**Visual crowding refers to a phenomenon whereby objects that appear in the periphery of the visual field are more difficult to identify when embedded within clutter. Pooling models assert that crowding results from an obligatory averaging or other combination of target and distractor features that occurs prior to awareness. One well-known manifestation of pooling is feature averaging, with which the features of target and nontarget stimuli are combined at an early stage of visual processing. Conversely, substitution models assert that crowding results from binding a target and nearby distractors to incorrect spatial locations. Recent evidence suggests that substitution predominates when target–flanker feature similarity is low, but it is unclear whether averaging or substitution best explains crowding when similarity is high. Here, we examined participants' orientation report errors for targets crowded by similar or dissimilar flankers. In two experiments, we found evidence inconsistent with feature averaging regardless of target–flanker similarity. However, the observed data could be accommodated by a probabilistic substitution model in which participants occasionally “swap” a target for a distractor. Thus, we conclude that—at least for the displays used here—crowding likely results from a probabilistic substitution of targets and distractors, regardless of target–distractor feature similarity.**

*n*= 1), 14 (

*n*= 2), or 16 (

*n*= 18) blocks of 64 trials for a total of 832, 896, or 1024 trials, respectively.

*μ*and concentration k with a uniform distribution with height

*ρ*: where

*x*is a vector of response errors in radians with range [−

*π*,

*π*].

*I*

_{0}is the modified Bessel function of the first kind of order 0:

*μ*(in radians) and

*k*(dimensionless units) correspond to the mean (center) and concentration of the von Mises distribution (respectively), and

*ρ*determines the height of a uniform distribution over the range (−

*π*,

*π*]. Concentration is a reciprocal measure of dispersion, and as

*k*increases, the von Mises begins to approximate a normal distribution with mean

*μ*and variance 1/

*k*(Abramowitz & Stegun, 1964; Fisher, 1993). Thus, we defined a measure of response variability as

*σ*= √

*k*− 1 with units in radians. Estimates of

*μ*and

*σ*were then converted to degrees by multiplying these values by (180/

*π*).

*σ*) should equal the precision of responses in the uncrowded condition (alternately, if the target and each distractor contributes an independent (noisy) orientation signal, then averaging could actually improve the precision of participants' responses). Conversely, any decrease in estimates of k (corresponding to a “broader” distribution of report errors and thus, lower precision) is inconsistent with feature averaging.

*μ*toward the pooled orientations of the distractors nearest to or furthest from fixation.

*τ*corresponds to the probability of reporting a distractor orientation; μ

*corresponds to the target orientation, and μ*

_{t}*corresponds to absolute distractor orientation relative to the target (i.e., 15°, 30°, 60°, or 90°);*

_{d}*k*and

*ρ*are as defined above. A trimodal distribution was used to account for the fact that in a given trial participants could report the orientation of the target, a counterclockwise flanker, or a clockwise flanker. For convenience, we assumed that targets and distractors would be reported with the same precision (

*σ*; estimated from

*k*using the method described above), and we fixed both μ

*and μ*

_{t}*at the target and flanker orientations, respectively, during fitting.*

_{d}*D*under this distribution, averaged over free parameters:

*L*(

*M*; Ω) =

*L*

*(*

_{max}*M*) = max

*L*(

*M*; Ω). Semicolons denote operations applied to the entire set of models M (i.e., m

_{1}, m

_{2}… m

_{n}). Subtracting

*L*(

_{max}*M*) avoids numerical problems by ensuring that the exponential in the integrand is of order 1 near the maximum likelihood value of Ω.

*p*(Ω

*|*

_{j}*M*), to be uniform over interval

*R*(see Table 1). That is, we iteratively computed the log likelihood of each model using all possible combinations of parameter values over the range

_{j}*R*(min) to

_{j}*R*(max) and selected the set of parameter values that maximized this quantity. Equation 5 thus becomes

_{j}μ | k | τ | ρ | |

Lower bound | −π/3 | 3 | 0 | 0 |

Upper bound | π/3 | 40 | 0.9 | 0.9 |

Increment | 0.01 | 0.1 | 0.01 | 0.01 |

*p*values as well as Bayes factors throughout the paper. Unlike traditional null-hypothesis significance testing, Bayesian analyses allow one to incorporate prior knowledge about the likely state of the world and to make precise statements about the likelihood of a hypothesis, including the null, given the observed data. Consider the following hypothetical scenario: We wish to compare a null model

*M*

_{0}with an alternative model

*M*

_{1}.

