# Nonlinear Dynamics of Reaction Time and Time Estimation during Repetitive Test

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

_{1}, SD

_{2}(SD—standard deviation), and the area of the fitting ellipse (AFE) (p < 0.01). We reported an underestimation of the time interval of 2 s during the VRT session of testing, with an average value of CV of VRT, the equivalent of the Weber fraction, of 15.21 ± 8.82%. (4) Conclusions: The present study provides novel evidence that linear and nonlinear analysis of RT and VRT variability during serial testing bring complementary insights to the understanding of complex neurocognitive processes implied in the task execution.

## 1. Introduction

_{1}, SD

_{2}, the area of the fitting ellipse (AFE), and the fraction between SD

_{1}and SD

_{2}(SD

_{1}/SD

_{2}). Practically, SD

_{1}represents the small axis of the geometrical elliptical representation of a time series of data and indicates their variation on a short time scale, while SD

_{2}is the large axis of the ellipse and shows the data variation on a long-term scale [19,20,21]. The SD

_{1}/SD

_{2}parameter, as the relative balance between SD

_{1}and SD

_{2}, reflects the clarity and linearity of the scatter pattern, with regards to the ratio between short- and long-term variabilities of the time series of data, associated with the degree of the system’s physiological disorder depth [22,23]. The resulted ellipse is the geometrical plot of a system with internal fluctuations, and the decrease of AFE reflects the concentration of data and a greater stability of the system. In reverse, a larger AFE indicates a system irregularity or poor control of the physiological variable [24].

_{1}, SD

_{2}, SD

_{1}/SD

_{2}, and AFE), taking into account some categorical variables of the subjects, such as age, sex, health status, and anxiety level.

## 2. Materials and Methods

#### 2.1. Study Design

#### 2.2. Data Acquisition

- Sex: male—1, female—2.
- Professional status: pupil/student—1, employee—2, unemployed—3, retired—4, household—5.
- SRH: excellent—1, very good—2, good—3, satisfactory—4, poor—5.
- SRA: not at all anxious—1, slightly anxious—2, moderately anxious—3, very anxious—4, extremely anxious—5.
- Laterality (hand used for writing): right—1, left—2.

#### 2.3. Statistical Analysis of Data

_{1}, SD

_{2}, SD

_{1}/SD

_{2}, and AFE, according to the following formulas [21,22,28,29]:

_{n}−x

_{n+1}) is the SD of the differences x

_{n}−x

_{n+1}from the string of data, and, respectively, SD(x

_{n}) represents the SD of x

_{n}.

## 3. Results

#### 3.1. Descriptive Statistics for the Study Group

#### 3.2. Reliability of the RT and VRT Serial Tests

#### 3.3. Correlation and Regression Analysis of Data

_{1}, SD

_{2}, and AFE and for the effect of the CV of VRT on SD

_{2}and AFE. Additionally, a medium effect size was reported for the effect of age on RT and for the effect of the CV of VRT on SD

_{1}. On the other hand, a small effect size was recorded for the effect of age on VRT, the effect of the CV of RT on SD

_{1}/SD

_{2}, and the effect of the CV of VRT on SD

_{1}/SD

_{2}.

## 4. Discussion

#### 4.1. The Study Implications

_{1}, SD

_{2}, SD

_{1}/SD

_{2}, and AFE).

_{1}, SD

_{2}, and AFE had a consistently closer relationship with the CV of RT and with the CV of VRT (p < 0.01). The best regression models were for the relationship between the CV of RT and SD

_{1}, which explains 62% of the variability in SD

_{1}(Table 5), and for the relationship between the CV of VRT and SD

_{2}, which explains 56% of the variability in SD

_{2}(Table 6).

