MEASUREMENT THEORY AND ITS
APPLICATIONS
FRED S. ROBERTS
RUTGERS UNIVERSITY
1
MEASUREMENT
Measurement has something to do with numbers.
Representational theory of measurement:
Assign numbers to “objects” being measured in such a way that
certain empirical relations are “preserved.”
In measurement of temperature, we preserve a relation “warmer
than.”
In measurement of mass, we preserve a relation “heavier than.”
2
A: set of objects
R : binary relation on A
aRb $ a is warmer t han b
aRb $ a is heavier t han b:
f :A ! R
aRb $ f (a) > f (b):
R could be preference. Then f is a utility function (ordinal utility
function).
In mass, there is more going on: There is an operation
of
a±b
combination of objects and mass is additive:
means a
combined with b:
f (a ± b) = f (a) + f (b):
3
Homomorphisms
Relational System:
A = (A; R 1 ; R 2 ; :::; R p ; ±1 ; ±2 ; :::; ±q )
A set
R±ii relation on A (not necessarily binary)
operation on A (binary)
A
Homomorphism
from
0 ; ±0 ; ±0 into
B = (B ; R 0 ; R 0 ; :::; R
; :::; ±0 )
1
2
p 1 2
q
f :A ! B
is a function
such that
(a1 ; a2 ; :::; am ) 2 R i $
i
(f (a1 ); f (a2 ); :::; f (am )) 2 R i0
i
f (a ±j b) = f (a) ±0j f (b):
4
(This only makes sense if Ri and
R0
i
are both
m i ¡ ar y:)
)
Examples of Homomorphisms
Example 1:
a´ b
Let a mod 3 be the number b in {0,1,2} such that
(mod 3).
Let A = {0,1,2,…,26} and
aRb $ a mod 3 > b mod 3.
(R;>):
Then f(a) = a mod 3 is a homomorphism from (A,R) into
f(11) = 2, f(6) = 0, ...
5
Example 2:
A = {a,b,c,d}
R = {(a,b),(b,c),(a,c),(a,d),(b,d),(c,d)}
f(a) = 4, f(b) = 3, f(c) = 2, f(d) = 1.
(R;>):
f is a homomorphism from (A, R) into
Second homomorphism: g(a) = 10, g(b) = 4, g(c) = 2, g(d) = 0.
Example 3:
A = {a,b,c}, R = {(a,b),(b,c),(c,a)}
f : (A;R) ! (R;>):
There is no homomorphism
aRb! f (a) > f (b); bRc! f (b) > f (c)
Therefore, f(a) > f(c):
But cRa  f(c) > f(a).
6
Example 4:
A = {0,1,2,…}, B = {0,2,4,…}
A homomorphism
f : (A;>;+) ! (B;>;+) :
f(a) = 2a
a > b $ f (a) > f (b)
f(a + b) = f(a) + f(b).
Second homomorphism: g(a) = 4a:
A homomorphism does not have to be onto.
Example 5:
f : (R;>;+) ! (R+;>;£) :
A homomorphism
f(a)
= e!a f (a) > f (b)
a > bÃ
f (a + b) = ea+ b = ea eb = f (a) £ f (b):
7
Temperature
Measurement: homomorphism
(A;R)
! (R;>)
Mass Measurement:
(A;R;±)
! (R;>;+): homomorphism
Since negative numbers do not arise(R
in+the
case of mass, we can think
;>;+):
of the homomorphism as being into
The Two Problems of Representational Measurement Theory
A
Representation Problem: Find conditions on (necessary)
and
A
sufficient for the existence of a homomorphism from into .
Subproblem: Find an algorithm for obtaining the homomorphism.
8
A
A
If f is a homomorphism from into , we call ( , ,f) a
scale.
A representation theorem is a solution to the representation problem.
A
Uniqueness Problem: Given a homomorphism from
unique is it?
into , how
A uniqueness theorem is a solution to the uniqueness problem.
A representation theorem gives conditions under which measurement
can take place. A uniqueness theorem is used to classify scales and to
determine what assertions arising from scales are meaningful in a
sense we shall make precise.
9
The Theory of Uniqueness
Admissible Transformations
An admissible transformation sends one acceptable scale into
another.
Cent igrade ! F ahr enhei t
K ilograms ! P ounds
In most cases one can think of an admissible transformation as
defined on the range of a homomorphism.
A
Suppose f is a homomorphism from into .
' ±f
' : f (A) ! B
is called an admissible
transformation of f if
is
A
again a homomorphism from into  .
10
Cent igrade ! F ahr enhei t :
' (x) =
9
x + 32
5
K ilograms ! P ounds :
' (x) = 2:2x
Example:
(R;>) ! (R;>) by f (x) = 2x:
It is a homomorphism:
a > b $ f (a) > f (b):
' : f (R ) ! R by ' (x) = 2x + 10:
 is admissible:
' ± f (x) = 4x + 10:
a > b $ ' ± f (a) > ' ± f (b):
11
Can every homomorphism g be obtained from a homomorphism f
by an admissible transformation of scale?
No. (Semiorders)
A = f 0; 1=2; 10g; aRb $ a > b+ 1:
B = f 0; 1=2; 10g; aSb $ a > b+ 1:
Two homomorphisms (A,R) into (B,S):
f(0) = f(1/2) = 0, f(10) = 10
g(0) = 0, g(1/2) = 1/2, g(10) = 10.
g = ' ±f :
' : f (A) ! B
There is no
such that
A homomorphism f is called irregular if there is a homomorphism g
not attainable from f by an admissible transformation.
12
Irregular scales are unusual and annoying. We shall assume (until
later) that all scales are regular. Most common scales of
measurement are. In the case where all scales are regular, we can
develop the theory of uniqueness by studying admissible
transformations.
A classification of scales is obtained by studying the class of
admissible transformations associated with the scale. This defines
the scale type. (S.S. Stevens)
13
Some Common Scale Types
Class of Adm.
Transfs.
' (x) = ®x, ® > 0
Scale Type
Example
ratio
Mass
Temp. (Kelvin)
Time (intervals)
Loudness (sones)?
Brightness (brils)?
interval
Temp(F,C)
Time (calendar)
IQ tests (standard
scores?)
' (x) = ®x + ¯, ® > 0
14
Class of Adm.
Transfs.
x ¸ y $ ' (x) ¸ ' (y)
Scale Type
Example
ordinal
Preference?
Hardness
Grades of leather,
wool, etc.
IQ tests (raw scores)?
absolute
Counting
strictly increasing
' (x) = x
15
Theorem
(Roberts and Franke 1976) If every homomorphism from
A
into  is regular and f and g are homomorphisms, then f is a
ratio scale iff g is a ratio scale. Same for interval, ordinal, absolute,
etc.
However: This fails in the irregular case.
16
Meaningful Statements
In measurement theory, we speak of a statement as being meaningful
if its truth or falsity is not an artifact of the particular scale values
used.
Assuming all scales involved are regular, one can use the following
definition (due to Suppes 1959 and Suppes and Zinnes 1963).
Definition: A statement involving numerical scales is meaningful if
its truth or falsity is unchanged after any (or all) of the scales is
transformed (independently?) by an admissible transformation.
17
If some scales are irregular, we have to modify the definition:
Alternate Definition: A statement involving numerical scales is
meaningful if its truth or falsity is unchanged after any (or all) of the
scales is (independently?) replaced by another acceptable scale.
We shall usually disregard the complication.
(But it does arise in practical examples, for example those involving
preference judgments under the semiorder model.)
Even this alternative definition is not always applicable in some
situations. However, it is a useful definition for a lot of situations and
we shall adopt it.
18
“This talk will be three times as long as the previous talk.”
Is this meaningful?
We have a ratio scale (time intervals).
(1)
f(a) = 3f(b).
This is meaningful' if
f is a ratio scale. For, an admissible
(x) = ®x, ® > 0
transformation is
. We want (1) to hold iff
(' ± f )(a) = 3(' ± f )(b):
(2)
But (2) becomes ®f (a) = 3®f (b):
(3)
(1) Ã ! (3) since®> 0:
19
“The high temperature today was three times the high
temperature yesterday.”
Meaningless. It could be true with Fahrenheit and false with
Centigrade, or vice versa.
In general:
For ratio scales, it is meaningful to compare ratios:
f(a)/f(b) > f(c)/f(d)
For interval scales, it is meaningful to compare intervals:
f(a) - f(b) > f(c) - f(d)
For ordinal scales, it is meaningful to compare size:
f(a) > f(b)
20
“I weigh 1000 times what the Statue of Liberty weighs.”
Meaningful. It involves ratio scales.
It is false no matter what the unit.
Meaningfulness is different from truth.
It has to do with what kinds of assertions it makes sense to make,
which assertions are not accidents of the particular choice of scale
(units, zero points) in use.
21
Average Performance
Study two groups of individuals, machines, algorithms, etc. under
different conditions.
f(a) is the time required by a to finish a task.
Data suggests that the average performance of individuals in the first
group is better than the average performance of individuals in the
second group.
a1 , a2 , ..., an
b1 , b2 , ..., bm
(1)
individuals in first group
individuals in second group.
1 Xn
1 Xm
f (ai ) >
f (bi )
n
m
i= 1
i= 1
We are comparing arithmetic means.
22
Statement (1) is meaningful iff for all admissible transformations of
scale , (1) holds iff
1 Xn
1 Xm
' ± f (ai ) >
' ± f (bi )
n
m
(2)
i= 1
i= 1
Performance = length of time, a ratio scale. Thus,
so (2) becomes
Xn
Xm
1
n
(3)
1
®f (ai ) >
m
i= 1
®> 0
Then
implies
(1) $ (3)
' (x) = ®x, ® > 0
®f (bi )
i= 1
. Hence, (1) is meaningful.
23
,
Note: (1) is still meaningful if f is an interval scale. For example,
the task could be to heat something a certain number of degrees and
f(a) = temperature increase produced by a.
Here,
(4)
' (x) = ®x + ¯, ® > 0
. Then (2) becomes
1 Xn
1 Xm
[®f (ai ) + ¯] >
[®f (bi ) + ¯]
n
m
i= 1
i= 1
This readily reduces to (1).
However, (1) is meaningless if f is just an ordinal scale.
24
To show that comparison of arithmetic means can be meaningless for
ordinal scales:
Suppose f(a) is measured on an ordinal scale, e.g., 5-point scale:
5=excellent, 4=very good, 3=good, 2=fair, 1=poor. In such a
scale, the numbers don't mean anything, only their order matters.
Group 1: 5, 3, 1 average 3
Group 2: 4, 4, 2 average 3.33 (greater)
5! 100, 4! 75, 3! 65, 2! 40, 1! 30
Admissible transformation:
.
New scale conveys the same information. New scores:
Group 1: 100, 65, 30 average 65 (greater)
Group 2: 75, 75, 40 average 63.33
25
Thus, comparison of arithmetic means can be meaningless for ordinal
data.
Of course, you may argue that in the 5-point scale, at least equal
spacing between scale values is an inherent property of the scale. In
that case, the scale is not ordinal and this example does not apply.
On the other hand, comparing medians is meaningful with ordinal
scales: To say that one group has a higher median than another group
is preserved under admissible transformations.
26
Importance Ratings/Performance Ratings
Suppose each of n individuals is asked to rate each of a collection of
alternative variables, people, machines, policies as to their relative
importance. Or we rate the falternatives
on different criteria or against
i (a)
different benchmarks. Let
be the rating of alternative a by
individual i (under criterion i). Is it meaningful to assert that the
average rating of alternative a is higher than the average rating of
alternative b? A similar question arises in performance ratings,
loudness ratings, brightness ratings, confidence ratings, etc.
(1)
1 Xn
1 Xn
f i (a) >
f i (b)
n
n
i= 1
i= 1
27
If each
(2)
fi
is a ratio scale, then we consider for  > 0,
1 Xn
1 Xn
®f i (a) >
®f i (b)
n
n
i= 1
i= 1
(1) Ã ! (2)
Clearly,
, so (1) is meaningful.
f 1 , f 2 , ..., f n
Problem:
might have independent units. In thisf case, we
i
want to allow independent admissible transformations of the . Thus,
we must consider
(3)
1 Xn
1 Xn
®i f i (a) >
®i f i (b)
n
n
i= 1
It is easy to see that there are
(1) is meaningless.
i= 1
®i
so that (1) holds and (3) fails. Thus,
28
Motivation for considering different
f 1 (a) = wei ght
®i
:
f 2 (a) = hei ght
n = 2,
of a,
of a. Then (1) says that the
average of a's weight and height is greater than the average of b's
weight and height. This could be true with one combination of weight
and height scales and false with another.
Conclusion: Be careful when comparing arithmetic mean importance
or performance ratings.
29
In
p this context,
p it is safer to compare geometric means (Dalkey).
n ¦
p
n ¦
all
n f (a)
i= 1 i
n
i= 1
> n ¦ n f i (b) $
i= 1
p
®i f i (a) > n ¦ n ®i f i (b)
i= 1
,
®i > 0
fi
Thus, if each
is a ratio scale, if individuals can change importance
rating scales (performance rating scales) independently, then
comparison of geometric means is meaningful while comparison of
arithmetic means is not.
30
Application
Roberts (1972, 1973). A panel of experts each estimated the relative
importance of variables relevant to energy demand using the
magnitude estimation procedure. If magnitude estimation leads to a
ratio scale -- Stevens presumes this -- then comparison of geometric
mean importance ratings is meaningful. However, comparison of
arithmetic means may not be. Geometric means were used.
31
Magnitude Estimation by One Expert of Relative Importance for Energy
Demand of Variables Related to Commuter Bus Transportation in a
Given Region
Variable
Rel. Import. Rating
1. No. bus passenger mi. annually
80
2. No. trips annually
3. No. miles of bus routes
4. No. miles special bus lanes
5. Average time home to office
100
50
50
70
6. Average distance home to office
7. Average speed
8. Average no. passengers per bus
65
10
20
9. Distance to bus stop from home
10. No. buses in the region
