Fuzzy Set Theory in Medicine

NOT M)R QUOTATION 'WITHOUT PERMISSION OF THE AUTHOR

FUZZY SET

THEORY

IN

MEDICINE

Klaus-Peter Adlassnig

J u n e 1984 CP-84-22

C o l l a b o r a t i v e P u p e r s r e p o r t work which h a s not been performed solely a t t h e International l n s t i t u t e f o r Applied Systems Analysis a n d which h a s received only limited review. Views o r opinions expressed herein do n o t necessarily r e p r e s e n t those of t h e Institute, i t s National Member Organizations, o r o t h e r organizations supporting che work.

INTFXNATION IKSTITUTE

FOR

APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

PREF'ACE

In this paper, Klaus-Peter Adlassnig, a participant in t h e 1983 Young Scientists' Summer Program, shows t h a t fuzzy s e t theory seems to be a suitable basis for t h e development of a computerized medical diagnosis and treatment-recommendation system. He describes a medical expert system of this type, CADIAGZ, developed a t t h e University of Vienna, and outlines some results obtained during testing.

Decision m a k n g is often characterized by a high degree of fuzziness and uncertainty. This may reside in the imperfect and complex n a t u r e of human information processing and/or in t h e decision systems them- selves. I t may lie in t h e generation of possible options, t h e formation of criteria by which the options are judged, the prediction of t h e effects of possible decisions, and/or t h e level of understanding of t h e underlying processes.

This paper represents a contribution to research in t h e field of com- puterized decision support, and was carried out as part of t h e Interactive Decision Analysis Project.

ANDRZEJ WlERZBICKl Chainnan

System and Decision Sciences

Fuzzy set theory has a number of properties t h a t make it suitable for formalizing the uncertain information upon which medical diagnosis and t r e a t m e n t is usually based.

Firstly, it allows us t o define inexact medical entities as fuzzy sets.

Secondly, i t provides a linguistic approach with an excellent approxima- tion to texts. Finally, fuzzy logic offers powerful reasoning methods capable of drawing approximate inferences.

These facts suggest t h a t fuzzy s e t theory might be a suitable basis for the development of a computerized diagnosis and treatment- recommendation system. This is borne out by trials performed with the medical expert system

CADIAGZ,

which uses fuzzy set theory to formal- ize medical relationships.

F'UZ2X SET THEORY IN MF,DIClNE

maus- Peter Adlassnzg

Department of Medical Computer Sciences, University of Vienna.

Garnisongasse 13, A-1090 Vienna, Austria

It is widely accepted t h a t the information available to the physician about his patient and about medical relationships in general is inherently uncertain.

Nevertheless, the physician is still quite capable of drawing (approximate) con- clusions from this information. This paper describes an attempt to provide a formal model of this process using fuzzy set theory, and implement it in the form of a computerized diagnosis and treatment-recommendation system.

In medicine, the principle of "Measuring everything measurable and trying to make measurable that which has not been measurable so far" (Galileo) is still practiced, although its fundamental limitations have been recognized dur- ing the course of this century. We now know t h a t all real-world knowledge is characterized by:

incompleteness (implying that the human process of cognition is infinite) inaccuracy (as stated in Heisenberg's Uncertainty Principle)

inconsistency (anticipated by Godel's Theorem).

Fuzzy s e t theory, which was developed by Zadeh [I], makes it possible to define inexact medical entities as fuzzy sets. It offers a linguistic approach which represents a n excellent approximation to medical texts [2,3]. In addi- tion, fuzzy logic provides powerful reasoning methods capable of making approximate inferences [4,5]. These facts suggest t h a t fuzzy s e t theory might be a suitable basis for the development of a computerized diagnosis and treatment-recommendation system [6]. Tests carried out with the medical expert system CADIAG-2 [7-91 are described which show that this is indeed the case.

