Signal Detection Theory

The classical psychophysicists believed in fixed thresholds

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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. 1 2 3 4 5 6 7 8 9

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• “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” threshold

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The measured threshold

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• 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).

Produces

Signal Strength

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Representation of Variable Neural Activity

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 light50% of trials - nothing

Thought Experiment

A record from a cat’s retinal ganglion cell showing the rate ofspike firing as a function of the presence or absence of a stimulus

There is spontaneous nonzero level activity even without a stimulus

Ganglion Cell Output

No Stimulus

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Output

Ganglion Cell Output

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Trial Number

Stimulus target

Signal +Noise

Noise

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Average Number of Impulses in 100 ms Interval

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( 3) ( 3)

Standard deviation = ( 3)

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.

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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 DistributionsPlots showing the probability that any given perceptual effect is caused by noise (no signal is presented) or by signal plus noise (signal is presented)

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Subjective intensity of the stimulus

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N S+N

On each trial the subject must decide whether no signalwas 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”.

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Criterion

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Criterion (Conservative)

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Criterion (Liberal)

According to SDT one can separate

sensitivity and the criterion

Sensitivity is conceptualized as the separationin the means of the noise and signal+noise distributions

Sensitivity is expressed as dd-prime)

The criterion is expressed as (Beta)

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Subjective intensity of the stimulus

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Differences in sensitivity mean differences inThe separation of the noise and signal+noise distributions

Low Sensitivity

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Differences in sensitivity mean differences inThe separation of the noise and signal+noise distributions

High Sensitivity

To represent differences in sensitivity and criterion placementa 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 MatrixRESPONSE

SIGNAL“YES” “NO”

PRESENT

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Hit Miss

FalseAlarm

CorrectRejection

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Subjective intensity of the stimulus

Noise only

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Subject says:

‘Yes’

FALSE ALARMS

CORRECTREJECTIONS

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Subjective intensity of the stimulus

Signal + Noise

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HITSMISSES

Conceptualizing an ROC Curve

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Proportion of false alarms

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Liberal Criterion

Conceptualizing an ROC Curve

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Neutral Criterion

Conceptualizing an ROC Curve

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Conservative Criterion

The criterion placement can be manipulated byexpectations and outcome payoff

RESPONSESIGNAL YES NO

PRESENT

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0.75 (Hit)

0.75 (Correct Rejection)

0.25 (Miss)

0.25 (False Alarm)

Signal present 50% of the time

RESPONSESIGNAL YES NO

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Signal present 90% of the time

RESPONSESIGNAL YES NO

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Signal present 10% of the time

Examples of possible Outcome Matrices for different payoffs:

RESPONSE

SIGNAL YES NO

PRESENT

ABSENT

win $10

win $1lose $1

lose $1

Liberal ResponseCriterion

RESPONSE

SIGNAL YES NO

PRESENT

ABSENT win $10

win $1 lose $1

lose $1

Strict ResponseCriterion

.98 .02

.90 .10

.10 .90

.01 .99

(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’ssensitivity?

It has no effect.

Observer sensitivity (d’) is related to the distancebetween the centres (means) of the Noise andSignal + Noise probability distributions.

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d’ d’

Small dprime (d’)weak signal orlow observer sensitivity

Large dprime (d’)strong signal orhigh observer sensitivity

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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.