" fightIlS

Date

 

This is to certify that the

thesis entitled

ASSESSING UNCERTAINTY IN MEDICAL
DIAGNOSIS BY FOUR PROBABILITY
MODELS

presented by

Raywin Rufus Huang

has been accepted towards fulfillment
of the requirements for
Department of

_P_ILLD___degree in mung ,Personnel
Services and

Educational Psychology
(Statistics & Research

Design
W, MM

Major professor

 

June 14, 1977

 

0-7639

ii

(3 Copyright by
RAYWIN RUFUS HUANG

1977

ASSESSING UNCERTAINTY IN MEDICAL DIAGNOSIS

BY FOUR PROBABILITY MODELS

BY

Raywin Rufus Huang

AN ABSTRACT OF A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Counseling, Personnel Services
and Educational Psychology

1977

ABSTRACT

ASSESSING UNCERTAINTY IN MEDICAL DIAGNOSIS
BY FOUR PROBABILITY MODELS

BY

Raywin Rufus Huang

Uncertainty plays a pernicious role in medical
diagnosis. This dissertation defines uncertainty as not

having knowledge of the relational structure of the disease

 

outcome and a set of symptoms in the true state of nature.
Conditional probability is used as the fundamental measure
of uncertainty. Four probability models, namely (1) the
Bayesian model, (2) the Binary Regression model, (3) the
Logistic Discrimination model, and (4) the Entropy Minimax
Pattern Discovery model, are presented as well as their
mathematical algorithms for generating the conditional
probability of a disease outcome given a set of symptoms.
An algorithm is also developed to simulate different classes
or levels of uncertainty within the structure of the diag-
nostic problem. Each model is applied to eachclass to
derive its parameters and each model is cross-validated

to an equivalent sample for the purposes of (l) determining
the stability of each model's estimated parameters in terms

of sensitivity, specificity, and predictive value, and (2)

Raywin Rufus Huang

to model the clinical situation where the physician is
cross-validating his set of strategies to new cases on the
basis of prior information. Each model is also evaluated in
terms of a utility function, losses and gains. Some special
classes of the uncertainty structure are also simulated and
each model is evaluated by the same methodology and with the
same evaluation indices. The models are then applied to
different relational structures and are then evaluated in
terms of sensitivity, specificity, predictive value and
utility function. These results are then compared to prior
findings. The findings of this dissertation are as follows:

1. Overall, sensitivity increases for all models as
the correlation with the disease outcome increases.

2. There is a "hump" or convex effect for sensitivity
for all models except the Bayesian (B), Bayesian with the
Bahadur's expansion (BB), and the Entropy Minimax Pattern
Discovery (EMPD) models, in situations where the symptoms
have a low correlation with the disease outcome. That is,
the maximum sensitivity is not when the intercorrelation
between the symptoms is greatest but when the symptoms are
moderately intercorrelated! This phenomenon did not appear
in situations where the symptoms have a high correlation
with the occurrence of the disease. In fact, sensitivity
increases as the intercorrelations increase under the

latter situation.

Raywin Rufus Huang

3. The values for sensitivity did not differ among
models in situations where highly interrelated symptoms
are also highly related to the occurrence of the disease.

In other words, when the relational structure is highly
correlated, it does not matter which model one uses if
sensitivity is chosen as a criterion for selection models.

4. The "pit" or concave effect of specificity across
binary regression models occurs when, given those situations
where the symptoms are highly correlated with the disease
outcome, the intercorrelations between the symptoms increase.
This means that specificity is at a minimum when the
symptoms are moderately related.

5. The "hump" or convex effect is also found for
predictive values in the same way as the sensitivity index,
that is, when the symptoms have a low correlation with the
occurrence of the disease.

6. With the presence of a suppressor symptom, it does
not matter what measure one uses as a criterion for select—
ing models as all models perform the same for all prediction
efficiency indices.

7. If a model is chosen with the criterion as having
the best sensitivity, it is at a cost of losing specificity
and vice versa. In other words, there are ng_models that
have the best of both indices for all classes considered in

this dissertation. The statement holds when one looks

Raywin Rufus Huang

across classes and within classes of problems. This also
means that there is no single model that performs consis-
tently better for each class or across classes in terms of
sensitivity and specificity.

8. A decision function analysis was performed.
Penalty (negative) weights were given for the two diagnostic
errors (i.e., Type I and Type II) and no credit was given to
the correct diagnosis. The binary least square model (BLS)
and the binary weighted least square model (BWLS) showed the
smallest loss when the symptoms had a low correlation with
the disease's occurrence but themselves had high intercor-
relations. However, when considering gains, with credits
given to the correct diagnosis, but the same penalty weights,
the Bayesian model (B) had the most gain when the intercor-
relations among the symptoms were 1ow but the correlation
between the symptoms and outcome was high. The logistic
discrimination model (LD) had the most gain when the symp-
toms had a low correlation with each other but had a high
correlation with the occurrence of the disease outcome.
The LD model also had the most gain when the symptoms were
moderately interrelated with each other and the symptoms
had a low correlation with the disease. If one disregards
the intercorrelation among symptoms, the LD model had the
highest gain whether the symptoms had a high or low corre-

lation with the occurrence of the disease. That is, the

Raywin Rufus Huang

best model to use to maximize gain in the absence of
knowledge about the relationship among and between symptoms
and disease outcomes, is the LD model.

Implications and applications of these findings to

diagnostic problem solving are also presented.

TO MY BELOVED WIFE, RITA

AND SON, WINSON

iii

ACKNOWLEDGMENTS

I would like to express my sincere and deepest
appreciation to the following individuals who have played
a major role in one way or the other in the making of this
dissertation.

To Dr. Maryellen McSweeney who has unselfishly offered
her time and effort throughout my graduate academic career
at Michigan State University. She has not only enriched my
academic experience but has also set me the example of a
person with modesty and humility.

To Dr. Howard Teitelbaum whose kindness expressed to
my account in numerous ways, has given me great encourage-
ment and moral support throughout my graduate career. He
has been more than simply an adviser and friend and I have
enjoyed tremendously and with great respect working with
him.

To Dr. Arthur Elstein for his many stimulating and
enjoyable discussions which brought much delightful inspi-
ration to this dissertation. He has led me to new levels
of intellectual capabilities.

To Dr. James Potchen who has offered me much valuable
advice in making this dissertation relevant to a real

clinical setting.

iv

To Dr. William Schmidt for his assistance in
statistical methodology of this dissertation which I
am most grateful.

To my beloved wife, Rita, whose love and patience
act as a buoy to the process of the making of this
dissertation.

To my father, Rufus Huang, who has given me great
moral encouragement and support throughout my academic
experience.

To Grace Rutherford who has so professionally typed
and prepared this manuscript.

And to my Lord Jesus Christ, who has provided me
the strength and will to overcome the hurdles towards
the making of this dissertation.

It is my hOpe that these efforts so kindly and
graciously offered to me by the above individuals will
not go in vain and I will see that their qualities will
be passed on to others in the manner they have been so

generously passed on to me.

TABLE OF CONTENTS

Page
LIST OF TABLES O O O O O O O 0 O O O O O O O 0 O O 0 ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . xii
Chapter
I. INTRODUCTION . . . . . . . . . . . . . . . . l
Uncertainty in Medicine . . . . . . . . . . 8
The Structure of Uncertainty . . . . . . . 11
Quantification of Uncertainty . . . . . . . 13
The Diagnostic Situation and the
Diagnostic Problem . . . . . . . . . . . 18
The Purpose and Strategy of the Study . . . 18
II. PROBABILITY MODELS . . . . . . . . . . . . . 22
Probabilistic Explanation . . . . . . . . . 22
Bayesian Model (B) . . . . . . . . . . 22
Pattern Recognition . . . . . . . . . . . . 28
Binary Regression . . . . . . . . 28
Ordinary Least Squares (BLS) . . . 28
Weighted Least Squares (BWLS) . . . 33
Ridge Regression (BR) . . . . . . 3S
Weighted Ridge Regression (BRWLS) . 36
Logistic Discrimination (LD) . . . . . 37
The Entropy Minimax Pattern Discovery
(EMPD) o o o o o o o o o o o o o o o o o 4 0
III 0 SIMULATION O O O O O O O O O O 0 O O O O O I 46
The Simulation Computer Routine . . . . . . 49
IV. DATA ANALYSIS . . . . . . . . . . . . . . . . 56
Special Classes . . . . . . . . . . . . . . 91
Clinical Application . . . . . . . . . . . 96

vi

Chapter Page

V. SUMMARY AND DISCUSSION . . . . . . . . . . . 101
Further Recommendations . . . . . . . . . . 105
Clinical Implications . . . . . . . . 106

Schema for Application of the Models
to a Diagnostic Problem . . . . . . . 108
An Example of Breast Cancer . . . . . . 114

Quantiative Models in Medical
Decision Making . . . . . . . . . . . 116
Appendix

A. THE RELATIONAL STRUCTURE OF THE POPULATION,
THE GENERATED SAMPLE AND THE SPLIT SAMPLE . . 120

B. THE ESTIMATED PARAMETERS FOR EACH
PROBABILITY MODEL FOR EACH CLASS . . . . . . 125

C. THE ESTIMATED DIAGNOSTIC PROBABILITIES FOR
EACH 29 PATTERN FOR EACH PROBABILITY MODEL

FOR EACH CLASS 0 O O I O O O O O O O O O O O 127

D. THE PREDICTIVE INDICES OF EACH MODEL FOR
EACH CLASS 0 o o o o o o o o o o o o o o o o 13 1

E. THE PREDICTIVE INDICES OF EACH MODEL WHEN
PREDICTING TO A DIFFERENT RELATIONAL
STRUCTURE C O O O O O O C O O O O O O O O O O 13 5

F. THE RELATIONAL STRUCTURE FOR EACH SPECIAL
CI‘ASS O O O O O O O O O C O O O I O O O O O O 140

G. THE ESTIMATED DIAGNOSTIC PROBABILITIES AND
PREDICTIVE INDICES FOR EACH MODEL WITHIN
EACH SPECIAL CLASS . . . . . . . . . . . . . 142
H. BRAIN SCAN EVALUATION QUESTIONNAIRE . . . . . 149

I. THE RELATIONAL STRUCTURE FOR THE BRAIN
SCAN STUDY 0 o o o o o o o o o o o o o o o o 153

J. THE RELATIONAL STRUCTURE FOR THE SPLIT
SAMPLES OF THE BRAIN SCAN STUDY . . . . . . . 154

vii

Appendix

OF THE SYMPTOMS ARE LOW .

OF THE SYMPTOMS ARE HIGH

K.
STUDY
L.
M.
BIBLIOGRAPHY

viii

THE ESTIMATED PARAMETERS AND PREDICTIVE
INDICES OF EACH MODEL FOR THE BRAIN SCAN

GRAPH FOR THE PREDICTIVE INDICES FOR EACH
PROBABILITY MODEL WHEN THE INTERCORRELATION

GRAPH FOR THE PREDICTIVE INDICES FOR EACH
PROBABILITY MODEL WHEN THE INTERCORRELATION

Page

155

156

159

162

LIST OF TABLES

Table Page

1.1 The Possible Distribution of Cases by Both
Disease and Symptom Outcome . . . . . . . . . . 14

1.2 The Frequency Distribution of Cases by
Disease Outcome and Symptomatic Patterns . . . 17

2.1 Summary of Probability Models . . . . . . . . . 45
3.1 The Simulated Structure of Uncertainty . . . . 53
4.1 Test of Fit for Sample Variance-Covariance

Matrices With the Specified Population

Variance-Covariance Matrices . . . . . . . . . 57

4.2 Test of Equivalence of Split-Samples
Variance-Covariance Matrices . . . . . . . . . 59

4.3 Test of Severity of Multicollinearity for
Sample I of Each Class . . . . . . . . . . . . 61

 

4.4 Exact Probabilities, P(D ii) for Each Class
of the First Sample . . . . . . . . . . . . . . 62

4.5 Comparison of Models in Terms of
Discrepancy Indices . . . . . . . . . . . . . . 65

4.6 Possible Outcome of a Diagnostic Decision . . . 68

4.7 The Class Where Each Model Has the Highest
and Lowest Predictive Indices . . . . . . . . . 75

4.8 Comparison Across Models Within Class . . . . . 77
4.9 Performances of Each Model in Terms of
Prediction Indices When the Symptoms Are

Lowly Correlated With the Disease and
Their Ranking . . . . . . . . . . . . . . . . . 78

ix

4.14

4.15

Performances of Each Model in Terms of
Prediction Indices When the Symptoms Are
Highly Correlated With the Disease and
Their Ranking . . . . . . . . . . . . . . .

Performances of Each Model in Terms of
Prediction Indices for Pooled Situation
and Their Ranking . . . . . . . . . . . . .

Performances of Models Relative to the
Predictive Indices Across Correlational
Patterns of Disease and Symptoms . . . . .

in Terms of Losses for Each
Each Class . . . . . . . . .

Utility Table
Model and for

in Terms of Gains for Each
EaCh Class C O O O O O O O 0

Utility Table
Model and for

The Best Model in Terms of Predictive
Efficiency Indices Under Different
Relational Structural Situation . . . . . .

The Performance of the Probabilistic Models
in Terms of Losses When Cross Validating to
a Different Relational Structure . . . . .

The Performance of the Probabilistic Models

in Terms of Gains When Cross Validating to
a Different Relational Structure . . . . .

Test of Equivalence for Special Classes . .

Test of Equivalence for Sub-Samples for
Special Classes . . . . . . . . . . . . . .

The Models That Perform Relatively the Best
in Terms of Predictive Indices . . . . . .

Loss Function for Various Models for Special

Classes 0 I O O O O O O O O O O O O O O I O

Gain Function for Various Models for Special

Classes 0 O O O I O O O O O O O I O O O O 0

Test for Equivalence and Multicollinearity

Page

79

80

81

84

86

88

90

90

92

93

93

95

95
98

5.2

Prediction Indices for Various Models for

Brain Scan . . . . . . .

Decision Function Values for Various Models

on Brain Scan . . . . . .

Decision Table in Choosing Models With
Respect to Prediction Index and Kind of
Cross-Validated Relational Structure

Final Decision by Clinical Intuition and

Quantitative Prediction .

xi

0

Page

99

100

104

108

Figure

LIST OF FIGURES

Page
The Relational Structure Between Attributes
and the Occurrence of an Outcome and Among
Attributes in the True State of Nature . . . . 3
Illustration of the Relationship of a
Single Symptom With Two Diseases . . . . . . . 6

Representation of the Physician's Prior
Knowledge of a Disease With Its Symptom . . . 10

The Structure of Uncertainty . . . . . . . . . 12

The Relational Structure of a Disease and
p symptom O O O O O O O O I O O O O O I O O O 47

The Covariance Structure of a Disease and
p symptoms 0 O O O O O O O O O I O O O O O O O 4 8

Possible Outcome of a Diagnostic Decision . . 68
Flow Diagram Indicating the Events Involved

in Completing a Single Brain Scan

Questionnaire . . . . . . . . . . . . . . . . 97

The Relationship Between Quantitative Model
Decision and Clinical Intuition . . . . . . . 106

Schema for Application of the Models to a
Diagnostic Problem . . . . . . . . . . . . . . 113

xii

CHAPTER I

INTRODUCTION

Uncertainty is a root of indecision. It is like a
disease to the process of decision making and it curtails
human performances. Swet (1961) found that performance in
signal detection by human subjects decreased as the amount
of uncertainty increased. Uncertainty, however, is defined
in many ways. Webster (1974) defined it as a quality or
state of being indefinite, indeterminate, problematical,
dubious and fitful. Bowman (1964) defined it as a situ-
ation characterized (either objectively or subjectively)

by incomplete predictability of alternative events.

 

Cohen (1973) distinguished two categories of

uncertainty, namely the intrinsic and the extrinsic.

 

 

Intrinsic uncertainties arise from imprecicion, ambiguity,
and limitation of the data on which the decision is to be
made. Extrinsic uncertainties refer to the failure on the
part of the data interpreter in translating the data,
otherwise known as "observer error." Kaplan (1964), on
the other hand, defined two different kinds of uncertainty.
One kind is £i§k_where there is a knowledge of a law that

operates in nature but involves a purely random element.

The outcome, despite a given probability, remains unassured.

The other kind of uncertainty is referred to as statistical

 

ignorance where the law of Operation itself is unknown.

 

Ignorance arises not necessarily because of non-specifiable
circumstances but rather because there is a lack of the
occurrence of enough significant outcomes so that deter-
ministic probabilities can be assigned to these outcomes.
Kaufmann (1968) classified levels of uncertainty according
to the degree of knowledge available. One level is £227

structural uncertainty, that is, when the states of the

 

system are unknown at any point in time. Structural uncer-

 

tainty occurs when the state of the system, despite being
known in general, is not known at any given time. The
condition when the states of the system with its laws of
probability are known at any time, is called chance.

Certainty is a state where the system is known and it

 

can be described at any point of time.
This dissertation defines uncertainty as not having

the knowledge of the structure of the relationship between

 

 

 

 

an outcome and certain sets of conditions and the inter-

 

relationships among_the attributes in the condition

 

 

 

(Figure 1.1). The term condition is referred to as a
universal space in which the attributes are its elements.
Predictors, indicators, attributes, independent variables,

and exogeneous variables will be used synonymously with

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symptoms and/or signs whereas dependent variable, the
criterion and endogeneous variables will be used synony-
mously with the disease outcome. The terms variables or
attributes will refer, in general, to both disease outcome
and symptoms. The term relational structure will refer to
the relations between the disease outcome and the symptoms
and the interrelations among the symptoms.

One underlying assumption of this definition of
uncertainty is that there exists a well defined and
structural deterministic relationship between an occurrence
of a disease and certain sets of conditions in the true
state of nature. This has two implications. One implying
that certain sets of conditions precede the disease and are

causative agents to the disease outcome. The other impli-

 

cation is the set of conditions are subsequent to the

 

disease outcome and are purely symptomatic in nature.

 

This stipulated definition of uncertainty also leads to

the formulation of the uncertainty principle which states

 

 

that only when full knowledge of this (true state of nature)

 

 

relational structure is obtained, can the outcome of any

 

 

diseases be stochastically predicted without error with

 

reference to the given known conditions. Hence, when only

 

 

partial or imperfect knowledge of this relational structure
is obtained, uncertainty arises and thus leads to random

guesses.

The paradox of this principle is that even when full
knowledge is gained, which demands the collection of
exhaustive information relating to the disease, the
prediction of a specific disease outcome is still subject
to error. This is due to the complex relations among
variables as illustrated in Figure 1.2. For instance,
symptom S is related to both disease D1 and D2 (denoted
by SD1 and SDZ)° The absence of the symptom, S, is also
related to the outcome of both of these diseases denoted
by SDl and SDZ. Adding to this complexity of relationship,
the presence of the symptom is not necessarily related to
the outcome of either Dl or D2 denoted by S51 and 852.
Hence, the presence or absence of the symptom, 8, could
not determine exactly the occurrence of either D1 or D2
for a single case. Error is, therefore, an inescapable
consequence. Nonetheless, this principle holds over a

large number of cases. That is, when the relational

structures are known, the prediction of the proportion

 

of cases having the disease will be without error. But
it should be noted that this is accomplished only when
full and perfect knowledge is obtained.

This imperfect and incomplete knowledge of the
relational structure is caused by the complexity of the
relational structure itself which leads to the difficulty

of obtaining this knowledge as Hammond et a1. (1975) noted:

Knowledge of the environment is difficult to
acquire because of casual ambiguity and because
of the probabilistic intangled relations among
environmental’variables. (emphasis mine)

 

In spite of this difficulty, partial knowledge of
the environment and its relational structure can still
be gained from samples from the complex state of nature.
These samples constitute imperfect information about the
universal relational structure. This sample information,
unfortunately, leads to inferences about the state of
nature in probabilistic terms such as "likely," probable,"
"perhaps," or "maybe" which constitute many human beliefs.
Inference of the true state of nature becomes then an art
of estimation. These probabilistic beliefs prompted

Tversky (1974) to describe uncertainty as an essential

 

element of the human condition. It should be noted that

 

prediction and inferences are used synonymously.

The ambiguity of the structural relationship of
the true state of nature constitutes uncertainty, and
this ambiguity is due to partial knowledge arising from
insufficient information. This, in turn, is primarily due
to the complexity of the true relational structure and
secondarily due to methodological limitations in obtaining

full and complete information about the structure.

Uncertainty in Medicine

 

Disease is defined as literally meaning "lack of ease"
or the pathological condition of the body that presents a
group of symptoms peculiar to it and which sets the condi-
tion apart as an abnormal entity differing from other normal
or pathological body states (Taber, 1970) and symptom simply
denotes the manifestation of the disease. Medical diagnosis
is then an art of identifying the correct disease with
reference to certain set of symptoms as Wakefield (1955)
remarked: "Diagnosis is the art and the science of
recognizing the presence of the absence of disease from
signs, symptoms. . . ."

The Dorland's Illustrated Medical Dictionary defined

 

diagnosis as the art of distinguishing one disease from
another.

In a different perspective, medical diagnosis is also
an art of probabilistic inferences or prediction. "Medicine
is a science of uncertainty and an art of probability," was
the dictum of Sir William Osler (Bean, 1950). Lusted (1968)
further remarked the following: "The uncertainty about the

correlation of signs, symptoms and disease makes medical

 

diagnosis a matter of probability."
Engel and Davis (1963) distinguished five orders
(levels) of medical uncertainty with variation of etiology

within each order. They are presented as follows:

1. Diagnosis of the First Order: the diagnostic
situation where the disease is considered to be
well defined and the etiology of the disease is,
in most instances, clear and the disease picture
does not vary much from person to person or from
environment to environment.

2. Diagnosis of the Second Order: diagnosis with
well defined etiology but the disease picture
has greater variability from patient to patient
and from environment to environment.

3. Diagnosis of the Third Order: the diagnosis is
clearly descriptive and the etiology is unknown.

4. Diagnosis of the Fourth Order: .the general type
of reaction is recognized but the specific cause
is not known and individual and environmental
variation occurs.

5. Diagnosis of the Fifth Order: the diagnosis is
based on the constellation of signs and symptoms
which comprise the disease picture. However,
the etiology of the disease is unknown.

