Predictions of the Demixing Model.

A. Model outline. The Demixing Model assumes that the observer receives noisy signals from multiple items (color patches S1 and S2, left). These signals are mixed in a common representational space (middle), so stimulus features (e.g., color) must be reconstructed by attributing signals to items, i.e., “demixing” them. Even an optimal solution is imperfect: some signals generated by one item are attributed to the other (right; dot outline – true source, fill color – attributed source; uncertainty in assignments is omitted for clarity). As a result, the estimated stimulus values (dashed outlines) systematically shift relative to the true ones (opaque circles), producing behavioral biases. B-D. Non-trivial predictions of the model emerge when the two stimuli differ in noise: the less noisy item is attracted toward the noisier one, while the noisier item is repelled away from the less noisy one. B. These predictions can be intuitively understood by considering the overlapping distributions of signals coming from the two stimuli (dashed lines). Samples (dots) from the noisier stimulus (orange) that fall close to samples from the less noisy one (blue) are likely to be attributed to the less noisy item (fill color – attributed source, outline – true source). As a result, the estimated distributions (solid lines) shift: the noisier stimulus is repelled from the less noisy one, while the latter is attracted to the former. C. The resulting average bias pattern across different similarity levels shows more attraction for the less noisy stimulus and more repulsion for the noisier one when the noise levels of the two items are unequal (“unequal noise”). This difference is more pronounced than when the target and non-target have the same noise level (“equal noise”). D. When the target noise level is kept constant, varying the non-target noise level leads to more attraction or repulsion depending on whether the target has a lower or a higher noise level relative to the non-target. Note that the absolute magnitude and direction of biases also depend on other factors (see Methods and Fig. S1).

Experimental procedure and results of Experiments 1-2.

A. All experiments in this paper used the same general procedure. Each trial started with a brief fixation, followed by two mosaic stimuli composed of small squares with varying hues. The hues of each stimulus were drawn from a normal distribution centered on a randomly selected hue. After a short delay, a cue (white circle) indicated which item to report. Participants reported the average hue of the cued item. In Experiments 2–4, cue presentation and report were then repeated for the second item (not shown). B. Overall performance in both experiments changed as a function of target noise, whereas non-target noise (and thus the noise equality condition) did not affect error magnitude. Bars show average absolute errors; error bars show 95% confidence intervals. C-D. Biases in Experiments 1–2 as a function of target–non-target dissimilarity, target noise level, and the match between target and non-target noise. In the equal noise condition, target and non-target had the same noise level. In the unequal noise condition, non-targets had low noise when targets had high noise, and vice versa. C. Continuous bias estimates, with the human data shown as solid lines and model fits as dashed lines; shaded areas show 95% confidence intervals for the mean bias. Gray regions indicate dissimilarity ranges with a significant interaction between target noise and noise equality. D. Average biases in the significant clusters in C for each participant (empty circles), their distributions within each condition (slabs), and condition means with 95% confidence intervals (solid circles and bars). Positive bias corresponds to attraction, negative to repulsion.

Biases in Experiment 3 as a function of dissimilarity and the noise levels.

A. Average biases. In the equal noise condition, target and distractor had the same noise level. In the unequal noise condition, distractors had low noise when targets had high noise, and vice versa. Replicating previous results, in the unequal noise condition low-noise targets were more strongly attracted than high-noise targets, whereas in the equal noise condition the opposite pattern was observed. B. Error probabilities as a function of error magnitude. To test whether attractive biases were driven by swap errors, we computed an asymmetry index (difference in probability of attractive minus repulsive errors, y-axis) for each error magnitude (x-axis). That is, for each possible error magnitude (the absolute value of response deviation from the stimulus), we measured how much more likely responses were to fall toward the non-target than away from it. The interaction between target noise and noise equality is already present for relatively small errors, which is inconsistent with swap errors being the main driving factor. Positive bias and asymmetry index correspond to attraction, negative to repulsion. Lines show mean asymmetry with shaded areas indicating 95% confidence intervals; gray regions indicate dissimilarity ranges with a significant interaction between target noise and noise equality.

Biases in Experiment 4 as a function of dissimilarity and the relative noise levels.

