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APPENDIX B: DETECTION SENSITIVITY AND RESPONSE BIAS 37
APPENDIX B: DETECTION SENSITIVITY AND
RESPONSE BIAS
Classical psychophysical methods have as their goal the determination of a stimulus threshold. Thresholds
measured can be detection, discrimination, recognition, and identification. The concept of threshold actually has
two meanings, one empirical and one theoretical. Empirically a threshold is the stimulus energy that will allow
the observer to perform a task (detection, discrimination, recognition, or identification) at some criterion level of
performance (75 percent correct, for example). Sensitivity is defined as the reciprocal of the threshold value.
The classical concept of detection threshold, as represented in the high threshold model of detection,
hypothesizes that there is a stimulus level below which the stimulus has no effect (as if the stimulus were not
there) and above which the stimulus is perceived. The classical psychophysical methods (the method of limits,
the method of adjustment, and the method of constant stimuli) developed by G.T. Fechner (1860) were deigned
to infer the stimulus value corresponding to the theoretical threshold from the observed detection performance
data. In this sense, the stimulus threshold is the stimulus energy that exceeds the theoretical threshold with a
probability of 0.5. Until the 1950s the high threshold model of detection dominated conceptualization of the
detection process and provided the theoretical basis for the psychophysical measurement of thresholds.
In the 1950s a major theoretical advance was made by combining detection theory with statistical decision
theory. Actual detection performance was conceived to be based on two separate and independent processes: a
sensory process and a decision process. The sensory process transforms the physical stimulus energy into some
sort of internal representation, and the decision process makes a decision based on this representation to say
“yes, the stimulus was present” or “no, the stimulus was not present” (in the simplest case). Each of these
separate processes is characterized by at least one parameter: the sensory process by a sensitivity parameter and
the decision process by a response criterion or response bias parameter. It was further realized that estimates of
thresholds made using any of the three classical psychophysical methods confounded the sensitivity of the
sensory process with the response criterion of the decision process. In order to measure these two separate
characteristics, one needs two measures of detection performance. Not only must one measure the probability
that the observer says “yes” when a stimulus is present (the hit rate: HR) but also one must measure the
probability that the observer says “yes” when a stimulus is not present (the false alarm rate: FAR). Under certain
assumptions, these two performance measures, the hit rate and the false alarm rate, may then be used to estimate
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detection sensitivity and decision criterion.
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APPENDIX B: DETECTION SENSITIVITY AND RESPONSE BIAS 38
HIGH THRESHOLD MODEL OF DETECTION
The specific way in which the hit rate and the false alarm rate are used to derive detection sensitivity and
response criterion depends on the specific model one adopts for the sensory process and the decision process.
Some of these different models and how to distinguish among them are discussed by Krantz (1969). Assuming
the high threshold model leads to the following measures of sensory process sensitivity and decision process
response criterion:
g = FAR (2)
where p is the probability that the stimulus will exceed the hypothetical threshold and g is the response bias
(called guessing rate in the high threshold model). Equation 1 is the widely used correction-for-guessing
formula. Extensive research testing the validity of the prediction of the high threshold model has led to its
rejection as an adequate description of the detection process and the conclusion that neither Equations 1 nor 2
succeeds in separating the effects of sensitivity and response bias (Swets, 1961; Swets et al., 1961; Krantz, 1969;
Green and Swets, 1974).
One important characteristic of any detection model is its prediction of the relationship between the hit rate
and the false alarm rate as the observer changes the response criterion. This plot of HR against FAR is called an
ROC curve (receiver operating characteristic). By algebraic rearrangement of Equation 1, the high threshold
model of detection predicts a linear relationship between HR and FAR in the ROC curve:
HR = p + (1-p) * FAR (3)
where p is the sensitivity parameter of the high threshold sensory process. This predicted ROC curve is
shown in Figure 17.
