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Ch 20 Scale Types. May 23, 2011 presented by Tucker Lentz. Depth disclaimer Presentation only goes up to Theorem 7. 108. What is the key to measurement? 1. Rich empirical structure 2. Symmetry. 109. Symmetry. “By symmetry, one means that the structure is isomorphic to itself...”
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Ch 20Scale Types • May 23, 2011 • presented by Tucker Lentz
Depth disclaimer • Presentation only goes up to Theorem 7
108 What is the key to measurement? 1. Rich empirical structure 2. Symmetry
109 Symmetry “By symmetry, one means that the structure is isomorphic to itself...” self-isomorphisms are called automorphisms
109 Stanley Smith Stevens • 1906-1973 • American psychologist who founded Harvard's Psycho-Acoustic Laboratory • 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?”
110 “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?”
110f “...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 ....”
111f Problems for Stevens 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.
113 Stevens’ Classification of Scale Types
115 Formal Definitions A is a non-empty set (possibly empirical entities, possibly numbers) J is the index set, non empty, usually integers ∀j ∈ J, Sj is a relation of finite order on A A = ⟨A, Sj⟩j ∈ J is a relational structure
115 Formal Definitions If one of the Sj is a weak or total order, we use ≿ A = ⟨A, ≿, Sj⟩j ∈ J is a weakly or totally ordered relational structure If A is a subset of Re and the weak or total order Sj we used ≿ for is ≥, then we write ℛ = ⟨R, ≥, Rj⟩j ∈ J and call it an ordered numerical structure
115 Formal Definitions 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
115 Formal Definitions • Numerical Representation: • A is a totally ordered structure • ℛ is an ordered numerical structure • A is isomorphic to ℛ
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)
116 N-point Uniqueness N-1 is the largest number of points at which any two distinct automorphic transformations may agree
116 Homogeneity and Uniqueness (in general) 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
116 Scale Type M is the largest degree of homogeneity N is the least degree of uniqueness (M, N) is the scale type
117 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 Sj, or the scale type is (∞,∞)
118 For theorem 3 we need some more definitions F is a function or “generalized operation” on An 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
118f Theorem 4 1-point uniqueness means that two translations can have at most 1 point in common.
119-122 Real Relational Structures
123 Homogeneous, Archimedean Ordered Translation Groups “[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).
123 Theorem 7
124 Theorem 7
124 Theorem 7
124f Theorem 7 “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...”
