Thresholds and slopes of a fitted psychometric function

Computes, per posterior draw and condition, the stimulus intensity at which a psychometric_function() fit reaches a target response probability, the slope of the psychometric function there, and the guess and lapse rates, including the lapse rate as Wichmann and Hill (2001) define it.

Usage

psychometric_threshold(
  fit,
  p = 0.75,
  conditions = NULL,
  probs = c(0.025, 0.975),
  ...
)

Arguments

fit: A bmmfit object returned by bmm() with a psychometric_function() model.
p: Numeric vector of target response probabilities, each strictly between 0 and 1. The default 0.75 is the conventional 2AFC threshold.
conditions: A data frame of predictor values at which to evaluate the parameters. The default uses every unique combination of the population-level predictors in the data, and of the guess or lapse column when a rate is fixed per row. A user-supplied data frame must have a column for each of these and no other: the thresholds are population-level, so a grouping column such as a participant id is refused.
probs: Numeric vector of length 2. Lower and upper quantiles for the credible interval in the summary. Default c(0.025, 0.975).
...: Further arguments passed to brms::posterior_linpred(), such as draw_ids.

Value

A data frame of class "bmm_psychometric_threshold" with one row per draw, measure and condition, and columns measure ("threshold", "slope", "guess", "lapse", "lapse_wh"), p (NA for the three rates, which do not depend on it), value, .draw and the condition columns. lapse_wh is lapse * (1 - guess), the gap between the upper asymptote and 1, which psignifit and Palamedes report as the lapse rate. Attribute summary holds the posterior mean, the credible interval and the share of undefined draws per measure, p and condition.

Details

The threshold at target probability \(p\) solves \(\gamma + (1 - \gamma)(1 - \lambda) F(x) = p\). It exists only between the lower asymptote, guess, and the upper asymptote, 1 - lapse * (1 - guess). A draw whose asymptotes do not enclose p has no threshold: its threshold and slope are NA, the summary reports the share of such draws as undefined, and the function warns. Asking for 0.75 in a 2AFC task with guess = 0.5 is the usual case; asking for 0.5 is not, because no 2AFC observer falls below chance.

The slope is \(d\psi / dx\) at the threshold, in probability per unit of intensity.

Thresholds, slopes and rates come from the population-level effects only (re_formula = NA in brms::posterior_linpred()): group-level effects are left out, and re_formula cannot be passed through .... The closed form needs guess and lapse constant along the intensity axis, so a fit whose formula predicts either by the intensity is refused.

References

Wichmann, F. A., & Hill, N. J. (2001). The psychometric function: I. Fitting, sampling, and goodness of fit. Perception & Psychophysics, 63(8), 1293–1313. doi:10.3758/BF03194544

Examples

if (FALSE) { # \dontrun{
dat <- expand.grid(contrast = c(0.01, 0.02, 0.04, 0.08, 0.16),
                   cond = c("a", "b"))
dat$n_trials <- 60L
dat$n_correct <- rpsychometric_function(nrow(dat), dat$contrast,
                                        midpoint = ifelse(dat$cond == "a", 0.03, 0.05),
                                        width = 2, guess = 0.5, lapse = 0.02,
                                        n_trials = 60, sigmoid = "weibull")
fit <- bmm(
  bmf(midpoint ~ 0 + cond, width ~ 1, lapse ~ 1), dat,
  psychometric_function("n_correct", "contrast", "n_trials", sigmoid = "weibull",
                        guess = 0.5),
  cores = 4, backend = "cmdstanr"
)
thr <- psychometric_threshold(fit, p = c(0.75, 0.9))
thr

# the threshold difference between the conditions, per draw
wide <- subset(thr, measure == "threshold" & p == 0.75)
diff_ab <- wide$value[wide$cond == "b"] - wide$value[wide$cond == "a"]
quantile(diff_ab, c(0.025, 0.5, 0.975))
} # }