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The Rasch Model from the Perspective of the Representational Theory of Measurement Andrew Kyngdon Theory Psychology 2008; 18; 89 DOI: 10.1177/0959354307086924 The online version of this article can be found at: http://tap.sagepub.com/cgi/content/abstract/18/1/89

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The Rasch Model from the Perspective of the Representational Theory of Measurement Andrew Kyngdon METAMETRICS, INC. ABSTRACT. Representational measurement theory is the dominant theory of measurement within the philosophy of science; and the area in which the theory of conjoint measurement was developed. For many years it has been argued the Rasch model is conjoint measurement by several psychometricians. This paper critiques this argument from the perspective of representational measurement theory. It concludes that the Rasch model is not conjoint measurement as the model does not demonstrate the existence of a representation theorem between an empirical relational structure and a numerical relational structure. Psychologists seriously interested in investigating traits for quantitative structure should use the theory of conjoint measurement itself rather than the Rasch model. This is not to say, however, that empirical relationships between conjoint measurement and the Rasch model are precluded. The paper concludes by suggesting some relevant research avenues. KEY WORDS: axiomatic conjoint measurement, homomorphism, Rasch model, real numbers, representation theorem

The dominant theory of measurement within the philosophy of science is the representational theory of measurement or representational measurement theory (hereafter referred to as RMT; Krantz, Luce, Suppes, & Tversky, 1971; Luce, Krantz, Suppes, & Tversky, 1990; Luce & Suppes, 2002; Mundy, 1986, 1987a, 1987b; Narens, 1985, 2002; Suppes, Krantz, Luce, & Tversky, 1989; Swoyer, 1991). An informal RMT definition of measurement is as follows: A theory of measurement consists of a precise specification of how a scale is formed. ... The representational theory really comprises many related theories of measurement. What they have in common is that they require a scale to be a set of structure preserving mappings (e.g., a set of isomorphisms or

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homomorphisms) from the qualitative or empirically based structure into a structure from pure mathematics. (Narens, 2002, p. 757)

That is, some naturally occurring structures can be mapped into the real numbers (a structure of pure mathematics) in such a way that the real numbers exactly represent those natural structures. RMT argues the real numbers do not exist in the same universe that natural structures exist; and instead argues that numbers are ‘abstract’, non-empirical set-theoretic objects (Hersh, 1998; Resnick, 1997; Shapiro, 1997). This philosophy is known as Platonism, which advances that ‘mathematical entities exist outside space and time, outside thought and matter, in an abstract realm outside of any consciousness, individual or social’ (Hersh, 1998, p. 9). More specifically, RMT argues real numbers are pure sets constructed out of the empty set according to the first-order language propositions contained in the Zermelo–Fraenkel set theory axioms (Narens, 2002). These axioms underpin current understanding of the real numbers in modern mathematics (Davis & Hersh, 1981; Maddy, 2001; Moore, 2001; Shapiro, 1997); however, there are severe and unresolved problems contained within them (Cohen, 1963; Gödel, 1986).1 (See Fraenkel, 1953/1976, for a thorough, technical presentation or Takeuti & Zaring, 1982, for an introductory one.) The Platonism of RMT places the theory in an interesting position. As measurement involves both the real numbers and natural structures, RMT must present some argument as to how the real numbers, as Platonic realm non-empirical objects, relate to the empirical world of natural structures given the brute fact that measurement of some natural structures is indeed possible. Moreover, any such argument must be conceptualized such that it can be empirically falsified if RMT is to be considered scientific, given that not all natural structures are measurable. That is, any relation proposed to hold between measurable structures and the real numbers must have features which differentiate it from the relations which may possibly hold between numbers and non-measurable and purely ordinal structures. This necessitates the prior formal characterization of the situations in which a real number representation can be sustained. This is achieved in the following manner. The initial premise of RMT is that the natural world consists of observable objects and events (Krantz et al., 1971; Narens, 2002). The behaviour of these, together with the observable relations holding upon them, may manifest what RMT calls surfacelevel structures. The objects, events and the relations comprise distinct sets in the mathematical sense of that term. Thence via the Zermelo–Fraenkel axiom of pairing, both these sets form another set called an empirical relational structure (hereafter referred to as ‘empirical structure’). The adjective ‘empirical’ is used given that in any empirical situation the relational structure observed will consist of only a restricted set of experimentally

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manipulable objects or events, given the limitations of human cognitive and sensory-motor capacities. An empirical structure set A contains both the set of all relevant objects or events (A) and the set of all relations which hold upon them (R); such that the empirical structure is of the form A = 〈A, R〉. The set A is called the domain of A or the domain of discourse of A. Any empirical structure is not a pure set like a real number as the elements of an empirical structure are sets whose members are natural objects, events or relations.2 The formal characterization of an empirical structure is conducted via the process of axiomatization, which involves the coherent organization of the propositions that are made concerning the behaviour of an empirical structure, especially the relations. Any axiom system within RMT usually (but not always) proposes that the relations holding amongst empirical objects reflect those which hold amongst the real numbers themselves. As such, many axiomatic characterizations of empirical structures are similar to the axioms of quantity given by Hölder (Michell & Ernst, 1901/1996, 1901/1997). Marley and Luce (1998), for example, propose such an axiomatic characterization for utility theory. If the propositions advanced by the axioms are supported by empirical evidence, it is concluded that the empirical structure sustains a representation into the real numbers. This is best illustrated by example. Imagine that there exists a collection of rigid steel rods. Each rod is under the same conditions of temperature and pressure; and each rod is able to be manipulated within the bounds of human abilities (that is, each rod is neither extremely heavy nor long nor too hot nor cold). These constitute a collection of observable objects (A) which can be physically manipulated such that a set of relations (R) can be observed to hold over them. For example, the relation ‘rod a is longer than rod b’ is directly observable and can be represented symbolically as “a
b”. The operation of concatenation (physical combination) allows us to manipulate the rods to produce rods of other lengths. It can thus be hypothesized that the length surface structure of the steel rods forms the empirical structure set A = 〈A, R, ◦〉, where A is the set of all such rods, R is the set of relations holding upon the elements of A, and ◦ denotes the operation of concatenation. If A is a measurable empirical structure, the following axioms must hold: E1. E2. E3. E4. E5. E6.

