Sensitivity on the noise, signal, or root-mean-square scale

Re-expresses the posterior sensitivity of a fitted SDT model against a different reference standard deviation. As long as sdratio is fixed at its default, d is the familiar $d'$ and all three scales below are the same number. Once sdratio is estimated, d is $d_a$, which measures the separation of the two evidence distributions in units of their root-mean-square SD, and the same separation can also be read against the noise SD ($d_N$, the classical $d'$) or against the signal SD ($d_S$). This function returns any of the three as posterior draws, so contrasts and intervals can be computed on whichever scale a literature reports.

Usage

sdt_sensitivity(
  fit,
  measure = c("da", "dn", "ds"),
  conditions = NULL,
  probs = c(0.025, 0.975),
  ...
)

Arguments

fit: A bmmfit object returned by bmm() from an SDT model.
measure: Character vector naming the scales to return: "da" (root-mean-square SD, the estimated parameter), "dn" (noise SD), and/or "ds" (signal SD). Defaults to all three.
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. Lower and upper quantiles for the credible interval (default c(0.025, 0.975)).
...: 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_sensitivity" with columns measure, value, .draw, and any condition columns. The object carries a summary attribute (measure, mean, lower, upper, plus condition columns), and probs, model_class, dist and conditions attributes.

Details

The model places the noise distribution at $-\delta/2$ with SD $\sigma_N = 1$ and the signal distribution at $+\delta/2$ with SD $\sigma_S = \exp(\mathrm{sdratio})$, so $\delta$ is the separation in noise-SD units. Dividing that separation by each reference SD gives

$$d_N = \delta / \sigma_N, \quad d_S = \delta / \sigma_S, \quad d_a = \delta / \sqrt{(\sigma_N^2 + \sigma_S^2)/2}.$$

The estimated parameter is $d_a$, hence $\delta = d_a \sqrt{(1 + \sigma_S^2)/2}$ and

$$d_N = d_a \sqrt{(1 + \sigma_S^2)/2}, \qquad d_S = d_a \sqrt{(1 + \sigma_S^2)/2} \,/\, \sigma_S.$$

All three coincide when sdratio is 0 (equal variance), which is the case for sdt_mafc() and for any fit that leaves sdratio at its default of 0. The conversion is applied draw by draw, so the returned intervals propagate the joint posterior uncertainty in d and sdratio rather than combining point estimates.

Only $d_a$ is invariant to which distribution is treated as the reference. $d_N$ and $d_S$ are not comparable across conditions that differ in sdratio: two conditions with identical discriminability can show a large, confidently estimated $d_N$ difference. Prefer $d_a$ for contrasts, and use $d_N$/$d_S$ for comparison with published values. When sdratio is estimated but constant across the conditions being compared, all three differ by one common factor per draw and give the same contrasts up to scale.

The criterion and rating thresholds are not converted: they stay on the noise-SD axis whichever sensitivity scale you report. Under the Gumbel distributions with sdratio estimated, none of the three is the AUC-equivalent index; use auc_sdt() to compare discriminability on the probability scale.

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"
)

# d_a (the fitted d), d_N and d_S, converted draw by draw
sdt_sensitivity(fit)
sdt_sensitivity(fit, measure = "dn", draw_ids = 1:500)
} # }