2.4.3. Fitting Ur to confidence probability distributions and τ to meta-d’

The parameter τ determines how many time steps pass after the initial perceptual decision is made, during which evidence continues to accumulate before a confidence judgment is formed. Larger values of τ correspond to more evidence accumulation prior to confidence rating and therefore higher values of meta-d’ (i.e., confidence ratings that carry more information about task accuracy). Thus, empirically observed values of meta-d’ can serve as a guide for appropriate values of τ.

In each data set simulation, we fit τ to the meta-d’ value of only one representative data point, and used this value of τ for all subsequent simulations of that data set. This approach ensured that τ was held constant across all conditions. Importantly, as a consequence, the fitting procedure guaranteed a close fit to meta-d’ in only one data point, and simulated meta-d’ values at all other data points were unconstrained by further parameter fitting and instead arose as a consequence of the value of τ fitted to the representative meta-d’ value.

To accomplish the fit, we performed simulations at 10 evenly spaced values of τ and fitted a quadratic polynomial to the resulting meta-d’ vs τ curve, which provided an excellent fit across a range of meta-d’ values from ~ 0 –d’ (S1C and S1D Fig). Using the fitted polynomial equation, we could solve for what value of τ yielded the value of meta-d’ to be fitted for the single fitted data point.

In order to compute meta-d’, continuous confidence values (C_x or C_δ, depending on the model being used) first had to be converted to a 4-point rating scale (corresponding to the 4-point confidence rating scale used in all three data sets to be fitted), which in turn required specifying the values of the confidence thresholds U_r. For each simulation, we set U_r such that the probability distribution of simulated confidence ratings across all trials of all conditions exactly matched the corresponding empirical probability distribution. More formally, we computed U_r as

where C corresponds to C_x or C_δ, depending on the model being used, quantile(C, p) returns the quantile of the distribution C corresponding to the cumulative probability p, and P_data(conf = i) is the empirical probability distribution of confidence ratings.

As noted above, full details of the model fitting procedures are provided in S1 Text.