For all a, b and c in A,
is both transitive and connected. For all a, b, c and d in A, if a
b and c
d, then a ◦ c
b ◦ d. For all a and b in A, a ◦ b
a and a ◦ b
b. For all a, b and c in A, a ◦ (b ◦ c) ~ c ◦ (a ◦ b). For each a and b in A, if a
b then there exists c such that b ◦ c
a. For any a, b in A, there exists an integer n such that a(n)
b.

If and only if all elements of A behave in the manner proposed by axioms E1 to E6 with respect to ‘
’, then a special type of algebraic relation known

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as a homomorphism exists between A = 〈A, R, ◦〉 and the numerical relational structure ℜ = 〈Re+, S, +〉, where ‘Re+’ is a pure set of positive real numbers, ‘S’ represents the set of relations holding over the pure set of real numbers, and ‘+’ the operation of numerical addition. This homomorphism maps the objects in A into the positive real numbers in Re+; the empirical relation ‘
’ into the numerical relation ‘>’ (‘greater than’) and the empirical operation of ‘◦’ into the numerical operation of ‘+’ in such a manner that ‘>’ and ‘+’ preserve the characteristics of their empirical counterparts (Krantz et al., 1971). Let φ be a real valued function. The following representation theorem holds: a
b ⇔ φ(a) > φ(b)

(1)

where ‘⇔’ represents a homomorphic mapping. Expression 1 is best illustrated by example. Let a, b, c and d be steel rods in A that satisfy axioms E1 to E6 such that a
b
c
d. Let 2, 7, 9 and 17 be elements in Re+. The representation theorem of Expression 1 means that: φ(a) = 17 φ(b) = 9 φ(c) = 7 φ(d) = 2

Expression 1, however, cannot uniquely represent A = 〈A, R, ◦〉 unless there exists φ ′ such that: φ ′(a) = 431.8 φ ′(b) = 228.6 φ ′(c) = 177.8 φ ′(d) = 50.8

where φ ′ = αφ, α = 25.4

(2)

Equation 2 is the uniqueness theorem (Krantz et al., 1971) for the representation theorem given in Expression 1. Expression 2 is known as a positive similarities transformation (Krantz et al., 1971), which means that any representation of an empirical structure satisfying axioms 1 to 6 is unique up to multiplication by a positive real number constant. In the above example, this constant is 25.4, which is obtained when φ ′ is measuring the steel rods in millimetres whilst φ is measuring them in inches. The Theory of Conjoint Measurement Axioms E1–E6 pertain only to those empirical structures which can sustain the operation of concatenation. That is, these axioms can only pertain to those surface-level structures whose behaviour can be ascertained by physical manipulation of the objects that possess these surface-level structures. There

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are, however, only a limited number of additive empirical structures, such as length and mass, which can do this. Hence it has been a long-term aim of RMT to conceptualize an empirical structure whose additivity can be proposed in the most general terms possible. This has largely become synonymous with finding an additive empirical structure without resort to concatenation. By far the best known discovery of RMT is the theory of conjoint measurement (referred to hereafter as ‘conjoint measurement’; Luce & Tukey, 1964). Consider two sets of objects, V and X, where V = {t, u, v} and X = {x, y, z). These sets are disjoint as they do not share any common elements. Furthermore, suppose the elements of these sets cannot be physically manipulated like steel rods and so therefore are not amenable to testing via axioms E1 to E6. This does not mean that they cannot be measured, although of course this is a possibility. They may be able to be measured if they relate to a third variable, C, in certain ways. The elements of V and X can pair to form the set C. The elements of C are the ordered pairs (t, x), (t, y), (t, z), (u, x), (u, y), (u, z), (v, x), (v, y), (v, z), and hence C is the Cartesian product of V and X. C = 〈V × X,
~〉 is a conjoint measurement empirical structure if and only if the elements of C satisfy the following axioms: C1.

C2.

C3.

C4.

C5.

