The method of constant stimuli was used in all experiments. The surface difference (
NΔ) was calculated as the difference between the number of the dots in the two surfaces (
N1,
N2) divided by the total number of dots (
Equation 1). The surface difference was zero, if both surfaces had the same number of dots; −100%, if all of the dots were in the second surface (
N2); and +100% if all of the dots were in the first surface (
N1).
The surface difference was −60, −40, −20, 0, 20, 40, and 60% except for Experiment 7. Here the standard surface had 64, 114, or 165 dots. The inducing surface had also 64, 114, or 165 dots. The surface difference between the standard and the comparison surface, which had to be compared by the observers, was −30%, −20%, −10%, 0%, 10%, 30%, and 60%. In this experiment, the psychometric functions were fitted to the number of dots in the comparison surface, in all other experiments to the surface difference between the two overlapping surfaces.
Cumulative Gaussian functions were fitted to the data, using the psignifit toolbox (Wichmann & Hill,
2001). The point of subjective equality (PSE) was defined as the mean of the cumulative Gaussian function. The just noticeable difference (JND) was defined as the standard deviation of the cumulative Gaussian function. Since we could not fit psychometric functions to all conditions in Experiment 1, we used a different analysis for this experiment. Here we used the proportion of backward choices and more numerous choices at a surface difference of zero. These proportions were arcsine-square-root transformed before they were submitted to statistical procedures. All
p values in ANOVAs are reported with Greenhouse–Geisser correction.
To quantify the magnitude of the directional biases in Experiments 1, 2, and 6, we calculated the root mean square error (RMSE) of the PSEs relative to the average PSE across all directions. The RMSE is zero, if the PSEs are identical for all directions and 100%, if PSEs are maximally different for all directions. For the proportion of choices in Experiment 1 the RMSE ranges from 0% to 50%.
To quantify the influence of the numerosity of the inducing surface (N
I) on the perceived numerosity of the standard surface (N
S) in Experiment 7, we fitted a model, which adds or subtracts a proportion (
a) of the dots in the inducing surface to the dots in the standard surface (
Equation 2).
Separate parameters a were determined for the two conditions in which the standard surface was in the front or in the back.