11. No. stops, home to office
50
20
20
32
Performance Evaluation
Fleming and Wallace (1986) make a similar point about performance
evaluation of computer systems. A number of systems are tested on
different benchmarks. Their scores on each benchmark are
normalized relative to the score of one of the systems. The
normalized scores of a system are combined by some averaging
procedure. If the averaging is the arithmetic mean, then the statement
that one system has a higher arithmetic mean normalized score than
another system is meaningless: The system to which scores are
normalized can determine which has the higher arithmetic mean.
33
However, if geometric mean is used, this problem does not arise
(under reasonable assumptions about scores).
f i (u)
= performance of system u on benchmark i.
F is a “combined score” such as arithmetic mean.
F ( f 1 ( u ) , ...,
f 1 (x)
f n (u) )
f n (x)
> F ( f 1 ( v) , ...,
f 1 (x )
f n (v) )
f n (x)
(normalized score relative to system x)
F ( f 1 ( u ) , ...,
f 1 (y)
f n (u) )
f n (y)
< F ( f 1 ( v ) , ...,
f 1 (y)
f n (v) )
f n (y)
(normalized score relative to system y)
34
System
BenchMark
R
M
Z
1
417
244
134
2
83
70
70
3
66
153
135
4
39449
33527
66000
5
72
368
369
35
Scores Normalized to System R
R
M
Z
1
1.00
0.59
0.32
2
1.00
0.84
0.85
3
1.00
2.32
2.05
4
1.00
0.85
1.67
5
1.00
0.48
0.45
Arith.
Mean
1.00
1.01
1.07
36
Scores Normalized to System M
R
M
Z
1
1.71
1.00
0.55
2
1.19
1.00
1.00
3
0.43
1.00
0.88
4
1.18
1.00
1.97
5
2.10
1.00
1.00
Arith.
Mean
1.32
1.00
1.08
37
Geometric Means
Geom. Mean of Scores Normalized Relative to R
R
1.00
M
0.86
Z
0.84
Geom. Mean of Scores Normalized Relative to M
R
1.17
M
1.00
Z
0.99
Similar situation if we test students on different benchmarks
38
How Should we “Average” or “Merge” Scores?
Sometimes arithmetic means are not a good idea. Sometimes
geometric means are. Are there situations where the opposite is the
case? Or some other method is better? Can we lay down some
guidelines about when to use what merging procedure?
u = F (a1 ; a2 ; :::; an )
a1 , a2 , ..., an
are scores.
F is an unknown merging function.
Approaches to finding acceptable merging functions:
(1) axiomatic
(2) scale types
(3) meaningfulness
39
An Axiomatic Approach
Theorem (Fleming and Wallace). Suppose
following properties:
F : Rn ! R+
+
has the
(1). Reflexivity: F(a,a,...,a) = a
F (a1 ; a2 ; :::; an ) = F (a¼( 1) ; a¼( 2) ; :::; a¼( n ) )
(2). Symmetry:
for all
¼ of f 1; 2; :::; ng
permutations
(3). Multiplicativity:
F (a1 b1 ; a2 b2 ; :::; an bn ) = F (a1 ; a2 ; :::; an )F (b1 ; b2 ; :::; bn ):
Then F is the geometric mean. And conversely.
40
A Functional Equations Approach Using Scale Type or
Meaningfulness Assumptions
Unknown function
u = F (a1 ; a2 ; :::; an ):
ai
Luce's idea: If you know the scale types of the and the scale type
of
ai u and you assume that an admissible transformation of each of the
leads to an admissible transformation of u, you can derive the
form of F.
Example:
scale.
a1 , a2 , ..., an
are independent ratio scales, u is a ratio
F : Rn ! R+
+
41
F (a1 ; a2 ; :::; an ) = u !
F (®1 a1 ; ®2 a2 ; :::; ®n an ) = ®u:
®1 > 0, ®2 > 0, ..., ®n > 0, ® > 0 and ® depends on ®1 , ®2 , ..., ®n .
Thus, we get the functional equation:
(*)
F (®1 a1 ; ®2 a2 ; :::; ®n an ) =
A(®1 ; ®2 ; :::; ®n )F (a1 ; a2 ; :::; an );
A(®1 ; ®2 ; :::; ®n ) > 0
F : Rn ¡ ! R+
+
Theorem (Luce 1964):
If
is continuous and satisfies
¸ > 0, c1 , c2 , ..., cn
(*), then there are
so that
F (a1 ; a2 ; :::; an ) = ¸ ac1 ac2 :::acn :
1
2
n
42
(Aczél, Roberts, Rosenbaum 1986): It is easy to see that the
assumption of continuity can be weakened to continuity at a point,
monotonicity, or boundedness on an (arbitrarily small, open) ndimensional interval or on a set of positive measure. Call any of these
assumptions regularity.
Theorem (Aczél and Roberts 1989): If in caddition
F =satisfies
=
c
=
::.
cn = 1=n
1
2
reflexivity and symmetry, then = 1 and
, so
F is the geometric mean.
Aside: Solution to (*) without Regularity (Aczél, Roberts, and
Rosenbaum):
F (a1 ; a2 ; :::; an ) = ¸ ¦ n M i (ai )
i= 1
M i > 0, M i (xy) = M i (x)M i (y), ¸ > 0
43
a1 , a2 , ..., an
Second Example:
unit; u is a ratio scale.
are ratio scales, but have the same
New Functional Equation:
(¤¤)F (®a1 ; ®a2 ; :::; ®an ) = A(®)F (a1 ; a2 ; :::; an );
A(®) > 0
Regular Solutions to (**) (Aczél, Roberts, and Rosenbaum):
F (a1 ; a2 ; :::; an ) = ac f (
1
f > 0, arbitrary regular function on
c arbitrary constant.
a2
a
, ..., n )
a1
a1
R n¡
1
(n = 1: f is a positive constant)
44
Solutions to (**) without Regularity (Aczél, Roberts, and
Rosenbaum):
a2
an
F (a1 ; a2 ; :::; an ) = M (a1 )f ( , ...,
)
a1
a1
f > 0, arbit rary funct ion
M > 0, M (xy) = M (x)M (y):
Third Example:
interval scale
a1 , a2 , ..., an
are independent ratio scales, u is an
New Functional Equation:
(***)
F (®1 a1 ; ®2 a2 ; :::; ®n an ) =
A(®1 ; ®2 ; :::; ®n )F (a1 ; a2 ; :::; an )+
B (®1 ; ®2 ; :::; ®n );
A(®1 ; ®2 ; :::; ®n ) > 0
45
Regular solutions to (***) (Aczél, Roberts, and Rosenbaum):
F (a1 ; a2 ; :::; an ) = ¸ ¦
n ac i
i= 1 i
+ b
¸ , b, c1 , c2 , ..., cn arbit rary const ant s
OR
Xn
F (a1 ; a2 ; :::; an ) =
ci loga + b
i
i= 1
b, c1 , c2 , ..., cn arbit rary const ant s.
46
Solutions to (***) without Regularity (Aczél, Roberts, and
Rosenbaum):
F (a1 ; a2 ; :::; an ) = ¸ ¦ n M i (ai ) + b
i= 1
M i > 0, M i (xy) = M i (x)M i (y)
¸ , b arbit rary const ant s
OR
Xn
F (a1 ; a2 ; :::; an ) =
L i (ai ) + b
i= 1
L i (xy) = L i (x) + L i (y)
b arbit rary const ant .
47
Sometimes You Get the Arithmetic Mean
a1 , a2 , ..., an
Fourth Example:
interval scales with the same unit and
independent zero points; u an interval scale.
Functional Equation:
(****)
F (®a1 + ¯1 ; ®a2 + ¯2 ; :::; ®an + ¯n ) =
A(®; ¯1 ; ¯2 :::; ¯n )F (a1 ; a2 ; :::; an )+
B (®; ¯1 ; ¯2 ; :::; ¯n );
A(®; ¯1 ; ¯2 :::; ¯n ) > 0
Solutions to (****) without Regularity Assumed (Aczél, Roberts, and
Xn
Rosenbaum):
F (a1 ; a2 ; :::; an ) =
¸ 1 , ¸ 2 , ..., ¸ n , b
¸ i ai + b
i= 1
arbitrary constants
48
Note that all solutions are regular.
Theorem (Aczél and Roberts):
(1).
P If in addition F satisfies reflexivity, then
n
i= 1
¸ i = 1, b = 0:
(2).
in addition F satisfies reflexivity and symmetry, then
¸ i = If
1=n
for all i, b = 0, i.e., F is the arithmetic mean.
49
Meaningfulness Approach
While it is often reasonable
to assume you know the scale type of the
a1 , a2 , ..., an
independent variables
, it is not so often reasonable to
assume that you know the scale type of the dependent variable u.
However, it turns out that you can replace the assumption that the scale
type of u is xxxxxxx by the assumption that a certain statement
involving u is meaningful.
50
a1 , a2 , ..., an
Back to First Example:
are independent ratio scales.
Instead of assuming u is a ratio scale, assume that the statement
F (a1 ; a2 ; :::; an ) = kF (b1 ; b2 ; :::; bn )
a1 , a2 , ..., an , b1 , b2 , ..., bn
is meaningful for all
get the same results as before:
and k > 0. Then we
Theorem (Roberts and Rosenbaum 1986): Under these hypotheses
and regularity,
F (a1 ; a2 ; :::; an ) = ¸ ac1 ac2 :::acn :
1
2
n
If in addition F satisfies reflexivity and symmetry, then F is the
geometric mean.
51
MEASUREMENT OF AIR POLLUTION
Various pollutants are present in the air:
Carbon monoxide (CO), hydrocarbons (HC), nitrogen oxides (NOX),
sulfur oxides (SOX), particulate matter (PM).
Also damaging: Products of chemical reactions among pollutants. E.g.:
Oxidants such as ozone produced by HC and NOX reacting in
presence of sunlight.
Some pollutants are more serious in presence of others, e.g., SOX are
more harmful in presence of PM.
Can we measure pollution with one overall measure?
52
To compare pollution control policies, need to compare effects of
different pollutants. We might allow increase of some pollutants to
achieve decrease of others.
One single measure could give indication of how bad pollution level
is and might help us determine if we have made progress.
Combining Weight of Pollutants:
Measure total weight of emissions of pollutant i over fixed period of
time and sum over i.
e(i,t,k) = total weight of emissions of pollutant i (per cubic meter)
over tth time period and due to kth source or measured in kth
location.
X
A(t; k) =
e(i ; t; k)
i
53
Early uses of this simple index A in the early 1970s led to the
conclusion that transportation is the largest source of air pollution,
accounting for over 50% of all pollution, with stationary fuel
combustion (especially by electric power plants) second largest. Also:
CO accounts for over half of all emitted air pollution.
Are these meaningful conclusions?
A(t; k) > A(t; k 0)
X
A(t; kr ) >
X
A(t; k)
k6
= kr
X X
e(i ; t; k) >
t ;k
e(j ; t; k)
t ;k j 6
=i
All are meaningful if we measure all e(i,t,k) in same units of mass
(e.g., milligrams per cubic meter) and so admissible transformation
means multiply e(i,t,k) by same constant.
54
These comparisons are meaningful in the technical sense.
But: Are they meaningful comparisons of pollution level in a practical
sense?
A unit of mass of CO is far less harmful than a unit of mass of NOX.
EPA standards based on health effects for 24 hour period allow 7800
units of CO to 330 units of NOX. These are Minimum acute toxicity
effluent tolerance factors (MATE criteria).
Tolerance factor is level at which adverse effects are known. Let t(i)
be tolerance factor for ith pollutant.
Severity factor: t(CO)/t(i) or 1/t(i)
55
One idea (Babcock and Nagda, Walther, Caretto and Sawyer): Weight
the emission levels (in mass) by severity factor and get a weighted
sum. This amounts to using the indices
Degree of hazard:
1 e(i ; t; k)
t (i )
and the combined index
Pindex:
B (t; k) =
P
1 e(i ; t; k)
i t (i )
Under pindex, transportation is still the largest source of pollutants,
but now accounting for less than 50%. Stationary sources fall to
fourth place. CO drops to bottom of list of pollutants, accounting for
just over 2% of the total.
56
These conclusions are again meaningful if all emission weights are
measured in same units. For an admissible transformation multiplies
t and e by the same constant and thus leaves the degree of hazard
unchanged and pindex unchanged.
Pindex was introduced in the San Francisco Bay Area in the 1960s.
But, are comparisons using pindex meaningful in the practical sense?
Pindex amounts to: for a given pollutant, take the percentage of a
given harmful level of emissions that is reached in a given period of
time, and add up these percentages over all pollutants. (Sum can be
greater than 100% as a result.)
57
If 100% of the CO tolerance level is reached, this is known to have
some damaging effects. Pindex implies that the effects are equally
severe if levels of five major pollutants are relatively low, say 20% of
their known harmful levels.
Severity tonnage of pollutant i due to a given source is actual
tonnage times the severity factor 1/t(i).
Data from Walther 1972 suggests the following. Interesting exercise
to decide which of these conclusions are meaningful.
58
1. HC emissions are more severe (have greater severity tonnage) than
NOX emissions.
2. Effects of HC emissions from transportation are more severe than
those of HC emissions from industry. (Same for NOX.).
3. Effects of HC emissions from transportation are more severe than
those of CO emissions from industry.
4. Effects of HC emissions from transportation are more than 20 times
as severe as effects of CO emissions from transportation.
5. The total effect of HC emissions due to all sources is more than 8
times as severe as total effect of NOX emissions due to all sources.
59
Social Networks and Blockmodeling
In studying
M = social
(m i j ) networks,
m i ja common approach is to start withi t ha
matrix
measures the friendship of the
th
jwhere
individual for the . M is used to derive such conclusions as:
* the degree to which friendship is propagated through the group (if k
is a friend of j and j is a friend of i, is k a friend of i?)
* the degree to which friendship is reciprocated (if j is a friend of i,
is i a friend of j?)
* grouping of individuals into friendship cliques or clusters or blocks.
60
Batchelder:
m i j Unless attention is paid to the types of scales used to
measure
, conclusions about propagation, reciprocation, and
grouping can be meaningless.
Simple Example: -Transitivity
The social network is called -transitive ( > 0) if
[m i j ¸ ±]& [m j k ¸ ±] ! m i k ¸ ±, i 6
= k:
1
M =
1
2
3
2 3
 10 8
4  12
1 2 
is 5-transitive.
61
Suppose each individual makes judgments of friendship on a ratio
scale. Even if all the ratio scales are the same, the conclusion of transitivity is meaningless:
1
½M = 1
2
3

