**Visual heading estimation is subject to periodic patterns of constant (bias) and variable (noise) error. The nature of the errors, however, appears to differ between studies, showing underestimation in some, but overestimation in others. We investigated whether field of view (FOV), the availability of binocular disparity cues, motion profile, and visual scene layout can account for error characteristics, with a potential mediating effect of vection. Twenty participants (12 females) reported heading and rated vection for visual horizontal motion stimuli with headings ranging the full circle, while we systematically varied the above factors. Overall, the results show constant errors away from the fore-aft axis. Error magnitude was affected by FOV, disparity, and scene layout. Variable errors varied with heading angle, and depended on scene layout. Higher vection ratings were associated with smaller variable errors. Vection ratings depended on FOV, motion profile, and scene layout, with the highest ratings for a large FOV, cosine-bell velocity profile, and a ground plane scene rather than a dot cloud scene. Although the factors did affect error magnitude, differences in its direction were observed only between participants. We show that the observations are consistent with prior beliefs that headings align with the cardinal axes, where the attraction of each axis is an idiosyncratic property.**

*absence of a feeling of self-motion*, and 7 =

*compelling sensation of self-motion*.

^{3}and dots had a size of 5 cm. Dot density in the plane was 50 dots/m

^{3}and dots had a size of 2.5 cm. Dimensions, size, and density of the dots were based on pilot sessions. They were chosen so that their size, number, and the resulting impression of depth were comparable across conditions. Dots in both scenes were bright gray on a dark gray background to improve the efficacy of the stereoscopic display and prevent ghosting.

*v*(

*t*) was

*ω*= 2

*π f*,

*f*= 0.5 Hz, and maximum velocity

*v*

_{max}= 0.15 m/s. Both motion profiles lasted 2 s and both had a displacement of 0.15 m.

^{4}× 16 × 3 = 768 experimental trials.

*r*and stimulus heading

*θ*(in radians):

*i*is the imaginary unit. We interpret values further away from the fore-aft axis than the stimulus heading angle as overestimations.

*ν*over

*r*as obtained in the

_{n}*n*repetitions of each experimental condition as

*ν*unnecessarily. Conditions for which not enough observations remained to estimate the standard deviation were also excluded from further analyses.

*π*rad). Figure 3 below illustrates this observation. To test this formally, we fitted a unimodal and a bimodal model to the data and compared their fits using the adjusted Akaike Information Criterion (AIC

_{c}). In the unimodal model, responses

*r*were considered a Von Mises distributed random variable:

*μ*and concentration parameter

*κ*as

*θ*is the heading angle. The model parameters for

*μ*,

*β*

_{0}, and

*β*

_{1}reflect the constant error in heading estimates. Specifically, they reflect a constant offset and the strength of the bias, respectively. The parameters for

*κ*,

*γ*

_{0}, and

*γ*

_{1}reflect the constant and periodic part of the variable error. This model was adopted from (de Winkel et al., 2015, 2017). The corresponding density function is specified as

*μ*was shifted by 180° (

*π*rad), with parameter P

_{inv}that specifies the weight of the nonshifted component. The density function is

*β*

_{0},

*β*

_{1},

*γ*

_{0},

*γ*

_{1}, and P

_{inv}were treated as free parameters, and their values were estimated by minimizing the model negative log-likelihood (i.e., by the method of maximum-likelihood; Fisher, 1925). Parameter

*β*

_{1}, indicative of the direction of constant errors, was smaller than 1 (indicating underestimation of heading angle) for five out of 18 participants (Participants 4, 5, 8, 9, and 18), and larger than 1 (indicating overestimation) for the others. All estimated coefficients are available in Supplementary Table S1.

_{c}(Supplementary Table S2) scores indicated that responses should indeed be treated as coming from a bimodal distribution for 15 out of 18 participants. Within this group, the minimum difference between AIC

_{c}scores was 89.83, providing decisive evidence in favor of the bimodal model (

*p*= 3.12 × 10

^{–20}; Motulsky & Christopoulos, 2004). For the remaining participants (Participants 1, 12, and 14), responses were unimodal. The percentage of shifted trials varied between individuals, ranging from 0% to 24.22% (186 out of 768 trials), with an average of 3.16%, and totaled 485 out of 13,824 trials.

*β*

_{1}near 0, which implies that responses were not related to the stimulus. These two participants were excluded from further analysis.

*df*) that correspond to the omission of paths that specify interaction effects between heading and the factors on vection. The path model structure is presented in Figure 4.

*θ*and errors in heading estimation, the value of

*θ*was transformed into sin 2

*θ*before being used in the analysis of component membership and constant error, and into

*θ*variables and each of the factors were included to specifically address the hypotheses on the nature of errors in heading estimation.

*θ*as well as the interaction terms were initially included to predict both the dependent variable and the vection rating. However, these models did not provide a very good fit and suggested that vection rating was not affected by these variables. Because these relations were also not specifically hypothesized, they were not included in the version of the models presented here.

*ν*are calculated for the group of participants who overestimated heading, for the condition where the scene has a level of 1 (dot cloud), and the other factors have level 0. The example uses only coefficients that were found to be significant in the analysis. Constant errors

*θ*. The analysis of

*ν*was performed on log-transformed data (

*ν*); therefore, we performed the inverse operation to calculate the standard deviations in degrees. For the present example, the function describing the variable error is:

_{T}*ν*= exp(2.031 – 0.240 + (0.337 + 0.358)

*df*(5 × 2) and 1536 observations. The

*χ*

^{2}test indicated that the null hypothesis that the model fits the theory should not be rejected:

*χ*

^{2}(10) = 5.750,

*p*= 0.836. Averaged over participants, the regression for component membership yielded an

*R*

^{2}of 0.135 (individual values: 0.048, 0.223) and a residual variance of 1.138 (values: 1.013, 1.263); the regression for vection yielded an

*R*

^{2}of 0.065 (values: 0.003, 0.123) and a residual variance of 1.070 (values: 0.718, 1.422). Overall model fit indices indicated that the model fitted the data well: Root Mean Square Error of Approximation (RMSEA) = 0.000, Confirmatory Fit Index (CFI) = 1.000, Standardizes Root Mean Square Residual (SRMR) = 0.003 (Kline, 2005).

