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Origins of Signal Detection Theory

Origins of Signal Detection Theory. Problem in Psychophysics Thresholds: is sensitivity discrete or continuous? Sensitivity confounded with response bias. Thresholds. Solution: detection theory (engineering ). Signal Detection Theory. Response. Yes. No. Signal.

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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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