全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用:99美元

查看量下载量

相关文章

更多...

Measures of Interindividual Variation: Definition and Computation

DOI: 10.4236/ojs.2026.162004, PP. 47-88

Keywords: Descriptive Measures, Variability, Interindividual Differences, Confidence Intervals, Sampling Distribution

Full-Text   Cite this paper   Add to My Lib

Abstract:

Measures of interindividual variation are rarely used or studied in descriptive/inference statistics, despite constituting a third approach to describing data variability, alternative to ranges (the difference between two extreme measures of position) and to the mean or median of the distances of the data from a measure of central tendency. The aim of this article is to present point and interval estimation (asymptotic and bootstrap) for four absolute measures of variation based on interindividual differences and one relative measure, to develop an R script for their computation, and to demonstrate the performance of the script using two examples. One example is based on a large sample (n = 1000) drawn from a bounded continuous distribution with negative skewness (PERT (0, 8, 10)), which may simulate data from a knowledge–ability test; the other is based on a medium-sized sample (n = 100) drawn from a bounded discrete distribution with positive skewness (BN (n = 10, p = 0.25)), which may simulate the recording (present/absent) of a behavior in independent observations. Both distributions are slightly platykurtic and non-normal. With the measure based on the median of interindividual differences, difficulties arise when generating its sampling distribution for discrete data, but not for continuous data. The absolute measures converge to a normal distribution, whereas the relative measure does not. It is concluded that the script enables the appropriate computation of interindividual variation statistics. Its use is recommended, and further investigation of these measures through simulation studies is encouraged.

