Continuous Dual-Process Signal Detection Theory Model

Continuous Dual-Process Signal Detection Theory Model

Usage

sdt_cdp(
  response = "",
  stimulus,
  n_new,
  n_old,
  dist = "normal",
  threshold_type = c("parsimonious", "equidistant", "log_distance"),
  links = NULL,
  ...
)

Arguments

response

An optional common prefix for the response count columns (default "", i.e. the bare canonical names new1, know2, ... that aggregate_sdt_cdp_data() and rsdt_cdp() produce).

stimulus

The name of the variable coding the stimulus type. Must be coded as 0 (new/lure) and 1 (old/target).

n_new, n_old

Integer numbers of "new" and "old" confidence levels. Together with response they determine the response count columns and the threshold parameters.

dist

The noise distribution. Only "normal" is currently supported (the CDP model is inherently Gaussian).

threshold_type

Character. Threshold parameterization on the strength axis: "parsimonious" (default) and "equidistant" use a single spacing parameter; "log_distance" (Paulewicz & Blaut, 2022) estimates the n_new + n_old - 2 distances between adjacent thresholds freely, each as a deltaN parameter on the log scale (the log width of the interval between thresholds N and N + 1, as in sdt_rating()). criterion stays on the old/new boundary, threshold n_new.

links

A named list of link functions for the parameters, e.g. links = list(drec = "log"). Only dfam, drec, criterion and rcrit can be set, to "log", "softplus", "logit", "probit" or "identity", the links the model can invert inside its formula. sigmar, rho, kcrit and the threshold parameters keep their identity links, because their fixed values and the model's own transformations (exp(), tanh()) assume it. A positive-range link ("log", "softplus") on criterion or rcrit states that the criterion is positive: the sampler then starts next to 0 on the native scale and cannot reach negative values.

...

used internally for testing, ignore it

Value

An object of class bmmodel

Details

• Domain: Recognition Memory

• Task: Old/New Recognition with Remember/Know Judgments

• Name: Continuous Dual-Process Signal Detection Theory (CDP)

• Citation:

  • Wixted, J. T., & Mickes, L. (2010). A continuous dual-process model of remember/know judgments. Psychological Review, 117(4), 1025-1054. https://doi.org/10.1037/a0020874

• Requirements:

  Provide aggregated data with one row per cell (unique combination of predictors and stimulus class) and one integer count column per response category:

  • new1 ... new<n_new> for 'new' judgments

  • know<k> and remember<k> (and optionally guess<k>) for 'old' judgments, with k on the unified confidence scale n_new+1 ... n_new+n_old

  • an optional common column prefix is set via response

  • stimulus: 0 = new/lure, 1 = old/target

Use aggregate_sdt_cdp_data() to build these columns from long-format (trial-level) data

• Parameters:

  • dfam: Familiarity sensitivity: target mean on the familiarity axis, in lure-SD units (d' on that axis, where targets and lures share SD 1)

  • drec: Recollection sensitivity: target mean on the recollection axis, in lure-SD units. The target SD is exp(sigmar), so once sigmar is estimated drec is not the d_a the other SDT models report as d

  • criterion: Response bias: old/new boundary on the strength (F+R) axis

  • spacing: Threshold spacing: controls distance between adjacent thresholds (exp(spacing) ensures positive spacing)

  • rcrit: Remember criterion: threshold on the recollection axis

  • sigmar: Log SD of the recollection target distribution, so exp(sigmar) is the SD and 0 means SD = 1

  • rho: Familiarity-recollection correlation on an unconstrained scale; tanh(rho) is the correlation and 0 means independent processes

  • kcrit: Know criterion: threshold on the familiarity axis that splits Know from Guess. Active only when the data include 'guess' counts

• Fixed parameters:

  • sigmar = 0

  • rho = 0

  • kcrit = -100

• Default parameter links:

  • dfam = identity; drec = identity; criterion = identity; spacing = identity; rcrit = identity; sigmar = identity; rho = identity; kcrit = identity

• Default priors:

  • dfam:

    • main: normal(1, 1)

    • effects: normal(0, 0.5)

    • sd: exponential(1)

  • drec:

    • main: normal(1, 1)

    • effects: normal(0, 0.5)

    • sd: exponential(1)

  • criterion:

    • main: normal(0, 1.5)

    • effects: normal(0, 0.5)

    • sd: exponential(2)

  • spacing:

    • main: normal(0, 0.5)

    • effects: normal(0, 0.3)

    • sd: exponential(2)

  • rcrit:

    • main: normal(0, 1)

    • effects: normal(0, 0.5)

    • sd: exponential(2)

  • sigmar:

    • main: normal(0, 0.5)

    • effects: normal(0, 0.3)

    • sd: exponential(2)

  • rho:

    • main: normal(0, 0.5)

    • effects: normal(0, 0.3)

    • sd: exponential(2)

  • kcrit:

    • main: normal(0, 1)

    • effects: normal(0, 0.5)

    • sd: exponential(2)

The continuous dual-process model (Wixted & Mickes, 2010) assumes two continuous memory signals per item, Familiarity (F) and Recollection (R), with correlation tanh(rho). Old/new confidence is read off the aggregate strength S = F + R; the Remember/Know judgment splits "old" responses on R (against rcrit); an optional Know/Guess split uses F (against kcrit).