2

REAL-worn KNOWLEDGE

Precision exists only through abstraction. Abstraction m a y be defined a s t h e ability of human beings to recognize a n d select t h e relevant properties of real-world phenomena a n d objects. This leads t o t h e construction of conceptual models defining a b s t r a c t classes of phenomena and objects. However, in actual fact every real-world phenomenon and object is of course unique.

Abstract models of real-world phenomena and objects s u c h a s mathemati- cal s t r u c t u r e s (circle, point, etc.), equalities (a

=

b

+

c ) a n d propositions (yes, no) a r e artificial constructs. They represent ideal s t r u c t u r e s , ideal equalities and ideal propositions.

Nevertheless, despite these caveats, abstraction forms t h e basis of human thought, and human knowledge is its result.

2.1 Incompleteness

Abstraction, however, is not a static concept. The process of abstraction is continuous and is constantly producing new results. The s e t of properties of real-world phenomena and objects under consideration is continually being enlarged and c h a n g e d Knowledge is therefore always a n d necessarily incom- plete.

2.2 Inaccuracy

Unlimited precision is impossible in t h e real world Anything said t o be

"precise" can only be considered as "precise t o a certain extent".

The pursuit of maximum precision is still an important aim in science.

Galileo, who is often credited with being t h e father of t h e quantitative scientific experiment, was certainly responsible for many scientific advances through his philosophy of "Measuring everything measurable and trying to make measur- able t h a t which has not been measured so far", although t h e limitations of this approach should be recognized

Heisenberg's Uncertainty Principle [ l o ] states t h e limits t o a c c u r a t e meas- u r e m e n t very clearly. Of course, t h e Principle applies only t o the world of microphenomena and microobjects, but its philosophical implications go further. It shows t h a t n a t u r e is fundamentally indeterministic. And it seems meaningless to ask whether nature inherently lacks determinism or whether uncertainty stems only from experimentation.

2.3 Inconsistency

Abstraction does not always lead to the same results, which in turn are not always interpreted in the same way. "Knowledge" may differ according to nation, culture, religion, social status, education, etc., and information from different sources may therefore be inconsistent. To eliminate inconsistency from the information system is only possible in limited systems, and Godel's Theorem [I 11 clearly demonstrates that contradictions within a system cannot be eliminated by the system itself.

3 KEDICAL INFDRMATION

In medicine, it is not necessary to deal with microphenomena and microobjects to run into t h e problems of incompleteness, uncertainty and inconsistency. The lack of information, and its imprecise and sometimes con- tradictory nature, is much more a fact of life in medcine than in, say, the phy- sical sciences. These problems have to be taken into account in every medical decision, where they may have important, even vital consequences for the object of medical attention, the patient.

3.1 Information about

the

_patient

Data about t h e patient can be divided into a number of different categories that are all characterized by an inherent lack of certainty.

Medical h i s t o r y o j the patient

The medical history of the patient is given by the patient himself. It is highly subjective and may include simulated, exaggerated or understated symptoms. Ignorance of previous diseases in himself or his family, failure to mention previous operations and general poor recollection often raise doubts about a patient's medical history in the mind of the doctor. On the other hand. however, the information that finally leads to the correct diag- nosis is very often found here.

2. Physical ezamination

The physician subjects the patient to a physical examination from which he obtains more or less objective data. But of course, physicians can make mistakes, overlook important indications or fail to carry out a complete examination. Furthermore, they may misinterpret other indcations because the boundary between normal and pathological status is not always clearly defined.

3. Results of laboratory t e s t s

The results of laboratory tests are considered t o be objective data. How- ever, measurement errors, organizational problems (mislabelling samples, sending them t o the wrong laboratory, etc.) or improper behavior on t h e part of t h e patients prior t o examinations c a n lead t o imprecise and some- times even totally incorrect data. Again, t h e boundaries between normal and pathological results a r e generally not strict: t h e r e are always border- line values t h a t cannot be said t o be either normal or pathological.