References

[1]  Hinton, P.R. (2024) Statistics Explained. 4th Edition, Routledge.
https://doi.org/10.4324/9780429353260
[2]  Alabi, O. and Bukola, T. (2023) Introduction to Descriptive Statistics. In: Kumar, S., Ed., Recent Advances in Biostatistics, IntechOpen, 1-12.
https://doi.org/10.5772/intechopen.1002475
[3]  Kvålseth, T.O. (1988) Measuring Variation for Nominal Data. Bulletin of the Psychonomic Society, 26, 433-436.
https://doi.org/10.3758/bf03334906
[4]  San Martin-Castellanos, R., Espinosa-Gil, L. and Fernández-Pedreira, L. (1990) Psicoestadística Descriptive. 2nd Edition, Pirámide.
[5]  Casella, G. and Berger, R. (2024) Statistical Inference. 2nd Edition, Chapman and Hall/CRC.
https://doi.org/10.1201/9781003456285
[6]  Merce, E. (2009) Pearson’s Coefficient of Variation, an Erroneous History in Assessing the Degree of Significance of the Mean Value. Bulletin UASVM Horticulture, 66, 295 300.
[7]  Pearson, K. and Filon, L.N.G. (1898) Mathematical Contributions to the Theory of Evolution. IV. On the Probable Errors of Frequency Constants and on the Influence of Random Selection on Variation and Correlation. Proceedings of the Royal Society of London, 62, 173-176.
https://doi.org/10.1098/rspl.1897.0091
[8]  Sokal, R.R. and Rohlf, F.J. (1995) Biometry: The Principles and Practice of Statistics in Biological Research. 3rd Edition, Freeman and Company.
[9]  Henze, N. (2024) Limit Theorems for U-Statistics. In: Henze, N., Ed., Asymptotic Stochastics, Springer, 103-141.
https://doi.org/10.1007/978-3-662-68923-3_8
[10]  Korolyuk, V.S. and Borovskich, Y.V. (2013) Theory of U-Statistics (Mathematics and Its Applications, Vol. 273). Springer Science & Business Media.
[11]  Hoeffding, W. (1948) A Class of Statistics with Asymptotically Normal Distribution. The Annals of Mathematical Statistics, 19, 293-325.
https://doi.org/10.1214/aoms/1177730196
[12]  Wendler, M. (2011) Bahadur Representation for U-Quantiles of Dependent Data. Journal of Multivariate Analysis, 102, 1064-1079.
https://doi.org/10.1016/j.jmva.2011.02.005
[13]  Silverman, B.W. (1998) Density Estimation for Statistics and Data Analysis. Routledge.
https://doi.org/10.1201/9781315140919
[14]  Ichimura, H. and Newey, W.K. (2022) The Influence Function of Semiparametric Estimators. Quantitative Economics, 13, 29-61.
https://doi.org/10.3982/qe826
[15]  Stuart, A. and Ord, K. (1994) Kendall’s Advanced Theory of Statistics. Volume 1. Distribution Theory. 6th Edition, Edward Arnold.
[16]  Andrianov, I.V. and Awrejcewicz, J. (2024) Asymptotic Methods for Engineers. CRC Press.
https://doi.org/10.1201/9781003467465
[17]  Clark, J.M. and Warr, R.L. (2023) Bootstrapping through Discrete Convolutional Methods. Applied Stochastic Models in Business and Industry, 40, 144-160.
https://doi.org/10.1002/asmb.2809
[18]  Van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge University Press.
https://doi.org/10.1017/cbo9780511802256
[19]  Leadbetter, R.M., Lindgren, G. and Rootzén, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer.
https://doi.org/10.1007/978-1-4612-5449-2
[20]  Efron, B. and Tibshirani, R.J. (1993) An Introduction to the Bootstrap. Chapman & Hall.
[21]  Microsoft Corporation (2024) Microsoft Excel (Versión 2409) [Software]. Microsoft.
https://www.microsoft.com/es-mx/microsoft-365/get-started-with-office-2024
[22]  R Core Team (2025) R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing.
https://www.R-project.org/
[23]  Estabrook, R., Grimm, K.J. and Bowles, R.P. (2012) A Monte Carlo Simulation Study of the Reliability of Intraindividual Variability. Psychology and Aging, 27, 560-576.
https://doi.org/10.1037/a0026669
[24]  Kohoutová, L., Atlas, L.Y., Büchel, C., Buhle, J.T., Geuter, S., Jepma, M., et al. (2022) Individual Variability in Brain Representations of Pain. Nature Neuroscience, 25, 749-759.
https://doi.org/10.1038/s41593-022-01081-x
[25]  Chen, L., Zhernakova, D.V., Kurilshikov, A., Andreu-Sánchez, S., Wang, D., Augustijn, H.E., et al. (2022) Influence of the Microbiome, Diet and Genetics on Inter-Individual Variation in the Human Plasma Metabolome. Nature Medicine, 28, 2333-2343.
https://doi.org/10.1038/s41591-022-02014-8
[26]  Giorgi, F.M., Ceraolo, C. and Mercatelli, D. (2022) The R Language: An Engine for Bioinformatics and Data Science. Life, 12, Article 648.
https://doi.org/10.3390/life12050648
[27]  Wehrspohn, U. and Ernst, D. (2022) When Do I Take Which Distribution? A Statis-tical Basis for Entrepreneurial Applications. Springer.
https://doi.org/10.1007/978-3-031-07330-4
[28]  Schultzberg, M. and Ankargren, S. (2022) Resampling-free Bootstrap Inference for Quantiles. In: Arai, K., Ed., Proceedings of the Future Technologies Conference (FTC) 2022, Volume 1. FTC 2022 2022, Springer, 548-562.
