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Psychological Review
1995. Vol. 102,No. 2,396-408
Copyright 1995
by theAmerican Psychological Association, Inc.
0033-295X/95/S3.00
A
Measurement-Theoretic Analysisof the
FuzzyLogic ModelofPerception
Court
S.
Crowther
University
of
California,
Lo s
An geles
William
H.Batchelder
University
of California,
Irvine
Xiangen Hu
University ofM emphis
Th e
fuzzylogic model
of
perception (FLMP)
is
analyzed
from
a
measurement-theoretic perspective.
FLMP
has an
impressive history
of fitting factorial
data, suggesting that
its
probabilistic form
is
valid.Theauthorsraise
questions about
theunderlyingprocessing
assumptions
ofFLMP. Although
FLMP parameters
are
interpreted
asfuzzy
logic truth values,
the
authors demonstrate that
for
sev-
eral
factorial designs widely used
in
choice experiments, most desirable fuzzytruth value properties
fail to
hold under permissible rescalings, suggesting that
th e fuzzy
logic interpretation
may be un-
warranted. The authors show that FLMP's choice rule is equivalent to a version of G.
Rasch's
(1960)
item response theory model, and the nature of FLMP measurement scales is transparent when stated
in
this
form.
Statistical
inference
theory exists
for the
Rasch model
and its
equivalent forms.
In
fact,
FLMP
can be
reparameterized
as a
simple 2-category logitmodel, thereby
facilitating
interpretation
of
its measurement scales and allowing
access
to commercially available software for performing
statistical inference.
Th e
fuzzy
logic modelo fperception
(FLMP)
is anapproach
to multicomponent, factorial pattern recognition experiments.
It ha sbeen applied inmany areas ofhuman information pro-
cessing,including speech perception (e.g., Massaro, 1987;Mas-
saro & Oden, 1980;Oden & Massaro, 1978)and letter percep-
tion
(Massaro &Hary, 1986; Oden, 1979).Furthermore, the
model
has
been much discussed, debated,
and
compared
to
other
models (e.g., Cohen & Massaro, 1992; Massaro,
1989;
Massaro & Friedman, 1990; Oden, 1988). The model makes
strong assumptions about the underlying processing events that
occur
when an individual must classify a factorially defined
stimulusintoone ofseveral response categories.A t thehearto f
themodelis theassumption that astimulusiscompared, fea-
Court
S .
Crowther, Department
of
Linguistics, University
of Califor-
nia,L os
Angeles; William
H.
Batchelder, Department
of
Cognitive Sci-
ences, UniversityofCalifornia,Irvine; XiangenHu ,DepartmentofPsy-
chology,Universityo fMemphis.
Portions of this article were presented at the 25th annual meeting of
the
Society
for
Mathematical Psychology, Stanford University, Palo
Alto,
California, August
22,
1992
(Crowther
& Hu, 1992). We
grate-
fully
acknowledge comments
from
Jean-Claude Falmagne, Christolf
Klauer,
EceKumbasar,R. Duncan Luce, and David M.Rieferon ear-
lier drafts.
The
researchpresented
in
this article
was
supported
by a
National Science Foundation (NSF) training grant
to the
Institute
for
Mathematical Behavioral Sciences at University of California, Irvine;
by
a
National Institutes
of
Health training grant
to the
Phonetics Labo-
ratory at
Un iversity
of
California,
Los Angeles; and by NSF Grant SBR-
9309667.
Correspondence concerning this article should be addressed to Wil-
liam
H. Batchelder, Department of Cognitive Sciences, Social Sciences
Tower,
UniversityofCalifornia, Irvine, California 92717. E-mailm ay
be sent via Internet to [email protected].
tureb y
feature,
withprototypes representing each relevantr e-
sponse category. The results of these comparisons are said to be
fuzzy logic truth values," indicating the degree ofmatch of
each stimulusfeature
to a
corresponding prototype feature.
The
fuzzytruth values thenarerepresentedbyparametersi n aprob-
abilistic model that predicts the classification probabilities for
each stimulus.
In
this article
w e
examine
the
probabilistic classification pro-
cess of FLMP from a measurement-theoretic perspective. Our
analysis has substantial consequences for the model, some posi-
tive
an d
some negative.
In
particular,
w e
show that
the
fuzzy
logicparameter values cannot berecovered uniquelyfrom th e
classification
probabilities. Although it is possible to set up
scales of measurement on the basis of the classification proba-
bilities,
these scalesfailto
satisfy
theproperties neededt ojustify
their interpretation
as
fuzzy logic truth value scales.
We
also
showthat thebasic probability formula ofFLMP isidentical
with
that of the
well-known
model of item response theory de-
veloped byRasch
(1960)
an dstudied
extensively
bypsychomet-
ricians, and invarious equivalent forms,in thefoundationsof
measurement literature. Neither the Rasch formulation nor the
others that w ecover have been analyzed previouslyintermsof
FLMP.
