Yes/No Signal Detection Theory Model

Estimates sensitivity (d, which is $d'$ under equal variance and $d_a$ once sdratio is estimated) and the decision criterion from yes/no detection or old/new recognition counts. The criterion is estimated either as criterion, measured from the mean of the noise distribution, or as bias, measured from the midpoint between the noise and signal means; the formula names the one to estimate.

Usage

sdt_yn(
  response,
  stimulus,
  n_trials,
  dist = c("normal", "gumbel_min", "gumbel_max", "logistic"),
  links = NULL,
  ...
)

Arguments

response

The name of the variable in the dataset containing the count of "old"/"signal" responses for each cell.

stimulus

The name of the variable in the dataset coding the stimulus type: 0 (noise/new) and 1 (signal/old). Logical, and factor or character columns holding "0"/"1", are coerced automatically; anything else (e.g. "noise"/"signal" labels) must be recoded by hand, since bmm cannot guess which level is the signal. dsdt_yn() and rsdt_yn() take the same column but are stricter, accepting only numeric or logical input.

n_trials

The name of the variable in the dataset containing the total number of trials for each cell. It may differ from cell to cell. Trial-level data also works: keep one row per trial, with the column named by response holding 0 or 1 and the column named by n_trials holding 1s.

dist

The noise distribution assumed for the latent evidence variable, given here by its cumulative distribution function. One of:

"normal" (default): Gaussian SDT, $\Phi(x)$
"gumbel_min": smallest-extreme-value SDT, $1 - \exp(-\exp(x))$ (complementary log-log)
"gumbel_max": largest-extreme-value SDT, $\exp(-\exp(-x))$ (log-log, as in evd::pgumbel)
"logistic": logistic SDT, $1 / (1 + \exp(-x))$

links

A named list of link functions for the parameters, one entry per parameter you want to change, e.g. links = list(d = "log"). Only d, criterion and bias can be set. sdratio keeps its log link, because the model's default of an equal-variance SD ratio is stored as the 0 that the log link maps to 1; read on any other link that same 0 is a different ratio, and on an identity link it is a ratio of zero that makes the signal trials' evidence infinite.

...

used internally for testing, ignore it

Value

An object of class bmmodel

Details

Domain: Perception & Recognition Memory
Task: Yes/No Detection or Old/New Recognition
Name: Signal Detection Theory (Yes/No)
Citation:
- Green, D. M., & Swets, J. A. (1966). Signal detection theory and psychophysics. Wiley.
Requirements:

Provide pre-aggregated data with the following columns:

Response counts (response): number of 'old'/'signal' responses
Stimulus type (stimulus): 0 = noise, 1 = signal
Number of trials (n_trials): total trials per cell

Parameters:
- d: d_a sensitivity (= d' when sdratio is fixed): the distance between the signal and noise distributions in units of their root-mean-square SD
- criterion: Decision criterion, measured from the mean of the noise distribution in noise-SD units; the false-alarm rate depends on it alone
- bias: Alternative to criterion: the decision criterion measured from the midpoint between the noise and signal means (0 = neutral, positive = conservative). Use one of the two
- sdratio: SD ratio signal/noise, log link (0 = equal SDs): summary() prints the log of this ratio, not the ratio itself, so exp() a posterior value to read it
Fixed parameters:
- sdratio = 0
Default parameter links:
- d = identity; criterion = identity; bias = identity; sdratio = log
Default priors:
- d:
  - main: normal(1, 1)
  - effects: normal(0, 0.5)
  - sd: exponential(1)
- criterion:
  - main: normal(0.5, 1.5)
  - effects: normal(0, 0.5)
  - sd: exponential(1)
- bias:
  - main: normal(0, 1.5)
  - effects: normal(0, 0.5)
  - sd: exponential(2)
- sdratio:
  - main: normal(0, 0.3)
  - effects: normal(0, 0.3)
  - sd: exponential(2)

Which sensitivity measure `d` is

d is the familiar $d'$ whenever sdratio keeps its default of 0 (an SD ratio of 1, equal variance), which is every fit that does not give sdratio a formula. The rest of this section only matters once you estimate sdratio.

