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A THEORY OF SIGNAL DETECTIONBASED UPON HYPOTHESIS ANALYSIS
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Authors Fobes, James L.
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FOBES, James Lewis, 1946-A THEORY OF SIGNAL DETECTION BASED UPON HYPOTHESIS ANALYSIS.
The University of Arizona, Ph.D., 1975 Psychology, experimental
Xerox University Microfilms , Ann Arbor, Michigan 48106
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.
A THEORY OF SIGNAL DETECTION BASED
UPON HYPOTHESIS ANALYSIS
by
James Lewis Fobes
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF PSYCHOLOGY
In Partial Fulfillment of the Requirements For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
19 7 5
THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
I hereby recommend that this dissertation prepared under my
direction by James Lewis Fobes .
entitled A THEORY OF SIGNAL DETECTION BASED UPON
"HYPOTHESIS ANALYSIS
be accepted as fulfilling the dissertation requirement of the
degree of DOCTOR OF PHILOSOPHY
n ̂ f . pL, , H /1 A S~~ f Dissertation Directory Date
\
After inspection of the final copy of the dissertation, the
follovring members of the Final Examination Committee concur in
its approval and recommend its acceptance-.-'*
Hh hf
4/ ihf -y/g/76
This approval and acceptance is contingent on the candidate's
adequate performance and defense of this dissertation at the
final oral examination. The inclusion of this sheet bound into
the library copy of the dissertation is evidence of satisfactory
performance at the final examination.
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.
SIGNED:
This dissertation is dedicated to my wife
Jacqueline, and to my major professor James King, in
appreciation of their continuous encouragement and support.
iii
ACKNOWLEDGMENTS
I would like to acknowledge the contributions of:
Dr. James E. King whose assistance was invaluable in all
stages of this dissertation; other major commitee members
Drs. Sigmund Hsiao and Ronald H. Pool; minor committee
members Drs. Terry C. Daniel and Dennis L. Clark; Theresa
Burton and Beth Wenzel who assisted in data collection;
Evan Thomas who made conceptual contributions; Charles
Davison who aided with equipment maintenance; and Robert
Dylan who cautioned that nothing is revealed. This research
was supported by Training Grant MH-112 86 from the United
States Public Health Service.
iv
TABLE OF CONTENTS
Page
LIST OF TABLES vi
LIST OF ILLUSTRATIONS viii
ABSTRACT lx
INTRODUCTION 1
THEORETICAL CONCEPTUALIZATION AND ANALYTICAL
PROCEDURE 9
METHOD 25 Subjects . . . 25 Apparatus . , 25 Procedure 26
RESULTS 29
DISCUSSION 39
APPENDIX A. FORMULAS FOR ESTIMATION OF HYPOTHESES . . 52
APPENDIX B. FORMULAS FOR ESTIMATION OF K 61
REFERENCES 67
y
LIST OF TABLES
Table Page
1. Probability of Detection and Nondetection States for Stimuli Intersected with Attention and Nonattention States, as a Function of Procedure 11
2. Probability of Response Outcome for Detection and Nondetection States as a Function of Procedure 11
3. Probability of Response Outcome for Stimuli Intersected with Attention and Non-attention States, as a Function of Procedure . 12
4. Characteristics of Hypotheses Involving Detection States , . . . . 14
5. Characteristics of Hypotheses not Involving Detection States 16
6. Thirty-Two Unique Problem Sequences 17
7. Signal Level Presentation Order 28
8. Percentage of Correct Responses as a Function of Procedure and Signal Level , 29
9. Proportion of Variance Explained as a Function of Procedure and Signal Level , « , , 34
10. Sensitivity as a Function of Procedure and Signal Level 35
11. Probability of Detection and Nondetection States per Trial Given Attention, as a Function of Procedure and Stimulus 35
12. Sensitivity Estimates Obtained with Two-Alternative Forced-Choice, and Values Predicted from Yes-No, as a Function of Signal Level 37
vi
vii
LIST OF TABLES—Continued
Table Page
13. Probability of Entering an Attention State per Trial, and of Zero, One, Two, and Three Attention States per Problem, as a Function o f Procedure a nd Signal Level . . . . 37
LIST OF ILLUSTRATIONS
Figure Page
1. Proportional Strengths of Hypotheses Involving Detection States, as a Function of Procedure and Signal Level 30
2. Proportional Strengths of Hypotheses not Involving Detection States, as a Function of Procedure and Signal Level 31
viii
ABSTRACT
This dissertation consisted of the description and
^application of a new theory of detection processes. The
sensory portion featured detection of noise as well as
signal stimuli and an absolute threshold for recognition of
these stimuli. This recognition threshold for noise or
signal was never exceeded by the other stimulus alone. It
was also assumed that an observer either attended or failed
to attend to the stimulation presented on each trial. When
the observer entered an attention state, either a detection
state or a nondetection state resulted. If a nonattention
state occurred, a detection state was precluded.
The theory's response portion assumed that the com
bination of an attention and detection state resulted in
correct responding, the combination of an attention and non-
detection state resulted in random responding, and that a
nonattention state resulted in nonrandom responding that was
determined by a response bias. The false alarms that
occurred were considered to be independent of the sensory
mechanism and to have resulted from either random responding
accompanying the combination of a nondetection and attention
state or from a response bias.
Therefore for the yes-no procedure model, an
attention state in the presence of a signal resulted in
ix
X
either a detection of the signal with correct responding or
a nondetection state accompanied by random responding.
Likewise, when an attention state occurred in the presence
of noise, the observer entered a detect noise state or a
nondetect state. For the two-alternative forced-choice pro
cedure model, an attention state resulted in either a detec
tion of the difference between signal and noise with correct
responding or a nondetection state accompanied by random
responding. For both psychophysical procedures, a non-
attention state always resulted in responding determined by
a response bias.
This theory was applied to data obtained by pre
senting three-trial brightness discrimination problems of
varying difficulty to capuchin monkeys. The outcomes of
these problems were assumed to be combinations of states
attention and detection, attention and nondetection, or non-
attention that occurred on each trial of a problem. These
states resulted in manifestations that reflected various
modes of systematic responding which permitted estimation of
their relative occurrence with an adaptation of the hypoth
esis and analysis technique.
This analytical procedure was used to determine the
strength of three modes of detection and four sequentially
dependent response biases. These values were then used to
determine the sensitivity index that was expressed in terms
of the probability of detection given attention. This
sensitivity index was considered to be independent of non-
sensory response factors and its method of calculation
included estimation of the probability of an attention state
per trial as well as the probabilities of the various
possible combinations of attention and nonattention states
on three-trial problems.
Support was provided for this theory which featured
a variety of information that was not included in previous
approaches. Specifically, a more detailed description of
detection processes was provided that included estimation of
the amount of responding due to specific nonsensory response
biases. This conceptualization formed the basis for non-
parametric estimation of sequential biases whose effects
were isolated from the nonparametric sensitivity index.
Further, the theory was not tied to a specific family of
receiver operating characteristic curves. A concept of
sensory thresholds was also included and a procedure was
suggested for determination of absolute recognition
thresholds that are not confounded with the effects of
response bias.
INTRODUCTION
Various classical psychophysical methods have
afforded considerable information concerning the relation
ship between perception and physical characteristics of
stimulation (Guilford, 1954). Typically, however, the usage
of these techniques has emphasized only the percentage of
correct responses with each stimulus value. With such pro
cedures, the reputed sensitivity of the subject, based upon
this single variable, confounded detection with response
biases. While variability and reduced discrimination
accuracy due to nonsensory response biasing factors have
been recognized for some time, attempts to deal with such
biases have generally been inadequate, particularly so with
single stimulus psychometric methods. Frequently, the
attempt to control response biases consisted of either
extended training of subjects or experimental designs such
as counterbalancing (Engen, 1971).
A common psychophysical approach to estimating
sensitivity involved detection tasks that favored almost
exclusive presentation of a stimulus to be detected, which
was designated as signal (S), while trials containing only a
stimulus designated as noise (N) were infrequently presented.
When subjects incorrectly reported the presence of a S on
the few catch trials containing only N that were presented,
1
subjects were typically advised to pay increased attention.
Such false alarms were typically not included in data
analysis. A more recent procedure involved more frequent
presentation of trials containing only a N stimulus and the
proportion of false alarms on these trials was utilized in
an attempt to obtain the proportion of detections corrected
for guessing (Blackwell, 1953, 1963) . However, the pro
cedure for correction resulted in sensitivity estimates that
were not independent of response biases; the assumption of
statistical independence between the proportion of detec
tions corrected for guessing and false alarms was unjusti
fied (Green and Swets, 1966; Swets, 1973).