This dissertation will consider Engel and Davis' last
order of diagnostic certainty.

Engel and Davis (1963) concluded their thesis by
stating the following:

Thus, inherent in every diagnosis is a factor

of uncertainty, greater in some and less in

others. These uncertainties are partially

related to our imperfections of knowledge

concerning health and disease7_ (emphasis
mine)

 

 

Consider the situation where a patient is at the
physician's office showing a set of symptoms or signs.
The physician has a prior knowledge of the disease as
represented in a disease-symptom matrix, something like

Figure 1.3, with p symptoms and N patients. The ones and

10

 

_ Condition
(Disease) “———_——‘
Outcome S
l 52 S3 0 Q 0 Sp
I
Patient 1 l ' l 0 0 l
I
l
Patient 2 0 : 0 l 1 0
l
I
I
l
I
l
. . ' . .
I
Patient N l : l O 1 l

 

Figure 1.3 Representation of the Physician's Prior Knowledge
of a Disease With Its Symptom.

zeroes represent the presence or absence of the disease or

symptoms. It is worthy to note that only two possible

 

disease outcomes will be considered in this dissertation,
namely, the presence of a disease, denoted by D and the
absence of the disease, denoted by 5, and that emphasis
is placed on discrete symptoms. Such form of diagnosis

is referred to as symptom diagnosis (Rinalde et al., 1963)

 

or a diagnosis of the fifth order. The physician, based
on this prior information can proceed with the medical

diagnosis in two possible ways. One way is by pgobabilistic

 

explanation and may be schematized as follows:

 

From the matrix, the probability for disease D
to have symptom or symptoms S is high. The
patient has symptom or symptoms S, (therefore
it is highly probable that) the patient has
disease D.

11

This is known as the laws pf ppobabilistic form

 

(Hempel, 1966); the explicans implies the explicandum
not with deductive certainty but with near certainty or
with high probability.

The other way is by pattern recognition: that is

 

to say, the selection of a number of possibilities which
come nearest to explaining the signs or symptoms. The
process of matching the disease with symptoms was noted

by Harvey and Bordley (1970). Alternatively, the physician
considers the process which enumerates in orderly fashion
the various diseases which give rise to particular signs

or symptoms .

These two methods represent two diagnostic paradigms
but the final diagnosis, by either method, is still
characterized in the form of "odds," "risk," and "chances."
Medical uncertainties undoubtedly play a detrimental role
in human welfare. It is a challenge to assess these

uncertainties in the hope of reducing them.
The Structure of Uncertainty

Since the relational structure in the true state of
nature is unknown, the main problem is how can "uncertainty"
be conceptualized such that it can be systematically and
formally investigated? The key to this problem is by

theoretically partitioning the relational structure into

12

possible exhaustive states. This is done by arbitrarily
dividing the degree of relationship among attributes into
categories and likewise the degree of relationship with
the disease. Figure 1.4 shows one way of partitioning
uncertainty into these possible classes.

The number of dividing levels is totally at the
discretion of the investigator. As the number of levels
increases, the structure of uncertainty is increasingly
defined. The mixture of the classes also constitutes

states of uncertainty.

Intercorrelation of the Symptoms

 

 

 

 

Low Medium High
Low I II III

Correlation with .
the Disease Medium IV VI VI
High VII VIII Ix

 

 

 

 

 

Figure 1.4 The Structure of Uncertainty.

Hence, uncertainty is "captured" into a well defined

bounded framework, making assessment possible.

l3

Quantifipation of Uncertainty

 

Bearing in mind with the above uncertainty structure,
a step is taken further to derive a quantitative measure.
Since the occurrence of any event cannot be determinis-
tically defined, the occurrence of any event can only be
stochastically derived. This means that with certain sets
of a known condition, the occurrence of an event appears
only n% of the time or n times out of a hundred. The
(lOO-n)% times that the event does not occur with relation
to the set of conditions is either due to imperfect or
partial knowledge from insufficient information or due
to error.

In deriving a quantified measure for uncertainty,
consider a disease, D, has n number of cases in a popu-
lation of size N. Assuming equally likely outcomes, the

probability of D occurring in this pOpulation is simply:
P(D)==n/N (1)

For a given sign or symptom, S, the probability that

D will occur conditioned upon the occurrence of S is:

P(DIS) = P(DnS)/P(S) (2)

14

where P(DnS) is the probability that both the disease and
the symptom will occur and P(S) is the probability that S
will occur in the population of size N. The probability,

P(DIS), is known as the conditional probability or posterior

 

 

probability. In this dissertation, it will be referred to

as diagnostic probability. Equation 2 can be elaborated by

 

the following 2 x 2 matrix as illustrated in Table 1.1:

Table 1.1

The Possible Distribution of Cases by Both
Disease and Symptom Outcome

 

 

Symptom S
l 0
Disease 1 n1 n2 4
2n=~
1
D 0 n3 n4 i=1

 

 

 

 

where n1 is the frequency or number of cases having the
symptom and the disease, n2 is the number of cases having
the disease but no symptom, n3 is the number of cases having
the symptom but not the disease, and n4 is the number of
cases not having either the symptom or the disease. Hence,
assuming equally likely events, the above probabilities can

be written as:

15

P(DnS) = nl/N (3)
P(D) = (nl+n2)/N (4)
P(S) = (n1-+n3)/N (5)

Hence, equation (2) can be written as follows:
P(DIS) = nl/(nl+n3) (6)

Extending to p number of signs or symptoms, the
probability that D will occur conditioned upon the

occurrence of the symptoms will be:

P(DISl,S .,Sp)==P(DnS nS n...nSp)/P(SlnS

1 2 n...nSp) (7)

2"' 2

where P(51“32”"'“Sp) is the probability that the symptoms
jointly occur. The left side of the term of equations (2)
and (7) can be interpreted as the probability of occurrence
of the disease given the occurrence of the symptom or p
symptoms. Let 5 and SE denote the absence of the disease
and the ith symptom, respectively. Then P(DIS) would be
the probability of the disease's not occurring given the
absence of the symptom. Likewise, for P(DISl,SZ,...,Si...Sp)
would be the probability of the disease's not occurring
given the occurrence of (p-l) symptoms and the absence

of the ith symptom. It is worthy to note that for p number

of signs or symptoms, there will be 2p number of possible

combinations or patterns. Let k=1...2p denote one of the

16

possible patterns and let ék denote the vector of the

pattern, then equation (7) can be rewritten as:

pmlggk) = P(Dn)_(k)/P(§k) (8)

With more than one symptom, the situation can be

presented as in Table 1.2. The probability that the disease

will occur given the 5k pattern and assuming equally likely

events is:

P(Dlxk) = mlk/(mlk-ImZR)

(9)
The conditional probability of the symptom(s) given
the disease, P(SID) or P(kuD) can be written for a single
symptom as:
P(SID) = P(SnD)/P(D) (10)
or in the case of p symptoms as:
P(xle) = P(xknD)/P(D) = mlk/Ml (11)

These probabilities are used to derive the diagnostic
probabilities with respect to the base rate of the disease
which will be presented in the following chapter. P(SID)
is also known as the likelihood probability.

The conditional probabilities derived from equations

(2) and (8) are exact probabilities. They are derived

directly from allocating observed cases according to the

disease outcome and the symptom pattern outcomes.

17

Table 1.2

The Frequency Distribution of Cases by Disease

Outcome and Symptomatic Patterns

 

 

 

 

 

Disease
Symptom Pattern D '5
51 = (Sl'SZ'°"'Sp) m11 m21
£2 = (Sllszl'°°lsp) mlz “122
5k ‘ (Sl'SZ'°"'sp) m1k m2k
5h = (Sl'SZ'°'°'Sp) 1h 2h
M1 M2

 

111..
13

3
H.
II

the total number of cases for the ith disease;

h = 2P where p is the number of symptoms.

the number of cases having the ith disease outcome
and the jth pattern;

and

18

The Diagnostic Situation and the
Diagnostic ProBIem

 

 

There are at least two paradigmatic ways of diagnosing
under uncertainty: (1) the probabilistic explanation and
(2) pattern recognition.

The two paradigms can be illustrated as follows.
Probabilistic explanation can be seen in a physician's
checking off the diseased and non-diseased cases in a
set of new patients based on his prior experience and the
manner used to integrate this information or the method
used for diagnosis. Pattern matching can be seen in the
attempt of a disease clinic to detect the high risk group
for a particular disease with respect to certain symptoms

or signs. This latter situation is known as mass screening.

 

One example of mass screening would be the detection of
breast cancer.
With reSpect to these two paradigms, the crucial

question to be raised is, which way i§_better? Better

 

will be considered in terms of diagnostic accuracy and
in terms of utility, losses or gains in dollars, or

mortality.

The Purpose and Strategy of the Study

 

The purpose of this dissertation is to answer this
question of diagnosing under uncertainty through quanti-

fication methodology. Quantitative methods or probability

19

models are chosen because they are a set of systematic
and formal procedures capable of deriving an Optimal
solution from a complicated entanglement of variables
in the true state of nature.

This dissertation will investigate the performance
of four different statistical models in assessing uncer-
tainty. The paradigms and associated probability models

are:

 

Paradigm Models

A. Probabilistic Bayesian
explanation

B. Pattern l. Binary Regression
recognition a. Ordinary Least Squares

b. ‘Weighted Least Squares
c. Ridge Regression
d. Weighted Ridge
2. Logistic Discrimination
3. Entropy Minimax Pattern
Discovery
The strategy of this study begins by simulating
the structure of uncertainty by a set of mathematical
algorithms shown in Figure 1.4. Each simulated class
with a fixed population is randomly divided into equal
halves, one representing the prior information available
and the other half representing the "unknown." Although

the two halves have the same relational structure statis-

tically, the second half is still referred to as the

20

"unknown." Statistical models are then applied to the
first sample to derive its parameters and these parameters
are used in turn to predict the outcome in the second
"unknown" sample. This latter procedure is known as

cross-validation. This step will assess the stability

 

of the estimates from a statistical point of view. It is
also analogous to the practice of medicine where a physician.
is constantly cross-validating his decision algorithms when
a conclusion is reached after examining two patients pre-
senting with the same sign and symptoms. How well each
model cross-validates is measured by a set of efficiency
indices. The values of each index will indicate the
accuracy and error of each model. Each model is also
evaluated in terms of utility or worth. The efficiency
indices and utility measures are then compared with each
other to determine the best model under different degrees
of uncertainty. The models will also be examined under
different relational structures between the two halves of
each pOpulation.

The above procedure will deal with the following
questions.

1. Which probability model has the best performances

in terms of efficiency indices and utility across

 

classes pf uncertainty?

21

2. Which probability model performs the best in terms

of efficiency indices and utility within each

 

class pf uncertainty?

 

The remainder of this dissertation is organized in the
following manner. Chapter II presents the description and
derivation of the probability models used for this study.
Chapter III derives and discusses the algorithm used to
generate the simulated data employed in deriving the
estimates for each model. Chapter IV presents the data
analysis and results of applying each probability model
for each degree of uncertainty. The analysis is in terms
of efficiency indices and utility functions. The models
are then cross-validated with the same and with different
relational structures between samples. Chapter V presents
general findings and recommendations for further research.
An implication of this study to decision making in a real

medical setting is also discussed in Chapter V.

CHAPTER II

PROBABILITY MODELS

The probability models which follow the two main
paradigms of medical diagnosis to derive the diagnostic

probabilities are selected in this thesis are as follows:

 

Paradigm Models

A. Probabilistic Bayesian
explanation

B. Pattern l. Binary Regression
recognition a. Ordinary Least Squares

b. Weighted Least Squares
c. Ridge Regression
d. Weighted Ridge

2. Logistic Discrimination

3. EntrOpy Minimax Pattern
Discovery

A description of each model within each paradigm is

now presented.

Probabilistic Explanation

 

Bayesian Model (B)

 

This model was originated by Rev. Thomas Bayes (1763)

and is simply formulated as:

22

23

P (D) P (sin)

_ _ 2.1
Punp(shn-+Punptshn ( )

P(DIS) =

where the probabilities are explained in the previous
chapter. In the situation of determining the diagnostic
probability when the symptom, S, is not present. The

probability becomes:

IND)PdflD)

 

P(Dls) = . (2.2)
Punp(§hn-+Pdip(§fii
However,
P(SID) = 1-P(sID) (2.3)
and
p(§|5) = 1=p(s|E) (2.3.1)

so that equation (2.2) can be rewritten as:

P(Dhn = P(D)Cl-P(Shfl) . (2.4)

pm) (1-P(s|D)) +p(‘6) (1 -p(s|1'5))
For convenience in computation, a new variable, a, is
defined to associate with the symptom. The new variable,
a, will take a value of 1 if the symptom is present and 0
if it is absent. Hence, equation (2.1) and equation (2.4)

are combined as follows:

Pans = Pm)mebn+%1-aHl-Pmbnl (25)

pm) (aP(SID)+(1-a) <1-p(s|o) I+PIBI (ap<s|3)+(1-a) (1—P(s|'5))

24

In extending to the situation of two symptoms,

equation (2.1) becomes:

Punp(s hnp(s hms )
P(nlslnsz)= 1 _? ¥_ __ (2.6)
P(D)P(SllD)P(SZIDnSl)+P(D)P(SllD)P(SZIDnsl)

 

and for p number of symptoms and letting Dl==D and D2==D,

the formula becomes:

P(D1)P(51|01)P(52|Dlnsl)...P(spln ns n...ns )

 

 

 

 

_ l 1 p-l
P(Dl|slnszn...nsp) — 2
i:lP(Di)P(sllni)P(szIninsl)...P(splninsln...nsp_1)
(2.7)
When the symptoms are independent, equation (2.7)
becomes:
p(D )P(s In )P(s In )...p(s In )
_ 1 1. 1 2 1. p 1.
P(Dllslnszn...nsp) — 2 (2.8)
'5 P‘Di’P(s1lDi’P(52|Di)"°P(Sp|Di)
1-1
or simply written as:
P
P(D1)jE1 p(sjlol)
P(Dllslnszn...nsp) = 2 p (2.9)
2P(D.) IIP(S,hk)
i=1 1 j=1 3

Similarly, when the symptoms are not present, the

diagnostic probability becomes:

25

 

 

p
P(Dl) jgl (l-P(Slel))
P(91I51“52“°°°“Sp) = 2 p I (2.10)
2 P(D.) H (1-p(s. 0.))
i=1 1 j=l 3 1

In combining equation (2.9) and equation (2.10), let

A denote the complete set of aj (i.e., A = {al,a2,...,ap}).
The diagnostic probability becomes:
P
P(Dl) H ((ajp(sjlol)-+(1-aj)(1-P(sjlnl)))
P(nllA) = 2 p (2.11)
aieA 2 PUL)H

1

In deriving the diagnostic probability for various
combinations of the symptoms, say 5k, the formula can be

written simply as:

P(D )P( In )
_ 1 5k 1
P(Dilék) ’ 2
.2 p(nin(§k|ni)

1=l

 

(2.12)

The Bayesian model has two underlying assumptions.
They are:
1. Independence among the symptoms--the occurrence of
one symptom is not related to other symptoms.

2. The set of diseases, D in this dissertation

it
i=1,2, is exhaustive and mutually exclusive—-

the diseases are distinct from each other.

26

In adjusting the model to correlated symptoms, Scheinok
(1972) proposed a solution by using the Bahadur's distribu-
tion (1961) and this is presented as follows. Let P(xk)
denote P(Dlxk). Then for p number of independent symptoms,

P(xk) becomes:

p Xi l‘Xi
P(xk) = H al (l-—a) (2.13)
i=1
=IN(§k)
where ai is the marginal probability or base rate for
symptom i and xi takes a value of either "1" or "0"
depending on the presence or absence of the symptom,
respectively. Let:
t
2.1 = (xi —ai)/(ai(l -ai)) (2014)

where i=1...p and 21 becomes the standardized variable for
the ith symptom with the following property:

2 ~ N(0, 1).
Then the following correlation parameters of second, third,

..., pth order can be defined as:

r.. = E (2.2.)
13 P 1]

rijh= Ep(zizjzh)

27

where Ep denotes expectation with respect to the

distribution of P(xk) which is presented as:

P(gkI = P' (5k)f(xk) (2.15)

where

z (2.16)

f(_)_gk) = 1 +2 ri,z.z.+ 2 rijkzizjzk+”'+2r1,2...nzlzz°°' p

i<j l 3 i<j<k

From Table 1.1, the probability that the pattern 5k given D

is estimated by:

P(Ele) = MIR/Ml = ck (2.17)

Bailey (1965) suggested an alternative estimator as follows:

P*(§k|D) = (mlk-I1)/(Ml-I2) = a; (2.18)

Then in estimating the correlations for D, the following is

used:
M1
rij = l/Mlg=1 zilzjfi
M1
rijk = 1/M1£:1 zizzjlzkz
M1

r12...p = l/Mliil 21222£°°°zp£

28

These correlation estimates are then substituted into
equation (2.16) and the probability estimates into equation
(2.14). P(xk) is derived and this is substituted into
equation (2.12) to obtain the desired diagnostic
probability.

These computations can become tedious even with a small
number of symptoms. Davies (1972), however, demonstrated
that with correlated symptoms, the diagnostic probability
is simply the proportion of patients having the disease,
out of the total having the symptom pattern, xk, which is
the exact formula of equation (9) in the previous chapter.

Further references on the Bayesian model and its
application can be found in studies by Warner et al. (1961),
Fraser and Franklin (1974), Overall et a1. (1963), Lusted
(1968), Vanderokas (1967), Barnoon and Wolfe (1972),
Cornfield (1967), Hall (1967), Parker (1967), Schmidt
(1971), and Gustfatason (1969).

Pattern Recognition

 

Binary Regression

Ordinary Least Squares (BLS). For p number of

 

symptoms, the binary regression model is formulated as

follows:

Y. = a+ 2 b.S,.+e. (2.19)
1 j=l j 1

29

where

Y. = the disease of the ith patient (i=1,...,N) and
takes on a value of one or zero depending on

the presence or absence of the disease outcome,

respectively;

a = constant term;

b. = regression weight or coefficient for the jth
symptom;

S.. = the jth symptom for the ith patient (j=l,...,p)
and it takes on a value of one or zero depending
on the presence or absence of the symptom for the
ith patient, respectively; and

e. = random error of the ith patient with the
following property:

e ~ N(0,02).

The matrix formulation of the binary regression model is

as follows:
Y = $3 + E (2.HL1)

where

Y = vector of disease outcome of ones and zeroes of
dimensions N x l;

S = matrix of symptoms with ones or zeroes of
dimensions N x (p-+l) including the constant
term;

B = vector of regression parameters, including the
constant term, of dimensions (p-+1) x l; and

E = vector of random errors of dimensions N x l with
the following property;

E ~ N(0, 102)

30

The objective of the binary regression model is to
minimize the sum of squared errors, E'E. This is done by

reformulating equation (2.19.1) as:

E = Y - SB. (2.19.2)

Premultiplying both sides of equation (2.19.2) by its
irrespective transpose to get positive squared errors,

the above equation becomes:

E'E (Y-SB)'(Y-SB)

Y'Y-2Y'SB+B'S'SB (2.19.3)

Differentiating equation (2.19.3) with respect to B and
setting the resultant matrix equation equal to 0 yields

BE'E

TBS—'- = -zs'y +ZS'SB = 0 (2.19.4)

Replacing B by p to denote an estimated parameter, the

previous step provides the normal equations as follows:

 

(S'S)p = s'y (2.19.5)

If the p normal equations are independent, (S'S) is non—

singular, and its inverse, (S'S)-1, exists. Then equation

(2.19.5) can be rewritten as:

g = (s'S)'ls'Y (2.19.6)

31

The solution g has the following properties:

1. It is an estimate of B which minimizes the sums
of squares error E'E irrespective of any distri-
butional prOperties of the errors.

2. The elements of p_are linear functions of the

observations Y1, Y2,...,Y and provide unbiased

n
estimates of the elements of B, irrespective of
distributional properties of the errors.

The procedure from equation (2.19.2) to the derived estimate

2 in equation (2.19.6) is known as the ordinary least

 

sguares procedure. Since both the disease outcome and
symptoms take on value of ones or zeroes, Haase (1976)
found that the prediction outcome (2) which is derived

from (2.19.1) as:

§==s p
or

I = b0.+s*2f
where:

S* = a N x p matrix of symptoms;
b = estimated constant term; and

b* = a pxl vector of estimated regression weights
without the constant term,

is in fact the diagnostic probability. Hence, for a given

pattern, say xk, the diagnostic probability is given by:

mung) = b0+b_*§k . (2.20)

32

It is very important to note at this point that the
estimated diagnostic probability's value is highly
dependent upon the values of the estimated regression
weights. These weights are derived from the matrix (5'8)-1
and vector S'Y.

The ordinary least squares solution to the binary
regression model is inappropriate when the errors have
unequal variances or are intercorrelated. In the former
case, the variance-covariance matrix of the errors is a
diagonal matrix with unequal diagonal elements. In the
latter case, the off-diagonal elements of the variance-
covariance matrix are non—zero values so that the matrix
is still symmetric but no longer diagonal. Weighted least
squares with a properly estimated variance-covariance matrix
can be used to correct for either or both of these problems.

For cases in which the observations are highly inter-
correlated, or multicollinear, the matrix (S'S) approaches
singularity. In such situations, the variances of the
estimated regression weights become highly unstable,
resulting in a highly unstable binary regressiOn equation
which is sensitive to changes in the data set. Ridge
regression is an appropriate technique to use in this
situation.

In some instances the estimated diagnostic probability,

P(Dlxk), can become greater than one, an overestimate, or

33

become less than zero, an underestimate for all three
forms of binary regression. As noted earlier, the values
of the estimated diagnostic probability depend on the values
of the estimated regression weights which are also subject
to underestimation and overestimation. In either case, the
diagnostic probability is reset to one if it is greater than
one and reset to zero if it is less than zero.

The description of the two forms of solution to the
ordinary least square binary regression will be presented
in the following two sections. Further reference on the
ordinary least squares binary regression can be found in
Draper and Smith (1966), Cohen and Cohen (1975) and
Kerlinger and Pedhazur (1973). References on the appli-
cation of the binary regression model in medicine can be
found in Feldstein (1966) and Elwood and Mackenzie (1971).

