Modeling Dependence Relationships of Anthropometric Variables Using Copula Approach
American Journal of Theoretical and Applied Statistics
Volume 9, Issue 5, September 2020, Pages: 245-255
Received: Sep. 3, 2020; Accepted: Sep. 21, 2020; Published: Oct. 22, 2020
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Funmilayo Westnand Oshogboye Saporu, National Mathematical Centre, Kwali, Abuja, Nigeria
Isaac Esbond Gongsin, Department of Mathematical Sciences, University of Maiduguri, Maiduguri, Nigeria
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Copula model is introduced in modeling the co-dependence structures of anthropometric variables-Body mass index (BMI), Abdominal circumference, Adiposity and Percent body fat-because it can capture monotonic dependence. Four copula-based Kumaraswamy-epsilon distributions are derived and used to determine the best fit to the anthropometric data, these are new. These are the Gaussian, Clayton, Frank and Gumbel copulas. Clayton model provided the best fit in four bivariate pairs-BMI and Percent body fat, BMI and Abdominal circumference, Adiposity and Abdominal circumference and Abdominal circumference and Percent body fat-while Gaussian is best for BMI and Adiposity pair and Frank is best for Adiposity and Percent body fat pair. Copula-based Kendall’s tau and tail dependence are used as estimates for measuring the strength of the co-dependence. The results strongly recommend the use of BMI as an anthropometric index for estimating human body composition of adiposity. However for individuals with BMI values in the two extreme tails, their adiposity should be measured directly. The results do not find any suitable anthropometric indices for estimating percent body fat and therefore is recommended that for such epidemiological research, percent body fat should be measured directly. The results also clearly show that the Kendall’s tau and the corresponding Pearson correlation coefficient estimates are largely at variance whenever the co-dependence structure cannot be described as linear dependence. This can prompt contradictory conclusions. It is therefore suggested that for such research, whenever Pearson correlation coefficient method is in use, a coefficient of determination of a minimum of 75% should be obtained before any anthropometric index can be recommended for body composition substitution.
Anthropometric Index, Body Composition, Correlation Matrix, Inference Function for Margin, Kendall’s Tau, Kumaraswamy-epsilon Distribution, Monotonic Dependence
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Funmilayo Westnand Oshogboye Saporu, Isaac Esbond Gongsin, Modeling Dependence Relationships of Anthropometric Variables Using Copula Approach, American Journal of Theoretical and Applied Statistics. Vol. 9, No. 5, 2020, pp. 245-255. doi: 10.11648/j.ajtas.20200905.18
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Aas, K.: Modelling the dependence structure of financial assets: A survey of four copulas, Norwegian Computing Center, Applied Research and Development, Note no.: SAMBA/22/04 (2004).
Akindele, M. O., Phillips, J. S. and Igumbor, E. U.: The relationship between percent body fatage and body mass index in overweight and obese individuals in an urban African setting, Journal of Public Health in Africa 2016; volume 7: 515, pp 15-19 (2016) https://doi:10.4081/jphia.2016.515
Arbel, Y., Birati, E. Y., Shapira, I., Finn, T., Berliner, S. & Rogowski, O.: Comparison of Different Anthropometric Measurements and Inflammatory Biomarkers. International Journal of Inflammation, vol 2012, article ID 124639, pp 1-5 (2012) doi: 10.1155/2012/124639.
Assessing Central Obesity: Waist Circumference (2009), accessed January 25, 2020.
Balkau, B., Deanfield, J. E., Després, J-P., Bassand, J-P., Fox, K. A. A., Smith, S. C., Barter, P., Tan, C-E., Gaal, L. V., Wittchen, H-U., Massien, C., Haffner, S. M.: A Study of Waist Circumference, Cardiovascular Disease, and Diabetes Mellitus in 168 000 Primary Care Patients in 63 Countries, Circulation. 2007; 116: 1942-1951. DOI: 10.1161/CIRCULATIONAHA.106.676379, available at
Bergman, R, N., Stefanouski, D. & Buchanan, T. A.: A better index of body adiposity. Obesity, vol 19, no. 5, pp 1083-1089 (2011).
