Decision making as a model 4. Signal detection: models and measures

1.Hard work Get several points on ROC-curve by inducing several criteria (pay-off, signal frequency) Compute hit rate and false alarm rate at every criterion Many trials for every point! Measure or compute A using graphical methods Nice theorems, but how to proceed in practice?

Certainly Certainly no signal a signal Variant: numeric (un)certainty scale: implies multiple criteria – consumes many trials too

2.Rough approximation Area measure for one point: A' Hits False Alarms F H Average of those two areas: A' = 1-H F 1 - ¼ F H (H-F)(1+H-F) = ½ + ¼ H(1-F) Iff H>F

Comparable measure for criterion/bias: Grier’s B'' H(1 - H) – F(1 – F) B'' = sign(H - F) H(1 - H) + F(1 – F) if H = 1 - F then B'' = 0 if F = 0, H≠ 0, H≠1 then B'' = 1 if H = 1, F≠0, F≠1, then B'' = -1 H F FALSE ALARM RATE HIT RATE B''= -.4 B''= -.07 B''=.07 B''=.4 B''= 0 Isobias curves

3.Introducing assumptions Even when several points ara available, they may not lie on a nice curve Then you might fit a curve, but which one? Every curve reflects some (implicit) assumptions about distributions Save labor: more assumptions  less measurement (but the assumptions may not be justified)

Simplest model: noise and signal distributions normal with equal variance One point (P H, P FA pair) is sufficient Normal distributions are popular (there are other models!)

Example: in an experiment with noise trials and signal trials these results were obtained: Hit rate:.933, False Alarm rate.309 (.067 misses and.691 correct rejections) Normal distributions: via corresponding z-scores the complete model can be reconstructed:

z.309 =.5 z.933 = distance: d´ = 2 measure for “sensitivity” h f h β = ---- =.37 f Measure for bias/criterion

several d'-s and corresponding ROC-curves

Gaussian models: preliminary Standaard normal curve M=0, sd = 1 Transformations: Φ(z)  P Φ -1 (P) or: Z(P)  z see tabels and standard software 1 φ(z)= e -z 2 /2 √2π z Φ (z) = - ∞ φdx ∫ z P

PHPH P FA zHzH z FA Roc-curve P H = f(P FA ) Z-transformation ROC-curve P  z z H = f(z FA ) λ Nice way to plot several (P FA, P H ) points -

Regression line z H = a 1 z FA + b 1, Minimize (squared) deviations z H : Underestimation of a Compromise: average of regression lines Z H = ½(a 1 +1/a 2 )Z FA + ½(b 1 +b 2 /a 2 ) Plotting with regression line? Regresionline z FA = a 2 z H + b 2 Minimize (squared) deviations z FA : Overestimation of a

zHzH z FA d'd' Equal variance model: z H = z FA + d' d' = z H –z FA z-plot ROC 45° P FA = 1- Φ(λ), = Φ(-λ), z FA = -λ P H = 1 – Φ(-(d' - λ)) = Φ(d' – λ), z H = d' – λ d'd' 45 ° 0 λ

f β = h/f = φ(z H )/φ(z F ) To get symmety a log transformation is often applied: log β = log h – log f = ½(z 2 FA – z 2 H ) Criterion/bias: h 1 -z 2 / 2 φ(z) = e (standard-normal) √ (2π) 1 -z H 2 / 2 φ(z H ) = e √ (2π) 1 -z FA 2 / 2 φ(z FA ) = e √ (2π) z FA 2 – z H Divide: = e

f h λ c z H + z FA c = Alternatve: c (aka λ center ), distance (in sd) between middle (were h=f) and criterion c = -(d ' /2 – λ) z FA = -λ d' = z H - z FA z H – z FA 2z FA c =

β c Isobias curves for β en c