![Fechner’s idea is essentially a probability
statement
equally often noticed differences are equal
(unless always or never noticed)
Fechner’s idea essentially states that
(a) - (b) = (c) - (d)
if and only if
P(a≽b) = P(c≽d)
This is equivalent to “Fechner’s problem”
For which increasing continuous and F
P(a≽b) = F[ (a) - (b)]](https://image.slidesharecdn.com/signaldetectiontheory-090922032700-phpapp02/85/Psychophysics-33-320.jpg)
![For non-fixed π there are several solutions to
Fechner’s problem
Fechner’s problem:
For which increasing continuous and F
P(a≽b) = F[ (a) - (b)]
holds?](https://image.slidesharecdn.com/signaldetectiontheory-090922032700-phpapp02/85/Psychophysics-40-320.jpg)
![For non-fixed π there are several solutions to
Fechner’s problem
Fechner’s problem:
For which increasing continuous and F
P(a≽b) = F[ (a) - (b)]
holds?
Thurstone’s Law of Comparitive
Judgement Cases III and V](https://image.slidesharecdn.com/signaldetectiontheory-090922032700-phpapp02/85/Psychophysics-41-320.jpg)
![For non-fixed π there are several solutions to
Fechner’s problem
Fechner’s problem:
For which increasing continuous and F
P(a≽b) = F[ (a) - (b)]
holds?
Thurstone’s Law of Comparitive
Judgement Cases III and V
Bradley-Terry-Luce model](https://image.slidesharecdn.com/signaldetectiontheory-090922032700-phpapp02/85/Psychophysics-42-320.jpg)
Psychophysics
- 1. Psychophysics relates response probabilities to stimulus intensities, tries to make subjective sensation measurable and relate physical and psychological scales
- 2. There are three main theories for scaling physical properties in terms of psychological sensation Signal Detection Theory Weber and Weber-Fechner Law Steven’s Power law
- 3. There are three main theories for scaling physical properties in terms of psychological sensation Signal Detection Theory Sensitivity to the presence of a signal Weber and Weber-Fechner Law Steven’s Power law
- 4. There are three main theories for scaling physical properties in terms of psychological sensation Signal Detection Theory Sensitivity to the presence of a signal Weber and Weber-Fechner Law Psychologically equal units Steven’s Power law
- 5. There are three main theories for scaling physical properties in terms of psychological sensation Signal Detection Theory Sensitivity to the presence of a signal Weber and Weber-Fechner Law Psychologically equal units Steven’s Power law People can quantify equal units
- 6. Signal Detection theory formalizes an ideal obser ver of faint signals Signal Detection Theory Sensitivity to the presence of a signal Weber and Weber-Fechner Law Psychological equal units Steven’s Power law People can quantify equal units
- 7. Signal Detection Theory postulates a signal threshold (criterion ) that must be exceeded Signal Detection Signal + Inherent Noise d' Inherent Noise -2 0 2 4 6 signal amplitude
- 8. Signal Detection Theory postulates a signal threshold (criterion ) that must be exceeded Signal Detection ! Signal + Inherent Noise d' Inherent Noise -2 0 2 4 6 signal amplitude
- 9. Signal Detection Theory postulates a signal threshold (criterion ) that must be exceeded Signal Detection ! d' Hit Rate False Alarm Rate -2 0 2 4 6 signal amplitude
- 10. Signal Detection Theory postulates a signal threshold (criterion ) that must be exceeded Signal Detection ! d' Hit Rate False Alarm Rate -2 0 2 4 6 signal amplitude pHit = P (X > β|S) = 1 − Φ(β − d ) pFA = P (X > β|S) = 1 − Φ(β)
- 11. The Receiver Operating Characteristic cur ve relates hit rates to false alarm rates Signal Detection ! d' Hit Rate Receiver Operating Characteristic False Alarm Rate hit rate 1.0 -2 0 2 4 6 ROC curve 0.8 signal amplitude Hit Rate 0.6 pHit = P (X > β|S) = 1 − Φ(β − d ) pFA = P (X > β|S) = 1 − Φ(β) 0.4 0.2 choose FA rate (by setting !) 0.0 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate
- 12. The Receiver Operating Characteristic cur ve relates hit rates to false alarm rates Signal Detection ! d' Hit Rate Receiver Operating Characteristic False Alarm Rate hit rate 1.0 -2 0 2 4 6 ROC curve 0.8 signal amplitude Hit Rate 0.6 pHit = P (X > β|S) = 1 − Φ(β − d ) pFA = P (X > β|S) = 1 − Φ(β) 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate
- 13. The Receiver Operating Characteristic cur ve relates hit rates to false alarm rates Signal Detection ! d' Hit Rate Receiver Operating Characteristic False Alarm Rate hit rate 1.0 -2 0 2 4 6 ROC curve 0.8 signal amplitude Hit Rate 0.6 pHit = P (X > β|S) = 1 − Φ(β − d ) pFA = P (X > β|S) = 1 − Φ(β) 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate
- 14. The Receiver Operating Characteristic cur ve relates hit rates to false alarm rates Signal Detection ! d' Hit Rate Receiver Operating Characteristic False Alarm Rate hit rate 1.0 -2 0 2 4 6 ROC curve 0.8 signal amplitude Hit Rate 0.6 pHit = P (X > β|S) = 1 − Φ(β − d ) pFA = P (X > β|S) = 1 − Φ(β) 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate
- 15. The Receiver Operating Characteristic cur ve relates hit rates to false alarm rates Signal Detection Receiver Operating Characteristic hit rate 1.0 ! d' ROC curve 0.8 Hit Rate 0.6 Hit Rate False Alarm Rate 0.4 0.2 -2 0 2 4 6 0.0 signal amplitude 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate pHit = P (X > β|S) = 1 − Φ(β − d ) pFA = P (X > β|S) = 1 − Φ(β) β = Φ−1 (1 − pFA ) =⇒ pHit = 1 − Φ(Φ−1 (1 − pFA ) − d )
- 16. The shape of the ROC cur ve depends on d’ Signal Detection Receiver Operating Characteristic 1.0 ! 5 d' 1. 0.8 '= d Hit Rate 0.6 Hit Rate False Alarm Rate 0.4 0.2 -2 0 2 4 6 0.0 signal amplitude 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate pHit = P (X > β|S) = 1 − Φ(β − d ) pFA = P (X > β|S) = 1 − Φ(β) β = Φ−1 (1 − pFA ) =⇒ pHit = 1 − Φ(Φ−1 (1 − pFA ) − d ) d = 1.5
- 17. The shape of the ROC cur ve depends on d’ Signal Detection Receiver Operating Characteristic 5 1.0 2. ! '= 5 d' d 1. 0.8 '= d 5 0. Hit Rate '= 0.6 Hit Rate d 0 '= d False Alarm Rate 0.4 0.2 -2 0 2 4 6 0.0 signal amplitude 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate pHit = P (X > β|S) = 1 − Φ(β − d ) pFA = P (X > β|S) = 1 − Φ(β) β = Φ−1 (1 − pFA ) =⇒ pHit = 1 − Φ(Φ−1 (1 − pFA ) − d ) d = 1.5 =⇒ d = 2.5, 1.5, 0.5, 0
- 18. False Alarm and Hit Rates determine ROC cur ve and d’ Receiver Operating Characteristic S ¬S 5 1.0 2. '= 5 d 1. 0.8 '= Y 231 18 d 0. 5 Hit Rate '= 0.6 d 0 N 27 223 '= d 0.4 pFA = 18 / (223+18) =0.075 0.2 pHit = 231 / (231+27) =0.895 0.0 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate β = Φ−1 (1 − pFA ) =⇒ pHit = 1 − Φ(Φ−1 (1 − pFA ) − d )
- 19. False Alarm and Hit Rates determine ROC cur ve and d’ Receiver Operating Characteristic S ¬S 5 1.0 2. '= 5 d 1. 0.8 '= Y 231 18 d 0. 5 Hit Rate '= 0.6 d 0 N 27 223 '= d 0.4 pFA = 18 / (223+18) =0.075 0.2 pHit = 231 / (231+27) =0.895 0.0 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate β = Φ−1 (1 − pFA ) =⇒ pHit = 1 − Φ(Φ−1 (1 − pFA ) − d )
- 20. False Alarm and Hit Rates determine ROC cur ve and d’ Receiver Operating Characteristic S ¬S 5 1.0 2. '= 5 d 1. 0.8 '= d Y 231 18 0. 5 Hit Rate '= 0.6 d 0 '= d N 27 223 0.4 0.2 pFA = 18 / (223+18) =0.075 0.0 pHit = 231 / (231+27) =0.895 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate β = Φ−1 (1 − pFA ) =⇒ pHit = 1 − Φ(Φ−1 (1 − pFA ) − d ) ⇐⇒ d = Φ−1 (1 − pFA ) − Φ−1 (1 − pHit )
- 21. False Alarm and Hit Rates determine ROC cur ve and d’ Receiver Operating Characteristic S ¬S 5 1.0 2. '= 5 d 1. 0.8 '= d Y 231 18 0. 5 Hit Rate '= 0.6 d 0 '= d N 27 223 0.4 0.2 pFA = 18 / (223+18) =0.075 0.0 pHit = 231 / (231+27) =0.895 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate β = Φ−1 (1 − pFA ) =⇒ pHit = 1 − Φ(Φ−1 (1 − pFA ) − d ) ⇐⇒ d = Φ−1 (1 − pFA ) − Φ−1 (1 − pHit ) d’ = -1(1-0.075) - -1(1-0.895) = 2.697