Weighted Least Squares (BWLS). As stated earlier,

 

when the error variance is heterogeneous, it is necessary
to amend the estimation procedure by using the weighted
least squares method. The key step is to transform the

outcome Yi' to a new variable, 2 such that it satisfies

it
the condition of homogeneous error variances. The new

transformed model becomes:

2 = SQ+F (2.22)

34

where

Z = vector of new transformed observation of
dimensions (N x 1);

Q = vector of new weights transformed from vector B
of (2.19.1). This vector has dimensions
((p+l) X l);

S = matrix of symptoms of ones and zeroes with
dimensions (N){(p+1); and

F = vector of errors of the new transformed model
of dimensions (N)(l) with the following
properties:

F ~ N(0, I02)

Computationally, the weighted least squares can be
performed by the two stage method (Neter and Wasserman,
1974) and is described in the following manner:

1. obtain the estimated outcome (§i) for each patient

or case by the ordinary least squares method; and

2. define a new variable for each case, wi, by:
wi = 1/(§i(1 -i?i))
obtaining a diagonal matrix, W, of rank (N){N).

The new regression weights or the "weighted" regression

weights, g, become:

g_= (s‘ws>'ls'wy (2.23.1)

where g is the estimate of Q.
Hence, the estimated diagnostic probability for pattern,

xk, for the weighted least squares model is given by:

35

P(Dlxk) = qO-Igka (2.24)

where

q0 = the constant term estimated by the weighted
regression model; and

g* = vector of new regression weights estimated by
the weighted regression model.

Ridge Regression (BR). In the situation of highly

 

correlated symptoms, the variances of the regression
estimates become highly unstable when derived by the
ordinary least squares procedure. The estimated values

of the regression weights will change with slight changes
in the data set. Thus, there is difficulty in determining
the contribution of each symptom to the outcome of the

disease. Therefore, it is necessary to stabilize the

 

variance of the regression estimates. One technique is

by the ridge regression method (Hoerl, 1964, 1970a, 1970b;
Marquardt & Snee, 1975). The method consists of adding a
constant term, c where c lies between 0.1 and 1.0, to the

estimation procedure,

y = (s's+c:)’1 5'? (2.24.1)

and at a certain point of c, say c*, the variance of the

estimated regression weights become stabilized. That is,
let V(b;.) be the variance of the ith estimate regression
weight at point cj and e be a predefined amount of change

such that the following condition holds:

36

{V(b?, )-v(b?.)}se .
+1 1]

I]

Then when the variance of estimated regression weights is
plotted against the various values of c, the following
properties will be found:

1. at a certain value of c, say c*, the variances of
the regression weights will all stabilize and have
the general characteristics of an orthogonal
system;

2. the weights will not have unreasonable values with
respect to the symptom for which they represent
rates of change; and

3. any weights with apparently incorrect signs at
c==0 will have changed to have the proper signs.

The curve connecting the points for all values of c is

known as the ridge trace, and the above properties not only

 

hold for a single estimate for a single symptom but for all
p estimates.

Weighted Ridge Regression (BRWLS). This model

 

combines ridge regression and the weighted least squares
method. The resulting weighted ridge coefficients or
weights are substituted into the first stage of weighted
least squares instead of the ordinary least squares regres-
sion weights. The second stage of the weighted least

squares procedure remains the same. This model is intended

37

to correct both for heterogenous error variances and highly

interrelated symptoms.

Logistic Discrimination (LD)

 

A second pattern recognition method maximizes the
relationship between the presence or absence of the disease
and a linear combination of the symptoms. This method was
develOped by R. A. Fisher (1936) and is commonly known as

linear discriminant analysis. The description of this

 

technique can be found in Morrison (1967), Tatsuoka (1971),
Van de Geer (1971), Timm (1974), and Bock (1975).
Conventional discriminant analysis only applies to
variables that are continuous, so when the variables are
dichotomous in nature it becomes inappropriate. Anderson

(1972a, 1972b, 1973, 1974) proposed logistic discrimination

 

which was originally introduced by Cox (1966) as a solution
to this problem. Essentially, the mathematical representa-
tion of the model is equivalent to the binary regression

model which can be presented as follows:

Y=SB+E

where

a vector of observations or disease outcomes with
ones and zeroes of dimensions (N x 1):

Y

S = a matrix of symptoms of ones and zeroes of dimen-
sions (N1c(p+l)), including the constant term;

38

B = vector of discriminant weights of dimension
(p+1xl); and
E = vector of random error of dimensions (N)rl).

However, the algorithm in deriving the estimated
discriminant weights is different from that used in the
binary regression model. In developing the mathematical
algorithm for this model, let aij represent the ith row

and jth column of the matrix S. Then thu diagnostic

probability of an outcome, say D, is given by (Cox, 1970):

aiB
e
P(Dls) =-————-— (2.25)
l + ealiB
and its complement is:
p(BlS) = ———l—7§- (2.26)
l + eal

The above two equations can be rewritten as the log odds

ratio as:

A = log ——-l—P(D S) = a.B . (2.27)

i e P(Bls) 1

Then the likelihood of Y1, Y2""’YN independent dichotomous

outcomes is:

 

 

N p

H eaiB exp( 2 Bsts)

1:1 = N s=1 (2.28)
n (1+eaiB) H (1+eaiB)

i=1 i=1

39

where

Thus, the log likelihood of the above equation becomes

N

P a:B
L(B) = 2 BsTs - 2 log(l+e 1 ) (2.30)
s=l i=1

The solutions for the estimated parameters are derived by

the Newton-Raphson iterative numerical procedure as

described by Bock (Bose, 1970) which is illuatrated

as follows: Let e be a criterion value to step iteration
Pi be the value of the parameters at the ith iteration;

Ai be the increment value at the ith iteration

then

Ai = -(I)-1F (2.31)

where I is the matrix of the second derivative and F is the
vector of first derivatives of equation (2.30) with respect
to B (Cox, 1970). The increment is, therefore,simply the
product of the negative of the inverse of the second deriva—
tive and the vector of the first derivatives. The values

for the parameters at the (i-+l)th iteration are:

P. = F.+~A.; (2.xn
1+1 1 1

40

the iteration stops when the vector of F is less than or
equal to e (i.e., F se), and the final I is the solution
for the estimated discriminant weights B. These estimated
weights are then substituted into equation (2.25) to derive
the estimated diagnostic probabilities.
There are two key assumptions to this model. They are:
l. the populations, the diseased and the non-diseased

populations, are multivariate normal with equal

 

variance-covariance matrices; and

2. the populations are multivariate independent and

dichotomous in nature.

Other techniques for deriving the solution besides the
Newton-Raphson solution can be found in Walker and Duncan
(1967) and Jones (1975).

Application of this model to medicine can be found in
Truett et a1. (1967), Halperin et a1. (1971) and Hartz and

Rosenberg (1975).

The EntrOpy Minimax Pattern Discovery (EMPD)

 

The term entrOpy refers to the statistical measure of
uncertainty. This method was developed from information
theory by Christenson (1967, 1968, 1972, and 1973). The
key concept of this method is to define symptom subsets
that are capable of acting as predictors of a disease

outcome. If the presence or absence of a symptom

41

contributes significantly to a change in the probability

of a given outcome, it will be classified as a determinant
of the outcome of the disease. The term determinant does
not imply deterministic or causative in nature. The purpose
of this model is to minimize uncertainty. The model assumes
that the measurement of uncertainty has the following
properties:

1. uncertainty is a continuous function of the
probabilities of various outcomes;

2. greater relative weights are given to occurrence
of rare events than to occurrence of common events
because rare events convey more information than
events that agree with previous prediction; and

3. additivity-—the uncertainty associated with two
or more independent sources is just the algebraic
sum of uncertainty associated with each taken
separately.

Given the above properties and give n possible out-
comes, each with probability of occurrence Pn' Shannon and
Weaver (1964) postulated the measure for the average infor-
mation per outcome for the discrete case is

n
H = E(-logzP(x)) = - 1:1 PilogzPi (2.33)

where

42

The function H is maximized when Pi = l/n for all i.
To derive the maximum of the above equation, take the

derivative of H with respect to Pi

3H

——-= —(logze-IlogZPi)-+(1og2e-+1ogZPn) (2.34)
8Pi = -1092(Pi/Pn)
Setting
3H _ 0
5Pi '

equation (2) becomes

n
H = - 2 (l/n)logz(l/n) = logzn (2.35)

max i"- l

The attributes can be partitioned into cells and can be
repartitioned into sets of disjointed cells whose sum fill
the space. This repartitioning of cells is referred to as
screening. Hence, the probability of an outcome for a given

cell and jth screening is given by

P(Dlith cell, jth screening) = PDIij
The measure of uncertainty for the ith cell and jth

screening becomes

H.. = -£ PDIij logzPD

13 13

43

Summing across outcomes, the measure of uncertainty for

the jth cell is

 

k D
H, = - 2 P.. 2 P .. log P .. (k==no. of cells) (2.36)
3 i=1 1] d=l dIIj 2 dlij
where
n..-+u..
P = 1] 13 .
ij n-tu. '
J
P = nijk‘+wijk
dlij nij+wij '
n = total number of events in the sample;
ni'k = number of events with outcome D in the ith
3 cell and jth screening;
ni. = total number of events across D for the ith
3 cell and jth screening;
wijk = theoretical number of outcome event; and

1 = the total sum of theoretical event for both
3 happening and non-happening outcome.

The ratios have the following meanings:

—$lE-= priori probability of the D outcome in the
wij jth cell; and

= priori probability of finding even in the
j ith cell.

44

The measure Hj determines how successful the
information is in separating the outcome into individual
cells in the feature space (i.e., the amount by which the
screening has reduced the average uncertainty in predicting
an outcome given a set of attributes).

The "best" screening that partitioned the feature
space is the one that minimized uncertainty or entropy.
The final results are the probabilities of outcome for
various patterns of attributes.

These models may be summarized in Table 2.1.

The following chapter will present the theoretical
foundation and the algorithm to simulate each individual
class of the uncertainty structure when the levels of the

correlation have been predefined as presented in Figure 1.4.

45

 

 

 

 

 

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xcEficwe Smouucm .m
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21 +928 w+ mm u 883“ 83:38: .8
0+ mamw+ mm H cofimmmumwu 0393 .o
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o mmucsom
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cofluwcmoomu
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m>amsaoxm adamsuse coflmcmmxm m.uscccmm
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a a and
w>Hmsaoxo haamsuss .AQVmA.Q_me H
mvcm w>Humgmcxm mHm mmmmmwimmu 0:9 oN N " COHUMCMmem
unoccmmoccfl xaamsuse mum meoumewm .H Aaovmflao_wvm amamoamm vaunwawncnoum .4
mmfluummoum\mcoflumEdmm< Ammo: mEmz emwcmumm

 

 

maopoz muwHwnmnoum no kHMEEdm

H.N OHDMB

CHAPTER III

SIMULATION

Prior to simulating the classes in Figure 1.3, consider
the situation for a single symptom. The probability that it
will have 111 number of occurrences in a pOpulation size of

N is simply:

P(n ) = ——§-'——P“1 (1-p)n2 (3.1)
1 n1! n2!

where N = nl-I-n2 and where P is the marginal proportion of

the symptom. This is known as the binomial distribution

 

(Hasting and Peacock, 1975). It is noted that the values
of n1 can be greater or equal to zero and less than or
equal to N. Extending this to p number of mutually exclu-
sive symptoms, the joint distribution of the symptoms,

, where n. is the number of occurrences for the

P J
jth symptom with marginal pr0portion Pj' is (Johnson, 1969)

nlpnzpoo on

as follows:

P
n°
P(n1,n ..,n ) = N! H (Pj J/nj!) (3.2)

'0
2 p j=l

where n.j 20 and N = 3 nj. This distribution is known as
i=1
the multinomial distribution.

 

46

47

Since the symptoms are not necessarily mutually
exclusive, attention should be given to their intercorre-
lation and also the correlation of each with the occurrence
of the disease. This can be represented in the following
matrix (Figure 3.1), Ri' where i denotes the ith class as

presented in Figure 1.3:

U
U)
(I)
O
U)
U)
U)

 

 

1 2 j 3' p
I
l
D
:
S -----| ------------------- —I
1 I
I
S2 I
I
° I
' I
' l
I = R
8j rs.d I rs.s 1
3 J j'
’ I
' l
° I
S I
P I

Figure 3.1 The Relational Structure of a Disease and p Symptom.

where rds. is the correlation between the disease outcome
3

and the jth symptom and r (j fj'), is the intercor-

. o I
835].

relation between the jth and j'th symptom. Since the
disease and the symptoms are dichotomous with only ones

and zeroes, denoting their presence and absence,

48

respectively, the correlations are phi-coefficients. In
terms of probability, this coefficient can be represented

as follows:

1
_ _ - _ _ ’2
rdsj ‘ ¢phi - (P(DnSj) P(D)P(Sj))/((P(D)(l P(D))(P(Sj)(l P(Sj)) (3.3)

Given the marginal proportions or base rates of the

disease and the symptoms, Pd and Ps-' respectively, the
3
above correlation matrix is reformulated into a variance

and covariance matrix, I, as in Figure 3.2:

D s s . . . s.s. ... S
I 1 2 33' P
I
D I
add I ads
|
--..— -.- ——————————————————— J
I S;
S1 ' “
\
32 5“ ‘.‘
I \\ ‘\ a
. l ‘ ‘\ 5.5.
o ' \‘ ‘\ J].
I \\ \‘
o I \ \
I \\ ‘s
S. \ 4‘ =
J : \?S.S_ \ $1
_ .3 J “
I ‘ \
o ' ‘\ \\J
\
o ' \
\
s 1 ‘.
P I I

 

 

Figure 3.2 The Covariance Structure of a Disease and p Symptoms.

49

where
add = Pd(1-Pd);
a = P (l-P ); and
. . S
5353 3 J
*5
ads ‘ rds.(addas.s.) ' and
J J J
a r ( )%
- a a o

S. S. . S S.

383' 353' S] 353'

It should be noted that a term of the form, P(l-P), is the

variance of the disease or the symptom.

The Simulation Computer Routine

 

A computer program was written by Scheifly (1974), to
generate a multivariate continuous distribution with a given
mean vector and variance—covariance matrix. In modifying
the program to generate the multinomial distribution, the
steps comprising this generation are as follows:

1. Generation of independent random variables which

are uniformly distributed between zero and one.

2. The generated uniform variates are then combined

to form normal deviates with zero means and with

the identity matrix for the covariance matrix.

50

3. The normal deviates are then generally transformed
to obtain the desired structure of means and
variance-covariance structure.

4. The resultant matrix is then transformed back into
probability terms and each variate is assigned a
one or zero according to whether the probability
is greater or less than the marginal probability
of that variable.

The following description elaborates the above steps.

The uniform random variate is generated by the mixed

congruential method (Mihram, 1972). This technique can be

represented by the following equation:

U = (aUk_

k -+c) (mod m) k=l,2,...

1

where a and c are constants, Uk is the kth recursion, and
U0 is known as the seed set in the initial recursion. The
residual is then divided by P. The values of a, c, and P
are chosen as to maximize the period of the generator that
produce numbers which behave as if they are random. In
terms of these three constants, the kth pseudorandom

variate in the sequence is given by

Uk = akU0-+c(ak-l)(a-l) (mod m) (3.5)

51

The generated sequence of uniform variates are then
converted to normal variates by the Teichroew's technique
(Knuth, 1968) which is an approximation of the inverse of
the probability function for the standard normal distribu-
tion. His procedure generates 12 independent random
variables, U1,U2,...,U12, uniformly distributed between

zero and one. Then, R is defined as follows:

R = (Ul-IU -I...-+U12-6)/4. (3.6)

2

The normal variate, z, is then approximated by

z = ((((alR2+a2)R2+a3)R2+a4)R2+a5)R (3.7)
where

a1 = .029899776;

a2 = .008355968;

a3 = .076542912;

a4 = .252408784; and

3.949846138.

This 2 is only a point in the N x p matrix. In order to
obtain the total entries of the matrix, this 2 is generated
N x p times. The result matrix is z of dimensions N x p.
In transforming the matrix 2 to the desired matrix, X,
which is distributed with the given mean vector, p, and
variance-covariance matrix, I, the following linear

transformation is necessary:

52

X=TZ+E
N'P
where
TT' = I (T is the cholesky factor of I).

In transforming the entries of the matrix X into

discrete entries, the following procedure is used:
2.. = (xij--ui)/Ji (3.9)

where ui is the given mean and Ji is the ith given standard
deviation. The new variable, zij’ is converted into a
probability or the area under the standard normal curve

by numerical approximation according to the following

equation:
*
2 . * 2
P(zg.) =/ 13 l - 81:213. dz (3.10)
3 (21!)

By the rejection method (Hasting and Peacock, 1975),
the entry on the ith column and jth row is assigned a zero

or one according to the following rule:

1 if 1>(z‘!",)<Pi
y = 13 (3.11)

o if p(z‘.'.) >p.
13 1

where Pi is the given marginal proportion of the ith symptom

or the disease.

53

In this dissertation, the marginal proportion or the
base rate for the disease is set at 0.2 and the marginal
prOportions for the symptoms, of which there are three, is
set at 0.5. Given such marginal prOportions, the maximum
positive correlation between symptoms and disease is 0.5
and among symptoms is 1.0. The number of cases, N, is
set at 300.

Given the above, Table 3.1 represents the partitioning
of uncertainty. It must be noted carefully that this table
is a reformulation of Figure 1.4 (page 10). Because the
maximum correlation between symptoms and the disease is
0.5, the medium and the high categories will be absorbed

under the label "High."

Table 3.1

The Simulated Structure of Uncertainty

 

 

Intercorrelations of the Symptoms

 

 

Low Medium High
0.00-0.30 0.31-0.50 0.51-1.00
Low I II III
Correlation 0.00-0.30
with the
disease High IV V VI

0.31-0.50

54

It should be noted that the resultant matrix of Y*
of zero and one entries is generated from an underlying
continuous distribution and the correlations computed

from his matrix are in fact tetrachoric correlations.

 

The relationship between the phi-coefficient, pij' and

the tetrachoric coefficient, pij, between the ith and jth
symptom, is developed by Pearson (1900) and cited by Lord

and Novick (1974) as:

Oio'pi' 1 1
= . _ .2 __ 2 _ 2 _ . 3
(vi) (qu pij+2 Yinpij+6 ”i 1’ ”j “913'

'9

2L 2_ 2_ .
-+24 Yin(Yi 3)(Y_j 3’pij

...L... “_ 2 Io_ 2 '5
4-120 (Yi 6Yi+3)(Yj 6Yj-I3)pij-+... (3.12)

where Y1 and Yj are the cutoff points for the ith and jth

variable, respectively, and oi and oj are its standard

deviations. In the special case where Pi = Pj = 0.5, the
relationship is simplified as:
pij = sin ("Dij/Z) (3.13)

The following chapter will present the analysis on the
generated samples by the above simulation algorithm. The
analysis will include (1) the statistical test of equivalence
between the generated sample and the predefined population

for each individual class in the uncertainty structure,

55

(2) the statistical test of equivalence between the
randomly split samples for each individual class in the
uncertainty structure, (3) the statistical test of severity
of multicollinearity for each generated class, (4) the
evaluation of each probabilistic model in terms of dis-
crepancy indices, (5) the evaluation of each probabilistic
model in terms of prediction indices, (6) the evaluation

of relative performance of each probabilistic model in
terms of utility functions, losses, and gains, (7) the
evaluation of relative performance of each probabilistic
model for three special classes of the uncertainty structure
with the same evaluation indices and utility function, and

finally (8) an application to a set of real data.

CHAPTER IV

DATA ANALYS I S

The population correlation matrix, RP' and the
variance-covariance matrix, Ip, are defined and shown
in Appendix A. The sample variance-covariance matrices,
$3, and correlation matrices, Rs' were then obtained by
the generation routine described in Chapter III, also shown
in Appendix A. The sample variance-covariance matrices
are then tested to determine if they are statistically
equivalent to the population matrices. This procedure

translates into testing the following hypothesis:

The test statistic used (Morrison, 1976) is as follows:

--1
L = v(1oge|th-1oge|tsl+tr tstp -p) (4.1)-

where p is the number of symptoms plus the disease outcome
and v is equal to (N-l) where N is the population size.

The test statistic, L, is distributed as a chi-square

56

57

variate with k(p(p-+1)) degrees of freedom if the null

hypothesis is true.

suggested the scaled

statistic as:

For moderate N, Bartlett (1954) has

 

 

 

1
6 (MD (2p+1 2/(p+1))}L (4.1.1)
as an improvement on the chi-square approximation. The
results of the tests are presented in Table 4.1
Table 4.1
Test of Fit for Sample Variance-Covariance
Matrices With the Specified Population
Variance-Covariance Matrices
(N = 300; p = 4)
Significance
Probability
Class L L' df P
I 11.03 10.98 10 .50
II 10.32 10.27 10 .50
III 11.05 11.00 10 .50
IV 9.97 9.93 10 .50
V 10.48 10.43 10 .50
VI 12.47 12.41 10 .25

 

*Significant at the 0.5 level.

58

From these results, the sample variance-covariance
matrices are not significantly different from the specified
population variance-covariance matrices.

Each class of 300 cases was then shuffled and randomly
divided into two equal sub-samples of 150 each, called
Sample I and Sample II. The resultant variance-covariance
matrices of the "split" samples for each class are also

shown in Appendix A. The hypothesis tested becomes:

The test statistic used (Morrison, 1976) is as follows:

2 2
M = :1 ni loge |tp,| -1121 ni ti (4.1.2)

where n1 is equal to (Ni -l), Ni is the sample size of the
ith sample and tp, is the pooled matrix of Ilanuitz. Box

(1949) has found that if the scale factor,

 

 

2p2+3p-1 1 1
c=1- —- , 4.1.3
6(P+1) (2 n1 ) ( )

2
i=1 3 n

59

is introduced into equation (4.1.2) (i.e., G)(M), GM is
approximately distributed as a chi—square variae with
degrees of freedom equal to k(p(p-+l)). The results of
the tests for equivalence between split samples are

presented in Table 4.2.