2
.5
2
3
5

1
4
6

is not 5-transitive.
62
How we present data can
m i jdetermine the meaningfulness of
conclusions. Suppose
is not the numerical estimate of i's
friendship for j, but the relative ranking by i of j among the other
individuals.
In the above example M, suppose we use only the ranks: n for first
place, n-1 for second place, etc. We get
1
M' = 1
2
3
2

2
2
3
3

3
2
3

This is 2-transitive. Even if the scales are only ordinal, this M' is
unchanged even when M changes. Thus, -transitivity of M' is a
meaningful conclusion. (Batchelder)
63
Blockmodeling
The process of grouping individuals into blocks is called
blockmodeling. A widely used algorithm in blockmodeling is the
CONCOR algorithm of Breiger, Boorman, and Arabie and Arabie,
Boorman, and Levitt. In CONCOR, the matrix M is used to obtain a
similarity measure between individuals i and j, using either column
by column or row by row product moment correlations. The
partition into blocks is based on the similarities.
Batchelder: If the friendship rating scales are independent interval
scales, then the column by column product moment correlation
obtained from M can change. Thus, the blocks can also change.
64
Hence, the blocking conclusions from a procedure such as CONCOR
can be meaningless.
ci j
E.g.: Suppose
is the Pearson column by column
c0j product moment
correlation obtained from matrix M and suppose i is the same
correlation obtained from M', where
m 0j = ®i m i j + ¯i , ®i > 0:
i
Let
0
2 2 1
2
0 6 3
M= 2
4 0 5
1
1 3 0
®1 = ®2 = ®3 = 1 ®4 = 3 ¯1 = ¯2 = ¯3 = ¯4 = 0
c12 = 1
If
,
,
, then
c0 2 = ¡ 1
and
1
. This is as dramatic a change as possible.
65
rij
Note: Batchelder
shows that if
is the product moment correlation
t
h
i
j th
between the
and
rows of M, then under the transformation
m 0j = ®i m i j + ¯i , ®i > 0;
i
rij
is invariant. Thus, conclusions from M based on row by row
correlations are meaningful.
66
Meaningfulness of Statistical Tests
(Marcus-Roberts and Roberts)
For more than 40 years, there has been considerable disagreement on
the limitations that scales of measurement impose on statistical
procedures we may apply. The controversy stems from the
foundational work of Stevens (1946, 1951, 1959, ...), who developed
the classification of scales of measurement we have given and then
provided rules for the use of statistical procedures which provided that
certain statistics were inappropriate at certain levels of measurement.
The application of Stevens' ideas to descriptive statistics has been
widely accepted, but the application to inferential statistics has been
labeled by some a misconception.
67
Descriptive Statistics
P = population whose distribution we would like to describe
We try to capture some of the properties of P by finding some
descriptive statistic for P or by taking a sample S from P and
finding a descriptive statistic for S.
Our previous examples suggest that certain descriptive statistics are
appropriate only for certain measurement situations. This idea was
originally due to Stevens and was popularized by Siegel in his wellknown book Nonparametric Statistics (1956).
68
Our examples suggest the principle that arithmetic means are
“appropriate” statistics for interval scales, medians for ordinal scales.
The other side of the coin: It is argued that it is always appropriate
to calculate means, medians, and other descriptive statistics, no matter
what the scale of measurement.
Frederic Lord: Famous football player example. “The numbers don't
remember where they came from.”
I agree: It is always appropriate to calculate means, medians, ...
But: Is it appropriate to make certain statements using these
descriptive statistics?
69
My position: It is usually appropriate to make a statement using
descriptive statistics if and only if the statement is meaningful. Why?
A statement which is true but meaningless gives information which is
an accident of the scale of measurement used, not information which
describes the population in some fundamental way.
So, it is appropriate to calculate the mean of ordinal data, it is just not
appropriate to say that the mean of one group is higher than the mean
of another group.
70
Inferential Statistics
Over the years, Stevens' ideas have also come to be applied to
inferential statistics -- inferences about an unknown population P.
They have led to such principles as the following:
(1). Classical parametric tests (e.g., t-test, Pearson correlation,
analysis of variance) are inappropriate for ordinal data. They should
be applied only to data which define an interval or ratio scale.
(2). For ordinal scales, non-parametric tests (e.g., Mann-Whitney U,
Kruskal-Wallis, Kendall's tau) can be used.
Not everyone agrees. Thus: Controversy
71
My View:
The validity of a statistical test depends on a statistical model, which
includes information about the distribution of the population and
about the sampling procedure. The validity of the test does not
depend on a measurement model, which is concerned with the
admissible transformations and scale type.
The scale type enters in deciding whether the hypothesis is worth
testing at all -- is it a meaningful hypothesis? The issue is: If we
perform admissible transformations of scale, is the truth or falsity of
the hypothesis unchanged?
72
A Brief Analysis of These Principles
Inferences about an unknown population P.
Measurement model for P defines the scale type, i.e., tells admissible
transformations.
Null hypothesis
H0
about P.
Suppose an admissible transformation
 transforms scale values.
' (P ) = f ' (x) : x 2 P g:
We get a new population
We also get a transformed Hnull
hypothesis
:
0
all scales and scale values in
' (H 0 )
by applying  to
73
Example:
H0 :
mean of P is 0.
Measurement
' (x) = exmodel: P ordinal
Then
is admissible.
' (H 0 ) :
' (0) = e0 = 1:
mean of (P) is
Note that if
Also,
P = f ¡ 2, -1, 0, 1, 2g
, then
' (P ) = f e¡ 2 , e¡ 1 , e0 , e1 , e2 g
H0
and so
is true.
' (H 0 )
is false.
Thus, a null hypothesis can be meaningless.
Summary of my position: The major restriction measurement theory
places on inferential statistics is on what hypotheses are meaningful.
We should be primarily concerned with testing meaningful hypotheses.
74
Can we test meaningless hypotheses? Sure. But I question what
information we get outside of information about the population as
measured.
More details: Testing
H0
about P :
1). Draw a random sample S from P.
2). Calculate a test statistic based on S.
3). Calculate
probability that the test statistic is what was observed
H0
given
is true.
H0
4). Accept or reject on the basis of the test.
Note: Calculation of probability depends on a statistical model,
which includes information about the distribution of P and about the
sampling procedure. But, the validity of the test depends only on the
statistical model, not on the measurement model.
75
Thus, you can apply parametric tests to ordinal data, provided the
statistical model is satisfied, in particular if the data is normally
distributed. (Thomas showed that ordinal data can be normally
distributed; it is a misconception that it cannot be.)
Where does the scale type enter? In determining if the hypothesis is
worth testing at all. i.e., if it is meaningful.
For instance, consider ordinal data and
H 0 : T he mean is 0.
The hypothesis is meaningless. But, if the data meets certain
distributional requirements such as normality, we can apply a
parametric test, such as the t-test, to check if the mean is 0.
76
The result gives us information about the population as we have
measured it, but no “intrinsic” information about the
population since its truth or falsity can depend on the particular choice
of scale.
So, what is wrong with using the t-test on ordinal data? Nothing. The
test is o.k. However, the hypothesis is meaningless.
Some Observations on Tests of Meaningful Hypotheses
H0
meaningful. We can test
it in various ways. We cannot expect
H0
that two different tests of
will lead to the same conclusion in all
cases, even forgetting about change of scale. For example, the t-test
can lead to rejection and the Wilcoxon test to acceptance.
77
But what happens if we apply the same test to data both before and
after admissible transformation? The best situation is if the results of
the test are invariantHunder admissible changes of scale. More
precisely: Suppose H0 is a meaningful hypothesis about
population P. Test 0by drawing a random sample S from P. Let
(S)
be the set of all (x) for x in S. We hope for a test T for
H0
with the following properties:
(a). The statistical model for T is satisfied by P and also by
(P) for all admissible transformations  as defined by the
measurement model for P.
H0
(b) T accepts with
S at level of significance  if and
' (H 0sample
)
only if T accepts
with sample (S) at level of significance
.
78
There are many examples in the statistical literature where conditions
(a) and (b) hold. We give two examples.
Example 1: t-test
Measurement model: interval scale
H0
mean is
¹0
Statistical model: normal distribution
Admissible transformations: (x) = kx+c, k>0
(a). Such  take normal distributions into normal distributions
79
(b) The test statistic is
p
t = [X ¡ ¹ 0 ]=(s= n)
where n = sample size,
X
= sample mean,
s2
= sample variance.
Transform by (x) = kx+c.
New sample mean is
' (H 0 ) :
mean is
kX + c
; new sample variance is
k 2 s2 :
' (¹ 0 ) = k¹ 0 + c:
New test statistic for
' (H 0 ) :
p
[kX + c ¡ (k¹ 0 + c)]=(ks= n) = t:
The test statistic is the same, so (b) follows.
80
Example 2: Sign Test
Measurement model: ordinal scale
H0 :
median is
M0
(meaningful)
Statistical model: Literature of nonparametric statistics is unclear.
Most books say continuity of probability
density
is needed.
P r (x
= M 0 )function
= 0
In fact, it is only necessary to assume
.
(a)
P r (x = M 0 ) = 0 $ P r (' (x) = ' (M 0 )) = 0
( is strictly increasing)
(Note that strictly increasing  also take continuous data into
continuous data, since there are only countably many gaps)
81
M0
(b) Test
statistic: Number of sample points above
or number
M0
below
, whichever is smaller. This test statisticMdoes not change
0
when
is replaced by
' (M 0 ) a strictly increasing  is applied to S and
. Thus, (b) follows.
What if Conditions (a) or (b) Fail?
What if (a) holds, (b) fails?
We might still find test T useful, just as we sometimes find it useful to
apply different tests with possibly different conclusions to the same
data. However, it is a little disturbing that seemingly innocuous
changes in how we measure things could lead to different conclusions
under the “same” test.
82
What if
H0
is meaningful but (a) fails?
This could happen Hif P violates the statistical model for T. We might
0
still
be
able
to
test
using T: Try to find admissible  such that
' (P )
satisfies the
apply T to test
H 0 statistical model
H 0for T and 'then
' (H 0 )
(H 0 )
. Since
is meaningful,
holds iff
, so T applied to
H0
' (P )
gives a test for
.
H0
For instance, if
is meaningful, then even for ordinal data, we can
seek a transformation which normalizes the data and allows us to
apply a parametric test.
83
What
if conditions (a) or (b) fail and we still apply test T to test
H0
?
There is much empirical work on this, especially emphasizing the
situation where (a) and (b) fail because we have an ordinal scale but
we treat it like an interval scale. I don't think measurement theory has
much to say about this situation. But good theoretical work does, and
more is needed.
84
Summary of My Views on the Limitations on Statistical Procedures
Imposed by Scales of Measurement
(1). It is always “appropriate” to calculate means, medians, and other
descriptive statistics. However, the key point is whether or not it is
“appropriate” to make certain statements using these statistics.
(2). If we want to make statements which capture something
fundamental about the population being described, it is appropriate to
make statements using descriptive statistics if and only if the
statements are meaningful.
85
(3). The appropriateness of a statistical test of a hypothesis is just a
matter of whether the population and sampling procedure satisfy the
appropriate statistical model, and is not influenced by the properties of
the measurement scale used.
(4). However, if we want to draw conclusions about a population
which say something basic about the population, rather than
something which is an accident of the particular scale of measurement
used, then we should only test meaningful hypotheses, and
meaningfulness is determined by the properties of the measurement
scale.
86
(5). A meaningful hypothesis can be tested by several different tests,
which may or may not give the same conclusion even on the same
data. It can also be tested by applying the same test to both given
data and data transformed by an admissible transformation of scale,
and in the best situation the two applications of the test will give the
same conclusion. So long as the hypothesis is meaningful, it can also
be tested by transforming the data by an admissible transformation
and applying a test which might not be appropriate for the data in its
initial form.
87
The First Representation Problem: Ordinal Utility Functions
(R ; > ):
Homomorphism
from (A,R) into
f :A ! R
aRb $ f (a) > f (b)
R = warmer than, louder than, preferred to, ...
If preference, then f is an ordinal utility function.
Definition: We say that (A,R) is a strict weak order if it satisfies the
following two conditions:
(1). Asymmetry: aRb
!
~bra
!
(2). Negative Transitivity: ~aRb & ~bRc
~aRc
88
Example:




