1. Origins of Signal Detection Theory
Problem in Psychophysics
Thresholds: is sensitivity discrete or continuous?
Sensitivity confounded with response bias

2. Thresholds
Solution: detection theory (engineering)

3. Signal Detection Theory
Response
Yes No
Hit (H)
Miss (M)
False Alarm (FA)
Correct Rejection (CR)
Signal
State of the World
Assumes two possible states of the world: signal and noise
(see Figure 1). This assumes a crisp, traditional set: events
either are or are not signals.
Noise

4. Assumptions of Signal Detection Theory
Noise is always present (i.e. in the nervous system and/or in the signal generating system)
The noise is normally distributed with σ2 = 1
For Gaussian model
When a signal is added to the noise, the distribution is shifted upward along the sensory dimension. Variance remains constant (equal variance model).

5. Assumptions of Signal Detection Theory
Observers are both sensors and decision makers
To evaluate the occurrence of an event, observers adopt a decision criterion
Sensitivity and Response Bias are independent
Statistical
Theoretical
Empirical

6. Distribution of Noise and Signal + Noise

7. Sensitivity d’ = zH - zF d’ Task Person
> very easy very sensitive
moderately easy moderately sensitive
moderately difficult moderately insensitive
< very difficult very insensitive

8. Relation of d’ to Other Statistics
If μn=0 and σn=1 (i.e., if the N distribution is unit normal) then the
ROC function, in its most general form, is

9. Testing a Mean From One Distribution

10. Relation of d’ to t-test

11. Comparing Means from Two Distributions

12. Standardized Mean Difference Effect Size

13. Response Bias  = f(SN)/f(N) c = -.5(zH + zF) Lenient: 0-1 Unbiased: 1
Conservative: 1- 8
c = -.5(zH + zF)
Lenient: <0
Unbiased: 0
Conservative: >0

14. Three values of  1 2 3
Sensory magnitude (X)
P(event|x) N SN

15. Receiver Operating Characteristic (ROC)

16. ROC Curves: Sensitivity

17. ROC Curves: Response Bias

18. ROC Curve in Z-score form1 ZH 1
ZFA

19. ROC for σ2N = σ2SN 3 ZH -3 -3 3
ZFA

20. What is Independence? Statistical: P(A|B)=P(A) PB|A)=P(B)
Theoretical/Logical: β can vary independently of d’
Empirical: experimental evidence is consistent with the independence assumption (e.g. Form of empirical ROC)

21. Three values of  1 2 3
Sensory magnitude (X)
P(event|x) N SN

22. ROC Curve in Z-score form1 ZH 1
ZFA

23. ROC for σ2N = σ2SN 3 ZH -3 -3 3
ZFA

24. What if both the mean and variance Change?

25. Alternative Sensitivity Measures
Az: Area under the ROC (e.g., see Swets,1995, ch. 2-3; Swets & Pickett, 1982)
Range: from .5—1.0
Underlying distributions can have unequal variances
Assumes that the underlying distributions can be monotonically transformed to normality
ZH= a + bZF

26. Area under the ROC

27. ‘Non-parametric’ Measures: Sensitivity
Not really non-parametric: No distribution assumed, but follows a logistic distribution (Macmillan & Creelman, 1990)

28. ‘Non-parametric’ Measures: Response Bias
For applications to vigilance, see
See, Warm, Dember, & Howe (1997)

29. What if the Situation is More Complex?
Response 1 2 3 4 5 6 7
State of the World

30. Identification and Categorization1
 2
 3
 4
 5
 6
Response 1 2 3 4 5 6 7 x

31. Fuzzy Logic Traditional Set Theory: A ∩ A = 0
Fuzzy Set Theory: A ∩ Ā ≠ 0
One assigns non-binary membership, or degrees of
membership, to classes of events (fuzzification).
Traditional Set Theory: Set membership is mutually exclusive and
exhaustive: elements either are or are not members of a set.
Fuzzy Set Theory: Events can simultaneously be and not be
members of a set.

32. Elements of Fuzzy Signal Detection Theory
Events can belong to the set “signal” (s) to a degree ranging
from 0 to 1
Events can belong to the set “response” (r) to a degree ranging

33. Computation of FSDT Measures
Select mapping functions for signal & response dimensions
Assignment of degrees of membership to the four outcomes
(H, M, FA, CR) using mixed implication functions.
Compute fuzzy Hit, Miss, False Alarm, and Correct Rejection
Rates
Compute detection theory measures of sensitivity and response
bias

34. 1. Mapping Functions
To assign degrees of (s, r) membership to events, all possible states of the world and each possible response must be evaluated using a mapping function. For the set s a mapping function relates the signal value (s) to a variable (or set of variables) that describe the state of the world (see Figure 3).
A mapping function for the set r relates the response value (r) to a response
variable (or set of variables). For instance, a mapping function could be based upon confidence ratings of signal presence, a technique used in traditional SDT (Green & Swets, 1966;MacMillan & Creelman, 1991).
Note that the mapping function can be discrete or continuous, and can be derived empirically, theoretically, or based upon legal or industry standards. Either s or r (or both) can be fuzzy. The measurement challenge is to map s to the actual states-of-the-world. The derivation of the function may be theoretically based, but the reliability and validity of these functions should be established empirically.

35. 2. Assignment of Set Membership to Categories
Mixed Implication Functions
H = min (s,r)
M = max (s-r, 0)
FA = max (r-s, 0)
CR = min (1-s, 1-r)
When r>s, some degree of FA membership will result, since the response is stronger than the actual degree of signal. Similarly, when r<s, some degree of miss membership results, since the response is less than the degree of “signalness” of the event.
●Performance is best when r=s, as the observer’s responses are well mapped to the degree of signal present. This constitutes an ideal observer.
●Note that Parasuraman et al. (2000) did not claim these functions to be the only possible implication functions. The form of the mapping function can change with the domain of interest.

36. 3. Computation of Fuzzy Hit and False Alarm Rate
H= Σ(Hi)/ Σ(si) for i=1 to N
M = Σ(Mi)/ Σ(si) for i =1 to N
FA = Σ(FAi)/ Σ(1-si) for i=1to N
CR = Σ(CRi)/ Σ(1-si) for i= 1 to N
Once fuzzy hit and false alarm rates are computed, the standard formulas for SDT indices are applied

37. Truth Table for FSDT Data
Hit FA
Miss CR
.83
1.00
.17
.50
.33
.67
.16

38. ‘Perfect’ Performances r
Hit FA
Miss
C’R
.83
1.00
.17
.50
.33
.67
.16

39. ‘Hitness’ and ‘False Alarmness’
Miss CR
.83
1.00
.17
.50
.33
.67
.16

40. ‘Hit and Miss’ s r
Hit FA
Miss CR
.83
1.00
.17
.50
.33
.67
.16

41. Fuzzy Stimulus and Response: Duration Discrimination1 2 3 4 5 6 7
20 msec
200
220
240
260
280
300
320
80 msec
360
440
520
600
680

42. Comparison of Fuzzy and Crisp ROC Curves

43. Comparison of Fuzzy and Crisp ROC Curves

47. Response Time as a Function of Degree of Stimulus Criticality
1100
1000
900
Response Time (msec)
800
Transition hh
700 hl
600 lh
500 ll 1 2 3 4 5 6 7 1
Stimuli

48. Reaction Time as a Function of Stimulus Value: 80 msec Discrimination

49. Reaction Time as a Function of Stimulus Value: 20 msec Discrimination