Althoughour analysis directly challenges the interpretation
ofFLMP parameters asfuzzy truth values, it helps to explain
whythe model
frequently
does a good job of fitting data in fac-
torial pattern classification experiments. Indeed, someof the
equivalent formulations have been quite successful in analogous
applications, and they havethe added benefit of having been
studied extensively in the psychometrics literaturefroma statis-
tical standpoint. Thus, rather than questioning the ability of
FLMP
to fit
data,
our
article calls attention
to the
need
for
more
396
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MEASUREMENT-THEORETIC
ANALYSIS
OF FLMP
397
work
justifying theprocessing interpretation of themodel.To
facilitate
this
and to aid
others
in
using
the
model,
we
demon-
strate
how to
reformulate FLMP
as a
simple
logit
model.
In
its
reformulated version, statistical inference
for
FLMP
can be
conducted
by
standard, log-linear statistical
software
packages
that can perform parameter estimation, goodness of fit, and hy-
pothesis testing.
Description
of the
Model
According
to
FLMP, conjoined features comprise prototypes
stored
in
long-term memory.
The
recognition process involves
matchingfeatures
that are perceived to be present in a stimulus
to the prototypes in long-term memory. For letter perception,
the
prototypes
are
letters (Massaro
&
Hary,
1986), and the
fea-
tures are visual featuresofthe letters. For speech perception, the
featuresareacousticandperhaps
visual features
of the
speech
signal, and the prototypes are syllables (Oden & Massaro,
1978).FLMP includes three processing stages:featureevalua-
tion, feature integration, and pattern classification. In the fea-
ture evaluation stage,
it is
assumed
that
sources of
information
corresponding
to
eachfeature
are
evaluated independently
for
thedegree
to
which
they
match
featuresof the
prototypes. Dur-
ingthefeatureintegration stage,featurevaluesare combined,
and the degree to which the resultant feature combination
matches
each relevant prototype is determined. During the pat-
ternclassificationstage,therelative goodnessofmatch between
each feature conjunction
and
each relevant prototype
is
deter-
mined
using a
formuladescribed later
in
Equation
1.
FLMP assumes that
the
output
of thefeature
evaluation
and
feature integration stages
are
continuous,
but the final
stage
(pattern classification) isdiscretein the sense that the individ-
ual will classify the
pattern
as a
token
of an
available category
according
to a probability distribution
over
the categories. The
modelpostulates that duringfeatureevaluation, each stimulus
feature
is
assigned
a
fuzzytruth value" (Goguen, 1969; Zadeh,
1965)
in the [0,1 ] intervalreflecting"the degree to which each
relevant feature is present" (Massaro&Hary, 1986,p.124).
Fuzzy
truth values
are
used
in the
model because they
. . .providea
naturalrepresentation
ofthe
degree
ofmatch.Fuzzy
truth
values
lie between 0 and 1, correspondingto a
proposition
being completely
false
and completely true.The value 0.5corre-
sponds
to a completely am biguous
situation,whereas
0.7wouldbe
more true than
false
and so on. (Massaro&Friedman,
1990,
pp .
231-232)
During feature integration, thefuzzytruth valuesassigneddur-
in gthefeatureevaluation stagearecombined,and thedegreeto
which they
match prototypes
is
assessed.
Typically,the
model
is
applied
to
data
from
straightforward
factorial
categorization experiments, where each stimulus
is
constructed
by
conjoining
onelevel
from
each
factor.
Forexam-
ple,
consideratwo-factor experimentinwhich individualsare
askedto
classify
each stimulus(Q,Oj)intoone of two
response
categories,T, orT
2
,whereQ
e
C =
{
d,...,C/}andOj
e
O
= {O,,...,Oj}.
It is
assumed that
CandO
represent
two
different
factors,
with
/and
/levels,respectively, and the model
postulatesI
+ Jparameters,
C i
and
o
j,with0
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398
C. CROWTHER, W. B A TCH ELD ER, AND X. HU
ingF2-F3."In
most
applications,
conjunction
isimplemented
in F L M Pby the
multiplication operator. Thus,
the
above pro-
totypedefinitionscan be expressed as:
/b a / :
q o j a n d
For the
pattern classification stage,
in
terms
of the
above form u-
lation,
th e
probability that
th e
participant classifies stimulus
(
Q
,
Oj) as ba is given by specializing Equa tion 1 as:
p ( b a |Q , O j )
=
C i O j
In
this article,
w e focus on
Equation
1 and
related equations
as mathematical models for factorial categorization experi-
ments.
As
such,theobservable quantitiesforthemodelare the /
X /category
proportions, Py ,representingth eobserved relative
frequency with which stimulus
( C
i?
Oj) is
categorized
as T j .