 
  



  
  
$
aRb
a is to the right of b
Representation Theorem:
Suppose A is a finite set. Then (A,R) is
(R ; > )
homomorphic to
iff (A,R) is a strict weak order.
This says that our example with the 's is the most general strict
weak order on a finite set.
89
Application:
Preferences Among Composers
i ; j ent ry = 1 i® i is preferred t o j
Mo H
Br
Be
W
Ba
Ma S
Mo
0 1
1
0 0
0
1 1
H
0 0
1
0 0
0
1 0
Br
0 0
0
0 0
0
0 0
Be
1 1
1
0 1
0
1 1
W
0 1
1
0 0
0
1 1
Ba
1 1
1
0 1
0
1 1
Ma
0 1
1
0 0
0
0 0
S
0 1
1
0 0
0
1 0
Mo = Mozart, H = Haydn, Br = Brahms, Be = Beethoven, W =
Wagner, Ba = Bach, Ma = Mahler, S = Strauss
Is there an ordinal utility function?
How does one tell if (A,R) is strict weak?
90
Proof of the Representation Theorem for A Finite:
Necessity: straightforward
Sufficiency:
Let f (x) = jf y : xRygj:
If (A,R) is strict weak, then f is a homomorphism.
Suppose aRb. Then ~bRa. Hence,
~aRy implies ~bRy, i.e., bRy
¸
implies aRy. It follows that f(a)
f(b). By asymmetry, ~bRb.
Since aRb, it follows that f(a) > f(b).
Suppose
~aRb. Then ~bRy implies ~aRy, so aRy implies bRy, so
¸
f(b)
f(a), i.e., ~f(a) > f(b).
91
Comment: Given any (A,R), we can define f this way. Then, if
there is a homomorphism at all, f is one.
Application to our Example:
Calculate row sums. Then f(x) = row sum of row x.
If the matrix is rearranged so that the alternatives are listed in
descending order of row sums (with arbitrary ordering in case of ties),
then we can test if f is a homomorphism by checking that there are
1's in row x for all those y with f(y) < f(x). This should give a
block of 1's in each row from some point to the end.
92
Mo
H
Br
Be
W
Ba
Ma
S
Row Sum
Mo
0
1
1
0
0
0
1
1
4
H
0
0
1
0
0
0
1
0
2
Br
0
0
0
0
0
0
0
0
0
Be
1
1
1
0
1
0
1
1
6
W
0
1
1
0
0
0
1
1
4
Ba
1
1
1
0
1
0
1
1
6
Ma
0
1
1
0
0
0
0
0
1
S
0
1
1
0
0
0
1
0
3
Mo = Mozart, H = Haydn, Br = Brahms, Be = Beethoven, W = Wagner, Ba = Bach,
Ma = Mahler, S = Strauss
Be
Rearranged
Matrix
Ba
W
Mo
S
H
Ma
Br
Be
0
0
1
1
1
1
1
1
Ba
0
0
1
1
1
1
1
1
W
0
0
0
0
1
1
1
1
Mo
0
0
0
0
1
1
1
1
S
0
0
0
0
0
1
1
1
H
0
0
0
0
0
0
1
1
Ma
0
0
0
0
0
0
0
1
Br
0
0
0
0
0
0
0
0
93
Are the axioms reasonable?
Prescriptive vs. descriptive.
As descriptive axioms for warmer than or preferred to:
Asymmetry: Seems ok for both.
Negative transitivity: Seems ok for warmer than. However, for
preference, consider the following case. We choose on the basis of
price, unless prices are close, in which case we choose by quality.
Suppose a is higher quality than b, which is higher quality than c,
with a higher in price than b and b higher in price than c. If a
and b are close in price and so are b and c, but a and c are not,
then we would choose a over b and b over c, but c over a.
Thus, ~cRb and ~bRa, but cRa.
94
Strict Simple Orders
Let (A,R) be a strict weak order.
Define E on A by
$
aEb
~aRb & ~bRa
(A,R) strict weak implies (A,E) is an equivalence relation. If R is
interpreted as preference, then E is interpreted as indifference. We
are indifferent between a and b iff we prefer neither to the other.
Let A* be the set of equivalence classes. Define R* on A* by
$
aR*b*
aRb
95
Then, R* is well-defined and is strict weak. (A*,R*) is called the
reduction of (A,R) by the equivalence relation E.
(A*,R*) has no ties. A strict weak order with no ties allowed is
called strict simple. (A*,R*) tells us how to order the equivalence
classes. It can be thought of as the relation “strictly to the right of”
on a set of points on a line:
     
96
Example:
A = f 0; 1; 2; :::, 10g;
$
aRb
a mod 3 > b mod 3.
Then (A*,R*) is given by:
f 2; 5; 8gR ¤ f 1; 4; 7; 10gR ¤ f 0; 3; 6; 9g
Formally, we say that a binary relation (A,R) is a strict simple
order if it satisfies the following conditions:
(1). Asymmetry
!
(2). Transitivity: aRb
aRc
(8a&=
6 bRc
b)
(3). Completeness:
(aRb or bRa)
Theorem: If (A,R) is a strict weak order, then (A*,R*) is a strict
simple order.
97
Theorem: If A is a finite set, then there is a 1-1 homomorphism
(R ; > )
from (A,R) into
iff (A,R) is a strict simple order.
Weak Orders
(R ; ¸ )
is not a strict simple order. It violates asymmetry: We can
have aRb and bRa. Similarly, “weakly to the right of” on a set of
points is not a strict weak order for the same reason.