Weak order. Given C = 〈V × X,
~〉 and the ordered pairs (t, x) and (u, x), then V and X are weakly ordered if and only if: • For t and u in V and (u, x)
~ (t, x) then u
~ t. • For X, y
x is defined similarly. ~ • The relation ‘
~’ is transitive and connected. Independence. The relation ‘
~’ upon V × X is independent if and only if: • For t and u in V and x in X then (u, x)
~ (t, x) is implied for every element w in X such that (u, w)
(t, w). ~ • For x and y in X and v in V then (v, y)
~ (v, x) implies for every element s in V that (s, y)
(s, x). ~ Double cancellation. The relation ‘
~’ upon V × X satisfies if and only if for every t, u and v in V and x, y and z in X then: If (u, x)
~ (t, y) and (v, y)
~ (u, z) therefore (v, x)
~ (t, z) Solvability. The relation ‘
~’ upon V × X is solvable if for any three of the four elements t and u in V and x and y in V, the fourth exists such that the inequality (u, x)
~ (t, y) is solved such that (u, x) ~ (t, y). Archimedean condition. Let there exist the elements t, u, v and s in V and x, y, z and w in X. If u − t ≺ s − v and y − x ≺ w − z, then

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for any natural number n, V and X are Archimedean if and only if n(u – t)
~ s − v and n(y − x)
~ w − z. If all axioms C1–C5 hold,3 then for t and v in V, x and z in X there exist real valued functions φV and φX such that: (v, x)
~ (t, z) ⇔ φV(v) + φX(x) ≥ φV(t) φX(z)

(3)

Thus a representation theorem from both sets of objects V and X into the real numbers is possible as the direct consequence of the relation ‘
~’ holding upon the Cartesian product of V and X rather than upon V and X in isolation. Thus empirical structures not capable of concatenation yet of sufficient complexity can sustain a real valued representation. The uniqueness of the conjoint measurement representation theorem is as follows. If φ V′ and φX′ are two other such real valued functions with the same property, then there exist real valued constants α > ◦, βX and βY such that: φ V′ = αφ V + βV and φ X′ = αφ X + βX

(4)

Thus the conjoint measurement representation theorem is unique up to affine (linear) transformations. In psychometric terms, conjoint measurement enables the interval scale measurement (Stevens, 1951) of the relevant structures. (The mathematical proof is available in Krantz et al., 1971, pp. 264–266.) Suppose there exists the conjoint measurement empirical structure C = 〈A × X,
~〉 that is solvable and Archimedean (Luce, 1987, Theorem 5.2). Moreover, suppose that X and A were themselves the domains of other empirical structures X = 〈X,
~〉 whose translations formed Archimedean ~〉 and A = 〈Α,
ordered mathematical groups (Luce, 1986). Luce (1987) found that if X and A were Dedekind complete, the following representation for C = 〈A × X,
~〉 held: (a,x) ⇔ φ(a)ψρ(x)

(5)

where a is in A, x is in X and φ,ψ being real valued functions into the set of the positive real numbers Re+, with ρ being a constant in the set of real numbers Re. The importance of this result is it demonstrated that examples of derived measurement (Campbell, 1920) and dimensional analysis in physics were instances of conjoint measurement with multiplicative composition rules (Luce, 1971, 1978). For example, the formula for density D = MV−1, where M is the mass and V is the volume of an object, is such an example of conjoint measurement, as is Newton’s Second Law of Motion F = MA1, where F is the force of an object, M is its mass and A is its acceleration. Additionally, Luce’s work found that if X and A were capable of supporting ratio scale representations in their own right (that is, they both satisfied axioms E1–E6), then a product representation for C did not have to be postulated, given that this followed both from the uniqueness of the representations sustained by A and X and the fact that A and X distributed in C.

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The Rasch (1960) Model and Conjoint Measurement Item response theory is the dominant field within psychometrics (Borsboom, Mellenbergh & van Heerden, 2003; Embretson & Reise, 2000). Representational measurement theory and conjoint measurement have had very little impact upon psychometrics (Cliff, 1992; Narens & Luce, 1993; Ramsay, 1975, 1991; Schwager, 1991). However, when Michell (2000) labelled psychometrics a ‘pathological science’ for failing to investigate psychological attributes for quantitative structure, Borsboom and Mellenbergh (2004) rejected Michell’s charge in part with the claim the Rasch (1960) model is ‘a probabilistic variant of the additive conjoint measurement model’ (p. 110). This claim, however, is not new. For over 30 years it has been advanced by many psychometricians that the Rasch model is conjoint measurement or a probabilistic version thereof (Andrich, 1988; Barrett, 2003; Bond & Fox, 2001; Borsboom & Mellenbergh, 2004; Brogden, 1977; Embretson & Reise, 2000; Fischer, 1995; Fisher, 2003; Green, 1986; Karabatsos, 2001; Keats, 1967; Kline, 1998; Perline, Wright, & Wainer, 1979; Roskam & Jansen, 1984; Scheiblechner, 1999; Wright, 1996, 1999). It is a proposition strongly believed to be true, as evidenced by the following: …only the Rasch model fulfils the conjoint measurement conditions fully, and so is often preferred in applications where measurement-scale properties are deemed very important. (Embretson & Reise, 2000, pp. 149–150) Since educational and psychological data are generally not perfectly reliable, the absence of an error theory has limited the usefulness of conjoint measurement models. One striking exception is the widespread use of Rasch models. (Green, 1986, p. 141) Of critical importance is the realisation that the currently fashionable Rasch item response theory is also an empirical instantiation of the conjoint additivity axioms (Perline et al., 1979). That is, the construction of a latent variable using Rasch item analysis is no less than the empirical test of quantitative structure for that latent variable. (Barrett, 2003, p. 429) Apparently not many psychologists are aware that the Rasch (1960) model is a practical realization of conjoint measurement. (Perline et al., 1979, p. 237)