4. R e d s obtained b y histological, X-ray, d t r a s o n i c ezamznations, e t c .

These results again depend on correct interpretation by medical or other staff. Such findings are often crucial because they frequently indicate invasive therapy. In many cases, consideration of uncertainty is part of t h e evaluation procedure, for example in cell counts, cell determination.

picture analysis, etc.

3.2 Information about medical relationships

Medical knowledge consists of medical descriptions and assertions t h a t a r e incomplete and uncertain. It has been built u p s t e p by step, a n d is based partly on theoretical studies (in areas such as anatomy and physiology) and partly on almost purely empirical observations (made in t h e course of su.rgery, for exam- ple). Medical knowledge may be said t o comprise knowledge about causal rela- tionships based in theory, statistical information, pure definitions and personal judgement.

To add to t h e problem, t h e elements considered t o form medical relation- ships differ according t o place and time, vary between medical schools and in some cases have not been studied t o any significant extent.

3.3 Medical inference

This is t h e process by which t h e physician uses his medical knowledge t o infer a diagnosis from the symptoms displayed by t h e patient, his lab test results and medical history. It is a complex and almost uninvestigated process in which the physician is obviously able t o work with uncertain and imprecise sets of data. To some extent it is a subconscious activity, which is why it is often called an art.

4 MEDICAL EXPEXT SYSI'EM

CADIAG-2

CADIAG2 (a Computer-Assisted DIAGnosis system) is intended to be a n active assistant to t h e physician in diagnostic situations. In this way t h e experience, creativeness and intuition of the physician may be supplemented by t h e information-based computational power of the computer. The general s t r u c t u r e of CADIAG-2 is shown in 1.

4.1 Representation of medical information

CADIAG2 considers four classes of medical entities:

symptoms, indications, t e s t results, findings ( S i ) diseases, diagnoses (Dj)

intermediate combinations (ICk) symptom combinations ( S C I ).

Symptoms Si take values

6

in [ O , l ]

u

$. The value pq indicates t h e degree t o which t h e patient exhibits symptom Si (a value of $ implies t h a t symptom Si has not yet been studied). In the language of fuzzy set theory.

4

expresses t h e grade of membership of the patient's symptom manifestation Si.

An example of this mode of representation is given in Table 1.

A binary fuzzy relationship

RpS c

n x C is then established, defined by pRm(Pp.Si)

=hi

for patient P q , where

Pq ~n =

fP1

,....

P,] and Si E C

=

(S1

,..., smj.

Diseases or diagnoses also take values in [O, 11

u

$. Fuzzy values 0.00

<

PD

<

1.00 represent possible diagnoses while t h e values p~~

=

1.00 and

j

pDj

=

0.00 correspond t o confirmed and excluded diagnoses, respectively. Diag- noses which have not yet been considered take t h e value p~

=

$. Formally, a

j

relationship

RpD

c

n

^{x A} is established. defined by mpg(Pq .Dj)

=

kgj for patient

I=*,

where

Dj

E A

= ID1 ...., Dn

j.

Intermediate combinations (fuzzy logical combinations of symptoms and diseases) were introduced to model the pathophysiological states of patients;

symptom combinations are combinations of symptoms, d s e a s e s and intermedi- a t e combinations. Both entities take their values 11% and hq (respectively) in [0,1]

u

$, where $ implies t h a t the actual value h a s not yet been determined.

Table 1. An example of t h e representation of medical knowledge.

Quantitative Symptom Fuzzy

value value

Potassium,

&, ⁼

^0.00

greatly decreased

Potassium,

4 =

0.00

decreased Measured

potassium level of 5.3 mmol/l

Fuzzy - interpreter

Potassium, normal

Potassium, &,

=

0.60 increased

Potassium,

& =

0.00 greatly increased

Symptoms

The relationship

Rpsc

C

n x ^K

is defined by

pRPC(PQ,SCl) = pscI

for patient

PQ,

where

SCl

E K

= ISC I,....SCt

formally describes t h e symptom combinations observed in t h e p a t i e n t (both t h e presence and absence of symptoms a r e regarded a s observations).

