## 4 Results
### Compression Effect Quantification across Rendering Methods  
The influence of the distance-rendering method and of the environmental context (visual spatial boundary and volume) on the compression effect inherent to auditory distance perception was quantified with power functions, similarly to what is usually employed in studies implying the quantification of psychometric distance functions [13,29]. Power functions of the following form were fit (linear fit on logarithmic values) to the geometric mean reported distances with each rendering method:  
\[D=k*d^{a} \tag{5}\]  
where \(D\) is the perceived auditory distance, \(d\) is the sound-source distance, \(k\) a constant, and \(a\) a power-law exponent. For comparison purposes, the reported distances that are considered in the fitting result from a linear conversion of scaled reports made by participants on the visual analogue scale. The maximum of the visual scale (0% on the slider corresponds to the minimal distance of 0 meter and 100% to the maximum possible distance) was 7 m. The fitting parameters, \(k\) and \(a\), were used as measures of auditory distance perception accuracy. The exponent \(a\) indicates the amount of nonlinear compression (\(a<1\)) or expansion (\(a>1\)) and is equivalent to the slope when representing reported distances on a logarithmic scale. The constant \(k\) indicates the amount of linear compression (\(k>1\)) or expansion (\(k>1\)) in the function and is equivalent to the intercept when representing reported distances on a logarithmic scale. The fitting parameters \(k\) and \(a\) quantify the amount of compression in auditory distance perception: the closer they are to 1, the less compression there is. Hence, the fitting parameters are mentioned in the following as compression coefficients. The cross-correlation coefficient \(R^{2}\) can be used to assess the fit's quality. For each rendering method, a positive correlation was found between a fitted power function and the distances reported by each participant for the rendering method based on measurements (mean\((R^{2})=0.86;SD=0.12\)) and for the envelope-based model (mean\((R^{2})=0.84\); \(SD=0.11\)). The intensity-based model had a moderately positive correlation (mean\((R^{2})=0.73\); \(SD=0.28\)). The mean value of \(R^{2}\) obtained on reported distances of stimuli generated by the intensity-based model is significantly lower than the mean obtained by the envelope-based model. In that respect, stimuli rendered with the intensity-based model tend to produce higher intrasubject variability. To further investigate the similarity between the rendering method based on measurements and the envelope-based model at an individual level in both groups, each participant's compression coefficients were compared. In Figure 4, the individual compression coefficients \(a\) and \(k\) collected among the two groups are contrasted between those estimated for each model (ordinate) and for the rendering method based on measured  
Figure 3: Estimated room volume of participants in each group (CW group: \(M=37.5\) m\({}^{3}\); FW group: \(M=81.7\) m\({}^{3}\)). BRIRs (abscissa). For each distribution of dots, a linear regression analysis was run. In all eight analyses, the null hypothesis \"the coefficient of the regression slope is equal to zero\" was rejected (\(p<0.001\)). The distributions evaluating individual similarity between the envelope-based model and the measurements produce regression slopes close to 1, while the distributions related to the intensity-based model produce regression slopes significantly inferior to 1. The distribution of compression coefficients \(a\) and \(k\) illustrates the presence of a similarity, in terms of auditory compression effect, between the envelope-based model and the rendering method based on measurements.