 
  



  
  
98
This suggests that there should be concepts of weak order and simple
order analogous to the concepts of strict weak order and strict simple
order and bearing the same relation to these concepts as  does to >.
Here are the formal definitions:
(A,R) is called a weak order if it satisfies the following conditions:
(1). Transitivity
(8a; b)
(2). Strong Completeness:
(aRb or bRa)
Think of “weakly preferred to” or “at least as warm as”.
99
(A,R) is called a strict simple order if it satisfies:
(1). Transitivity
(2). Strong Completeness
!
(3). Antisymmetry: aRb & bRa a = b.
Theorem:
If A is finite, then there is a homomorphism from (A,R)
(R ; ¸ )
into
iff (A,R) is a weak order.
Theorem: If
(R ;A¸ )is finite, then there is a 1-1 homomorphism from
(A,R) into
iff (A,R) is a simple order.
100
If (A,R) is a weak order, define a binary relation E on A by:
aEb
$
aRb & bRa
Then (A,E) is an equivalence relation. Again, we can define a binary
relation R* on the set A* of equivalence classes, with
a*R*b*
$
aRb
hen (A*,R*) is well-defined and is a simple order.
101
Generalizations of the Representation Theorem to Infinite Sets of
Alternatives.
Cantor's Theorem(R
(1895):
If A is a countable set, then (A,R) is
;>)
homomorphic to
iff (A,R) is a strict weak order.
This theorem fails for arbitrary infinite sets.
Use the lexicographic ordering of the plane:
A= R£ R
(a; b)R(s; t) $ [a > s or (a = s& b > t)]:
This is strict weak, but there is no homomorphism:
f(a,1) > f(a,0).
 rational number g(a) with f(a,1) > g(a) > f(a,0).
g is 1-1 function from! reals into rationals:
a>b
g(a) > f(a,0) > f(b,1) > g(b)
102
Birkhoff-Milgram Theorem
(A,R) binary relation, B  A.
Say B is order-dense in A if a,b  A-B, aRb and ~bRa imply
that  c  B such that aRcRb.
(R ; > )
Theorem (Birkhoff-Milgram): (A,R)
iff
¤ ) homomorphic to
(A ¤ ; Ris
(A,R) is a strict weak order and
has a countable set which is
order-dense.
103
A = [0; 1] [ [2; 3], R = >
(A ¤ ; R ¤ ) = (A; R)
Q  A is countable, order-dense
A = [0; ¼] [ [2¼; 3¼], R = >
(A ¤ ; R ¤ ) = (A; R)
Q QA\ isAnot
order-dense
[ f ¼; 2¼g
But
is.
A ¤=; R
{0,1},
R=
¤ ) = (A;
(A
R)>
A is countable, order-dense
A = R £ R (a; b)R(s; t) $ a > s
,
(a; b)E (s; t) $ a = s:
Note:
The set of equivalence classes with rational first component is
countable and order-dense.
104
Uniqueness Results
(R ; > )
Uniqueness Theorem: Every homomorphism from (A,R) into
is regular and every homomorphism defines an ordinal scale.
Meaningfulness:
f(a) > f(b) is meaningful.
f(a) = 3f(b) is not meaningful.
f(a) = f(b) > f(c) - f(d) is not meaningful.
We do not get an interval scale of temperature as we might have
wanted. We'll return to this point.
105
What Representations Lead to Ordinal Scales?
(x 1 ; x 2 ; :::; x M )
Let M be a positive integer. An M-tuple
M¹ = f 1; 2; :::; M g from
:
R
corresponds to a strict weakxorder
of
rank
first
>
x
i
,
i
,
...,
i
j
i
1 2
p
all i such that we never have
;
having
ranked
,
xj > xi :
j 6
= i , i , ..., i
1 2
p
rank next all i such that for no
, is
Thus, (3,9,3,2,8) has 2 ranked first, 5 ranked second, 1,3 tied for
third, 4 last.
M¹
TM
R
Suppose (x
 ; is
a:::;
strict
weak order on . Let
on consist of all
x
;
x
)
M
M-tuples 1 2
whose
corresponding
¼=
f (1; 2)g,
T 2 = f (x; y) :ranking
x > yg (strict weak
©
¼
order)
is y)
.: xE.g.:
. If  = ,
T 2 = f (x;
= yg: If
¼
¼
106
R
Theorem (Roberts 1984): Suppose S is an M-ary relation on and
(R ; S)
f is a homomorphism from
which defines a regular
jf (A)j(A,R)
¸ M into
ordinal scale. Suppose
. Then S is of the form
S = (T M ) 0 [ (T M ) 0 [ :::;
¼1
¼2
M¹
(T M ) 0
¼1 , ¼2
¼i
where
,
...,
are
all
the
strict
weak
orders
of
and
is
;:
TM
¼i
or
For example, if M = 2, then
¼1 = f (1; 2)g ¼2 = f (2; 1)g
,
,
¼3 = ;
. So
S = T M [ T M i s ¸ on R
¼1
¼3
S = TM [ TM is 6
= on R :
¼1
¼2
107
TM [ TM
Note that the condition is necessary but not sufficient:
not lead to an ordinal scale.
¼1
¼2
does
It is an open question to
characterize those S so that
(Rcompletely
; S)
homomorphisms into
lead to ordinal scales. It is also an open
question to obtain similar results for homomorphisms into
(R ; S1 ; S2 ; :::; Sp ; ±1 ; ±2 ; :::; ±q ):
A ;related
open question: If [f TisM a homomorphism from (A,R) into
(R
S)
¼
i
and S is of the form
, when is the statement f(a) > f(b)
meaningful? (Partial results: Harvey and Roberts 1989; Roberts and
Rosenbaum 1994.)
108
Second Representation Problem: Extensive Measurement
(A; R; ±)
(R ; > ; + )
Consider the homomorphisms from
into
representation is known as extensive measurement.
Motivation: R is “heavier than”,
“preferred to” and combination.
±
. This
is combination. Also:
Historically, scales were called extensive if they were at least additive
and a scale was not considered acceptable unless it was at least
extensive.
Representation Theorem (Hölder
Every Archimedean ordered
(R ; > ; +1901):
):
group is homomorphic to
109
Axioms for an Archimedean Ordered Group
(A; R; ±)
(A; ±)
(1).
is a group.
(2). (A,R) is a strict simple order.
(3). Monotonicity:
aRb $ (a ± c)R(b± c) $ (c ± a)R(c ± b)
(4). Archimedean:
If e is the group identity and aRe,
then there is a positive integer n
a ± a ± a ± ::
such that naRb. (Here, na is
. n times)
Which of the axioms are necessary?
(1). Associativity, identity, inverse are not.
(2). Strict simple order is not; there could be ties.
(3). Monotonicity is.
(4). The Archimedean axiom as stated is not; it doesn't even make
sense without e.
110
(A; R; ±)
Definition:
is an extensive structure if it satisfies the
following conditions:
(1). Weak Associativity:
[a ± (b± c)R(a ± b) ± c]
~
[(a ± b) ± cRa ± (b± c)]
&
~
(2). (A,R) is a strict weak order.
(3). Monotonicity.
(4). Modified Archimedean Axiom:
(na ± c)R(nb± d):
If aRb, then there is a positive integer n such that
(For the latter axiom, think of n(a-b) > (d-c).)
111
(A; R; ±)
Theorem (Roberts and Luce 1968):
is homomorphic to
(R ; > ; + )
(A; R; ±)
iff
is an extensive structure.
Are the axioms reasonable? Descriptive?
For mass: OK
For preference:
Strict weak could be questioned. (Recall price-quality example.)
Weak associativity could be questioned if combination allows
physical interaction:
(a = flame, b = cloth, c = fire retardant)
112
Monotonicity could be questioned:
a = black coffee,
b = candy bar, c = sugar
a ± cRb± c
bRa and
Modified Archimedean:
ana=±$1,
lifed?as a cripple, d is a long and healthy life. Is
c b = $0, c isnb±
ever R to
Uniqueness Theorem (Roberts
and Luce
1968): Every
(A; R; ±)
(R ; > ; + )
homomorphism from
into
is regular and defines a
ratio scale.
Note: This gives the desired scale type for mass.
Note: f(a) = 3f(b) is meaningful.
113
Question: What representations
A
into  give rise to ratio scales?
(R ; S)
Roberts (1984) showed that (A,R) into
never gives rise to a
ratio scale. Thus,  requires at least two relations or at least one
operation. Can we get a ratio scale without any operations in ?
114
Third Representation Problem:
Difference Measurement
Define  on
R
by
¢ (x; y; u; v) $ x ¡ y > u ¡ v:
LetDD
relation on A. Consider the representation
(A;
) ! be(Ra; quaternary
¢ ):
A homomorphism f satisfies
D (a; b; c; d) $ f (a) ¡ f (b) > f (c) ¡ f (d):
This representation is called algebraic difference measurement.
Motivation: Temperature measurement. D is comparison of
differences in warmth. Related motivation: Utility with comparison
of differences in value.
115
Representation Theorems: Sufficient conditions were given by
Debreu (1958), Scott and Suppes (1958), Suppes and Zinnes (1963),
Kristof (1967), Krantz, Luce, Suppes, and Tversky (1971), ...
Necessary and sufficient conditions when A is finite were given by
Scott (1964).
A certain solvability condition plays an important role in the
representation theorem of Krantz, Luce, Suppes, and Tversky and is
basic for the corresponding uniqueness theorem.
Definition: (A,D) satisfies solvability if whenever ~D(s,t,a,b) and
~D(x,x,s,t), there are u,v  A such that ~D(a,u,s,t), ~D(s,t,a,u),
~D(v,b,s,t) and ~D(s,t,v,b).
116
What this says: f (a) ¡ f (b) ¸ f (s) ¡ f (t) ¸ 0
9u; v 2 A
implies
such that
f (a) ¡ f (u) = f (s) ¡ f (t) and
f (v) ¡ f (b) = f (s) ¡ f (t):
Solvability is not a necessary condition for the representation.
Uniqueness Theorem (Suppes and Zinnes 1963, Krantz, Luce,
(R ; ¢ Suppes,
)
Tversky 1971). Every homomorphism from (A,D) into
is
regular. Moreover, if (A,D) satisfies solvability, then every
homomorphism defines an interval scale.
Note: This gives temperature as an interval scale.
117
Example to Show Hypothesis of Solvability is Needed:
A = f 0; 1; 3g
D (x; y; u; v) $ x ¡ y > u ¡ v
f(x) = x, all x; g(0) = 1, g(1) = 2, g(3) = 8
(R ; ¢ )
Then f and g are homomorphisms from (A,D) into
.
g = ' ±f
If
, then (0) = 1, (1) = 2, (3) = 8 and we cannot
find  >0,  so that (x) = x+. Thus, f is not an interval
scale.
Comment: Solvability fails.
Open Question: What representations
scales?
A
into  lead to interval
118
Fourth Representation Problem:
Conjoint Measurement
Very important in the history of measurement theory was the
introduction of conjoint measurement as an example of a fundamental
measurement structure different from extensive measurement.
Conjoint measurement is concerned with multidimensional
A = A 1 £ A 2 £ ::. £ A n :
alternatives. The set A has a product structure
(a1 ; a2 ; :::; an )
In economics, we can think
of an alternative
as a market
A1
basket. In perception,
could be intensities of sounds presented to
A2
the left ear and
could be intensities of sounds presented to the right
ear.
119
A1
A2
In testing, set could be a set of subjects and set a set of
test items. In studying response strength, we could take a combination
of drive and incentive. And so on.
We have a binary relation on A interpreted as “preferred
f i : A ito”,
! R“sounds
louder”, “performs better”, etc. We seek functions
so that
(a1 ; a2 ; :::; an )R(b1 ; b2 ; :::; bn ) $
f 1 (a1 ) + f 2 (a2 ) + ::: + f n (an ) >
f 1 (b1 ) + f 2 (b2 ) + ::: + f n (bn ):
Sufficient conditions for the representation were given by Debreu
[1960] and Luce & Tukey [1964];
necessary and sufficient conditions
Ai
by Scott [1964] in case each is finite.
120
Semiorders
Suppose R is preference.
Recall that indifference corresponds to the relation:
$
aIb
~aRb & ~bRa
Suppose f is an ordinal utility function:
$
aRb
f(a) > f(b)
Then
aIb
$
f(a) = f(b)
This implies that indifference is transitive.
121
Arguments against transitivity of indifference (or more generally of
“tying” in a measurement context) go back to Armstrong in the 1930's,
Wiener in the 1920's, even Poincaré (in the 19th century).
One argument against transitivity of indifference: The Coffee-Sugar
example (Luce 1956).
Let 0* = cup of coffee with no sugar, 5* = cup of coffee with 5 spoons
of sugar.

