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...”
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.
Stevens’ Classification of
Scale Types
113
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
Homogeneity and
Uniqueness (in general)
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
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.
119122
Real Relational Structures
Homogeneous, Archimedean
Ordered Translation Groups
123
“[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...”
END