- 22. To summarize SDT: the ROC cur ve, and d’ are Signal Detection Receiver Operating Characteristic hit rate 1.0 ! d' ROC curve 0.8 Hit Rate 0.6 Hit Rate False Alarm Rate 0.4 0.2 -2 0 2 4 6 0.0 signal amplitude 0.0 0.2 0.4 0.6 0.8 1.0 False Alarm Rate pHit = P (X > β|S) = 1 − Φ(β − d ) pFA = P (X > β|S) = 1 − Φ(β) β = Φ−1 (1 − pFA ) =⇒ pHit = 1 − Φ(Φ−1 (1 − pFA ) − d ) =⇒ d = Φ−1 (1 − pFA ) − Φ−1 (1 − pHit )
- 23. Fechner obtained logarithmic relation bet ween psy- chological and physical intensity from Weber’s law Signal Detection Theory Sensitivity to the presence of a signal Weber and Weber-Fechner Law Psychological equal units Steven’s Power law People can quantify equal units
- 24. Weber suggested Just Noticeable Differences as the unit of psychological intensity Weber procedure: Present t wo stimuli differing on one dimension (e.g., weights w1 and W2) Subject indicates which is heavier Modify W2 until π = P(W1≼ W2) = 0.75 ∆(W1) = W2-W1 is called JND
- 25. Weber’s fraction holds that JND’s are proportional to physical stimulus intensity Weber fraction (1860) 60 candles require 1 candle for one JND 120 candles require 2 candles for one JND 300 candles require 5 candles 600 require 10, etc. ∆(I) =k I
- 26. Weber’s fraction holds that JND’s are proportional to physical stimulus intensity Weber’s law ∆(I) =k ∆(I) = k I I holds for many stimuli and intensity ranges Dimension k Taste (salt) 0.083 inaccurate for very low Brightness 0.079 and very high intensities Loudness 0.048 Fechner: ∆(I) = k I + c Line length 0.029 Heaviness 0.020
- 27. Weber’s is applied in photo enhancement soft ware Weber’s law implies that brighter pixels in an image need more enhancement than darker pixels original enhanced using Weber’s law
- 28. Fechner postulated Weber’s JND as a psychological yard stick Weber’s law ∆(I) = k I Fechner sought a unit that would allow quantification of psychological intensity on par with quantification of physical intensity ¹ Luce & Galanter (1963)
- 29. Fechner postulated Weber’s JND as a psychological yard stick Weber’s law ∆(I) = k I Fechner sought a unit that would allow quantification of psychological intensity on par with quantification of physical intensity reasoned that JND’s would do but not only for π = 0.75: ¹ Luce & Galanter (1963)
- 30. Fechner postulated Weber’s JND as a psychological yard stick Weber’s law ∆(I) = k I Fechner sought a unit that would allow quantification of psychological intensity on par with quantification of physical intensity reasoned that JND’s would do but not only for π = 0.75: equally often noticed differences are equal (unless always or never noticed)¹ ¹ Luce & Galanter (1963)
- 31. Fechner’s idea is essentially a probability statement equally often noticed differences are equal (unless always or never noticed)
- 32. Fechner’s idea is essentially a probability statement equally often noticed differences are equal (unless always or never noticed) Fechner’s idea essentially states that (a) - (b) = (c) - (d) if and only if P(a≽b) = P(c≽d)
- 33. Fechner’s idea is essentially a probability statement equally often noticed differences are equal (unless always or never noticed) Fechner’s idea essentially states that (a) - (b) = (c) - (d) if and only if P(a≽b) = P(c≽d) This is equivalent to “Fechner’s problem” For which increasing continuous and F P(a≽b) = F[ (a) - (b)]
- 34. For fixed π JND’s as a unit imply a logarithmic intensity relationship Weber’s Law states ∆(I) = k I 1 JND I=a a ⟼ a + ∆(a) = a+ka = (1+k) a 5 4 3 2 1 0 a
- 35. For fixed π JND’s as a unit imply a logarithmic intensity relationship Weber’s Law states ∆(I) = k I 1 JND I=a a ⟼ a + ∆(a) = a+ka = (1+k) a (1+k)a 2⟼ (1+k)a + ∆((1+k)a) JND 5 = (1+k)a+k(1+k)a 4 = (1+k)²a 3 2 1 0 a (1+k)²a
- 36. For fixed π JND’s as a unit imply a logarithmic intensity relationship Weber’s Law states ∆(I) = k I 1 JND I=a a ⟼ a + ∆(a) = a+ka = (1+k) a (1+k)a 2⟼ (1+k)a + ∆((1+k)a) JND 5 = (1+k)a+k(1+k)a 4 = (1+k)²a 3 (1+k)²a ⟼ (1+k)²a + ∆((1+k)²a) 2 3 JND 1 0 = (1+k)³a a (1+k)²a(1+k)³a
