Area under the ROC curve from a fitted SDT model

Computes the posterior area under the ROC curve (AUC). For Gaussian and Gumbel (min or max) equal-variance binary SDT the AUC is available in closed form from the d draws; otherwise it is obtained by trapezoidal integration of the model-implied curve (for rating fits, the curve swept from the posterior of d and sdratio, lifted by Ro/Rn for version = "dpsdt", so the AUC is the area under the predicted mixture ROC and not \(\Phi(d/\sqrt{2})\); for version = "metad" it is the type-1 curve). The returned AUC is always the area under the full curve, not the trapezoid of the discrete operating points or the K-1 rating thresholds; for binary multi-criteria fits it is one value per curve.

Usage

auc_sdt(
  fit,
  conditions = NULL,
  probs = c(0.025, 0.975),
  criterion_points = NULL,
  ...
)

Arguments

fit: A bmmfit object returned by bmm() from an SDT model.
conditions: Optional data frame of predictor values at which to evaluate the model. Column names must match predictor variables used in the formula. If NULL (default), unique predictor combinations are derived from the data.
probs: Numeric vector of length 2. Quantiles for the credible interval (default c(0.025, 0.975)).
criterion_points: Optional. Control of the binary multi-criteria behaviour. NULL (default) auto-detects predictors that vary the criterion only (not d/sdratio) and treats their levels as operating points on one curve. Pass a character vector of column names to force that classification, or FALSE to disable it (one separate curve per predictor combination). Ignored for rating models.
...: Additional arguments passed to brms::posterior_linpred(), such as draw_ids to use a subset of the posterior draws. ndraws is refused: each parameter is drawn by its own call, so random subsets would not match across parameters.

Value

A data frame of class "bmm_sdt_auc" with columns AUC, .draw, and any condition columns, plus a summary attribute (AUC_mean, AUC_lower, AUC_upper), model_class, dist and conditions attributes.

Details

The closed form is used only when sdratio is fixed at 0, where d (which is \(d_a\)) equals \(d'\); every fit with sdratio estimated or fixed away from 0 takes the numerical route, so the AUC is invariant to the sensitivity parameterization.

Analytical formulas (equal variance, where \(d_a = d'\)): normal EV-SDT \(AUC = \Phi(d'/\sqrt{2})\); Gumbel-min and Gumbel-max EV-SDT \(AUC = \mathrm{logistic}(g')\).

Examples

if (FALSE) { # \dontrun{
# Three base-rate conditions shift the criterion, which identifies sdratio
dat <- expand.grid(id = 1:20, stimulus = c(0L, 1L),
                   condition = c("liberal", "neutral", "strict"))
dat$n_trials <- 100L
# simulated as bias, measured from the midpoint between the means (0 = boundary at the midpoint)
criteria <- c(liberal = -0.5, neutral = 0, strict = 0.5)
dat$n_old <- rsdt_yn(nrow(dat), dat$n_trials, dat$stimulus, d = 1.5,
                     bias = criteria[as.character(dat$condition)],
                     sdratio = 1.3)

fit <- bmm(
  formula = bmf(d ~ 1, criterion ~ 0 + condition, sdratio ~ 1),
  data = dat,
  model = sdt_yn(response = "n_old", stimulus = "stimulus",
                 n_trials = "n_trials"),
  cores = 4,
  backend = "cmdstanr"
)

auc <- auc_sdt(fit)
auc
plot(auc)
} # }