#### 4.2. Limitations of the Study

#### 4.3. Future Research Directions

## 5. Conclusions

_{1}, SD

_{2}, and AFE (p < 0.01). Finally, our findings suggested an underestimation of the time interval of 2 s that was imposed for repetitive reproduction for the VRT session of testing. The average value of CV of VRT (15.21 ± 8.82%) during the repetitive short-time estimation test, with visual stimuli, as an equivalent of the WF, confirms the hypothesis of the scalar property of temporal representation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CV | the coefficient of variation |

RT | reaction time |

VRT | virtual reaction time |

SD | standard deviation |

AFE | area of the fitting ellipse |

WB | Weber fraction |

SRH | self-reported health |

SRA | self-reported anxiety |

RRE | Relative Reproduction Error |

R | Pearson’s coefficient of correlation |

SE | standard error |

F | test for overall significance for the linear model |

p | level of statistical significance |

β0 | the intercept coefficient |

β1 | the regression coefficient |

95%LB | lower bound of the 95% confidence interval |

95%UB | upper bound of the 95% confidence interval |

n | group size |

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**Table 1.**Descriptive parameters of the experimental group, based on the questionnaire answers (n = 180).

Variable | Age Years | SRH | SRA |
---|---|---|---|

Mean | 31.61 | 2.32 | 1.77 |

SD | 13.56 | 0.80 | 0.79 |

**Table 2.**Descriptive parameters of the experimental group, based on the RT session of testing (n = 180).

Variable | RT ms | CV % | SD_{1}ms | SD_{2}ms | AFE ms ^{2} | SD_{1}/SD_{2} |
---|---|---|---|---|---|---|

Mean | 263.94 | 33.04 | 80.60 | 87.76 | 28,070.77 | 0.93 |

SD | 69.17 | 15.91 | 42.90 | 47.59 | 36,014.82 | 0.16 |

_{1}, SD

_{2}, AFE, SD

_{1}/SD

_{2}, Poincaré plot descriptors; n, group size.

**Table 3.**Descriptive parameters of the experimental group, based on the VRT session of testing (n = 180).

Variable | VRT ms | CV % | RRE % | SD_{1}ms | SD_{2}ms | AFE ms ^{2} | SD_{1}/SD_{2} |
---|---|---|---|---|---|---|---|

Mean | 1540.80 | 15.21 | −22.96 | 170.88 | 265.34 | 191,331.54 | 0.69 |

SD | 592.63 | 8.82 | 29.63 | 109.59 | 189.00 | 305,745.98 | 0.24 |

_{1}, SD

_{2}, AFE, SD

_{1}/SD

_{2}, Poincaré plot descriptors; n, group size.

Variable | Age | Sex | SRH | SRA | RT | CV RT | SD_{1} RT | SD_{2} RT | AFE RT | SD_{1}/SD_{2} RT | VRT | CV VRT | RRE | SD_{1} VRT | SD_{2} VRT | AFE VRT | SD_{1}/SD_{2} VRT |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Age | 1.00 ^{b} | ||||||||||||||||

Sex | −0.12 ^{a} | 1.00 ^{a} | |||||||||||||||

SRH | 0.15 ^{a}* | 0.14 ^{a} | 1.00 ^{a} | ||||||||||||||

SRA | −0.12 ^{a} | 0.11 ^{a} | 0.29 ^{a}* | 1.00 ^{a} | |||||||||||||

RT | 0.49 ^{b}* | 0.13 ^{a} | 0.12 ^{a} | −0.04 ^{a} | 1.00 ^{b} | ||||||||||||

CV RT | −0.15 ^{b}* | −0.05 ^{a} | −0.06 ^{a} | 0.14 ^{a} | −0.24 ^{b}* | 1.00 ^{b} | |||||||||||

SD_{1} RT | 0.10 ^{b} | 0.03 ^{a} | 0.00 ^{a} | 0.14 ^{a} | 0.32 ^{b}* | 0.79 ^{b}* | 1.00 ^{b} | ||||||||||

SD_{2} RT | 0.15 ^{b}* | 0.03 ^{a} | −0.03 ^{a} | 0.10 ^{a} | 0.46 ^{b}* | 0.71 ^{b}* | 0.92 ^{b}* | 1.00 ^{b} | |||||||||

AFE RT | 0.19 ^{b}* | 0.03 ^{a} | −0.01 ^{a} | 0.12 ^{a} | 0.51 ^{b}* | 0.61 ^{b}* | 0.92 ^{b}* | 0.93 ^{b}* | 1.00 ^{b} | ||||||||

