Bivariate Copula-based Linear Mixed-effects Models: An Application to Longitudinal Child Growth Data
DOI:
https://doi.org/10.5540/tema.2019.020.01.37Keywords:
bivariate copula, linear mixed-effects model, longitudinal growth data, time-varying dependenceAbstract
Multiple longitudinal outcomes are common in public health research and adequate methods are required when there is interest in the joint evolution of response variables over time. However, the main drawback of joint modeling procedures is the requirement to specify the joint density of all outcomes and their correlation structure, as well as numerical difficulties in statistical inference, when the dimension of these outcomes increases. To overcome such difficulty, we present two procedures to deal with multivariate longitudinal data. We first present an univariate approach, for which linear mixed-effects models are considered for each response variable separately. Then, a novel copula-based modeling is presented, in order to characterize relationships among the response variables. Both methodologies are applied to a real Brazilian data set on child growth.
References
Akaike, H. (1977). On entropy maximization principle. In: Applications of Statistics, P. R. Krishnaiah (ed.), North-Holland, Amsterdam, 27-41.
Black, R. E., Victoria, C. G., Walker, S. P., Bhutta, Z. A., Christian, P., and de Onis, M. (2013). Maternal and child undernutrition and overweight in low-income and middle-income countries. Lancet, 382, 427-451.
Byrd, R. H., Lu, P., Nocedal, J., and Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM J. Scientific Computing, 16, 1190-1208.
Carpenter, J., and Bithell, J. (2000). Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians. Statistics in Medicine, 19, 1141-1164.
Catalano, P., and Ryan, L. (1992). Bivariate latent variable models for clustered discrete and continuous outcomes. Journal of the American Statistical Association, 87, 651-658.
Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, 141-152.
Cherubini, U., Luciano, E., and Vecchiato, W. (2004). Copula Methods in Finance. Chichester: Wiley.
Cho, H. (2016). The analysis of multivariate longitudinal data using multivariate marginal models. Journal of Multivariate Analysis, 143, 481-491.
Cox, D., and Wermuth, N. (1992). Response models for mixed binary and quantitative variables. Biometrika, 79, 441-461.
Diggle, P. J., Heagerty, P., Liang, K.-Y., and Zeger, S. L. (2002). Analysis of Longitudinal Data. Oxford: Oxford University Press, 2nd Edition.
Domma, F., Giordano, S., and Perri, P. F. (2009). Statistical modeling of temporal dependence in financial data via a copula function. Communications in Statistics - Simulation and Computation, 38, 703-728.
Efron, B., and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. New York: Chapman & Hall.
Embrechts, P., McNeil, A., Straumann, D. (2002). Correlation and dependence in risk management: properties and pitfalls. In: Risk Management: Value at Risk and Beyond, M. A. H. Dempster (ed.), Cambridge University Press, Cambridge, 176-223.
Fang, K.-T., Kotz, S., and Ng, K.-W. (1990). Symmetric Multivariate and Related Distributions. London: Chapman & Hall.
Fieuws, S., and Verbeke, G. (2006). Pairwise fitting of mixed models for the joint modeling of multivariate longitudinal profiles. Biometrics, 62, 424-431.
Fitzmaurice, G., Davidian, M., Verbeke, G., Geert, M. (2009). Longitudinal Data Analysis. Handbooks of Modern Statistical Methods. New York: Chapman & Hall/CRC.
Fitzmaurice, G., and Laird, N. (1995). Regression models for a bivariate discrete and continuous outcome with clustering. Journal of the American Statistical Association, 90, 845-852.
Frank, M. J. (1979). On the simultaneous associativity of F(x, y) and x + y - F(x,y). Aequationes Mathematicae, 19, 194-226.
Frees, E. W., and Wang, P. (2006). Copula credibility for aggregate loss models. Insurance Mathematics & Economics, 36, 360-373.
Genest, C., and Boies, J. C. (2003). Detecting dependence with Kendall plots. The American Statistician, 57, 275-284.
Genest, C., and Favre, A. C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12, 347-368.
Giles, L. C., Whitrow, M. J., Davies, M. J., Davies, C. E., Rumbold, A. R., Moore, V. M., et al. (2015). Growth trajectories in early childhood, their relationship with antenatal and postnatal factors, and development of obesity by age 9 years: results from an Australian birth cohort study. International Journal of Obesity, 39, 1049-1056.
Gueorguieva, R. V., and Agresti, A. (2001). A correlated probit model for joint modeling of clustered binary and continuous responses. Journal of the American Statistical Association, 96, 1102-1112.
Gumbel, E. J. (1960). Bivariate exponential distributions. Journal of the American Statistical Association, 55, 698-707.
Huybregts, L., Becquey, E., Zongrone, A., Le Port, A., Khassanova, R., Coulibaly, L., et al. (2017). The impact of integrated prevention and treatment on child malnutrition and health: the PROMIS project, a randomized control trial in Burkina Faso and Mali. BMC Public Health, 17, 237.
Ida, A., Ishimura, N., Nakamura, M. (2014). Note on the measures of dependence in terms of copulas. Procedia Economics and Finance, 14, 273-279.
