Utility functions for Signal Detection Theory

Compute sensitivity and criterion from hit and false alarm rates for different SDT distribution families. A single (hit, false alarm) pair cannot identify the signal-to-noise SD ratio, so both quantities are the equal-variance values; see sdt_yn() for the unequal-variance model.

Usage

sdt_d(
  hit_rate,
  fa_rate,
  dist = c("normal", "gumbel_min", "gumbel_max", "logistic")
)

sdt_criterion(
  hit_rate,
  fa_rate,
  dist = c("normal", "gumbel_min", "gumbel_max", "logistic")
)

Arguments

hit_rate

Numeric. Proportion of hits (P("old" | signal)).

fa_rate

Numeric. Proportion of false alarms (P("old" | noise)).

dist

Character. The distribution assumed for the latent evidence, given here by its cumulative distribution function:

• "normal" (default): Gaussian, \(\Phi(x)\)

• "gumbel_min": smallest extreme value, \(1 - \exp(-\exp(x))\) (the complementary log-log distribution)

• "gumbel_max": largest extreme value, \(\exp(-\exp(-x))\) (the log-log distribution, as in evd::pgumbel)

• "logistic": \(1 / (1 + \exp(-x))\)

Value

sdt_d returns the distance between the signal and noise distributions on the latent evidence axis, obtained by inverting the decision rule "respond old when the evidence exceeds the criterion": \(Q(1 - FA) - Q(1 - H)\), where \(Q\) is the quantile function of dist. For dist = "normal" this reduces to the familiar \(d' = \Phi^{-1}(H) - \Phi^{-1}(FA)\). Because one operating point implies equal variance, this matches the d parameter of sdt_yn() whenever sdratio is at its default.

sdt_criterion returns the criterion (response bias) on the centred, noise-standardized evidence axis used by sdt_yn(), where the noise and signal distributions sit at -d'/2 and +d'/2: \((Q(1 - FA) + Q(1 - H)) / 2\). For dist = "normal" this reduces to the familiar \(-(\Phi^{-1}(H) + \Phi^{-1}(FA)) / 2\).

References

Green, D. M., & Swets, J. A. (1966). Signal detection theory and psychophysics. Wiley.

See also

sdt_yn(), whose d and criterion parameters these two functions compute in closed form from observed rates: sdt_d() returns the same quantity as the d parameter and sdt_criterion() the same quantity as criterion, on the same axis, whenever sdratio is at its default.

Examples

# Compute d from hit and false alarm rates (Gaussian SDT)
sdt_d(hit_rate = 0.8, fa_rate = 0.2, dist = "normal")
#> [1] 1.683242

# The extreme-value analogue
sdt_d(hit_rate = 0.75, fa_rate = 0.25, dist = "gumbel_min")
#> [1] 1.572534
# Compute criterion from hit and false alarm rates
sdt_criterion(hit_rate = 0.8, fa_rate = 0.2, dist = "normal")
#> [1] 0