A. Average biases. Target noise level was kept constant while non-target noise varied, so the target could have higher, equal, or lower noise than the non-target. In line with the model predictions, targets were repulsed more when they had higher noise than the non-target and attracted more when they had lower noise. Points show mean biases with 95% confidence intervals (bars). B. As in Experiment 3, we estimated an asymmetry index in error probability (y-axis) for errors of different magnitudes (x-axis). That is, for each possible error magnitude (the absolute deviation of the response from the target), we calculated the difference in the probability of errors toward the non-target versus away from it. In the 45° bin, which showed the largest attractive bias in the lower-noise condition, the effect of target noise is already present for relatively small errors, inconsistent with swap errors being the main driving factor of the differences between conditions. Positive bias and asymmetry index correspond to attraction, negative to repulsion. Lines show mean error asymmetry with shaded areas indicating 95% confidence intervals; gray regions indicate clusters with a significant effect of target noise.

Predictions of the Demixing Model for varying levels of noise for the mixture components.

Each column shows the average bias for the case when stimuli noise in the unequal noise case corresponds to the values shown at the top. In the equal noise case, the two components have equal noise levels corresponding to one of these two values. Color corresponds to the target noise level, which is either the same (‘equal noise’) or different (‘unequal noise’) from the non-target.

Example bias estimating procedure for a single participant in a single condition.

A The response probability density (solid line) in the case of biased responses is often asymmetric. This asymmetry is clearly visible when comparing a symmetric version of the same distribution computed by mirroring the positive side of the distribution to the negative values (dashed line). The difference between the two distributions is shown with a shaded area. B-C When dissimilarity between target and non-target varies, it might affect the strength and direction of biases. To obtain a continuous estimate of bias as a function of dissimilarity, we computed asymmetry in response probabilities for each 1° step of dissimilarity using weighted probability density estimates, highlighted here for four arbitrary points (1, 25, 50, and 75°). The weights were computed as a Gaussian distribution with the mean at the dissimilarity point and a standard deviation of 20°. B. The weights of individual response errors (points), with more transparent points indicating lower weights. C. The resulting bias curve as a function of dissimilarity, with 4 red points corresponding to four panels in B.

Biases within the first and the second response.

In Experiments 2 and 3, we found that response order (first vs. second response) affects the overall balance between attraction and repulsion but does not interact with the other factors. In Experiment 2, the data from the first and the second response were used to create bias curves, which were then subject to the same analysis as the combined bias curves in the main text. The same two clusters with the interaction between target noise and noise equality as in the combined analyses were identified (from 20° to 73°, pperm = .057; from 101° to 162°, pperm = .031). However, no significant clusters that involved response order or its interactions with other factors were observed. Additional analyses indicated that, overall, biases in Experiment 2 were more attractive in the second response (F(1, 15) = 17.72, p < .001, η2G = .17), but, as the figure shows, that overall shift did not change the effect of target noise or noise equality. In Experiment 3, we also repeated the analyses in the main text with the response order factor included. We found the same two interactions reported in the overall analysis, between target noise and noise equality (F(1, 22) = 29.19, p < .001, η2G = .02) and between noise equality and the similarity bin (F(2, 36) = 3.66, p = .026, η2G < .01). The effect of response order was also significant, with more attraction for the second response (F(1, 22) = 22.04, p < .001, η2G = .06), but it did not interact with other factors. In sum, the second response shows more attraction, but the key interaction between the target noise and noise equality does not depend on it. Lines show the mean bias, and bars and shaded regions show 95% CIs.

Biases in Experiment 3 for small (<10°) errors.

As an additional test of the role of swap errors, we also repeated the analysis of biases but using only the errors below 10° in absolute value, which cannot be interpreted as swap errors even in the 20° similarity bin. Including only small errors in the analysis did not change the observed effects pattern, suggesting that the effect of target noise and noise equality on attractive biases in Exp. 3 is not likely to be explained by swap errors. Similar to the analyses of the full dataset, reported in the main text, we observed an interaction between the target noise level and noise equality (F(1, 22) = 7.82, p = .011, η2G = .03), further qualified by a three-way interaction with the similarity bin (F(2, 44) = 3.80, p = .030, η2G = .03).