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
• 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
    
• ​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
      
• ​​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'
    
• ​d' is independent from 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
      
  • ​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
    

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?