Techniques for the separation of certain response
biases and the sensitivity estimate were considerably ad
vanced by a theory of decision making based upon the trans
lation of that portion of statistical decision theory
dealing with situations in which a choice is made between
two alternatives on the basis of an observed event (Smith
and Wilson, 1953; Tanner and Swets, 1954). This approach
conceptualized the detection process as a problem of S-to-N
ratio (Tanner and Swets, 1954; Swets, Tanner, and Birdsall,
1961; Green and Swets, 1966; Swets, 1973). When a sensory
event occurred that was due to either N or to S, the observer
was thought to choose between normal distributions of N and
S along which the effects of stimulation continuously varied
as a single dimension of increasing magnitude. Therefore,
3
the sensory effect of an observation was thought to arise,
with a specific probability, from either the N or the S
probability density function.
It was further assumed that observers assigned con
ditional probabilities that a particular sensory effect
arose from N and that it arose from S. The continuum of
sensory effects was thus characterized as a continuum of
likelihood ratio values, each of which expressed the likeli
hood that a particular observation arose from the S distri
bution relative to the likelihood that it arose from the N
distribution. Thus, on the basis of the likelihood ratios
that varied in magnitude on the basis of their sensory
effect, or some monotonic function of the ratios, observers
were thought to decide which of two populations was sampled.
The actual choice between populations was considered to be
determined by a decision rule involving whether or not a
particular observation's likelihood ratio value exceeded
some fixed response criterion or response threshold value
along the continuum of increasing likelihood ratio values.
For example, if the likelihood ratio value of the sample was
below the criterion likelihood ratio value, the subject
indicated that the observation came from the N distribution;
samples with corresponding likelihood ratio values above the
critical value resulted in an indication that the observa
tion was from the S distribution.
4
The d' index of S detectability or sensitivity was
defined as the difference between the means of the N and S
density functions, expressed in terms of standard deviation.
Both this d1 sensitivity estimate and the response threshold
value (3) were estimated with the conditional probability of
saying S given (|) S and of saying S|N. For d1, a table of
normal curve functions gave the z score corresponding to the
conditional probability of saying S|N. This z value indi
cated the distance along the dimension of sensory effects
that the criterion was above the mean of the N density
function. Similarly, the z score corresponding to the con
ditional probability of saying s|s indicated the distance
that the criterion was below the mean of the S density
function. The value of d' equaled the combination of these
two z scores and was computable regardless of the location
of the criterion along the dimension of sensory effects; d'
was independent of the location of the criterion. The
measure 3 was the ratio of the ordinate of the S density
function to the ordinate of the N density function, at the
location of the response criterion. The numerator and de
nominator in this ratio were estimated from the two condi
tional probabilities (S|S and S|N) with a table of ordinates
of the normal curve.
Thus, d' was considered to be a function of de
tectability and monotonically related to S strength. For a
given S strength, the location of the variable decision
5
criterion along the continuum of likelihood ratio values has
been found to be affected by such things as the information
an observer has concerning relevant variables of the situa
tion and situational goals. However, the method of deter
mination of d' resulted in an estimate of sensitivity that
was relatively unaffected by certain types of biases; d' was
relatively independent of: (1) instructions (response
criteria from "be careful" to "be careless," or simultaneous
usage of multiple response criteria), (2) the a priori
probability of a S being presented (expectations), (3) the
values and costs associated with various decision outcomes
(motivation), and (4) the psychophysical procedure used to
estimate sensitivity (single stimulus or forced-choice).
However, the generality of these findings has been ques
tioned (Pike, 1973).
In addition to these biases, various types of
sequential response dependencies have been reported (Senders
and Soward, 1952; Verplank, Collier, and Cotton, 1952;
Howard and Bulmer, 1956; Speeth and Mathews, 1961;
Carterette and Wyman, 1963; Kinchla, 1964; Parducci, 1964;
Parducci and Sandusky, 1965; Haller, 1969; Baumstiler, 1970;
Sandusky, 1971; Sandusky and Ahumada, 1971). The lack of
response independence between trials can reasonably be
expected to affect the assumed stability of the response
criterion and accordingly the estimated value of d' (Pastore
and Scheirer, 1974). While signal detection theory (SDT)
6
has provided for an estimate of sensitivity that was rela
tively unaffected by certain response biases that affect the
response criterion, it did not include a technique for
dealing with sequential dependencies nor did it estimate the
strength of specific response biases (Green and Swets, 1966).
Further, this approach to detection systematically
excluded a sensory threshold with a lower limit on sensi
tivity. Thus, rather than assisting in the determination of
sensory thresholds unaffected by response biases, some
exponents of the SDT under discussion concluded that a
determination of the level at which a threshold may possibly
exist is neither critical nor useful. Such a view ignores
the intent of a considerable body of literature.
This dissertation consists of the description of a
proposed theory of sensitivity as well as its application
with two psychophysical procedures. The sensory portion of
the theory includes detecting or not detecting N stimuli as
well as S stimuli, and the occurrences of an attention (A)
state is though to be accompanied by either a detection (D)
state or a nondetection (D) state. If a nonattention (A)
state is entered, a D state cannot occur. A sensory
threshold for recognition of both N and S is assumed, with
continuous gradation of sensation above the threshold and
no gradation below. The recognition threshold for N or S
is thought to be never exceeded by the other stimulus alone.
7
The response portion of the theory assumes that the
intersection (A) of an A and D state results in correct
responding, A A D states result in random responding, and a
A state results in responding that is determined by a
response bias. False alarms are considered to be independent
of the sensory mechanism and result from either random re
sponding accompanying A A D states or from a .A state with
its ensuing response bias responding.
As an illustration of the proposed theory, data from
three-trial brightness discrimination problems, presented by
two psychophysical procedures, were analyzed by a hypothesis
(H) analysis technique similar to Levine's (1965) Method II.
This analytical technique was used to determine the relative
proportion of four response biases and three modes of D
states. The strengths of these hypotheses (Hs), in combina
tion with other values to be described, were then used to
estimate the sensitivity parameter K, the probability of
D|A. The calculation procedure to determine sensitivity
also resulted in estimation of the probabilities of partic
ular combinations of A, A, D, and D states on three-trial
problems. Further, estimation of all measures was accom
plished without assumptions concerning possible probability
distributions of the sensory effects of S or N.
An important advantage to the proposed approach
resulted from the inclusion of sequential responding infor
mation and nonsensory response .bias emerged as
8
multidimensional. These dimensions included: (a) the
tendency to perseverate with yes (Y) or no (No) responses
in yes-no (Y-N) tasks, or with a particular position in two-
alternative forced-choice (2AFC) tasks, (b) the tendency to
alternate responses, as well as (c) the win-stay:lose-shift
and win-shift:lose-stay syndromes. These particular non-
sensory factors are considered to be especially appropriate
for subhuman observers. The type and amount of these
response biases were specifically estimated and their
effects were isolated from the sensitivity estimate. A
technique is also suggested for estimation of sensory recog
nition thresholds which are not confounded with the effects
of response bias.
The theory's empirically testable consequences that
have been investigated indicate support for the theory.
This is noteworthy in that the theory includes a high-
threshold concept of absolute sensory recognition thresholds.
THEORETICAL CONCEPTUALIZATION AND ANALYTICAL PROCEDURE
The proposed theory of sensitivity assumes that S as
well as N stimuli may be detected or not detected, and a
momentarily varying, absolute sensory threshold for the
recognition of S or N stimuli is proposed. The recognition
threshold for N and S stimuli is thought to feature con
tinuous gradation of sensation above the threshold and no
gradation below. The recognition threshold for S is thought
to be never exceeded by N alone and the recognition threshold
for N is thought to be never exceeded by S alone. Thus,
observers can detect or not detect only that which is
presented.
The sensory portion of the theory also assumes that
on each trial subjects enter a state of either A or A to
stimulation. For the Y-N model, the occurrence of an A
state in the presence of S is thought to result in either a
Dg state or a D state. Likewise, the occurrence of an A
state in the presence of N results in the subject entering
state DN or state D. It is assumed that the probability
(P) of (D IS A A) = P(D |N A A) = K,. In the model for ̂ IN -L
2AFC, a D state is entered when the subject detects the
difference between the S and N conditions, with the
P(D|A) - ̂ 2' For both models, the occurrence of an A state
9
10
precludes a D state. A proposed relationship between and
K2 will be discussed later.
These assumptions concerning the sensory portion of
the theory are presented in Table 1 which depicts the
probability of a D or D state on each trial given a
particular stimulus condition A A or A states. The Y-N
technique involves presentation of only one stimulus (S or
N) per trial. Therefore, subjects may evidence Dg A A or
D A A when S is presented and DN A A or D A A when N is
presented. Conversely, the A state always results in a D
state. The 2AFC procedure entails presentation of both
stimuli (S and N) per trial and, with the spatial 2AFC used
(King and Fobes, 1973), S occurred on either the right (R)
or left (L).side. With this procedure, subjects may or may
not enter a state of D relative to the difference between S
and N. The A state in 2AFC also results in a D state.
Table 2 depicts the response aspects of the theory.