Response format. The model is fit to aggregated counts, like sdt_rating(): one row per cell (e.g. participant x stimulus class x condition) and one integer count column per response category. The columns follow a fixed naming scheme – new1 ... new<n_new> for "new" judgments, and know<k> / remember<k> (optionally guess<k>) for "old" judgments, where k runs over the unified confidence scale n_new + 1 ... n_new + n_old (1 = most confident "new", K = most confident "old") – so the constructor only needs the numbers of confidence levels, not a long vector of column names. An optional common prefix is set via response (e.g. response = "cdp_" for columns cdp_new1, cdp_know2, ...). Use aggregate_sdt_cdp_data() to build these columns from long-format (trial-level or count) data; rsdt_cdp() generates them directly. The numbers of "new" and "old" confidence levels need not be equal (e.g. the 1-new / 5-old scale of Rotello et al., 2005). Whether the Know/Guess split is modelled is driven by the data: include guess<k> columns to fit the three-way Remember/Know/Guess model.

Variants via the formula. By default sigmar, rho, and kcrit are fixed (equal recollection variance, independent F and R, no Know/Guess split), recovering the classic independent-process CDP. Free them through the formula:

• sigmar ~ 1 estimates unequal recollection variance.

• rho ~ 1 estimates the within-item F-R correlation; rho also accepts predictors (e.g. rho ~ condition), bounded to (-1, 1) via an internal tanh. This structural, within-item correlation is distinct from a between-subject correlation of random effects, which is available separately through brms syntax such as (1 |p| id) on any parameter.

• kcrit ~ 1 estimates the Know/Guess criterion. The data must then contain the guess<k> count columns, and data with those columns must free kcrit: with kcrit at its fixed value every Guess response has probability 0, so bmm() refuses that combination. The fixed value -100 is a sentinel that switches the split off; dsdt_cdp() and rsdt_cdp() use kcrit = NULL for the same purpose.

In the three-way model a Guess response in confidence bin k is possible only while rcrit + kcrit exceeds the lower threshold of that bin, because Guess needs R below rcrit and F below kcrit, so S = F + R below their sum. Below that boundary the Guess category has probability 0, and when rcrit (or kcrit) varies across participants the posterior explores it: hierarchical Remember/Know/Guess fits can report divergent transitions near this boundary without biased estimates. If they do, raise adapt_delta (0.95 or 0.99), give rcrit a tighter group-level prior, or check which participants' Guess counts in the top bins drive it. The Remember/Know model has no such boundary.

The kernel's gradient is wrong, with the value right, on two measure-zero sets that init = 0 starts on (a threshold exactly on the strength mean, and rcrit == kcrit); bmm's own initial values avoid them, and a fit started on them walks off in its first step.

Sensitivity scales

dfam and drec are the target means \(\mu_F\) and \(\mu_R\) of Wixted and Mickes (2010), each in units of the lure SD on its own axis, so estimates can be compared with the values they report. On the familiarity axis targets and lures both have SD 1, and dfam is \(d'\) there. On the recollection axis the target SD is exp(sigmar), so drec is standardized by the lure SD alone. With sigmar fixed at 0 it is \(d'\) as well; with sigmar estimated it is not the balanced index \(d_a\) that sdt_yn() and sdt_rating() report as d.

This matters when drec is compared across conditions whose sigmar differs: a difference in the recollection SD alone can then appear as a credible difference in drec, the problem the sensitivity section of sdt_yn() describes for \(d'\). To compare such conditions, compute the balanced recollection sensitivity from the posterior draws as drec / sqrt((1 + exp(2 * sigmar)) / 2), or give sigmar the same formula in every condition being compared.

Neither dfam nor drec is the discriminability of the old/new decision, which is read off the aggregate strength \(F + R\) and depends on both sensitivities, on sigmar and on rho.

When no Remember/Know split is available (confidence ratings only), use sdt_rating() instead.

References

Wixted, J. T., & Mickes, L. (2010). A continuous dual-process model of remember/know judgments. Psychological Review, 117(4), 1025–1054. doi:10.1037/a0020874

Paulewicz, B., & Blaut, A. (2022). The general causal cumulative model of ordinal response. PsyArXiv. doi:10.31234/osf.io/e7a3x

Rotello, C. M., Macmillan, N. A., Reeder, J. A., & Wong, M. (2005). The remember response: Subject to bias, graded, and not a process-pure indicator of recollection. Psychonomic Bulletin & Review, 12(5), 865–873. doi:10.3758/BF03196778

Examples

if (FALSE) { # \dontrun{
# Simulate a Remember/Know data set (3 new + 3 old confidence levels) for
# 20 subjects: rsdt_cdp() returns the count columns sdt_cdp() expects
dat <- expand.grid(id = 1:20, stimulus = c(0L, 1L))
thresholds <- c(-1.1, -0.5, 0, 0.6, 1.3)
dat <- cbind(dat, rsdt_cdp(nrow(dat), 200, dat$stimulus,
                           dfam = 0.8, drec = 1.0,
                           thresholds = thresholds, rcrit = 0.7,
                           n_new = 3))

model <- sdt_cdp(stimulus = "stimulus", n_new = 3, n_old = 3)

fit <- bmm(
  formula = bmf(dfam ~ 1, drec ~ 1, criterion ~ 1,
                spacing ~ 1, rcrit ~ 1),
  data = dat, model = model, backend = "cmdstanr"
)

# Estimate unequal recollection variance and the F-R correlation
fit_uv <- bmm(
  formula = bmf(dfam ~ 1, drec ~ 1, criterion ~ 1,
                spacing ~ 1, rcrit ~ 1, sigmar ~ 1, rho ~ 1),
  data = dat, model = model, backend = "cmdstanr"
)
} # }