https://doi.org/10.1007/978-3-031-18461-1_36
[29]  Wegner, L. and Wendler, M. (2024) Robust Change-Point Detection for Functional Time Series Based on U-Statistics and Dependent Wild Bootstrap. Statistical Papers, 65, 4767-4810.
https://doi.org/10.1007/s00362-024-01577-7
[30]  Efron, B. and Narasimhan, B. (2025) Packages ‘Bcaboot’. Bias Corrected Bootstrap Confidence Intervals.
https://cran.r-project.org/web/packages/bcaboot/bcaboot.pdf
[31]  Mokhtar, S.F., Md Yusof, Z. and Sapiri, H. (2023) Confidence Intervals by Bootstrapping Approach: A Significance Review. Malaysian Journal of Fundamental and Applied Sciences, 19, 30-42.
https://doi.org/10.11113/mjfas.v19n1.2660
[32]  Grün, B. and Miljkovic, T. (2023) The Automated Bias-Corrected and Accelerated Bootstrap Confidence Intervals for Risk Measures. North American Actuarial Journal, 27, 731-750.
https://doi.org/10.1080/10920277.2022.2141781
[33]  Efron, B. and Narasimhan, B. (2020) The Automatic Construction of Bootstrap Confidence Intervals. Journal of Computational and Graphical Statistics, 29, 608-619.
https://doi.org/10.1080/10618600.2020.1714633
[34]  Wu, S., Zhu, X. and Wang, H. (2023) Subsampling and Jackknifing: A Practically Convenient Solution for Large Data Analysis with Limited Computational Resources. Statistica Sinica, 33, 2041-2064.
https://doi.org/10.5705/ss.202021.0257
[35]  D’Agostino, R.B. (1970) Transformation to Normality of the Null Distribution of G1. Biometrika, 57, 679-681.
https://doi.org/10.1093/biomet/57.3.679
[36]  Anscombe, F.J. and Glynn, W.J. (1983) Distribution of the Kurtosis Statistic b2 for Normal Samples. Biometrika, 70, 227-234.
https://doi.org/10.1093/biomet/70.1.227
[37]  Royston, P. (1993) A Toolkit for Testing for Non-Normality in Complete and Censored Samples. The Statistician, 42, 37-43.
https://doi.org/10.2307/2348109
[38]  Shapiro, S.S. and Francia, R.S. (1972) An Approximate Analysis of Variance Test for Normality. Journal of the American Statistical Association, 67, 215-216.
https://doi.org/10.1080/01621459.1972.10481232
[39]  D’Agostino, R.B., Belanger, A. and D’agostino, R.B. (1990) A Suggestion for Using Powerful and Informative Tests of Normality. The American Statistician, 44, 316-321.
https://doi.org/10.1080/00031305.1990.10475751
[40]  Canty, A., Ripley, B. and Brazzale, A.R. (2025) ‘Boot’: Bootstrap Functions.
https://doi.org/10.32614/CRAN.package.boot
[41]  Komsta, L. and Novomestky, F. (2022) ‘Moments’: Moments, Cumulants, Skewness, Kurtosis and Related Tests.
https://doi.org/10.32614/CRAN.package.moments
[42]  Gross, J. and Ligges, U. (2015) ‘Nortest’: Tests for Normality.
https://doi.org/10.32614/CRAN.package.nortest
[43]  Pouillot, R. Delignette-Muller, M.L., Denis, J.B., Chen, Y and Havelaar, A. (2024) ‘MC2D’: Tools for Two-Dimensional Monte-Carlo Simulations.
https://doi.org/10.32614/CRAN.package.mc2d
[44]  Clark, C.E. (1962) Letter to the Editor—The PERT Model for the Distribution of an Activity Time. Operations Research, 10, 405-406.
https://doi.org/10.1287/opre.10.3.405
[45]  American Psychiatric Association (2022) Diagnostic and Statistical Manual of Mental Disorders. 5th Edition, Text Revised, American Psychiatric Association Publishing.
https://doi.org/10.1176/appi.books.9780890425787
[46]  Matthys, W. and Schutter, D.J.L.G. (2022) Improving Our Understanding of Impaired Social Problem-Solving in Children and Adolescents with Conduct Problems: Implications for Cognitive Behavioral Therapy. Clinical Child and Family Psychology Review, 25, 552-572.
https://doi.org/10.1007/s10567-021-00376-y
[47]  Tukey, J.W. (1977) Exploratory Data Analysis. Addison-Wesley.
[48]  Chattamvelli, R. and Shanmugam, R. (2023) Descriptive Statistics for Scientists and Engineers (Synthesis Lectures on Mathematics and Statistics). 2th Edition, Springer.
https://doi.org/10.1007/978-3-031-32330-0
[49]  Nikolic, B. and Popovic, T. (2024) Variables, Data, and Descriptive Statistical Methods—Part I. Medicinski pregled, 77, 325-330.
https://doi.org/10.2298/mpns2412325n
[50]  Zhang, X., Astivia, O.L.O., Kroc, E. and Zumbo, B.D. (2023) How to Think Clearly about the Central Limit Theorem. Psychological Methods, 28, 1427-1445.
https://doi.org/10.1037/met0000448
[51]  Chernozhuokov, V., Chetverikov, D., Kato, K. and Koike, Y. (2022) Improved Central Limit Theorem and Bootstrap Approximations in High Dimensions. The Annals of Statistics, 50, 2562-2586.
https://doi.org/10.1214/22-aos2193
[52]  Bera, A.K. and Koley, M. (2023) A History of the Delta Method and Some New Results. Sankhya B, 85, 272-306.
https://doi.org/10.1007/s13571-023-00305-9

Full-Text

Contact Us

service@oalib.com

QQ:3279437679

WhatsApp +8615387084133