Brannsether, B., Eide, G. E., Roelants, M., Bjerknes, R. & Juliusson, P. B., Interrelationships between anthropometric variables and overweight in childhood and adolescence. American Journal of Human Biology 26: 502–510 (2014).
Carnicero, J. A., Ausin, M. C., & Wiper, M. P.: Non-parametric copulas for circular-linear and circular-circular data: An application to wind directions. Stochastic Environmental Research and Risk Assessment, 27, pp. 1991–2002 (2013).
Cordeiro, G. M. & de Castro, M.: A new family of generalized distributions, Journal of statistical computation and simulation. 81, 883-898 (2011)
de Michele, C., Salvadori, G., Vezzoli, R. & Pecora, S.: Multivariate assessment of droughts: Frequency analysis and dynamic return period. Water Resources Research, 49, pp. 6985–6994 (2013).
Dobra, A. & Lenkoski, A.: Copula Gaussian graphical models and their application to modeling functional disability data. Annals of Applied Statistics, 5, pp. 969–993 (2011).
Dombi, J., J´on´as, T. & T´oth, Z., E.: The Epsilon Probability Distribution and its Application in Reliability Theory. Acta Polytechnica Hungarica, Vol. 15, No. 1, pp 197-216 (2018).
Embrechts, P., McNeil A. & Strauman, D.: Correlation and Dependency in Risk Management, Preprint ETH Zurich (1999).
Fermanian, J-D.: Recent Developments in Copula Models, Econometrics, 5, 34 (2017) https://doi:10.3390/econometrics5030034
Genest, C., Rémilard, B. & Beaudoin, D.: Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics, 44, 199-213 (2009)
Gongsin, I. E. & Saporu F. W. O.: On the Construction of Kumaraswamy-Epsilon Distribution with Applications, International Journal of Science and Research, Volume 8 Issue 11, pp 1199-1204 (2019).
Hofert, M. & Scherer, M.: Collateralized Debt Obligation pricing with nested Archimedean copulas. Quantitative Finance (2011).
Joe, H.: Multivariate models and dependence concepts, Monographs on Statistics and Applied Probability, 73, Chapman & Hall, London (1997).
Joe, H.: Asymptotic efficiency of the two-stage estimation method for copula-based models, Journal of Multivariate Analysis, 94, 401–419 (2005).
Joe, H. & Xu, J. J.: The estimation method of inference functions for the margins for multivariate models, Department of Statistics, University of British Columbia, Technical Report, 166 (1996).
Jung, Y. S., Kim, J. K. & KIM, J.: New approach of directional dependence in exchange markets using generalized FGM copula function. Communications in Statistics-Simulation and Computation, 37, pp. 772–788 (2008).
Kim, J. M., Jung, Y. S. & Soderberg, T.: Directional dependence of genes using survival truncated FGM type modification copulas. Communications in Statistics-Simulation and Computation, 38, pp. 1470–1484 (2009)
Kojadinovic, I. & Yan, J.: Modeling Multivariate Distributions with Continuous Margins using the copula R Package. Journal of Statistical Software, vol. 34, Issue 9, pp 1-20 (2010)
Leong, I-F., Fang, J-J. & Tsai, M-J.: A feature-based anthropometry for garment industry, International Journal of Clothing Science and Technology, Vol. 25 No. 1, pp. 6-23 (2013) DOI: 10.1108/09556221311292183.
Li, D. X.: On Default Correlation: A Copula Function Approach, Journal of Fixed Income, 9: 43-54 (2000).
Lichtash CT, Cui J, Guo X, Chen Y-DI, Hsueh WA, et al.: Body Adiposity Index versus Body Mass Index and Other Anthropometric Traits as Correlates of Cardio-metabolic Risk Factors. PLoS ONE 8 (6): e65954 (2013) https://doi:10.1371/journal.pone.0065954
Lojowska, A., Kurowicka, D., Papaefthymiou, G., & Van Der Sluis, L.: Stochastic modeling of power demand due to EVs using copula. IEEE Transactions on Power Systems, 27, pp. 1960–1968 (2012).