Table 4.2

Test of Equivalence of Split-Samples
Variance-Covariance Matrices

 

 

 

(N1 = N2 = 150; p = 4)

SignifIcance

Probability
Class GM df P
I 1.39 10 .99
II 3.39 10 .99
III 6.44 10 .75
IV 1.55 10 .99
V 10.29 10 .50
VI 2.02 10 .99

 

Again the split-half samples for each class show no
statistical differences at the 0.5 level of significance.
From these results, it can be concluded that the similar—
ities between the specified pOpulation and the sample and

between split samples are statistically assured.

60

Before proceeding further into the analysis, the first
sample, Sample I, of each class is tested for the severity
of multicollinearity (i.e., highly interrelated symptoms).
This is achieved by testing the following hypothesis:

HO: IS'SI = 1

against the alternative:
H : Is's|< 1

where S is the matrix of symptoms of dimension (p)(p).
If S is a standardized matrix, then (S'S) will be a
correlational matrix and the testing hypothesis can be

reformulated as:

HO: (S'S) = I

against the alternative:

H1:(S'S)7‘I

where I is an identity matrix of the same rank. Barlett
(1950) formulated the following test statistic:

A = -((N-—1)-1/6(2pI-5))1oge|s's|

where A distributed as a chi-square variate with degrees
of freedom 8(p(p'-l)). The results for testing for

multicollinearity for each class are presented in Table 4.3.

61

Table 4.3

Test of Severity of Multicollinearity
for Sample I of Each Class

 

Significance
Probability
Class df P

I 7.94 3 .025*
II 54.04 3 .005*
III 149.98 3 .005*
IV 13.99 3 .005*
V 76.07 3 .005*
VI 174.16 3 .005*

 

*Significant at the 0.5 alpha level.

Each class shows the presence of multicollinearity,
even in Class I and Class IV which were supposed to have
low intercorrelated symptoms. This is not at all sur—
prising. Correlation coefficients of 0.16 are significantly
different from zero at the 0.05 level for 150 cases and 3
symptoms. Since most of the correlations among the symptoms
exceed that value, the situation represents a significant
multicollinear condition.

Since there are only three symptoms, there are 23 = 8
possible combinational patterns. The exact probability for
a disease to be positive (present) with pattern §k is

calculated as follows:

62

(number of patients with disease that has pattern 5k)

 

P(Dlfik) = (number of patients with pattern 5k)

The exact probabilities calculated from the preceding

formula for each class and pattern are presented in Table

 

 

 

4.4.
Table 4.4
Exact Probabilities, P(DI ) for Each
Class of the First Sample

Pattern I I I I I I IV V VI
111 .36 .41 .27 .58 .57 .43
110 .22 .13 .12 .21 .36 .20
100 .08 .14 .10 .00 .00 .00
001 .00 .07 .33 .00 .00 .00
011 .12 .24 .00 .00 .00 .00
101 .37 .57 .36 .17 .07 .14
010 .18 .15 .25 .06 .07 .00
000 .08 .03 .10 .00 .00 .00

 

In evaluating the discrepancies between the estimated
probabilities from the exact probabilities for each model,

the following three discrepancy measures are used:

63

1. Mean Square Deviation (MSD):

“ 2
p (pjk Pg)
MSDj = Z T (4.3)
i=1
where fiij denotes the estimated probability for the ith

pattern by the jth model and p; (where pfi = P(Dlxk) is the
exact probability for the kth pattern. It is worthy to note

that the numerator of equation (4.3) is the squared differ-

 

gpg§_between the model's estimates and the true probability.
Squaring the differences insures a positive value. Dividing
by the denominator which is the number of combinations or
patterns and summing across all 2p possible combinations,
equation (4.3) gives the mean square differences within a
class.

2. Weighted Mean Square Deviation (WMSD):

 

2p (g. -p"')2 ‘15.
k
mmng = 2: 3 2 3k (4.4)
k=1 2

Equation (4.4) is essentially the same as equation
(4.3) with the exception of the inclusion of the multiplier,
fijk, in the numerator. This means that more weight is given
to the higher probability estimates by the model. In other
words, higher probability estimates for the kth pattern by
the jth model are assumed to have greater squared differ-

ences and vice versa.

64

3. Misclassification Rate (MR):

2 2p A
MRj = 2: pjkdpd (4'5)
d=l k=l
where id is the marginal probability or the base rate of the
dth disease not occurring (i.e., 56 = l-Pd) where d is
equal to l and 2 in this dissertation. The term, fijkd, is
the estimated probability for the dth disease outcome. It

is noted that 8.

1jl = fiij = P(Dlxk) and p is simply equal

ij2
to (1-§ijl). Then the multiplication of the terms gives
the expected misclassification rate for the kth pattern in
the event that the dth disease does not occur. Summing
across all possible 2p patterns, equation (4.5) gives the
£2221 expected misclassification rate within a class.

The probability models with the exception of the
Bayesian model are then applied to the first sample to
obtain the estimated parameters for each symptom as shown
in Appendix B. These parameters are then used to derive
the conditional or diagnostic probabilities ij for each
class, pattern and model as presented in Appendix C. The
performances of these probability models in terms of the

above three indices are then obtained and they are presented

in Table 4.5.

65

Table 4.5

Comparison of Models in Terms of Discrepancy Indices

 

 

 

 

 

 

 

 

 

 

Classes
Discrepancy
Indices I R? II R III R R*b Iv R v R vr R Rfi*° R***d
SMD .000 1 .000 1 .000 1 1 .000 1 .000 1 .000 1 1 1
s MWSD .000 1 .000 1 .000 1 1 .000 1 .000 1 .000 1 1 1
MR .306 4 .335 6 .306 3 4.6 .278 2 .280 1 .255 1 1.3 2.9
SMD .006 4 .012 4 .012 4 4 .010 3 .016 s .013 .6 8
BLS MHSD .001 4 .003 4 .002 4 4 .002 3 .004 4 .002 3 3.6 3.8
MR .311 6 .325 4 .320 5 5 .304 4 .328 6 .309 5
sun .007 6 .012 4 .027 5 5 .014 4 .025 6 .017 6 5.3 5.15
BWLS uwso .001 4 .003 4 .009 7 5 .004 4 .007 6 .003 5 5 5
MR .312 7 .325 4 .355 7 6 .305 5 .335 7 .312 7 6.3 6.15
SMD .014 7 .017 7 .030 7 7 .020 5 .031 7 .014 4 5.3 6.65
BR MHSD .001 4 .004 7 .002 4 5 .005 5 .009 7 .002 3 5 5
MR .265 2 .290 1 .236 l 1.3 .328 6 .327 5 .306 4 5 3.15
SMD .006 4 .012 4 .027 5 4. .034 6 .013 4 .015 5 5 4.65
BRWLS nwsn .001 4 .003 4 .004 6 4.6 .007 6 .004 4 .003 5 s 4.8
MR .301 3 .320 3 .293 2 2.6 .269 l .326 4 .309 3.3 '2.9
SMD .005 3 .008 3 .009 3 3 .002 2 ~.000 .1 .000 l 2
LD msn . 000 1 . 002 3 . 001 2 2 . 000 1 . 000 1 . 000 1 1 . 5
MR .308 5 .315 2 .315 3.6 .293 3 .285 2 .256 2.3 2.9
sun .000 1 .000 1 .001 2 1.3 .042 7 .001 3 .024 7 5.6 ' 3.4
EMPD MWSD .000 l .000 l .001 2 1.3 .024 7 .000 1 .010 7 5 3.15
MR .031 l .339 7 .331 6 4.6 .328 6 .295 3 .303 3 4 4.3

 

3R I ranking of model within classes.

bR* - average ranking of model across classes I, II, and III (pool situation where the symptoms
are lowly correlated with the disease outcome).

cRn - average ranking of model across classes IV, B, and VI (pool situation where the symptoms
are highly correlated with the disease outcome).

dR**‘* - average ranking all classes.

66

In the situation in which the symptoms have a low
correlation with the disease outcome, the Bayesian model
(B) has the smallest mean squared deviation (MSD), followed
by the Entropy Minimax Pattern Discovery (EMPD) model. The
Binary Ridge (BR) model has the highest MSD. In terms of
having the smallest weighted mean squared deviation (WMSD),
the B model again ranks first with the EMPD model ranking
second. The Binary weighted least square regression model
(BWLS) ranks first in having the highest WMSD and also in
having the smallest misclassification rate (MR) when the
Binary Ridge regression (ER) is used as the solution.

In the situation where the symptoms are highly
correlated with the disease outcome, the logistic dis-
crimination model (LD) and the B models have the smallest
MSD and WMSD. The EMPD has the highest MSD and WMSD. The
B model ranks first in terms of having the smallest MR
followed by the LD model. The Binary weighted least squares
(BWLS), BLS and BR models, all three solutions to the BLS
model, have the highest MR, implying these solutions did
not improve the BLS model with respect to MR.

When the situations of (1) symptoms having high
correlation with the disease outcome and (2) the symptoms
having low correlations with the disease outcome are pooled,
the B model has the smallest MSD, WMSD, and MR followed by
the LD model. The BR, BWLS, and the BRWLS did not improve

the ranking of all three indices for the BLS model.

67

However, the main thrust of this dissertation is
diagnostic prediction. The term prediction entails a
different and perhaps future event. The indices discussed
so far address the guality of the probability models that
are available for building a basis for judgment. The
assessment of prediction will be done by cross-validating
each model on Sample II, the statistically equivalent
counterpart to Sample I. This notion is similar to the
process used by physicians of determining the best procedure
to treat a class of problems (model selection, Sample I) and
then evaluating its efficiency and correctness on new
patients with the same problem (cross—validating, Sample II).

The process of cross-validation can result in four
possible outcomes. They are:

l. the correct prediction or identification of a truly

diseased case, also known as true positives (TP).

2. the correct prediction or identification of a truly
non-diseased case, also known as true negatives
(TN).

3. the incorrect prediction or identification of a
truly non-diseased case as having the disease, also
known as false positives (FP).

4. The incorrect prediction or identification of a
truly diseased case as not having the disease,

otherwise known as false negatives (FN).

68

These outcomes can be represented in the following

figure (Figure 4.1):

True State
D H?)
B a?)
D
The Model
Diagnostic
Decision _. D (FN)
D
3 MN)

Figure 4.1 Possible Outcome of a Diagnostic Decision.

The above situation can also be reformulated into the

following table (Table 4.6):

Table 4.6

Possible Distribution of Cases by Model
Decision and the True Outcome

 

 

True State
D 3
Model's -
Decision - nl _ N
D FN(n3) TN(n4) i=1

 

 

 

 

69

where n1 and n4 are the numbers of patients with true
positives and true negatives, respectively, n2 and n3

are the numbers of patients with false positives and false
negatives, respectively. From Table 4.6, the following

indices henceforth termed as prediction indices are defined:

 

1. Sensitivity (SEN): The ability of the model to

 

predict the proportion of patients who truy have the disease.

The formula is:

SEN = nl/(n1-+n3).

The standard error of SEN is found to be:

A A 1 A
SE(SEN) = {(plql)/(nl + n3) )1 where p1 = nl/(nl + n3)

A

and q]. = (1'51).

Hence, the confidence interval for SEN becomes:

SEN i za ' SE(SEN) where 1 -0I - confidence level.

The greater the sensitivity, the greater the accuracy of
the model in predicting the occurrence of the disease.

2. Specificity (SPEC): The ability of the model to

 

predict the proportion of patients who truly do not have the

disease. The formula is:

SPEC = n4/(nL2+n )

4

The standard error of SPEC is found to be:

70

SE(SPEC) = {(§2§2)/(n2+n4)};5 where 62 n4/(n2-I-n4)

and q2 = (l-pz).
Hence, the confidence interval for SPEC becomes:
SPEC i 20 °SE(SPEC).

The greater the specificity, the greater the accuracy of

the model in predicting the non-occurrence of the disease.

3. Predictive value (PRED): The Proportion of

 

patients who truly have the disease among those predicted

by the model to have it. The formula is:
PRED = nl/(nl + n2)

The standard error of PRED is found to be:

SE(PRED) = i(’(f>3<’i3)/(nl+n2)}l5 where £53 nl/(nl+n2)

and a =(1-i3

3 ).

3
Hence, the confidence interval for PRED becomes:

PRED i” za ' SE(PRED) .

The greater the predictive value, the more accurate or
"precise" is the prediction of the model.

It should be noted that when the values for both SEN
and SPEC are one, it implies that the PRED is also one.
However, a PRED of one does pgp_necessarily imply a SEN
value of one or a SPEC value of one. A SEN value of zero

would imply a PRED value of zero and vice versa.

71

4. Type I error (E1): The proportion of patients
which the model predicted as not having the disease among

those who truly have the disease. The formula is:
E1 = r13/(nl +n3)

or simply the complement of SEN, i.e.,
E1 = l-SEN.

It can also be written in the form of a conditional
probability as:

El==P(§JDS)
where 55 is the model's diagnosis as not having the disease
and DS denotes the true state as having the disease. The
standard error of E1 is equivalent to the standard error

of SEN as E1 is the complement of SEN. Hence, the con-

fidence interval for E1 is simply:
El 1‘ za ' SE(SEN).

5. Type II error (E2): The proportion of cases

 

which the model predicts as having the disease when the

patients are in fact non-diseased. The formula is:
E2 = nZ/(n2 +n4) ,
or simply is the compliment of SPEC; i.e.,

E2 = 1 - SPEC.

72

Written as a conditional probability:
E2==PU%JDS)

where Dm is the model's diagnosis as having the disease

and Us denotes the true state as not having the disease.

The standard error of E2 is equivalent to the standard error
of SPEC as E2 is the complement of SPEC. Hence, the

confidence interval for E2 is simply:
E2 1 2a - SE(SPEC) .

The conventional rule in allocating patients with

pattern 5k as having the disease, D, or not having the

disease, 5, is as follows:

 

Diagnostic Rule Decision
P(Dlxk):>P(DI§k) Disease
P(Dlxk):>P(D|xk) Non-Disease
P(Dlxk)==P(D|xk) Equivocal

These decision rules, however, are arbitrary and they
are at the discretion of the decision maker. In this dis-
sertation, the criterion of allocation is chosen at n where
n is equal to the base rate of the disease in the first
sample. Hence, these decision rules are reformulated as

follows:

73

 

Diagnostic Rules Decision
P(Dlxk) 2n Disease
_ > _ ,
P(Dlxk) 0 Non D1sease

This has, in effect, eliminated the equivocal decision
and has the advantage of increasing sensitivity of the model
to detect diseased cases which have a low base rate. The
price of using this decision rule, however, is maximizing
the probability of identifying a case as diseased when, in
fact, it is non-diseased.

However, this is seen as better than identifying a case
as non-diseased when, in fact, it is a diseased case. The
reason for this is explained by Neyman (1950) as follows:

[If the patient is actually well, but the
hypothesis that he is sick is accepted, a

Type 2 error] then the patient will suffer some
unjustified anxiety and, perhaps, will be put to
some unnecessary expense until further studies

of his health will establish that any alarm about
the state of his chest is unfounded. Also, the
unjustified precautions ordered by the clinic

may somewhat affect its reputation. On the other
hand, should the hypothesis (of sickness) be true
and yet the accepted hypothesis be (that he is
well, a Type 1 error), then the patient will be

in danger of losing the precious opportunity of
treating the incipient disease in its beginning
stages when the cure is not so difficult. Fur-
thermore, the oversight by the clinic's specialist
of the dangerous condition would affect the clinic's
reputation even more than the unnecessary alarm.
From this point of view, it appears that the error
of rejecting the hypothesis (of sickness) when it is
true is far more important to avoid than the error
of accepEIfigthe hypofhesis (of illness) when it is
false. (1950, p. 270, emphasis added)

 

74

Increasing the opportunity of committing the former
error to reduce the risk of the latter error is one of the
pervasive and fundamental rules in medicine which may be
stated as: "When in doubt, continue to suspect illness."
The logic of this decision rule rests on two assumptions
(Scheff, 1963). They are:

1. Disease is usually a determinate, inevitably
unfolding process, which, is undetected and
untreated, will grow to a point where it endangers
the life or limb of the individual, and in the
case of contagious disease, the lives of others.

2. Medical diagnosis unlike legal judgment, is not
an irreversible act which does untold damage to
the status and reputation of the patient.

He further states that: "In light of these two
assumptions, it is far better for the physician to chance
a Type 2 error than a Type 1 error."

The results for the cross validation in terms of
these prediction indices for each model and for each
class are presented in Appendix D. These results are
then re-tabulated in Table 4.7 to show those classes
within each model with the highest and lowest values

for SEN, SPEC, and PRED.

The Class Where Each Model Has the Highest
and Lowest Predictive Indices

75

Table 4. 7

 

 

 

 

 

 

Highest Lowest

Models SEN SPEC PRED SEN SPEC PRED
Bayesian VI IV IV I III III
Bayesian w/Bahadur

Binary VI IV IV I I I I I I I
Binary:

Ordinary Least Squares V IV IV I I I

Weighted Least Squares V IV IV I I I

Ridge Regression IV,V III IV III V III

Weighted Ridge V IV IV-VI I I I I
Logistic Discrimination VI IV IV I III I
Entropy Minimax Pattern

Discovery VI IV IV I III III

Class VI is the class where the B, BB, LD, and EMPD

models have the highest SEN, and Class V is the

the BLS and its solutions have the highest SEN.

class where

In terms of

SPEC, all models with the exception of BR have the highest

SPEC in Class IV where all models also have the highest

PRED. Hence, Class IV, V, and VI, where the symptoms

are highly related to the disease outcome, are optimal

situations for these models in terms of the three predictive

efficiency indices.

76

All models with the exception of BWLS have the lowest
SEN for Class I, and Class III has the lowest SPEC for B,
BB, LD, and EMPD. Class III also has the lowest PRED for
B, BB, BWLS, and EMPD, and Class I has the lowest PRED for
BLS, BR, and LD. Class I and Class III, where the symptoms
are lowly correlated with the disease outcome, are "pit"
situations for these models in terms of these indices as
all models have the lowest values in this class. From the
above results, the B, BB, LD, and EMPD models perform
similarly in having the highest and lowest predictive
efficiency indices.

To determine the relative performance of these models
within each class, Table 4.8 is reformulated. The binary
regression models have the highest SEN across all classes
except Class III. The BR model has the lowest SEN in
Classes I, II, and III, and BRWLS model has the lowest
SEN in Class IV, and LD and B models have the lowest SEN
for Classes V and VI. In terms of having the highest SPEC,
the BR model performs the best in Classes I to III and the
LD model from IV to VI. In terms of PRED, BRWLS performs
the best in Classes I and III while the LD model performs

most optimally in Classes II, IV, V, and VI.

77

 

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78

Those classes where the symptoms have low correlation
with the disease outcome are now considered. Table 4.9
shows the values and ranking for the three predictive
indices across models with their condition. The BLS and
BWLS models have the highest SEN but have the lowest SPEC.
The BR model has the lowest SEN but ranks first in having
the highest SPEC. The B, BB, LD, BRWLS, and EMPD have the

best predictive value.

Table 4.9

Performances of Each Model in Terms of Prediction Indices
When the Symptoms Are Lowly Correlated With

the Disease and Their Ranking

 

 

 

Mode 1 SEN Rank SPEC Rank PRED Rank
Bayesian .76 3 .61 4 .32 1
BB .76 3 .61 4 .32 l
BLS .78 l .54 8 .29 7
BWLS .78 l .56 7 .30 6
BR .31 8 .84 l .21 8
BRWLS .73 7 .62 2 .32 1
LD .75 6 .63 3 .32 1

EMPD .76 3 .61 4 .32 1

 

79

When the symptoms are highly correlated with the
disease, Table 4.10, the BR model has the highest SEN,
implying that the ridge solution improves the ordinarily
BLS in SEN but at the price of losing SPEC. The B has the
lowest SEN but has the highest SPEC and PRED. The LD and

EMPD share in having the highest PRED.

Table 4.10

Performances of Each Model in Terms of Prediction Indices
When the Symptoms Are Highly Correlated With

the Disease and Their Ranking

 

 

 

Model SEN Rank spec Rank PRED Rank
Bayesian .86 7 .79 1 .49 1
BB .88 5 .79 1 .49 1
BLS .96 2 .64 7 .37 7
BWLS .92 3 .65 6 .38 5
BR .98 1 .64 7 .38 5
BRWLS .90 4 .66 5 .37 7
LD .86 7 .79 l .49 1

EMPD .88 5 .78 4 ".49 1

 

80

When the conditions of high and low intercorrelated
symptoms are pooled, the BLS model has the highest SEN
and the BR model has the lowest SEN and PRED but with the
highest SPEC as shown in Table 4.11. The BWLS model has
the lowest SPEC. The BB model ranks first in having the
highest PRED followed by the B, LD, and EMPD models.

Summarizing the above results, Table 4.12 is

formulated, as can be seen below.

Table 4.11

Performances of Each Model in Terms of Prediction Indices
for Pooled Situation and Their Ranking

 

 

 

Model SEN Rank SPEC Rank PRED Rank
Bayesian .81 5 .70 3 .40
BB .82 3 .70 3 .41
BLS .87 l .69 6 .33
BWLS .85 2 .61 8 .34
BR .65 8 .74 l .30
BRWLS .81 5 .64 7 .34
LB .80 7 .71 2 .40

EMPD .82 3 .70 3 .40

 

81

Table 4.12

Performance of Models Relative to the Predictive
Indices Across Correlational Patterns of
Disease and Symptoms

 

 

 

 

 

 

 

Correlation With the Disease Outcome
Intercorrelation Among Symptoms
Low High
Low High Pooled Low High Pooled Overall
SEN BLS, B, BB, BLS, BR BB, BLS, BLS BLS
BWLS EMPD BWLS BWLS , BR,
BRWLS ,
EMPD
SPEC BR BR BR LD LD LD , B , BB BR
PRED BRWLS BRWLS B, BB , LD LD LD , BB , B , BB
BRWLS , EMPD
LD, EMPD II

 

 

 

 

 

 

 

A key question could be asked as to the cost of using
these models in each class (i.e., when the symptoms have low
correlation with the disease outcome or when the symptoms
are highly correlated with the disease outcome). In exam-
ining these models to answer this question, the following
table represents the consequences for various outcomes.

This table is also known as the utility_matrix.