It is assumed that a D state always results in a correct
response and that a D state always results in random
responding. An A state is thought to always result in a D
state that is accompanied by nonrandom responding which is
determined by a response bias. For both D A A and A
states, outcomes are uncorrelated with S and N and the
probability of a correct response is one-half when the a
priori probability of S is one half. False alarms (saying
S|N) are considered to be independent of the sensory
11
Table 1. Probability of Detection and Nondetection States for Stimuli Intersected with Attention and Non-attention States, as a Function of Procedure
Y-N 2AFC
State State
Event DS °N D Event Ddifference D
S A A K1 0 1 - K1 s (L) A A K2 1 K2
N A A 0 K1 1 - K1 s (R) A A k2 1 - K2
S A A 0 0 1 s (L) A A 0 1
N A A 0 0 1 s (R) A A 0 1
Table 2. Probability of Response Outcome for Detection and Nondetection States as a Function of Procedure
Y-N 2AFC
State Correct Incorrect State Correct Incorrect
DS 1 0 Ddifference 1 0
dn 1 0
D 1/2 1/2 D 1/2 1/2
12
mechanism and are assumed to arise from either random
responding accompanying states D A A or from a response bias
accompanying state A.
Table 3 depicts the probabilities of response out
comes for a particular stimulus A A as well as A. These
probabilities are based upon the results of matrix multi
plication of Table 1 times Table 2. When an A state occurs,
a correct response results with probability (K + l)/2 and an
incorrect response occurs with probability (1 - K)/2. Again,
an A state always results in a D state with outcomes un-
correlated with the presence or absence of S. The proba
bility of a correct or incorrect response is one-half when
the a priori probability of S equals one-half. It should
also be noted that the values of K determined with the Y-N
and 2AFC procedures will generally not be the same.
Table 3. Probability of Response Outcome for Stimuli Intersected with Attention and Nonattention States, as a Function of Procedure
Y-N 2AFC
Outcomes
Correct Incorrect
S A A S (L) A A (K + l)/2 (1 - K)/2
N A A S (R) A A (K + l)/2 (1 - K)/2
S A A s (L) A A 1/2 1/2
N A A S (R) A A 1/2 1/2
13
The results of three-trial problems are assumed to
be combinations of states D A A, D A A, and A that occur on
each trial. Further, these states result in manifestations
that reflect various modes of systematic responding which
can be referred to as Hs. The unfortunate nature of the
term H with respect to unintentional connotations is
acknowledged, but it is used in order to be consistent with
Levine (1959, 1965). Thus, Hs are considered to be
sequences of internal states that determine behavior and
are manifested by a specificable pattern of responses to
selected patterns of stimulation. As such, the H is <
regarded as a dependent variable.
The Hs that include at least one D state on a
three-trial problem are listed in Table 4. If D states
occur on all three trials of a problem, the only possible
manifestation is three correct responses (+++), as indi
cated for H A in Table 4. With a D state on exactly two
trials, two correct responses result on these trials. The
outcomes of H B will also include correct as well as
incorrect responses (-) for the one trial on which states
D A A or A occur. Therefore, outcomes to sequences involv
ing two D states include: +++, ++-, +-+, and - + +.
Likewise for H C, exactly one D state occurs and the out
comes include correct and incorrect responses for the two
trials on which states D A A or A occur. Possible
14
Table 4. Characteristics of Hypotheses Involving Detection States
State H Sequence Manifestation
A: 3 Ds D D D + + +
B: 2 Ds D D DAA + + + or + + —
or D D A.
D DAA D + + + or + - +
or DA D.
DAA D D + + + or - + +
or A D D
C: 1 D D DAA DAA, + + + , + — + ,
D A A , + + - or + - -
D A DAA,
or D DAA A.
DAA D DAA, + + + / - + + ,
A D A , + + - or - + -
A D DAA,
or DAA D A.
DAA DAA D, + + + , — + + ,
A A D, + - + or - - +
A DAA D,
or DAA A D
15
manifestations of C include all sequences except - - - ,
since at least one correct response results with the single
D state.
The Hs that do not involve a D state on any of a
problem's trials are depicted in Table 5. Response bias Hs
are defined when an A state is entered on all three trials
of a problem and the residual H is defined by all other
sequences of A and Astates not involving D. The manifesta
tions of these Hs are specified in terms of response
sequences, where I is the type of response (Y or No) on the
f irst trial of a problem and 0 is the other type of response.
The results of three-trial problems are entirely
specified by the stimulus, response, and outcome sequences
depicted in.Table 6. The eight possible stimulus sequences
can be grouped into symmetrical pairs, viz, SSS and NNN,
SSN and NNS, SNS and NSN, as well as SNN and NSS. In
Table 6 these pairs are represented by the sequences XXX,
XXV, XVX, and XW, where X is the stimulus (S or N) on the
f irst trial and-V is the other stimulus. The outcome of a
particular stimulus sequence, in terms of being correct or
incorrect, is determined by the response sequence (III, 110,
101, or 100) evidenced. The lower case letters in each cell
in Table 6 are the symbols that indicate the Hs that can
result in each particular outcome given those stimulus and
response sequences. The number in each cell will be used
for reference purposes.
Table 5. Characteristics of Hypotheses not Involving Detection States
H Definition State Sequence Manifestation
D: Triple Response Repetition
Type of Response Stays the Same for Three Consecutive Trials
A A A I I I
E: Double Response Repetition
Type of Response Stays the Same for Two Consecutive Trials
A A A I I 0 or I 0 0
F: Win-Stay: Lose-Shift
Response Type the Same as Preceding Rewarded Trial
A A A I+I+I, I+I-O
1-0+0 or I-O-I
G: Win-Shift: Lose-Stay
Response Type Opposite from Preceding Rewarded Trial
A A A I+O+I, 1+0-0,
I-I+O or I-I-I
R: Residual Sequences Not Defined Above
A DAA
DAA DAA DAA,
DAA, DAA A DAA,
All Sequences
r
DAA DAA A, A A DAA,
A DAA A or DAA A A
Table 6. Thirty-Two Unique Problem Sequences
17
Stimulus Sequence
Response Sequence Stimulus Sequence I I I I I 0 I 0 I 10 0
X X X 1) d g r
2) c e
- - + r
3) - + c r
- 4) — + + b 2c e f r
5) + + + a 3b 3c d f r
6) b 2c
+ + -e r
7) + -b 2c r
+ 8) + - -c e g r
X X V 9) - - + c d g r
10) e r
_ _ _ 11) - + b 2c r
+ 12) - + -c e f r
13) + + -b 2c d f r
14) a 3b r
+ + + , 3c e
15) + -c r
16) + - + b 2c e g r
X V X 17) - + -c d r
18) b 2c r
- + + e g
19) - -f r
- 20) + c e r
21) + - + b 2c d r
22) c e
+ - -f r
23) + + a 3b 3c r
+ g
2 4 ) + + — b 2c e r
X V V 25) - + + b 2c d r
26) c e
- + -g r
27) - -c f r
+ 28) e r
29) + - -c d r
30) b 2c
+ - + e f
31) + + -b 2c g r
32) + + + a 3b 3c e
18
Table 6 indicates that a given response sequence is
always consistent with more than a single H. For example,
the response sequence I-I-I- is consistent with Hs d, g, and
r in cell number one. Although a given response sequence is
always consistent with more than one H, estimation of the
strength of each H can be accomplished by a technique
similar to Levine's (1965) Method II. However, the particu
lar Hs presented here, as well as their solution equations,
differ from Levine's. An additional important difference
from Levine's procedure is the present use of weighting
coefficients in Table 6 which reflect the l ikelihoods of
different response sequences when one or two D states occur
on a problem. As indicated for H B in Table 4, when two D
states occur on a problem, either two or three correct out
comes can result depending upon whether a correct response
occurs on the trial involving states D A A or A. Therefore,
the possible manifestations are: +++, ++-, +-+, and
- + +. If all manifestations of two D states are assumed to
be equally l ikely, then + + + is three times as l ikely to
occur as any other manifestation of B. Therefore, b is
assigned a coefficient of three in those cells containing
+ + + and assigned a coefficient of one in all cells con
taining exactly two correct outcomes. The value of b is
equal to the probability that H B occurs and is manifested
by, for example, +•
19
Likewise for C, al l outcomes with exactly one D
state are possible and include: + + +, + + - , + - +, - + +,
- - + , + - - , a n d - + T h e + + + s e q u e n c e i s a s s i g n e d a
coefficient of three since it is three times as l ikely as
- - or + - given H C. Sequences ++-,+-+,
and - + + are similarly assigned a coefficient of two since
they are twice as l ikely as +, - + - , or + - - .