Typically,the / +J
fuzzytruth value parameters,
th eqa n do
J 5
are estimated from the Py.Because the estimates of theqa nd
O j ,
which
represent
the
influence
o r
effect
of theexperimental
factors,
aredetermined
from
th ecategory choice proportions,
FLM P can be
seen
as
im pleme nting probabilistic con joint m ea-
surement(see, e.g., Falm agne,
1985),
and it leads to a scale of
measurement for
each
factor. Typically,
Massaro
and his co-
workers estimate
the
parameters
by
using
the
STEPITminimi-
zation
algorithm (Chandler,
1969),
and the estimates of the
q
an d
Oj
are a se t of values, c, and O j ,
that min imize
the
root mean
squared
deviation
( R M S D )
between
the
predicted probabilities
in
E O
T
ation 1 and the observed proportions:
R M S D
=
IX.J
2)
ModelIdentifiability
an d
Measurement
In
this section
we
raise questions about
the
interpretability
of
th e
model's parameters
as
fuzzy logic truth values. Such
an
interpretation
requires that the estimates have certain proper-
ties
such as those cited earlier from Massaro and Friedman
(1990) .
F urthermore, M assaro
an d
Cohen suggested that the
param eter value s can be used to determine the relative contri-
bution
of each source and to ascertain the psychophysical rela-
tionship
between the stimu lus source and the perceptual
conse-
quence
(1983,p. 759) .If
this were
so,
then uniqueness
m ay be
anecessary propertyfor the estimates. By uniqueness we re-
fer to the
condition that there exists only
one set of
parameters
(i.e.,
on e
solution)
that best fits the data in the sense of mini-
mizingth e
R M SD
in
Equation
2. If
there were more than
on e
set
o f
parameter values that provided identical, good
fits to the
data, then it would be impossible, without making additional
assumptions or conducting further experiments, to determine
which
set of parameter values reflected the true
"perceptual
consequence of the stimuli under study.
Are the
Estimates Unique?
Let us
consider
the
uniqueness
of the
estimates obtained
in a
two-factor,
two-category FLMP. A lthoug h there
are / X/ob-
servable
response proportions,and
only
/ +Jparameters to be
estimated, uniqueness is not necessarily guaranteed (see
Bamber& vanSanten,
1985).
In the following,wedemonstrate
that
the estimates that min imiz e RM SD are not unique. The
uniqueness
question
is clarified by
Theorem
1,
which shows
that if
Equation
1 is
satisfied
by any set of
parameter
values,
then
there
are
infinitely
many other sets
of
parameter values
that
satisfy the
same
factorial
p robabilities.
Theorem1
L et
p j j ( C j , O j )
begivenby
Equation
1,
forsome
0
< C i ,
Oj < 1,
1
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MEASUREMENT-THEORETIC ANALYSIS
OF
F L MP
399
not
identifiable
2
for a
two-factor, two-category expe rimen t
(cf.
Riefer & Batchelder, 1988). Thepractical
implication
ofthis
lackofidentifiabilityis
that
any set of
proportions, Py , obtained
in
afactorial designcan be fit equallywellby
infinitely
m a n y
setsof
parameters
in the
sense
of
m inimizing
the
R M S D .
Th e
samecan besaidfor anyother classicalg oodness-of-fit criterion
such
as
m axim um likelihood estimation, m inim um chi-
squared estimation, and others discussed by Read and Cressie
( 1 9 8 8 )a nd Batchelder ( 1 9 9 1 ) . In other words, foreachset of
parameters obtained using the RGR (Equation 1),for any B >
1 the
transformations
in
Equations
3 and 4 will
generate
a
new set of
parameters, within
the
unit interval, that yields
an
identical
fit to thedata. Therefore , becausetheestimatesare not
unique
in the
above sense,
it is not
reasonable
to
report
an d
interpret
a pa rticular set of parameters re turn ed by STEPIT
(o r
an y other optimization algorithm) as estimates of the
fuzzy
truth values that correspond to the"true" perceptual conse-
quence
of the
stim ulus. Such estimates
are
arbitrary
an d
depend
on arbitrary featuresof thealgorithm , suchas itsstarting values
used in theestimation procedure.
Oden(1979)
applies FL M P em bodied
in
Equation
1 to
data
from aletter recognition experiment. Whenfitting thedata,h e
took steps to avoid the n onu niq uen ess of parameter estimates;
for
example,".. .the scale un it w as set by choosing a value for
(3 ,such thatth e rangeofparameters isapproximately equalfor
both
factors...
(Oden,
1979,p.347) .In our
terms,
/ ? ,=
( 1
-
Oj ) /0 j ;thus, it is clear that the lack of uniqueness of the param-
eter estimateswa sacknowledgedin theFLMP literature. How-
ever,
m any more recent
fits of factorial
data make
n o
m ent ion
ofsetting the scale unit; nevertheless, tables of scale values are
presented
an d
described
as fuzzy
truth values
a nd
sometimes
areplotted ag ainst physical mea suresas inpsychophysicalfunc-
tions
(e.g., M assaro,
1987;Massaro&
Cohen, 1983 ,1993;M as-
saro
&
Friedman,
1990;
M assaro
&
Oden, 1980).
