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Distribution functions for Yes/No SDT

Source: R/distributions.R
sdt_yn_dist.Rd

Density and random generation for the yes/no signal detection theory model, where the response is the number of "old"/"signal" responses out of a fixed number of trials (a binomial likelihood).

Usage

dsdt_yn(
  n_old,
  n_trials,
  stimulus,
  d,
  criterion,
  sdratio = 1,
  dist = c("normal", "gumbel_min", "gumbel_max", "logistic"),
  log = FALSE
)

rsdt_yn(
  n,
  n_trials,
  stimulus,
  d,
  criterion,
  sdratio = 1,
  dist = c("normal", "gumbel_min", "gumbel_max", "logistic")
)

Arguments

n_old

Integer vector. Number of "old"/"signal" responses.

n_trials

Integer vector. Total number of trials per cell.

stimulus

Numeric or logical vector (0/1). Stimulus type: 0 = noise, 1 = signal.

d

Numeric. Sensitivity: \(d'\) when sdratio is 1, and otherwise the balanced index \(d_a\), the separation between the two distributions divided by the root-mean-square of their SDs (see sdt_yn()). The separation in noise units is d * sqrt((1 + sdratio^2) / 2).

criterion

Numeric. Response bias (decision boundary location), on the noise-standardized axis, i.e. in units of the noise distribution's SD.

sdratio

Numeric. Ratio of signal to noise standard deviations (default 1, i.e., equal variance). Must be positive, and is on the natural scale — see the section below before reusing a fitted value.

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": \(1 / (1 + \exp(-x))\)

log

Logical. If TRUE, returns log-density (default FALSE).

n

Integer. Number of observations to generate. n_trials, stimulus, and the model parameters are recycled to this length.

Value

dsdt_yn returns the (log-)density (binomial probability). rsdt_yn returns an integer vector with the number of "old"/"signal" responses per observation.

Parameter scales

As everywhere in bmm, these functions take their arguments on the natural scale, while the parameters sdt_yn() estimates are on their link scale. d and criterion have identity links and carry across unchanged, but sdratio has a log link: a fitted sdratio of 0.375 is a ratio of exp(0.375) = 1.455, and passing 0.375 here instead asks for a signal distribution 2.7 times narrower than the noise. That is a legal value and raises no error, so exponentiate first.

References

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

Examples

# Density of yes/no SDT data
dsdt_yn(n_old = 80, n_trials = 100, stimulus = 1,
        d = 1.5, criterion = 0.2)
#> [1] 0.01136708

# Vectorized over observations
dsdt_yn(n_old = c(30, 80), n_trials = c(100, 100),
        stimulus = c(0, 1), d = 1.5, criterion = 0.2,
        log = TRUE)
#> [1] -7.463009 -4.477034

# Unequal variance from a fitted model: sdt_yn() reports sdratio on its log
# link, so exponentiate before passing it here
dsdt_yn(n_old = 80, n_trials = 100, stimulus = 1,
        d = 1.5, criterion = 0.2, sdratio = exp(0.375))
#> [1] 0.005538898
# Generate yes/no SDT data for a design
dat <- expand.grid(id = 1:20, stimulus = c(0L, 1L))
dat$n_trials <- 100L
dat$n_old <- rsdt_yn(nrow(dat), dat$n_trials, dat$stimulus,
                     d = 1.5, criterion = 0.2)
head(dat)
#>   id stimulus n_trials n_old
#> 1  1        0      100    14
#> 2  2        0      100    19
#> 3  3        0      100    14
#> 4  4        0      100    13
#> 5  5        0      100    11
#> 6  6        0      100    21