 

82

 

 

True State
D 5
Model ' s D w11 w12
Diagnostic ._
Action D w21 w22

 

 

 

 

where wij represents the arbitrary weight given to each
outcome. These weights can be in the form of mortality or
cost in dollars. They could either be gain--(positive in
value) or a loss--(negative in value) or zero (neither gain

or loss). Hence, a decision function, E(D), is defined for

 

the mth model as follows:

2
= *A

2
i=1
where p; denotes the marginal probability or base rate of
the disease. In this dissertation, pi = P(D) and P* = P(D).

2
fiijm is the probability of patients having the disease
predicted by the mth model for the ith and jth outcome.

To emphasize the Type 1 and Type 2 error differences,
let us assume the following weights: w21 = -2, w12 = -l,
and w11 = w22 = 0, which means that the penalty for com-
mitting a Type 1 error is twice as costly as that for
committing a Type 2 error, and there is no credit or gain

given to the right diagnosis. A loss function can now be

83

formulated since there will be only loss and no gains.

This can be shown in the following matrix.

True State

 

 

D D
Model's D 0 -1
Diagnostic
Action D. _2 0

 

 

 

 

The model's performance in terms of this loss function is
presented in Table 4.13.

The LD model has the least loss followed by the B
and BR models, and the BLS model has the most loss when
the symptoms are lowly correlated with the disease outcome.
The B and LD models share in having the least loss and the
BB model has the greatest loss when the symptoms are highly
correlated with the disease outcome. For both situations,
the B and LD models again have the least loss with the BLS
and BWLS models having the greatest loss.

If credits are given to the right diagnosis by setting
w11 = 2, w22 = 1, then the amount of credit given to a
correct diagnosis of a truly diseased patient is worth
twice as much as a correct diagnosis of a truly non-diseased
patient. The matrix below references the result of setting

w12 = -l and w22 = l.

84

 

 

 

 

 

 

 

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85

 

 

True State
0 E
Model's D 2 '1
Diagnostic
Action -—
D -2 1

 

 

 

 

In terms of gain, as shown in Table 4.14, the LD model
has the most gain followed by the EMPD, BRWLS, and B models
with the BR model having the least gain when the symptoms
are lowly correlated with the disease outcome. And when
the symptoms are highly correlated with the disease outcome,
the B and LD models share the highest gain with BRWLS having
the least gain. Again, combining the conditions where (l)
the symptoms have a low correlation with the disease and
(2) the symptoms have a high correlation with the disease
produces the following results. The LD model had the most
gain with the B model ranking second. The binary regression
‘models have the smallest gain. These results show that
solutions resulting from the binary regression model and
the Bayesian model which are intended to correct for highly
interrelated symptoms did not improve significantly in
reducing loss nor in increasing gain.

The summary of these resultant losses and gains are

presented in the following matrix:

86

 

 

 

 

 

 

 

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87

 

 

 

 

 

 

Least Loss Best Gain
Low correlation LD LD
with the disease
High correlation
with the disease LD' B LD, B
=
Pooled LD, B LD

 

 

It should be borne in mind that the above evaluation
is contigent on the choice of weights and the fact that the
second sample has the same relational structure as the
first. It is now of interest to see how the models would
perform when the estimated parameters are applied to a
second sample that has a different relational structure
than the first, keeping the weights constant. This situ-
ation would represent the case when a sample of information
is gained and the relational structure is derived based on
that sample and assumed to hold for all subsequent samples.
That is, the derived parameters from this initial sample
are "blindly" generalized to a second sample which has an
unknown relational structure. The results of using one
class to generalize to another class in terms of the
prediction indices are shown in Appendix E.

In Table 4.15, the rows represent the knowledge of
the sample equivalent to the class and the columns represent
the model that has the highest SEN, SPEC, and PRED across

the classes. These classes have a different structure from

88

Table 4.15

The Best Model in Terms of Predictive Efficiency Indices
Under Different Relational Structural Situation

 

 

 

Highest Highest Highest
Situation SEN SPEC PRED
A BB EMPD LD, EMPD, B
B BB EMPD LD
C BLS LD B , LD
D BLS LD LD, B

 

the sample relational structure from which they are
developed. For simplicity, only four classes were chosen,
namely, Class I, III, IV, and VI. These classes permit
comparisons of the effects of low versus high intercor-
relations among symptoms and low versus high correlations
with the disease. The case of intermediate intercorre-
lations among the symptoms was ommited. Let the following
notation represent these situations:
A. Prior knowledge of the relational structure of I
and predicting across relational structure III, IV,
and VI.
B. Prior knowledge of the relational structure of III
and predicting across relational structures I, IV,

and VI.

89

C. Prior knowledge of the relational structure of IV
and predicting across the relational structures I,
III, and VI.

D. Prior knowledge of the relational structure of VI
and predicting across relational structures, I,
III, and IV.

In terms of the predictive efficiency indices, the
results are shown in Table 4.15.

The BB model has the highest SEN with the EMPD model
having the highest SPEC and LD with the highest PRED for
situations A and B. The BLS model has the highest SEN and
the LD model has the highest SPEC and PRED along with the
B model in situations C and D.

In terms of utility, the B, LD, and EMPD models have
the least loss for situation A. The LD model has the least
loss for situation B. The B, BB, and LD models have the
least loss in situations C and B, and the LD model has the
least loss in situation D. Across all four situations, the
LD model has the least loss followed by the B and BB models.
This is shown in Table 4.16.

The BB model has the most gain in situation A with the
BLS model having the least gain. The LD model has the most
gain in situation B and also in situation C along with the
B model. The B and LD models also have the most gain in

situation D. This is shown in Table 4.17.

90

Table 4.16

The Performance of the Probabilistic Models
Terms of Losses When Cross Validating to
a Different Relational Structure

 

 

 

 

 

 

 

 

 

 

 

 

 

Prior
Knowledge B BB BLS LD R EMPD R
I .28 .29 .36 .28 l .28 1
III .46 .42 .43 .35 1 .46 4
IV .32 .32 .34 .32 l .35 5
VI .32 .33 .38 .32 l .34 4
.34 .34 .37 .32 l .36 4
Table 4.17
The Performance of the Probabilistic Models
Terms of Gains When Cross Validating to
a Different Relational Structure
Prior.
Knowledge B BB BLS LD R EMPD R
I .64 .71 .48 .64 2 .64 2
III .28 .36 .34 .50 l .28 4
IV .55 .54 .52 .55 l .50 5
VI .56 .54 .43 .56 l .52 4
.51 .54 .44 .56 1 .48 4

 

91

These results can be summarized as follows:

Least Loss Highest Gain

 

 

 

 

A LD, EMPD, & B BB
B LD LD
C B, BB, & LD B 8 LD
D LD,I3 LDII3

 

 

 

 

Special Classes

 

Besides the above Six classes generated, three
"special" classes were generated to have the following

properties.

1. Mixed Class: The mixture of the six classes,

 

i.e., there are highly correlated symptoms and also lowly
correlated symptoms and some are highly correlated or lowly
correlated with the disease outcome.

2. Suppressor Class: The presence of a symptom which

 

is highly correlated with other symptoms but has low corre-
lation, near zero, with the disease outcome, i.e., if ith
is the symptom, then, rij = high and riD = 0. This symptom

is known as the suppressor symptom (Lubin, 1957; Conger and

 

Jackson, 1972).

3. High Correlated Class: An extreme class of high

 

correlation among symptoms and high correlation with the

disease.

92

The population and sample variance and covariance
matrices for these special classes are presented in
Appendix F. Using the same test of equivalence, the

following results were obtained (Table 4.18).

Table 4.18

Test of Equivalence for Special Classes

 

 

 

 

 

 

Class L L' d.f. p

 

Mixed 152.04 151.32 10 .005*
Suppressor 4.83 4.81 10 .999
High correlation 36.94 36.76 10 .025*

 

*Significant at the 0.05 level.

Despite the statistical lack of equivalence between the
papulation and sample variance-covariance matrices for the
mixed and high correlation classes, the two classes still
represent the intended situations and hence, would not be
a major concern for later analysis and interpretation.

The three samples were then randomly split into two
equal halves and the two sub-samples were then tested for

equivalence as shown in Table 4.19.

93

Table 4.19

Test of Equivalence for Sub-Samples for Special Classes

 

 

 

Significance
Probability
Class 2 d.f. P
Mixed 1.56 10 .995
Suppressor 15.77 10 .10
High correlation 2.86 10 .99

 

The estimated parameters were derived from the first
sample as before and they were cross-validated with the
second sample. The performances in terms of the prediction
indices are presented in Appendix G. Table 4.20 shows the

summary results for the special classes.

Table 4.20

The Models That Perform Relatively the Best in
Terms of Predictive Indices

 

==

 

Highest Highest Highest
Class SEN SPEC PRED
Mixed All B, BB, LD, B, BB, LD,
EMPD EMPD
Suppressor All All All

High correlation All B, BB, LD B, BB, LD

 

94

In terms of decision function values and using the same
weights as given before, Table 4.21 and Table 4.22 Show the
values of loss and gains, respectively.

In all three special classes, all models surprisingly
perform the same in terms of sensitivity! In terms of
having the highest SPEC and PRED, in the mixed class, all
models except the binary regression models, BLS, BWLS, and
BR, perform the same. In the suppressor class, all models
have the same performances in terms of SEN, SPEC, and PRED.
In the high correlation class, all models have the same SEN,
and B, BB, LD models have the highest SPEC and PRED.

In terms of loss, all models except the binary regres-
sion models have the same amount of loss in the mixed class.
In the suppressor class, all models have the same amount of
loss. In the high correlation class, the B, BB, and LD
models have the least loss while the BR model has the
most loss.

In terms of gain, the B, BB, LD, and EMPD models have
the most gain in the mixed class. In the suppressor class
there is no difference in gain for all models. In the high
correlation class, the EMPD model has the most gain while

the BWLS model has the least gain.

95

Table 4.21

Loss Function for Various Models
for Special Classes

 

 

 

 

 

 

 

 

 

 

Models
C las s B BB R BLS R BWLS R ER LD EMPD
Mixed .19 .19 l .26 5 .33 7 .31 .19 .19
Suppressor .40 .40 1 .40 l .42 1 .42 .42 .42
High cor-
relation .30 .30 1 .32 5 .36 7 .34 .30 .31
Table 4.22
Gain Function for Various Models
for Special Classes
Models
Class B BB R BLS R BWLS R BR LE EMPD
Mixed .82 .82 l .68 5 .53 7 .57 .82 .82
Suppressor .36 .36 1 .36 l .36 l .36 .36 .36
High cor-
relation .60 .60 2 .55 5 .48 7 .52 .60 .88

 

96

Clinical Application

 

The data selected for application are from a study on
brain scans by Potchen (1975) from July 1974 to June 1975
at Johns Hopkins Hospital, Maryland. The procedure involved
in his study is shown in Figure 4.2. The instrument that
was used is presented in Appendix H. The patients with the
given symptoms were recorded on a questionnaire and these
were given to physicians to determine the probability of
having an abnormal scan for each patient and whether a brain
scan was necessary. The final results were confirmed by the
brain scan when the patient was referred for such action.
In this application, only those patients that were referred
for brain scan were used. There are altogether 86 patients
in which 8 patients had abnormal brain scans which means
tumor growth in the brain. Since the application is about
symptomic diagnosis, only the signs and symptoms were
selected. They are:

l. headaches;

2. seizure;

3. cortical deficit;

4. motor deficit;

5. sensory abnormality; and

6. visual field defect.

97

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98

The normality or the abnormality of the final brain
scan will be considered as the disease outcome, the 86
patients will be considered as the "population" of brain
tumor suSpected patients with the given Six symptoms—-the
set of conditions. It should be borne in mind that with
such few patients, the following results can only be con-
sidered a pilot study or preliminary investigation for the
models.

With the same procedure, the 86 patients were split
into two equal halves of 43. Each half having 4 abnormally
scanned patients. The variance-covariance matrix for the
"population" and the samples are shown in Appendix J. The
test of equivalence for the split samples and test for

multicollinearity are presented in Table 4.23.

Table 4.23

Test for Equivalence and Multicollinearity

 

 

Significance

Probability
Test 2 d.f. P
Equivalence .001 28 .99

Multicollinearity 20.65 15 .25

 

99

The estimated parameters are shown in Appendix F.
The decision point n is set at .10 (n = 8/86 = .10). The
results of computing the prediction indices for the models
are shown in Table 4.24. From the table, the weighted ridge
solution surprisingly improves the sensitivity of the ordi-
nary least squares binary model. The entropy model has the
highest specificity with the binary model having the least
specificity. However, the binary model has the highest

predictive value.

Table 4.24

Prediction Indices for Various Models for Brain Scan

 

 

 

 

Models
Indices B BB BLS BWLS BR BRWLS LD EMPD PSPa
SEN .00 .50 .50 .25 .75 .75 .25 .00 .75
SPEC .90 .95 .54 .65 .69 .41 .74 .97 .41
PRED .00 .00 1.00 .06 .20 .12 .09 .00 .09

 

aPSP= physician subjective probability derived from category IV,
section la, on the questionnaire as shown in Appendix H.

With respect to the values of the decision function
with the weights as given on page 82, Table 4.25 shows the
values for both the loss and gain for various models.for

the brain scan data.

100

Table 4.25

Decision Function Values for Various
Models on Brain Scan

 

 

 

 

Models
Function B as BLS BWLS BR BRWLS LD EMPD Pspa
Loss .29 .14 .51 .46 .33 .58 .38 .03 .25
Gain .52 .81 .08 .17 .44 .07 .34 .84 .63

 

aPSP = physician subjective probability derived from category IV,
section la, on the questionnaire as shown in Appendix H.

Hence, from the results, the EMPD model is the best
model in terms of utility; the least loss and the most
gain, in screening or predicting brain scan patients.

What are the significant findings from the analyses
in this chapter? What can these models have to offer for
diagnostic problem-solving? How do these models relate to
a real clinical setting? And how can one go about using
these probabilistic models for diagnostic problem solving?
These issues and other important issues will be discussed

in the following chapter.

CHAPTER V

SUMMARY AND DISCUSSION

The significant findings in this thesis may be
summarized as follows:

1. Overall, sensitivity increases for all models as
the correlation with the disease outcome increases.

2. There is a "hump" or convex effect for sensitivity
for all models except the Bayesian (B), Bayesian with the
Bahadur's expansion (BB) and the Entropy Minimax Pattern
Discovery (EMPD) models, in situations where the symptoms
have a low correlation with the disease outcome. That is,
the maximum sensitivity is not when the intercorrelation
between the symptoms is greatest but when the symptoms are
moderately intercorrelated as shown in Appendix L. This
phenomenon did not Show in situations where the symptoms
have a high correlation with the occurrence of the disease.
In fact, sensitivity increases as the intercorrelations
increase under the latter situation as shown in Appendix M.

3. The values for sensitivity did not differ among
models in situations where highly interrelated symptoms are
also highly related to the occurrence of the disease. In

other words, when the relational structure is highly

101

102

correlated, it does not matter which model one uses if
sensitivity is chosen as a criterion for selection models.

4. The "pit" or concave effect of specificity across
binary regression models occurs when, given those situa-
tions where the symptoms are highly correlated with the
disease outcome, the intercorrelations between the symptoms
increase. This is also shown in Appendix M. This means
that specificity is at a minimum when the symptoms are
moderately related.

5. The "hump" or convex effect is also found for
predictive values in the same way as the sensitivity index,
that is, when the symptoms have a low correlation with the
occurrence of the disease as shown in Appendix L.

6. With the presence of a suppressor symptom, it does
not matter what measure one uses as a criterion for select—
ing models as all models perform the same for all prediction
efficiency indices.

7. If a model is chosen with the criterion of having
the best sensitivity, it is at a cost of losing specificity
and vice versa. In other words, there are pp_models that
have the best of both indices for all classes considered
in this dissertation. The statement holds when one looks
across classes and within classes of problems. This also
means that there is pp_single model that performs consist-
ently better for each class or across classes in terms of

sensitivity and Specificity.

103

8. A decision function analysis was performed.
Penalty (negative) weights were given for the two diag-
nostic errors (i.e., Type 1 and Type 2) and no credit is
given to the correct diagnosis. The binary least square
model (BLS) and the binary weighted least square model
(BWLS) showed the smallest loss when the symptoms had a
low correlation with the disease's occurrence but themselves
had high intercorrelations. However, when considering gains,
with credits given to the correct diagnosis, but the same
penalty weights, the Bayesian model (B) had the most gain
when the intercorrelations among the symptoms was low but
the correlation between the symptoms and outcome was high.
The logistic discrimination model (LD) had the most gain
when the symptoms had a low correlation with each other
but had a high correlation with the occurrence of the
disease outcome. The LD model also had the most gain
when the symptoms were moderately interrelated with each
other and the symptoms had a low correlation with the
disease. If one disregards the intercorrelation among
symptoms, the LD model had the highest gain whether or
not the symptoms had a high or low correlation with the
occurrence of the disease. That is, the best model to use
to maximize gain in the absence of knowledge about the
relationship among and between symptoms and disease

outcomes, is the LD model.

104

9. Summarizing the above results, the following
table (Table 5.1) can be formulated. The columns denote
the kind of relational structure cross—validated and the
rows represent the criterion for selecting models. For
example, assume one wants to cross-validate under the
assumption of an unknown relational structure and also
chooses specificity (SPEC) as the criterion. One would
go to the intersection of column two (Unknown) and the
second row (SPEC) and conclude that the B model should

be used.

TafleSJ

Decision Table in Choosing Models With Respect to
Prediction Index and Kind of Cross—Validated

Relational Structure

 

Criterion for

 

Selecting Models Same unknown
SEN BLS BLS
SPEC BR B
PRED BB, LD, EMPD LD
LOSS LD, B LD

GAIN EMPD, LD LD

 

105

Further Recommendations

 

The purpose of this thesis was to demonstrate how

different statistical models perform when applied to

different relational structures and under differing degrees

of uncertainty. The following are some recommendations for

further research:

1.

Vary the base rates of the disease and the symptoms
and determine the changes of the prediction
efficiency indices for various models.

Increase the number of disease categories beyond
the two that were considered in this dissertation
(i.e., D1, D2, ... Dd).

Change the direction of the intercorrelation among
the symptoms and with the occurrence of the disease
to negative and determine the changes in prediction
efficiency indices for various models.

Vary the decision rules and determine the changes
in prediction efficiency indices for various models.
For symptoms that have high intercorrelations,
combine symptoms to form "factors" by means of
factor analysis and principal components techniques
and use these generated factors or components to

predict the occurrence of the disease.

106

Clinical Implications

 

What sort of implications do these models and the
findings have for an empirical clinical setting? First
of all, these models are attempts to quantify uncertainty.
They are a set of mathematical algorithms to generate
indices from a complicated universe in order to enable
decision-making to be less difficult and to be more effec-
tive. They are ppp_meant to replace the human decision
maker but rather to supplement the decision process. They
act as an additional source of information for the decision
maker. The model's relationship with the human decision

maker may be illustrated as in Figure 5.1.

 

   
    

  
 
 
   
 

Quantitative
Models

 

 

 

 

   
 

Clinical
Decision

Clinical
Intuition

   

. Clinical
Uncertainty Action

 

 

 

 

Figure 5.1 The Relationship Between Quantitative Model Decision
and Clinical Intuition.

107

After one obtains the additional information from the
quantitative models, one can choose the following three
alternatives: (1) ignore the prediction made by the
quantitative models and follow clinical information,

(2) modify the clinical impression on the basis of the
information provided by the quantitative models, or (3)
abandon clinical intuition in favor of the quantitative
choice. It should be borne in mind that the final and full
responsibility of medical diagnosis lies on the physician
and not on a set of mathematical algorithms, regardless of
which of the three alternatives is chosen.

When the physician's clinical intuition is in agree-
ment with the quantitative prediction, there is no problem
and the quantitative prediction is seen as "reinforcing"
clinical intuition. However, when clinical intuition is
in disagreement with the quantitative prediction, the
physician should weigh all the evidence by objectively
examining the validity of his own intuition and the validity

{pf the assumptions of the quantitative models to generate

 

the prediction. If the model's assumptions are violated,
then he should take alternative (1) (i.e., abandon the
quantitative prediction and follow his own intuition).
However, if the physician feels that for some reason his
clinical intuition is somehow suspect, then it is recom-

mended that he take alternative (3) (i.e., abandon his own

108

intuition and follow the model's prediction). Again the
physician must bear the responsibility of abandoning his
own intuition and abiding by the quantitative prediction.
For all clinical decisions, if the crux is to determine
whether the patient has a disease, and there is doubt,
deSpite all possible evidence gathered by both the human
decision maker and the models, it is better to diagnose the
patient as having the disease. This follows the axiom, "If
in doubt, diagnose illness." The above situations can be

illustrated in the following table (Table 5.2).

TfifleSJ

Final Decision by Clinical Intuition
and Quantitative Prediction

0|

 

 

D
Clinical D D D
Intuition - —-
D D D

 

 

 

 

Schema for Application of the Models
to a Diagnostic Problem

 

 

To apply these probabilistic models to a diagnostic
problem, the following steps should be taken:
1. Select the disease to be identified.

2. Identify the set of signs or symptoms which are

thought to occur jointly with the disease. That

is, in effect, similar to identifying the signs or

109

symptoms which are related to the occurrence of the
disease without implying causality between the
symptoms and the occurrence of the disease.

Collect all available cases of the set of signs or
symptoms. It is to be noted that the frame-of-
reference for the data collection is with respect
to the set of signs or symptoms and not with
respect to the occurrence of the selected disease.
Hence, the collected data will include those cases
that the selected disease and those cases that have
other diseases or no diseases of interest.

From the collected data and for each individual
case, code a one (1) if the case shows the presence
of the selected disease and code a zero (0) if the
case shows other diseases or no disease. Likewise,
use the same scheme of coding with the signs or
symptoms for each individual case. The resultant
coded data will resemble the data matrix shown in
Figure 1.3.