- 37. For fixed π JND’s as a unit imply a logarithmic intensity relationship Weber’s Law states ∆(I) = k I 1 JND I=a a ⟼ a + ∆(a) = a+ka = (1+k) a (1+k)a 2⟼ (1+k)a + ∆((1+k)a) JND 5 = (1+k)a+k(1+k)a 4 = (1+k)²a 3 (1+k)²a ⟼ (1+k)²a + ∆((1+k)²a) 3 JND 2 1 0 = (1+k)³a 4 JND a (1+k)²a(1+k)³a (1+k)⁴a (1+k)³a ⟼ (1+k)⁴a
- 38. For fixed π JND’s as a unit imply a logarithmic intensity relationship n JND In general: I= (1+k)n-1a ⟼ (1+k)na To get n (the number of JND’s) take log1+k log1+k(I) = n log1+k(1+k) + log1+k(a) 5 or 4 3 log1+k(I/a) = n (JND’s) 2 1 or in natural logarithms 0 a (1+k)²a(1+k)³a (1+k)⁴a b ln(I / a) = n (JND’s)
- 39. For non-fixed π there are several solutions to Fechner’s problem
- 40. For non-fixed π there are several solutions to Fechner’s problem Fechner’s problem: For which increasing continuous and F P(a≽b) = F[ (a) - (b)] holds?
- 41. For non-fixed π there are several solutions to Fechner’s problem Fechner’s problem: For which increasing continuous and F P(a≽b) = F[ (a) - (b)] holds? Thurstone’s Law of Comparitive Judgement Cases III and V
- 42. For non-fixed π there are several solutions to Fechner’s problem Fechner’s problem: For which increasing continuous and F P(a≽b) = F[ (a) - (b)] holds? Thurstone’s Law of Comparitive Judgement Cases III and V Bradley-Terry-Luce model
- 43. Thurstone’s Law of Comparative Judgement Case V postulates noise on pair wise compared stimuli Thurstone (1927) proposed that (non- physical)sensations are stochastic in that on trial t t(a) = (a) + eat t(b) = (b) + ebt
- 44. Thurstone’s Law of Comparative Judgement Case V postulates noise on pair wise compared stimuli Thurstone (1927) proposed that (non- physical)sensations are stochastic in that on trial t t(a) = (a) + eat t(b) = (b) + ebt eat and ebt are Normal with Var(eat - ebt) = 1
- 45. Thurstone’s Law of Comparative Judgement Case V postulates noise on pair wise compared stimuli Thurstone (1927) proposed that (non- physical)sensations are stochastic in that on trial t t(a) = (a) + eat t(b) = (b) + ebt eat and ebt are Normal with Var(eat - ebt) = 1 Law of Comparative Judgement: P(a≽b) = P( t(a) ≥ t(b)) = P( (a)- (b) ≥ eat -
- 46. Stevens argued against “equal probability means equal subjective intensity” and found power law Signal Detection Theory Sensitivity to the presence of a signal Weber and Weber-Fechner Law Psychological equal units Stevens’ Power law People can quantify equal units
- 47. Stevens argued against “equal probability means equal subjective intensity” and found power law Stevens pointed out that Fechner’s assumption is arbitrary and permits in a fundamental way only logarithmic relations That his method produced relations inconsistent with Fechner’s Law
- 48. Stevens proposed that subjects could quantitatively estimate the magnitude of a physical stimulus Stevens’ magnitude estimation procedure: Provide “modulus” (baseline) stimulus (e.g., loud tone, electric shock) and call it 100 Have subject estimate intensity of subsequent stimuli on scale 0 ... 100 e.g. if tone sounds half as loud rate it 50 More efficient than Weber’s procedure!
- 49. Stevens’ magnitude estimation method yields power relations for some stimuli Stevens’ Power Law = k Ib Dimension b Taste (salt) 1.3 Brightness 0.33 Loudness 0.6 Line length 1.0 Heaviness 1.45
- 50. Stevens’ Power Law relates psychological intensities bet ween modalities A strong argument in favor of Steven’s power law is “cross-modal matching”: Present t wo stimuli differing in one dimension (e.g., size) Let subject adjust second dimension (e.g., brightness) to match the ratio in the first dimension A strong prediction is linear relationship with predictable slope (from log k1I1b = log k2I2b’)
- 51. All these methods have been very influential and are subject of concurrent theory development There is more to these theories than presented here and in the book Both theories are fundamental to measurement theory and very detailed mathematical foundations have been developed (e.g., Falmagne, 1985) None of them consider context effects which matter