0*

5*
We have a preference between 0* and 5* -- we are not indifferent.
122
Arguments against transitivity of indifference (or more generally of
“tying” in a measurement context) go back to Armstrong in the 1930's,
Wiener in the 1920's, even Poincaré (in the 19th century).
One argument against transitivity of indifference: The Coffee-Sugar
example (Luce 1956).
Let 0* = cup of coffee with no sugar, 5* = cup of coffee with 5 spoons
of sugar.

0*

5*
We have a preference between 0* and 5* -- we are not indifferent.
Add sugar one grain at a time. We are indifferent between the first and
the last.
123
Second argument against transitivity of indifference: Pony-bicycle
Example (Armstrong 1939):
pony I bicycle
pony I bicycle+bell
not bicycle I bicycle+bell
The semiorder model we shall discuss is intended to account for the
first kind of problem, but not the second.
124
The Idea: Find a function f so that
aRb
$
f(a) is “sufficiently larger” than f(b)
Let  > 0 be a threshold or just noticeable difference. Find a
function f on A so that
aRb
Define >  on
R
$
f(a) > f(b) + 
by
x > ± y $ x > y + ±:
We are seeking a homomorphism from (A,R) into
(R ; > ± ):
125
Note that if such a homomorphism f exists for some  > 0, then
another f ' exists for all  ' > 0: Take f ' = ( '/  )f.
Definition (Luce 1956): (A,R) is a semiorder if it satisfies the
following conditions:
(1). Irreflexivity: !
(2). aRb & cRd !
(3). aRb & bRc
~aRa
aRd or cRb
aRd or dRc.
Representation Theorem (Scott and Suppes 1958). Suppose (A,R) is
a binary relation(R
on; >a finite
set A. Then there is a homomorphism
)
±
from (A,R) to
iff (A,R) is a semiorder.
126
Necessity of the conditions:
(1). trivial
( 
)
(3).
(
)



f(c)
f(b)
f(a)



f(d)
f(c)
f(a)



f(b)
f(a)
f(c)
(2). If f(a)  f(c):
(
)
If f(a)  f(c):
( 
)
127
Idea of the Proof
A = {1,2,3,4,5}
R = {(1,3), (1,4), (1,5), (2,4), (2,5), (3,5)}
Is this a semiorder? Not easy to check by the axioms.
However: It is easy to construct f.
First, find the order of embedding.
Finding the Order:
1R3, not 2R3  f(1) > f(2).
1R3, not 1R2  f(2) > f(3).
3R5, not 4R5  f(3) > f(4).
3R5, not 3R4  f(4) > f(5).