The analysis of systematic patterns of response
sequences begins with categorization of the frequencies
with which each of the 32 unique sequences in Table 6
occurs. The relationship between these resulting 32
frequencies and the probability of each H is determined by
the assumptions that: (a) the Hs are mutually exclusive,
(b) their effects are additive, and (c) al l Hs whose
probability exceeds zero are included. Summing across any
row in Table 6 results in the statement
a + 6b + 12c + 2d + 4e + 2f + 2g + 8r = 1, (1)
where a = A, 6b = B, 12c = C, 2d = D, 4e = E, 2f = F, 2g =
G, and 8r = R, Line (1) may also be stated as
A + B + C + D + E + F + G + R = l , ( 2 )
Therefore, each term in the two above equations may be
interpreted as a probability or proportion with the terms in
Equation (2) being the proportions of three*-trial sequences
accounted for by the corresponding H.
20
The relative frequencies with which each of the 32
three-trial sequences occur can be used to estimate the
probabilit ies in Equations (1) and (2). For example, the
theoretical probability of three incorrect outcomes given
the stimulus sequence XXX (cell number one) equals the
sum of the probabilit ies of the associated Hs since the Hs
are assumed to be mutually exclusive and additive.
P t(- - - |X X X) = d + g + r. (3)
The theoretical probability of all 32 sequences in Table 6
may similarly be expressed by a particular l inear combina
tion of certain H probabilit ies.
The obtained frequencies of each three-trial
sequence may then be used to estimate the probability of
each H. Continuing with the sequence from cell number one,
P (- - - | X X X) = n(X- X- X-)/n(X X X), (4)
where n(X-X-X-) is the frequence with which X~X*-X- occurs
and n(X X X) is the frequency with which XXX occurs. To
estimate the theoretical from the observed probability, i t
follows from Equations (3) and (4) that
d + g + r = n(X- X- X-)/n(X X X). (5)
Levine's (1965) Method II may then be used to
evaluate the probability of Hs in such a way as to minimize
the sum of the squared differences between the theoretical
and obtained cell frequencies (see Levine, 1959). While a
21
complete derivation of H analysis solution formulas is
presented in Appendix A, a brief description of the pro
cedure for obtaining solutions is i l lustrated by the solu
tion of H A.
From the 3 2 equations, those containing A include
(.cell #5) a+3b + 3c + d + f + r = Q^,
(cell #14) a+3b + 3c + e + r =
(cell #23) a+3b+3c+g+r= , and
(cell #32) a + 3b + 3c + e + r = (6)
where each Qa is the observed proportion of the particular
outcome sequence. It should be noted that in the present
case each stimulus sequence was presented equally often.
These equations in l ine (6) involving A totalled
EQ = 4a + 12b + 12c +d+2e+f+g+ 4r. (7) a.
Applying Equation CI) and solving for a = A results
in
2EQa = 8a + 24b + 24c + (1 - a - 6b - 12c) , ( .8)
which reduces to
2£Q - 1 = 7a + 18b + 12c a
(9)
which may be rewritten
a = (2EQa - 18b - 12c - l) /7 CIO)
22
The solution in l ine (10) for A is thus found to be a
function of Hs B and C. Applying Equation (1) and solving
for B and C in an analogous manner results in
b = (l /2EQb - a - 6c - l) /6 (11)
and
c = (2EQc - a - 6b - 7)/12. (12)
These three l inear equations (10, 11, 12) with three
unknowns may then be solved by Cramer's rule with these
resulting unique solutions
A = a = l /2ZQa - l/2IQb + l /2£Q c - 1, (13)
B = 6b = 6(l/4£Qb - l/12IQa - 5/12EQ c - 1), (14)
C = 12c = 12 (1/3EQ - 1/12XQ, - 1). (15) —~ C JD
The solution for D is a function of the EQ^ as well
as E. These two equations with two unknowns may be simul
taneously solved with the unique solutions
D = 2d = 2(l/4ZQd + l /8IQ e - 1/2), (16)
and
E = 4e = 4(3/16EQ e + l /8ZQd - 1/2). (17)
The solutions for F and G are similarly a function
of each other with the unique solutions
23
F = 2f = 2(3/16ZQ f + 1/16ZQ - 1/4) (18)
and
G = 2g = 2(3/16EQ + 1/16ZQ- - 1/4). y
(19)
Finally, from Equation (2), R is simply a residual,
Once the H strengths are obtained, they may then be
used to estimate the sensitivity parameter K, the P(D|A).
This is accomplished by solution of the f ive equations
l isted below which contain five unknowns (K, M^, t , and
T ) . V a l u e s f o r A, B, C, D, E, F, G, and R are H strengths
and the remaining terms are defined as:
T = P (Ah A An + 1 A An + 2) , K = P (D | A) , M1 = P (A A A)
+ P (A A A) + P (A A A) , M2 = P (A A A) + P (A A A)
+ P (A A A) , t = P(An) , (J) = P(A|D) = (t - tK) / (1 - tK) ,
0 = P(A|D) = (1 - t) /( l - tK) , hits = P(Y|S), and
false alarms = P(Y|N).
A = TK3 , (1)
B = 3TK2(1 - K) + M-^K2 , (2)
C = 3TK (1 - K) 2 + 2M ] ,K(1 - K) + M2K, (3)
R = L = ( A + B + C + D + E + F + G ) . ( 2 0 )
24
T = A + B ( p + C ( p 2 + R{{l-{ [30(j)]/ l - [ (D + E + F + G) /
( D + E + F + G + R ) ] } } } , ( 4 )
Hits - False Alarms = tK. (5)
The derivation of these equations is contained in Appendix B.
Since the equations containing K are l inear when K is
constant, a computer may be used to obtain solutions for
values of K from 0.0 to 1.0 for Equations (1), (2), (3), and (5) .
While this set has an infinite number of solutions, only
one particular solution gives a value of T in Equations
(1), (2), and (3) that is equal to the value of T in
Equation (4), for a particular value of K. This value of
K defined the solution selected.
METHOD
Subj ects
Four adult male capuchin monkeys (Cebus apella)
served as subjects. These feral animals had extensive prior
experience with sameness-difference learning-set (King and
Fobes, in press, 1975) and 2AFC brightness discriminations
( K i n g a n d F o b e s , 1 9 7 3 ) .
Apparatus
The apparatus featured stimulus presentation behind
a one-way-screen (7.5 by 6.5 cm) through which subjects
viewed stimuli only during i l lumination behind the screen
with a 4 0-watt frosted incandescent l ight. Stimuli con
sisted of block mounted pigmented papers (3.8 by 3.8 cm)
previously presented in the 2AFC investigation of brightness
discrimination (King and Fobes, 1973).
Stimuli became visible at the start of each trial
and i l lumination automatically ceased following: (a) a
correct Y response of interrupting a photocell beam, which
was recessed and centered in front of the screen, before
f ive seconds had elapsed (a "go" condition); (b) a correct
No response of not breaking the photocell beam for five
seconds (a "no-go" condition); or (c) an incorrect response.
Correct responses were followed by a one second tone and
dispensation of one-half raisin into a receptacle mounted
below the screen,
25
Procedure
Sixteen three-trial problems a day were presented
throughout pretraining, training, and testing. Eight
stimulus sequences of S and N were used throughout and each
of these stimulus sequences (viz. SSS, NNN, SSN, NNS, SNS,
NSN, SNN, and NSS) appeared twice daily in random order with
the restriction that no stimulus was presented for more than
three consecutive trials. The S condition consisted of a
grey stimulus and a white stimulus and the N condition con
dition consisted of a pair of white stimuli . Subjects were
pretrained by being rewarded for a Y response to a black
next to a white stimulus and for a No response to two side-
by-side white stimuli . This Y-N pretraining continued until
subjects achieved 90 per cent correct responding with both
types of responses for two consecutive days,
Subjects were then trained with a t itration pro
cedure that involved the same series of stimuli (white to
dark grey in 60 increments) previously presented in the
2AFC phase. This training (King and Fobes, 1973) began with
a brightness discrimination between the darkest grey-white
and white-white. After three consecutively correct
responses of a Y to grey-white with a particular intensity
of grey, the intensity of grey was increased by one step;
failure to achieve three consecutively correct responses
resulted in a one step decrease in the intensity of the
grey. Subjects were thus tested to determine the grey
27
stimulus intensity around which responding stabilized.
Three intensity steps below this point of stable responding
was arbitrarily selected as the base intensity value of
grey for each individual subject. The log^Q intensity
difference between each subject's base grey and white
was then divided by three in order to create three sub
jectively equal proportions (Guilford, 1954), and a third
of the intensity difference was selected as A grey for that
subject. Varying around a mean A intensity of 0.062 foot/
candle, the intensity values of three previously presented
greys that were included in the present investigation were:
(a) 1/2A, (b) A, and (c) 2A. Since the base value was
individually determined for each subject the actual inten
sity of A differed among subjects.
In the present investigation, this t itration pro
cedure was presented as a training task with a Y-N pro
cedure. To assure that Y-N performance had stabilized,
t itration continued until each subject responded correctly
with a Y to grey-white and a No to white-white on a problem
consisting of a discrimination between its 2A grey and white
versus white and white.