When one actually measures the HR and FAR pairs in a detection experiment using different degrees of
response bias, one obtains a bowed-shaped ROC curve shown by the filled circles in Figure 17. This curve,
which one actually obtains in experiments is quite different from the straight line relationship predicted by the
high threshold model and is one of the bases for rejecting that model.
SIGNAL DETECTION THEORY
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A widely accepted alternative to the high threshold model is the signal detection model. This model does
not contain the concept of a sensory threshold (Swets, 1961). It assumes that the sensory process has a
continuous output based on random Gaussian noise and that when a signal is present, it adds to that noise. The
sensitivity of the sensory
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APPENDIX B: DETECTION SENSITIVITY AND RESPONSE BIAS 39
process is expressed as the difference between the mean output under no signal condition and that under signal
condition: d• (d-prime). The decision process is assumed to hold a single decision criterion (in more elaborate
versions of this model, multiple criteria are possible). This decision criterion is based on the output of the
sensory process. If the output of the sensory process equals or exceeds the decision criterion, the observer says
“yes, the signal was present.” If the output of the sensory process is less than this criterion, the observer says “no,
the signal was not present.” The decision criterion may be expressed in several ways. One is beta, the ratio
formed by the likelihood that the observed output of the sensory process was due to a signal being present
divided by the likelihood that the output was due to the signal being absent. Another measure is xc, the critical
value of the sensory process output used as the decision criterion. If one assumes that the probability
distributions describing the output of the sensory process are normal Gaussian distributions of equal variance,
then d• and xc are calculated from the HR and FAR in the following way:
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FIGURE 17 Hit rate as a function of false alarm rate. The filled circles are HR-FAR pairs from a detection
experiment, forming a bowed-shaped ROC curve. The straight line is the ROC curve predicted by the high
threshold model of detection.
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APPENDIX B: DETECTION SENSITIVITY AND RESPONSE BIAS 40
d• = ZHR − ZFAR (4)
xc = ZFAR (5)
where ZHR and ZFAR are the z-score transforms, based on the normal distribution, of the HR and FAR
probabilities. The ROC curve predicted by the signal detection model is shown in Figure 18 along with the
empirical data shown in the previous figure. The signal detection prediction is in accord with the observed data.
The data shown in Figure 18 correspond to d• = 1.0. All ROC curves predicted by this model are anchored at the
0,0 and 1,1 points on the graphs. Each different value of d• generates a different ROC curve. For d• = 0, the ROC
curve is the positive diagonal extending from 0,0 to 1,1. For d• greater than 0, the ROC curves are bowed. As d•
increases, so does the bowing of the corresponding ROC curve.
An algebraic rearrangement of Equation 4 leads to this relationship between HR and FAR:
ZHR = d• + ZFAR (6)
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FIGURE 18 Hit rate as a function of false alarm rate. The filled circles are the same data as in Figure 18. The
smooth curve is the ROC curve predicted by the equal-variance Gaussian signal detection theory.
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APPENDIX B: DETECTION SENSITIVITY AND RESPONSE BIAS 41
Equation 6 predicts that when HR and FAR are plotted as z-scores instead of probabilities, the bowed-
shaped ROC curve shown in Figure 18 will be a straight line, because equation 6 is in the form of a linear
equation. This predicted ROC curve is shown in Figure 19, along with the data from the previous figures.
Sensitivity is generally a relatively stable property of the sensory process, but the decision criterion used by
an observer can vary widely from task to task and from time to time. The decision criterion used is influenced by
three factors: (1) the instructions to the observer; (2) the relative frequency of signal and no-signal trails (the a
priori probabilities); and (3) the payoff matrix, the relative cost of making the two types of errors (false alarms
and misses) and the relative benefit of making the two types of correct responses (hits and correct rejections).