The Rasch (1960) model takes its name from the Danish statistician Georg Rasch (1901–1980). Consider an item, i, from a test of a certain intellectual ability, say spelling. This item has a certain difficulty which can be denoted δi. A test examinee, v, has a certain ability in spelling that is denoted βv. If v attempts this item and gives a correct answer, v receives a score of 1 for that item. If v gives an incorrect answer, v receives a score of 0. Given the item’s difficulty (δi) and the person’s ability (βv), the probability (Pr) that v gets item i correct is given by the following formula:

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Prfxvi = 1g =

& PSYCHOLOGY 18(1)

eðβv − δi Þ 1 + eðβv − δi Þ

(6)

The probability that v gets the item incorrect is given by: 1 1 + eðβv − δi Þ

Prfxvi = 0g =

(7)

The number ‘e’ in both equations is the base of the natural logarithms, which has a value of approximately 2.718. The term ‘xvi’ denotes the score of person v on item i. Three assumptions characterize the Rasch model and unidimensional, cumulative item response theory models (Junker & Sijtsma, 2001, p. 212; Karabatsos & Sheu, 2004, pp. 110–111): 1.

Local independence: If X is a vector of item response (x = 1,…, I) and θ is a unidimensional latent trait, the probability of the vector of item response (conditioned on θ) is equal to the product of the probabilities of response to each item (also conditioned on θ) such that (Embretson & Reise, 2000, Eqn 9.1): I

Pr(X = x|θ ) = Π

i=1

2.

3.

Pri (Xi = x|θ )

Monotonicity: If βa and βb are the locations of persons a and b upon the structure θ, Item Response Functions (IRFs) are strictly increasing, continuous monotonic functions such that Pri(βa) ≤ Pri(βb) if and only if βa < βb for items i = 1, …, I. Unidimensionality: The latent trait ? takes values from a subset of the pure set of real numbers.

Given the attestations of the psychometricians quoted above, it is interesting to note that the argument the Rasch model is conjoint measurement has never been subject to critical appraisal from the perspective of RMT. There are two ways in which the Rasch model has been considered to be conjoint measurement. Both of these will now be addressed. The Rasch Model as Conjoint Measurement via Analogy with Derived Physical Measurement Equation 5 is the representation theorem for conjoint measurement with a multiplicative composition rule (Luce, 1986, 1987). This revealed that derived measurement and dimensional analysis in physics were conjoint measurement empirical structures. Noting the multiplicative relationships between physical quantities, Andrich (1988), Fischer (1995) and Rasch (1960) argued that test performance was a measurement derived from the abilities of persons and the difficulties of items. The generality of Luce’s

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(1987) work revealed this was a plausible proposition; however, instead of the investigation of the conjoint measurement axioms, Andrich (1988), Fischer (1995) and Rasch (1960) rested their argument on an analogy. Andrich (1988) made this explicit: The parametric structure. Following the lead from the laws of physics, we specify a simple or additive structure for relating B to D. Beginning with the multiplicative one, let: Lni = Bn / Di; Bn > 0, Di > 0. (p. 25)

That is, person n’s performance on item i (Lni) is the ratio of n’s intellectual ability (Bn) to i’s difficulty (Di) if both Di and Bn are positive real numbers. Both Andrich (1988) and Michell (2003) argued that if a real valued function f = x/1 + x exists, this ratio is mapped into the set of probabilities such that: PrfLni g =

Bn =Di 1 + Bn =Di

(8)

Taking natural logarithms of Lni = Bn / Di transforms the inverse multiplicative relationship into a subtractive one such that ln(Lni) = ln(Bn) − ln(Di). If ln(Bn) = βn and ln(Di) = δi, then Equation 8 becomes the additive Rasch model (Equations 6 and 7). Assuming statistical independence, consider two items i and j with difficulties Di and Dj, respectively. Let person n attempt both items. Given that either i or j may be correct, the ratio of the probability of i being correct and j incorrect (Pr {i, j}) to the probability of j being correct to i (Pr {j, i}) being incorrect can be calculated using Equation 8 such that: Prfi, jg Bn =Di
= Prfj, ig Bn D j

!


! ð1 + Bn =Di Þ 1 + Bn Dj 1=D D 
 =
i = j ð1 + Bn =Di Þ 1 + Bn Dj 1 Dj Di

(9)

This is the concept of specific objectivity (Rasch, 1977), where the comparison of the difficulties of two items is independent of the abilities of persons; and the comparison of the abilities of any two persons is independent of any two items. Fischer (1995) argued: The favourite example used by Rasch (1960, 1967) to illustrate the meaning of the objectivity of comparisons is based on the joint definition and measurement of mass and force in classical mechanics. … According to the Second Newtonian Axiom (force = mass × acceleration), an observed acceleration is proportional to the force exerted on the object, and inversely proportional to the object’s mass, Avi = Mv−1 Fi (2.39) The comparison of any two objects Ov and Ow with respect to their masses Mv and Mw is easily carried out by means of the quotient Avi / Awi = Mv−1 Fi / Mw−1 Fi = Mw / Mv (2.40)

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which is independent of Fi. The quotient (2.40) can alternatively be replaced, upon a logarithmic transformation, by the parameter difference ln (Avi/Awi) = ln Mw – ln Mv. (p. 27)