Testing consisted of three phases. In each phase,
subjects were tested with one of the three (1/2A, A, and 2A)
grey-white S levels and white-white N for 12 days. Each
phase was preceded by two days of training with the
particular intensity of grey to be tested, The order in
28
which each of these grey-white S conditions was presented is
depicted in Table 7 and was determined for each subject by
random assignment of one of the six permutations of three
quantities.
Table 7. Signal Level Presentation Order
Phase
Subject 1 2 3
1 1/2A 2A A
2 2A A 1/2A
3 A 2A 1/2A
4 A 1/2A 2A
Thus, on each trial subjects were presented with one
of two stimulus alternatives and could respond with one of
two response alternatives. The S consisted of a condition
wherein white was presented next to one of three grey
stimuli that varied in brightness. The appropriate response
to indicate S detection was a Y response of interrupting a
photocell beam within five seconds. The N condition con
sisted of white-white stimuli and a No response of not
breaking the beam before f ive seconds had elapsed indicated
N detection.
RESULTS
Table 8 depicts the percentage of correct responses
with each S intensity level of grey for both the present Y-N
procedure and the previous 2AFC technique (King and Fobes,
1973). Percentages of correct responses were virtually
identical for the two procedures and were higher for A and
2A than for the 1/2A condition.
Table 8. Percentage of Correct Responses as a Function of Procedure and Signal Level
Y-N 2AFC
1/2A ' A 2A 1/2A A 2A
74% 83% 83% 72% 80% 81%
Figures 1 and 2 depict the proportional strength of
Hs which was determined with the computational formulas
presented in the Theoretical Conceptualization and Analyti
cal Procedure Section as well as in Appendix A. Figure 1
depicts the obtained proportion of Hs that involved one or
more D states in a three-trial sequence, as a function of
procedure and S level. With both procedures, the proportion
of three D states per problem (A) increased as the task
30
Y-N 2AFC
.5 r
.4
x I— g 3 LU cr h~ CO
< .2 O I— QC O Q_
§ J Q_
0
0 O B D • • c ®
-•
1
1/2 A A
BRIGHTNESS
2A
Figure 1. Proportional Strengths of Hypotheses Involving Detection States, as a Function of Procedure and Signal Level
31
.5
4
X I— -2 o .O -Z. LLI or h-i f )
< 2 o i-a: o Q_ 0 01 Q_
. 2
Y-N 2 AFC • 9 D • • O O E • •
• — — p S
O -o G
A A R A -A
1 1/2 A A
BRIGHTNESS 2A
Figure 2. Proportional Strengths of Hypotheses not Involving Detection States, as a Function of Procedure and Signal Level
32
became easier with decreasing intensities of grey, the
proportion of two D states per problem (B) showed an
inverted U shaped curve and the proportion for a single D
state per problem (C) showed a U shaped curve.
Figure 2 depicts the obtained proportions of Hs not
involving any D states, as a function of procedure and S
level. The triple response repetition (D) and residual (R)
Hs accounted for the bulk of these D state manifestations.
The proportions for double response repetition (E), win-
stay : lose-shift (F) , and win-shift:lose-stay (G) were
negligible. The D H displayed a U shaped curve that was
more pronounced with the Y-N technique and the other
prominent H, R, decreased as the brightness of grey
decreased.
In Table 5, E was defined as response repetition for
two consecutive trials of a problem (I00 or 110). An addi
tional type of sequential dependency consists of response
alternation (101). While D and response repetition or
alternation (E2) can be estimated by H analysis, no unique
solution exists for D with both response repetition and
alternation. Therefore, D and response alternation were
solved assuming response repetition to be zero while D and
response repetition were solved assuming alternation to be
zero. Response repetition was found to be of greater
magnitude than alternation and is therefore presented here.
It should be further noted that formulas for D and
33
alternation differ from those presented for D and response
repetition.
To check the validity of the theoretical approach
underlying the H analysis, predicted frequencies were
calculated for each of the 3 2 unique combinations of
stimulus, response, and outcome sequences in Table 6. These
frequencies were based upon the overall strength of each H
during the testing by a given procedure with a particular
S level (see Levine, 1965) . A measure of the amount of
variability among the 32 obtained cell frequencies accounted
for by the Hs in the model was obtained from the following
formula, where CK and are the observed and predicted
proportions of cell i , and M is the mean of the observed
cell proportions.
3 2 1 - 2 ( 0 i - P i ) 2 / ( 0 i - M ) 2 .
i=l
This statistic is similar to Levine's (1965) Proportion of
Variance Explained (P.V.E.) but differs sl ightly since
Levine obtained the predicted frequencies by his Method I in
such a way that each predicted frequency was based upon data
which did not include the corresponding obtained frequency.
Both Levine's P,V,E, and this modified version are thought
2 to be fairly comparable to r_ and thus give an indication
of the amount of between cell variance predictable by the
a d d i t i v e e f f e c t s o f t h e H s m e a s u r e d ( L e v i n e , 1 9 5 9 ) .
34
Table 9 depicts the P.V.E. values determined. These high
values are evidence that all nonnegligible l is were included
in the analysis and that the Hs combined additively to
determine the frequency of each of the 32 possible combina
tions depicted in Table 6.
Table 9. Proportion of Variance Explained as a Function of Procedure and Signal Level
Y-N 2AFC
1/2 A A 2A 1/2A A 2A
. 9 7 . 9 9 . 9 9 . 9 7 . 9 9 . 9 8
The computed strengths of the Hs were then used in
the K estimation formulas presented in the Theoretical
Conceptualization and Analytical Procedure Section as well
as in Appendix B. The values for K determined as a
function of procedure and brightness level of S are
depicted in Table 10. For both procedures, K increased as
the discriminations became easier.
In order to relate K resulting from a 2AFC pro
cedure to K estimated by a Y-N task in such a way as to be
able to predict one from the other, all possible outcomes
for an attentional trial were considered as depicted in
Table 11, Values in this table are based upon the
35
Table 10. Sensitivity as a Function of Procedure and Signal Level
Y-N 2AFC
1/2A A 2A 1/2A A 2A
. 8 7 . 9 1 1 . 0 0 . 8 2 . 8 7 . 9 5
Table 11. Probability of Detection and Nondetection States per Trial Given Attention, as a Function of Procedure and Stimulus
y-n 2AFC
S n Probability S n Probability
DS |a K1 Ds |a °n 1a k2(l - K2^
dn |a K1 Ds 1a dn 1 a k2(i - k2)
°S |a 1 l
h
Ds 1a dn 1a (k2)2
°n |A 1 " K1 °s 1a 1a (i - k2)2
36
assumption that the probability of detecting S or N in a
2AFC task is equal to the probability of detecting these
same events in a Y-N task. It is further assumed that the
subject will enter a D state in the 2AFC task whenever S,
N, or both are detected. Therefore, i t follows that
K2 = 2K1(1 - K^) + (I
37
Table 12. Sensitivity Estimates Obtained with Two-Alternative Forced-Choice, and Values Predicted from Yes-No, as a Function of Signal Level
K 1/2A A 2A
Obtained .82 .87 .95
Predicted .83 .87 .96
Table 13. Probability of Entering an Attention State per Trial, and of Zero, One, Two, and Three Attention States per Problem, as a Function of Procedure and Signal Level
State Sequence
Y-N 2AFC State
Sequence 1/2A A 2A 1/2A A 2 A
A Per Trial . 54 . 71 . 65 . 55 . 69 . 6 6
3 As Per Problem . 4 0 , 5 6 . 4 9 . 4 0 . 4 9 . 4 6
2 As Per Problem . 0 6 - . 0 5 . 25 . 2 6 . 1 5 . 22
1 A Per Problem .19 . 2 5 . 1 2 . 08 . 22 .19
3 As Per Problem . 3 5 . 24 . 1 4 . 26 . 1 4 . 1 3
two, and three A states per problem for each procedure and
S level. These values were obtained with the solution of
the equations for K estimation presented in the Theoretical
Conceptualization and Analytical Procedure Section. The
values of these various A and A states across S levels were
comparable between procedures.
DISCUSSION
The data reported in the results section were pro
vided to i l lustrate an application of the proposed theory
with two different testing procedures, rather than to
provide an experimental comparison of these particular
experimental designs. Therefore, tests for statistical
significance of the differences were not included. The pro
posed theory contains a three-state discrete sensory model
which assumes that on each trial subjects either attend or
do not attend to the S or N stimulation presented. If sub
jects enter an A state on a given trial , they also enter
either a D or a D state. An A state is always assumed to
result in a D state. In addition, the sensory portion of
the theory is unique in applying the threshold concept to
the sensory effects of both N and S stimuli , with the assump
tion that P(D |s A A) equals P(D|N A A). This is considered
reasonable since the designation of what is S and what is N
is totally arbitrary and detecting one of the stimuli to be
discriminated is just as l ikely as detecting the other.