The fuzzy logical connectives are defined as Follows:

Conjunction:

min (z1,z2) if z 1 E [0,1] and z 2 E [0,1]

z l A z 2 = if z l

=

$ and/or z2

=

$

The following relationships between medical entities are considered in CADIAG-2:

symptom-disease relationships (S,Dj) z 1

v z 2 =

symptom combination-disease relationships (SCIDj) symptom-symptom relationships (S, Sj)

disease-disease relationships (DiDj).

I

max (zl.z2) if z1 E [0,1] and z2 E [0,1]

( = I if z l E [0,1] and z2

=

$

= 2 if z l = $ I and z 2 € [0,1]

$ if z l

=

$ and z 2

=

$I

These relationships are characterized by two parameters:

frequency of occurrence ( 0 )

strength of confirmation (c).

For a relationship between medical entities X and Y (where X and Y may be symptoms, diseases or symptom combinations), the frequency of occurrence describes the frequency with which X occurs when Y is present. Similarly, t h e strength of donfirmation reflects the degree to which the presence of X implies the presence of Y.

The relationships between medical entities are given in the form of rela- tionship rules with associated relationship tupels. The general formulation of these rules is:

IF

(premise) TEEN (conclusion)

WITH

( o , c )

.

The relationship tupels ( o , c ) contain either numerical fuzzy values b, and k, or linguistic fuzzy values A, and

& ,

or both [3].

The definitions of t h e linguistic values and A,, t h e fuzzy intervals t h a t they cover a n d t h e i r representative numerical values a r e given in Table 2.

Representative numerical values a r e necessary in order t o make fuzzy infer- e n c e s possible (see Section 4.2). The way in which t h e linguistic fuzzy values, t h e fuzzy numerical intervals and their representative numerical values were chosen is described in m o r e detail in refs. 8 and 9. Some examples of relation- s h i p rules are given below.

Table 2. Linguistic fuzzy values, numerical intervals a n d representative nu- merical values describing frequency of o c c u r r e n c e a n d s t r e n g t h of confirmation.

Frequency of occurrence Strength of confirmation

Value Interval Represent- Value Interval Represent-

%

ative A, ative

value ^)A, value A

Always

Almost always Very often Often Medium Seldom Very seldom Almost never Never

Always

Almost always Very strong Strong Medium Weak Very weak Almost never Never

[1.00.1.00]

[0.99,0.98]

[0.97.0.83]

[0.82,0.68]

[0.67,0.33]

[0.32,0.18]

[O. 17,0.03]

[0.02,0.01.]

l0.00.0.001

- - - - - - - - -

Unknown

4

# Unknown # #

Ezample 1

IF (ultrasonic of pancreas is pathological) THEN (pancreatic carcinoma)

WITH (0.75

=

often, 0.25

=

weak) Ezample 2

IF (tophi) THEN (gout)

WITH (0.25

=

seldom, 1.00

=

always)

IF (lower back pain A limitation of motion of the lumbar spine A &min- ished chest expansion A male patient A age between 20 and 40 years)

THEN

(ankylosing spondylitis)

WITH

(-, 0.90

=

very strong)

The values p, and p, are interpreted as the values of the fuzzy relation- ships between premises and conclusions:

Si

Dj

(occurrence relationship)

RasD C C X A

SiDj (confirmation relationship)

RhD c C x A

SCI

Dj

(occurrence relationship)

GCD c

K

x A

SCI

Dj

(confirmation relationship)

FSCD c

K

x A

Si S, (occurrence relationship)

C C X C

Si S, (confirmation relationship)

RhS c

C

x C

DiDj (occurrence relationship) RODD

c A X A

DiDj (confirmation relationship)

aD ^{c A X A}

4.2 Puzzy logical inference

The compositional inference rule proposed by Zadeh [4] and introduced into medical diagnosis by Sanchez [12,13] is adopted as an inference mechan- ism. I t accepts fuzzy descriptions of the patient's symptoms and infers fuzzy descriptions of t h e patient's condition by means of the fuzzy relationships described in the previous section.