W e
also w ere
unable
to find an y explicit characterization of the scale typ e
of fuzzy
truth values
for
F L M P
in the
literature. Massaro
an d
Friedman(1990)
dodiscuss scalesingeneral;forexample,
The outcome of the first stage, evaluation,can bedescribedby a
scale
value,
whichin general we denote asxfor agiven inform ation
source
X,
....Weassume that
x
is a real number on aninterval
scale
that
is measured in some sort of"currency," such as truth
value, probability, activation, energy,
or
strength. (Massaro
&
Friedman, 1990,
p.
227)
The
scale
defined in
Theorem
1 is
clearly
not an
interv al scale;
furthermore, quan tities like probability an d tru th value could
no t be on interval scales because rescalings could violate the
constraint that they
are
confined
to the
[0,
1]
un it interval.
A s
wewill
see, the arbitrariness in the particular scale for FLM P in
Theorem
1 can
have several serious consequences
for the
fuzzy
logic
interpretation
of the
estimated scale values.
Equations 1,3, and 4 im ply that there are twoscales,one for
each
factor,
andne xt, Corollary 1shows that theyareuniqueup
to the setting of a single level of one of the factors to an arbitrary
valuein (0,
1).
In
other w ords,
in the
terms
of the
Massaro
an d
Friedman
(1990,
pp .231-232)quote
cited
earlier, a given level
of one of the factors can be arbitrarily assigned a fuzzy truth
value
such as 0.3, 0.5, or 0.7, and this assignment determines
allof theother values.
Corollary
1.
Suppose that Equation
1
holds
for
some
pa-
rameters
and< 0 j>.Let xe(0, 1)bearbitraryandarbi-
trarily
pick
any
particular
c
k
(or
Oi).
Then there
is
exactly
on e
se tofparameters,
that
satisfies Eq uations 1 and 5
a n d c
k
=
x(orc>i
= x) .
Proof.
Witho ut lossofgen erality, pick C
k
e
C. First, note
thatfor all 0 < x < 1,
B ( x )
=
x ( l - c
k
)
7)
Note from Equation
3that
l + B ( x ) ( l - C k )
an dfurther note
that no other B > -1
will yield
c
k
= x.
From
Theorem 1 ,
all
parameter values consistent with Equation
1 are
obtained from Equations 3 and 4; thus, the B (x) in Equation 7
yieldstheuniq ue scales
andwith
c
k
= x.
Tosee the
consequences
of
Theorem
1 and
Corollary
1,let us
return to the audiovisual integration example
from
Massaro
an dCohen(1983)that wasdiscussed earlier.Todetermine the
fuzzy t rut h values assigned to the stim ulus features, M assaro
and
Cohen
(1983)estimated a setofparameter
values
that
min-
imizedR M SD s between predicted and observed data. Averaged
over all 14con ditions and all six participants, the R M SDw as
0.015,
suggesting to the a uthors that the model fit the data well.
Fo rillustrative purposes, F igure 1(open circles) showstheesti-
mated parameter values returnedfrom STEPIT fo rparticipant
numbersix
reported
in
Massaro
an d
Cohen
( 1 9 8 3 ) .
Figure
1
compares th e original estimated fuzzy truth values (open
circles)
for
each level
of the
acoustic
(top panel) an d
visual
(bottom
panel)
factors, together with three new sets of
fuzzy
truth values that result when Equations 4 and 3,respectively,
areappliedto thereported STEPIT output using three
different
values of the
scale parameter,
B.
Each
of
these
four
curves
is
equally supported by the pattern classification data in that
STEPIT could have produced
any one of
them
as
minimiz ing
th e
R M SD
in
Equation
2.
Furthermore, they suggest quite
different, contradictory stories about the psychophysical rela-
tionship
between
the
acoustic
an d
visual factor
levels and the
corresponding fuzzy truth values.Forexam ple, consider Level
3of the aco ustic factor in F igure 1.The reported estim ation run
yielded
the
value 0.28. Interpreting this parameter value
as a
fuzzy truth valuein thespiritof the quote cited in our intro-
duction from Massaro an d Friedman (1990), we would at-
tribute a relatively low degree of tru th to the proposition, the
feature 'fallingF2-F3'
is
present."
However, by transform ing
2
A
probabilistic model
fo r
categorical
data
assigns
toeach
value
o f
its parameters,9
e
fl, aunique
probability
distribution,p(0),overthe
categories,where }is the
parameter
space. Statisticians refer to such a
model
as
(globally) identifiable,
if corresponding to
every
distinct
pair
ofparameters
aredistinct d istributions, thatis,
for
all
0|,0
2
n
,
*i
2
implies p(*i) p(8
2
).Twomodels are said to be equivalent if their
sets
of
probability distribution s
are
identical. Nonequivalent m odels
are
potentially distinguishable by
data.
