To examine whether serial dependence was present in the orientation estimation, we first calculated the relative orientation. The relative orientation was the difference in the actual orientation between the previous first trial (
n − 1
th trial,
n = 2, 3, etc.) and the current trial (
nth trial). Then, the estimation bias was the mean estimation error of the current trials in each relative orientation. For example, the −170° relative orientation had two possibilities: the difference between the −85° previous trial and the 85° current trial (i.e., −85° to 85° = −170°) or the difference between the −80° previous trial and the 90° current trial (i.e., −80° to 90° = −170°), so the orientation bias was the mean of orientation errors of 85° and 90° current trials. Like the previous studies (e.g.,
Frische & Whitney, 2014), we fitted the estimation bias (EB) as a first derivative of Gaussian function (DoG) of the relative orientation (RO), given as:
\begin{eqnarray}
{\rm{EB}} &\;=& {{\rm{\alpha }}_{EB}} \times {\rm{\ w}} \times RO \times \sqrt 2 /{e^{ - 1/2}}\nonumber\\
&& \times {e^{\left( { - {{\left( {w \times RO} \right)}^2}} \right)}} + \varepsilon\quad
\end{eqnarray}
in which α
EB indicates the amplitude of the DoG curve. In addition, a positive α
EB indicates that the perceived orientation is biased toward the previously seen orientation, showing an attractive serial dependence; in contrast, a negative α
EB indicates that the perceived orientation is repelled away from the previously seen orientation, showing a repulsive serial dependence. To test the significance level of α
EB, a bootstrapping test was conducted. Specifically, we sampled the data with replacement for 10,000 times and fitted the sampled data with
Equation 1, which generated a distribution of α
EB and got its 95% confidence interval (CI). If 0 was smaller than the lower bound of the 95% CI, then α
EB was positive, indicating an attractive serial dependence; vice versa.