When the signal and noise distributions have different widths, their separation only becomes dimensionless after choosing an SD to divide by. bmm then reports $d_a$, the separation divided by the root-mean-square of the two SDs: $$d_a = \sqrt{2}\,\delta / \sqrt{1 + r^2},$$ where $\delta$ is the separation in noise-SD units and $r$ is the SD ratio, exp(sdratio). This weights the two distributions equally, and it is the measure Simpson and Fitter (1973), Macmillan and Creelman (2005), and Mickes et al. (2007) recommend under unequal variance. The classical noise-standardized index is $d_N = \delta = d_a \sqrt{(1 + r^2)/2}$, so a published $d'$ from an unequal-variance analysis is larger than d when $r > 1$: by 13% at $r = 1.25$ and by 33% at $r = 1.6$. sdt_sensitivity() returns $d_N$ and the signal-standardized $d_S$ from a fitted model, and summary() notes when d is $d_a$.

d is $d_a$ rather than $d_N$ because only $d_a$ is comparable across conditions or subjects that differ in sdratio: two conditions that are equally discriminable can show a large, confidently estimated difference in $d_N$. When sdratio is estimated but constant across the conditions you compare, the two indices differ by one common factor and give the same contrasts up to scale.

Units of the other parameters. criterion and bias are not rescaled: both are in noise-SD units, so under unequal variance d (in root-mean-square SD units) and the criterion are on different scales, and a ratio such as bias / d mixes them.

Extreme-value distributions. For dist = "normal", $d_a$ is also the AUC-equivalent index, $d_a = \sqrt{2}\,\Phi^{-1}(\mathrm{AUC})$, so it carries the same information as 2AFC accuracy. For "gumbel_min" and "gumbel_max" that identity holds only under equal variance. With sdratio estimated, $d_a$ keeps its balanced geometry but drifts away from the AUC-equivalent index as the SD ratio moves away from 1 (for "gumbel_min" at $\delta = 1.5$ and $r = 2$, $d_a$ is 31% larger), so compare such fits on the AUC rather than on d.

Because d is a short name, a column called d in your data that is also used as a predictor will collide with this parameter; bmm() warns when that happens.

Choosing between `criterion` and `bias`

Both parameters place the same decision boundary on the noise-standardized evidence axis, and they differ only in the point they measure it from:

criterion is the distance from the mean of the noise distribution. The false-alarm rate depends on it alone: $FA = 1 - F(criterion)$.
bias is the distance from the midpoint between the noise and signal means: 0 is the neutral point, positive values a conservative bias toward "new"/"noise". Under equal variance it is the index usually written $c$.

They are related by $bias = criterion - \delta / 2$, where $\delta = d \sqrt{(1 + r^2)/2}$ is the separation in noise-SD units. Write one of them in the formula; a formula that names neither estimates criterion ~ 1, and one that names both is refused. sdt_criterion() and sdt_bias() compute the two from a single hit and false-alarm rate.

The choice matters whenever d or sdratio differs between rows that share a criterion term, because a linear formula holds constant the parameter you write. criterion ~ 1 with d ~ 1 + strength says the observer keeps one boundary while targets get stronger; bias ~ 1 with the same d formula says the boundary moves up by half of every change in separation. When the criterion and d share condition-wise (factor) predictors and sdratio does not vary within those cells, or share predictors while d keeps its identity link and sdratio is the same in every row, the two give the same likelihood and their coefficients differ by $\delta / 2$. Otherwise they are different models. Even with the same likelihood, the posteriors differ through the default priors: criterion takes normal(0.5, 1.5) for its intercept, or for each cell under ~ 0 + condition, and exponential(1) for its random-effect SDs, while bias takes normal(0, 1.5) and exponential(2).

Use criterion when the lures, and the observer's boundary, do not depend on the manipulation of d: the classic recognition design in which weak and strong targets are tested against one shared set of lures under the same criteria. Its lure rows have no strength level, and with criterion they need none, because the false-alarm rate does not involve d. Use bias when you want to hold, or compare, the observer's position relative to the two distributions, e.g. a bias contrast between groups that differ in sensitivity.

bias is measured from the midpoint, which is the point where the two densities are equal – the optimal boundary for equal base rates and payoffs – only for the symmetric distributions ("normal", "logistic") under equal variance. Under unequal variance, or for the Gumbel distributions, bias = 0 is the midpoint, not the optimum.

Identifying `sdratio`

sdratio needs a design that supplies more than one operating point, and whether it gets one depends on which criterion parameter the formula holds constant.