Joe, H. (1997). Multivariate Models and Dependence Concepts. London: Chapman & Hall.
Joe, H. (2014). Dependence Modeling with Copulas. Boca Raton: Chapman & Hall.
Joe, H., and Xu, J. (1996). The estimation method of inference functions for margins for multivariate models. Technical Report 166, Department of Statistics, University of British Columbia.
Lambert, P., and Vandenhende, F. (2002). A copula-based model for multivariate non-normal longitudinal data: Analysis of a dose titration safety study on a new antidepressant. Statistics in Medicine, 21, 3197-3217.
Littell, R. C., Pendergast, J., and Natarajan, R. (2000). Modelling covariance structure in the analysis of repeated measures data. Statistics in Medicine, 19, 1793-1819.
Liu, X., Daniels, M. J., and Marcus, B. (2009). Joint models for the association of longitudinal binary and continuous processes with application to a smoking cessation trial. Journal of the American Statistical Association, 104, 429-438.
Manner, H., and Reznikova, O. (2012). A survey on time-varying copulas: Specification, simulations, and application. Econometric Reviews, 31, 654-687.
Merialdi, M., Widmer, M., Gulmezoglu, A. M., Abdel-Aleem, H., Bega, G., Benachi, A., et al. (2014). WHO multicentre study for the development of growth standards from fetal life to childhood: the fetal component. BMC Pregnancy Childbirth, 14, 157.
Molenberghs, G., and Verbeke, G. (2005). Models for Discrete Longitudinal Data. New York: Springer.
Nelsen, R. B. (2006). An Introduction to Copulas. 2nd ed., New York: Springer.
Patel, R., Tilling, K., Lawlor, D. A., Howe, L. D., Bogdanovich, N., Matush, L., et al. (2014). Socioeconomic differences in childhood length/height trajectories in a middle-income country: a cohort study. BMC Public Health, 14, 932.
Prentice, A. M., Nabwera, H., Unger, S., and Moore, S. E. (2016). Growth monitoring and the prognosis of mortality in low-income settings. The American Journal of Clinical Nutrition, 103, 681-682.
Prudhon, C., Langendorf, C., Roederer, T., Doyon, S., Mamaty, A. A., Woi-Messe, L., et al. (2017). Effect of ready-to-use foods for preventing child undernutrition in Niger: analysis of a prospective intervention study over 15 months of follow-up. Maternal & Child Nutrition, 13, 1.
R Core Team (2016). R: A language and environment for statistical computing (version 3.3.2). R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/.
Sammel, M., Ryan, L., and Legler, J. (1997). Latent variable models for mixed discrete and continuous outcomes. Journal of the Royal Statistical Society, Series B, 59, 667-678.
Scherdel, P., Dunkel, L., van Dommelen, P., Goulet, O., Salaun, J. F., Brauner, R., et al. (2016). Growth monitoring as an early detection tool: a systematic review. The Lancet Diabetes & Endocrinology, 4, 447-456.
Schirmacher, D., and Schirmacher, E. (2008). Multivariate dependence modeling using pair-copulas. Technical report, Society of Acturaries: 2008 Enterprise Risk Management Symposium, April 14-16, Chicago.
Schwarz, G. E. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461-464.
Schweizer, B., and Wolff, E. F. (1981). On nonparametric measures of dependence for random variables. The Annals of Statistics, 9, 879-885.
Scott, S. L., James, G. M., and Sugar, C. A. (2005). Hidden markov models for longitudinal comparisons. Journal of the American Statistical Association, 100, 359-369.
Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publications de l'Institut de Statistique de l'Université de Paris, 8, 229-231.
Sun, J., Frees, E. W., and Rosenberg, M. A. (2008). Heavy-tailed longitudinal data modeling using copulas. Insurance Mathematics & Economics, 42, 817-830.
Verbeke, G., Fieuws, S., Molenberghs, G. (2014). The analysis of multivariate longitudinal data: A review. Statistical Methods in Medical Research, 23, 42-59.
Verbeke, G., and Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. New York: Springer-Verlag.
Verbeke, G., Molenberghs, G., Rizopoulos, D. (2010). Random effects models for longitudinal data. In: Longitudinal Research with Latent Variables, K. Montfort, H. HO, A. Satorra (eds.), Springer-Verlag, New York, 37-96.
Victora, C. G., Adair, L., Fall, C., Hallal, P. C., Martorell, R., Richter, L., et al. (2008). Maternal and child undernutrition: consequences for adult health and human capital. Lancet, 371, 340-357.
West, B. T., Welch, K. B., and Galecki, A. T. (2014). Linear Mixed Models: A Practical Guide Using Statistical Software. 2nd ed., Boca Raton: Chapman & Hall.
Whelan, N. (2004). Sampling from Archimedean Copulas. Quantitative Finance, 4, 339-352.
Xu, J., and Mackenzie, G. (2012). Modelling covariance structure in bivariate marginal models for longitudinal data. Biometrika, 99, 649-662.
Yan, J. (2006). Multivariate modeling with copulas and engineering applications. In: Handbook in Engineering Statistics, H. Pham (ed.), Springer-Verlag, New York, 973-990.
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