Model-implied latent decision-variable distributions

Computes the noise and signal latent decision-variable densities implied by a fitted sdt_yn() or sdt_rating() model, together with the response criterion (binary) or the K-1 confidence thresholds (rating). This is the canonical signal detection picture: two evidence distributions separated by the sensitivity and cut by one or more decision boundaries.

Usage

latent_sdt(
  fit,
  conditions = NULL,
  n_grid = 200,
  probs = c(0.025, 0.975),
  collapse = NULL,
  show_competitors = FALSE,
  ...
)

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.
n_grid: Integer. Number of points on the evidence-axis grid at which each density is evaluated (default 200).
probs: Numeric vector of length 2. Lower and upper quantiles for the credible band on the criterion/threshold locations (default c(0.025, 0.975)).
collapse: Control of which predictor dimensions are collapsed into one panel. NULL (default) auto-detects: predictors of the criterion/thresholds only (and the set size) are collapsed; predictors of d/sdratio are faceted. Pass a character vector of column names to force those columns to collapse, or FALSE to facet every dimension (one panel per combination).
show_competitors: Logical (default FALSE). For sdt_mafc/sdt_ranking only, additionally overlay the density of the maximum of the m - 1 distractor samples (one curve per set size) – the "effective competitor" the target must beat – which shifts rightward as m grows and visualises why accuracy falls with set size. The overlay shows the set sizes in the fitted data, whatever conditions requests. Ignored for sdt_yn/sdt_rating.
...: 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_latent" with columns x (the evidence axis), density, distribution ("noise" or "signal"), and any faceting (density) condition columns. A lines attribute holds the decision-boundary positions (columns position, lower, upper, marker, level, plus condition columns), or NULL for the criterion-free sdt_mafc/sdt_ranking models. When show_competitors = TRUE, a competitors attribute holds the max-of-distractors densities. An extra attribute carries the response-process parameters of the sdt_rating() versions as a posterior summary (parameter, mean, lower, upper, plus condition columns): for version = "dpsdt", Ro and Rn on the probability scale; for version = "metad", the M-ratio (mratio) and meta-d' (metad); NULL for version = "standard". The object also carries probs, model_class, dist, is_rating and conditions (the conditions of the density panels).

Details

Densities are evaluated at the posterior-mean parameters of each condition, on the evidence axis the model's criterion is measured on: noise at 0 with unit SD and signal at sep with SD exp(sdratio), where sep = d * sqrt((1 + exp(sdratio)^2) / 2) is the separation in noise-SD units. A fit that estimated bias has its boundaries drawn on the same axis, at bias + sep / 2. d is \(d_a\), measured in root-mean-square SD units, so sep equals d whenever sdratio is fixed at 0. The decision boundaries additionally carry a posterior credible band on their location. The area beyond a boundary equals the corresponding hit/false-alarm rate for every noise distribution: the likelihood evaluates the survivor function of the evidence distribution at the boundary, which is exactly that area.

Available for all four SDT models. sdt_yn() draws the criterion and sdt_rating() the K-1 confidence thresholds as boundary lines. sdt_mafc() and sdt_ranking() have no response criterion – the decision is a max/rank rule over the m alternatives – so only the densities are drawn (no boundary lines); there the noise density represents each of the m - 1 distractor alternatives and the signal density the target.

Because the densities depend only on d/sdratio, predictors that vary the criterion/thresholds only (e.g. a base-rate manipulation, as in broeder_schuetz_2009_e3) leave the densities unchanged: by default they are collapsed into a single panel with their several boundaries overlaid and colour-coded, rather than shown as repeated identical panels. Predictors that vary d/sdratio produce distinct density panels (faceted). For sdt_mafc/sdt_ranking the set size is likewise collapsed unless it predicts d. Use collapse to override the auto-detection.

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"
)

latent <- latent_sdt(fit)
latent
plot(latent)
} # }