# Signal Detection Theory Signal Detection Theory (SDT) is a framework for understanding how observers detect weak signals in the presence of noise, separating two independent factors: **sensitivity** (ability to discriminate signal from noise) and **response bias** (tendency to say "yes" or "no"). Developed in the 1950s from radar and communications engineering, SDT revolutionized [[Psychophysics]] by recognizing that detection isn't just about sensory acuity—it's also a decision process influenced by expectations, payoffs, and caution. A radiologist scanning for tumors, a security guard watching for threats, and a witness identifying a suspect all face signal detection problems. The theory models detection as comparing an internal response to a criterion: if the response exceeds the criterion, say "signal present." This produces four outcomes: hits (correctly detecting signal), misses (failing to detect), false alarms (saying "signal" when none), and correct rejections. The key insight: you can't minimize both misses and false alarms simultaneously—there's a fundamental tradeoff. Sensitivity (d') measures how well signal and noise distributions are separated; criterion (β or c) measures where the observer sets their threshold. SDT applies beyond perception to medical diagnosis, memory recognition, lie detection, and any situation requiring decisions under uncertainty. ## The Signal Detection Model ``` ┌─────────────────────────────────────────────────────────────┐ │ SIGNAL DETECTION THEORY │ ├─────────────────────────────────────────────────────────────┤ │ │ │ Two overlapping distributions: │ │ │ │ NOISE ONLY SIGNAL + NOISE │ │ ╱╲ ╱╲ │ │ ╱ ╲ ╱ ╲ │ │ ╱ ╲ ╱ ╲ │ │ ╱ ╲ ╱ ╲ │ │ ╱ ╲ ╱ ╲ │ │ ────╱──────────╲──────────╱──────────╲──── │ │ │ │ │ │ ← CRITERION (β) │ │ │ │ │ ◄──────────► │ │ d' (sensitivity) │ │ │ │ LEFT of criterion: Say "No signal" │ │ RIGHT of criterion: Say "Signal present" │ │ │ └─────────────────────────────────────────────────────────────┘ ``` ## Four Outcomes | | Signal Present | Signal Absent | |---|----------------|---------------| | **Say "Yes"** | HIT ✓ | FALSE ALARM ✗ | | **Say "No"** | MISS ✗ | CORRECT REJECTION ✓ | ## Key Measures | Measure | What It Captures | Formula | |---------|------------------|---------| | **d' (d-prime)** | Sensitivity, discriminability | z(Hit rate) - z(False alarm rate) | | **β (beta)** | Response bias (likelihood ratio) | Height of noise dist / Height of signal dist at criterion | | **c (criterion)** | Response bias (location) | -0.5 × [z(Hit) + z(FA)] | | **Hit rate** | P(say yes \| signal) | Hits / (Hits + Misses) | | **False alarm rate** | P(say yes \| no signal) | FA / (FA + CR) | ## The Tradeoff ``` ┌─────────────────────────────────────────────────────────────┐ │ ROC CURVE (Receiver Operating Characteristic) │ ├─────────────────────────────────────────────────────────────┤ │ │ │ Hit Rate │ │ │ 1.0 │ ●●●●● │ │ │ ●●● │ │ │ ●●● Higher d' │ │ │ ●● (more sensitive) │ │ │ ●● │ │ │ ●● Lower d' │ │ │ ●● (less sensitive) │ │ │ ●● │ │ 0.0 │●●──────────────────────────────── │ │ └────────────────────────────────► │ │ 0.0 1.0 │ │ False Alarm Rate │ │ │ │ Moving along curve = changing criterion (bias) │ │ Different curves = different sensitivity (d') │ │ │ └─────────────────────────────────────────────────────────────┘ ``` ## Factors Affecting Criterion | Factor | Shifts Criterion | |--------|-----------------| | **High signal probability** | Liberal (more "yes") | | **High cost of misses** | Liberal (more "yes") | | **High cost of false alarms** | Conservative (more "no") | | **Payoff for hits** | Liberal | | **Instructions** | Either direction | ## Applications | Domain | Signal | Noise | |--------|--------|-------| | **Medical diagnosis** | Tumor present | Normal tissue variation | | **Eyewitness memory** | Seen before | Similar face | | **Radar/sonar** | Enemy aircraft | Clutter, birds | | **Spam filtering** | Spam email | Legitimate email | | **Quality control** | Defective product | Acceptable variation | | **Lie detection** | Deception cues | Normal variation | ## SDT in Memory Research | Response | Old Item | New Item | |----------|----------|----------| | **"Old"** | Hit | False alarm | | **"New"** | Miss | Correct rejection | ## Advantages Over Classical Psychophysics | Classical Threshold | Signal Detection | |--------------------|------------------| | Single threshold exists | No fixed threshold | | Sensitivity = threshold | Separates sensitivity from bias | | Ignores decision factors | Models decision process | | Binary detection | Gradual, probabilistic | ## Key Figures | Person | Contribution | |--------|--------------| | John Swets | SDT applications to psychology | | David Green | SDT for psychoacoustics | | Wilson Tanner | Early SDT development | | R. Duncan Luce | Mathematical foundations | ## References - Green, D.M. & Swets, J.A. *Signal Detection Theory and Psychophysics* (1966) - Macmillan, N.A. & Creelman, C.D. *Detection Theory: A User's Guide* (2005) - https://en.wikipedia.org/wiki/Signal_detection_theory ## Related - [[Psychophysics]] - [[Decision Making]] - [[Perception]] - [[Memory]]