SD_{1}/SD_{2} RT | −0.17 ^{b}* | 0.06 ^{a} | 0.02 ^{a} | 0.16 ^{a}* | −0.31 ^{b}* | 0.24 ^{b}* | 0.24 ^{b}* | −0.10 ^{b} | 0.02 ^{b} | 1.00 ^{b} | |||||||

VRT | −0.21 ^{b}* | −0.01 ^{a} | 0.03 ^{a} | 0.07 ^{a} | −0.09 ^{b} | 0.04 ^{b} | −0.01 ^{b} | −0.03 ^{b} | −0.06 ^{b} | 0.01 ^{b} | 1.00 ^{b} | ||||||

CV VRT | −0.03 ^{b} | −0.15 ^{a}* | −0.04 ^{a} | 0.03 ^{a} | 0.17 ^{b}* | 0.00 ^{b} | 0.14 ^{b} | 0.13 ^{b} | 0.16 ^{b}* | 0.06 ^{b} | −0.12 ^{b} | 1.00 ^{b} | |||||

RRE | −0.21 ^{b}* | −0.01 ^{a} | 0.03 ^{a} | 0.07 ^{a} | −0.09 ^{b} | 0.04 ^{b} | −0.01 ^{b} | −0.03 ^{b} | −0.06 ^{b} | 0.01 ^{b} | 1.00 ^{b} | −0.12 ^{b} | 1.00 ^{b} | ||||

SD_{1} VRT | −0.15 ^{b}* | 0.11 ^{a} | 0.04 ^{a} | 0.06 ^{a} | 0.12 ^{b} | −0.01 ^{b} | 0.10 ^{b} | 0.10 ^{b} | 0.09 ^{b} | 0.04 ^{b} | 0.57 ^{b}* | 0.49 ^{b}* | 0.57 ^{b}* | 1.00 ^{b} | |||

SD_{2} VRT | −0.14 ^{b} | 0.02 ^{a} | −0.02 ^{a} | 0.03 ^{a} | 0.11 ^{b} | 0.03 ^{b} | 0.12 ^{b} | 0.12 ^{b} | 0.11 ^{b} | 0.02 ^{b} | 0.49 ^{b}* | 0.75 ^{b}* | 0.49 ^{b}* | 0.76 ^{b}* | 1.00 ^{b} | ||

AFE VRT | −0.14 ^{b} | 0.05 ^{a} | 0.02 ^{a} | 0.06 ^{a} | 0.11 ^{b} | 0.03 ^{b} | 0.11 ^{b} | 0.14 ^{b} | 0.11 ^{b} | −0.03 ^{b} | 0.46 ^{b}* | 0.61 ^{b}* | 0.46 ^{b}* | 0.90 ^{b}* | 0.87 ^{b}* | 1.00 ^{b} | |

SD_{1}/SD_{2} VRT | 0.06 ^{b} | 0.08 ^{a} | 0.15 ^{a}* | 0.06 ^{a} | −0.01 ^{b} | −0.08 ^{b} | −0.07 ^{b} | −0.09 | −0.08 ^{b} | 0.05 ^{b} | 0.02 ^{b} | −0.27 ^{b}* | 0.02 ^{b}* | 0.16 ^{b}* | −0.35 ^{b}* | −0.07 ^{b} | 1.00 ^{b} |

_{1}, SD

_{2}, AFE, SD

_{1}/SD

_{2}, Poincaré plot descriptors;

^{a}, Spearman’s rank correlation coefficient;

^{b}, Pearson’s correlation coefficient; *, p < 0.05 was considered statistically significant (2-tailed); n, group size.

**Table 5.**Model summary, ANOVA report and coefficients for simple linear regression analysis—age versus RT and VRT (n = 180).

Variable | R | R Square | Adjusted R Square | SE | F | p | β0 | SE | p | 95%LB | 95%UB | β1 | SE | p | 95%LB | 95%UB |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

RT | 0.49 | 0.24 | 0.23 | 60.65 | 54.84 | 0.001 | 185.67 | 11.49 | 0.001 | 162.99 | 208.36 | 2.48 | 0.33 | 0.001 | 1.82 | 3.14 |

VRT | 0.21 | 0.04 | 0.04 | 581.16 | 8.13 | 0.005 | 1829.62 | 110.15 | 0.001 | 1612.25 | 2046.98 | −9.14 | 3.20 | 0.005 | −15.46 | −2.81 |

**Table 6.**Model summary, ANOVA report and coefficients for simple linear regression analysis—CV of RT versus Poincaré plot descriptors (n = 180).