Fischer concluded: Although the original scales B and D are only ordinal by nature of the psychological notion of ‘ability’ and ‘difficulty’, the transformations are highly specific and unique except for linear transformations. … It applies to the case of force and mass in physics (namely ln Avi = ln Fi – ln Mv) in the same way as to ability and item difficulty in psychology. (p. 31)

In other words, Fischer concluded that because logarithmic transformations result in additive relationships between derived measurements in physics, it follows that person ability and item difficulty are structures of sufficient complexity to sustain representation theorems into the real numbers that are unique up to affine (linear) transformations, given that logarithmic transforms of physical variables hold ‘in the same way as to ability and item difficulty in psychology’ (p. 31). That is, person ability and item difficulty are concluded to be additive structures on the sole basis of conducting a transformation upon the relationship between these structures. This relationship, however, is held by analogy to derived measurement via the concept of specific objectivity. Andrich and Fischer argued the Rasch model (Equations 6 and 7) presents a ‘subtractive’ relationship between person ability and item difficulty loginterval unit parameters in exactly the same way that would hold for derived physical measurements. Given this subtractive relationship on a log-interval scale, it can be transformed into a multiplicative one via a power transformation to give Equation 8. On the basis of this transformation and specific objectivity, it is inferred that person ability and item difficulty must form a conjoint measurement structure with a multiplicative composition rule like mass and acceleration do with respect to force. Hence the conclusion that the Rasch model is conjoint measurement. Andrich (1988) and Fischer (1995) asserted that additive interval measurements of person ability and item difficulty are given by the Rasch (1960) model. In order for this proposition to be true, however, it must first be demonstrated that the premise underlying it is true (namely, test performance is a multiplicative conjoint measurement structure comprising of person ability and item difficulty). From the perspective of RMT, this must involve the demonstration of test performance being a derived measurement of person ability and item difficulty in the same way that force is of mass and acceleration. Luce’s (1987) work suggests one way to proceed would be to test person ability and item difficulty with axioms E1–E6. If person ability and item difficulty empirical structures did indeed satisfy these axioms, then they could be considered sufficiently complex to support ratio scale representations into the real numbers. If further experimental evidence showed that test

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performance was a function of an inverse multiplicative relationship between person ability and item difficulty, a subtractive relationship could be obtained by conducting logarithmic transformations. Specifying the probability function (Andrich, 1988) as the logistic cumulative would lead to the Rasch model. At present, there is no evidence to suggest that person ability and item difficulty are structures that behave according to axioms E1–E6. Hence the Rasch model cannot be considered as conjoint measurement via an analogy with derived measurement in physics.

The Rasch Model as Conjoint Measurement via Ordinal Relations upon Probabilities Rasch (1960) argued that person ability and item difficulty induce specific ordinal relations to hold upon probabilities of responding to test items: …a person having a greater ability than another person should have the greater probability of solving any item of the type in question, and similarly, one item being more difficult than another one means that for any person the probability of solving the second item correctly is the greater one. (p. 117)

It has been argued that given the presence of such relations over response probabilities when modelled using the Rasch model, the Rasch model is conjoint measurement (Brogden, 1977; Embretson & Reise 2000; Perline et al., 1979; Scheiblechner, 1999). Karabatsos (2001) outlined the reasoning underlying this argument. Let B = 〈β1, β2,…, βv,… βV 〉 be an ordered finite set of Rasch model person parameter estimates, with β1 being the lowest real logit-valued parameter and βV the greatest. Let ∆ = 〈δ1, δ2,…, δi,… δI 〉 be a finite set of Rasch item parameter estimates ordered in the same manner as the elements of B. Let the elements of Β define the row vectors of a simple matrix and let the elements of ∆ define the column vectors. Let the cell values of this matrix be the probabilities P = (Prvi)V×I of correctly responding to items 1, 2,…, I. The matrix of probabilities P can be defined such that P = f(B,∆), where f is the simple logistic function. Hence via f, the numerical relation ‘≥’ will always hold upon P if the relevant items have been empirically demonstrated to fit the Rasch model (Embretson & Reise, 2000). Hence a Rasch structure P = 〈B×∆, ≥〉 is analogous to the conjoint measurement empirical structure C = 〈X × Y,
~〉 and so hence the argument the Rasch model is conjoint measurement. From the perspective of RMT, such an argument is incorrect. As the ‘latent trait’ θ is assumed to be a subset of the real numbers (Junker & Sijtsma, 2001, p. 212; Karabatsos & Sheu, 2004, p. 110), θ is a Platonic realm pure set (Narens, 2002). Moreover, sets of Rasch person and item estimates are proper subsets of θ (B, ∆ ⊂ θ). As these sets too consist of Platonic realm pure sets