*M*

_{0}states that the true magnitude of the difference between two conditions is 0 whereas

*M*

_{1}states that the true magnitude of the difference is nonzero, and our a priori uncertainty about the true magnitude of the effect is a normal distribution over a plausible range of possible effect sizes. The a priori plausibility of model

*M*

_{1}to

*M*

_{0}is given by

*p*(

*M*

_{1})/

*p*(

*M*

_{0}). Given this ratio and the observed data, Bayes rule allows one to calculate the posterior odds of each model given the data:

*p*(

*D*|

*M*

_{1})/

*p*(

*D*|

*M*

_{0}) is called the Bayes factor (bf), and it describes the amount by which the data have changed the prior odds. For example, a bf of 10 indicates a 10-to-1 change in the prior odds favoring the alternative model. Conversely, a bf of 0.1 indicates a 10-to-1 change in the prior odds favoring the null model (see Edwards, Lindman, & Savage, 1963, and Kass & Raftery, 1995, for further information).

*μ*,

*σ*, and

*ρ*during crowded and uncrowded trials. These are shown in Table 2. Recall that in each crowded trial three distractors were tilted clockwise relative to the target in each trial while the remainder were tilted counterclockwise. If the orientations of the targets and distractors are pooled (averaged) at an early stage of processing (e.g., Parkes et al., 2001), then the effect of the distractors should cancel, and performance should equal levels observed in the uncrowded condition. To evaluate this possibility, estimates of

*μ*,

*σ*, and

*ρ*returned by the averaging model were subjected to separate one-way ANOVAs with condition (uncrowded, ±15°, ±30°) as the sole within-subjects factor. This analysis revealed a significant effect of condition on

*σ*(

*p*= 2.84e − 08, bf = 2.90e + 06; Figure 4C) and on

*ρ*(

*p*< 2.19e − 11, bf = 3.22e + 08; Figure 4E), but not on

*μ*(

*p*= 0.27, bf = 0.36; Figure 4A). Post hoc comparisons (repeated measures

*t*tests) revealed robust differences between estimates of

*σ*for the uncrowded and ±15° conditions (

*p*= 0.002, bf = 19.00), and the uncrowded and ±30° conditions (

*p*= 1.39e − 06, bf = 13,256). Estimates of

*σ*were also significantly different across the ±15° and ±30° conditions (

*p*= 0.003, bf = 103). Identical comparisons on

*ρ*revealed robust differences between the uncrowded and ±15° conditions (

*p*= 4.80e − 08, bf = 2.94e + 05), and the uncrowded and ±30° conditions (

*p*= 1.56e − 07, bf = 9.91e + 04). The difference between the ±15° and ±30° conditions was not significant (

*p*= 0.29, bf = 0.38).

μ | σ | ρ | |

No flankers | −1.37 [−3.60– −2.37] | 19.78 [16.58–22.90] | 0.08 [0.05–0.12] |

±15° | −2.00 [−3.09– −0.91] | 23.45 [21.73–25.30] | 0.26 [0.20–0.32] |

±30° | −0.97 [−2.29–0.40] | 27.63 [25.48–19.77] | 0.28 [0.22–0.34] |

*μ*to be systematically shifted away from 0 and toward the mean of the three distractors nearest or furthest from fixation in each trial. To evaluate this possibility, we conducted separate analyses in which we extracted the orientations of the three distractors either nearest or furthest from fixation in each ±15° and ±30° trial. We then sorted each participant's report errors into one of two bins, depending on whether the mean orientation of the flankers were tilted counterclockwise or clockwise relative to the target (data were pooled across ±15° and ±30° trials to ensure adequate statistical power). We then estimated

*μ*within each bin using Equation 1. False-discovery-rate-corrected repeated-measures

*t*tests revealed no effect of mean distractor rotation direction (i.e., clockwise or counterclockwise relative to the target) on

*μ*,

*σ*, or

*ρ*; both

*p*s > 0.70 and both bfs < 0.27. Qualitatively similar results were obtained when we included all target–distractor tilts in this analysis.

