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Webers law and Fechners law

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(Sensory Threshold) (Absolute Threshold) (Difference Threshold, Difference Limen, DL) (Just Noticeable Difference, JND) 50% JND

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(Sensory Threshold) (Method of constant stimuli) (Method of limits) (Adjustment method)

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1/7 8-10 100% 100%

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2/7

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3/7 (psychometric function) % 3-7 3-7

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4/7 Fitting: , 50% % z 3-8 3-1 3-8 3-1 fitting: ( 3-1, 3-2) z a + b set z = 0, ? 0 a/b

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5/7 (standard stimulus, St) (comparison stimulus, Co) St 3-5 Co 100% Co 100%

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6/7 ( , time error) 3-9 3-9 (space error)

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7/7 3-10 3-10 (point of subjective equality, PSE) 50% (CE) CE PSE St DL u f -1 (0.75) PSE DL l PSE f -1 (0.25) z a + b DL u (0.67 a)/b DL l ( 0.67 a)/b

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1/5 3-2 3-2 (ascending series) (ascending trial) (descending series) (descending trial)

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2/5 3-2 3-2 (error of habituation) (error of expectation)

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3/5 3-3 3-3 St Co Co Co Co Co St Co

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4/5 3-3 3-3 (lower limen) (upper limen) PSE PSE PSE

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5/5 (up-and-down method) (staircase method) (Threshold tracking method) 3-11 3-11

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1/2

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2/2 Co St (Method of average error) 3-12 3-12 PSE PSE St

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Webers law and Fechners law Webers Law 1/3 / c c c Weber (Weber fraction) 3-16 3-16 3-17 3-17 3-17 3-17 3-18 3-19 3-18 3-19

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Webers law and Fechners law Webers Law 2/3 / c c Modified Webers Law /( a) c c( a) / c (1 a/) a: a/ 0 a/>>0

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Webers law and Fechners law Fechners Law 3/3 3-20 3-20 Webers Law () Fechners Law: k log / c (1/c) ( /) (1/c) c 1 c / 1 c (1/)d d k log

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(Signal Detection Theory) ROC

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1/2 4-1 4-1 (hit) (miss) (false alarm) (correct rejection) 4-2 4-3 4-5 4-2 4-3 4-5 (payoff) (payoff matrix) 4-4 4-4

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2/2 & (response bias) P( /S) P(D /S) 1 P(D/S) P( /N) P(D/S) P( /S) P( /N) / 1 P( /N) ROC (receiver-operating characteristic) 4-6 4-6 4-7 4-7

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1/2 (noise) N, SN f N (x), f SN (x) X: S f N (x), f SN (x) 4-8 4-8 (likelihood ratio) l (x) f SN (x)/ f N (x)

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2/2 = f SN (C)/ f N (C) l (x) SN l (x) < N 4-9 4-9 (expected value) l (x) EV P i V i 4-1 4-1 EV P( /SN) P(SN) V SN (1 P( /SN) ) P(SN) V SN P( /N) P(N) V N (1 P( /N)) P(N) V N P(N)(V N V N ) / P(SN)(V SN V SN ) 4-6 4-6

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ROC 1/2 N, SN ROC 4-11 4-11 (detectability) d = (M SN M N )/ N d ROC 4-12 4-12 d = z(+/SN) z(+/N) ROC d P(+/SN), P(+/N) P(+/SN) z(+/SN) ; P(+/N) z(+/N) d

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ROC 2/2 = f SN (C)/ f N (C) ROC f SN (C)/ f N (C) = P(+/SN)/P(+/N) ROC P(+/SN) f SN (C) ; P(+/N) f N (C)

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1/3 (experimental session) P(SN) 300 (yes-no procedure) 4-16 4-17 4-16 4-17

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2/3 (observation intervals) (forced choice procedure)

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3/3 (confidence rating) ROC 4-18 4-18 4-7 4-7

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1/2 (scale)

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2/2 (nominal scale) (identity) (ordinal scale) (interval scale) y=a+bx (ratio scale) y=bx

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1/8 (confusion scales) DL

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2/8 (rank order scales) 5-1 5-1

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3/8 (rank order scales)

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4/8 DL DL Webers Law jnd 5-1 5-1 jnd 5-2 5-3 5-2 5-3 0 10 (=0), 0 3 1 10 3 13 (=1), 1 3.1 2 13 3.1 16.1 (=2)

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5/8 (paired comparison scales) Thurstone n(n-1)/2 F f xy x y 5-2 5-2

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6/8 (paired comparison scales) p xy (1/N) f xy (1/N) (N f yx ) 1 P yx f xy p xy z xy (N ) (discriminal process) 5-4 5-4 (discriminal dispersion)

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7/8 (paired comparison scales) 5-4 5-4 (discriminal difference) B A B A 0 S B S A B A 0 S A S B B A ean B A Var Var( B ) Var( A ) 2rVar( B )Var( A ) B A z BA Var

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8/8 (paired comparison scales) 5-4 5-4 1. B A r 0 2. Var( A ) Var( B ) 1 B A z BA 2 F p z transposed z 5-3 5-3 2 5-4 5-4

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5-5 5-5 5-6 5-7 5-8 5-9 5-5 5-5 5-6 5-7 5-8 5-9 (method of single stimuli) (absolute judgment) (3-11 ) 5-6 5-10 5-11 5-6 5-10 5-11 (anchor stimulus) 5-12 5-12

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1/3 1930 Fechner (dB)

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2/3 (method of fractionation), Stevens (1936): sone 5-13 5-13 5-14 5-15 5-14 5-15

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3/3 A 1 ( a 1 ) 1 A i (a i ) i 1 i a 1 a i 10 20

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(power law) Stevens k a 5-17 5-17 log log k a log 5-18 5-18 1 / 2 ( 1 / 2 ) a