With criterion, the boundary must move among rows that share the same d: give criterion a predictor that shifts it while d stays put, such as a base-rate, payoff, or confidence manipulation (d ~ 1, criterion ~ 0 + condition; see broeder_schuetz_2009_e3). A criterion that is the same in every row is fixed by the false-alarm rate, and each hit rate then fixes only $(\delta - criterion) / r$, so sdratio trades against d. A change in sdratio is matched by shifting the separation of every cell by one common amount and rescaling it by one common factor, and whenever the formula for d can make that change, sdratio returns its prior. A factor can, under the identity and the log link alike (d ~ 0 + condition or d ~ 1 + condition), because it gives every cell a value of its own. An intercept with a numeric predictor can under the identity link (d ~ 1 + strength), because the intercept shifts and the slope rescales. sdratio is identified only when the formula constrains d across cells beyond that. A numeric predictor without an intercept (d ~ 0 + strength) holds d proportional to strength. It identifies sdratio when strength takes at least two nonzero values and the criterion lies clearly away from the noise mean (for "normal", a false-alarm rate clearly away from .5); at a criterion of 0 it does not. The estimate then rests on the assumption that d is zero where strength is zero, not on a second operating point in the data. A numeric predictor under a log link on d barely identifies sdratio, and in practice sdratio returns its prior. bmm() warns for every criterion that is the same in every row, so it also warns for d ~ 0 + strength. Giving d and criterion the same condition-wise predictors (d ~ 0 + condition, criterion ~ 0 + condition) fails too, because each condition then has two rates for two parameters of its own, and no warning catches it. A random effect such as criterion ~ 1 + (1 | id) moves the boundary between subjects and does not trigger the warning, but it informs sdratio only weakly, through the assumption that the subject effects on d and on the criterion are independent.

With bias, any predictor that moves the operating point supplies what is missing, including a sensitivity manipulation such as d ~ 0 + condition with bias ~ 1: holding the bias constant moves the criterion with the separation, and that movement traces the ROC. The identification then rests on that assumption about the observer. The design in which every parameter is intercept-only with no random effects cannot work: a single (hit, false-alarm) pair is two numbers for three unknowns, and bmm() warns for it. Neither can bias and d with the same condition-wise predictors, for the reason given above, and that design raises no warning.

Sampling converges and Rhat is fine in every unidentified case, so the warnings are the only signal. They count formula terms, not how far the terms move the operating point, so a predictor with a small effect carries little information without a warning. Leaving sdratio at its default is always identified.

Reading `sdratio` and carrying it to `dsdt_yn()`/`rsdt_yn()`

As in every bmm model, the parameters the model estimates are on their link scale, while the distribution functions take their arguments on the natural scale. sdratio has a log link, so summary() reports $\log r$ whereas dsdt_yn() and rsdt_yn() expect the ratio $r$ itself (their default is 1, equal variance). Exponentiate before carrying a posterior value across:

r <- exp(as_draws_matrix(fit)[, "b_sdratio_Intercept"])

A posterior mean of sdratio = 0.22 is a ratio of exp(0.22) = 1.25. Passing 0.22 straight to rsdt_yn() instead asks for a signal distribution 4.5 times narrower than the noise — a legal value that raises no error, and the one mistake worth checking for in a posterior predictive check written by hand. d, criterion and bias have identity links, so they carry across unchanged.

The zROC slope reported in the recognition-memory literature is the reciprocal of that ratio, 1 / exp(sdratio), so a sdratio posterior mean of 0.375 is a zROC slope of 0.69.

The same log link applies going the other way: a constant you supply yourself, whether as bmf(sdratio = ) or through a hand-written brms::set_prior(..., dpar = "sdratio"), is read on it too. bmf(sdratio = 1) does not fix a ratio of 1 — it fixes exp(1) = 2.72, and bmm() raises no warning; a fixed ratio of 1.25 needs bmf(sdratio = log(1.25)). Both sdratio defaults default_prior() reports — normal(0, 0.3) on the intercept and exponential(2) on the random-effect SDs — are on that same scale, unannotated.