f(5)
0

f(4)
.5

f(3)
1.2

f(2)
1.7

f(1)
2.3
with  = 1
128
Uniqueness Problem for Semiorders
There can be irregular
R = > 1 homomorphisms from (A,R) to
A = {0,1/2,10},
.
(R ; > 1 ) :
Two homomorphisms from (A,R) into
f(0) = 0, f(1/2) = 0, f(10) = 10
g(x) = x, all x.
There
g =transformation
' ±f :
' : f (A)is! noRadmissible
so that
(R ; > ± )
.
So, the “usual” theory of scale type does not apply.
There are various ways around this.
129
The Method of Perfect Substitutes
Define an equivalence relation E on A by:
aE b $ (8c)(aRc $ bRc & cRa $ cRb):
In this case, a and b are perfectRsubstitutes
for each other with
¤
respect Ato¤ the relation R. Define
on the collection of equivalence
classes by:
a¤ R ¤ b¤ $ aRb:
R¤
If (A,R) is a semiorder, is well-defined and is again a semiorder.
Moreover, every homomorphism is now 1-1 and regular.
130
A similar trick of canceling out the perfect substitutes relation E
always leads to regular representations for equivalence classes.
We
is the scale type of a homomorphism from
(A ¤ ;can
R ¤ ) now ask:
(R ; >What
)?
±
into
None of the common scale types applies and
no succinct characterization of admissible transformations is known.
A Uniqueness Result
Suppose
Define W on A by:
aRb $ f (a) > f (b) + ±:
aW b $ f (a) > f (b).
Then (A,W) is a strict weak order.
131
Theorem (Roberts 1971): The order (A,W) is essentially unique. The
only changes allowed are to permute elements a and b which are
equivalent under the perfect substitutes relation.
Meaningfulness
One
still ask, given a homomorphism f from (A,R) into
(R ; >can
)
±
, if the conclusion f(a) > f(b) is meaningful in the broader
sense of invariance under all possible homomorphisms. It isn't always.
In the example above, we have g(1/2) > g(0), but not f(1/2) > f(0).
However, it is easy to see that if f is a homomorphism, f(a) > f(b) is
a meaningful conclusion for all a, b if for all x  y, ~xEy.
132
Indifference Graphs
Consider the indifference representation corresponding to the
semiorder representation:
aI b $ jf (a) ¡ f (b)j · ±:
Define
J±
on
R
by:
xJ ± y $ jx ¡ yj · ±:
We seek a homomorphism from (A,I) into
(R ; J ±) :
Graph-theoretic formulation: G = (V,E), V = A, edge a to b iff
aIb. Assign numbers to vertices of G so that two vertices are
adjacent iff their corresponding numbers are close (within ).
133
Here is an example of such a representation on a graph:
1.1
=1
0
.7
1.6
2.5
Definition: We say that
is an indifference graph if there is a
(R ; J(A,I)
):
±
homomorphism into
134
Examples of Graphs that are Not Indifference Graphs
C4
Similarly, C5, C6, ...
135
Representation Theorem (Roberts 1969). G = (A,I) is an indifference
graph iff it has none of the above graphs as an induced subgraph.
Uniqueness:
Irregularity can again be a problem. Now, the perfect substitutes
relation is equivalent to:
aE b $ (8c)(aI c $ bI c)
(Assuming that aIa for all a, we have aEb iff a and b have the
same closed neighborhoods.)
Define
(A ¤ ; I ¤ )
I¤
a¤ I ¤ b¤ $ aI b
A¤
on(R ; Jby
. Then all homomorphisms from
)
±
into
are regular, but the class of admissible
transformations is not known.
136
aW b $ f (a) > f (b)
Again,
semiorders.
is essentially unique, as in the case of
Meaningfulness
f(a) > f(b) is never meaningful for indifference graphs since -f is
again a homomorphism. However, some ordinal conclusions are
meaningful.
Say y is between x and z if x<y<z or z<y<x. Betweenness is a
significant observation in psychophysical and perceptual experiments
and in judgments of worth or value. When is the conclusion that f(b)
is between f(a) and f(c) meaningful?
137
Theorem (Roberts 1984). Suppose
(R ;AJ ±is
) finite and f is a
homomorphism from (A,I) into
. Then the conclusion that
f(b) is between f(a) and f(c) is meaningful for all a,b,c in A iff
(A,I) is connected as a graph and for all x y in A, ~xEy.
It is an open question to generalize this result beyond indifference
graphs and determine for what representations the conclusion that f(b)
is between f(a) and f(c) is meaningful for all a,b,c.
138
Interval Orders and Measurement without Numbers
Let f satisfy the semiorder representation:
aRb $ f (a) > f (b) + ±:
Let
J (a) = [f (a) ¡ ±=2, f (a) + ±=2]:
Think
of J(a) as a range of possible values (utilities) for a. Let
Â
J J' mean that x > y for all x  J, y  J'. We have
aRb $ J (a) Â J (b):
(*)
The intervals J(a) all have the same length. What if we allow
different lengths? When does there exist an assignment of an interval
J(a) to each a  A satisfying (*)?
139
This is a representation problem in the spirit we have been discussing.
It is measurement without numbers.
Definition: (A,R) is an interval order if it satifies the first two
conditions in the definition of a semiorder, i.e.,
(1). Irreflexivity
(2). aRb & cRd  aRd or cRb.
Representation Theorem (Fishburn 1970): Suppose (A,R) is a binary
relation and A is a finite set. Then there is an assignment J of real
intervals satisfying (*) iff (A,R) is an interval order.
140
Uniqueness Results
No one has formulated a theory of admissible transformations of
interval assignments. However, there are some uniqueness results.
There are four types of relations among intersecting intervals J(a)
and J(b) that we would like to distinguish:
(1) J(a) ____________
J(b) _______
(2) J(b) ______________
J(a) _____
(3) J(a) ___________
(4) J(b) ___________
J(b) ____________
J(a) _________
(1)
(2)
(3)
(4)
J(a)
J(b)
J(a)
J(b)
“overreaches” J(b)
“overreaches” J(a)
“weakly precedes” J(b)
“weakly precedes” J(a).
141
One interesting question is to try to determine when the conclusion
that J(a) overreaches J(b) is meaningful. These questions arise in
economics when the intervals are ranges of possible values of a
commodity. They also arise in archaeological seriation when these
are intervals of time during which a particular type of artifact was in
use. In the latter context, Skrien (1980,1984) developed an O(n3)
algorithm for determining if these conclusions are meaningful.
142
Interval Graphs
Consider the indifference version of the representation (*) for
preference:
aI b $ J (a) \ J (b) 6
= ;:
The graph-theoretic formulation of this is to assign a real interval to
each vertex of a graph so that two vertices are adjacent iff their
intervals overlap. If we can find such an assignment, we say that the
graph is an interval graph.
Early representation theorems for interval graphs were due to
Lekkerkerker and Boland (1962), Gilmore and Hoffman (1964), and
Fulkerson and Gross (1965).
143
Interval graphs have numerous applications, in archaeology,
developmental psychology, molecular biology, scheduling, traffic
phasing, ecology, computer science, etc.
Uniqueness Questions
Uniqueness questions similar to those about interval orders arise. For
example,
when are conclusions about overreaching and weak
Â
precedence meaningful? WeÂhave an additional question here: How
unique are conclusions J(a) J(b)? These are never meaningful,
Â
since we can always “reverse” all intervals. However, induces a
transitive orientation on the complement of the graph in question, and
the uniqueness of transitive orientations has been widely studied.
144
Higher Dimensional Intersection Graphs
Consider again
aI b $ J (a) \ J (b) 6
= ;:
What if the J(a)'s are other objects, such as boxes, cubes, spheres,
arcs on a circle, etc.? One can ask representation and uniqueness
questions about these situations. For the most part, these questions are
unsolved. For instance, we still don't know a representation theorem if
the J(a) are rectangles in the plane (with sides parallel to the
coordinate axes), or squares.
145
Partial Orders
Every semiorder and every interval order is asymmetric and transitive.
In general, a binary relation (A,R) that satisfies these two conditions is
called a strict partial order.
Proper containment on a set of sets is an example of a strict partial
order. Strict partial orders often arise when we state preferences among
alternatives
that have several dimensions or aspects. i tWe
might let
f i (a)
h
measure the value of alternative a on the
dimension,
i=
1,2,...,n. We might prefer a to b iff a rates higher on every
dimension, i.e.,
($$)
aRb $
[f 1 (a) > f 1 (b)]& [f 2 (a) > f 2 (b)]& ... & [f n (a) > f n (b)]:
146
If R is defined this way for real-valued functions fi, then
(A,R) defines a strict partial order.
The converse problem is important in utility theory and elsewhere.
Suppose we are given a strict partial order (A,R). For Rgiven n, can
we find functions f1, f2, ..., fn, each mapping A into
, so that for
all a,b, ($$) holds? This is a representation problem in the spirit of
measurement theory.
It is not hard to show that for all strict partial orders on finite sets A,
we can find such functions for n sufficiently large. The smallest
such n is called the dimension of the strict partial order.
147
There is a second notion of dimension that is used in the literature
(more commonly than this one). Szpilrajn [1930] showed that for
every strict partial order (A,R) and for every a,b in A so that ~aRb
and ~bRa, there is a strict simple order (A,S) so that R  S
and so that aSb. It follows that R can be written as the intersection
of strict simple orders on A. The smallest number of such strict
simple orders is called the dimension of the strict partial order.
For every d > 2, the two notions of dimension agree. However, strict
weak orders have dimension one by the first definition and two by the
second.
148
Example:
A = f a; b; c; dg, R = f (a; b), (b; c), (a; c), (a; d)g
This is a strict partial order. It is the intersection of two strict simple
orders: the order ranking a over b over c over d and the order
ranking a over d over b over c.
We can also find two functions f1 and f2 satisfying ($$):
f1(a) = 4, f1(b) = 3, f1(c) = 2, f1(d) = 1,
f2(a) = 4, f2(d) = 3, f2(b) = 2, f2(c) = 1.
149
Partial Orders vs. Strict Partial Orders
We shall use the term partial order for the binary relations related to
strict partial orders in the same way that weak orders are related to
strict weak orders. Thus, proper containment on a set of sets defines a
strict partial order, while containment defines a partial order.
Formally, we say that (A,R) is a partial order if it is reflexive,
transitive, and antisymmetric.
150
Subjective Probability
When we say that one event seems more probable than another, we
are making subjective comparisons of probabilities. One can ask when
such comparisons are consistent with a probability measure.
(X ; E)
Formalization of this question: We
say that
is an E
E
algebra of events if elements of
are subsets of X and
is closed
under union and complementation. We say that a function p on an
algebra of events is a finitely
E additive probability measure if p is a
real-valued function from so that
(1). p(A)  0
(2). p(X)
p(A [ =B1) = p(A) + p(B ) if A \ B = ; :
(3).
151
Â
(X ; E)
Suppose
 is an algebra of events and is a binary relation on
E
, with A B interpreted to mean that A is (subjectively) judged
more probable than
B. Does there exist a finitely
additive probability
(X ; E)
Â
measure
which “preserves” , i.e., so that for all
B 2 E p on
A,
,
(%)
A Â B $ p(A) > p(B ):
Such a measure p is called a subjective probability measure. We can
ask for representation and uniqueness theorems in the spirit of
representational measurement theory.
152
Â
The study of the comparative probability relation goes back at least
to Bernstein (1917). The following necessary conditions for the
representation (%) go back to Bruno de Finetti (1937).
de Finetti Axioms
(E; Â )
(1).
 X Â;
(2).
is a strict weak order.
and
Aº ;
(3). Monotonicity: If
, where A  B means that ~B
A\ B = A\ C= ;
Â
A.
, then
B Â C $ A [ B Â A [ C:
153
The axioms are clearly necessary.
That they are not sufficient was
E
shown by Savage in 1954 for infinite and by Kraft, Pratt, and
E
Seidenberg in 1959 for finite.
Representation Theorems
Scott (1964) gave necessary and sufficient conditions in the case
E
where is finite and Suppes and Zanotti (1976) gave necessary and
E
sufficient conditions for arbitrary .
Perhaps historically one of the most interesting non-necessary axioms
is the following:
154
(X ; E)
We say that
has an n-fold almost uniformEpartition if there
is a collection of disjoint subsets X1, X2, ..., Xn in whose union is
X and such that whenever 1  k < n, the union of no k of the Xi is
more probable than the union of any k+1 of them.
We say that
A» B $
~
[A Â B ]
&~
[B Â A]
(X ; E)
An n-fold uniform partition
of
is a collection of disjoint
E
Xi » Xj
subsets X1, X2, ..., Xn in whose union is X and such that
for all i, j.
Every n-fold uniform partition is almost uniform.
155
(X ; E)
Representation Theorem
(Roberts 1979). Suppose
is an
Â
E
algebra of sets and is a binary relation on
. The following
axioms are sufficient to prove the existence of a measure of subjective
probability:
(1). (E
The
¤ ; Âde
¤ ) Finetti axioms.
(2).
has a countable, order-dense subset.
E
(3). For all n, there is an n-fold almost uniform partition of
.
E
These sufficient conditions imply that
is infinite. Luce (1967)
E
gave important sufficient conditions that do not require that
be
infinite.
156
Uniqueness Results
(X ; E)
Â
Theorem (Roberts 1979). Suppose
is an algebra of sets and
E
is
binary relation on . Suppose that for all positive integers n,
(X a; E)
has an n-fold almost uniform partition.
Then if p and p' are
(X ; E; Â )
two measures of subjective probability on
, p = p'. In other
words, p is an absolute scale.
Intuition: Finite Additivity
plays
to additivity in
((A;
R; ±) a! role
(R ;analogous
> ; + ))
extensive measurement
. The uniqueness
theorem for extensive measurement can be modified to show that
p' = p,  > 0. But p(X) = p'(X) = 1, so  = 1.
157
Unfortunately, the theorem fails without an added hypothesis like
existence of n-fold almost uniform partitions for all n.
E = 2X
Example: X = {a,b},
,
f a; bg  f ag  f bg  ; :
;
Then p({a,b}) = 1, p( ) = 0, p({a}) = , p({b}) =  is a measure of
subjective probability so long as  > ,  = 1- .
Other conditions sufficient for getting an absolute scale were given for
X finite by Fishburn and Odlyzko [1989] and Van Lier [1989];
necessary and sufficient conditions by Fishburn and Roberts [1989].
Other conditions for absolute scales, based on the additional concept
of probabilistic independence, were given by Suppes and Alechina
[1994].
158
Probabilistic Consistency
If judgments are made repeatedly, individuals are often inconsistent.
For example, a subject might prefer a to b once, b to a later; or
say that a is warmer than b once, b is warmer than a later. Let
pab represent the proportion of times that a is preferred to b, or a is
judged warmer than b, etc.
Then p is a function from A into [0,1]. Call (A,p) a forced choice
pair comparison system (fcpcs) if for all a,b in A, pab + pba = 1.
We try to capture the idea that the individual, while being inconsistent,
is being consistent in a probabilistic sense.
159
Definition:
A fcpcs satisfies the weak utility model if there is
f :A ! R
so that for all a,b in A,
pab > pba $ f (a) > f (b):
Representation and Uniqueness Theorems: Define W on A by
aW b $ pab > pba :
Then (A,p) satisfies
(R ; >the