*θ*and constant errors brought about by the different levels of the factor. The model was fitted to the data of two groups separately, where the groups correspond to participants who were found to either over- (Group 1) or underestimate (Group 2) heading in the exploratory analysis, excluding trials where responses appeared shifted. Combined, this yielded a total of 10

*df*(5 × 2) and 13,339 (Group 1: 9,758, 13 participants; Group 2: 3581, five participants) observations. The

*χ*

^{2}test indicated that the null hypothesis that the model fits the theorized relations should not be rejected:

*χ*

^{2}(10) = 6.562,

*p*= 0.766. Overall model fit indices indicated that the model fitted the data well: RMSEA = 0.000, CFI = 1.000, SRMR = 0.003 (Kline, 2004).

*R*

^{2}of 0.141 and residual variance of 236.870; for Group 2 these values were 0.038 and 249.270. The regressions for vection yielded

*R*

^{2}values of 0.080, 0.013 and residual variances of 1.560, 2.576, for Groups 1 and 2, respectively.

*θ*had a coefficient of 9.847. Interaction effects with

*θ*were observed

*θ*and its interactions ranged between 3.004 and 12.202. This indicates positive bias, regardless of experimental condition. For Group 2, the coefficient for

*θ*was not significant, but interactions were found between

*θ*and FOV (–1.509), disparity (–4.050) and scene (–2.079). The composite coefficient for

*θ*and its interactions thus ranged –7.638 to 0.00, indicating that this group underestimated heading in some conditions, and did not show apparent bias in others. Notably, the coefficient for the factor scene had an opposite sign in the latter group. No mediating effects of vection were found.

*p*values is available in Supplementary Table S4.

*ν*were log-transformed before analysis to make their distribution approximately normal. The transformed variable is designated

*ν*. The model was again fitted separately to the data of participants who over- (Group 1) and underestimated (Group 2), as per the exploratory analysis. Combined, there were a total of 10

_{T}*df*(5 × 2) and 4,536 (Group 1: 3,312, 13 participants; Group 2: 1,224, five participants) observations. The

*χ*

^{2}test indicated that the null-hypothesis that the model fits the theorized relations should not be rejected:

*χ*

^{2}(10) = 4.736,

*p*= 0.908. Overall model fit indices indicated that the model fitted the data well: RMSEA = 0.000, CFI = 1.000, SRMR = 0.003 (Kline, 2004).

*R*

^{2}and a residual variance of 0.039 and 0.721, respectively, and for Group 2, these values were 0.046 and 0.866; the regression for vection yielded

*R*

^{2}and residual variances for Group 1 of 0.129, 0.958 and of 0.022, 1.512 for Group 2.

*θ*was found to affect the size of the error in Group 1, with a coefficient of 0.337, and an interaction effect between

*θ*and scene was present in both groups (0.358, 0.776, for Groups 1 and 2, respectively). This indicates that for those who overestimate heading angle, a heading dependency of variable errors is visible for all factors, and it is amplified for the dot cloud scene. For the group of participants who underestimated heading angle, a heading dependency of variable errors was only visible for the dot cloud scene. No mediating effect of vection was found.

*p*values is available in Supplementary Table S5.

*θ*):

*κ*. The prior expresses beliefs on how probable the occurrence of any stimulus is. The general shape of the prior was derived from previous research showing that visual estimation of orientation is biased toward the cardinal axes (Girshick, Landy, & Simoncelli, 2011). Specifically, we used an equal-weights convex combination of four Von Mises functions, with peaks (component means) aligned with the north (N), east (E), south (S), and west (W). Here north and south lie on the fore-aft axis, with north equivalent to straight ahead; and east and west lie on the interaural axis. The peakedness of each component, as expressed by concentration parameters

_{x}*κ*(with

_{i}*i*= N, E, S, W), was allowed to vary. Because biases in heading estimation are symmetrical around the origin, we imposed the additional constraint that

*κ*

_{E}=

*κ*

_{W}. The general shape of the prior is visualized in Figure 6.

*κ*

_{E,W}= 11.05) and a minor contribution of the north peak (

*κ*

_{N}= 0.86); for those who underestimated heading angle, the data were best described using a prior with peaks on the fore-aft axis (

*κ*

_{N}= 4.67;

*κ*

_{S}= 3.44).

_{c}≥ 59.19; median

*κ*

_{E,W}= 5.99, average

*R*

^{2}= 0.372); a model with peaks on the fore-aft axis best explained the data for three other participants (ΔAIC

_{c}≥ 8.50; medians

*κ*

_{N}= 4.05,

*κ*

_{S}= 3.22, average

*R*

^{2}= 0.092). For the remaining 11 participants, the data were best described by a model with four peaks (ΔAIC

_{c}≥ 10.47; medians

*κ*

_{E,W}= 5.90,

*κ*

_{N}= 2.23,

*κ*

_{S}= 0.37, average

*R*

^{2}= 0.222). The patterns of bias resulting from the different priors are illustrated in Figure 7.

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