Modelo en Lógica Difusa de Percepción

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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


2/13

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


3/13

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


4/13

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 ).


5/13

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.


6/13

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


7/13

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*


8/13

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:


9/13

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,


10/13

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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