) weak utility model iff (A,W) is
homomorphic to
. Moreover, if f is a function satisfying the
weak utility model, then f defines a (regular) ordinal scale.
160
Definition:
A fcpcs satisfies the strong utility model if there is
f :A ! R
so that for all a,b,c,d in A,
pab > pcd $ f (a) ¡ f (b) > f (c) ¡ f (d):
Representation and Uniqueness Theorems: Define D on A by
D (a; b; c; d) $ pab > pcd :
Then (A,p) satisfies
(R ; ¢the
) strong utility model iff (A,D) is
homomorphic to
. Moreover, if (A,D) satisfies the solvability
axiom, then f is a (regular) interval scale.
The open uniqueness
questions for difference measurement, the
(A; D ) ! (R ; ¢ )
representation
, are of equal interest for the strong
utility model.
161
Stochastic Transitivity Conditions
We now state some conditions on the pab values that are related to the
weak and strong utility models. They are probabilistic versions of
transitivity.
Weak Stochastic[pTransitivity
(WST):
¸
1=2]&
[p
¸ 1=2] ! pac ¸ 1=2:
ab
bc
Moderate Stochastic Transitivity (MST):
[p ¸ 1=2]& [p ¸ 1=2] ! pac ¸ minf pab; pbc g
ab
bc
Strong Stochastic Transitivity (SST):
[pab ¸ 1=2]& [pbc ¸ 1=2] !
pac ¸ maxf pab; pbc g
162
Theorem: WST is equivalent to the weak utility model.
Theorem: The Strong Utility Model implies SST, but they are not
equivalent.
The pony-bicycle example mentioned in connection with semiorders
gives an argument against SST and hence against the strong utility
model:
Suppose
We have
But
pbi cy cl e;pon y ¼ 1=2
.
pbi cy cl e+ bel l ;bi cy cl e = 1
pbi cy cl e+ bel l ;pon y ¼ 1=2
.
.
163
Homogeneous Families of Semiorders
In experiments with loudness judgments, we often take a to be
louder than b (or beyond threshold) if it is judged to be louder a
sufficiently high percentage of the time, like 3/4. Given a number
  [1/2,1), we can define a binary relation R on A by:
aR ¸ b $ pab > ¸ :
Luce [1958,1959] showed that under certain conditions, each (A,R)
is a semiorder.
For instance, SST implies this conclusion. Thus,
F = f (A; R ¸ ) : ¸ 2 [1=2; 1)g
under SST,
is a family of semiorders.
164
We have noted that for a semiorder (A,R), the order in which the(R ; > )
±
elements are mapped into the reals under a homomorphism into
is essentially unique. This order is sometimes called the compatible
weak order.
We shall call a family of semiorders on the same underlying set
homogeneous if there is one weak order compatible with all of the
F
semiorders in the family. It turns out that under SST,
is a
homogeneous family of semiorders. This is another condition of
probabilistic consistency.
The converse is also true under one additional assumption.
165
Theorem (Roberts 1971): Suppose (A,p) Fis a fcpcs and pab  1/2
for all a,b. Then (A,p) satisfies SST iff
is a homogeneous
family of semiorders.
Actually, the result holds under the weaker hypothesis:
pab = 1=2 ! (8c)(pac = pbc ):
166
Meaningfulness of Conclusions from Combinatorial Optimization
(Roberts 1990)
What conclusions about optimal solutions or algorithms are meaningful?
Shortest Path Problem
1
a
x
2
y
20
What is the shortest path from x to y?
167
Meaningfulness of Conclusions from Combinatorial Optimization
(Roberts 1990)
What conclusions about optimal solutions or algorithms are meaningful?
Shortest Path Problem
1
a
x
2
y
20
What is the shortest path from x to y?
Solution: x, a, y
168
But what if weights are measured on an interval scale?
An admissible transformation takes the form x  x +  , 
> 0. Take  = 2,  = 100.
102
a
x
104
y
140
169
But what if weights are measured on an interval scale?
An admissible transformation takes the form x  x +  , 
> 0. Take  = 2,  = 100.
102
a
x
104
y
140
Now the shortest path is x, y.
The original conclusion was meaningless.
It is meaningful if weights are measured on a ratio scale.
170
Linear Programming
The shortest path problem can be formulated as a linear
programming problem. The above discussion illustrates the
following point: The conclusion that z is an optimal solution in a
LPP can be meaningless if the cost parameters are measured on an
interval scale.
171
Minimum Spanning Tree
Given a connected weighted graph, what is the spanning tree
with a total sum of weights as small as possible?
8
a
b
70
54
63
22
c
53
e
12
10
d
172
a
b
8
70
54
63
22
c
53
e
12
10
d
optimal solution
173
What if we change weights, say by an admissible transformation for
interval scales?
Say we take x  x +  with  = 2,  = 100. We get the
following weighted graph.
174
What if we change weights, say by an admissible transformation for
interval scales?
Say we take x  x +  with  = 2,  = 100. We get the
following weighted graph.
116
a
b
240
226
208
144
c
206
e
124
120
d
175
a
240
226
208
206
e
144
c
b
116
124
120
d
optimal solution
176
The optimal solution is the same. Is this an accident?
177
The optimal solution is the same. Is this an accident?
No. It is a corollary of the result that Kruskal's algorithm (the greedy
algorithm) gives a minimum spanning tree.
Kruskal's Algorithm
1. Order edges in increasing order of weight.
2. Examine edges in this order. Include an edge if it does not form a
cycle with edges previously included.
3. Stop when all vertices are included.
Corollary: The conclusion that a given spanning tree is minimum is
meaningful even for ordinal scales.
178
Note that analysis of meaningfulness is something like sensitivity
analysis. However, the perturbations studied do not have to
be small. We are investigating the sensitivity (or invariance) of
conclusions under certain kinds of changes.
179
Frequency Assignment and T-Coloring
The following problem arises from questions of channel assignment.
Let G = (V,E) be a graph and T be a set of nonnegative integers.
Find an assignment f of a positive integer f(x) to each vertex x in G
so that
f x; yg 2 E ! jf (x) ¡ f (y)j 2= T:
Such an f is called a T-coloring of G.
In the application:
V = set of transmitters;
E means conflict;
f(x) is channel assigned to transmitter x;
T is a set of disallowed separations between channels assigned to
conflicting transmitters. E.g.., T = {0}, T = {0,1}.
180
T = {0,1,4,5}
Find a “greedy” solution by always choosing the lowest
possible channel.
181
T = {0,1,4,5}
1
Find a “greedy” solution by always choosing the lowest
possible channel.
182
T = {0,1,4,5}
1
3
Find a “greedy” solution by always choosing the lowest
possible channel.
183
1
T = {0,1,4,5}
9
3
Find a “greedy” solution by always choosing the lowest
possible channel.
184
Here is a “better” solution:
1
7
4
185
In what sense is this better? In the sense that
S(f ) = max jf (x) ¡ f (y)j
x ;y 2 V
is smaller.
S(f) is called the span of f.
The minimum S(f) over all T-colorings f of G is called the
span of G.
(Note that it is NP-complete to compute the optimal span.)
Widely studied question: For what graphs does the greedy algorithm
obtain the optimal span?
186
What if numbers in T change by a scale transformation? Is the
answer to this widely studied question meaningful?
Consider the case of ratio scales. Then we can have transformations
t t,  > 0.
In this case, even for complete graphs Kn, the conclusion that the
greedy algorithm obtains the optimal span is meaningless.
Take the above example with  = 2. Then
T = {0,2,8,10}
The greedy algorithm does obtain the optimal span:
187
T = {0,2,8,10}
1
188
T = {0,2,8,10}
1
2
189
T = {0,2,8,10}
1
5
2
190
Aside
It is not as easy to give an example of a complete graph Kn and set T
and number  such that the greedy algorithm obtains the optimal
span for Kn with T but not with T.
One example is K6 with
T = f 0; 1; 2; 3; 4g [ f 8g [ f 10; 11; 12g [ f 31; 32g
and  = 2.
The greedy T-coloring uses the channels 1, 6, 15, 20, 29, 34.
191
Proof of optimality:
Consider any T-coloring g with g(1) < g(2) < g(3) < g(4) < g(5) <
g(6).
Without loss of generality, take g(1) = 1.
f 0; 1; 2; 3; 4g ½ T ! g(j + 1) ¸ g(j ) + 5
Then
g(j + 2) ¸ g(j ) + 10
,
.
f 10; 11; 12g ½ T ! g(j + 2) ¸ g(j ) + 13
But
.
g(3) ¸ 14, g(5) ¸ 27, g(6) ¸ 32
Thus,
.
But 32-1 = 31, 33-1 = 32, so g(6)  34.
QED
192
With  = 2,
®T = f 0; 2; 4; 6; 8g [ f 16g [ f 20; 22; 24g [ f 62; 64g:
The greedy T-coloring uses the channels 1, 2, 11, 12, 29, 30.
A better T-coloring uses the channels 1, 2, 13, 14, 27, 28.
193
What is meaningful?
Theorem (Cozzens and Roberts 1991): The conclusion that the greedy
algorithm obtains the optimal span for all complete graphs is
meaningful if elements of T are measured on a ratio scale.
For example, if T = {0,1,2, ... , r}, this conclusion is true for all
complete graphs. Even true if T = {0,1,2, ..., r}  S where S
contains no multiple of r+1.
194
Comment: The theory of meaningfulness needs to be applied with
care.
In the T-coloring application, limiting the set T to integers could be
problematical in the interpretation.
Suppose for example T = {0,1} and frequencies are measured in
kilohertz (kHz). Then frequencies at distance 1 kHz are not allowed
on conflicting transmitters. Neither, of course, are frequencies at
distance 1/2 kHz.
However, if we take  = 1000, then the new units are Hertz (Hz)
and now frequencies at distance 500 Hz = 1/2 kHz are allowed on
conflicting channels. This changes the problem.
195
Scheduling with Priorities and Lateness/Earliness Penalties
(Roberts 1995, Mahadev, Pekec, and Roberts 1997, 1998)
The following problem came from the Military Airlift Command.
n items (equipment, people) are to be transported from an origin to a
destination.
Item i has a desired arrival time di. (For simplicity, di is
a positive integer.) Transportation time is constant.
For a given arrival time, there is available a certain capacity c for
transportation (# of seats, cargo space, etc.) Note constant c.
196
au
i
A schedule u assigns a positive integer arrival time
to each
au
item, subject to the constraint that the number of i that can equal a
given integer is at most c.
Penalties are applied -- for late arrivals and perhaps for early
arrivals. Penalties depend upon desired and scheduled arrival times
and upon priorities. Different items have different status or priority.
Let ti be a measure of the priority of item i.
197
Some Penalty Functions
Pe(u;t,d,c) is the penalty with schedule u if t = (t1,t2,...,tn), d =
(d1,d2,...,dn) and c is the capacity.
P1 (u; t; d; c) =
P
n
t jau
i= 1 i i
¡ di j
.
This penalty is summable in Xthe
sense that
n
Pe =
ge (t i ; au ; di ):
i
i= 1
It is also separable in the sense that
³
ge (t; a; d) =
h e ( t ) f e ( a;d)
1
h e ( t ) f e ( a;d)
if
if
2
Here,
f 1 (a; d) = ja ¡ dj
,
h1 (t) = h1 (t) = t
1
2
a¸ d
a< d
.
198
Other
examples of summable, separable penalty functions have
he (t) = he (t) = t
1
2
and fe(a,d) defined as follows:
P2 (u; t; d; c) : f 2 (a; d) = ja ¡ dj
P3 (u; t; d; c) : f 3 (a; d) = 1
if a > d and 0 otherwise.
if a  d and 0 otherwise.
P4 (u; t; d; c) : f 4 (a; d) = ° ja ¡ dj
if a  d and |a - d| otherwise
P5 (u; t; d; c) : f 5 (a; d) = ja ¡ dj 2
A summable example is P6(u;t,d,c): g6(t,a,d) = 1 if a  d + F(t) and
0 otherwise.
199
Problem: Find an optimal schedule, i.e., one which minimizes the
penalty.
Analogous Problem in Manufacturing
n tasks to perform;
task i has desired completion time di;
c processors;
constant processing time;
tasks have different priorities
This problem arises in "just-in-time" production: Toyota, Boeing,
etc.
200
I will not discuss algorithms for solving the optimization problem:
minimize
P . (Many things are known. For example if
he (t) = he (t) e= 1
1
2
and fe(a,d) = |a - d| and c = 1, Garey, Tarjan, and
Wilfong [1988] show that the problem is NP-complete if
transportation times can vary but polynomial if they are constant as in
our problem.)
I will ask: Is the conclusion that schedule u is optimal a meaningful
conclusion? That is, does optimality still follow if we change how we
measure priorities?
201
Example 1
Suppose priorities are measured on a ratio scale. Consider the
statement:
(1) Pe(u;t,d,c)  Pe(v;t,d,c) for all schedules v for the scheduling
problem t,d,c
Let  > 0. Does (1) imply
(2) Pe(u;t,d,c)  Pe(v;t,d,c) for all schedules v for the
scheduling problem t,d,c
Here, t = (t1,t2,...,tn) means (t1, t2,..., tn).
202
Suppose that Pe is summable and separable. Suppose that
he (®t) = K (®)he (t) for
i
(For example,
i
he (t) = t
i
K (®) > 0:
satisfies this condition.) Then
Pe(w;t,d,c) = K()Pe(w;t,d,c).
Since K() > 0, (1)  (2). We conclude that under our
assumptions, the assertion that schedule u is optimal is meaningful
under ratio scale priority measurement.
203
Example 2:
he (t) = he (t) = 2t ¡
1
Pe summable and separable, fe(a,d) = |a-d|,
Xn
Thus,
P (u; t; d; c) =
2t i ¡ 1 jau ¡ d j
e
i
2
1
.
i
i= 1
Then meaningfulness can fail under ratio scale priority measurement.
t = (1,2,2,2), d = (1,2,2,2), c = 1.
Consider u defined by au = (1,2,3,4). Then
Pe(u;t,d,c) = h1(2) + 2h1(2) = 6.
It is easy to check that any other schedule v has penalty  6, so u is
optimal.
204
However, consider  = 2. Then t = (2,4,4,4).
Pe(u;2t,d,c) = h1(4) + 2h1(4) = 24.
Let av be given by (4,1,2,3). Then
Pe(v;2t,d,c) = 3h1(2) + h2(4) + h1(4) = 22.
Thus, u is no longer optimal after a change of unit.
205
Constant Arrival Times
d = (d,d,...,d)
Observation: Suppose
P is summable and separable
he (t)and symmetric
he (t) = hee (t)
2
in the sense that 1
. Suppose also that i
is increasing in
t and fe(a,d) = |a - d|. Then a greedy algorithm gives the only
optimal solution and this remains the case after any ordinal scale
(strictly increasing) transformation -- i.e., the conclusion of optimality
is meaningful under ratio, interval, and ordinal scales.
206
Example 3
The conclusion we have stated is false without symmetry.
fe(a,d) = |a - d| p
he (t) = t 2 , he (t) =
1
2
t
t = (1/4,1/9,1), d = (2,2,2), c = 1.
Then the optimal schedule u is given by au = (3,4,2).
Pe (u; t; d; c) = he (1=4) + 2he (1=9) = 1=16 + 2=81
1
1
.
207
In comparison, if av = (3,1,2), then
Pe (v; t; d; c) = he (1=4) + he (1=9) = 1=16 + 1=3
1
2
.
However, let  = 36. Then t = (9,4,36) and
Pe (u; ®t; d; c) = he (9) + 2he (4) = 81 + 32
1
1
Pe (v; ®t; d; c) = he (9) + he (4) = 81 + 2
1
2
.
.
Thus, u is no longer optimal.
208
Nonconstant Arrival Times
Observation: Suppose
he (t) = he (t) = t
1
2
, fe(a,d) = |a - d|.
As noted above in Example 1, optimality is meaningful for ratio scale
priority measurement with this penalty function. However, we note
that it fails for interval scale priority measurement. Are interval scales
relevant to priority measurement in scheduling? Yes. Utility or worth
or loss of goodwill are not necessarily ratio scales and could very well
be interval or ordinal scales.
t = (9,9,9,1), d = (2,2,2,1), c = 1.
au = (2,1,3,4); Pe(u;t,d,c) = 21.
It is easy to check that u is optimal.
209
However, let  = 1/8,  = 7/8. Then t+ = (2,2,2,1) and
Pe(u; t+,d,c) = 7.
If av = (2,3,4,1), then
Pe(v; t+,d,c) = 6.
210
A Special Case of Interval Scale Priority Measurement and
Nonconstant d
Assumptions:
Let d = (d,d,...,d,k), with 0 < d < k  n. Let c = 1.
he (t)
Suppose Pe is summable and separable with
1,2, and with fe(a,d) = D|a - d|, D > 0.
Assume that
i
increasing in t, i =
he (®t + ¯) = K (®; ¯)he (t) + L (®; ¯);
i
i
K (®; ¯) > 0:
Theorem: Then the conclusion that u is an optimal schedule is
meaningful if priorities are measured on an interval scale.
211