Eyewitness Identification: A Signal Detection Theory Lab (2D)

Signal Distributions

Mean separation Δμ (true signal) 1.50

Distribution spread σ 1.00

Decision criterion c 0.50

Lineup instructions

True d′ = Δμ / σ

1.50

Effective criterion

0.50

Live Areas (analytical)

Hit-rate area H

—

False-alarm area F

—

Run Witnesses

Hit Miss False alarm Correct rejection

Empirical H

—

Empirical F

—

Empirical d′ = z(H)−z(F)

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Witnesses run

ROC Curve

How it works

Each witness's memory produces a continuous strength of recognition when they view a lineup member. Signal Detection Theory models this as two overlapping Gaussians on that strength axis:

Lure (innocent suspect):  strength ~ N(0, σ)
Target (guilty suspect):  strength ~ N(Δμ, σ)

The witness says "that's them" whenever the strength exceeds their internal decision criterion c. This single rule produces four outcomes, and their probabilities are the real areas under the two curves split by the criterion line, computed from the normal CDF Φ:

Hit rate  H = 1 − Φ((c − Δμ) / σ)
False-alarm rate F = 1 − Φ(c / σ)
True d′ = Δμ / σ

Given empirical H and F measured from real simulated trials, discriminability can be recovered without ever being told the true Δμ or σ:

d′(estimated) = Φ⁻¹(H) − Φ⁻¹(F)

Drag the criterion line on the chart, or use its slider — the shaded hit/miss/FA/CR areas and the ROC marker update instantly from the real overlap integral, not a lookup table.
Biased instructions shift only the effective criterion (witnesses become "trigger-happy"), never Δμ or σ. Run witnesses under both settings at the same Δμ: the hit and false-alarm rates change, but the recovered empirical d′ stays the same — the classic finding that suggestive lineup procedures inflate false identifications through response bias, not through any real gain (or loss) in memory discriminability.
The ROC curve is built by sweeping c across its whole range and plotting the resulting (F, H) pair at every point — every point on that curve shares the same Δμ/σ, so moving along it is exactly what changing the criterion does to a witness's operating point.

Real-world relevance: this exact H/F/d′ framework is what forensic-psychology researchers (e.g. Wells, Loftus, Steblay) use to compare lineup procedures and argue for reform in police identification protocols.

Lure N(0, σ) Target N(Δμ, σ) Criterion c (drag me)