The response model is partly deterministic in that a
detect S state and a detect N state are assumed to always
result in a correct response. It is also partly probabil
istic since the probability of a correct response with a D
state equals the a priori probability of S being presented.
40
With Y-N presentation, an A state in the presence of S
results in either correct responding with P( + | s A A) =
( K + l ) / 2 o r i n c o r r e c t r e s p o n d i n g w i t h P ( - | s A A) =
( 1 - K ) / 2 . S i m i l a r l y , a n A s t a t e i n t h e p r e s e n c e o f N a l s o
results in either correct responding with P(+|N A A) =
( K + l ) / 2 o r i n c o r r e c t r e s p o n d i n g w i t h P ( - | N A A ) =
( 1 - K ) / 2 . W i t h 2 A F C p r e s e n t a t i o n , a n A s t a t e r e s u l t s i n
correct responding with P(+|s A A) = (K + l ) /2 or incorrect i_j
responding with P(- |S A A) = (1 - K)/2. A A state with J -L
both Y-N and 2AFC always results in a D state accompanied by
responding which is determined by a response bias.
The advantage of the present approach is thought to
be the variety of information that i t provides which was not
included in previous approaches (Blackwell , 1953, 1963;
Luce, 1963a, 1963b; Green and Swets, 1966). Specifically,
the proposed theory is thought to result in a highly de
tailed description of the detection process which includes
an estimation of the amount of responding due to specific
nonsensory response biases. This analysis forms the basis
for an estimate of sensitivity that is independent of
response bias including the effects of sequential response
dependencies. Both the sensitivity estimate and response
bias measures are nonparametric and a technique is sug
gested for the estimation of sensory thresholds that are not
confounded with the effects of response bias. Further, this
theory includes the concept of a high-threshold and is not
41
tied to a specific family of receiver operating character
istics (ROC) curves.
Initially, the probabilit ies of internal state sequences
assumed to be manifested by systematic responding during Y-N
and 2AFC problems are estimated by the relative proportions
of the various Hs analyzed. This determination is based upon
H analysis of the results of three-trial problems when
results are assumed to reflect combinations of states
D A A, D A A, and A that occur on each trial . The Hs that
involve various modes of D states provide a nonparametric
estimation of the proportion of one (A), two (B), and three
(C) D A A states per problem. Nonparametric estimation of
the type and amount of specific nonsensory response factors
is provided by the proportions of response bias Hs. The
lack of response independence between trials that is
examined is based upon a conceptualization of response bias
as multidimensional. These dimensions include (a) the
tendency to perseverate responses for two (E) or three (D)
trials, (b) the tendency to alternate responses (E^), as
well as (c) the win-stay:lose-shift (F) and win-shift:lose-
stay (G) syndromes.
The accuracy of prediction for the degree of
responding accounted for on the basis of H analysis was
quantitatively described by the P.V.E. estimations. The
P.V.E. values presented in Table 9 indicate that the H
analysis procedure quite accurately estimated the patterns
J
of frequencies observed. Changes in the occurrence of the
proportions of the various Hs as a function of such
variables as testing procedure and S level may then be
examined as depicted in Figures 1 and 2.
The H strengths were used to estimate the sensi
tivity index that is expressed in terms of the P(d|a).
Evidence that this index is independent of response biases
including those of sequential response dependencies will be
discussed shortly. In addition, the method of calculation
of the sensitivity index also results in determination of
the probability of an a state per trial as well as the prob
ability of one, two, and three a states and three a states
on three-trial problems. These combinations of a and a
states may also be examined as a function of such variables
as testing procedure and S intensity, as depicted in Table
13.
In l ine with the concept of a sensory recognition
threshold, the present procedure includes the following
technique for estimation of sensory thresholds in such a
way as to be unconfounded with a range of nonsensory
response biases. This procedure consists of replacing the
proportion of correct responses to various stimulus inten
sities with probability values of K. The threshold would
then be defined by the stimulus magnitude corresponding to a
value of K equal to .5. This technique is presently being
43
accessed by i ts use for the determination of Macaque visual
acuity thresholds (Fobes, in preparation, 1975) .
Nonparametric est imation of sensit ivity and response
bias avoids the diff icult ies with assumptions accompanying
parametric d' est imation with SDT procedures. The measures
of sensit ivity and response bias used with the theory pre
sented here do not depend upon any concept of underlying
density functions describing the effects of sensory events .
Nor do they depend upon accompanying assumptions concerning
normality and variance as does the frequently used d' tech
nique of Green and Swets (1966) . Their procedure for cal
culat ion of the sensit ivity est imate depends upon whether
the N and S density functions are assumed to be normal with
equal or unequal variances. Thus, the present approach is
in l ine with increasing interest in nonparametric est imators
of sensit ivity and response bias (Grier, 19 71; Hammerton and
Altham, 197i; Richardson, 1972) .
The nature of the variance assumption that i s made
in a given case is usual ly based upon the degree of symmetry
of an P.OC curve. Such a curve connects plots of the P (Y|S)
on the ordinate and the P(Y|N) on the abscissa as the P(Y)
responses i s varied with the S level held constant. How
ever with the present data, the empirical ROC curve can not
be determined s ince the P(Y) responses was varied between
rather than within S levels . Thus, distribution l inked d 1
can not be calculated on these data s ince neither the
44
distribution nor the appropriate variance assumption can be
determined. This does not pose a problem for the est imation
of nonparametric K.
The concept of a sensory threshold that i s rarely
or never exceeded in the absence of the S has been specif i
cal ly crit ic ized by some advocates of SDT. In their
analysis of the high-threshold concept, Green and Swets
(1966) compared SDT with a specif ic version of high-
threshold theory that was advanced by Blackwell (1953, 1963) .
Blackwell 's high-threshold theory included a sensory
threshold that was thought to be rarely i f ever exceeded in
the absence of S and below which sensory events were in-
discriminable from one another. While the theoretical pro
portion of "true" false alarms [(Y|N)*3 was assumed to
approximate zero, empirical proportions of false alarms
greater than zero were assumed to result from a Y response
to some sensory events that fai led to exceed the sensory
threshold. Since subthreshold sensory events were con
sidered to be indist inguishable from each other, these Y
responses were guesses and were correct by chance. There
fore, the obtained proportion of hits was assumed to con
s ist of the proportion of "true" hits [(Y |s)*], the value
of which depended upon S strength, plus a guessing factor
modif ied by the opportunity for guessing.
P(Y|S) = P(Y|S)* + P(Y|N) [ l - P(y|s)*3 ( l )
A procedure was therefore used to obtain the pro
portion of "true" hits corrected for guessing which adjusted
the obtained hits according to the obtained false alarms
that were taken as an index of the amount by which the hit
rate was inf lated by guessing. This correct ion for chance
success was a rearrangement of l ine (1) and took the form
P(Y|S)* = P(Y|S) - P(Y|N) / 1 - P(Y|N). (2)
This correct ion attempted to normalize the obtained psycho
metric function by e l iminating the proportion of false
alarms from the proportion of obtained hits . Upon deter
mination of the P(Y|S)* for each st imulus magnitude, a
psychometric function was plotted and the st imulus magnitude
corresponding to a .5 P(Y|S)* was selected as the threshold
value. Since the guessing mechanism which produced false
alarms was thought to operate only in the absence of a
sensory basis for a response, the procedure for correct ion
of chance success assumed ff iat the P(Y|S)* and the P(Y|N)
were independent.
This assumption of the stat ist ical independence
between "true" hits and false alarms has been shown to be
false by several l ines of evidence. One approach consisted
of a determination of the degree of correlat ion between the
p(y |S)* and the P(y|n ) . Stat ist ical ly s ignif icant correla
t ion coeff ic ients for these measures were found to be on the
order of .90 (Green and Swets , 1966) . Green and Swets also
noted other diff icult ies with Blackwell 's theory. Thresh
olds determined with Y-N and 2AFC procedures were not con
s istent with each other. In addit ion, the equation in l ine
(1) expressed the P(Y|S) as a linear function of the P(Y|N).
Thus, the theoretical ROC curve based upon Blackwell 's
version of high-threshold theory was a straight l ine from
P(Y|S)*, P(Y|N)* through P(Y|S ) = 1 , P(Y|N ) = 1 . The vast
majority of published ROC data are not adequately f i t ted by
a straight l ine. Although Green and Swets (1966) have
successful ly argued against Blackwell 's (1953, 1963) de
fect ive version of high-threshold theory, their evidence
does not automatical ly inval idate al l formulations that
include a high-threshold concept, in contrast to their
strong implication to the contrary.
