Glossary
Signal Detection Theory
What is signal detection theory?
Signal detection theory separates how well you distinguish a signal from noise from how readily you report that the signal is present.
Imagine listening for a faint tone. Saying “yes” more often might catch more real tones, but it could also produce more false alarms. More yeses alone do not show better hearing.
Four possible outcomes
- Hit: A signal is present, and you report it.
- Miss: A signal is present, and you do not report it.
- False alarm: No signal is present, but you report one.
- Correct rejection: No signal is present, and you report none.
Sensitivity and criterion
The hit rate is the proportion of signal-present trials with a hit. The false-alarm rate is the proportion of signal-absent trials with a false alarm. In a common model with equal-variance normal distributions, sensitivity is calculated as d′ = z(hit rate) − z(false-alarm rate), where z converts each rate to a standard-normal score.
The criterion is the response threshold. Expectations and the consequences of a mistake can affect it.
Criterion and rate formulas
In the same equal-variance normal model, a common convention is c = −[z(hit rate) + z(false-alarm rate)] / 2. Positive c indicates a more conservative response threshold; negative c indicates a more liberal one.
Rearranging the d′ and c equations gives hit rate = Φ(d′/2 − c) and false-alarm rate = Φ(−d′/2 − c), where Φ is the standard-normal cumulative distribution function. These formulas use that convention for c; they are not independent of the model assumptions.
The ROC curve
A receiver operating characteristic, or ROC, plots hit rates against false-alarm rates across thresholds. One pair of rates gives one point, not a whole measured curve. Rating and forced-choice tasks can also be analyzed with signal detection theory.
Example: detecting a possible threat
A security system can lower its threshold to flag more possible threats. That may also flag more harmless events. Checking both hits and false alarms helps distinguish better discrimination from a greater willingness to issue an alert.
Why it matters
The framework helps distinguish discrimination from response policy in perception, memory, and classification tasks. Estimates depend on the study design, data, and model assumptions; rates of zero or one require special handling. The analysis describes a tradeoff. It does not choose the right tradeoff for you.