Define an uncertainty structure by dividing the
magnitudes of the intercorrelations among the signs
or symptoms into levels and likewise with the mag- I
nitudes of the correlation of the signs or symptoms

with the occurrence of the disease. Then label,

110

numberically or alphabetically, the cells or
classes in the uncertainty structure. The above

two procedures will result in the following figure:

Intercorrelation Among Symptoms

Level 1 Level 2 Level 3

 

 

 

Level 1 I II III
Correlation
of Symptoms Level 2 IV V VI
With Disease

Level 3 VII VIII IX

 

 

 

 

 

It should be noted that the levels need not be of
equal intervals.

From the new coded data matrix, compute all pos-
sible pairwise correlations among the symptoms and
the correlations between the symptoms and the occur-
rence of the disease by using the phi-coefficient
formula (Cohen and Cohen, 1976), obtaining the
correlational matrix as shown in Figure 3.1. The
computed correlational matrix constitutesthe
relational structure of the disease and the
symptoms.

Identify the cell or class where the computed

relational structure is the closest in value with

the results in step 5. This is done by either

111

"eyeballing" the values of the computed
relational structure along with the values of
the correlations in each individual classes in
the uncertainty structure and selecting the
class that bears the most resemblance (a purely
subjective judgment) to the computed relational
structure, or

performing a statistical test of equivalence
between the classes and the computed relational
structure. This is, in effect, testing the

following hypothesis:

H : = .
0 Rs Rc1

against the alternative,

H1: Rs 7‘ RC1

where Rci = the ith class in the uncertainty
structure. It is worthy to note that the
values in the correlation matrix, Rci’ are the
median values of the two intervals of the ith
class (i.e., if the level of correlation among
symptoms is 0.5 to 0.7 and the level of corre-
lation of symptoms and the disease is 0.0 to
0.30 for the ith class, the median values for

the correlational matrix, R are 0.6 and

ci'

0.15, respectively).

112

RS = the computed relational structure from

the collected data.
Such test of equicorrelation patterns of relational
structures can be found in Morrison (1976, p. 276).

8. Select a criterion, sensitivity, specificity, or

predictive value, according to the following rule:

 

 

Criterion

Situation SeIécted
The consequences of committing a
Type 1 error is more serious than
committing a Type 2 error Sensitivity
The consequences of committing a
Type 2 error is more serious than
committing a Type 1 error Specificity
The consequences of committing both Sensitivity
Type 1 and Type 2 errors are of no or
difference Specificity

9. Use Table 4.8 where the rows represent the crite-
rion to be selected and the columns represent the
classes of the uncertainty structure. The inter-
section of the rows and the columns represents the
probability model or models that perform relatively
the best with reSpect to the selected criterion in
that particular class.

10. Use that model for diagnostic prediction for the
selected disease in maximizing the chosen
criterion.

The summary of the above ten steps is represented in

Figure 5.2.

113

Select the Disease
to be diagnosised

Identify the iymptoms
relating to the
occurrence of the
disease

 

Addition of___’ Collect those cases
New cases that have the set of
symptoms I

Compute the relational
structure of the disease
and the sym toms

Define the uncertainty
structure for the disease
and the symptoms

Identify the class which
the computed relational
structure best fits

Select the criterion
according to the nature
of the diagnostic problem

From table identify the
model that performs the
best with respect to the
criterion in that class

Use the model for
diagnostic prediction

Figure 5.2 Schema for Application of the Models to a Diagnostic
Prdblem.

114

An Example in Breast Cancer

Consider the problem of detecting breast cancer which
is one of the major causes of death among women. It is
widely recognized that the early detection of this cancer
will reduce its mortality rate. However, the term "early"
has equivocal meanings as it denotes the absence of any
signs or symptoms at the onset stage of the cancerous growth.
The only "signs" or "symptoms" for such an early detection
are sociological cues: the patient's familial history of
breast cancer, the patient's pregnancy and menarche history,
and other cues which are not directly related to the cancer.

These cues are known as risk factors and they constitute the

 

physician's index-of—suspicion. The diagnostic problem is

 

then to use these risk factors to identify the high risk
group of patients as having breast cancer. The risk factors
that are known to be highly related to the occurrence of
breast cancer are (1) age, (2) socioeconomic status, (3) age
at menarche, (4) age at pregnancy, (5) age at menOpause,

(6) familial history of breast cancer, and (7) number of
pregnancies. Gather all available cases that have these
risk-factors. An excellent data source would be from mass
screening centers. A portion of the collected cases will

be confirmed breast cancer cases (coded as ones) and the

other portion of cases will be non-confirmed breast cancer

115

cases (coded as zeroes). Each risk factor is dichotomized
by setting a subjective cut-off point and coding a value

of one if the value of the risk-factor exceeds the cut-off

point and coding a value of zero if the value of the risk—

factor lies below the cut—off point. The next step is then
to define an uncertainty structure. The following matrix

is one possible definition of the uncertainty structure:

Intercorrelations Among the
Risk Factors

<O.20 0.21-0.50 3>O.50

 

 

 

Correlations :>O'20 I II III
of the Risk 0.21- IV V VI
Factors with 0.50

Breast Cancer <10.50 VII VIII Ix

 

 

 

 

 

Then compute all possible pairwise correlations among the
risk-factors and the correlations between the risk-factors
and the occurrence of breast cancer, thereby deriving the
relational structure of the risk factors and breast cancer.
Using the relational structure and the uncertainty structure,
identify the class of the relational structure by either
strategy as mentioned in Step 7. Misclassifying a breast
cancer case as non-breast cancer case (Type 1) has more
serious consequences than classifying a non-breast cancer

case as a cancer case (Type 2 error), since the former

116

action means later detection and delayed therapy which might
lead to death; consequently, sensitivity is preferable to
specificity as a criterion for selecting models. Finally,
from Table 4.8, with sensitivity as the criterion and the
class that has been identified for the relational structure,
say hypothetically Class IV, the binary ridge regression
model is the best model relative to the other models, for
identifying the high risk breast cancer group.

It should be borne in mind that Table 4.8 is generated
from the assumption that the base rate for the selected
disease is 0.2 and the base rates for the symptoms are 0.5.
However as further studies which use the same methodology
as this dissertation, investigate the effects of varying
the base rates for the disease and the symptoms, this
assumption can be relaxed.

Quantitative Models in Medical
Decision Making

 

 

The use of quantitative models can achieve three main
objectives which are merits of the models in their own
right. They:

1. Combine probabilistic reasoning and uncertainty

of the data in a formal explicit system rather
than by intuition to achieve more efficient and

consistent information processing.

117

2. Provide a systematic processing of uncertainty

 

that takes account of all available information

for decision making and find the optimal weighting

 

combination of symptoms, ensuring that each

 

contributes properly to the disease outcome.

3. Develop formulae, rules, or strategies for optimal

 

 

consistent information processing in the presence
of uncertainty.

There are three areas in which quantitative models can
assist in better medical decision making. They are (1)
teaching tools, (2) patient management, and (3) public
policy. These areas might be considered as follows:

1. Teaching Tool: Elstein (1976) has noted that

 

strategies for different degrees of uncertainty have been
made explicit by quantitative models. Hence, they can
become a learning device for the novice in finding strat-
egies and rules for identification of a disease. Consider
the detection of breast cancer. The problem is to find the
"high risk" group without referring every case for radiolog-
ical examination. Radiological examinations haveturned out
to be hazardous to health. Blair (1976) has found that
radiation has killed as many patients as breast cancer
itself. Yet, radiological examinations or techniques

remain the best device for detecting breast cancer despite

their potential hazards. Hence, the crux of the problem is

118

to (l) assume the patient has breast cancer and to refer
the patient for radiological examination knowing that
exposure to radiation is hazardous, a possible Type 2
error, or (2) assume the patient does not have breast
cancer with the danger of committing a Type 1 error. When
the explicit rules and strategies for making this crucial
decision have been generated by quantitative models, the
novice could learn from these rules to make his decision.

2. Patient Management: In situations of diagnostic

 

ambiguity, the physician has difficulty in taking clinical
action. But when rules and strategies for diagnosing the
disease have been made explicit, this information becomes
a frame-of—reference for diagnosis hence removing the
ambiguity of the situation. An excellent example for
patient management is the common symptom, headache.
MacBryde and Blacklow (1970) have listed fifteen diseases
associated with the symptom, headache, among which is brain
tumor. Each disease demands unique treatment and therapy.
The kinds of treatment range from administering an aspirin
to brain surgery. Each treatment procedure demands cost,
time, and potential hazard. The problem is to identify
the disease correctly in order to give the correct form

of patient management.

119

3. Public Policy Making: In the area of health care,

 

there are many decisions involving the expenditure of large
dollar amounts for public health programs. AS one instance,
a debate is current between the Department of Public Health
and the third party carriers as to whether physicians be
for "whole body" computed assisted tomography (CT) scans.
This is only symptomatic of the impact of technology on
medical diagnosis. The question becomes, how should one
and when should one use these expensive and sometimes
potentially hazardous diagnostic techniques. Further,

who will pay for it; how much will be paid; and how often
will these procedures be paid for, become a series of
questions that are entering into health policy. It is
anticipated that the application of the quantitative models
studied in this dissertation will help provide answers to
questions such as these. For example, if one can determine
the efficacy and correctness of a clinical diagnosis through
the use of quantitative models, then the procedures used to
reach the diagnosis would be strengthened and consequently,
be candidates for reimbursement. If, however, the weight
assigned to particular procedures is low, which would indi-
cate little or no contribution to the overall clinical
diagnosis, then the procedures needed to obtain the
information as to whether the symptom is present or

absent should be scrutinized for reimbursement.

APPENDICES

APPENDIX A

THE RELATIONAL STRUCTURE OF THE POPULATION,

THE GENERATED SAMPLE AND THE SPLIT SAMPLES

120

NOTE

The correlational matrices (R) and the variance-covariance
(2) matrices in this appendix and the following appendices
should be interpreted as follows:

where

a11'322'333'add =

81 82 S3 D
1 a11
a12 a22 (Symmetric)

 

 

1 if it is a correlational matrix and
the variances of symptom l, 2, and 3,
and the disease, respectively, if it is
a variance-covariance matrix;

— correlation between symptom 1 and

symptom 2 if it is a correlational matrix,
and the covariance of symptom 1 and
symptom 2 if it is a variance-covariance
matrix;

— correlation between symptom l and

symptom 3 if it is a correlational matrix,
and the covariance of symptom l and
symptom 3 if it is a variance-covariance
matrix;

— correlation between symptom 2 and

symptom 3 if it is a correlational matrix,
and the covariance of symptom l and
symptom 2 if it is a variance-covariance
matrix; and

ald’aZd'a3d

121

2,

= the correlation of the occurrence

of disease with symptom l, and

3, respectively, for the correlational
matrix and the covariance of the disease

with symptom 1,

2, and 3, respectively,

for the variance-covariance matrix.

Relational Matrices of Populations and Samples

 

w>

w>

 

 

 

 

Class
II III

1.00 1.00 1.00
0.20 1.00 0.40 1.00 0.60 1.00
0.20 0.20 1.00 0.40 0.40 1.00 0.60 0.60 1.00
0.20 0.20 0.20 1.00 0.20 0.20 0.20 1.00 0.20 0.20 0.20 1.00
1.00 1.00 1.00
0.10 1.00 0.40 1.00 0.56 1.00
0.13 0.14 1.00 0.35 0.33 1.00 0.67 0.62 1.00
0.21 0.10 0.11 1.00 0.24 0.24 0.31 1.00 0.23 0.18 0.23 1.00

IV V VI
1.00 1.00 1.00
0.20 1.00 0.40 1.00 0.60 1.00
0.20 0.20 1.00 0.40 0.40 1.00 0.60 0.60 1.00
0.40 0.40 0.40 1.00 0.40 0.40 0.40 1.00 0.40 0.40 0.40 1.00
1.00 1.00 1.00
0.15 1.00 0.31 1.00 0.56 1.00
0.23 0.21 1.00 0.44 0.42 1.00 0.67 0.61 1.00
0.41 0.34 0.32 1.00 0.45 0.41 0.37 1.00 0.43 0.44 0.42 1.00

 

122

Variance-Covariance Matrices of Population and Samples

 

 

 

 

 

Class
I II III
.250 .250 , .250
.050 .250 .100 .250 .150 .250
.040 .050 .250 .100 .100 .250 .150 .150 .250
.040 .040 .040 .160 .040 .040 .040 .160 .040 .040 .040 .160
.250 .250 .250
.020 .250 .100 .250 .140 .250
.030 .030 .250 .080 .080 .250 .168 .155 .250
.040 .020 .020 .150 .040 .040 .060 .160 .045 .037 .046 .154
IV v VI
.250 .250 .250
.050 .250 .100 .250 .150 .250
.050 .050 .250 .100 .100 .250 .150 .150 .250
.080 .080 .080 .160 .080 .080 .080 .160 .080 .080 .080 .60
.250 .250 .250 '
.040 .250 .080 .250 .140 .250
.060 .050 .250 .110 .100 .250 .170 .150 .250

.080 .060 .060 .150 .090 .080 .070 .160 .080 .080 .080 .150

 

123

Correlational Matrices of Sub—Samples

 

I!)

w>

W)

:0)

 

 

 

 

Class
I II III
1.00 1.00 1.00
0.16 1.00 0.43 1.00 0.52 1.00
0.15 0.09 1.00 0.30 0.35 1.00 0.63 0.60 1.00
0.23 0.13 0.11 1.00 0.29 0.22 0.29 1.00 0.16 0.10 0.18 1.00
1.00 1.00 1.00
0.03 1.00 0.38 1.00 0.60 1.00
0.11 0.17 1.00 0.41 0.32 1.00 0.72 0.64 1.00
0.18 0.08 0.10 1.00 0.20 0.26 0.33 1.00 0.29 0.27 0.28 1.00
IV V VI
1.00 1.00 1.00
0.21 1.00 0.41 1.00 0.59 1.00
0.36 0.26 1.00 0.49 0.36 1.00 0.68 0.60 1.00
0.36 0.43 0.34 1.00 0.49 0.44 0.32 1.00 0.43 0.43 0.40 1.00
1.00 1.00 1.00
0.10 1.00 0.22 1.00 0.55 1.00
0.12 0.16 1.00 0.39 0.49 1.00 0.67 0.65 1.00
0.46 0.26 0.30 1.00 0.42 0.38 0.42 1.00 0.48 0.46 0.45 1.00

 

124

Variance-Covariance Matrices

of Sub-Samples

 

‘91-)
...a

*)
N

 

 

 

 

Class
I II III
.250 .240 .240
.040 .250 .100 .240 .120 .240
.030 .040 .250 .070 .080 .250 .150 .150 .250
.040 .020 .020 .150 .050 .040 .050 .160 .030 .020 .030 .150
.250 .250 .250
.010 .250 .090 .250 .150 .250
.020 .040 .250 .090 .070 .240 .180 .160 .250
.030 .010 .020 .150 .040 .050 .060 .170 .050 .050 .050 .150
IV V VI
.250 .250 .240
.050 .250 .100 .250 .140 .250
.090 .060 .250 .120 .090 .250 .160 .150 .250
.060 .080 .060 .150 .090 .080 .060 .150 .080 .080 .070 .140
.250 .240 .250
.020 .240 .050 .250 .140 .250
.030 .040 .250 .090 .120 .250 .160 .160 .240
.080 .050 .050 .150 .080 .070 .080 .150 .090 .080 .080 .140

 

APPENDIX B

THE ESTIMATED PARAMETERS FOR EACH

PROBABILITY MODEL FOR EACH CLASS

Estimated Parameters Binary Regression (LS)

 

 

 

 

 

 

 

 

 

 

 

Class
I II III IV V VI
Constant .03 .03 .10 -.14 -.07 —.04
Symptom 1 .17 .14 .07 .27 .28 .17
Symptom 2 .07 .09 -.02 .19 .22 .18
Symptom 3 .06 .14 .11 .12 .03 .09
Estimated Parameters Binary Regression (Weighted LS)
Class
I II III IV V VI
Constant .07 .02 .11 -.27 .04 .02
Symptom l .16 .09 .05 .23 .28 .14
Symptom 2 .08 .14 .02 .24 .14 .17
Symptom 3 .01 .02 .09 .34 -.05 .03
Estimated Parameters Binary Regression (Ridge)
Class
I II III IV V VI
Constant
Symptom 1 .05 .12 .05 .19 .06 .12
Symptom 2 .11 .06 .01 .14 .16 .13
Symptom 3 .05 .12 .06 .10 .19 .09

 

126

Estimated Parameters Binary Regression (Weighted Ridge)

 

 

 

 

 

 

 

Class
I II III IV V VI
Constant
Symptom 1 .16 .15 .07 -.07 .18 .14
Symptom 2 .11 .10 .18 .13 .19 .16
Symptom 3 .06 .15 .06 .15 .04 .07
Estimated Parameters Logistic Discrimination (LD)
Class
I II III IV V VI
Constant -2.75 -3.07 -2.10 -6.78 —6.69 -9.25
Symptom 1 1.20 1.12 .52 3.48 3.27 5.87
Symptom 2 .53 .40 -.14 2.15 3.04 1.69
Symptom 3 .44 1.31 .75 1.46 .66 1.46

 

APPENDIX C

THE ESTIMATED DIAGNOSTIC PROBABILITIES
FOR EACH 2P PATTERN FOR EACH

PROBABILITY MODEL FOR EACH CLASS

127

 

 

 

Estimated Diagnostic Probabilities for 2p Possible Pattern for p = 3
Bayesian (B)
Class

Pattern I II III IV V VI
111 .35 .42 .27 .59 .60 .43
110 .21 .13 .10 .22 .35 .18
100 .08 .14 .09 .00 .00 .00
001 .00 .07 .26 .00 .00 .00
011 .14 .24 .00 .00 .00 .00
101 .38 .63 .35 .18 .06 .13
010 .18 .15 .25 .05 .06 .00
000 .08 .02 .10 .00 .00 .00

 

Estimated Diagnostic Probabilities for 2p Possible Pattern for p = 3
Binary Regression (BLS)

 

 

 

Class
Pattern I II III IV V VI
111 .33 .39 .29 .45 .47 .39
110 .27 .25 .18 .32 .43 .31
100 .20 .16 .18 .13 .22 .12
001 .09 .17 .22 .00 .00 .04
011 .17 .26 .22 .17 .18 .23
101 .26 .30 .29 .25 .25 .21
010 .11 .11 .11 .05 .15 .14

000 .03 .02 .11 .00 .00 .00

 

128

Estimated Diagnostic Probabilities for 2p Possible Pattern for p = 3
Binary Regression (BWLS)

 

 

 

Class
Pattern I II III IV V VI
111 .31 .39 .27 .54 .41 .34
110 .30 .25 .18 .19 .46 .33
100 .22 .16 .16 .00 .32 .16
001 .07 .17 .19 .06 .00 .05
011 .15 .26 .22 .30 .13 .22
101 .23 .30 .25 .29 .27 .19
010 .14 .11 .13 .00 .18 .19
000 .07 .02 .11 .00 .04 .02

 

Estaimted Diagnostic Probabilities for 2p Possible Pattern for p = 3
Binary Regression-Ridge (BR)

 

 

 

Class
Pattern I II III IV V VI
111 .22 .30 .12 .43 .42 .35
110 .16 .18 .06 .32 .23 .26
100 .05 .12 .05 .18 .06 .12
001 .05 .12 .06 .10 .19 .09
011 .17 .17 .07 .24 .36 .23
101 .10 .24 .11 .29 .26 .22
010 .11 .06 .01 .14 .16 .13

000 .00 .00 .00 .00 .00 .00

 

129

Estimated Diagnostic Probabilities for 2p Possible Pattern for p = 3
Binary Regression (BRWLS)

 

 

 

Class
Pattern I II III IV V VI
111 .34 .40 .31 .21 .42 .37
110 .27 .25 .25 .06 .37 .30
100 .16 .14 .07 .00 .18 .14
001 .06 .15 .06 .15 .04 .07
011 .18 .25 .24 .28 .24 .23
101 .22 .29 .13 .08 .23 .19
010 .11 .10 .18 .13 .19 .16
000 .00 .00 .00 .00 .00 .00

 

Estimated Diagnostic Probabilities for 2p Possible

Logistic Discrimination (LD)

Pattern for p = 3

 

 

 

Class
Pattern I II III IV V VI
111 .36 .44 .28 .65 .57 .44
110 .27 .17 .15 .29 .40 .15
100 .17 .12 .17 .05 .03 .03
001 .09 .14 .21 .00 .00 .00
011 .14 .20 .18 .05 .05 .00
101 .25 .34 .33 .18 .06 .13
010 .09 .06 .10 .01 .02 .00
000 .06 .04 .11 .00 .00 .00

 

130

Estimated Diagnostic Probabilities for 2p Possible Pattern for p = 3

Entropy Minimax Pattern Discovery (EMPD)

 

 

 

 

Class
Pattern I I I I I I IV V VI
111 .36 (.17) .41 (.25) .28 (.35) .58 (.23) .57 (.26) .44 (.36)
110 .24 (.09) .15 (.09) .17 (.03) .23 (.07) .38 (.07) .25 (.02)
100 .09 (.11) .15 (.09) .11 (.19) .01 (.02) .07 (.04) .06 (.01)
001 .03 (.02) .10 (.04) .37 (.08) .01 (.02) .01 (.02) .00 (.02)
011 .14 (.06) .25 (.09) .06 (.01) .58 (.02) .04 (.01) .44 (.02)
101 .38 (.10) .57 (.04) .37 (.08) .18 (.08) .07 (.04) .16 (.06)
010 .19 (.08) .18 (.05) .28 (.35) .08 (.04) .10 (.04) .00 (.02)
000 .09 (.11) .04 (.05) .11 (.19) .01 (.02) .01 (.02) .00 (.02)
Note: The entries in the parentheses are the entropy values (H) of the

particular pattern within the particular class.

APPENDIX D

THE PREDICTIVE INDICES OF EACH MODEL

FOR EACH CLASS

131

Predictive Indices

 

 

 

 

 

 

 

Bayesian

Class
Indices I II III IV V VI
SEN .65i.17 .81i.14 .82i.13 .78:.14 .90i.10 .92:.09
SPEC .66i.08 .63i.08 .S4i.08 .84i.06 .78:.07 .77i.07
PRED .3li.11 .37i.11 .29i.10 .52i.15 .50i.13 .47i.13
E1 .35i.l7 .l9i.14 .18i.13 .211.14 .lOi.10 .07i.09
E2 .34i.08 .37i.08 .46i.O8 .16i.06 .22i.07 .22:.07
Note: Results are reported as estimate i standard error.

