Extended Data Fig. 7. Consistency of individual perceptual thresholds across ‘yes-no SOA’, ‘yes-no threshold’ and ‘2IFC’ experiments (cf. Supplementary Table 4).

Left (columns 1-3): Pairwise correlations of perceptual window estimates (in ms) between different experiments (rows), separately for no sound, one sound and two sound contexts (columns). Right (column 4): Bayes factors show the evidence for HA relative to H0 plotted on a log10-scaled ordinate. Purple lines indicate thresholds for substantial evidence favouring H₀ (< 1/3) or H_a (> 3)⁷⁷.

Separately for each sound context, we assessed the pairwise correlations between the perceptual thresholds from the ‘yes-no SOA’, the ‘yes-no threshold’ and the ‘2IFC’ experiments. In the ‘yes-no SOA’ and the ‘2IFC’ experiments the thresholds were obtained from psychometric functions fitted to percent correct scores as a function of SOA. In the ‘yes-no threshold’ experiment, they were obtained from adaptive staircases. Critically, only the threshold obtained from the ‘2IFC’ paradigm can be interpreted as a measure of sensitivity, because it depends only on the variance parameter of the underlying signal distribution, but not on a particular criterion. By contrast, the threshold obtained from ‘yes-no SOA’ or ‘yes-no threshold’ experiments depends on observers’ criterion (or bias) and the variance of the underlying signal distribution.

Moreover, it is important to emphasize that the adaptive staircases of the ‘yes-no threshold’ experiment adjusted the SOA of the two flashes or sounds individually for each participant to match their “one flash” and “two flash” reports independently for the stimulus combinations: ‘2 flash + 0 sound’; ‘2 flash + 1 sound’; ‘1 flash + 2 sound’. Thus, while the thresholds from the ‘yes-no SOA’ experiment depended jointly on the ‘1 flash’ and the ‘2 flash’ conditions, the thresholds from the adaptive staircases depended either on the ‘1 flash’ or the ‘2 flash’ condition alone in a particular sound context. In the ‘0 sound’ and ‘1 sound’ conditions this difference is not critical for understanding the relationship between the thresholds for the ‘yes-no SOA’ and the ‘yes-no threshold’ experiments, because the ‘1 flash + 0 sound’ and ‘1 flash + 1 sound’ conditions do not vary across SOAs. Hence, we simply add the same constant to the accuracy values across different SOA levels. However, for the ‘2 sound’ context, the percent correct score of the ‘2 flash + 2 sound’ conditions varies across SOAs thereby affecting the shape of the psychometric function for the ‘yes-no SOA’ paradigm. Moreover, consistent with previous research³², we observed substantial inter-participant variability for the ‘1 flash + 2 sound’ condition of the ‘yes-no SOA’ experiment. Some observers almost never experienced the double flash illusion irrespective of SOA. Others experienced the double flash illusion on almost every trial despite SOAs of 200 ms. Some participants were more likely to perceive the double flash illusion for large relative to small SOAs. Further, in several participants the thresholds (or perceptual windows) estimated from the psychometric function were at the bounds. These patterns illustrate that threshold parameters cannot be reliably estimated for the ‘2 sound’ condition in yes-no paradigms, because the experience of the double flash illusion is very susceptible to various sorts of biases. By contrast, the ‘2IFC’ paradigm presents on each trial one flash and two flashes in separate intervals together with 0, 1 or 2 sounds (i.e. the sound is uninformative about the number of flashes), so that observers decisions are not affected by these biases.

In summary, because the thresholds obtained from the different experiments have been estimated differently, we would not expect strong correlations, particularly not for the ‘2 sound’ conditions, for which only the ‘2IFC’ task can estimate reliable thresholds.

Consistent with this conjecture, we observed strong pairwise correlations for the ‘0 sound’ and ‘1 sound’ contexts. The correlations were particularly strong between the thresholds from the ‘yes-no SOA’ and the ‘yes-no threshold’ experiments that are both influenced by sensitivity and bias. The thresholds for the ‘2IFC’ experiment that reflect only sensitivity correlate slightly less but still significantly with those from the yes-no experiments.

In contrast to the strong correlations in the ’0 sound’ and ‘1 sound’ conditions, the threshold parameters estimated for the ‘2 sound’ conditions did not correlate across the ‘yes-no SOA’, ‘yes-no threshold’ or ‘2IFC’ experiment as indicated by Bayes Factors (BFs; column 4; for more detailed statistics see Supplementary Table 4). This lack of consistency for the ’2 sound’ condition mainly arises from the fact that observers’ “two flash” reports on ‘1 flash + two sound’ trials is susceptible to shifts in criterion within and across participants (see above), so that threshold could be reliably estimated only in the ‘2IFC’ paradigm.