*r*

^{2}across participants and crowded conditions was 0.77; Figure 3C, D), we decided to examine the behavior of this model across the various crowded conditions. Estimates of

*σ*,

*τ*, and

*ρ*for each crowded condition (i.e., ±15° through ±90°) were subjected to a one-way ANOVA with target–flanker rotation as the sole within-subjects factor. These analyses revealed a significant effect of rotation on

*ρ*(

*p*< 2.47e − 07, bf = 5.48e + 04; Figure 4F), but no effect of rotation on either

*σ*(

*p*= 0.21, bf = 0.42; Figure 4D), or

*τ*(

*p*= 0.24, bf = 0.36; Figure 4B).

^{1}Inspection of Figure 3 suggests that increasing target–distractor rotations increased the likelihood of random responses but has no discernable effect on the likelihood of reporting a nontarget or the precision of participants' responses.

*μ*,

*σ*, and

*ρ*(for the averaging model) and

*σ*,

*τ*, and

*ρ*(for the substitution model) during crowded and uncrowded trials (separately for near and far trials) using the same logic outlined in Experiment 1.

*μ*in either the near or far condition (

*p*= 0.60 and 0.33, bf = 0.22 and 0.36, respectively). Conversely, tilt had a substantial effect on estimates of

*σ*during both near and far trials (

*p*= 3.30e − 05 and 8.45e − 04, bf = 560 and 61.29 for ±15° and ±60° trials, respectively). Post hoc analyses on

*σ*revealed substantially lower estimates during both ±15° and ±60° near trials relative to uncrowded trials (

*p*= 6.61e − 05 and 0.004; bf = 407.17 and 72.59, respectively), but the difference between ±15° and ±60° trials was not significant (

*p*= 0.93, bf = 0.24). Similar findings were observed for far trials. Specifically, we observed reliably lower estimates of

*σ*during ±15° and ±60° trials relative to uncrowded trials (

*p*= 2.94e − 06 and 0.004, bf = 6,697 and 11.55 for ±15° and ±60° trials, respectively). The difference between ±15° and ±60° trials was not significant (

*p*= 0.11; bf = 9.77). Finally, an ANOVA on estimates of

*ρ*revealed significant effects of target–distractor tilt during both near and far trials (

*p*= 3.18e − 06 and 3.50e − 11, bf = 4,881 and 6.71e + 08, respectively).

*σ*from uncrowded to both near and far crowded trials are inconsistent with a simple feature-averaging model in which items are averaged prior to reaching awareness. However, the current data can be accommodated by a substitution model with which targets and distractors are occasionally “swapped,” leading participants to report a distractor value as the target. Thus, we examined the behavior of this model in more detail. Specifically, separate 2 × 2 ANOVAs with target–flanker rotation (±15°, ±60°) and target–flanker distance (near, far) as within-participants factors were performed on estimates of

*σ*,

*τ*, and

*ρ*. For

*σ*, this analysis revealed a marginally significant main effect of distance (

*p*= 0.05) but no main effect of rotation (

*p*= 0.20) nor an interaction between these factors (

*p*= 0.76). A complementary Bayesian ANOVA revealed very modest support for a model containing only the main effect of distance (bf = 1.88 relative to the null hypothesis of no main effects nor an interaction).

*τ*revealed a main effect of target–flanker rotation, (

*p*= 7.42e − 04) but no main effect of distance (

*p*= 0.15) nor an interaction (

*p*= 0.77). A Bayesian ANOVA revealed strong support for a model containing only target–distractor rotation relative to a null model containing no main effects nor an interaction (bf = 1,919) but relatively weak support when compared to the next strongest model containing both main effects (bf = 1.72). Finally, the same analysis on estimates of

*ρ*revealed a main effect of rotation (

*p*= 4.33e − 04), a main effect of distance (

*p*= 8.17e − 05), and a significant interaction between these factors (

*p*= 1.18e − 04). A complementary Bayesian ANOVA revealed strong support for a model containing both main effects and their interaction (bf = 1.61e + 07 relative to a null model containing no effects and bf = 8.19 relative to the next strongest model containing both main effects but not their interaction). Visual inspection of Figure 7F suggests that random responses were more likely during near relative to far trials.

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