# A Probabilistic Classification Procedure Based on Response Time Analysis Towards a Quick Pre-Diagnosis of Student’s Attention Deficit

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Participants

#### 2.2. Experiments with DMDX

#### 2.3. Procedure for the Data Analysis

## 3. Results and Discussions

#### 3.1. Ex-Gaussian Analysis

_{C}, σ = ω y τ = t

_{0}, that is, in terms of the Gaussian and exponential functions parameters,

^{2}= σ

^{2}+ τ

^{2}, and the skewness 2τ

^{3}/S

^{3}[38]. In fact, one can characterize this distribution f(x) through its moments. One can consider moments of this distribution centered either at the origin (raw moments), or centered at the corresponding average (central moments). Thus,

#### 3.2. Classification Methodology

#### 3.3. Vector Criterion Based on the Ex-Gaussian Parameters

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Sroubek, A.; Kelly, M.; Li, X. Inattentiveness in attention-deficit/hyperactivity disorder. Neurosci. Bull.
**2013**, 29, 103–110. [Google Scholar] [CrossRef] [PubMed] - Clauss-Ehlers, C.S. (Ed.) Encyclopedia of Cross-Cultural School Psychology; Springer Science & Business Media LLC: Boston, MA, USA, 2010. [Google Scholar]
- American Psychiatric Association. American Psychiatric Association Diagnostic and Statistical Manual of Mental Disorders, 5th ed.; APA: Washington, DC, USA, 2013. [Google Scholar]
- Barkley, R.A. Attention Deficit Hyperactivity Disorder: A Handbook for Diagnosis and Treatment, 3rd ed.; Guilford: New York, NY, USA, 2006. [Google Scholar]
- DuPaul, G.J.; Volpe, R.J.; Jitendra, A.K.; Lutz, J.G.; Lorah, K.S.; Gruber, R. Elementary school students with AD/HD: Predictors of academic achievement. J. Sch. Psychol.
**2004**, 42, 285–301. [Google Scholar] [CrossRef] - Sonuga-Barke, E.; Koerting, J.; Smith, E.; McCann, D.C.; Thompson, M. Early detection and intervention for attention-deficit/hyperactivity disorder. Expert Rev. Neurother.
**2011**, 11, 557–563. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lavigne Cerván, R.; Romero-Pérez, J.F. The Attention Deficit Hyperactivity Dissorder; Ediciones Pirámide: Madrid, Spain, 2010; (In Spanish, EL TDAH). [Google Scholar]
- Pliszka, S. Practice parameter for the assessment and treatment of children and adolescents with attention-deficit/hyperactivity disorder. J. Am. Acad. Child Adolesc. Psychiatry
**2007**, 46, 894–921. [Google Scholar] [CrossRef] - Tarantino, S.C.; Mogentale, C.; Bisiacchi, P.S. Time-on-Task in Children with ADHD: An ex-Gaussian Analysis. Vincenza J. Int. Neuropsychol. Soc.
**2013**, 19, 820–828. [Google Scholar] [CrossRef] - Klein, C.; Wendling, K.; Huettner, P.; Ruder, H.; Peper, M. Intra-subject variability in attention-deficit hyperactivity disorder. Biol. Psychiatry
**2006**, 60, 1088–1097. [Google Scholar] [CrossRef] [PubMed] - Nigg, J.T.; Willcutt, E.G.; Doyle, A.E.; Sonuga-Barke, E.J. Causal heterogeneity in attention-deficit/hyperactivity disorder: Do we need neuropsychologically impaired subtypes? Biol. Psychiatry
**2005**, 57, 1224–1230. [Google Scholar] [CrossRef] [PubMed] - Willcutt, E.G.; Doyle, A.E.; Nigg, J.T.; Faraone, S.V.; Pennington, B.F. Validity of the executive function theory of attention-deficit/hyperactivity disorder: A meta-analytic review. Biol. Psychiatry
**2005**, 57, 1336–1346. [Google Scholar] [CrossRef] - Castellanos, F.X.; Sonuga-Barke, E.J.; Scheres, A.; Di Martino, A.; Hyde, C.; Walters, J.R. Varieties of attention-deficit/hyperactivity disorder-related intra-individual variability. Biol. Psychiatry
**2005**, 57, 1416–1423. [Google Scholar] [CrossRef] - Luce, R.D. Response Times: Their Role in Inferring Elementary Mental Organization; Oxford University Press: New York, NY, USA, 1986. [Google Scholar]
- Hockley, W.E.; Corballis, M.C. Test of serial scanning in item recognition. Can. J. Psychol.
