Computational models of perceptual decision-making and confidence judgements, grounded largely in statistical decision theory and SDT, have successfully accounted for a range of confidence related empirical data (
Kepecs et al., 2008;
Sanders et al., 2016;
Pouget et al., 2016;
Pleskac & Busemeyer, 2010). Here, we modeled perceptual decisions and confidence ratings within an extended SDT framework (
Maniscalco & Lau, 2012). This model assumes that, during yes/no detection or 2-AFC discrimination tasks, binary decisions are made by the comparison of internal evidence (indexed by a noisy decision variable [
dv]) with a decision criterion (
c). Across trials, evidence generated by each stimulus class (i.e., noise/signal, choice A/choice B) is sampled from a stimulus-specific normal distribution. The relative separation between the distributions (in standard deviation units) indexes the overall level of evidence available for the decisions (
dʹ) and, hence, how well the observer can discriminate between noise and signal or between choice A and choice B. On a given trial, the probability that the choice is correct is indexed by the absolute distance between
dv and
c (in an unbiased observer), and, hence, statistically optimal confidence judgements should reflect this computation (
Sanders et al., 2016;
Pouget et al., 2016). When a discrete confidence rating scale is used, the rating on a given trial is defined by where the
dv falls with respect to the so-called “type-2” criteria (
c2). The
c2 are response conditional, with separate criteria for the 2 possible choices (i.e., noise/signal, choice A/choice B). Overall, there are (k-1) × 2
c2, where k equals the number of confidence ratings available.
Figure 1B presents the model schematically for three differing levels of decision evidence: no evidence (left panel), weak evidence (middle panel), and strong evidence (right panel). The distributions and predicted effects in
Figure 1B–E were produced using code developed by
Urai et al., (2017) (
https://github.com/anne-urai/pupilUncertainty). The x-axis ranges from −15 to 15 in these examples, and
dʹ was set to 0.1 (no evidence), 1.58 (weak evidence), and 3.17 (strong evidence), whereas
c was always set to 0 (unbiased observer). The flanking
c2 were set at ±3 (conservative) and ±6 (liberal) for each. To formalize the predicted relationships between evidence strength, accuracy, and confidence (
Figure 1E), we simulated a normal distribution of
dv for one response (i.e.,
µ > 0) at each level of evidence strength. All samples from the simulated distribution were split into correct and error “choices” based on their position relative to
c. For each combination of evidence strength and choice, the level of confidence is
\begin{eqnarray*}
{\hbox{Confidence}} = \frac{1}{n}{\rm{\;}} \times \mathop \sum \limits_{i = 1}^n f\left( {\left| {d{v_i}} \right| - c|} \right)\end{eqnarray*}
where
f is the cumulative distribution function of the normal distribution
\begin{equation*}f\left( x \right) = \;\frac{1}{2}\;\left[ {1 + erf\left( {\frac{x}{{\sigma \sqrt 2 }}} \right)} \right]\end{equation*}
which transforms the distance between
dv and
c into the probability of a correct response (
Urai et al., 2017;
Lak et al., 2014). Ten million trials were simulated, and for each iteration a binary choice was computed along with its accuracy and corresponding level of confidence. Because response times are often taken as a proxy of decision confidence (with response times increasing as a function of decreasing confidence) (
Urai et al., 2017;
Sanders et al., 2016) the response time prediction (
Figure 1E) represents an inversion of the confidence prediction (
Figure 1D).