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In the theory of probabilistic measurement concepts of measurement are developed in a probabilistic framework to provide a theoretically sound treatment of variability of measurements. Concepts of probabilistic measurement evolved in the context of the behavioral and social sciences, for instance in psychophysics and utility theory. They diﬀer from the standard Gaussian theory of error as applied in physics, where at a theoretical level the underlying scales considered (e.g., length, mass) are conceived in a deterministic fashion. To deal with the variability of measurements (‘error’) that arise in practical applications, statistical concepts are then introduced in an adhoc manner on top of these deterministic concepts. In the Social and Behavioral Sciences, in contrast, variability (e.g., of psychophysical or preference judgments made by the subjects) usually is considered a feature of the domain considered. Therefore, concepts of probabilistic measurement aim at the theoretical (possibly axiomatic) foundation of the scales themselves in a probabilistic theoretical framework.

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Roughly, two traditions can be distinguished: the psychometric and the measurement-theoretic approach. The following account focuses mainly on the measurement-theoretic perspective which strives for suitable probabilistic counterparts to concepts of the representational theory of measurement.

Several general strategies of probabilization will be discussed. For brevity’s sake they will be explained using the examples of additive conjoint and algebraic diﬀerence measurement (Krantz et al. 1971).

## 1. Fechnerian and Thurstonian Scaling

This section brieﬂy outlines two basic approaches to the scaling of subjective magnitudes which played an important part in the development of probabilistic measurement.

Consider an experimental paradigm where a subject is asked to compare two stimuli out of a stimulus set A according to some criterion (e.g., loudness of tones, brightness of ﬂashes, subjective utilities of commodities). If in repeated comparisons of the same pair of stimuli the subject made the same judgments throughout, this would deﬁne a relation ˃̰ on A, with a ˃̰ b meaning that a is chosen over b. If ˃̰ fulﬁlled the (testable) axioms of a weak order this would give rise to the introduction of ordinal subjective scales u : A → Re satisfying u(a) ≥ u(b) if and only if a ˃̰ b.

However, ordinarily the subject’s judgments will vary between experimental trials, particularly for stimuli ‘close’ to each other. This variability can be accounted for by considering the corresponding system of choice probabilities p(a, b), where p(a, b) denotes the probability that a is chosen over b.

The weak-utility model assumes that the relation ˃̰ deﬁned by

a ˃̰ b if and only if p (a, b) ≥ 1/2

is a weak order giving rise to an ordinal scale u as described before. A necessary testable condition for this model is weak stochastic transitivity

If p (a, b) ≥ 1/2 and p (b, c) ≥ 1/2 then p (a, c) ≥ 1/2.

The weak-utility model is implied by the strongutility model, which postulates that p (a, b) is a measure of the strength-of-preference or discriminability in the sense that for some scale u

p (a, b) ≥ p (c, d)

if and only if u (a) ‒ u (b) ≥ u (c) ‒ u (d).

The strong-utility model is essentially equivalent to the Fechnerian model which is based on the assumption that there is a strictly increasing function F : Re Re and a scale u satisfying

p (a, b) = F [u(a) ‒ u (b )].

(for relations to Fechnerian psychophysics see Falmagne 1986).

A diﬀerent path was taken in the twenties by L. L. Thurstone. On his account, subjective magnitudes are not represented by single numbers but by real random variables. The basic assumption is that there is a family of jointly distributed random variables, Ua (a A) on a sampling space Ω that corresponds to diﬀerent experimental situations ω Ω, such that in situation ω the stimulus a is chosen over b if and only if Ua(ω) ≥Ub(ω). It is further assumed that for a suitable probability measure P

p (a, b ) = P( ω ɛ ΩǀUa(ω) ≥ Ub(ω) ) = : Pr[Ua ≥ Ub]

Models of that kind are called random utility models. An example of a testable necessary condition for this model is the so-called triangle inequality,

p (a, b) + p (b, c) ‒ p (a, c) ≤ 1.

Under certain conditions this model implies the Fechnerian model. For instance, Thurstone assumed that the random variables Ua are normally distributed; under the additional assumption that these random variables ar e pairwise independent and have equal variances σ² (Case V of Thurstone’s law of comparative judgment) it follows that a particular case of the Fechnerian model holds (with u(a) the expected value of Ua and F the cumulative distribution function of N (0, 2σ²)).

Note that the strict utility model is equivalent both to a special case of the Fechnerian and of the random utility model (compare Suppes et al. 1989, Chap. 17).

While these and other instantiations of the Fechnerian and the random utility model involve speciﬁc distributional assumptions, attention will be focused in the following on distribution-free approaches. The concepts that are introduced are probabilistic versions of standard concepts of the representational theory of measurement.