**1982**, 36, 189–212. [Google Scholar] [CrossRef] - Ratcliff, R.; Murdock, B.B. Retrieval processes in recognition memory. Psychol. Rev.
**1976**, 83, 190–214. [Google Scholar] [CrossRef] - Gmehlin, D.; Fuermaier, A.B.M.; Walther, S.; Debelak, R.; Rentrop, M.; Westermann, C.; Sharma, A.; Tucha, L.; Koerts, J.; Tucha, O.; et al. Intraindividual Variability in Inhibitory Function in Adults with ADHD—An Ex-Gaussian Approach. PLoS ONE
**2014**, 9, e112298. [Google Scholar] [CrossRef] - Adamo, N.; Hodsoll, J.; Asherson, P.; Buitelaar, J.K.; Kuntsi, J. Ex-Gaussian, Frequency and Reward Analyses Reveal Specificity of Reaction Time Fluctuations to ADHD and Not Autism Traits. J. Abnorm. Child Psychol.
**2019**, 47, 557–567. [Google Scholar] [CrossRef] - Burbeck, S.L.; Luce, R.D. Evidence from auditory simple reaction times for both change and level detectors. Percept. Psychophys.
**1982**, 32, 117–133. [Google Scholar] [CrossRef] - Heathcote, A.; Popiel, S.J.; Mewhort, D.J.K. Analysis of response time distributions: An example using the Stroop task. Psychol. Bull.
**1991**, 109, 340–347. [Google Scholar] [CrossRef] - Matzke, D.; Wagenmakers, E.-J. Psychological interpretation of the ex-Gaussian and shifted Wald parameters: A diffusion model analysis. Psychon. Bull. Rev.
**2009**, 16, 798–817. [Google Scholar] [CrossRef] - Shahar, N.; Teodorescu, A.R.; Karmon-Presser, A.; Anholt, G.E.; Meiran, N. Memory for Action Rules and Reaction Time Variability in Attention-Deficit/Hyperactivity Disorder. Biol. Psychiatry Cogn. Neurosci. Neuroimaging
**2016**, 1, 132–140. [Google Scholar] [CrossRef] - Leth-Steensen, C.; King Elbaz, Z.; Douglas, V.I. Mean response times, variability, and skew in the responding of ADHD children: A response time distributional approach. Acta Psychol.
**2000**, 104, 167–190. [Google Scholar] [CrossRef] - Navarro-Pardo, E.; Navarro-Prados, A.B.; Gamermann, D. Moret-Tatay, C. Differences between younger and older university students on lexical decision task: Evidence through an ex-Gaussian approach. J. Gen. Psychol.
**2013**, 140, 251–268. [Google Scholar] [CrossRef] - Lemus-Zúñiga, L.G.; Navarro-Pardo, E.; Moret-Tatay, C.; Pocinho, R. Serious games for elderly continuous monitoring. In Data Mining in Clinical Medicine; Springer: New York, NY, USA, 2015; pp. 259–267. [Google Scholar] [CrossRef]
- Moret-Tatay, C.; Moreno-Cid, A.; Argimon, I.I.D.L.; Quarti Irigaray, T.; Szczerbinski, M.; Murphy, M.; Vázquez-Martínez, A.; Vázquez-Molina, J.; Sáiz-Mauleon, B.; Navarro-Pardo, E.; et al. The effects of age and emotional valence on recognition memory: An ex-Gaussian components analysis. Scand. J. Psychol.
