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Origins of Signal Detection Theory

Learn about the foundations of Signal Detection Theory in psychophysics, exploring if sensitivity is discrete or continuous, the role of noise, and how sensitivity and response bias interact. Discover key assumptions and statistical measures used to evaluate detection processes.

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Origins of Signal Detection Theory

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  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 Signal State of the World 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 >3.5 very easy very sensitive 2.6-3.5 moderately easy moderately sensitive 1.6-2.5 moderately difficult moderately insensitive <1.5 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 Lenient: 0-1 Unbiased: 1 Conservative: 1- 8  = f(SN)/f(N) c = -.5(zH + zF) Lenient: <0 Unbiased: 0 Conservative: >0

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

  15. Receiver Operating Characteristic (ROC)

  16. ROC Curves: Sensitivity

  17. ROC Curves: Response Bias

  18. ROC Curve in Z-score form 1 ZH 0 0 1 ZFA

  19. ROC for σ2N = σ2SN 3 0 ZH -3 3 -3 0 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  2 1 3 P(event|x) N SN Sensory magnitude (X)

  22. ROC Curve in Z-score form 1 ZH 0 0 1 ZFA

  23. ROC for σ2N = σ2SN 3 0 ZH -3 3 -3 0 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 State of the World

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

  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 from 0 to 1

  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

  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)

  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

  37. Truth Table for FSDT Data

  38. ‘Perfect’ Performance

  39. ‘Hitness’ and ‘False Alarmness’

  40. ‘Hit and Miss’

  41. Fuzzy Stimulus and Response: Duration Discrimination

  42. Comparison of Fuzzy and Crisp ROC Curves Crisp Fuzzy

  43. Comparison of Fuzzy and Crisp ROC Curves Crisp Fuzzy

  44. Response Time as a Function of Degree of Stimulus Criticality 1100 1000 900 Response Time (msec) Transition 800 hh 700 hl 600 lh 500 ll 1 2 3 4 5 6 7 1 0 Stimuli

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

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

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