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.
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.
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")
#> [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] 0.8416212
# Compute the bias relative to the midpoint
sdt_bias(hit_rate = 0.8, fa_rate = 0.2, dist = "normal")
#> [1] 0