Does the asymmetry effect inflate the temporal expansion of odd stimuli?

We did not perform statistical analyses on TE because averaging the observed TEs would already introduce a bias. For example, assume that the PSE of two observers are 80 and 120 ms for a standard duration of 100 ms. Hence, the average CE for these two observers would be zero. By contrast assume that we average their TEs instead of their CEs, that is, (100/80 + 100/120)/2 = 25/24 > 1. Thus, the average CE is zero, the average TE would be larger than one. In fact, this bias would generally apply as can be seen by using Jensen’s inequality. Let x _i denote the PSE for participant i (i = 1…n) and note that the TE can be written as

\( f\left( {x_{i} } \right) = \frac{S}{{x_{i} }} \)

\( \frac{{\sum\limits_{i = 1}^{n} {f(x_{i} )} }}{n} \ge f\left( {\frac{{\sum\limits_{i = 1}^{n} {x_{i} } }}{n}} \right) \)

and hence

\( \frac{{\sum\limits_{i = 1}^{n} {\frac{S}{{x_{i} }}} }}{n} \ge \frac{S}{{{\raise0.7ex\hbox{${\sum\limits_{i = 1}^{n} {x_{i} } }$} \!\mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^{n} {x_{i} } } n}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$n$}}}}. \)

Thus, the mean TE for n participants is always larger than the TE of the mean PSE for n participants. As a consequence, averaging the TE introduces a bias that can mimic an overall oddball effect. We therefore only report the overall TEs as the standard duration divided by the mean PSE but do not perform an ANOVA on them.

These values are actually ideal. The actual standard durations employed in the experiment had to be in accordance with the refresh rate of the computer screen and were actually 73.3, 120, 193.3, 306.7, 493.3, 786.7, 1,253.3 and 2,000 ms. For the sake of clarity, however, we will use the ideal values in the main text. In general, we round all durations in the text to the nearest millisecond. .

The following values give the standard duration and in parentheses the corresponding comparison durations: 75 (33, 47, 60, 73, 87, 100, 113, 127, 140), 120 (40, 53, 67, 80, 93, 107, 120, 133, 147), 192 (33, 60, 87, 113, 140, 167, 193, 220, 247), 306 (67, 107, 147, 187, 227, 267, 307, 347, 387), 490 (213, 260, 307, 353, 400, 447, 493, 540, 587), 783 (347, 420, 493, 567, 640, 713, 787, 860, 933), 1,251 (813, 887, 960, 1,033, 1,107, 1,180, 1,253, 1,327, 1,400), and 2,000 (1,080, 1,233, 1,386, 1,540, 1,693, 1,847, 2,000, 2,153, 2,307). The comparison durations were chosen both to resemble the pattern of the values used by Tse et al. (2004) and to be compatible with the 150-Hz refresh rate of the monitor.

A separate ANOVA was performed on DL. As one should expect according to Weber’s law, DL increased with standard duration, F(7, 63) = 54.11, p < 0.001. The mean Weber fraction was 0.11 and this figure is consistent with the ones reported in the literature (e.g., Goodfellow, 1934; Grondin, 2001). Neither the main effect of procedure, F(1,9) = 0.16, p = 0.700, nor the interaction of the two factors, F(7,63) = 0.42, p = 0.732, were significant.

We conducted a third experiment (standard durations: 192, 783, and 1,251 ms), that further supports the robustness of the oddball effect. In this experiment (n = 12) the oddball was physically smaller than the standard. Specifically, this stimulus was a small stationary circle whereas the standards were stationary circles of a larger size. Since previous work (e.g., Long & Beaton, 1980; Thomas & Cantor, 1975) has shown that perceived duration increases with object size, it is possible that the effect observed in Experiment 1 and 2 is due to object size rather than to oddness. If this is the case, the effect should be eliminated or even be reversed in this experiment. However, the opposite pattern of results was found. Like in Experiment 2, CE increased with standard duration F(2,22) = 6.06, p = 0.022, and was significantly smaller than zero for standard durations of 786 and 1,251 ms (p’s < 0.05, two-tailed) but not for 192 ms. The temporal expansion factors were 1.00, 1.06 and 1.11 for standard durations of 192, 783, and 1,251 ms, respectively. This result replicates the finding from Tse et al. (2004) and from Experiment 2 that participants judge the duration of the oddball longer than the one of the standard, at least, when the duration of the standard is sufficiently long. We acknowledge that Tse et al. (2004) have already demonstrated the confounding effect of stimulus size to be rather unlikely. In their Experiment 2 they reversed oddball and standard stimulus, so that the oddball was of smaller size than the initial radius of the expanding standard. In this experiment and also with auditory stimuli, they still were able to reveal an oddball effect. Our third experiment, however, broadens the database for visual stimuli by reaffirming the oddball effect for small stationary oddballs compared to larger stationary standard stimuli.