M odel identifiability should not be
confused with model equivalence (see Bamber & van
Santen,
1985;
Bishop,
Fienberg,&
Holland,
1975;Riefer&
Batchelder, 1 988 ).
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400
C.
CROWTHER,
W .
B A T C H E L D E R ,
AND X. HU
ba
1
0.9
0.8 3
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
3 4 5
AcousticFactor Level
-O-
B =Orig.
-A- B = 0.96
B
B =1.57
-A- B = 11
ba
da
Visual Factor Level
Figure 1. Relative evidencefor thesyllable da for theacoustic(top panel)a nd visual
(bottom
pane l)
factors.
The
original scale values
from
Massaro
and
Cohen
( 1 9 8 3 )are
plotted with open circles (B
=
orig.),
and the
other markers represent scale values computed with three
different
choices
for B
using Equations
4
and 3,
respectively. Open triangle,
B =
-0.96;open square,
B =
1.57;
filled
triangle,
B =
11 .Orig.
=
original.
this value usingEquation4 andlettingB = 11 , w eobtain th e
value
0.82, which would suggestahigh degree oftruth to that
same proposition. Letting B = 1.57 would produce thevalue
0.5, indicating thatthetruthof theproposition is completely
ambiguous. As a result, unless constraintsareintroduced to
restrict FLMP parameters, theestimate of anyparticular pa-
rameterin a two-category,two-factor experiment cannotbe in-
terpreted as a fuzzy truth value in thesensedescribed in the
quote byMassaro and Friedman (1990, pp. 231-232) cited
earlier.
Thestatements in the Massaro and Friedman (1990, pp.
231-232)
quote pertainto theparameter valuesof asingle level
ofone o f the
factors. From
a
measurement-theoretic perspective
th e
statements
in the
quote
are not
meaningful.
Fo r
example,
in
Roberts
( 1 9 7 9 )
an dSuppesan dZinnes(1963) ,astatement
involvingscale values
is
said
to be
m ean ingfu l
3
in
case
it s
truth
value
i s
preserved under permissible
rescalings. In the
case
w e
3
Thedefinitionsof
meaningfulness
giveninRoberts
( 1 9 7 9 )
an dSup-
pes and
Zinnes
( 1 9 63 )are not
accepted
by all
measurement theorists
ascoveringal lapplications of the concept of
meaningfulness.
In
fact,
considerable foundational workhassince occurred toprovideamore
adequate senseof
m eaningfulness
(e.g., Luce, Krantz, Suppes,&Tver-
sky,
1990).However,thesituation in the current article is sufficiently
simple
thatthe
Suppes
and
Zinnes
( 1 9 63 )and
Roberts ( 1 9 7 9 ) defini-
tion
can be
regarded
as
adequate.
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MEASUREMENT-THEORETIC ANALYSIS OF FLMP
401
have
been discussing, statemen ts interpreting
the
scale value
of
a given level, as in the quote, can be rendered either true orfalse
byscale transformations, and hence they are not m eaningful in
the intended sense.
Rather than interpreting any particular value in isolation,
one might hope to m ake
meaningful
statements about the rela-
tive
values of some of the parameters. For example, consider a
stim ulus (C
k
,
O i)
in the
"ba"-"da"
exam ple discussed earlier.
On e m ight hope to m ake weaker statements, such as"C
k
gives
stronger support for
'ba'
than
does
Oi"One way to quantify
suchinterfactorcomparisonsis toasserttheproposition c
k
>
O i .
Unfortunately, as Corollary 2 shows, such statements are not
meaningful
in the
sense that rescalings
c an always
reverse ine-
quality relationships comparing specific levels of the two
factors.
Corollary
2. Suppose that
a set of
parameter values
determinesthep robabilitiesPij(q, O ; )throu gh Equation 1 .
For
a particular stimulus(C
k
,
Ot),
supposec
k
> Q I .Another set
of parameter values
satisfies Equations 1 and 5,
where
o* >c*.
Proof.
On the
basis
of Equations 3 and 4, w e note
that
5B
;
8)
(9 )
for
all 0
-1,
for 0 B
0
.
Corollary 2 shows that it is meaningless to com pare even or-
dinally
the
magnitudes
offuzzy
tr uth values between respective
levelsof the two factors. Infact,it follows by a min or extension
ofthepreceding argumentthatforany set ofproportions
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40 2
C. CROWTHER, W . B A T C H E L D E R , A N D X. H U
L ( o f ) =
L (
0j
)
+
log( l
( 1 3 )
forB>-l .
From Equations
12 and
13,
it is
clear that
the
ranges
of the
log-odds ratios
for
both factors can be ordina lly compared. Spe-
cifically,
define
an d
=
L(c
ma x
)-L(c
mi n
)
=
L( o
ma x
)-L(o
mi n
),
where
c
m ax
, c
m in
,0, *,an d
o
m in
aredefinedas
before.