Three such inference rules (compositions) are used to deduce the diseases D, suffered by patient

Pp

from the observed symptoms Si:

1. Composition for Si

Dj

confirmation:

defined by

2. Composition for Si Dj non-confirmation:

R ~ D

=

R P S O ( ~ - % D ) defin ed by

pRbD (Pq.Dj)

=

max min [IIRpS(Pq ,Si): 1 -/1 (Si*Dj)I

Sf R b

3. Composition for Si Dj without symptoms:

RJD

=

( l-Rps) 0 R i D defined by

%D (P,.Dj)

=

m a r m i n [ I - / L ~ ~ ~ ( P ~ . S ~ ) : /I (SinDj)I

Sf Rib

The following diagnostic results a r e obtained:

a diagnosis is confirmed i f pRh (Pq ,Dj)

=

1-00

a diagnosis is possible if O . l O s p R (Pq.Dj)s0.99

JD

The boundary value 0.10 is a heuristic value which rejects diagnoses with very low evidence.

a diagnosis is excluded if

Symptom combination-disease inferences (compositions 4,5 a n d 6) a r e carried out a n d interpreted in an analogous way. Symptom-symptom infer- ences (compositions 7, 8 and 9) a r e computed in order t o complete t h e patient's symptom patterns. Disease-disease inferences (compositions 10, 11 and 12) a r e also performed in order to confirm t h e underlying disease from t h e

presence of t h e secondary complaints or t o exclude entire areas of secondary complaints if a particular primary disease is absent.

4.3 Acquisition of medical bowledge

The knowledge acquisition system i s capable of acquiring information on medical entities and t h e relationships between them. In CADIAG2. relation- ships are stored a s numerical fuzzy values in t h e range [0,1]. Medical informa- tion can be acquired in two ways:

through linguistic evaluation by medical experts

by statistical evaluation of a data base containing medical data on patients with confirmed diagnoses.

Information on relationships can be gathered linguistically using predefined linguistic values t o determine p a r a m e t e r s such a s frequency of occurrence o and strength of confirmation c (cf. Table 2). Empirical, judge- m e n t a l and definitive knowledge may be acquired i n t h i s way.

CADIAG2 relationships have t h e important property t h a t they may be i n t e r p r e t e d statistically. The values of t h e frequency of occurrence po a n d t h e strength of confirmation pc may be defined a s follows:

where

F(S,

n

Dj)

-

absolute frequency of occurrence of S, a n d Dj F(Dj)

-

absolute frequency of o c c u r r e n c e of

D j

F(Si)

-

absolute frequency of o c c u r r e n c e of Si F(S,/ Dj)

-

conditional frequency of Si given Dj F(Dj/ Si)

-

conditional frequency of

Dj

given

S,.

With definitions (8) and (9), extended statistical evaluations of h o w n medi- cal relationships or a s yet unidentified relationships can be carried o u t using data on patients with confirmed diagnoses.

4.4

The

diagnostic process 4.4.1 Symptoms

The symptoms of t h e p a t i e n t can be e n t e r e d into CADIAG-2 in t h r e e ways (described in detail in 191):

(i) by n a t u r a l language i n p u t of symptoms Si

(ii) by n a t u r a l language i n p u t of keywords t h a t trigger whole groups of s y m p t o m s Si

(iii) by accessing a d a t a base containing t h e patient's d a t a a n d transferring information via a fuzzy i n t e r p r e t e r .