Variable | R | R Square | Adjusted R Square | SE | F | p | β0 | SE | p | 95%LB | 95%UB | β1 | SE | p | 95%LB | 95%UB |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

SD_{1} | 0.79 | 0.62 | 0.62 | 26.45 | 292.91 | 0.001 | 10.32 | 4.55 | 0.025 | 1.34 | 19.32 | 2.13 | 0.12 | 0.001 | 1.88 | 2.37 |

SD_{2} | 0.71 | 0.51 | 0.50 | 33.55 | 182.18 | 0.001 | 17.48 | 5.78 | 0.003 | 6.08 | 28.88 | 2.13 | 0.16 | 0.001 | 1.82 | 2.44 |

AFE | 0.61 | 0.37 | 0.36 | 28,713.49 | 103.61 | 0.001 | −17,295.44 | 4944.17 | 0.001 | −27,052.2 | −7538.71 | 1373.2 | 134.9 | 0.001 | 1106.97 | 1639.42 |

SD_{1}/SD_{2} | 0.24 | 0.06 | 0.05 | 0.16 | 10.78 | 0.001 | 0.85 | 0.03 | 0.001 | 0.79 | 0.9 | 0.002 | 0.001 | 0.001 | 0.001 | 0.004 |

_{1}, SD

_{2}, AFE, SD

_{1}/SD

_{2}, Poincaré plot descriptors; n, group size.

**Table 7.**Model summary, ANOVA report and coefficients for simple linear regression analysis—CV of VRT versus Poincaré plot descriptors (n = 180).

Variable | R | R Square | Adjusted R Square | SE | F | p | β0 | SE | p | 95%LB | 95%UB | β1 | SE | p | 95%LB | 95%UB |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

SD_{1} | 0.49 | 0.24 | 0.24 | 95.60 | 57.23 | 0.001 | 77.72 | 14.23 | 0.001 | 49.64 | 105.8 | 6.13 | 0.81 | 0.001 | 4.53 | 7.73 |

SD_{2} | 0.75 | 0.56 | 0.56 | 125.47 | 228.15 | 0.001 | 21.21 | 18.67 | 0.001 | −15.64 | 58.06 | 16.06 | 1.06 | 0.001 | 13.96 | 18.15 |

AFE | 0.61 | 0.37 | 0.36 | 243,925.9 | 103.23 | 0.001 | −127,906 | 36,301.7 | 0.001 | −199,543.1 | −56,268.9 | 20,995 | 2066 | 0.001 | 16,917 | 25,072.7 |

SD_{1}/SD_{2} | 0.27 | 0.07 | 0.07 | 0.23 | 14.08 | 0.001 | 0.8 | 0.03 | 0.001 | 0.74 | 0.87 | −0.007 | 0.002 | 0.001 | −0.011 | −0.003 |

_{1}, SD

_{2}, AFE, SD

_{1}/SD

_{2}, Poincaré plot descriptors; n, group size.

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**MDPI and ACS Style**

Iconaru, E.I.; Ciucurel, M.M.; Tudor, M.; Ciucurel, C.
Nonlinear Dynamics of Reaction Time and Time Estimation during Repetitive Test. *Int. J. Environ. Res. Public Health* **2022**, *19*, 1818.
https://doi.org/10.3390/ijerph19031818

**AMA Style**

Iconaru EI, Ciucurel MM, Tudor M, Ciucurel C.
Nonlinear Dynamics of Reaction Time and Time Estimation during Repetitive Test. *International Journal of Environmental Research and Public Health*. 2022; 19(3):1818.
https://doi.org/10.3390/ijerph19031818

**Chicago/Turabian Style**

Iconaru, Elena Ioana, Manuela Mihaela Ciucurel, Mariana Tudor, and Constantin Ciucurel.
2022. "Nonlinear Dynamics of Reaction Time and Time Estimation during Repetitive Test" *International Journal of Environmental Research and Public Health* 19, no. 3: 1818.
https://doi.org/10.3390/ijerph19031818