It is
mean-
ingful
to
assert that, say,
R C
>
R O , because from Equations 12
an d
13,
the valuesof R Ca nd RO do not depend on the valueof
B,an d
therefore
are
in varian t under rescaling.
What Measurement Scale Is Implied?
The lack of unique nes s of the param eter values in a statistical
modeldoesnot, in
itself,
ren der the m odel useless. For exam ple,
Luce's
(1959)
choice rule has been quite successful in describ-
ing
paired-comparison judgm entsfroma set of T V objects by the
rule
P,=
-
( 1 4 )
wherepyis the probability that object i i schosen over object
j in apaired com parison,and thescale values
V j ,v
j > 0,where
1
< i j
0,
shows that Luce's
choice,
model
is not identifiable. In
fact,
Suppes and Zinnes
( 1 9 6 3)
have
referred tothis typeofscale as an indi rect, ratio-scale mea-
surem ent. This sort
of
nonun iqueness
is
common
in
statisti-
ca l
models, particularly
the logit
models such
as the
Bradley-
Terry-Luce
model that is based on E quation 14, and n on-
un iqueness
does not
affect
the ir value or usefulnes s in ana-
lyzingand interpretin g
data.
So
far,
for the
two-factor, two-category FLMP,
it is
clear
from Theorem 1 and its consequences that the scale of mea-
surement
is
de te rmined
by one fixed
quantity,
B.
However,
the situation isdifferent from the choice rule in Equation 14
in tw oessentialrespects.First, tw oseparate
sets
ofscale va l-
ues are set up
indirectly from Equation 1 ,
one for
factor
C
and on e for factorO.As Corolla ry 2 establishes, if these two
sets
of
scale v alues
a re
regarded
as
being
in the
same cur-
rency (a ssuggested by
Massaro
&Friedm an, 1990,p .2 2 7 ) ,
and are
therefore merged onto
a
singlescale
of fuzzy
logic
t ru thvalues,
no t
even ordinal properties among
th e
scale val-
ues are preserved by rescaling. A second
difference
is that
Equations
3 and 4
show that
th e
formulas
fo r
rescaling each
factordo notassumefamiliar,conventionalformssuchas for
ratioo rin terv al scales.
Most of the properties we have considered concerning inter-
pretation
of the
parameters
as
fuzzy truth values have proven
not
meaningful
in the
sense that scale transformations
can de-
stroy them . A weaker interpretation offuzzy truth, asdiscussed
inan article by Goguen (1969, p.332)that is cited often in the
FL M P literature (see, e.g., Massaro
&
Oden, 1980), assumes
thatfuzzy
t ru th
valuesar e
scaled ordin ally.
InGoguen's
frame-
work,a
scale
of
"degree
o f
m em bership preserves
0 and 1 but
otherwise
is
only ordinal.C orollary
3
shows that FLM P satisfies
this property
as
long
as the two
factor scales
are
considered
separately.
Corollary3:intrascaleordinality. Suppose that a set of pa-
rameter values
determines
t he
probabilities Pjj(ci,
Oj )
through Equation
1.
Suppose
further
that
is
another
set ofparameter values that satisfy Equations1 and 5.Then
c
k
< Q
iff
c*
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MEASUREMENT-THEORETIC
ANALYSIS
OF
FL M P 403
16apply
to the
two-factor situation. Incorporating Equation 16
into
Equation 1yieldsthe
relationship
Pij(Q,Oj)
=
(17)
Equation 17is a
strong condition
and is
easily tested with
a chi
square using the one-factor relative frequencies as estimates of
the
Q
and
Oj .
If itdoesnothold, Equation 1 ofFLMPmaystill
bevalid,but thesingle-factor
data
willnot be
consistent
with
the
two-factorscales.
A
second issue concerns
the
phenomenological character
of
the
single-factor experiment itself.
To
perceive
the
stimuli
as in-
tended, bothfactors
may be
required
in
some cases.
For
exam-
ple, among the large number of factorial experiments in the
speech perception literature, many become problematic when
singlefactors
are
presented
in
isolation.
For
example,Crowther
(1993)tested
the
influence
of two
acoustic factors,voweldura-
tion
and firstformantoffset
frequency,
onstop consonant voic-
ingperception.If the firstformantoffset factorhad been pre-
sented in the absence of the vowel duration
factor,
the resultant
stimuli wouldnoteven
sound
likespeech.Furthermore, if the
vowel
durationfactorhad been presented in the absence of the
first
formant offsetfrequencyfactor,then
the
stimuli would
not
likelyhavebeen perceived as containing a voiced stop conso-
nant.
Sometimes
it is not
even feasible
to
produce stimuli
for
single-factor
trials.