Predictive Indices
Bayesian (BB)

Class
Indices I II III IV V VI
SEN .65 i .17 .81: .14 .82 i .13 .78 i .14 .90 i .10 .96 i .07
SPEC .66i .08 .631t .18 .54 i .08 .841r .06 .78i .07 .74: .07
PRED .31-41.11 .37i.11 .29i.10 .52i.15 .50:.13 .45i.12
E1 .34i.17 .191.14 .181.13 .211.14 .10:.10 .04i.07
E2 .34i .08 .37: .08 .46: .08 .16i .06 .21:r .07 .262: .07

132

Predictive Indices
Binary Regression (LS)

 

 

 

Class
Indices I II III IV V VI
SEN .72i.16 .83i.13 .78i.14 .93i.09 1.00:.00 .96i.07
SPEC .Sli.08 .54i.08 .58i.08 .75i.07 .53i.08 .651.08
PRED .25i.09 .33i.10 .30:.10 .45:.12 .34i.10 .38i.11
El .27i.16 .l6i.13 .22i.l4 .07i.09 .OOi.00 .04i.07
E2 .49i.08 .45:.08 .42:.08 .25i.07 .47i.08 .35.+..08

 

Note: Results are reported as estimate 1 standard error.

Predictive Indices
Binary Regression (WLS)

 

 

 

Class
Indices I II III IV V VI
SEN .72i.16 .84i.13 .78i.l4 .82i.13 1.00:.00 .96i.07
SPEC .51 i .08 .54 i .08 .64 i .08 .75 i .07 .53 i .08 .69 i .08
PRED .251.07 .33i.10 .33i.11 .42i.13 .34i.10 .40:.11
E1 .27i.16 .16i.13 .22:.14 .18i.13 .OOi.00 .04I.07
E2 .49i.08 .46i.08 .36i.08 .251.07 .47i.08 .31i.08

 

133

Predictive Indices
Binary Regression-Ridge (BR)

 

 

 

 

 

 

 

Class
Indices I II III IV V VI
SEN .271L .15 .66i.17 .00: .08 1.001L .00 .00i.00 .96i.07
SPEC .84i.06 .68:.08 1.00:.00 .66i.08 .601.08 .65i.08
PRED .29i.17 .35:.12 .00.+..00 .40:.11 .38i.10 .38.+_.11
El .72i.15 .34i.l7 1.00:.00 .00i.00 .00i.00 .04t.07
E2 .15t.06 .32 $.08 .00i.00 .33 i.08 .40:.08 .35:.08
Note: Results are reported as estimate standard error.

Predictive Indices
Binary Regression (BRWLS)

Class
Indices I II III IV V VI
SEN .65:.17 .84i.13 .7li.16 .7Si.15 1.00:.00 .96i.07
SPEC .66i.08 .54i.08 .67i.08 .72 $.08 .601.08 .651.08
PRED .321.1l .33i.10 .33i'.1l .38i.12 381.10 .381.11
E1 .34i.l7 .15i.13 .29i.16 .25i.15 .OOi.00 .O4i.07
E2 .34i.08 .46i.08 .331.08 .28i.08 .39i.08 .35i.18

 

134

Predictive Indices
Logistic Discrimination (LD)

 

 

 

 

 

 

 

Class
Indices I II III IV V VI
SEN .65i.17 .81:.14 .78:.l4 .78t.l4 .90:.10 .92i.09
SPEC .66t.08 .63i.08 .6li.OB .84i.06 .78i.07 .77i.07
PRED .31i.ll .37i.11 .32i.10 .52i.15 .50i.13 .47i.13
E1 .34i.17 .l9i.14 .22i.14 .21i.14 .10i.10 .07i.09
E2 .34i.08 .37i.08 .39i.08 .16:.06 .21i.07 .221-.07
Note: Results are reported as estimate i standard error.

Predictive Indices
Entropy Minimax Pattern Discovery (EMPD)

Class
Indices I. II III IV V VI
SEN .65i.l7 .81:.l4 .821.13 .78:.14 .90:.10 361.07
SPEC .66 i .08 .63 i .08 .54 i .08 .84 i .06 .78 i .07 .74 i .07
PRED .31i.11 .37i.11 .29i.10 .52t.15 .501.13 .45:.12
El .343:.17 .18i.14 .18i.13 .21:.14 .10:.10 .04i.07
E2 .34i.08 .371.08 .461.08 .l6i.06 .21i.07 .261.07

 

APPENDIX E

THE PREDICTIVE INDICES OF EACH MODEL

WHEN PREDICTING TO A DIFFERENT

RELATIONAL STRUCTURE

135

Prediction from Prior Knowledge of One Relational Structure to

Another Different Relational Structure

Bayesian (B) Model

 

 

 

 

 

 

Class
Prior
Knowledge I III IV VI Average
SEN .65 .78 .93 .96 .89
SPEC .66 .66 .74 .70 .70
I PRED .31 .34 .45 .41 .40
El .35 .22 .07 .03 .11
E2 .34 .34 .25 .29 .29
SEN .55 .82 .86 .92 .78
SPEC .65 .54 .66 .70 .67
III PRED .28 .29 .37 .40 .35
El .45 .18 .14 .08 .22
E2 .35 .46 .34 .30 .33
SEN .45 .71 .78 .92 .69
SPEC .76 .67 .84 .74 .72
IV PRED .30 .33 .52 .45 .36
E1 .55 .28 .21 .08 .31
E2 .24 .33 .16 .26 .28
SEN .45 .71 .78 .92 .65
SPEC .76 .67 .83 .77 .75
VI PRED .30 .33 .52 .47 .38
E1 .55 .28 .22 .07 .35
E2 .24 .33 .17 .22 .25

 

Prediction from Prior Knowledge of One Relational
Another Different Relational Structure
Bayesian W/Baduhur (BB) Model

136

Structure to

 

 

 

 

 

 

Prior
Knowledge I III IV VI Average

SEN .65 .78 .96 .96 .90

SPEC .66 .66 .66 .69 .67

I PRED .31 .34 .39 .41 .38
E1 .34 .21 .03 .03 .09

E2 .34 .34 .33 .30 .32

SEN .62 .82 .89 .96 .82

SPEC .54 .54 .53 .60 .56

III PRED .25 .29 .30 .35 .30
El .38 .18 .11 .04 .18

E2 .46 .46 .47 .40 .44

SEN .44 .71 .78 .96 .70

SPEC .76 .70 .84 .73 .73

IV PRED .30 .36 .52 .44 .37
El .55 .29 .21 .03 .29

E2 .23 .30 .16 .26 .26

SEN .45 .71 .85 .96 .67

SPEC .76 .70 .79 .74 .75

V PRED .30 .36 .48 .45 .38
El .55 .29 .14 .04 .33

E2 .23 .30 .21 .26 .25

 

137

Prediction from Prior Knowledge of One Relational Structure to

Another Different Relational Structure
Binary Regression (BLS) Model

 

 

 

 

 

 

Class
Prior
Knowledge I III IV VI Average
SEN .76 .78 .92 .96 .89
SPEC .51 .59 .60 .62 .60
I PRED .55 .31 .35 .35 .34
El .24 .22 .07 .03 .11
E2 .49 .41 .39 .37 .39
SEN .62 .78 .86 .92 .80
SPEC .51 .58 .51 .65 .56
III PRED .23 .30 .28 .37 .29
El .38 .22 .14 .08 .20
E2 .48 .42 .49 .35 .44
SEN .65 .78 .93 .96 .80
SPEC .65 .66 .75 .70 .67
IV PRED .31 .34 .45 .41 .35
El .34 .22 .07 .03 .20
E2 .34 .34 .25 .29 .32
SEN .72 .78 1.00 .96 .83
SPEC .52 .62 .66 .65 .60
VI PRED .26 .32 .40 .38 .33
El .27 .21 .00 .04 .16
E2 .47 .38 .33 .35 .39

 

138

Prediction from Prior Knowledge of One Relational Structure to

Another Different Relational Structure
Entropy Minimax Pattern Discovery (EMPD) Model

 

 

 

 

 

 

Class
Prior
Knowledge I III IV VI Average
SEN .65 .78 .93 .96 .89
SPEC .66 .66 .75 .70 .70
I PRED .31 .34 .45 .41 .40
El .34 .22 .07 .04 .11
E2 .34 .34 .25 .30 .30
SEN .62 .82 .89 .96 .72
SPEC .54 .54 .53 .60 .56
III PRED .25 .29 .30 .35 .30
El .38 .18 .11 .04 .28
E2 .46 .46 .47 .40 .44
SEN .45 .71 .78 .92 .69
SPEC .76 .67 .84 .74 .72
IV PRED .30 .33 .52 .45 .36
El .55 .28 .21 .08 .31
E2 .24 .33 .16 .26 .28
SEN .45 .71 .78 .96 .65
SPEC .76 .67 .83 .74 .75
VI PRED .30 .33 .52 .45 .38
E1 .55 .28 .22 .04 .35
E2 .24 .33 .17 .26 .25

 

139

Prediction from Prior Knowledge of One Relational Structure to
Another Different Relational Structure
Logistic Discrimination (LD) Model

 

 

 

 

 

 

Class
Prior
Knowledge I III IV VI Average

SEN .65 .78 .93 .96 .89

SPEC .66 .66 .74 .70 .70

I PRED .31 .34 .45 .41 .40
E1 .34 .22 .07 .03 .11

E2 .34 .34 .25 .29 .29

SEN .55 .78 .86 .92 .78

SPEC .65 .61 .66 .70 .67

III PRED .28 .32 .37 .40 .35
E1 .45 .22 .14 .08 .22

E2 .35 .39 .34 .30 .33

SEN .45 .71 .78 .96 .71

SPEC .76 .70 .84 .74 .73

IV PRED .31 .36 .52 .44 .37
E1 .55 .29 .21 .03 .29

E2 .23 .30 .16 .25 .26

SEN .27 .71 .60 .92 .53

SPEC .84 .72 .92 .77 .83

VI PRED .29 .37 .63 .47 .43
El .72 .28 .39 .07 .46

E2 .15 .27 .08 .22 .17

 

APPENDIX F

THE RELATIONAL STRUCTURE FOR

EACH SPECIAL CLASS

140

Correlational and Variance-Covariance Matrices for Special Classes

 

 

 

 

Class
Mixed Suppressor High Correlation
1.00 1.00 1.00
0.20 1.00 0.40 1.00 0.80 1.00
0.40 0.60 1.00 0.01 0.10 1.00 0.80 0.80 1.00
0.50 0.50 0.20 1.00 0.09 0.40 0.20 1.00 0.50 0.50 0.50 1.00
1.00 1.00 1.00
0.55 1.00 0.36 1.00 0.70 1.00
0.27 0.37 1.00 0.01 0.10 1.00 0.77 0.71 1.00
0.46 0.48 0.26 1.00 0.08 0.37 0.15 1.00 0.46 0.47 0.41 1.00
.250 .250 .250
.200 .250 .100 .250 .400 .250
.100 .125 .250 .002 .020 .250 .400 .400 .250
.090 .100 .040 .160 .020 .080 .040 .160 .100 .100 .100 .160
.250 .250 .250
.140 .250 .090 .250 .350 .250
.070 .100 .250 .003 .030 .250 .380 .350 .250
.100 .100 .050 .150 .018 .070 .030 .160 .080 .080 .070 .140

 

141

Correlational and Variance-Covariance Matrices
for Special Classes for Split-Samples

 

 

 

 

Class
Mixed Suppressor High Correlation
1.00 1.00 1.00
R 0.49 1.00 0.24 1.00 0.76 1.00
l 0.24 0.34 1.00 0.08 0.14 1.00 0.84 0.76 1.00
0.46 0.48 0.25 1.00 0.10 0.46 0.04 1.00 0.48 0.47 0.45 1.00
1.00 1.00 1.00
R 0.59 1.00 0.49 1.00 0.78 1.00
2 0.29 0.40 1.00 0.06 0.06 1.00 0.83 0.78 1.00
0.49 0.50 0.28 1.00 0.08 0.28 0.27 1.00 0.45 0.46 0.42 1.00
.250 .250 .250
t .120 .250 .060 .250 .380 .250
1 .060 .090 .250 .020 .030 .240 .420 .380 .250
.090 .090 .040 .150 .020 .090 .008 .160 .090 .090 .080 .140
.250 .250 .250
t .140 .250 .120 .250 .400 .250
2 .070 .100 .250 .010 .020 .250 .400 .400 .250
.090 .090 .050 .150 .010 .050 .050 .150 .080 .080 .070 .140

 

APPENDIX G

THE ESTIMATED DIAGNOSTIC PROBABILITIES AND

PREDICTIVE INDICES FOR EACH MODEL

WITHIN EACH SPECIAL CLASS

142

Estimated Parameters and Estimated Diagnostic Probabilities
for 2p Number of Patterns of Special Classes
Bayesian (B)

 

 

 

Class
Pattern Mixed Suppressor High Correlation
111 .54 .40 .43
110 .42 .42 .30
100 .00 .00 .00
001 .00 .04 .00
011 .00 .35 .00
101 .00 .00 .00
010 .00 .45 .00
000 .00 .01 .00

 

Estimated Parameters and Estimated Diagnostic Probabilities
for 2P Number of Patterns of Special Classes

Bayesian W/Baduhur (BB)

 

 

 

Class
Pattern Mixed Suppressor High Correlation
111 .5250 .4000 1.4262
110 .4118 .4118 .2500
100 .0000 .0000 .0000
001 .0000 .0476 .0000
011 .0000 .3333 .0000
101 .0000 .0000 .0000
010 .0000 .4444 .0000

000 .0000 .0400 .0000

 

143

Estimated Parameters and Estimated Diagnostic Probabilities
for 29 Number of Patterns of Special Classes

Binary Regression (BLS)

 

 

 

 

 

Estimated Class
Parameters Mixed Suppressor High Correlation
Constant -.08 .04 -.02
Symptom 1 .22 —.01 .20
Symptom 2 .25 .37 .17
Symptom 3 .06 —.02 .04
Pattern Mixed Suppressor High Correlation

111 .45 .38 .39

110 .39 .40 .35

100 .14 .03 .18

001 .00 .02 .01

011 .23 .39 .19

101 .19 .01 .22

010 .17 .41 .15

000 .00 .04 .00

 

.fl ‘

I’ln‘n' K '0.-

29.

 

56““

0 2"- '-(_M_.II r'

 

144

Estimated Parameters and Estimated Diagnostic Probabilities
for 2p Number of Patterns of Special Classes

Binary Regression (BWLS)

 

 

 

 

 

Class

Estimated
Parameters Mixed Suppressor High Correlation
Constant .11 .04 .005
Symptom 1 .12 -.O4 .22
Symptom 2 .23 .38 .15
Symptom 3 -.13 .00 .002
Pattern Mixed Suppressor High Correlation

111 .32 .38 .37

110 .45 .38 .33

100 .22 .00 .22

001 .00 .04 _ .00

011 .20 .42 .15

101 .09 .00 .24

010 .34 .42 .15

000 .11 .04 .00

 

145

Estimated Parameters and Estimated Diagnostic Probabilities
for 2p Number of Patterns of Special Classes
Binary Regression-Ridge (BR)

 

 

 

 

 

Classes

Estimated
Parameters Mixed Suppressor High Correlation
Constant
Symptom 1 .06 .00 .12
Symptom 2 .17 .21 .12
Symptom 3 .16 .06 .09
Pattern Mixed Suppressor High Correlation

111 .39 .23 .34

110 .23 .21 .25

100 .06 .00 .12

001 .16 .02 .09

011 .33 .23 .21

101 .22 .02 .21

010 .17 .21 .12

000 .00 .00 .00

 

146

Estimated Parameters and Estimated Diagnostic Probabilities
for 29 Number of Patterns of Special Classes

Logistic Discrimination (LD)

 

 

 

 

 

‘ Class

Estimated
Parameters Mixed Suppressor High Correlation
Constant -19.15 -3.54 -l7.40
Symptom 1 9.35 -.08 8.13
Symptom 2 9.44 3.27 8.17
Symptom 3 .46 -.16 .80
Pattern Mixed Suppressor High Correlation

111 .52 .37 .57

110 .41 .41 .25

100 .00 .03 .00

001 .00 .02 .00

011 .00 .39 .00

101 .00 .02 .00

010 .00 .43 .00

000 .00 .03 .00

 

147

Estimated Parameters and Estimated Diagnostic Probabilities
for 2P Number of Patterns of Special Classes

Entropy Minimax Pattern Discovery (EMPD)

 

 

 

 

Class

Pattern Mixed Suppressor High Correlation
111 .53 (.26) .40 (.19) .43 (.40)
110 .42 (.11) .41 (.11) .31 (.02)
100 .00 (.02) .01 (.02) .06 (.01)
001 .00 (.02) .05 (.09) .006 (.02)
011 .03 (.02) .35 (.09) .43 (.02)
101 .00 (.02) .01 (.02) .06 (.01)
010 .03 (.02) .45 (.05) .006 (.02)
000 .00 (.02) .05 (.09) .006 (.02)

Note: The entries in the parentheses are the entropy values (H) for

that particular pattern within that particular class.

148

Predictive Indices for Special Classes for Various Models

 

 

 

 

 

 

 

 

 

Class
Models Indices Mixed Suppression High Correlation
SEN 1.001 .00 .76$ .15 1.00$ .00
SPEC .761 .07 .601 .08 .631 .08
B PRED .491 .12 .311 .10 .371 .11
E1 .001 .00 .241 .15 .001 .00
E2 .24 1 .07 .401 .08 .37 1 .08
SEN 1.001 .00 .76$ .15 1.00$ .00
SPEC .761 .07 .60 $ .08 .63 $ .08
BB PRED .49$.12 .31$.10 .37$.11
El .00 1 .00 .24 $ .15 .00 $ .00
E2 .24 1 .07 .40 $ .08 .37 $ .08
SEN 1.00 1 .00 .76 $ .15 1.00 $ .00
SPEC .67 1 .08 .60 1 .08 .59 $ .08
BLS PRED .411.11 .311.10 .35$.10
E1 .001.00 .241.15 .00$.00
E2 .331.08 .401.08 .41$.08
SEN 1.001.00 .76$.15 1.00$.00
SPEC .58 1.08 .60$.08 .55 $.08
BWLS PRED .35$.10 .31$.10 .33-$.10
El .00$.00 .24$.15 .00i.00
E2 .42 $ .08 .40 $ .08 .45 i .08
SEN 1.00 1 .00 .76 $ .15 1.00 $ .00
SPEC .61 1 .08 .60 $ .08 .58 $ .08
BR PRED .371.10 .31$.10 .34$.1o
E1 .00 1 .00 .24 $.15 .00 $.00
E2 .39 1.08 .40 $.08 .42 $.08
SEN 1.00 $.00 .76 $.15 1.00 $.00
SPEC .76 1.07 .60 1.08 .63 $.08
LD PRED .49 1.12 .31 1.10 .37 $.11
E1 .00 1.00 .24 1.15 .00 $.00
E2 .24 1.07 .41 1.08 .37 $.08
SEN 1.00 1.00 .76 1.15 1.00 1.00
SPEC .76 $.07 .60 $.08 .61 $.08
EMPD PRED .49 $.12 .31 $.10 .36 $.10
E1 .00 $.00 .24 1.15 .00 $.00
E2 .24 $.07 .41 $.08 .39 $.08

Note: Results are reported as estimate i standard error.

APPENDIX H

BRAIN SCAN EVALUATION QUESTIONNAIRE

149

 

 

 

 

 

 

 

COO-2427-5
I. Eatiggt Identification
JHH History Number
Patient Name
Sex: ----- 0 Male
5 0 Female
2
Age: Fill in or use JHH Patient
Identification Card.
0 Outpatient
O Inpatient
11. W
1. Physician Filling Out Form
2. Date
.. 3. Is the decision to do a brain 0 yes
3 scan based (in part) on the (3 no
0 results of another diagnostic
D procedure?
If yes, what was it?
m
g 4. Has the patient had a: yes no normal abnormal
It
E Lumbar Puncture ------ O O 0 0
g EEG -------------- O O O O
9 Skull x-ray -------- O O O 0
f3 Arteriorgram ------- O O 0 0
5 Echo ------------ O O O O
a
g . . .
9 III. n 1 M v
m
g 1. Efficacy:
E The use of a diagnostic procedure is motivated by
g efficacy if the outcome of the procedure could con-
g tribute to-or effect-a change in the course of the
a patient's disease.
g 2. Defense:
0
m A procedure is being used defensively if its use is
E motivated by either potential peer incrimination or
5 legal responsibility.
m

3. Innovation-Curiosity

Innovation-Curiosity is the principal motivating
force if the objective in ordering a procedure is
simply to find out what the result will be for this
particular case. It may even help the patient.

 

150

COO-2427-5
IV. Historical Data
1. Headache ———————————— Oyes Ono

a) Duration — -- - ——--- O<1 Week
01 Week to 1 Mo.
0 1 Mo. to 3 Mo.
0>3 Mo.

b) Continuity -------- 0 Continuous
O Intermittent

c) Severity ———————— O Mild
0 Moderate
0 Severe

d) Location if diffuse-- -— O Bilateral
O Unilateral

e) Location if focal ----- Retroorbital
Frontal
0 Temporal
O Parietal
O Occipital

2. Seizure ----------- 0 yes 0 no
a) Number of Episodes - - - - 0 Single (First)

0 Multiple > 10
0 Multiple < 10 (Longstanding)

b) Location --------- O Generalized
0 Focal
c) Type ----------- 0 Major Motor

0 Minor Motor
0 Temporal Lobe

O Other
d) Is Seizure Pattern - --- 0 yes
Changing? 0 no

e) Pertinent Family History 0 yes
of Seizures 0 no
0 unknown

3. Neoplasm ——————————— 0 yes 0 no Osuspect ‘
a) Location ---------- 0 Brain
0 Lung
0 Breast
O Other

b) Pertinent Family History Oyes
of Neoplasm 0 no
0 unknown

4. History of Trauma ------- Oyes Ono

V.

151

Physical Examination

1.