Natural language i n p u t of symptoms Si s u c h a s "high fever", "increased GOT" or "blood stool positive" is achieved by a symptom s e a r c h algorithm with a n embedded word s e g m e n t a t i o n algorithm t h a t allows t h e use of synonyms and abbreviations, orthographic variants a n d different p a r t s of speech.

Input of keywords s u c h a s "present complaints", "previous complaints",

"blood count" a n d "ultrasonic" c a u s e s whole sections of t h e symptom t h e s a u r u s t o be displayed. Subsequently, fuzzy values c a n be linked with t h e s e symptoms by t h e physician.

The existence of a d a t a base which already contains t h e patient's s y m p t o m s suggests t h e a u t o m a t i c t r a n s f e r of information from t h e d a t a base t o CADIAG-2. During t h i s transfer, t h e d a t a i s passed through a fuzzy i n t e r p r e t e r which contains i n s t r u c t i o n s about t h e assignment of fuzzy values t o observa- tions, lab t e s t r e s u l t s a n d even simple alphanumeric texts.

After t h e p a t i e n t ' s s y m p t o m s have been collected, syrnptom-symptom inferences a r e p e r f o r m e d The s y m p t o m list contains all necessary i t e m s of data, including fuzzy value, origin (measured; inferred), predefined s y m p t o m class (routine; specially requested; invasive o r expensive), numerical value, u n i t s a n d d a t e of observation. The list of symptoms is t h e n checked for con- tradictions.

4.4.2 S y m p t o m combznntions

I n t e r m e d i a t e combinations of symptoms a r e evaluated in t h e next step.

Having passed t h e consistency check, fuzzy values for all symptom combina- tions a r e computed. The resulting lists a r e now a s complete a s possible a n d do n o t contain a n y contradictions.

4 . 4 . 3 C o n f i r m e d d i a g n o s e s

The fuzzy values p g

=

1.00, i.e., confirmed diagnoses

D,

for patient P q , a r e f

identified using t h e following equation:

4 . 4 . 4 Q c l u d e d d i a g n o s e s

The fuzzy values PD

=

0.00, i.e., excluded diagnoses

D,

for patient P q . a r e i

identified using:

Disease-disease relationships now allow t h e inference of f u r t h e r diagnoses (confirmed or excluded):

4 . 4 . 5 h s s i b l e d i a g n o s e s

Method 2. f u z z y values p g j s u c h t h a t 0.10 r 1 a 0.99 indicate possible diag- f

noses. These a r e d e t e r m i n e d a s foilows:

Method 2. Because the value p~ calculated by (13) is independent of t h e rules

j

t h a t c a n be used to define Dj, a powerful heuristic function i s introduced which considers t h e number of c r i t e r i a present which suggest but do not confirm disease Dj, and t h e n calculates t h e corresponding number of points PN The

D j ' values of PNDj a r e helpful in judging between t h e various possible &agnoses, although t h e ultimate aim should be t o obtain a confirmed diagnosis. The n u m b e r of points PN is calculated as follows:

D i

~ro,

=mall[@ R$D (Pq.Dj): pRjD(Pq.Dj); p (P .D - ) ] i f . RdB q I

where m' is t h e number of symptoms exhibited by t h e patient t h a t occur in t h e definition of

Dj,

and a

+ B =

1.00. We generally take

a =

0.09 and /3

=

0.91, i.e., t h e s t r e n g t h of confirmation h a s ten times more influence than t h e frequency of occurrence on the value of PN

Dl'

' O . l O r r R h ( ~ , , ~ , ) s 0.99 and/ or

0 . 1 0 g p (Pq . D j ) s 0.99 (13) R ~ D

and/ or

0 . 1 0 s p (P .D ) s 0 . 9 9 R P j

4 . 4 . 6 &planation o f diagnostic results

The physician's acceptance of CADlAG's diagnoses depends strongly on t h e ability of CADIAGZ t o explain its diagnostic output. On request, t h e information supporting confirmed diagnoses, excluded diagnoses and possible diagnoses is presented; this takes the form of t h e names of the medical entities, t h e i r definitions, their m e a s u r e d and fuzzy values, and t h e i r relationships to t h e diagnostic output.

