The psychometric function is the fundamental curve of psychophysics, mapping stimulus intensity (or stimulus difference) to the probability of a particular response. It is typically a sigmoid (S-shaped) curve that transitions from chance performance at low intensities to perfect performance at high intensities, with the steepness of the transition reflecting the observer's sensitivity.
Parameterization
α = threshold (location parameter, typically 75% correct in 2AFC)
β = slope (inversely related to variability of underlying process)
γ = guess rate (0.5 for 2AFC, lower for yes/no)
λ = lapse rate (accounts for stimulus-independent errors)
The sigmoid function F is commonly chosen as the cumulative normal (probit), logistic, or Weibull distribution. The Weibull function is popular because its parameters have clean interpretations and it accommodates both detection and discrimination tasks. The choice of sigmoid rarely matters much for well-sampled data.
Fitting and Estimation
Psychometric functions are typically fit using maximum likelihood estimation or Bayesian methods. The threshold α is usually defined as the stimulus level producing a criterion performance level (e.g., 75% correct in 2AFC). The slope β reflects the observer's internal noise: steeper slopes indicate less noise and more precise discrimination. The lapse rate λ accounts for finger errors and inattention, and is typically constrained to small values (0–0.05).
Modern fitting packages (Palamedes, psignifit, quickpsy) provide bootstrap or Bayesian confidence intervals for all parameters and goodness-of-fit diagnostics.