Cortical Deficit ———————

a) State of Consciousness
(Indicate by an X anywhere
on the scale)

b) Generalized Deficit - —1—-
If "Other” what is it?

C) Focal Deficit -------
If yes, what is it?

Motor Deficit ---------

a) If yes: Location

Lateralization
Severity
b) Ataxia __________
Type?
c) Involuntary Movement-—--—-
Type?
d) Reflex Abnormality -----
Type?
e) Abnormal Gait -------
Sensory Abnormality -------

If yes, what is it?
Visual Field Defect -------
If yes, what is it?

Alteration of Brain Stem----
Function Including Eye

'Movements

If yes, what is it?

(Dyes
C)no

Mun-WNH

O Dementia
COther Abnormality

COO-2427-S

Normal

Abnormal

 

()yes
()no

 

0 yes
C)no

 

1

 

UibUNH

()yes
0110

Mild

Severe

 

()yes
()no

 

Oyes
()no

 

0 yes
()no

Oyes
()no

 

 

Oyes
()no

 

0 Yes
C)no

 

1152

C00-2427-S

VI. Prospective Outcomes

1. Subjective Probabilities (Mark anywhere on the scale, probabil-
ities need not total 100%)

a) In your opinion what is the
probability that this brain 0% 20% 40% 60%
scan will be normal E 4' { {

0% 100%

b) With what probability do
you suspect each of the
following diagnosis?

Note:‘ Probabilities need not
total 100%

N
O
as

40% 60% 80% 100%

 

 

Subdural Hematoma

Vascular Malformation

 

F — —— o
'7 1 w
--1— c-u— ~1—
-— -_ —0—
—L— dh- ——
nun—- u—(h—v u-Iu—

Stroke (e.g., TIA, Hemorrhage
or Infarction)

 

Cerebral Infection

 

Cerebral Tumor (Primary)
Cerebral Tumor (Metastasis)

Other Pathology

 

—— —— Inq— -—
our-— --11-— 1.1-m— —-+-—

c) What is your Presumtive
Diagnosis?

 

d) What do you feel the odds Certain Even Remote
are that your diagnosis 10:1 1:1 1:10
is correct? L. I J

._- l 1...:fi

e) Will you alter your manage-
ment of this patient if the
result of this brain scan is:

(i) Normal Oyes 0 no

 

(ii) Abnormal Oyes Ono

Taking total motivation to request brain scan as 100 - How do you dis-
tribute your subjective motivation over the following reasons (as
defined above) for requesting this examination?

Efficacy

Defense
Innovation-Curiosity
Other

flflfifi

Total 100

APPENDIX I

THE RELATIONAL STRUCTURE FOR

THE BRAIN SCAN STUDY

L~I - ----r
5

'1! '.-‘."

“1' '

 

«fl-au: us . l I
.

 

153

Correlation Matrix and Base Rates of the Symptoms
and the Disease for Brain Scan

 

 

Headache 1.000

Seizure -.006 1.000

Cort. def. -.256 -.079 1.000

Motor Def. -.081 -.172 .318 1.000

Sen. ab. .054 -.035 -.151 .215 1.000

Visual -.052 -.113 .151 .031 -.190 1.000

Outcome -.015 -.130 .215 .225 .097 .064 1.000
Pheadache = '52 Pseizure = '31
Pcort. def. = '31 Pmotor def. = '30
Psen. ab. = '24 Pvisual = '17

P = .10
outcome

 

 

 

APPENDIX J

THE RELATIONAL STRUCTURE FOR THE SPLIT SAMPLES

OF THE BRAIN SCAN STUDY

 

 

154

 

 

 

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OmH. OHO.I mOO.I OmO. OMO.I ONO.I OVH. omO.l ONO. ONO. OHO.I NOO.
OhH. OHO. OVO.I OmO. OmO. OON. OhO. ONO.I OmO.l NOO.I
OON. OOO. OhO.l OHO.I ONN. OmO. NOO. ONO.I
OHN. OOO.I OmO.l ONN. OmO. OOO.I
OON. OHO. OmN. OHO.I
OmN. OmN.
OO.H mo. 0H.l Wm. Hm. NO.I mO.l 50. mm. NH. NH. MN.I mO.
OO.H mO.l mo.l ON. OH.I NH.I OO.H NN.! 00. mo. mO.l HO.
OO.H mo. HN.I mH. mH. OO.H VM. OH.I MN.I HO.I
OO.H mv. mm.l 00.! OO.H mH. HO. $0.!
OO.H Om.l MN.I OO.H NH. mN.l
OO.H mo. OO.H MO.I
OO.H OO.H
HH mamemm H THQEMm

 

mmHmEMm uflamm now sumo swomnaflmum How mooauumz

OUCMHHMKIOUIOUCMflHMNr mug fiOHUMH GHHOU

APPENDIX K

THE ESTIMATED PARAMETERS AND PREDICTIVE INDICES

OF EACH MODEL FOR THE BRAIN SCAN STUDY

I!

"" .Illfisalj 2):.

 

Estimated Parameters for Brain

155

Scan by Various Models

 

 

 

..' . l‘l—‘fl

 

 

 

 

 

Models

Symptom BLS BWLS BR BRWLS LD

Constant -.02 -.06 .00 -1l.23
Headache .06 .13 .03 1.96 .58
Seizure -.09 .41 -.07 -6.38 —8.57
Cortical deficit .12 -.29 .12 .39 1.58
Motor deficit -.03 .08 .06 -2.15 -.47
Sensory abnormal .24 .15 .05 .25 9.89
Visual defect .12 .09 .02 -.05 9.16

Prediction Indices for Various Models for Brain Scan
Models

Indices BB BLS BWLS BR BRWLS LD EMPD
SEN .00 .50 .25 .75 .75 .20 .00
SPEC .95 .54 .61 .69 .41 .74 .97
PRED .00 .10 .06 .20 .11 .09 .00
E1 1.00 .50 .75 .25 .25 .75 1.00
E2 .05 .46 .38 .31 .59 .26 .02

W’suu! 0'4“; . ..'-

 

APPENDIX L

GRAPH FOR THE PREDICTIVE INDICES FOR EACH

PROBABILITY MODEL WHEN THE INTERCORRELATION

OF THE SYMPTOMS ARE LOW

SENSITIVITY INDEX

156

PREDICTIVE EFFICIENCY INDEX

 

.9000

.700001

.50000

.80000'

.10000

T

 

-.1000

J

<)-
r-
m-
1-
1-
..
....

J

    

 

 

 

0 ' 1.&m

1
r

2110
CLHSSES

3.000

 

 

LEGEND
-9- B

-B- 88
-+- BLS
-)(- BWLS
+ BR
+ BRWLS
* L0
+ EMPD

 

 

Y
SPECIFICFT

.900

0700

.500!

13000

~1000

~01000t

SPECIFICPIY

PREDICTIVE EFFICIENCY INDEX

157

 

.90000-

.700001-

.60000+>

.30000

.10000

T

r

 

“01000

J l
I

)4

l l
1

 

1\.

 

 

1110

24h0
CLHSSES

1
fi

3.000

 

 

 

LEGEND
-€- 8

-B- 88
+ BLS
-)(- BWLS
-6- BR
+ BRNLS
* L0
+ EMPD

 

 

Tm 7'1. 4. f

 

PREDICTIVE VRLUE

.7000

.5000

~3000

~1000

‘~1000

PREDICTIVE VHLUE

158

PREDICTIVE EFFICIENCY INDEX

 

.70000-

.60000-

030000"

T

.10000

 

“.1000

j

 

 

 

 

1.000

2.360
CLRSSES

j
T

3.000

 

 

MEET
-e— B

-e— 88
-1— BLS
~-)(-- BHL\$
+ SR
+ BRWLS
+ LD
41- EMPD

 

 

APPENDIX M

GRAPH FOR THE PREDICTIVE INDICES FOR EACH
PROBABILITY MODEL WHEN THE INTERCORRELATION

OF THE SYMPTOMS ARE HIGH

 

flew-g

SENSITIVITY INDEX

159

PREDICTIVE EFFICIENCY INDEX
Q -

 

 

.90000

1

.700001

.50000

1

I

.30000'

r

.100001

 

 

-.1000 4 I i
3.000 4.000

 

l
I

5 .300“ 6.000
CLASSES

 

 

LEQ£N0
-9- B

-B- 80
+ BLS
-)(- BWLS
+ BR
+ BRWLS
* [.0
+ EHPD

 

 

 

w“ Emu-..- _L-

 

SPECIFPCITY

160

PREDICTIVE EFFICIENCY INDEX

.9000

 

.70000-

.50000-

1'

.300001-

010000"

 

-' o r000

 

1

 

 

 

 

3.000

4Em1

5.300
CLRSSES

l
j

6 .000

 

 

EGEND
-6- B

-B- 88
-+- 0L3
-X- BHLS
+ BR
+ BRWLS
* L0
+ EMPD

 

 

PREDICTIVE VRLUE

161

PREDICTIVE EFFICIENCY INDEX

 

.9000

.70000

i

060000?

 

P

 

 

r

.30000

1

. 10000

 

 

' 5.000 8.000
CLHSSES

”01000 I
3.000 4.000

 

4L

 

 

LEQQND
-9- B

-B- BB
-+- BLS
96- BHLS
+ BR
+ BRWLS
-X- L0
+ EI'IPD

 

 

 

BIBLIOGRAPHY

BIBLIOGRAPHY

Anderson, J. A. Separate sample logistic discrimination.
Biometrika, 1972, 52, 19-34.

 

Anderson, J. A. Logistic discrimination with medical
application. In T. Cacoullos (Ed.), Discriminative
Analysis and Application. New York: AcadefiIc Press,
1973.

 

 

II\uI.r I.

Anderson, J. A. Diagnosis by logistic discrimination
function; further practical problems and results.
Applied Statistics, 1974, 23(3), 397-404.

 

 

Anderson, J. A., Whaley, K., Williamson, J., & Buchanan,
W. W. A statistical aid to the diagnosis of
kertoconjunctivitis sioca. Quarterl Journal
g£_Medicine, New Series, 1972, El, 1¥5-189.

Anderson, T. W. An introduction to multivariate statistical
analysis. New York: Wiley and Sons, 1958.

 

 

Bahadur, R. R. A representation of the joint distribution
of responses to n dichotomous items. In H. Solomon
(Ed.), Studies in item analysis and prediction.
Stanford: StanfEId—University Press, 1961,
pp. 1568-1588.

 

Bailey, N. T. J. Probability methods of diagnosis based
on small samples. Biology and Medicine. London:
Her Majesty's Stationery Office, I965.

Bartlett, M. S. A note on multiplying factors for various
chi-squared approximations. Journal of the Royal
Statistical Society, 1954, 16, 296-2987

 

 

 

Barnoon, 8., & Wolfe, H. Measuring the effectiveness of_
medical decisions. SpringfiéId: Cfiarles C. Thomas,

I972.

 

 

Bayes, T. Essays towards solving a problem in the doctrine
of chances. Biometrika, 45, 293~315.

 

162

163

Bean, W. B. (Ed.). Aphorisms from Osler's bedside teaching
and writings. New York: Sohuman, I950.

 

Blair, J. C. Mammography: A contrary view. Annals of
Internal Medicine, 1976, 23, 77-84. —_

Bock, R. D. Estimating multinomial response relation.
In Boss (Ed.), Essa s 23 probability and statistics.
Chapel Hill: University of North Carolina Press,
1970.

 

Bock, R. D. Multivariate statistical methods 32 behavioral
research. New York: McGraw-HIIl Book Co., 1975.

 

 

Bowman, M. J. Risk and uncertainty. In J. Gould and
W. Kalb (Eds.), 2 dictionary of the social science.
New York: The Free Press, 1964.

 

 

Box, G. E. A general distribution theory for a class of
likelihood criteria. Biometrika, 1950, 33, 317-346.

 

Christensen, R. A. A general approach to pattern discovery.
Technical Report, 1967, 22, University of California,
BerkeIey, Computer Center.

 

Christensen, R. A. Pattern discovery program for analyzing
qualitative and quantitative data. Behaviour Science,
1968, 33, 423.

 

Christensen, R. A. Entro minimax method 92 pattern
discove and proBaBiIity determination. New York:
ArtHur D. Little, Inc.,’I972.

 

Christensen, R. A. Entropy to minimax: A non-bayesian
approach to probability estimation from empirical
data. Paper presented at the International Conference
on Cybernetics and Soc., Boston, November 1973.

Cohen, J. Psychological probability. Cambridge: Schenkman
Publis ing Co., 3.

Cohen, J., & Cohen, P. Applied regression analysis for the
behavioral sgience. New Jersey: Lawrence Erlbaum
Assn. Inc. PublIShers, 1975.

Conger, A. J., & Jackson, D. N. Suppressor variables,
prediction, and the interpretation of psychological
relationships. Educational and Psychological

Measurement, 1972, 32, 579-599.

 

 

 

\‘Mi._‘,f

 

164

Cornfield, J. Bayes theorem. Review of the International
Statistical Institute, 1967, 33, 34}

 

 

Cox, D. R. Some procedures associated with the logistic
qualitative response curve. In F. N. David (Ed.),
Research pa er in statistics: Festschrift for J.
Neyman. London?— WiIey, 1966, pp. 55-7I. —_—'—

 

Cox, D. R. The analysis of pinary data. London: Methuen,
1970. "’

 

 

David, A. P. Properties of diagnostic data distribution.
Biometrics, 1976, 32, 647-658.

 

Davies, P. Symptom diagnosis using bahadur's distribution.
International Journal Bio-Medical Computing, 1972, 3,
307-312. —

 

 

Dorland's Illustrated Medical Dictionary. Philadelphia:
W. B. Saunder, 1974.

 

Draper, N. R., & Smith, H. gpplied regression analysis.
New York: John Wiley & Sons, 1966.

 

Elwood, J. H., & Mackenzie, G. The measurement and
comparison of infant mortality risk by binary multiple
regression analysis. Journal 22 Chronic Disease, 1971,
93-106.

 

Elstein, A. S. Clinical judgment: Psychological research
and medical practice. Science, 1976, 194, 696-700.

Engel, R. L., & Davis, B. J. Medical diagnosis: Present,
past, and future. Archives 92 Internal Medicine, 1963,
112' 512-519.

.Ir

 

Feldstein, M. S. A binary variable multiple regression
model of analysing factors affecting perinatal
mortality and other outcomes of pregnancy. 'Journal
92 Royal Statistical Society, Ser. A, 1966, 223, 6I-73.

 

 

Fisher, R. A. The use of multiple measurements in taxonomic
problems. Annals g£_Eugenics, 1936, 3, 376-386.

Fraser, P. M., & Franklin, D. A. Mathematical models for
the diagnosis of liver disease. Quarterl Journal 22
Medicine, New Series, 1974, 169, 73-88.

165

Galen, R., & Gambino, S. R. Beyond normality: The
predictive value and efficiency 92 medical diagnosis.
New York: Wiley, 1975.

Gustafson, D. H. Evaluation of probabilistic information

processing in medical decision making. Or anic
Behavior and Human Performance, 1969, 3, 20-34.

Haase, R. The use of multiple regression analysis to
generate conditional probabilities about the occurrence

of events. Educational and Psychological Measurement,

Hall, G. H. Clinical application of bayes' theorem.
Lancet, 1967, 2, 555.

Halperin, M., Blackwelder, W. C., & Verter, J. I.
Estimation of the multivariate logistic risk function:
A comparison of the discriminant function and maximum
likelihood approaches. Journal of Chronic Disease,
1971, 23, 125-158. ' “— "

 

Hammond, K. R., Thomas, T. R., Brehmer, B., & Steinmann,
D. 0. Social judgment theory. In M. F. Kaplan and
S. Schwartz (Eds.), Human 'ud ment and decision
processes. New York: Acagemic Press Inc., 1975.

 

 

Hartz, S., & Rosenberg, L. The computation of maximum
likelihood estimates for the multiple logistic risk
function for use with categorical data. Journal of
Chronic Disease, 1975, 2g, 421-429. “"‘ “ ‘—

 

Harvey, A. M., & Bordley, J. B. Differential giagnosis.
Philadelphia: W. B. Saunders Co., I970.

 

Hasting, N. A. J., & Peacock, J. B. Statistical
distribution. London: Butterworfhs, 1974.

 

 

Hempel, C. G. Philosophy g£_natura1 science. Englewood
Cliffs, N.J.: Prentice- aII, I9 .

Hoerl, A. E. Application of ridge analysis to regression
problem. Chemical Engineering 2£ogress, 1962, 33(3),
54-59.

 

Hoerl, A. E., & Kennard, R. W. Ridge regression: Biased
estimation for nonorthogonal problems. Technometrics,
1970a, 32, 55-67.

 

 

:‘K’Jm‘jd‘iLou . 1

166

Hoerl, A. E., & Kennard, R. W. Ridge regression:
Applications to nonorthogonal problems. Technometrigs,
1970b, 22, 69-82.

 

Johnson, N. Discrete distribution. Boston: Houghton
Mifflin Co.,CI969.

 

Jones, R. Probability estimation using a multinomial
logistic function. Journal 92 Statistical Comp.
Simulation, 1975, 3, 3I5-329.

 

 

Kaplan, A. The conduct 2: inguiry. San Francisco:
Chandler PuSIisHing Co., .

Kaufmann, A. The science of decision making, New York:
McGraw-Hill Book Co.,-I968.

 

 

Kerlinger, F. N., & Pedhazur, E. J. Multiple regression
3g_behavioral research. New York: Holt, RInehart
& Winston, Inc., 1973.

Knuth, D. The art of com uter ro rammin : Seminumerical
algoriEHfis (VoIT 2). Cambridge: Adgison-Wesley PuBZ
Co., 1969.

 

 

Kuhn, T. S. The structure of scientific revolutions.
Chicago: University oI—Cfiicago Press, I962.

 

Lord, F. M., & Novick, M. R. Statistical theories of mental

test scores. Cambridge: AddISon-Wesley PuBliSEing Co.,

 

Lubin, A. Some formulae for use with suppressor variables.
Educational and Psychological Measurement, 1957, 32,
286-296.

Lusted, L. Introdugtion Eg_medica1 decision making.
Springfield: CharIes C. Thomas,l968.

 

 

MacBryde, C. M., & Blacklow, R. 8. Signs and symptoms.
Philadelphia: J. B. Lippincott Co., I970.

Marquardt, D. W., & Snee, R. D. Ridge regression in
practice. The American Statistician, 1975, 23(1), 3-20.

 

Mihram, G. A. Simulation: Statistical foundations and
methodology. New York: Academic Press, 1972.

 

 

 

Morrison, D. F. Multivariate statistical methods. New York:

McGraw-Hill BooE Co., I976.

 

 

 

167

Neter, J., & Wasserman, W. Applied linear statistical
models. Homewood, 111.: RiEHard D. Irwin Co., 1974.

Neyman, J. First course 33 statistics and probabilipy.
New York: Holt, 1950.

 

Overall, J. E., Manhattan, K., & Williams, C. Conditional
probability program for diagnosis of thyroid function.
Journal of American Medical Association, 1963, 183(5),

307- .

Parker, R. D., & Lincoln, T. L. Medical diagnosis using
bayes' theorem. Health Service Research, 1967, 2, 34.

 

Pearson, K. On the correlation of characters not

quantitatively measurable. Ro a1 Society Philosophical
Transactions, Ser. A, 1900, , 1- .

 

Potchen, E. J. Biologic considerations in anatomic imaging
with radionuclides. Energy ResearcH—and Development
Administration, Division of Biomedical and Environmen-
tal Research, Final Progress Report, July 1974-June
1975.

 

Rinaldo, J. A., Scheinok, P., & Rupe, C. E. Symptom
diagnosis: A mathematical analysis of epigastric
pain. Annals 22 Internal Medicine, 1963, 59(2),
145-154. '_—

 

Scheff, T. J. Decision rules, types of error and their
consequences in medical diagnosis. Behavioral Science,
1963, 3, 97-107.

 

 

Scheifly, V. M. Analysis 92 re eated measures data: A
simulation study. UnpuinsHea Hoctoral dissertation,

T974.

Scheinok, P. Symptom diagnosis: Bayes theorem and
bahadur's distribution. International Journal 22
Bio-Medical Computing, 1972, 3, 17.

 

 

Schonbein, W. Analysis of decisions and information in
patient management. In E. J. Potchen (Ed.), Current
concepts ip_radiology. St. Louis: C. V. Mos5y Co.,

Shannon, E. C., & Weaver, W. The mathematical theogy 92
communication. Urbana: University of Illinois Press,

m4.

 

168

Swets, J. A. Detection theory and psychophysics: A review.
Psychometrika, 1961, 23, 49-63.

 

Taber, C. W. Taber's cyclopedic medical dictionary.
Philadelphia: F. A. Davis Co., 1970.

 

Tatsuoka, M. M. Multivariate analysis ip_educationa1 and
psychological researcfi} New York: WiIey, 1971.

 

 

 

Tou, J. T., & Gonzalez, R. C. Pattern recognition
principles. London: Addison-Wesley PubliSfiing Co.,
1974.

 

 

Truett, J., Cornfield, J., & Kannel, W. B. A multivariate
analysis of the risk of coronary heart disease in
framingham. Journal of Chronic Disease, 1967, 20,
511-524. _— _—

 

Tversky, A. Assessing uncertainty. Journal g£_Ro a1
Statistical Association, Ser. B, 1974, pp. 1 8-159.

 

Van de Geer. Introduction pp multivariate ana1y§is for the
social science. San Francisco: W. H. Freeman and Co.,
1971.

 

 

 

 

Vanderplas, J. M. A method of determining probabilities for
correct use of bayes' theorem in medical diagnosis.
Computers Biomedical Research, 1967, 2, 215-220.

 

Wakefield, E. G. Clinical diagnosis. New York: Appleton-
Century-Crofts, Inc., .

Walker, 8. H., & Duncan, D. Estimation of the probability
of an event as a function of several independent
variables. Biometrika, 1967, 31, 167-179.

 

Warner, H. R., Toronto, A. F., Veasey, G., & Stephenson, R.
A mathematical approach to medical diagnosis. Journal
92 American Medical Association, 1961, 177, 177-I83.

 

Webster's New Collegiate Qictionary. Springfield, Mass.:
G. & C. Merriam Co., Publishers, 1974.