4 . 4 . 7 Proposals for further ezarnination o f the patient

One of the main objectives of CADIAG-2 is t o provide iterative consultations, s t a r t i n g with simple, easy-to-examine and cheap data. A number of possible diagnoses can usually be inferred from t h e s e data, and further examinations a r e then necessary t o confirm or exclude these hypotheses. CADIAC-2 uses the medical information stored in its data bank to propose what form t h e s e further

examinations should take. The symptoms selected for f u r t h e r study a r e clearly those which would confirm or exclude a particular diagnosis.

4 . 4 . 8 h e z p l a i n e d symptoms

The confirmed diagnoses and any remaining possible diagnoses should together explain any pathological symptom, indication or lab test result of t h e patient. Unexplained d a t a (usually) indicates f u r t h e r diseases t h a t should be investigated.

5.1 Rheumatic diseases

CADIAG-~/RHEUMA has undergone partial t e s t s with data from patients a t a rheumatological hospital. A study of 169 patients with rheumatoid arthritis, Sjogren's disease, systemic lupus erythematodes, Reiter's disease or scleroder- m i a showed t h a t CADIAG-2 obtained t h e correct diagnosis in 77.16% of t h e cases considered. This figure was calculated by comparing t h e clinical diagnoses established by t h e consultant a t t h e rheumatological hospital (assumed to be c o r r e c t ) with t h e confirmed diagnoses made by CADIAGZ. Most of t h e cases in which clinical diagnoses could not be confirmed fell into two classes:

(i) The patient was in hospital only temporarily t o check t h e efficacy of drugs already administered

(ii) The patient was in t h e early stages of one of t h e rheumatic diseases con- sidered; in almost all of these cases a possible diagnosis was suggested.

5.2 Pancreatic diseases

CADIAG2/PANCREAS was t e s t e d with data from 31 patients. The final clini- cal diagnoses of t h e s e patients h a d not been confirmed by histological examina- tion, but were nevertheless assumed to be correct.

Pancreatic carcinoma was confirmed twice. Confirmation was aided by t h e existence of a r e s u l t "Specific abnormal pancreatic biopsy", which h a s a s t r e n g t h of confirmation p,

=

1.00 for pancreatic carcinoma.

Possible hypotheses were generated for t h e o t h e r cases, and the heuristi- cally determined ~ u n l b e r of polnts was taken a s t h e basis for evaluation. The results a r e given in Table 3.

Table 3. Comparison of CADIAGZ possible d a g n o s e s with t h e clinical diagnoses.

Clinical diagnosis Percentage

of cases

CADIAG diagnosis with highest number of points 50.0

CADIAG diagnosis with second highest number of points 21.4 CADIAG diagnosis with third highest number of points 10.8 CADIAG diagnosis with fourth highest number of points 7.0

No CADIAG diagnosis 10.8

The a u t h o r gratefully a c h o w l e d g e s t h e contributions of G. Kolarz, M.D., a n d W. Scheithauer, M.D., in t h e medical documentation of r h e u m a t i c a n d pan- creatic diseases.

R E F z R m C F s

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H.

Bossel, S. Klaczko a n d N. Miiller (Eds.), @ s t e m s

'/Reory in t h e Social Sciences. Birkhauser Verlag, Easel a n d S t u t t g a r t , 1976, pp. 202-282.

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Med. h f o m . 8 (1983) 173-186.

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5.

R.E.

Bellman a n d L.A Zadeh. Local a n d h z z y Logics. Memorandum No.

ERL-M584, Electronics Research Laboratory, College of Engineering, University of California, Berkeley 94720, 11 May 1976.

6.

K.-P.

Adlassnig. A survey on medical diagnosis and fuzzy subsets. In M.M.

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