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of real numbers, B and ∆ are always structurally identical both to each other and to other pure sets of real numbers such as sets of probabilities. Recollect the RMT concept of empirical relational structure. The domains of such structures are spatio-temporally located objects, events or attributes (Krantz et al., 1971). With the conjoint measurement empirical structure C = 〈A × X,
~〉, the sets A and X contain such objects or events. With the Rasch structure P = 〈B×∆, ≥〉, however, the sets B and ∆ are sets of pure sets and thus do not contain empirical objects or events. Hence P = 〈B×∆, ≥〉 is not a conjoint measurement empirical structure, despite its similarity. From the perspective of RMT, the range of the Rasch model equation is a probability and is thus a real number and so hence is a pure set. The domain of the Rasch equation is the difference between two real numbers (being elements of the pure sets B and ∆), hence this difference itself is a real number. Thus the Rasch model is simply an isomorphism between two pure sets of structurally identical real numbers. Irrespective of whether the model fits any set of data or not, the model will isomorphically map probabilities (real numbers) into real numbers (the elements of B and ∆). Moreover, it has been shown the model does this for sets of data for which no underlying latent trait is assumed to exist (Wood, 1978). As it does not involve an empirical structure, the isomorphism of the Rasch model equation is not a representation theorem; therefore the Rasch model is not conjoint measurement. The item response theory models known as the two-parameter model (Birnbaum, 1968) and the three-parameter model (Lord, 1980) are not conjoint measurement for the same reason as the Rasch model. In these models, probabilities are homomorphically mapped into more than one set of real numbers, such as sets of ‘item discrimination’ and ‘guessing’ parameters in addition to the real valued difference between person ability and item difficulty parameters.

Investigating the Relationship between Conjoint Measurement and the Rasch Model From the perspective of RMT, the Rasch model is not conjoint measurement. Equations 6 and 7 simply map a set of real numbers (probabilities) into another set of real numbers (differences between logarithmic unit parameters). Moreover, Equations 6 and 7 do this for any given set of data (even one of coin-tosses [Wood, 1978]), irrespective of fit. In contrast, Equations 3 and 5 are representation theorems between empirical structures and the set of real numbers; and only hold if and only if the conjoint measurement axioms hold. These axioms may not hold for any given set of data. That the Rasch model is not conjoint measurement does not mean that a relationship between the two theories cannot exist. Force and mass are different structures which are nonetheless formally related (namely, Newton’s

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Second Law of Motion). Given that the Rasch model is a probabilistic numerical theory, a complete, formal investigation into a relationship between the model and conjoint measurement must identify exactly what kind of empirical structure probability represents. This, however, may not be possible. Difficulties present themselves even with the most basic of rules of probability. For disjoint events J and K, the probability union rule argues that Pr(J or K) = Pr(J) + Pr(K). Despite the additive relationship, this rule holds only for disjoint events. As there is no restriction to disjoint sets within axioms E1–E6, these axioms cannot characterize a probability empirical structure (Luce, 1997). Luce speculated events could be mapped into random variables such that expected values of these variables behave according to axioms E1–E6. However, Luce concluded: ‘The simple fact is that we do not know how to do this. All treatments of randomness with which I am familiar pre-suppose a numerical representation; there simply is no qualitative theory of the concept’ (p. 82). Moreover, whilst A and X must be disjoint in the conjoint measurement empirical structure C = 〈A × X,
~〉, satisfaction of the conjoint measurement axioms means the elements within both A and X sustain additive representations. Therefore a probability empirical structure is not a conjoint measurement empirical structure. Other research has encountered difficulties with probability empirical structures. Suppes and Alechina (1994) proved that the probability independence rule, Pr(J and K) = Pr(J) Pr(K), was only definable upon indicator functions of events; and not definable upon events themselves. Clark (2000) investigated indicator functions of events, however, and encountered difficulties with the uniqueness of representations. With the subjectivist concept of probability, axioms were proposed by de Finetti (1937/1964), but these only hold for sets of disjoint events (Adams, 1992; Luce, 1967). Moreover, they hold only for sets containing a very small number of disjoint events, perhaps less than five (Kraft, Pratt, & Seidenberg, 1959). Exactly how many they hold for, however, continues to be debated (see Conder & Slinko, 2004; and Fishburn, 1996). These results pose a serious problem in that events themselves and the relations holding over them cannot be taken as the empirical primitives of probability (Luce 1997, 2005; Suppes & Alechina, 1994). Hence it is not known exactly what kind of empirical structure probability represents. Therefore, a complete formal relationship between the Rasch model and conjoint measurement may not be possible, as it appears probability and structure cannot be unified at the foundational level (Luce, 2005). This does not, however, preclude empirical investigations, such as the kind conducted by Perline et al. (1979). The Rasch model argues that test performance is a non-interactive, additive function of ‘person ability’ and ‘item difficulty’, and therefore posits an argument that can be tested by conjoint measurement. Let P denote a set of persons, I the set of items and C, the Cartesian product of P and I. Let a, b and c in P be sets of persons and let x,