While Blackwell 's (1953, 1963) theoretical ROC curve
resulted in a poor approximation of ROC data, the theo
ret ical ROC curve result ing from the low-threshold theory
proposed by Luce (1963a, 1963b) occasional ly provides a good
f i t for empirical curves (Luce, 1963a; Green and Swets , 1966;
King and Fobes, in preparation, 1975) . Luce's (1963a, 1963b)
theory specif ied a sensory threshold that was frequently
exceeded in the absence of S; the theoretical proportion of
false alarms was greater than zero. As in Blackwell 's
(1953, 1963) theory, Luce proposed a guessing mechanism that
resulted in an observed proportion of false alarms that was
greater than the "true" proportion due to random Y responses
47
to some subthreshold sensory events . In addit ion, a
guessing mechanism was also proposed that resulted in an
obtained proportion of hits that was less than the "true"
proportion due to random N responses to some suprathreshold
sensory events . Luce's theoretical ROC curve consisted of
two linear segments. One segment went from a zero P(Y|S)
and P(y|n ) at the lower left corner of the ROC graph to the
point representing the calculated "true" probabil i ty that S
results in detect ion and the calculated "true" probabil i ty
that N alone results in detect ion. From this point , the
other segment went to a P(yjs) and P(Y|N) of one at the
upper r ight corner of the graph.
One of the problems with this low-threshold concept
i s that i t had an observer frequently detect ing something
than was not presented, that i s , the theoretical P(Dg|N) was
greater than zero. An addit ional diff iculty with Luce's
theory i s that while ROC curves of the predicted shape are
sometimes obtained, the most frequently reported empirical
shape of the best f i t t ing l ine connecting ROC data i s
curvi l inear, the shape predicted by SDT's theoretical ROC
curve (Green and Swets , 1966) .
Two general points emerge from a consideration of
theoretical and empirical ROC curves. The proposed theory
involves more components than did previous formulations and
consequently, s l ight emphasis i s placed upon the two con
dit ional probabil i t ies (hits and false alarms) that form the
48
basis of ROC data. Rather than being t ied to ROC curves of
a particular form, the proposed theory i s consistent with
any ROC curve according to the relat ionship
tK = P(Y|S) - P(Y|N) , (3)
which may be rearranged
P(Y|S) = tK + P(Y|N) , (4)
This reduced emphasis upon ROC data gives the present theory
more general i ty than those l inked to ROC curves belonging to
a particular family. Data f i t ted by the curve predicted by
Luce's theory have provided some diff iculty for SDT and the
f inding of data f i t ted by the curve predicted by SDT are
diff icult to account for with Luce's theory. The second
point i s that not a l l formulations of high-threshold theory
necessari ly imply ROC curves of the type predicted by
Blackwell 's version of high-threshold theory.
In addit ion to the effects of sequential response
bias , the proposed approach i s also thought to provide a
sensit ivity index that i s independent of other nonsensory
biases . I t was noted in the introduction that the SDT
approach of Green and Swets (1966) a lso provided a measure
of sensit ivity that was independent of the effects of some
nonsensory factors , although not independent of the effects
of sequential response bias . Specif ical ly , d' was rela
t ively independent of varied response criteria, the a priori
probabil i ty of S, and psychophysical procedure.
49
While ongoing research at The University of Arizona
has yet to be completed on al l of these areas, a con
siderable amount of data have been col lected which are
directed at the quest ion of independence of the sensit ivity
index K and mult iple response criteria. King and Fobes ( in
preparation, 1975) presented humans with visual problems by
both 2AFC and Y-N rat ing scale procedures. Subjects used a
confidence index that featured f ive categories of confidence
levels which varied from absolutely certain to just guessing.
For data that have been analyzed, calculated values of K
varied only 2.5% as a function of decis ion criteria. This
indicates that the sensit ivity est imate i s independent of
response biases . I t is noteworthy that d' was calculated
with these data and d' was found to vary systematical ly as
much as 15% as a function of decis ion criteria. This
evidence in support of the proposed theory is in addit ion
to that provided by this dissertat ion. That i s , the theory
is conceptual ly sound, est imated sensit ivity increases as
the S level increases , a reasonable profi le of Hs emerges,
and a relat ionship exists between sensit ivity est imates
obtained by several psychophysical procedures.
An invest igation featuring various a priori prob
abi l i t ies of S i s currently being completed with pigeons
(Thomas and King, in preparation, 1975) . As reported for
the d' sensit ivity index (Green and Swets , 1966) , the value
50
of the sensit ivity index K i s expected to remain constant as
the a priori probabil i ty of S i s varied.
A theory of detect ion processes should have gener
al i ty and not be bound to any s ingle psychophysical method.
Therefore, a complete theory should predict performance in
a variety of test s i tuations. Green and Swets (1966)
derived a predicted relat ionship between performance with
two test ing procedures using a constant S level; the area
under the Y-N rat ing ROC curve approximated the percentage
of correct responses with the forced-choice technique. The
present approach includes a proposed relat ionship between
the sensit ivity est imate result ing from the differing pro
cedures, however the init ial predict ion had to be modif ied.
I t i s not known i f this modif icat ion was necessitated by the
within subject design that tested al l subjects init ial ly
with the 2AFC procedure, or i f the usage and s ize of the
constant correct ion is a necessary part of the formula. In
addit ion, the comparison between K^ a n t^ was based upon a
spatial rather than upon the temporal forced-choice pre
sentation which was used by Green and Swets (1966) . This
issue may be resolved after al l the human rat ing data are
analyzed (King and Fobes, in preparation, 1975) . The com
parison of the sensit ivity est imate obtained by the two
psychophysical procedures wil l then include the typical
between groups, temporal 2AFC design.
51
The trend in contemporary psychophysics i s to con
sider a threshold in terms of a response continuum. This
has resulted in part from SDT's separation of sensit ivity
est imation and factors affect ing a response threshold. The
present paper describes an approach which also permits
separation of nonsensory bias and sensit ivity, while re
taining the concept of a sensory threshold. This theory is
promising at this stage and warrants addit ional examination
of i ts empirical ly testable consequences.
APPENDIX A
FORMULAS FOR ESTIMATION OF HYPOTHESES
Table 6 indicated that a given response sequence to
a particular st imulus pattern was not synonymous with a
s ingle H. Therefore, formulas were developed to est imate
the proportion of the observed frequency due to each H,
Assumptions concerning the relat ionship between Hs' proba
bi l i t ies and the observed frequencies of the 32 events in
Table 6 permit the statement
4a + 24b + 48c + 8d + 16e + 8f + 8g + 32r = 4 , (A. l )
which reduced to
a + 6b + 12c + 2d + 4e + 2f + 2g + 8r = 1 , (A.2)
where a = A, 6b = B, 12c = C, 2d = D, 4e = E, 2f = F, 2g =
G, and 8r = R.
The solutions for "a" involved the equations that
included "a" (cel ls 5, 14, 23, and 32) from the 32 equations
in Table 6 , These total led
EQ = 4a + 12b + 12c +d+2e+f+g+ 4r. (A.3)
In order to apply Equation (A.2) , i t was rewritten
2d + 4e + 2f + 2g + 8r = 1 - (a + 6b + 12c) . (A.4)
52
53
The ZQ in l ine (A.3) was then mult ipl ied by two, ci
2£Q = 8a + 24b + 24c + (2d + 4e + 2f + 2g + 8r) , (A.5) ci
and the term on the r ight s ide of Equation (A.4) was
substituted for the identical expression in l ine (A.5)
2EQ = 8a + 24b + 24c + (1 - a - 6b - 12c) . (A.6) ci
Solving for "a,"
a = (2£Q= - 18b - 12c - l ) /7 . (A.7) CI
The solution for "a" was thus found to be a function of the
obtained proportion of "a" as wel l as variables "b" and
" c . "
The equations containing "b" (cel ls 4 , 5 , 6 , 1 , 11,
13, 14, 16, 18, 21, 23, 24, 25, 30, 31, and 32) total led
= 4a + 24b + 36c + 4d + 8e + 4f + 4g + 16r. (A.8)
Applying l ine (A.2) ,
l /2ZQb = 2a + 12b + 18c + (1 - a - 6b - 12c) . (A.9)
Solving for "b,"
b = ( l /2ZQb - a - 6c - l ) /6 . (A.10)
The equations containing "c" (al l cel ls except 1 ,
10, 19, and 28) total led
£Q c = 4a + 24b + 48c + 7d + 14e + 7f + 7g + 28r. (A.11)
54
Applying l ine (A.2) ,
2/7£Q c = 8/7a + 48/7b + 96/7c + (1 - a - 6b - 12c) .