Forexample,one may beinterestedinstudy-
ingstimulus duration and stimulus intensity, but clearly these
factorscannot be isolated physically. Therefore, depending on
the
nature
of the
stimuli,
it may or may not be
possible
to
over-
come the nonuniqueness problem for the two-factor, two-cate-
gory FLMPbyincluding single-factor
trials.
Athird issue involves cases where
the
single-factor experi-
ment is
both possible
and
satisfiesEquation 17
to an
acceptable
degree. In this case, it is possible to postulate scale-invariant
fuzzy
logic truth values;
however,
that this scale is directly
linked
to
observable
proportions
questions
the
need
or
useful-
ness
of the fuzzy
logic interpretation.
At a
minimum, consider-
ableevidencefromother sources would be required to assume
that individuals are processingfuzzytruth values.
E f f e c t of
Adding More
Experimental Factors
Anotherpossiblewaytoavoid parameter nonidentifiabilityis
to include more than two experimental factors in the design
while
holding constant the number of response categories. This
wouldincrease the number of parameters to be estimated, but
(because the experiment isfactorial)it would also increase to a
greater extent
the
number
of
observable entities.
For
example,
consider a factorial experiment withNfactors and />2 levels
perfactorn. Then the number of
parameters
to beestimatedis
AT
2 /
n
, but the number of conditions for observed data is IT /
n
.
-'
n -i
Asn increases, the latter term increases fasterthan the former,
and,
in
fact,
the
ratio
of the
number
of
parameters
to the
num-
ber ofobservable
entitiesdecreases
tozero with increasing n.
Therefore, one might hope that adding experimental factors
mightavoid
the
problem
of
nonuniqueness
of
estimates.
Unfor-
tunately,
as
Corollary
4
shows, such
an
approach
not
only fails
to resolve the nonidentifiability problem, but instead worsens
the
situation
byincreasingthenumberofarbitrary scalevalues.
Corollary4.
Add a
third experimental factor,
U,
with
K
lev-
els, to the two-category, two-factor hypothetical experiment.
The resultant three-factor model is not identifiable.
Proof .
Expressing the three-factor experiment in the form
of
Equation
1,
Pijk(Ci,Oj,U
k
) =
Cj OjU
k
Cj
O j
U
k
+ ( 1 -
Ci)(
1 -
Oj)(
1 - U
k
)
=
1
-Uk)
u
k
, (18a)
where pyk(Cj ,
Oj,u
k
) is the
probability
o f
identifying stimulus
(Q, Oj,
U
k
)
as an
instance
ofT,.Let
DI,
D
2
,
D
3
> 0 be
such
D I
D
2
D
3
= 1.If thefollowingtransformations,
( l - c D _ ( l - C i )
cf
D,,
U8b)
and
u
k
'D
3
,
(18d)
aresubstitutedfortheir corresponding terms,theprobabilities
inEquation 18aremain invariant. This result follows immedi-
ately
by
inserting Equations 18b,
18c, and 18dinto
Equation
18a.
Corollary 4 shows that for a three-factor experiment, there
are three scales unique up to two arbitrary constants. Of course,
these scales are not fuzzy truth value scales; however,simple
algebraic manipulations such as in Theorem 1 can be per-
formed
to examine the implied fuzzy truth value parameter
scales for the factorlevels. It is easy to extend this result to an
Nfactor experiment, the consequence being that the N scales
underlyingthegeneralizationofEquation 18ahave
T V
- 1arbi-
trary constants
in
them.
E f f e c t ofAdding
More Response
Categories
Another
possible approach that avoidsnonidentifiabilityis to
glean more information fromthe experiment by increasing the
number of response categories while holding constant the num-
ber of
experimental factors.
To see how
this
might
be
done,
let
us expand the hypothetical two-factor experiment that involved
the use of two-response categories,
T,
andT
2
,to include four
response categories,T,,T
2
,
T
3
,
and
T
4
.
CohenandMassaro
(1992)provide
an
extensive discussion
of the
four-categorypar-
adigm.Inthis
case,
theprototypedefinitions may be
expressed
as follows:
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404
C. CROWTHER, W. BATCHELDER, AND X. HU
The probability ofidentifying stimulus
(Q,O j)
as an instance
of
categoryT, ,
T
2
, T
3
,
or
T
4
in this m odel is given by:
Cf Oj
CjOj
+
Cj
( 1 -Oj)+ ( 1 -
Ci)0j
+ ( 1 -
Ci)(
1 - Oj )
an d
sim ilarly
an d
= C j O j ,
P i j ( T
3
| C i , O j )= (l-
p
i j
(T
4
l c
i
, o
j
)=(l-
It iseasy to see that only the identity transformation on the
parameters leaves invarian t
the
four equations pij(T
k
|q, O j ) ,
k
= 1, 2, 3, 4.