Terms used on this page

noise-standardized axis — latent evidence expressed in units of the noise distribution's SD, with its origin at the noise mean. criterion is a position on this axis; bias is a distance on it, measured from the midpoint between the means.
$\delta$ (separation) — the distance between the signal and noise means, on that axis.
$d_N$ — the classical $d'$: $\delta$ itself, i.e. the separation in noise-SD units.
$d_a$ — the separation divided by the root-mean-square of the two SDs, which is what d reports. $d_a = d_N$ under equal variance.
operating point — one (false-alarm rate, hit rate) pair, i.e. one point of an ROC curve. One condition gives one operating point.
AUC — the area under that ROC curve, equivalently the probability that a random signal trial yields more evidence than a random noise trial. Obtain it from the posterior with pnorm(d / sqrt(2)) for dist = "normal".
main / effects / sd — the keys of the default priors shown in the model description above: main is the prior on the intercept, effects the prior on regression coefficients, and sd the prior on the standard deviations of the parameter's random effects.

References

Green, D. M., & Swets, J. A. (1966). Signal detection theory and psychophysics. Wiley.

DeCarlo, L. T. (1998). Signal detection theory and generalized linear models. Psychological Methods, 3(2), 186–205. doi:10.1037/1082-989X.3.2.186

Simpson, A. J., & Fitter, M. J. (1973). What is the best index of detectability? Psychological Bulletin, 80(6), 481–488. doi:10.1037/h0035203

Macmillan, N. A., & Creelman, C. D. (2005). Detection theory: A user's guide (2nd ed.). Erlbaum.

Mickes, L., Wixted, J. T., & Wais, P. E. (2007). A direct test of the unequal-variance signal detection model of recognition memory. Psychonomic Bulletin & Review, 14(5), 858–865. doi:10.3758/BF03194112

Examples

if (FALSE) { # \dontrun{
dat <- expand.grid(id = 1:20, stimulus = c(0L, 1L))
dat$n_trials <- 100L
dat$n_old <- rsdt_yn(nrow(dat), dat$n_trials, dat$stimulus,
                     d = 1.5, criterion = 0.9)

model <- sdt_yn(
  response = "n_old",
  stimulus = "stimulus",
  n_trials = "n_trials"
)

fit <- bmm(
  formula = bmf(d ~ 1, criterion ~ 1),
  data = dat,
  model = model,
  cores = 4,
  backend = "cmdstanr"
)

# Sensitivity and bias per participant
fit_re <- bmm(
  formula = bmf(d ~ 1 + (1 | id), criterion ~ 1 + (1 | id)),
  data = dat,
  model = model,
  cores = 4,
  backend = "cmdstanr"
)

# Unequal-variance yes/no SDT. sdratio needs more than one operating point:
# on the single-condition `dat` above it would not be identified.
# `model` already names the columns this dataset uses.
fit_uv <- bmm(
  formula = bmf(d ~ 1, criterion ~ 0 + condition, sdratio ~ 1),
  data = broeder_schuetz_2009_e3,
  model = model,
  cores = 4,
  backend = "cmdstanr"
)

# The same observer, with the boundary measured from the midpoint
fit_bias <- bmm(
  formula = bmf(d ~ 1 + (1 | id), bias ~ 1 + (1 | id)),
  data = dat,
  model = model,
  cores = 4,
  backend = "cmdstanr"
)

# Weak and strong targets against one shared set of lures, one criterion per
# subject: `strong` is 1 for strong targets and 0 for weak targets and lures
# (the lure rows' value does not matter, their false-alarm rate depends on
# the criterion alone)
lure_dat <- expand.grid(id = 1:20, cell = c("lure", "weak", "strong"))
lure_dat$stimulus <- as.integer(lure_dat$cell != "lure")
lure_dat$strong <- as.integer(lure_dat$cell == "strong")
lure_dat$n_trials <- 100L
lure_dat$n_old <- rsdt_yn(nrow(lure_dat), lure_dat$n_trials,
                          lure_dat$stimulus, d = 1 + 0.8 * lure_dat$strong,
                          criterion = 0.7)
fit_strength <- bmm(
  formula = bmf(d ~ 1 + strong + (1 | id), criterion ~ 1 + (1 | id)),
  data = lure_dat,
  model = model,
  cores = 4,
  backend = "cmdstanr"
)

# Simulating from a fitted unequal-variance model: the model reports
# sdratio on its log link, rsdt_yn() takes the ratio itself.
sdratio_posterior <- 0.375
rsdt_yn(2, 100L, c(0L, 1L),
        d = 1.3, criterion = 0.75, sdratio = exp(sdratio_posterior))
} # }