Ch 20 Scale Types May 23, 2011 presented by Tucker Lentz May 23, 2011 presented by Tucker Lentz

Depth disclaimer Presentation only goes up to Theorem 7

What is the key to measurement? 1. Rich empirical structure 2. Symmetry 108

“By symmetry, one means that the structure is isomorphic to itself...” self-isomorphisms are called automorphisms Symmetry 109

Stanley Smith Stevens American psychologist who founded Harvard's Psycho-Acoustic Laboratorypsychologist Harvard Stevens’ Power Law in psychophysics “In most cases a formulation of the rules of assignment discloses directly the kind of measurement and hence the kind of scale involved. If there remains any ambiguity, we may seek the final and definitive answer in the mathematical group structure of the scale form: in what ways can we transform its values and still have it serve all the functions previously fulfilled?” 109

“Why do not psychologists accept the natural and obvious conclusion that subjective measurements of loudness in numerical terms (like those of length or weight or brightness)... are mutually inconsistent and cannot be the basis of measurement?” 110

“...Measurement is not a term with some mysterious inherent meaning, part of which may have been overlooked by physicists and may be in course of discovery by psychologists...we cease to know what is to be understood by the term when we encounter it; our pockets have been picked of a useful coin....” 110f

1. Wasn’t interested in existence and uniqueness theorems 2. Limited his work to only a handful of transformations and didn’t ask what the possible groups of transformations are. 3. Failed to raise the question of “possible candidate representations that exhibit a particular degree of uniqueness” (?) 4. No proper justification for the importance of invariance under automorphisms has been provided. Problems for Stevens 111f

Stevens’ Classification of Scale Types 113

A is a non-empty set (possibly empirical entities, possibly numbers) J is the index set, non empty, usually integers ∀ j ∈ J, S j is a relation of finite order on A A = A, S jj ∈ J is a relational structure Formal Definitions 115

If one of the S j is a weak or total order, we use ≿ A = A, ≿, S jj ∈ J is a weakly or totally ordered relational structure If A is a subset of Re and the weak or total order S j we used ≿ for is ≥, then we write ℛ = R, ≥, R jj ∈ J and call it an ordered numerical structure Formal Definitions 115

Isomorphism: φ is 1-to-1 mapping between the structures Homomorphism: φ is onto, but not 1-to-1 Automorphism: φ is an isomorphism between A and itself Endomorphism: φ is a homomorphism between A and itself Formal Definitions 115

Numerical Representation: A is a totally ordered structure ℛ is an ordered numerical structure A is isomorphic to ℛ Formal Definitions 115

M-point Homogeneity M is the size of two arbitrarily selected sets of ordered points that can always be mapped into each other by one of our automorphisms (element in ℋ ) 115

N-1 is the largest number of points at which any two distinct automorphic transformations may agree N-point Uniqueness 116

Note that we have moved from ℋ to G. Homogeneous if at least 1-point homogeneous Unique if there is an upper bound on the number of fixed points distinct automorphisms can agree Homogeneity and Uniqueness (in general) 116

M is the largest degree of homogeneity N is the least degree of uniqueness (M, N) is the scale type Scale Type 116

Theorem 1 i) if M-point homogeneous, then (M-1)-point homogeneous ii) If N-point unique, then (N+1)-point unique iii) M ≤ N 117

Theorem 2 M < order of some S j, or the scale type is (∞,∞) 117

For theorem 3 we need some more definitions F is a function or “generalized operation” on A n F is a set of generalized operations A -invariance Algebraic closure of B under F 118

Theorem 3 Another way of getting at N-point uniqueness, relating uniqueness to invariance 118

Dilation & Translation Every automorphism is either a dilation or a translation The identity function is the only automorphism that is both Dilations have at least one fixed point 118

Theorem 4 1-point uniqueness means that two translations can have at most 1 point in common. 118f

Real Relational Structures

“[The Archimedean] concept has been defined up to now only in structures for which an operation is either given or readily defined, as in the case of difference or conjoint structures. For general relational structures one does not know how to define the Archimedean property. This may be a reasonable way to do so in general, as is argued at some length in Luce and Narens (in press). Homogeneous, Archimedean Ordered Translation Groups 123

Theorem 7 123

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“What is clear is that the usual physical representation involving units in no way depends upon extensive measurement or even on having an empirical operation. The key to the representation is for the structure lying on one component of an Archimedean conjoint structure to have translations that form a homogeneous, Archimedean ordered group...” Theorem 7 124f

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