**2014**, 55, 420–426. [Google Scholar] [CrossRef] [PubMed][Green Version] - Moret-Tatay, C.; Irigaray, T.Q.; Oliveira, C.; Argimon, I.I.D.L. Reaction Times as a dependent variable of memory: Future lines of research through an ex-Gaussian fit. Horiz. Neurosci. Res.
**2015**, 17, 60–71. [Google Scholar] - Hohle, R.H. Inferred components of reaction times as functions of foreperiod duration. J. Exp. Psychol.
**1965**, 69, 382–386. [Google Scholar] [CrossRef] - Borella, E.; de Ribaupierre, A.; Cornoldi, C.; Chicherio, C. Beyond interference control impairment in ADHD: Evidence from increased intraindividual variability in the color—Stroop test. Child Neuropsychol.
**2012**, 19, 495–515. [Google Scholar] [CrossRef] [PubMed] - World Medical Association. Declaration of Helsinki: Ethical Principles for Medical Research Involving Human Subjects. J. Am. Med. Assoc.
**2013**, 310, 2191–2194. [Google Scholar] [CrossRef] - Forster, K.I.; Forster, J.C. DMDX: A windows display program with millisecond accuracy. Behav. Res. Methods Instrum. Comput.
**2003**, 35, 116–124. [Google Scholar] [CrossRef] - Fan, J.; McCandliss, B.D.; Sommer, T.; Raz, A.; Posner, M.I. Testing the efficiency and independence of attentional networks. J. Cogn. Neurosci.
**2002**, 14, 340–347. [Google Scholar] [CrossRef] - Posner, M.I.; Dehaene, S. Attentional networks. Trends Neurosci.
**1994**, 17, 75–79. [Google Scholar] [CrossRef] - Posner, M.I.; Raichle, M.E. Images of Mind; Scientific American Library: New York, NY, USA, 1994. [Google Scholar]
- Lacouture, Y.; Cousineau, D. How to use MATLAB to fit the ex-Gaussian and other probability functions to a distribution of response times. Tutor. Quant. Methods Psychol.
**2008**, 4, 35–45. [Google Scholar] [CrossRef] - Ratcliff, R.; McKoon, G. The diffusion decision model: Theory and data for two-choice decision tasks. Neural Comput.
**2008**, 20, 873–922. [Google Scholar] [CrossRef] [PubMed] - Ratcliff, R. Group reaction time distributions and an analysis of distribution statistics. Psychol. Bull.
**1979**, 86, 446–461. [Google Scholar] [CrossRef] [PubMed] - Moret-Tatay, C.; Gamermann, D.; Navarro-Pardo, E.; Fernández-de-Córdoba-Castellá, P. ExGUtils: A python package for statistical analysis with the ex-Gaussian probability density. Front. Psychol.
**2018**, 9, 1–11. [Google Scholar] [CrossRef] - Levenberg, K. A method for the solution of certain non-linear problems in least squares. Q. Appl. Math.
**1944**, 2, 164–168. [Google Scholar] [CrossRef][Green Version] - Marquardt, D. An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math.
**1963**, 11, 431–441. [Google Scholar] [CrossRef] - Castellanos, F.X.; Sonuga-Barke, E.J.; Milham, M.P.; Tannock, R. Characterizing cognition in ADHD: Beyond executive dysfunction. Trends Cogn. Sci.
**2006**, 10, 117–123. [Google Scholar] [CrossRef] - Emond, V.; Joyal, C.; Poissant, H. Neuroanatomie structurelle et fonctionnelle du trouble déficitaire d’attention avec ou sans hyperactivité (TDAH) [Structural and functional neuroanatomy of attention-deficit hyperactivity disorder (ADHD)]. Encephale
**2009**, 35, 107–114. (In French) [Google Scholar] [CrossRef] [PubMed] - Reinhardt, M.C.; Reinhardt, C.A. Attention deficit-hyperactivity disorder, comorbidities, and risk situations. J. Pediatr. (Rio. J.)
**2013**, 89, 124–130. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hwang-Gu, S.L.; Chen, Y.C.; Liang, S.H.; Ni, H.C.; Lin, H.Y.; Lin, C.F.; Gau, S.S. Exploring the Variability in Reaction Times of Preschoolers at Risk of Attention-Deficit/Hyperactivity Disorder: An ex-Gaussian Analysis. J. Abnorm. Child Psychol.
**2019**. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The age distribution of the sample is shown in panel (

