order (i.e. one that is transitive, antisymmetric and strongly connected) holding upon the Cartesian product of A and X, $A \times X$ . Krantz et al. (1971) argued the triple $Q = \langle A, X, \succsim \rangle$ is an additive conjoint structure if and only if the following axioms hold:

1. WEAK ORDER. Suppose " $\succsim$ " is an independent relation upon $A \times X \,$ . A and X are weakly ordered if and only if:

For all $a, b \in A, a \succ b$ if and only if for some $x \in X, \left(a, x\right) \succ \left(b, x\right)$ ;
For all $x, y \in X, x \succ y$ is defined similarly;
The relation " $\succ$ " is transitive and connected.

2. SINGLE CANCELLATION. The relation " $\succsim$ " upon $A \times X$ satisfies single cancellation if and only if:

For all $a, b \in A, a \succsim b$ if and only if for all $x \in X, \left(a, x\right) \succsim \left(b, x\right)$ .
For all $x, y \in X, x \succsim y$ if and only if for all $a \in A, \left(a, x\right) \succsim \left(a, y\right)$ .

3. DOUBLE CANCELLATION. The relation " $\succsim$ " upon $A \times X$ satisfies double cancellation if and only if for every $a, b, c \in A$ and $x, y, z \in X$ , if $\left(a, y\right) \succsim \left(b, x\right)$ and $\left(b, z\right)\succsim \left(c, y\right)$ therefore $\left(a, z\right)\succsim \left(c, x\right)$ .

4. SOLVABILITY. The relation " $\succsim$ " upon $A \times X$ is solvable if for any three of the four elements, $a, b \in A$ and $x, y \in X$ , the fourth exists such that $\left(a, x\right)\sim \left(b, y\right)$ .

5. ARCHIMEDEAN CONDITION. Let $\mathit{I} \,$ be any set of consecutive integers (positive or negative, finite or infinite). The set $\big\{a_i | a_i \in A, i \in I \big\}$ is a standard sequence upon A if and only if there exist $x, y\in X$ such that $x \nsim y$ and for all $i, i + 1 \in I$ , then $\left(a_i, x\right)\sim \left(a_\left(i+1\right), y\right)$ . The condition is defined similarly upon X.

6. ESSENTIALNESS. Attribute A is essential if and only if there exists $a, b \in A$ and $x \in X$ such that not $\left(a, x\right)\sim \left(b, x\right)$ .

7. SYMMETRY. The relation " $\succsim$ " upon $A \times X$ is symmetric if and only if for $a, b \in A$ there exists $x, y\in X$ such that $\left(a, x\right)\sim \left(b, y\right)$ ; and for $x', y' \in X$ there exists $a', b' \in A$ such that $\left(a', x'\right) \sim \left(b', y'\right)$ .

[edit] Representation theorem

If in any empirical circumstance Axioms 1 to 7 hold then for $a, c \in A$ and $x, z \in X$ , there exist functions $\phi_A \,$ , $\phi_X \,$ from A and X into the real numbers $\langle \Re, \geqslant \rangle$ such that:

$\left(a, z \right)\succsim\left(c, x \right)\iff \phi_A \left(a\right) + \phi_X \left(z\right)\geqslant\phi_A \left(c\right) + \phi_X \left(x\right)$ .

If $\phi'_A \,$ and $\phi'_X \,$ are two other real valued functions satisfying the above expression, there exist $\alpha > 0, \beta_A \,$ and $\beta_X \,$ real valued constants satisfying:

$\phi'_A = \alpha \phi_A + \beta_A \,$ and $\phi'_X = \alpha \phi_X + \beta_X \,$ .

That is, $\phi'_A, \phi_A, \phi'_X \,$ and $\phi_X \,$ are measurements of A and X unique up to affine transformation (i.e. each is an interval scale in Stevens’ (1946) parlance). The mathematical proof of this result is given in Krantz, et al. (1971, pp. 261–266).

[edit] Explanation

Figure One: Graphical representation of the single cancellation axiom. It can be seen that a > b because (a, x) > (b, x), (a, y) > (b, y) and (a, z) > (b, z).

Arguably the most critical components of the theory of conjoint measurement are the cancellation axioms. Single cancellation is the lowest form of cancellation within Scott's (1964) hierarchy. Consider one of the attributes, A, that constitutes an additive conjoint system. Single cancellation states that if any two levels, a, b, of this attribute are weakly ordered, then this order holds irrespective of each and every level of the other attribute (X). The same holds for any two levels, x, y of X. Single cancellation is so-called because a single common element of two ordered pairs of elements cancel out to leave the same ordinal relationship holding on the remaining elements. Krantz, et al., (1971) originally called this axiom independence, as the ordinal relation between two levels of an attribute is independent of any and all levels of the other attribute. However, given that the term independence causes confusion with statistical concepts of independence, single cancellation is the preferable term. Figure One is a graphical representation of one instance of single cancellation.

