Ch 20Scale Types

May 23, 2011presented by Tucker Lentz

• Depth disclaimer

• Presentation only goes up to Theorem 7

What is the key to measurement?

1. Rich empirical structure

2. Symmetry

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“By symmetry, one means that the structure is isomorphic to itself...”

self-isomorphisms are called automorphisms

Symmetry109109

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?”

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“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?”

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“...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 ....”

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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 Stevens111f111f

Stevens’ Classification of Scale Types

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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

Formal Definitions115115

If one of the Sj is a weak or total order, we use ≿A = ⟨A, , S≿ j⟩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

Formal Definitions115115

Isomorphism: φ is 1-to-1 mapping between the structuresHomomorphism: φ is onto, but not 1-to-1Automorphism: φ is an isomorphism between A and itselfEndomorphism: φ is a homomorphism between A and itself

Formal Definitions115115

Numerical Representation:

A is a totally ordered structure

ℛ is an ordered numerical structure

A is isomorphic to ℛ

Formal Definitions115115

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

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

N-point Uniqueness116116

Note that we have moved from to ℋ G. Homogeneous if at least 1-point homogeneousUnique if there is an upper bound on the number of fixed points distinct automorphisms can agree

Homogeneity and Uniqueness (in general)

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M is the largest degree of homogeneityN is the least degree of uniqueness (M, N) is the scale type

Scale Type116116

Theorem 1

i) if M-point homogeneous, then (M-1)-point homogeneousii) If N-point unique, then (N+1)-point uniqueiii) M ≤ N

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Theorem 2

M < order of some Sj, or the scale type is (∞,∞)

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For theorem 3 we need some more definitionsF is a function or “generalized operation” on An

F is a set of generalized operationsA-invarianceAlgebraic closure of B under F

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Theorem 3

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

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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

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Theorem 4

1-point uniqueness means that two translations can have at most 1 point in common.

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Real Relational Structures

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“[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

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Theorem 7123123

Theorem 7124124

Theorem 7124124

“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 7124f124f

END