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y and z in I be individual items. Let ‘person ability’ and ‘item difficulty’ be surface-level relational structures that may weakly order the elements of P and I. Thus the existence of a conjoint measurement empirical structure C = 〈P × I,
~〉 could be hypothesised where C, is test performance. Consider the proportion of correct responses out of the total responses made to the items in I. In RMT terms, proportions are real numbers formed by natural number representations of discrete structures whose uniqueness is determined by the empirical procedure employed (namely, counting the number of correct responses and the total number of responses; Krantz et al., 1971; Luce & Suppes, 2002). Let the elements of a set of proportions comprise the cell values of a matrix corresponding to the empirical structure C = 〈P × I,
~〉. That is, each column in the matrix is a vector of proportions pertaining solely to x, y and z in I; and each row is a vector of proportions pertaining to a, b and c in P. Given the law of large numbers, the greater the number of observations, the closer these proportions will approximate probabilities. Let the matrix be permuted such that the magnitudes of the proportions are ordered from greatest to smallest or vice versa. It is possible that the conjoint measurement axioms could be tested through direct inspection of this permuted matrix; however, it is likely the proportions are adversely affected by extraneous factors such as fatigue, guessing, cheating and motivation given that experimental control over these factors is weak. Therefore it cannot be dismissed that the proportions are not contaminated with error; and so hence an entirely reasonable scientific conclusion is that direct inspection of the matrix is not advisable. Instead, the conjoint measurement axioms can be subjected to probabilistic test. Research in mathematical psychology has recently focused on the direct probabilistic testing of RMT axioms in their algebraic form, including the conjoint measurement axioms (Karabatsos, 2001, 2005, 2006; Karabatsos & Sheu, 2004; Karabatsos & Ulrich, 2002).4 The construction and testing of probabilistic numerical models of these axioms is no longer necessary. If the methodology devised by Karabatsos (2001) is applied to the proportions matrix, and if the results suggest all the conjoint measurement axioms have been supported, then it can be concluded in RMT terms that a representation theorem exists between the empirical structure C = 〈P × I,
~〉 and the set of the real numbers. That is, the surface structures of ‘person ability’ and ‘item difficulty’ holding over P and I are sufficiently complex to enable interval-scale representation into the real numbers (namely, Equation 3). Consider that all conjoint measurement axioms in this example are satisfied and the Rasch model is applied to the data obtained. Attention must be paid to the issue of model fit in whatever way fit may be assessed.5 For example, Embretson and Reise (2000) argued that ‘data that fit the Rasch model have the additivity property, which justifies interval-level measurement’ (p. 148). If this argument is correct, then the Rasch model will fit the data.

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However, Michell (2004) argued that tests of statistical model fit are not capable of distinguishing quantitative from merely ordinal psychological structures. Consider datasets which support the conjoint measurement independence axiom but reject double cancellation. Such datasets obtain from ordinal but not additive structures. If the argument of Embretson and Reise (2000) is true, then the Rasch model will not fit such datasets. If Michell’s (2004) argument is true, then the Rasch model may or may not fit. Datasets which violate all conjoint measurement axioms may also be investigated in addition to simulation studies. A critical area of research in quantitative psychology is individual differences (Luce, 1997). Empirical work in this vein has the potential to yield important results. For example, consider individuals whose responses violate the conjoint measurement axioms. Let their data be collated with data from individuals whose responses supported the conjoint measurement axioms, and data from individuals who supported only the independence axiom. The local homogeneity assumption (Ellis & van den Wollenberg, 1993) of item response theory psychometrics states that models have the same form within and between subjects (Borsboom et al., 2003). Assuming local homogeneity and given Embretson and Reise’s (2000) argument, would the Rasch model only fit those individuals who supported all conjoint measurement axioms? Or, given Michell’s (2004) argument, would the model also fit the data from those individuals who supported only the independence axiom of conjoint measurement? Would any of the ‘person fit’ indices investigated by Karabatsos (2003) indicate individual violation or support of the conjoint measurement axioms? These are questions that have yet to be examined. Simple, rote analyses of psychometric test data, however, may not reveal the underlying causal processes which these questions imply are involved. Contingent upon comprehensive investigations are the development and testing of explicit, substantive psychological theories of the item response process. Psychology is currently without such theory; hence the causal systems underlying individual performance on psychometric tests are not well understood. Calls for the development of substantive theory in psychometrics have been made (e.g. Borsboom et al., 2003; Goldstein & Wood, 1989); however, as Michell (2004) noted, psychometricians ‘studiously avoid that kind of theorizing’ (p. 125).

Conclusions Representational measurement theory is the dominant theory of measurement within the philosophy of science (Balzer, 1992; Narens, 2002; Schwager, 1991). Arguably its most important contribution to science has been the theory of conjoint measurement (Krantz et al., 1971; Luce & Tukey, 1964). It has been argued that the Rasch model is conjoint measurement by several psychometricians

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(Andrich 1988; Barrett, 2003; Bond & Fox, 2001; Borsboom & Mellenbergh, 2004; Brogden, 1977; Embretson & Reise, 2000; Fischer, 1995; Fisher, 2003; Green, 1986; Karabatsos, 2001; Keats, 1967; Kline, 1998; Perline et al., 1979; Roskam & Jansen, 1984; Scheiblechner, 1999; Wright, 1996, 1999). This paper critically evaluated this argument from the perspective of RMT and concluded it was incorrect. The Rasch model is not conjoint measurement as it cannot demonstrate the existence of a representation theorem between an empirical relational structure and a numerical relational structure. Possible relationships between the Rasch model and conjoint measurement, however, should be explored. Empirical work investigating the behaviour of the Rasch model upon datasets which satisfy all, some or none of the conjoint measurement axioms should be undertaken as a starting point. Model fit should be assessed. The effect of individual differences should also be investigated. It is highly doubtful, however, that manuscripts of such research would find a receptive audience in the mainstream psychometric literature (Cliff, 1992). Psychometrika and Applied Psychological Measurement contain little on this issue (Michell, 2004). The interest that Rasch adherents (e.g. Andrich, 1988; Embretson & Reise, 2000; Fisher, 2003; Wright, 1996, 1999) have displayed in attempting to link conjoint measurement to the Rasch model underscores the importance of conjoint measurement and fundamental measurement issues to this group of psychometricians. Indeed, Fisher (2003, p. 792) called these issues ‘crucial’. Perhaps only within the Rasch modelling field can the difficult empirical research into these crucial issues be conducted. Notes 1. The Zermelo–Fraenkel axioms avoid paradoxical definitions of sets (such as the famous Russell Paradox) and give a precise treatment of infinity. However, no unique model of them exists. Their intended interpretation was for uncountable sets (all real numbers), but the Löwenheim–Skolem theorem proved the Zermelo–Fraenkel axioms can also be interpreted as only referring to countable sets (such as the integers, rational and natural numbers). This does not confound conjoint measurement as the cancellation axioms hold for countable sets. Indeed, if double cancellation holds in unidimensional unfolding theory, positive integer representations are sustained (see Kyngdon, 2006). 2. Mundy (1987a) devised a second-order syntactic, naturalistic, Platonist RMT that he called TQ. However, he conceded that ‘from a purely formal viewpoint, elementary second order Platonist theories such as TQ are mathematically much weaker than any standard axiomatic set theory’ (p. 48). 3. An important caveat here is any empirical situation is finite and thus the ‘solvability’ and ‘Archimedean’ axioms (C4 and C5) can only be tested indirectly via the hierarchy of cancellation conditions (Scott, 1964). These axioms are not firstorder language concepts (see Luce & Narens, 1992; Mundy, 1987b). 4. This work is being led by George Karabatsos and his associates (Luce, 2005). The inference problems caused by the ordinal restrictions entailed by the conjoint