Satisfaction of the single cancellation axiom is necessary, but not sufficient, for the quantification of attributes A and X. It only demonstrates that the attributes concerned are at least weakly ordered. Informally, single cancellation does not sufficiently constrain the order upon the levels of Q to allow A and X to sustain the representation theorem stated above. For example, consider the ordered pairs (a, x), (b, x) and (b, y). If single cancellation holds then $\left(a, x\right)\succsim \left(b, x\right)$ and $\left(b, x\right)\succsim \left(b, y\right)$ . Hence via transitivity $\left(a, x\right)\succsim \left(b, y\right)$ . The relation between these latter two ordered pairs, informally a left-leaning diagonal, is determined by the satisfaction of the single cancellation axiom, as are all such relations upon Q.

Figure Two: A Luce - Tukey instance of double cancellation, in which the consequent inequality (broken line arrow) does not contradict the direction of both antecedent inequalities (solid line arrows), so supporting the axiom.

Single cancellation, however, does not determine the "right-leaning diagonal" relations. Even though by transitivity and single cancellation it was established that $\left(a, x\right)\succsim \left(b, y\right)$ , the relationship between (a, y) and (b, x) remains undetermined. It could be that either $\left(b, x\right)\succsim \left(a, y\right)$ or $\left(a, y\right)\succsim \left(b, x\right)$ and such ambiguity cannot remain unresolved. The cancellation conditions higher in Scott's (1964) hierarchy than single cancellation must be invoked to ascertain all right leaning diagonal relations upon Q.

The double cancellation axiom concerns a class of such relations in which the common terms of two antecedent weak inequalities cancel out to produce a third inequality. Consider the instance of double cancellation graphically represented by Figure Two. The antecendent inequalities of this particular instance of double cancellation are $\left(a, y\right) \succsim \left(b, x\right)$ and $\left(b, z\right)\succsim \left(c, y\right)$ . Given that:

$\left(a, y\right) \succsim \left(b, x\right)$ is true if and only if $a \oplus y \succsim b \oplus x$ ; and
$\left(b, z\right) \succsim \left(c, y\right)$ is true if and only if b $b \oplus z \succsim c \oplus y$ ,

it follows that $a \oplus y \oplus b \oplus z \succsim b \oplus x \oplus c \oplus y$ . Cancelling the common terms results in $\left(a, z\right)\succsim \left(c, x\right)$ . Hence double cancellation can only obtain when A and X possess additive structure; and so hence double cancellation is the "acid test" for the hypothesis of quantity in empirical investigations.

The number of instances of double cancellation is contingent upon the number of elements A and X contain. If there are n levels of A and m of X, then the number of instances of double cancellation is n! $\times$ m!. Therefore, if n = m = 3, then 3! $\times$ 3! = 36 instances in total of double cancellation. However, all but 6 of these instances are trivially true if single cancellation is true, and if anyone of these 6 instances is true, then all of them are true. One such instance is that shown in Figure Two. Michell (1988) calls this a Luce - Tukey instance of double cancellation. If single cancellation has been tested upon a set of data first and is established, then only the Luce - Tukey instances of double cancellation need to be tested. For n levels of A and m of X, the number of Luce - Tukey double cancellation instances is $\tbinom{n}{3}$ $\tbinom{m}{3}$ . For example, if n = m = 4, then there are 16 such instances. If n = m = 5 then there are 100. In any empirical situation, if the direction of the consequent inequality contradicts the direction of both antecedent inequalities, double cancellation is violated and it cannot be concluded that A and X possess additive structure (Michell, 1988).

Figure Three: An instance of triple cancellation.

The next two axioms involve the concept of continuity. Solvability means that for any three elements of a, b, x and y, the fourth exists such that the equation $ax = by\,$ is solved, hence the name of the axiom. The Archimedean axiom states that a difference between two levels of an attribute cannot be infinitely larger than any other difference. Consider $a, b, c, d \in A$ where $a - b \succ c - d$ . There exists a natural number n such that $n \left(c - d\right) \geqslant \left(a - b\right)$ , where $n \left(c - d\right)$ is a standard sequence $a_1, a_2, \ldots, a_\left(n + 1\right)$ such that $c = a_1\,$ and $d = a_2\,$ (Michell, 1990).

However, as they involve infinitistic concepts, the solvability and Archimedean axioms cannot be expressed nor implied in a first order language sentence (Luce & Narens, 1992). Hence they are not amenable to direct testing in any finite empirical situation. But this does not entail that these axioms cannot be empirically tested. Scott's (1964) finite set of cancellation conditions can be used to indirectly test these axioms; the extent of such testing being empirically determined. For example, if both A and X possess three levels, the highest order cancellation axiom within Scott’s (1964) hierarchy that indirectly tests solvability and Archimedeaness is double cancellation. With four levels it is triple cancellation (Figure 3). If such tests are satisfied, the construction of standard sequences in differences upon A and X are possible. Hence these attributes may be dense as per the real numbers or equally spaced as per the integers (Krantz, et al., 1971). In other words, A and X are continuous quantities.

Essentialness simply states both A and X must contain levels of differing magnitude. Symmetry means that any interval on one attribute (say, A), an interval on X can be found that is equivalent to the one in A.

[edit] Conjoint measurement in n - components

Conjoint measurement in n components is simply a generalisation by Krantz, et al., (1971) of the theory of conjoint measurement to n disjoint attributes, where n $\geqslant$ 3.