Mathematical Psychology
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Response Criterion

The response criterion in Signal Detection Theory determines the observer's decision rule — the point on the internal evidence axis above which 'signal present' is reported — and is mathematically independent of sensitivity.

c = −0.5 × [z(HR) + z(FAR)]

In Signal Detection Theory, the response criterion is the decision boundary that the observer uses to convert internal evidence into a response. Observations exceeding the criterion produce a "yes" (signal present) response; those below produce a "no" response. The criterion's location determines the observer's bias — their tendency to say "yes" versus "no" — and is mathematically independent of sensitivity (d′).

Measures of Criterion

Criterion Measures c = −0.5 × [z(HR) + z(FAR)]
β = f_n(z_HR) / f_n(z_FAR) = likelihood ratio at criterion

c = 0: unbiased (criterion midway between distributions)
c > 0: conservative (tendency to say "no")
c < 0: liberal (tendency to say "yes")

The measure c locates the criterion relative to the midpoint between the noise and signal distribution means. The measure β (beta) gives the likelihood ratio at the criterion point — the ratio of the signal distribution's height to the noise distribution's height. An optimal observer sets β equal to the ratio of payoffs and prior probabilities.

Factors Affecting Criterion

Criterion placement is influenced by signal probability (observers become more liberal when signals are frequent), payoff structure (higher rewards for hits shift the criterion liberally), and instructions (emphasizing accuracy vs. speed). Understanding criterion shifts is essential for interpreting changes in hit and false alarm rates: a shift in criterion changes both rates simultaneously, while a change in sensitivity shifts them differentially.

Interactive Calculator

Each row represents a trial: trial_type (signal or noise) and response (yes or no). Computes hit rate, false-alarm rate, d′, criterion c, and β.

Click Calculate to see results, or Animate to watch the statistics update one record at a time.

Related Topics

References

  1. Macmillan, N. A., & Creelman, C. D. (2005). Detection theory: A user's guide (2nd ed.). Lawrence Erlbaum Associates. https://doi.org/10.4324/9781410611147
  2. Green, D. M., & Swets, J. A. (1966). Signal detection theory and psychophysics. Wiley. https://doi.org/10.1901/jeab.1969.12-475
  3. Snodgrass, J. G., & Corwin, J. (1988). Pragmatics of measuring recognition memory: Applications to dementia and amnesia. Journal of Experimental Psychology: General, 117(1), 34–50. https://doi.org/10.1037/0096-3445.117.1.34
  4. Stanislaw, H., & Todorov, N. (1999). Calculation of signal detection theory measures. Behavior Research Methods, Instruments, & Computers, 31(1), 137–149. https://doi.org/10.3758/BF03207704

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