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measurement axioms (Falmagne, 1979; Iverson & Falmagne, 1985) seem to have been solved by this work. 5. Possible exceptions here are fit statistics based on Rasch model residuals. These suffer from dependencies between observed and expected scores and thus tend to underestimate model misfit (Karabatsos, 2000, 2001). References Adams, E.W. (1992). On the empirical status of measurement axioms: The case of subjective probability. In C.W. Savage & P. Erlich (Eds.), Philosophical and foundational issues in measurement theory (pp. 53–73). Hillsdale, NJ: Erlbaum. Andrich, D. (1988). Rasch models for measurement. Newbury Park, CA: SAGE. Balzer, W. (1992). The structuralist view of measurement: An extension of received measurement theories. In C.W. Savage & P. Erlich (Eds.), Philosophical and foundational issues in measurement theory (pp. 93–117). Hillsdale, NJ: Erlbaum. Barrett, P. (2003). Beyond psychometrics: Measurement, non-quantitative structure, and applied numerics. Journal of Managerial Psychology, 18, 421–439. Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In F.M. Lord & M.R. Novick (Eds.), Statistical theories of mental test scores (pp. 395–479). Reading, MA: Addison-Wesley. Bond, T., & Fox, C.H. (2001). Applying the Rasch model: Fundamental measurement in the human sciences. Hillsdale, NJ: Erlbaum. Borsboom, D., & Mellenbergh, G.J. (2004). Why psychometrics is not pathological: A comment on Michell. Theory & Psychology, 14, 105–120. Borsboom, D., Mellenbergh, G., & van Heerden, J. (2003). The theoretical status of latent variables. Psychological Review, 110, 203–219. Brogden, H.E. (1977). The Rasch model, the law of comparative judgement and additive conjoint measurement. Psychometrika, 42, 631–634. Campbell, N.R. (1920). Physics, the elements. Cambridge: Cambridge University Press. Clark, S.A. (2000). The measurement of qualitative probability. Journal of Mathematical Psychology, 44, 464–479. Cliff, N. (1992). Abstract measurement theory and the revolution that never happened. Psychological Science, 3, 186–190. Cohen, P.J. (1963). The independence of the continuum hypothesis, I. Proceedings of the National Academy of Sciences USA, 50, 1143–1148. Conder, M., & Slinko, A. (2004). A counterexample to Fishburn’s conjecture on finite linear qualitative probability. Journal of Mathematical Psychology, 48, 425–431. Davis, P.J., & Hersh, R. (1981). The mathematical experience. New York: Penguin. de Finetti, B. (1964). Foresight: Its logical laws, its subjective sources. In H.E. Kyburg (Ed. and Trans.) & H.E. Smokler (Ed.), Studies in subjective probability (pp. 93–158). New York: Dover. (Original work published 1937) Ellis, J.L., & van den Wollenberg, A.L. (1993). Local homogeneity in latent trait models: A characterization of the homogeneous monotone IRT model. Psychometrika, 58, 417–429. Embretson, S.E., & Reise, S.P. (2000). Item response theory for psychologists. Hillsdale, NJ: Erlbaum. Falmagne, J.C. (1979). On a class of probabilistic conjoint measurement models: Some diagnostic properties. Journal of Mathematical Psychology, 19, 73–88.

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ACKNOWLEDGEMENTS. This paper is based on material presented at the 12th Biennial International Objective Measurement Workshop (IOMW-XII), James Cook University, Cairns, Australia, July 2004. The author would like to thank A.A.J Marley, Paul Barrett and Denny Borsboom for their critical comments, discussions and encouragement upon a first draft of this paper; as well as three anonymous reviewers. ANDREW KYNGDON is Senior Research Scientist at MetaMetrics, Inc., of Durham, North Carolina, USA. His interests are in psychological measurement, and he has recently completed a three-part series on unidimensional unfolding theory for the Journal of Applied Measurement. ADDRESS: MetaMetrics Inc., 1000 Park Forty Plaza Drive, Suite 120, Durham, NC, 27713 USA. Email: [[email protected]].

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