SDTdist: Utility functions for Signal Detection Theory

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Utility functions for Signal Detection Theory

Description

Compute sensitivity, criterion and bias 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 all three 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")
)

sdt_bias(
  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 location of the decision criterion measured from the mean of the noise distribution, in noise-SD units, as the criterion parameter of sdt_yn() is: Q(1 - FA). It depends on the false-alarm rate alone. For dist = "normal" this is -\Phi^{-1}(FA).

sdt_bias returns the criterion measured from the midpoint between the noise and signal means instead, as the bias parameter of sdt_yn() is: (Q(1 - FA) + Q(1 - H)) / 2, which is sdt_criterion() - sdt_d() / 2. For dist = "normal" this is the familiar c = -(\Phi^{-1}(H) + \Phi^{-1}(FA)) / 2. Positive values are a conservative bias, toward "new"/"noise" responses.

References

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

See Also

sdt_yn(), whose d, criterion and bias parameters these functions compute in closed form from observed rates: sdt_d(), sdt_criterion() and sdt_bias() return the same quantities as the parameters of the same name 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")

# The extreme-value analogue
sdt_d(hit_rate = 0.75, fa_rate = 0.25, dist = "gumbel_min")
# Compute criterion from hit and false alarm rates
sdt_criterion(hit_rate = 0.8, fa_rate = 0.2, dist = "normal")
# Compute the bias relative to the midpoint
sdt_bias(hit_rate = 0.8, fa_rate = 0.2, dist = "normal")

bmm documentation built on Oct. 16, 2026, 9:08 a.m.