Thus, FLM P
for any
two-factor experiment that
supports
four
categories is identifiable. More generally, when
all
possible pro totypes are
defined
for theexperimen tal factors,
FLMPwillbe identifiable; that is, if there are n factors and all
2
prototypes
are
used, unique parameter estimates
can be ob-
tained. In
fact, it is
worth noting that
the
above equations
for
the complete four (and 2
n
) category experiments satisfy the
properties
of
general processing tree models described
in Hu
an d
Batchelder(1994), wh ich provides algorithm s
for
statisti-
calinference.
Many
applications ofFLM Pinvolvefourcategoriesand two
experimental factors. However,the nature of stimuli in some
categorization
experimen ts that lend themselves quite natu rally
to a bin ary choice paradigm m ay be such that it is not possible
toexpandthe design naturally to afour-choiceparadigm; that
is,
forsome experiments it is possible that there are stimulus
featurecombinations that
do not
correspond
to
naturalproto-
types that could
be the
basis
of a
response category. Further-
more, even whenallfourprototypes do correspond to natural
categories, the simplicityof the four equations for the model
does
n ot
argue,
a t
least
for us, for the
desirability
or
usefulness
ofpostulating that individua ls process
fuzzy
tru th values.
Fixingthe ValueofO neParameter
A
fourth,
potential remedy does not involve changes at the
experimentaldesign level,
bu t
ratherchanges
in the
parameter
estimation procedure. Considering
our
hypothetical two-cate-
gory,
two-factor experiment,if onewereto set thevalueof,say,
C i before passing the
data
through the parame ter estimation
procedure, then Corollary 1 shows that the parameter value
scales wo uld
be
determined
in the
sense that
a ll
of
the
parameter
estimates
are
constrained
to be
unique
by the
setting
of C i.In
fact, this was men tioned earlier as the course taken byOden
5
(1979)in a 6 X 6factorial letter perception experiment. Oden
claimsthat
the
generalshapes"
of the
curves relating
th e
levels
ofafactorto thescale valuesof thefactor(as in ourFigure 1
)
".
. .wouldnot begreatlyaffected bychangesin theparameter
scaleunit"
(
1
979,
p.
34 7
).However, it is not clear w hat is m eant
by general curve"shape." All three curves in both panels of
Figure
1 are
monotonic
an d
defined
b y a
single parameter,
al-
though
on em ight thin k their shapes
differ.
In anycase,ma n y
aspects of the relationship between any such curves can be un -
derstood
by
analyzing Equations
3 and 4. We
think
it is difficult
to maintainthatthe
"shape"
offunctionso f theQandOjdo not
changewiththescale parameter,B.
Fixing a parameter value to resolve the nonidentifiability
problem may entail negative consequences depending on just
howon ewantsto use theparam eter values.Inparticular,we see
no w ay to set a particular scale value a priori in a m anner that
guaranteesthepropertiesof thefuzzylogic interpretation of the
parameters
givenin thequotecited
from
Massaroan dFried-
m an
( 1990,
pp.
2 31
-232
)
.
Criteria such
as
equating parameter
valueranges usedbyOden (1979,Footnote 4)could alwaysbe
used; however,they have only arbitrary an d unsystematic im -
plications for the fuzzy
logic interpretation
of the
parameters.
On the other hand, it m ay be reasonable to fix a parameter at
acertain valuean dmaintainafuzzylogic interpretation incer-
tain circumstances.
Fo r
example,
if one had
sufficient reason
forbe lieving that thepsychological va lueof aparticular levelof
on eof thefactors should be com pletely ambiguous, then it
seems reasonable to set the corresponding
fuzzy
truth value to
0.5 before startingtheestimation procedure. Ou rsurveyof the
applications of FLM P did not yield m any situations with a fac-
tor level that was obviou sly completely am biguous, and even
in cases w here there w as such a level, one w orries that response
bias for one of the two categories in the presence of ambiguity
might enter and thus thwart this approach to the nonidentifi-
ability
problem . The next section provides some insight into the
typeofindirect m easurement thatisentailedbyEquation
1
.
FLMPand the
Rasch
(1960)
Model
Model Equivalence
It
turnsou t that Equation 1 ofFLMP isequivalentto aver-
sion ofRasch's (1960)item response theory m odel that is
well
known
topsychom etricians. Rasch'stwo-parameter model con-
cernsth ecase whe re/participantstakeatest with/items.The
Rasch model
is
perhaps
the
most popular among many item
response theory m odels. It is discussed an d an alyzed extensively
inthepsychometric literature(e.g.,Hambleton, Swaminathan,
&Rogers,
1991;
Lord, 1974, 1980), and agreat dealisknown
about its statistical theory. Let
, j 1 ifsubject i iscorrect onitem j
1J
J O otherwise,
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MEASUREMENT-THEORETIC ANALYSIS OF
F L M P
405
Batchelder and Romney(1989)desired a form of Equation
19
that constrained the parameters to the unit interval. In es-
sence, they showed that the continuo us tran sforma tions
aiMl+e^r
1
(20)
an d
yield
0
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