Yes/No Signal Detection Theory Model

Estimates sensitivity (d, which is $d'$ under equal variance and $d_a$ once sdratio is estimated) and response bias (criterion) from yes/no detection or old/new recognition counts.

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 and criterion 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: Response bias: location of decision boundary
- 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; sdratio = log
Default priors:
- d:
  - 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)
- 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 $r = 1.6$.

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 is not rescaled. It is the location of the decision boundary relative to the midpoint between the two distributions, in noise-SD units, so under unequal variance d (in root-mean-square SD units) and criterion (in noise-SD units) are on different scales, and a ratio such as criterion / 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.

Identifying `sdratio`

sdratio needs a design that supplies more than one operating point. A single (hit, false-alarm) pair is two numbers for three unknowns, so when every parameter is intercept-only with no random effects, sdratio ~ 1 returns its prior and d is pulled along the resulting ridge: sampling converges, Rhat is fine, and the profile likelihood over sdratio is flat to 1e-12. bmm() warns for that one design, which is the only shape that provably cannot work.

Any linear predictor that moves the operating point along the ROC supplies what is missing, and it need not sit on criterion: a sensitivity manipulation (d ~ 0 + condition, a study-time or strength manipulation with bias held constant) identifies sdratio just as a criterion manipulation does, and so does between-subject variation entering through a random effect such as criterion ~ 1 + (1 | id) — though, like any predictor, only in proportion to how far it actually moves the operating point: a random effect with little between-subject spread carries little information and will not trigger the warning above, because that warning counts formula terms, not how much they move the design.

A criterion manipulation is still the cleanest design, because it traces the ROC at fixed sensitivity: give criterion a predictor that shifts the decision boundary — a base-rate, payoff, or confidence manipulation — as in criterion ~ 0 + condition; see broeder_schuetz_2009_e3. 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 and criterion 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. criterion always lives on this axis.
$\delta$ (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 = 0.2)

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"
)

# 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 = 0.1, sdratio = exp(sdratio_posterior))
} # }