diff --git a/README.md b/README.md index fcff272..6261baa 100644 --- a/README.md +++ b/README.md @@ -25,6 +25,37 @@ corresponding model provides a better fit to the data. The following models are - Lognormal weighted evidence and visibility model +## Mathematical description of models +The computational models are all based on signal detection theory (Green & Swets, 1966). It is assumed that participants select a binary discrimination response \eqn{R} about a stimulus \eqn{S}. +Both \eqn{S} and \eqn{R} can be either -1 or 1. +\eqn{R} is considered correct if \eqn{S=R}. +In addition, we assume that there are \eqn{K} different levels of stimulus discriminability +in the experiment, i.e. a physical variable that makes the discrimination task easier or harder. +For each level of discriminability, the function fits a different discrimination +sensitivity parameter \eqn{d_k}. If there is more than one sensitivity parameter, +we assume that the sensitivity parameters are ordered such as \eqn{0 < d_1 < d_2 < ... < d_K}. +The models assume that the stimulus generates normally distributed sensory evidence \eqn{x} with mean \eqn{S\times d_k/2} +and variance of 1. The sensory evidence \eqn{x} is compared to a decision +criterion \eqn{c} to generate a discrimination response +\eqn{R}, which is 1, if \eqn{x} exceeds \eqn{c} and -1 else. +To generate confidence, it is assumed that the confidence variable \eqn{y} is compared to another +set of criteria \eqn{\theta_{R,i}, i=1,2,...,L-1}, depending on the +discrimination response \eqn{R} to produce a \eqn{L}-step discrete confidence response. +The number of thresholds will be inferred from the number of steps in the +`rating` column of `data`. +Thus, the parameters shared between all models are: +- sensitivity parameters \eqn{d_1},...,\eqn{d_K} (\eqn{K}: number of difficulty levels) +- decision criterion \eqn{c} +- confidence criterion \eqn{\theta_{-1,1}},\eqn{\theta_{-1,2}}, +..., \eqn{\theta_{-1,L-1}}, \eqn{\theta_{1,1}}, \eqn{\theta_{1,2}},..., +\eqn{\theta_{1,L-1}} (\eqn{L}: number of confidence categories available for confidence ratings) +How the confidence variable \eqn{y} is computed varies across the different models. + +### \strong{Signal Detection Rating Model (SDT)} +According to SDT, the same sample of sensory evidence is used to generate response and confidence, i.e., +\eqn{y=x} and the confidence criteria span from the left and +right side of the decision criterion \eqn{c}(Green & Swets, 1966). + ## Installation The latest released version of the package is available on CRAN via