Decision making as a model 3. Heavy stuff: derivation of two important theorems

Recap: two types of measures of sensitivity (independent of criterion:) 2.Area under ROC-Curve: A 1.Distance between signal and noise distributions cf. d'

Recap:four types of measures for criterion: 1. Likelihood ratio LR c = p(x c |S)/p(x c |N) = h/f (vgl β) h f 2. Position on x-axis (c) 3. Position in ROC-plot (left down vs. right up) 4. Slope of tangent at point of ROC (S) c

Relationships between measures of sensitivity Relationships between measures of criterion

Interpreting A Area theorem: A is equivalent with proportion correct answers in 2AFC-experiment: Given: 1 noise stimulus 1 signal (+noise) stimulus, Which is which? Makes sense. Important

Produce a formula for proportion correct in a 2AFC-experiment (P c ) Produce a formula for area under ROC-curve (A) Show that the formula for A looks like the formula for P c Show that formulas are identical. Approach:

P FA P H fnfnfnfn fsfsfsfs x 0 λ ∞ P H = ∫ f s (x)dx λ ∞ P FA = ∫ f n (x)dx λ = H(λ) = F A (λ) λ = F A -1 (P FA ) ROC-curve: P H = H(λ) = H[F A -1 (P FA )] Recap: In general: Specific model depends on f n and f s

fnfnfnfn fsfsfsfs x 0 λ ∞ P H = ∫ f s (x)dx λ ∞ P FA = ∫ f n (x)dx λ = H(λ) = F A (λ) Reinterpretation for 2A FC experiment: Two alternatives correspond with two points on the x-axis. Suppose λ is noise stimulus: if x n = λ P C = p(x s >x n ),p(x s >x n ) = H(λ) “summate” H(λ) for every λ, weighted for density of λ [= f n (λ)]: ∞ P C = ∫ H(λ)f n (λ)dλ -∞

P FA P H fnfnfnfn fsfsfsfs x 0 λ ∞ P H = ∫ f s (x)dx λ ∞ P FA = ∫ f n (x)dx λ = H(λ) = F A (λ) ROC-curve: P H [= H(λ)] as a function of P FA [ F A (λ)] ROC-curve: P H [= H(λ)] as a function of P FA [= F A (λ)] 1 A = ∫ H(λ)dF A (λ) 0 Area under Roc-curve: ∞ P C = ∫ H(λ)f n (λ)dλ -∞ A looks like P C ; A is P C ; can be proved

proof (optional): dF A (λ) d(λ) = -f n (λ) ∞ P C = ∫ H(λ)f n (λ)dλ -∞ dF A (λ) = -f n (λ)dλ Still two small chores: Limits of integration and minus sign f n (x)dx = 1 - f n (x)dx f n (x)dx = 1 - f n (x)dx ∞ λ ∫ -∞ λ ∫ 1 A = ∫ H(λ)dF A (λ) 0 -f n (λ)dλ -f n (λ)dλ

Limits: if F A (λ)=P FA = 0 then λ = ∞ if F A (λ)=P FA = 1 then λ = -∞ reverse: -H(λ)f n (λ)  H(λ)f n (λ) ∞ P C = ∫ H(λ)f n (λ)dλ -∞ ∫  ∫ 0 1 -∞ ∞ ∞ ∫  ∫ -∞ ∞ fnfnfnfn fsfsfsfs x 0 λ -∞   ∞ 1 A = ∫ H(λ)dF A (λ) 0 -f n (λ)dλ -f n (λ)dλ from Integration over F A to integration over λ ? ∫ - H(λ)f n (λ)dλ ?

P FA P H ROC-curve: P H as a function of P FA Every point of ROC-curve gives criterion/bias at that sensitivity Slope tangent at that point as measure for bias/criterium S =.49 dP H Slope S= dP FA Measures for criterion

fnfnfnfn fsfsfsfs x 0 λ ∞ P H = ∫ f s (x)dx λ ∞ P FA = ∫ f n (x)dx λ dP H dP H dP FA = dx dP FA dx d(1-P FA ) dP FA dP FA dP H = = f n, = - f n, also: = - f s …dx dx dx dx (chain rule) dP H dP H /dx S = = dP FA dP FA /dx - f s f s = f n f n = LR c Measures for criterion dP H from to dx dP FA

Intermezzo. A category of (older) alternative models. Finite state models: Measuring sensitivity.

Finite State models High threshold: Yes 1 α detect signal 1-α ηYes uncertain 1-ηNo 1 η Yes noise uncertain 1-η NoHit Hit HitMissFA cr P H = α +η(1-α) P FA = η

Hits False Alarms α P H = α +η(1-α) P FA = η η Theoretical ROC curve detect: Yes uncertain: η  Yes 1-η  No α “high threshold”

P H = α +η(1-α) P FA = η P H = α + P FA (1-α) P H = α + P FA - αP FA α – αP FA = P H - P FA α(1- P FA ) = P H - P FA P H – P FA α = P FA

P H = α +η(1-α) P m = (1-η)(1-α) P m (1-α) = (1-η) P m P H =α + η (1-η) η α = P H P m (1-η) Cf correction for guessing MC-questions: N-AFCG  Hits, F  misses η S c = G F (1-η)

Analogously: a low threshold model : Signal leads always to uncertain state noise leads with P = β to nondetect state (always NO) and else to uncertain state. hits False Alarms 1-β β Nondetect: No Uncertain: η  Yes 1-η  No

hits False Alarms A combined three state model N UD