Distribution functions for the psychometric function model

Density and random generation for the psychometric_function() model: the number of positive responses ("yes", "correct", "longer") out of n_trials at a stimulus intensity, a binomial with probability $$\psi(x) = \gamma + (1 - \gamma)(1 - \lambda) F(x),$$ where \(F\) is the sigmoid with its midpoint and width.

Usage

dpsychometric_function(
  x,
  intensity,
  midpoint,
  width,
  guess = 0,
  lapse = 0,
  n_trials = 1,
  sigmoid = c("normal", "logistic", "gumbel_min", "gumbel_max", "weibull", "lognormal",
    "loglogistic"),
  log = FALSE
)

rpsychometric_function(
  n,
  intensity,
  midpoint,
  width,
  guess = 0,
  lapse = 0,
  n_trials = 1,
  sigmoid = c("normal", "logistic", "gumbel_min", "gumbel_max", "weibull", "lognormal",
    "loglogistic")
)

Arguments

x

Integer vector. Number of positive responses, between 0 and n_trials.

intensity

Numeric vector. Stimulus intensity. Must be positive for the sigmoids that read it on the log scale ("weibull", "lognormal", "loglogistic").

midpoint

Numeric. Intensity at which the sigmoid \(F\) is 0.5, in the units of intensity. Positive for the log-scale sigmoids.

width

Numeric, positive. Distance on the intensity axis between the points where \(F\) is 0.05 and 0.95; for the log-scale sigmoids, in log units of intensity.

guess

Numeric in [0, 1). Guess rate, the lower asymptote: 0.5 in 2AFC, 1 / m in m-AFC.

lapse

Numeric in [0, 1). Proportion of trials on which the observer does not attend and responds at the guess rate. The upper asymptote is 1 - lapse * (1 - guess).

n_trials

Integer vector. Number of trials per observation (default 1, a single binary response).

sigmoid

The sigmoid \(F\); see psychometric_function().

log

Logical. If TRUE, returns the log-density (default FALSE).

n

Integer. Number of observations to generate. intensity, n_trials and the parameters are recycled to this length.

Value

dpsychometric_function() returns the (log-)density, a binomial probability. rpsychometric_function() returns an integer vector with the number of positive responses per observation.

Parameter scales

These functions take every argument on the natural scale, while psychometric_function() estimates width on the log scale and guess and lapse on the logit scale (and midpoint on the log scale for the log-scale sigmoids). Transform a fitted value back before passing it here.

References

Schütt, H. H., Harmeling, S., Macke, J. H., & Wichmann, F. A. (2016). Painfree and accurate Bayesian estimation of psychometric functions for (potentially) overdispersed data. Vision Research, 122, 105–123. doi:10.1016/j.visres.2016.02.002

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

# Probability of 7 correct out of 10 2AFC trials at the midpoint
dpsychometric_function(7, intensity = 1, midpoint = 1, width = 2, guess = 0.5,
                       n_trials = 10)
#> [1] 0.2502823

# Simulate a method-of-constant-stimuli experiment with a Weibull observer
dat <- data.frame(contrast = rep(c(0.01, 0.02, 0.04, 0.08, 0.16), 4))
dat$n_correct <- rpsychometric_function(nrow(dat), dat$contrast, midpoint = 0.04,
                                        width = 2, guess = 0.5, lapse = 0.02,
                                        n_trials = 40, sigmoid = "weibull")
head(dat)
#>   contrast n_correct
#> 1     0.01        19
#> 2     0.02        21
#> 3     0.04        24
#> 4     0.08        40
#> 5     0.16        40
#> 6     0.01        23