Louzada, F., Suzuki, A. K., Cancho, V. G., Prince, F. L. & Pereira, G. A.: The long-term bivariate survival FGM copula model: An application to a Brazilian HIV data. Journal of Data Science, 10, pp. 511–535 (2012).
Luke, A., Durazo-Arvizu, R., Rotimi, C., Prewitt, T. E., Forrester, T., Wilks, R., Ogunbiyi, O. J., Schoeller, D. A., McGee, D. & Cooper, R. S.: Relation between Body Mass Index and Body fat in Black Population Samples from Nigeria, Jamaica, and the United States, American Journal of Epidemiology, Vol. 145, No 7, pp 620-628 (1997).
Nadarajah, S., Afuecheta, E. & Chan, S.: A Compendium of Copulas. STATISTICA, anno LXXVII, n. 4 (2017).
Nelsen, R. B.: An Introduction to Copulas. Springer Verlag, New York (2006).
Pettere, G. & Kollo, T.: Risk modeling for future cash flow using skew t -copula. Communications in Statistics - Theory and Methods, 40, pp. 2919–2925 (2011).
Raimi, T. H. & Oluwayemi, I. O.: Anthropometric Correlates and Prediction of Body fat Measured by Bioelectric Impedance Analysis among Women. Annals of Medical and Health Sciences Research (7), pp 32-35 (2017).
Reddy, M. J. & Ganguli, P.: Risk Assessment of Hydroclimatic Variability on Groundwater Levels in the Manjara Basin Aquifer in India Using Archimedean Copulas. Journal of Hydrologic Engineering Vol. 17, 12, 1345-1357 (2012)
Salvadori, G. & de Michele, C.: Multivariate real-time assessment of droughts via copula-based multi-site Hazard Trajectories and Fans. Journal of Hydrology, 526, pp. 101–115 (2015).
Sklar, A.: Fonctions de Répartition à n Dimensions et Leurs Marges. Publications de l'Institut de Statistique de L'Université de Paris, 8, 229-231 (1959).
Söğüt, M., Altunsoy, K., Varela-Silva, M. I.: Associations between anthropometric indicators of adiposity and percent body fatage in normal weight young adults, Anthropological Review, Vol. 81 (2), 174–181 (2018). Available online at:
Sun, W., Rachev, S., Stoyanov, S. V. & Fabozzi, F. J.: Multivariate skewed Student’s t copula in the analysis of nonlinear and asymmetric dependence in the German equity market. Studies in Nonlinear Dynamics and Econometrics, 12 (2008)
US Department of Health and Human Services, National Institute of Health. WIN Weight Control Network: Understanding Adult Obesity. NIH Publication No. 01-3680.
Valdez, E. & Frees, E.: Understanding Relationships Using Copulas, Actuarial Research Clearing House, Proceedings, 32nd Actuarial Research Conference, August 6-8, 1997, 5 (1998).
Vinué, G.: Anthropometry: An R Package for Analysis of Anthropometric Data, Journal of Statistical Software, Volume 77, Issue 6, pp 1-39 (2017)
Xie, K., Li, Y. & Li, W.: Modeling wind speed dependence in system reliability assessment using copulas. IET Renewable Power Generation, 6, pp. 392–399 (2012).
Yu, J., Chen, K., Mori, J. & Rashid, M. M.: A Gaussian mixture copula model based localized Gaussian process regression approach for long-term wind speed prediction. Energy, 61, pp. 673–686 (2013).
Zeng, Q., Dong, S_Y., Sun, X_N, Xie, J. & Cui, Y.: Percent body fat is better predictor of cardiovascular risk factors than body mass index, Brazilian Journal of Medical and Biological Research. Vol. 45 (7), pp 591-600 (2012) Doi: 10.1590/S0100-879X2012007500059,
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