# The Random Step Method for Measuring the Point of Subjective Equality

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Participants

#### 2.2. Apparatus

#### 2.3. Algorithm

#### 2.4. Stimuli

#### 2.5. Procedures

#### 2.6. Data Analysis

## 3. Results

#### 3.1. Simulations

#### 3.2. Psychophysics

#### 3.3. Control Experiment

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Anaglyph representation of the stimulus (for representation only, the actual stimulus was presented using a polarized passive 3D screen). It consists of 200 Gabor patches that are oscillating horizontally and presented dichoptically. As an interocular phase shift is introduced, the stimulus is seen as a rotating cylinder in depth using 3D glasses. (

**b**) Schematic illustration of the direction of rotation of the cylinder. The phase difference in the oscillation of the Gabor patches between the eyes generates different percepts of the rotating direction of the motion-defined cylinder in depth. When the phase shift is negative, anticlockwise rotation is perceived; when the phase shift is positive, clockwise rotation is perceived; when the phase is null or close to zero, an ambiguous oscillation in the plane is perceived.

**Figure 3.**(

**a**) The distribution of test levels from the simulation of 50,000 trials with a noise level σ = 2. The black curve represents a Gaussian + offset fit. (

**b**) The psychometric function from the simulation is a function of test levels. The size of the circles indicates the number of trials tested at each level.

**Figure 4.**Simulations with random step (RS), constant stimuli (CS), and staircase (SC) methods under different noise levels (2, 3.75, and 5). (

**a**) average psychometric functions. RS: red circles; CS: green squares; SC: blue hexagrams. The size of the symbols indicates the number of trials applied at each test level. (

**b**) the distribution of trials from the RS, CS, and SC methods in one cycle. In each condition, 10,000 simulation cycles of 100 test trials were implemented with a pivot PSE of 13. SDs were estimated from bootstrapping each of those simulations.

**Figure 5.**(

**a**) normalized histogram of PSE estimates. (

**b**) normalized histogram of slope estimates. In each condition, 10,000 simulation cycles of 100 test trials were implemented under different noise levels (2, 3.75, and 5). The expected pivot of PSE was 13, marked out with a black dotted line.

**Figure 6.**(

**a**) normalized histogram of the estimated standard deviations (SD) of the PSE. (

**b**) normalized histogram of the estimated standard deviations (SD) of the slope. The failure rates shown on the right side of each histogram refer to invalid SD estimation. The mean of the standard deviation of the PSE (

**c**) and slope (

**d**) estimated from bootstrapping of the simulations as a function of the noise level. Error bars represent SD.

**Figure 7.**Psychometric functions of one representative subject for the 3 viewing conditions (−1ND, 0ND, and 1ND) using the RS (red) and CS (green) protocols, with 75 (top row), 150 (middle row) and 300 trials (bottom row). The size of the red circles represents the number of trials involved at each phase level in the RS method.

**Figure 8.**Distribution of the number of trials in the RS method for the different phase shift levels (not isometric) under the 3 viewing conditions (−1ND, 0ND, and 1ND) with 75 (top row), 150 (middle row), and 300 trials (bottom row). These figures were plotted from the same subject’s data in Figure 7. The red curve in each graph represented a Gaussian + offset fit to the distribution.

**Figure 9.**(

**a**) correlation of the PSE estimates of 4 subjects with the RS and CS methods. (

**b**) the Bland–Altman plot of the data between the RS and CS methods. (

**c**) correlation of the PSE estimates of 3 subjects between two repetitions within the RS method. (

**d**) the Bland–Altman plot of the data within the RS method. PSE estimates were from the 3 viewing conditions and the 3 sets of numbers of trials tested. Different markers represented individual subjects. Gray markers indicate 0ND condition. The black lines in the scatter maps represent the identity, and the red line represents linear regression.

**Figure 10.**Boxplots of the distribution of PSE estimates using RS and CS protocols as a function of viewing conditions (−1ND, 0ND, and 1ND) at (

**a**) 75, (

**b**) 150, and (

**c**) 300 trials. Horizontal lines and crosses in the boxes represent the median and mean values of the PSE estimates. Error bars represent the standard deviation of the PSE estimates.

**Figure 11.**(

**a**) Distribution of SDs of the PSE for 75, 150, and 300 trials between RS and CS protocols, respectively. (

**b**) Distribution of SDs of the PSE in total between RS and CS protocols. (

**c**) Distribution of SDs of the slope for 75, 150, and 300 trials between RS and CS protocols, respectively. (

**d**) Distribution of SDs of the slope in total between RS and CS protocols. Horizontal lines and crosses in the boxes represent the median and mean values of PSE estimates, respectively.

**Figure 12.**Boxplots of the distribution of PSE estimates using RS, CS, and SC protocols as a function of viewing conditions (−1ND, 0ND, and 1ND) at (

**a**) 15, (

**b**) 30, and (

**c**) 45 trials. Horizontal lines and crosses in the boxes represent the median and mean values of the PSE estimates. Error bars represent the standard deviation of the PSE estimates.

**Figure 13.**(

**a**) Distribution of SDs of the PSE for 15, 30, and 45 trials among RS, CS, and SC protocols, respectively. (

**b**) Distribution of SDs of the PSE in total among RS, CS, and SC protocols. (

**c**) Distribution of SDs of the slope for 15, 30, and 45 trials among RS, CS, and SC protocols, respectively. (

**d**) Distribution of SDs of the slope in total among RS, CS, and SC protocols. Horizontal lines and crosses in the boxes represent the median and mean values of PSE estimates, respectively. P-values less than 0.05 or 0.01 were flagged with one asterisk (*) or 2 asterisks (**), respectively.

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**MDPI and ACS Style**

Wang, P.; Reynaud, A.
The Random Step Method for Measuring the Point of Subjective Equality. *Vision* **2023**, *7*, 74.
https://doi.org/10.3390/vision7040074

**AMA Style**

Wang P, Reynaud A.
The Random Step Method for Measuring the Point of Subjective Equality. *Vision*. 2023; 7(4):74.
https://doi.org/10.3390/vision7040074

**Chicago/Turabian Style**

Wang, Penghan, and Alexandre Reynaud.
2023. "The Random Step Method for Measuring the Point of Subjective Equality" *Vision* 7, no. 4: 74.
https://doi.org/10.3390/vision7040074