These three factors can cause the observer to use quite different decision criteria at different times, and, if the
proper index of sensitivity is not used, changes in decision criteria will be incorrectly interpreted as changes in
sensitivity. Figure 20 shows the high threshold sensitivity index, p, for different values of decision criteria, for an
observer having constant sensitivity. The detection
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FIGURE 19 Z-score of the hit rate as a function of the Z-score of the false alarm rate. The same data from
Figure 19 replotted in Z-score coordinates. The predicted ROC curve and the data form a straight line.
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APPENDIX B: DETECTION SENSITIVITY AND RESPONSE BIAS 42
sensitivity, p, calculated from Equation 1, is not constant but changes as a function of decision criteria.
FIGURE 20 Detection sensitivity, p, calculated under the assumptions of the high threshold model (equation 1),
as a function of response bias, g (equation 2). Note that p is not independent of g, contrary to the assumption of
the high threshold model of detection. Also plotted is d′ (equation 4) as a function of response bias. Note that it is
unaffected by shifts in response bias.
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A widely used psychophysical procedure is the forced-choice paradigm, especially the two-alternative,
forced-choice (2AFC) paradigm. Because only one performance index is obtained from this paradigm, the
percentage correct, it is not possible to calculate both a detection sensitivity index and a response criterion index.
It is now understood, however, that detection performance in the 2AFC paradigm is equivalent to an observer
using an unbiased decision criterion and that the percentage correct performance can be predicted from signal
detection theory. Specifically, the percentage correct in a 2AFC detection experiment corresponds to the area
under the ROC curve obtained if the same stimulus were used in the yes-no signal detection paradigm (Green
and Swets, 1974; Egan, 1975). Calculation of d• from the 2AFC percent correct is also straightforward:
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APPENDIX B: DETECTION SENSITIVITY AND RESPONSE BIAS 43
d• = 2.00.5 * Zpc (7)
where Zpc is the z-score transform of the 2AFC percentage correct (Simpson and Fitter, 1973). The area
under the ROC curve for d• = 1.0, illustrated above, is 0.76 (the maximum area of the whole graph is 1.0).
SUMMARY
The classical psychophysical methods of limits, of adjustment, and of constant stimuli provide procedures
for estimating sensory thresholds. These methods, however, are not able to separate the independent factors of
sensitivity and decision criterion. Furthermore, there is no evidence to support the existence of sensory
thresholds, at least in the form these methods were designed to measure.
There are today two methods that allow one to measure an observer's detection sensitivity relatively
uninfluenced by changes in decision criteria. One method is the forced-choice paradigm, which forces all
observers to adopt the same decision criterion. The second method is based on signal detection theory and
requires that there be two types of detection trials: some containing the signal and some containing no signal.
Both detection sensitivity and response criterion may be calculated from the hit rates and false alarm rates
resulting from the performance in these experiments. Either of these methods may be used to measure the
contrast sensitivity function. “Threshold” contrast corresponds to the stimulus contrast giving rise to a certain
level of detection performance. A d• of 1.0 or a 2AFC detection of 0.76 is often used to define threshold, but
other values may be chosen as long as they are made explicit.
A comparison of contrast sensitivity functions measured by means of the method of adjustment and the two-
alternative, forced-choice method is reported by Higgins et al. (1984). The variability of the 2AFC measurements
is less than half of those made with the adjustment method. This reduction of measurement variability will
increase the reliability of the threshold measures and increase its predictive validity.
Although there are clear benefits of reducing the variability due to differences in decision criterions, the
cost effectiveness of these benefits must be evaluated on a case by case basis. Factors such as testing time, ease
of administration, ease of scoring, and cost must be carefully considered in relation to the desired reliability,
accuracy, and ultimate use to which the measurements will be put. Finally, it must be recognized that no
psychophysical method is perfect. Observers may make decisions in irrational ways; some may try to fake a loss
of sensory capacity. Care must be taken, regardless of the psychophysical method used to measure capacity, to
detect such behavior. A properly administered, conceptually rigorous psychophysical procedure will ensure the
maximum predictive validity of the measured sensory capacity.
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