Next, we pooled the response errors from all subjects and fitted the group data with a first derivative of a Gaussian function (DoG) (
Ceylan et al., 2021;
Collins, 2019;
Fischer & Whitney, 2014;
Fritsche & De Lange, 2019). The fits were separately conducted for the three types of trial pairs. The DoG is given by
\begin{eqnarray}
y = h + (x + b)awc{e^{ - {{(w(x + b))}^2}}}, \quad
\end{eqnarray}
where
y is the response error of the second trial of the trial pairs,
x is the orientation difference of the Gabor stimuli between the first and second trial of the trial pairs,
a is the amplitude of the DoG curve,
w is the width of the DoG curve,
h is the height,
b is the intercept, and
c is the constant
\(\sqrt 2 /{e^{ - 0.5}}\). The constant
c is chosen to ensure that the parameter
a numerically aligns with the peak amplitude of the DoG curve. The amplitude parameter
a represents the strength of serial dependence effects, indicating how much the orientation responses to the Gabor stimuli of the second trial of the trial pairs could be biased by the orientations of the Gabor stimuli on the first trial of the trial pairs. A positive
a amplitude indicates an attractive perceptual bias (i.e., serial dependence); a negative
a amplitude indicates a repulsive perceptual bias (i.e., a negative aftereffect) (
Gibson & Radner, 1937;
Thompson & Burr, 2009). The parameter
w was considered as a free parameter that we constrained to vary within a range of 0.02 to 0.07 (corresponding to the peaks of the DoG curve in the range of 10° to 35° orientation difference). The parameter
h is included to allow us to assess general response biases (independent of biases from stimulus history) (
Collins, 2019;
Collins, 2020;
Collins, 2021;
Collins, 2022). A positive or negative
h represents more responding biases of clockwise or counterclockwise orientations, respectively. In addition, the coefficient of determination (
R2) is used to evaluate how well the DoG model captures the data pattern.