**a**) and a fragment of a *.azk output file in panel (

**b**). In order to protect the privacy of the child who performed the experiment, the characters “TTTTT” have been used as a pseudonym for the actual name.

**Figure 2.**In panel (

**a**), a visual example of the attention network task (ANT) carried out in this work is shown whereas in panel (

**b**), the four cue conditions are included.

**Figure 5.**Probability distributions of the mode (

**a**) and of μ parameter in an ex-Gaussian representation of the data (

**b**).

**Figure 6.**Probability distributions of σ (

**a**) and τ (

**b**) parameters in an ex-Gaussian representation of the data.

**Figure 7.**Probability distributions of the candidates resulting from the probability distribution of the mode in panel (

**a**) and of the candidates resulting from the probability distribution of the parameter μ in panel (

**b**). The average curve has also been included.

**Figure 8.**Probability distributions of the candidates resulting from the probability distribution of σ (

**a**) and τ (

**b**) parameters of the ex-Gaussian probability density. The average curve has also been included.

**Figure 9.**Euclidean norm and the norm of the maximum applied to the vector defined in Equation (8) as a function of the student label. In the upper panel, students from 1 to 100 are shown, and in the lower panel, students from 101 to 200. The students labeled as 110 (M), 131 (F), 137 (M), 169 (F), 185 (M), and 189 (F) appear in this classification only. The letter “M” between parentheses stands for male and the letter “F” for female.

**Table 1.**Results of the classification for the distribution of the mode, $\mathsf{\mu}$, $\mathsf{\sigma}$ and $\mathsf{\tau}$ (columns from second to fifth) taking into account a 7% of world prevalence of ADHD in school-aged children. The rows show the mode, the probability percentages at both sides of the mode (%-PD), the splitting of the prevalence percentage (%-Prev.), corresponding number of children (No. Ch.), and the selected children in terms of labels.

Mode | $\mathsf{\mu}$ | $\mathsf{\sigma}$ | $\mathsf{\tau}$ | |||||
---|---|---|---|---|---|---|---|---|

Mode (ms) | 525.5 | 545.5 | 87.5 | 112.5 | ||||

L | R | L | R | L | R | L | R | |

%-PD | 44.4 | 55.6 | 45.0 | 55.0 | 16.5 | 83.5 | 14.9 | 85.1 |

%-Prev. | 3.1 | 3.9 | 3.2 | 3.8 | 1.2 | 5.8 | 2.0 | 5.0 |

No. Ch. | 6 | 7 | 6 | 7 | 2 | 11 | 2 | 11 |

Selected children (labels) | 2, 28, 34, 55, 77, 75, 130, 85, 102, 107, 134, 142, 159 | 28, 34, 55, 58, 75, 77, 80, 102, 106, 130, 163, 167, 184 | 5, 39, 41, 58, 71, 80, 85, 109, 130, 136, 141, 183, 184 | 5, 28, 34, 55, 58, 80, 109, 113, 133, 136, 163, 183, 187 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hernaiz-Guijarro, M.; Castro-Palacio, J.C.; Navarro-Pardo, E.; Isidro, J.M.; Fernández-de-Córdoba, P.
A Probabilistic Classification Procedure Based on Response Time Analysis Towards a Quick Pre-Diagnosis of Student’s Attention Deficit. *Mathematics* **2019**, *7*, 473.
https://doi.org/10.3390/math7050473

**AMA Style**

Hernaiz-Guijarro M, Castro-Palacio JC, Navarro-Pardo E, Isidro JM, Fernández-de-Córdoba P.
A Probabilistic Classification Procedure Based on Response Time Analysis Towards a Quick Pre-Diagnosis of Student’s Attention Deficit. *Mathematics*. 2019; 7(5):473.
https://doi.org/10.3390/math7050473

**Chicago/Turabian Style**

Hernaiz-Guijarro, M., J. C. Castro-Palacio, E. Navarro-Pardo, J. M. Isidro, and P. Fernández-de-Córdoba.
2019. "A Probabilistic Classification Procedure Based on Response Time Analysis Towards a Quick Pre-Diagnosis of Student’s Attention Deficit" *Mathematics* 7, no. 5: 473.
https://doi.org/10.3390/math7050473