SDT via GLM/GLMM

Why a probit regression gives you d′

The connection is exact rather than approximate.

Signal detection theory says the witness’s evidence is normally distributed with mean 0 for new items and mean d′ for old ones, with a criterion at some point \(c\). The probability of saying “old” is the area of that distribution above the criterion:

\[P(\text{said old} \mid \text{new}) = \Phi(-c), \qquad P(\text{said old} \mid \text{old}) = \Phi(d' - c)\]

A probit regression models exactly this form:

\[\Phi^{-1}\bigl(P(\text{said old})\bigr) = \beta_0 + \beta_1 \cdot \text{is old}\]

Matching the two, \(\beta_1 = d'\) and \(\beta_0 = -c\). The coefficient on item status is d′ — not an analogue of it.

The intercept needs one further step. With is_old coded 0/1, \(\beta_0\) is the value when is_old = 0, so it is the criterion measured from the new-item distribution. Coding is_old as −0.5/+0.5 instead centres it, and the intercept becomes the criterion measured from the midpoint of the two distributions — which is the conventional definition of \(c\). fit_sdt_glm() centres by default, which is why the criterion is only reported when center_isold = TRUE.

With a logit link instead, the same coefficient is a log odds ratio rather than d′. It answers the same question about discriminability but on a different scale, and should not be reported as d′.

The estimates move too, and predictably

The d′ estimates also differ across the four rows, and this is not noise. The GLM estimate is marginal (population-averaged): it describes the average response probability across all participants and items. The GLMM estimate is conditional (subject- and item-specific): it describes the effect for a participant and item at the average of the random effects.

For a probit link the two are related by a known attenuation. With only random intercepts, the marginal d′ is approximately the conditional d′ divided by \(\sqrt{1 + \sigma^2_{\text{subject}} + \sigma^2_{\text{item}}}\). With a random slope the variance of the random part differs between new and old items, so the probit-scale value for each kind of item is divided by its own factor \(\sqrt{1 + v(x)}\), where \(v(x) = \sigma^2_{\text{int}} + 2x\sigma_{\text{int,slope}} + x^2\sigma^2_{\text{slope}} + \sigma^2_{\text{item}}\) and \(x\) is \(-0.5\) for new and \(+0.5\) for old items (centred coding). The marginal d′ is the difference between the two scaled values:

fit_re <- structures[["GLMM: subject slope and item"]]
vc     <- lme4::VarCorr(fit_re)
v_subj <- vc[[".subject_id"]]
v_item <- vc[[".item_id"]][1, 1]
v_x    <- function(x) v_subj[1, 1] + 2 * x * v_subj[1, 2] +
  x^2 * v_subj[2, 2] + v_item

b    <- lme4::fixef(fit_re)
cond <- b[[".is_old_c"]]
pred <- (b[["(Intercept)"]] + 0.5 * cond) / sqrt(1 + v_x(0.5)) -
  (b[["(Intercept)"]] - 0.5 * cond) / sqrt(1 + v_x(-0.5))
marg <- stats::coef(structures[["GLM: no clustering"]])[[".is_old_c"]]

knitr::kable(
  data.frame(
    Quantity = c("Conditional d' (GLMM with random slope)",
                 "Random-effect variance, new items",
                 "Random-effect variance, old items",
                 "Predicted marginal d'",
                 "Marginal d' (GLM), observed"),
    Value = round(c(cond, v_x(-0.5), v_x(0.5), pred, marg), 3)
  ),
  row.names = FALSE,
  caption = "Marginal and conditional d' are different quantities related by a known attenuation. Simulated data."
)

The predicted marginal value and the observed one agree to within about one standard error of the GLM estimate. So the two models are not disagreeing about the data; they are answering slightly different questions. Neither is wrong, but they should not be compared as though interchangeable, and a paper should say which is being reported. The conditional estimate is often the one meant by “d′” in a cognitive model, since it refers to a typical participant rather than to an average over heterogeneity.

What is estimated

Standard errors for the GLM come from the usual Fisher information; for the GLMM they come from lme4’s Laplace approximation to the marginal likelihood.

Coding conventions

center_isold = TRUE (the default) codes item status as −0.5/+0.5 rather than 0/1. This does not change the coefficient on is_old, but it changes the intercept from “criterion relative to the new-item distribution” to “criterion relative to the midpoint”, which is the conventional \(c\). extract_sdt_metrics() returns the criterion only under centred coding, since under 0/1 coding the intercept is not \(c\).

Assumptions

1. Equal variance. The probit model as fitted assumes the old- and new-item evidence distributions have the same variance. The unequal-variance case — which is the empirically supported one in recognition memory — needs a model with a scale parameter, for which see fit_sdt_roc() in R/zroc_sdt.R, or the ordinal approach discussed below.
2. Random effects are normally distributed with mean zero, and independent of the predictors.
3. Crossed, not nested. Participants and items are crossed: every participant sees every item, or nearly so. (1|subject_id) + (1|item_id) (or (1 + is_old|subject_id) + (1|item_id) with a random slope) is correct for this. If items were nested within lists, the specification would differ.
4. Random intercepts only, unless random_slope = TRUE. Allowing d′ itself to vary across participants is often the more realistic model and requires more data to fit.

A limitation

This treats the response as binary — “old” or “new”. Recognition data usually carry confidence as well, and reducing a confidence rating to a binary response discards it. The natural extension is an ordinal (cumulative link) model with confidence as the ordered outcome, in which the coefficient on item status is still d′ and the thresholds are the criteria; allowing the scale to depend on condition then gives the unequal-variance model. That is not implemented here.