## 2. Probabilistic Diﬀerence and Additi e Conjoint Measurement

### 2.1 Fechnerian Scaling as Probabilistic Diﬀerence Measurement

Recall that a structure ‹A, ≥›, where ˃̰ is a weak ordering on A x A, is called an algebraic diﬀerence structure if there is representation u : A → Re such that

(a, b) ˃̰ (c, d)

if and only if u (a) ‒ u (b) ≥u (c) ‒ u (d).

If one deﬁnes a quaternary relation ˃̰ by

(a, b) ˃̰ (c, d) if and only if p (a, b) ˃̰ p (c, d),

then the strong-utility model holds if and only if the structure A, is an algebraic diﬀerence structure. Hence, Fechnerian scaling can be conceived as probabilistic diﬀerence measurement.

In consequence, axiom systems (Krantz et al. 1971, Chap. 4) for algebraic diﬀerence structures directly yield testable conditions that have to be satisﬁed by the choice probabilities for this model to apply.

### 2.2 Probabilistic Additi e Conjoint Measurement

By way of illustration, consider an experimental paradigm where subjects are asked to compare the binaurally induced subjective loudness of successively presented pairs (a, x), (b, y) of tones, with a and b being presented to the left ear and x and y to the right ear. Here two sets of stimuli A and X are referred to (tones presented to the left right ear) and subjects have to compare pairs (a, x) ɛ A x X of stimuli. A possible deterministic approach could be to assume that those judgments induce a weak ordering ˃̰on A x X and that the structure ‹A x X, ˃̰› is an additive conjoint structure, meaning that there are scales u : A → Re and v : X → Re such that

(a, x) ˃̰ (b, y)

if and only if u (a) + v (x) ≥ u (b) + v (y).

Again, in practice a probabilistic variant of this approach is preferable. This means that choice probabilities p(a, x; b, y) have to be considered. A probabilistic variant of the above approach can be obtained by assuming that the ordering deﬁned by

(a, x) ˃̰ (b, y) if and only if p (a, x; b, y) ≥ 1/2

yields an additive conjoint structure. As before, testable conditions can be directly derived from the standard axiomatizations of additive conjoint measurement. For example, the axiom of double cancellation translates into the condition

if p (a, x; b, y) ≥ 1/2 and p (c, y; a, z) ≥ 1/2,

then p (c, x; b, z) ≥ 1/2.

This model together with the condition p(a, x; b, y) ≤ p(d, z; b, y) ↔ p(a, x; d, z) ≤ 1/2 is equivalent to the existence of a representation of the form

p (a, x; b, y) = F [u (a) + v (x), u (b) + v ( y)]

where F is a function, strictly increasing in the ﬁrst and strictly decreasing in the second argument. Falmagne (1979) discusses a number of more speciﬁc cases of this model and their interrelations. Some of these combine concepts of algebraic-diﬀerence measurement (Fechnerian scaling) and additive conjoint measurement, a diﬀerent such combination being the additivediﬀerence model of preference (see Suppes et al. 1989, Chap. 17). An analogous probabilization of extensive measurement was developed in Falmagne (1980).

## 3. Mixture Models and Generalized Random Utility Models

A diﬀerent perspective is taken in some recent developments in which generalized distribution-free random utility models are considered and proven to be equivalent to models based on probabilistic mixtures of standard deterministic measurement structures (Niederee and Heyer 1997; Regenwetter and Marley in press). Being a generalization of Block and Marschak’s (1960) classical account of binary choice systems induced by rankings, it lends itself to a probabilization of a wide class of concepts of the representational theory of measurement. The underlying general principles are explained by considering the case of additive conjoint measurement. For simplicity, only two-factorial additive conjoint structures with ﬁnite domains A = X will be considered for which there is a representation u : A → Re satisfying (a, b) ˃̰(c, d ) if and only if u (a) + u (b) ≥ u (c) + u (d ).

### 3.1 Mixture Models

Assume choice probabilities p (a, b; c, d) to be given as considered above. The basic assumption underlying an additive conjoint mixture model is that in each experimental trial the subject is in a speciﬁc state corresponding to an additive conjoint structure. These structures may vary, each of them occurring with a certain probability. The states themselves will usually be unobservable, but they are assumed to determine the observable individual response (choice) in each situation.

Formally, this can be stated as follows. Let **Μ** be the set of all conjoint structures on A x A, that is, structures ‹A x A, ˃̰› with diﬀerent, and let P a probability measure on **M** describing the probability with which states, that is, conjoint structures, occur. The measure P is then said to explain the choice probabilities p(a, b; c, d ) if and only if

p (a, b; c, d) = P ({‹A x A, ˃̰› ɛ **M**ǀ(a, b ) ˃̰ (c, d)})

for all a, b, c, d A.