(A.12)
Solving for "c,"
c = (2IQC - a - 6b - 7) /12 (A.13)
These three solution equations (A.7, A.10, and A.13)
may be rewritten
2£Q - 1 = 7a + 18b + 12c a
1/220^ ~* 1 — a + 6b + 6c
2£Q c - 7 = a + 6b + 12c, (A.14)
and solved by the method of determinates where
7 18 12
A = 1 6 6
1 , 6 12
504 + 108 +72-72 ~ 216 - 252
144. (A.15)
Substituting 2£Q - 1, 1/2EQ, - 1, and 2EQ - 7 for the Ct iJ
coeff ic ients of "a" in the matrix of l ine (A.15) ,
55
2EQ -1 a
18 12
Aa = l /2EQb - l 6 6
2ZQ -7 c
6 12
72(2EQ a - 1) + 108(2£Q c 7) + 72(l /2EQb - 1)
- 72(2EQC - 7) - 216(l /2EQb - 1) - 36(2EQ a - 1)
= 36(2£Q a - 1) - 144(l /2EQb - 1) + 36(2ZQ c - 7)
(A.16)
Where a = Aa/A,
a = [ 3 6 (2EQ - 1) - 144(1/2EQ. - 1) + 36(22Q„ - 7J/144 a. JJ C
(A.17)
which reduced to
a = l /4(2ZQ a - 1) - ( l /2ZQb - 1) + 1/4C2EQ c - 7) . (A.18)
Therefore,
A =-=a = , l /2EQ a - l /2EQb + l /2EQ c - 1 (A.19)
Substituting 22Q - 1, 1/2ZQ, - 1, and 2£Q - 7 for 'a ' ~~^b ' -"«c
the coeff ic ients of "b" in the matrix of l ine (A.15) ,
b =
2X0,-1 12 cL
1 l /2EQb - l
1 2EQ c -7 12
= 84Cl/2EQb - 1) + 6(2EQ a - 1) + 12(2£QC - 7)
- 12 ( l /2£Qb - 1) - 12(2EQ a - 1) - 42(2ZQ c - 7)
56
= 72(l /2ZQb - 1) - 6(2EQ a - 1) - 30(2EQ c - 7) . (A.20)
Where b = Ab/A,
b = [72(l /2£Qb - 1) - 6(2£Q a - 1) - 30(2ZQ c - 7)] /144,
(A. 21)
which reduced to
b = l /2( l /2ZQb - 1) - 1/24(2£Q a - 1) - 5/24(2EQ c - 7) .
(A.22)
Therefore,
B = 6b = 6( l /4EQb - l /12EQ a - 5/12EQ c + 1) . (A.23)
Final ly , substituting 2EQ a - 1, l /2EQb - 1, and
2EQ - 7 for the coeff ic ients of "c" in the matrix of l ine c
(A,15) ,
Ac =
7 18 2£Q a~l
1 6 l /2EQb - l
2ZQC~7 1 6
= 42 (2£QC - 7) + 18U/2EQb 1 ) + 6 ( 2 E Q - 1 )
- 6 (2ZQ_ - 1) - 18(2EQ - 7) - 4 2(1/2EQK - 1)
24C2ZQ c - 7) - 24(l /2ZQb - 1) . (A.24)
Where c = Ac/A,
c = [24(2EQC - 7) - 24(l /2£Qb - 1) ] /144, (A. 25)
57
which reduced to
c = 1/6(2EQC - 7) - l /6( l /2ZQb - 1) . (A.26)
Therefore,
C = 12c = 12(1/3ZQ - l /2EQb - 1. (A.27)
The solution for D included the equations containing
"d" (cel ls 1 , 5 , 9 , 13, 17, 21, 25, and 29) which total led
= a + 6b + 12c + 8d + 2f + 2g + 8r. (A.28)
Applying l ine (A.2) ,
= 1 - (2d + 4e + 2f + 2g + 8r)
+ 8d + 2f + 2g + 8r. (A.29)
Solving for."d,"
d = (SQd - 4e - l ) /6 . (A.30)
Thus, "d" was found to be a function of the obtained pro
portion of "d" as wel l as variable "e."
The equations containing "e" (al l even numbered
cel ls) total led
£Q e = 2a + 12b + 24c + 16e + 4f + 4g + 16r. (A.31)
Applying l ine (A.2) ,
l /2IQ e = 1 - (2d + 4e + 2f + 2g + 8r) + 8e + 2f
+ 2g + 8r, (A.32)
58
Solving for "e,"
e = ( l /2ZQ e + 2d - l ) /4 . (A.33)
Lines (A.30) and (A.33) may be rewritten
£Qd - 1 = 6d - 4d
l /2ZQ e - 1 = 4e - 2d, (A.34)
and these two equations with two unknowns can be s imul
taneously solved. For "d," the £Qd - 1 = 6d - 4e and
l /2£Q e - 1 = 4e - 2d combined to
4d = EQd - 1 + l /2EQ e - 1, (A.35)
which reduced to
D = 2d = 2( l /4£Qd + 1/8ZQ - 1/2) . (A.36)
For "e," l /2ZQ e - 1 was mult ipl ied by a posit ive
number ( three) to prevent cancel lat ion of "e" by combination
of - 1 and l /2£Q e - 1. Therefore, £Qd - 1 = 6d - 4e and
3( l /2EQ e - 1) = 12e - 6d combined to
8e = 3( l /2ZQ e - 1) + EQd - 1, (A.37)
which reduced to
E = 4e = 4(3/16EQ e + l /8EQd - 1/2) . (A.38)
To solve for F, the equations containing "f" (cel ls
4, 5 , 12, 13, 19, 22, 27, and 30) were total led
59
£Q f = a + 6b + 12c + 2d + 4e + 8f + 8r. (A.39)
Applying l ine (A.2) ,
EQ f = 1 - (2d + 4e + 2f + 2g + 8r) + 2d + 4e + 8f + 8r.
(A. 40)
Solving for "f ,"
f = (£Q f + 2g - l ) /6 . (A.41)
Thus, "f" was found to be a function of the obtained
proportion of "f" as wel l as variable "g."
The equations containing "g" (cel ls 1 , 8 , 9 , 16, 18,
23, 26, and 31) total led
EQ = a + 6b + 12c + 2d + 4e + 8g + 8r. (A.42) g
Applying l ine (A.2) ,
£Q g = 1 - (2d + 4e + 2f + 2g + 8r) + 2d + 4e + 8g + 8r.
(A.43)
Solving for "g,"
g = (£Q„ + 2f - l ) /6 . (A.44) y
Lines (A.41) and (A.44) may be rewritten
ZQf - 1 = 6f - 2g
EQ - 1 = 6g - 2f , (A. 45) 9
and these two equations with two unknowns can be s imul
taneously solved.
60
For "f", 3(ZQ^ - 1) and EQ^ - 1 combined to
16f = 3(EQ f - 1) + EQ g - 1, (A.46)
which reduced to
F = 2f = 2(3/16EQ f + 1/16EQ - 1/4) . (A.47) l. y
For "g," 3(EQg - 1) and EQ^ - 1 combined to
16g = 3 (EQ - 1) + EQ f - 1, (A.48) y
which reduced to
G = 2g = 2(3/16EQ + 1/16EQ- - 1/4) . (A.49) H J-
APPENDIX B
FORMULAS FOR ESTIMATION OF K
The formulas for the est imation of K presented in
the Theoretical Conceptual izat ion and Analytical Procedure
sect ion involved the fol lowing components .
A = P(3Ds) = P(3Ds A 3As) (B.1)
B = P (2Ds) = P(2Ds A 3As) + P(2Ds A 2As) (B.2)
C = P ( I D )
= P ( I D A 3 A s ) + P ( I D A 2 A s ) + P ( 1 D A 1 A ) ( B . 3 )
D = obtained value of H D (B.4)
E = obtained value of H E (B.5)
F = obtained value of H F (B.6)
G = obtained value of H G (B.7)
R = 1 - ( A + B + C + D + E + F + G ) ( B . 8 )
Mj^ = P (2As A 1A)
= P ( An A An+1 A V2 1 + P ( An A Vl A W
+ P ( An A An+1 A W ( B - 9 )
M , = P ( 1 A A 2 A s )
= P ( A„ A An+X A + P ( An A A„+ l A An+2>
+ P ( A f l A n + 1 A A n + 2 * ( B . 1 0 ) n
61
62
t = P ( A n ) ( B . l l )
T = P ( A A A A A „ ( B . 1 2 ) n n+1 n+2
K = P ( D n | A n ) ( B . 1 3 )
63
+ p ( D n A V l A V 2 I 3 A S ) + P ( D n A V l A V 2 I 3 A S ) I
+ p ( An A Vl A An+2' P ( D„ A Vl A V2 lAn A An+1
A V 2 > + P ( A n A A n + 1 A V 2 > P ( D n A V l A V 2 l A n
A A A A , „ ) + P ( A A A . A A ) P ( D A D , n+1 n+2 n n+1 n+2 n n+1
A D . J A A A - A A , 0 ) . ( B . 2 0 ) n + 2 ' n n + 1 n + 2
Substituting for def init ions in (B.12) and (B.13) ,
B = 3TK 2 (1-K) + [P(An A An + 1 A An + 2> + P(An A An + 1
A An+2> + P ( An A An+1 A An+ 2 ) I K 2 - ( B" 2 1 )
Substituting for the definit ion in (B.9) ,
B = 3 T K 2 ( ( 1 - K ) + M
Recommended