Signal Detection Theory
Theory of Signal Detection resolves the problem of how to effectively separate sensitivity from response proclivity
- What is it and why do we use it?
- Theory provides a tool to use to separate out the effects of bias and sensitivity in psychophysical measurement
- sensitivity - what an organism can do in response to sensory stimuli
- response proclivity - what the organism characteristically does
- both of these are limited by the anatomy and physiology of the sensory systems
- Theory provides a tool to use to separate out the effects of bias and sensitivity in psychophysical measurement
- According to signal detection theory, perception in general is controlled by two basic internal processes
- sensory impression - on average is dependent on the intensity of the signal
- decision process - an evaluation of this sensory impression
- NOISE - a random disturbance that can be confused with signals, is always present inside humans
- any stimulus, even noise, produces a distribution of sensory impressions
- the sensory impression (internal response) produced by a stimulus differs across different presentations of that stimulus because the sensory system itself is not perfectly stable (it is noisy)
GOALS
- describe variability in threshold (normal bell-shaped curve)
- distinguish between noise and signal + noise probability distributions
- describe two principles of theory of signal detection
- define theory of signal detection terms
- hit, false alarm, miss, correct rejection, bias, sensitivity
- hit, false alarm, miss, correct rejection, bias, sensitivity
- know the application of theory of signal detection theory in audiology
Factors Affecting Responses
- The theory of signal detections provides the best approach to separate the effects of sensitivity from those of response bias
- What led to a "yes" or "no" decision as opposed to "what did the subject hear (or not hear)?"
- A hit occurs when the signal is present and the subject says "yes"
- A correct rejection occurs when the signal is absent and the subject says "no"
- The signal is present but the subject says "no." This is called a miss
- The signal is absent but the subject says "yes." Here a false alarm has occurred
- The traditional formulat to correct the hit rate for change success is
- p(hit) = p(hit) - p(false alarm) / 1 -p(false alarm)
- We assume that there is always some degree of noise present
- this may be noise in the environment, instrumentation noise, or noise due to the subject's moving around and fidgeting
- in other words, the subject must decide whether the stimulation in the auditory system is due to noise alone (N) or to signal-plus-noise (SN)
- this process may be represented by distributions along a decision axis
- The N and SN distributions show the probability functions of noise alone (N) and signal-plus-noise (SN)
- The separation between the N and SN curves thus becomes a measure of sensitivity
- This is an unbiased measure because the separation between the two curves is not affected by the subject's criteria for responding (biases)
- The separation is determined solely by the energy in the signals and the sensitivity of the auditory system
- This separation is measured in terms of a parameter called d prime (d')
- The value of d' is equal to the difference between the means and the N and SN distributions divided by their standard deviation
- The value of d' is equal to the difference between the means and the N and SN distributions divided by their standard deviation
- Criterion points for two degrees of overlapping of the noise alone N and signal-plus-noise (SN) distributions. The probabilities corresponding to the SN and N distributions at the criterion point are highlighted by brackets. Values of x below the criterion result in no decisions and those greater than the criterion yield yes decisions
- The first factor affecting the criterion may be expressed by the question "What is the probability that there is a noise alone compared to the probability that there is a signal-plus-noise for a given value of x?"
- The four possible outcomes of a "yes" or "no" response based upon a given criterion value (vertical line).
- Relationships between ROC curves (center of figure) and the
The Criterion; The case of observers
- Hits
- Misses
- False Alarm
- Correct Rejection
d - prime (d')
- d-prime is a characterization of performance (like % correct of response time) that is independent of the subject's criterion for answering "yes" or "no"
- referred to as "sensitivity"
- d' = meanSN - mean N/S
- (SN = signal + noise; n = noise alone)
- d' is the distance between the N and S+N curves in the bell curve
- d' is measured in standard deviations
- assumption: the two underlying distributions are normal with equal variance (i.e. both curves have the same standard deviation)
- d' is determined by both the separation and spread
- the separation of the two peaks and spread around each peak
- high noise = lots of overlap and a smaller d'
- low noise means not much overlap and a larger d'
- the separation of the two peaks and spread around each peak
- d' is independent from criterion
- the distance between the distributions can change regardless of the observer's criterion
- the distance between the distributions can change regardless of the observer's criterion
- RESPONSE CRITERION -- β/Beta
- β is the subject's criterion for deciding if stimulus is present or not
- β lies on the decision axis
- β reflects subject's confidence in their decision
- if energy is > β subject says yes if < β subject says no
- the decision about whether an input is on SN or N distributuion is usually made with respect to a preselected point along whatever decision axis the detector is using
- this point is the detector's response criterion
- the value of β depends on
- probability of N compared to probablility of SN
- compare N & SN distributions at any point on decision axis - this N/SN ratio varies as overlap of distributions (d') varies
- subject's knowledge of N & SN probabilities
- β also affected by value of correct and incorrect responses
- ideal observer - (conceptual) bases β on point where chance of error is minimized - minimum of misses and false alarms
- human observers respond to changes in a pirori probabilities by adjusting β in the same direction as an ideal detector but not always by same amount
- the criterion can change, without changing the separation between the two distributions
- probability of N compared to probablility of SN
- changing response criterion or bias when sensitivity doesn't change will increase (or decrease) hits and false alarms
- changing sensitivity when criterion doesn't change will increase (or decrease) hits and decrease (or increase) false alarms
- β is the subject's criterion for deciding if stimulus is present or not
More on Signal Detection Theory
- you can change the proportion of different responses by instructing the listener to change their criterion
- in one case you say respond only if they are sure they hear it
- in the other case you say respond even when it is very soft and you are not sure you hear it
- The Receiver Operating Characteristic Curve
- the ROC curve is traced out by plotting Hits against False Alarms as the criterion moves
- the criterion is made to move by changing the payoffs to the observer (increasing the reward for correct answers or increasing the penalty for wrong answers)
- ROC curves start at (0,0) and go up to (100,100) Why?
- If d-prime is zero, what is the shape of the ROC curve?
- If d-prime is large (e.g. 4 or larger) what is the shape of the ROC curve?