This concept is related closely to the following distribution-free random utility model.

### 3.2 Generalized Random Utility Models and Their Relation to Mixture Models

The underlying basic assumption is that a subject’s states correspond to sample points ω in the sampling space Ω and that for each a A there is a real random variable Ua such that in state ω the subject chooses (a, b) over (c, d) if and only if Ua(ω) + Ub(ω) ≥ Uc (ω) + Ud(ω). Again, it is assumed that states occur with certain probabilities.

Formally, let P be a probability measure on Ω. The measure P and a family Ua of random variables are then said to explain the choice probabilities p(a, b; c, d ) if and only if

p (a, b; c, d) = P ({ω ɛ ΩǀUa (ω) + Ub (ω) ≥ Uc (ω) + Ud (ω)}).

A probabilistic representation theorem can be proven which states that for each system of choice probabilities p(a, b; c, d ) there is a probability measure on **M** explaining these probabilities if and only if there is a probability measure on Ω and a family Ua of real random variables explaining these probabilities. Hence, the above additive conjoint mixture and random utility model explain the same systems of choice probabilities.

### 3.3 The General Case

The concepts and results outlined in the previous subsections can be extended to a wide class of relational structures, including linear orders, weak orders, semiorders, partial orders, equivalence relations, algebraic diﬀerence structures, extensive structures, and other classes of structures studied in the representational theory of measurement. In Niederee and Heyer (1997) a corresponding unifying general conceptual framework and general representation and characterization theorems are presented, which include the above results as special cases (cf. also Regenwetter and Marley (in press); see Characterization Theorems in Random Utility Theory for related developments).

## 4. Falmagne’s Concept of Random Additive Conjoint Measurement

A diﬀerent strategy of probabilizing the concept of additive conjoint measurement underlies Falmagne’s notion of random conjoint measurement. By way of illustration, consider again the above example of binaural loudness comparison. For simplicity, stimuli will be identiﬁed with suitable numbers (e.g., denoting the stimulus energy). The experimental paradigm is now modiﬁed in such a way that for stimuli a, x, y chosen by the experimenter, subjects are asked to determine a stimulus b with the property that (a, x) appears equally loud as (b, y). In a deterministic conjoint measurement framework, a plausible assumption would be that this is the case if and only if u (a) + v (x) = u (b) + v(y). Assume instead that for each triple a, x, y the settings b can be conceived as a random variable Ux,y(a) with uniquely deﬁned median mxy(a). Falmagne’s model then postulates that there are scales u, v such that

u [Uxy (a)] = v (x) ‒ v ( y) u (a) + ε xy(a)

where ε xy(a) is a random variable (denoting error) which has a unique median equal to zero. For such a representation to exist, certain testable conditions must be satisﬁed. For instance, it must hold that mxy[myz(a)] mxz(a).

## 5. Concluding Remarks

All of the above probabilistic approaches are derived from deterministic measurement concepts. While in Sects. 1 and 2 algebraic measurement structures are considered whose ordering is deﬁned in terms of (choice) probabilities, the mixture models of Sect. 3 refer to an ‘urn’ of algebraic measurement structures, each of which corresponds to a latent state. The concept of random conjoint measurement of Sect. 4 refers to a single algebraic structure where the experimental outcome is assumed to be pertubated by ‘error.’ Despite these fundamental diﬀerences in perspective, relationships can be established for certain special cases. This was illustrated in Section 1 by Thurstone’s Case V. (For the relation between probabilistic and random conjoint measurement see Falmagne 1986, Sect. 9.7.) Further research is needed on the relation among these approaches and the role of independence and distributional assumptions. In this connection it could be proﬁtable to combine these approaches with qualitative characterizations of distributions as found, for example, in Suck (1998).

As in the axiomatic tradition of the representational theory of measurement, emphasis is placed in all of these developments on testable conditions. In the present context, however, empirical tests of these conditions require suitable statistical procedures, many of which still need to be developed (for some results see Iverson and Falmagne 1985).

Which of the above concepts of probabilistic measurement, if any, is ‘the correct’ one for a given situation will certainly depend on the substantive area under consideration. This suggests that future developments of more speciﬁc concepts of probabilistic measurement should be tied to the corresponding substantive theories.

In the above description of problems and results of probabilistic measurement details had to be omitted. For a more detailed and comprehensive account of the material in Sects. 1, 2, and 4 see, for example, Falmagne (1986, Roberts (1979), and Suppes et al. (1989, Chaps. 16–7).

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