Signal Detection Theory

The classical psychophysicists believed in fixed thresholds Ideally, one would obtain a step-like change from no detection to detection as stimulus energy increases. We have seen, however, that in detection and discrimination tasks one does not obtain such a discontinuous function, but rather usually gets an S-shaped or ogive function.

“Classical” psychophysics is based on the assumption that there is a real, biologically-based, threshold and that the shape of the psychometric function is a consequence of moment-to moment- variability in the level of the threshold The “true” thresholdThe measured threshold

In an experiment using the method of constants, two observers obtain the the results shown below On the basis of these data it is reasonable to assume that they have equivalent sensitivities

In another experiment, two observers show a different pattern of results Question: Are the differences in the thresholds real, or can they be attributed to other factors?

It is possible that these observers are equally sensitive but for some reason have different thresholds

Although it is possible that one observer is more sensitive than the other, it is also possible that she is a more liberal responder; i.e. she is more likely to say “yes” to barely detectable stimuli In signal detection terms, she has a lower response criterion It is possible to determine if this is the case by conducting a signal detection experiment

According to Signal Detection Theory, observer sensitivity and decision criterion placement can be distinguished

Some Assumptions Signal Detection Theory began with the assumption that there is no such thing as a biologically based threshold Assumed that there was a continuum of sensation from low to high, even in the absence of stimulation When a signal is presented, it adds to the sensation level When an observer reports that he detects a stimulus he is simply making a decision as to whether his sensation level has exceeded some internal criterion that he has set.

Around the beginning of the 20th century researchers and theorists began to question the notion of a fixed threshold. One such theorist was Solomans (1900). A Precursor to SDT

The importance of variability in neural response emphasized by Solomans and others began a new era in the thinking about detection and discrimination tasks. SDT is a model of perceptual decision making whose central tenet is that perceptual performance is limited by inherent variability and as such requires a decision process.

Suppose you were monitoring the output of the activity of a ganglion cell in a cat's retina. You have to judge whether a weak light was presented or not on each trial. All of the information that you have, however, is the number of impulses in a 100 ms interval that was generated in response to a stimulus or not. 50% of the trials - a weak light 50% of trials - nothing Thought Experiment

A record from a cat’s retinal ganglion cell showing the rate of spike firing as a function of the presence or absence of a stimulus There is spontaneous nonzero level activity even without a stimulus

Noise

To do this task we would probably choose some value (criterion) such that if the number of impulses were equal to or greater than this value (e.g. 10) we would say a signal occurred and if less than this value we would say that it didn't. We would be wrong sometimes, but correct most of the time

Two distributions of importance according to SDT are the noise distribution (N) and the signal + noise distribution (S+N). It is common to illustrate these distributions as normal or Gaussian distributions with the same shape.

Probability Distributions Plots showing the probability that any given perceptual effect is caused by noise (no signal is presented) or by signal plus noise (signal is presented) Subjective intensity of the stimulus Probability NS+N

On each trial the subject must decide whether no signal was present (just Noise) or whether a signal was present (Signal + Noise). But probability distributions for N and S+N can overlap, therefore judgment is difficult. Subject sets a criterion level. (called beta =  ). If subjective intensity of stimulus is greater than criterion, subject says “Yes” If subjective intensity of stimulus is less than criterion, subject says “No”.

Subjective intensity of the stimulus Probability NS+N Criterion

Subjective intensity of the stimulus Probability NS+N Criterion (Conservative)

Subjective intensity of the stimulus Probability NS+N Criterion (Liberal)

According to SDT one can separate sensitivity and the criterion Sensitivity is conceptualized as the separation in the means of the noise and signal+noise distributions Sensitivity is expressed as d  d-prime) The criterion is expressed as  (Beta)

Subjective intensity of the stimulus Probability NS+N d

Subjective intensity of the stimulus Probability N S+N Differences in sensitivity mean differences in The separation of the noise and signal+noise distributions Low Sensitivity

Subjective intensity of the stimulus Probability N S+N Differences in sensitivity mean differences in The separation of the noise and signal+noise distributions High Sensitivity

To represent differences in sensitivity and criterion placement a Receiver Operating Characteristic Curve (ROC) is used

An ROC curve plots ‘Hits’ against ‘False Alarms’ Hit = indicating that a signal was present when it was False Alarm = indicating that a signal was present when it wasn’t

Outcomes of a Signal Detection Experiment Outcome Matrix RESPONSE SIGNAL “YES”“NO” PRESENT ABSENT HitMiss False Alarm Correct Rejection

When signal is not present Subjective intensity of the stimulus Noise only ‘No’ Subject says: ‘Yes’ FALSE ALARMS CORRECT REJECTIONS

When signal is present Subjective intensity of the stimulus Signal + Noise ‘No’ Subject says: ‘Yes’ HITS MISSES

Conceptualizing an ROC Curve Proportion of false alarms Proportion of hits Liberal Criterion

Conceptualizing an ROC Curve Proportion of false alarms Proportion of hits Neutral Criterion

Conceptualizing an ROC Curve Proportion of false alarms Proportion of hits Conservative Criterion

The criterion placement can be manipulated by expectations and outcome payoff RESPONSE SIGNAL YESNO PRESENT ABSENT 0.75 (Hit) 0.75 (Correct Rejection) 0.25 (Miss) 0.25 (False Alarm) Signal present 50% of the time

RESPONSE SIGNAL YESNO PRESENT ABSENT Signal present 90% of the time

RESPONSE SIGNAL YESNO PRESENT ABSENT Signal present 10% of the time

Examples of possible Outcome Matrices for different payoffs: RESPONSE SIGNAL YESNO PRESENT ABSENT win $10 win $1lose $1 Liberal Response Criterion RESPONSE SIGNAL YESNO PRESENT ABSENT win $10 win $1lose $1 Strict Response Criterion (Note: Signal strength is the same in both cases !).

Summary of Criterion effects. Probability distributions show how the proportion of hits and false alarms depends on the observer’s criterion level. How does the criterion level affect the observer’s sensitivity? It has no effect. Observer sensitivity (d’) is related to the distance between the centres (means) of the Noise and Signal + Noise probability distributions.

Some SDT Demos

Summary SDT is a theory developed to deal with the detection of weak signals where a significant decision component is involved. I haven't really shown you any calculation procedures, but it is quite simple to get estimates of d' and the criterion (  According to SDT these two aspects of the detection situation (sensitivity and criterion placement) can be distinguished and it is this aspect of the theory that lends it to some interesting applications beyond sensory psychology.