Abstract
Longitudinal studies often involve multiple mixed response variables measured repeatedly over time. Although separate modeling of these multiple mixed response variables can be easily performed, they may lead to inefficient estimates and consequently, misleading inferences. For obtaining correct inference, one needs to model multiple mixed responses jointly. In this paper, we use copula models for defining a multivariate distribution for multiple mixed outcomes at each time point. Then, we use transition model for considering association between longitudinal measurements. Two simulation studies are performed for illustration of the proposed approach. The results of the simulation studies show that the use of the separate models instead of the joint modeling leads to inefficient parameter estimates. The proposed approach is also used for analyzing two real data sets. The first data set is a part of the British Household Panel Survey. In this data set, annual income and life satisfaction are considered as the continuous and the ordinal correlated longitudinal responses, respectively. The second data set is a longitudinal data about heart failure patients. This study is a treatment–control study, where the effect of a treatment is simultaneously investigated on readmission and referral to doctor as two binary associated longitudinal responses.
Similar content being viewed by others
References
Aas, K., Czado, C., Frigessi, A., Bakken, H.: Pair-copula constructions of multiple dependence. Insurance Math. Econ. 44(2), 182–198 (2009)
Agresti, A.: Categorical Data Analysis, vol. 996. Wiley, New York (1996)
Agresti, A.: Analysis of Ordinal Categorical Data, vol. 656. Wiley, New York (2010)
Baghfalaki, T., Ganjali, M., Berridge, D.: Joint modeling of multivariate longitudinal mixed measurements and time to event data using a Bayesian approach. J. Appl. Stat. 41(9), 1934–1955 (2014)
Bandyopadhyay, S., Ganguli, B., Chatterjee, A.: A review of multivariate longitudinal data analysis. Stat. Methods Med. Res. 20(4), 299–330 (2011)
de Leon, A.R., Wu, B.: Copula-based regression models for a bivariate mixed discrete and continuous outcome. Stat. Med. 30(2), 175–185 (2011)
Diggle, P., Heagerty, P., Liang, K.Y., Zeger, S.: Analysis of Longitudinal Data. Oxford Unuiversity Press, Oxford (2002)
Fieuws, S., Verbeke, G.: Pairwise fitting of mixed models for the joint modeling of multivariate longitudinal profiles. Biometrics 62(2), 424–431 (2006)
Ganjali, M., Baghfalaki, T.: A copula approach to joint modeling of longitudinal measurements and survival times using Monte Carlo expectation-maximization with application to AIDS studies. J. Biopharm. Stat. 25(5), 1077–1099 (2015)
Greene, W.H.: Marginal effects in the bivariate probit model. NYU Working Paper No. EC-96-11 (1996)
Jiryaie, F., Withanage, N., Wu, B., de Leon, A.R.: Gaussian copula distributions for mixed data, with application in discrimination. J. Stat. Comput. Simul. 86(9), 1643–1659 (2015)
Kirk, D.B.: On the numerical approximation of the bivariate normal (tetrachoric) correlation coefficient. Psychometrika 38(2), 259–268 (1973)
Lambert, P., Vandenhende, F.: A copula-based model for multivariate non-normal longitudinal data: analysis of a dose titration safety study on a new antidepressant. Stat. Med. 21(21), 3197–3217 (2002)
Liang, K.Y., Zeger, S.L.: Longitudinal data analysis using generalized linear models. Biometrika 73(1), 13–22 (1986)
Masarotto, G., Varin, C.: Gaussian copula marginal regression. Electron. J. Stat. 6, 1517–1549 (2011)
Molenberghs, G., Lesaffre, E.: Marginal modeling of correlated ordinal data using a multivariate Plackett distribution. J. Am. Stat. Assoc. 89(426), 633–644 (1994)
Molenberghs, G., Verbeke, G.: Models for Discrete Longitudinal Data. Springer Series in Statistics. Springer, New York (2005)
Morrell, C.H., Brant, L.J., Sheng, S., Najjar, S.: Using multivariate mixed-effects models to predict hypertension. In: Joint Statistical Meetings-Biometrics Section-to include ENAR & WNAR (2003)
Panagiotelis, A., Czado, C., Joe, H.: Pair copula constructions for multivariate discrete data. J. Am. Stat. Assoc. 107(499), 1063–1072 (2012)
Reeves, J., MacKenzie, G.: A bivariate regression model with serial correlation. J. R. Stat. Soc. Ser. D (Stat.) 47(4), 607–615 (1998)
Reinsel, G.: Multivariate repeated-measurement or growth curve models with multivariate random-effects covariance structure. J. Am. Stat. Assoc. 77(377), 190–195 (1982)
Reinsel, G.: Estimation and prediction in a multivariate random effects generalized linear model. J. Am. Stat. Assoc. 79(386), 406–414 (1984)
Rencher, A.C.: Methods of Multivariate Analysis, vol. 492. Wiley, New York (2003)
Rochon, J.: Analyzing bivariate repeated measures for discrete and continuous outcome variables. Biometrics 52(2), 740–750 (1996)
Roy, J., Lin, X.: Latent variable models for longitudinal data with multiple continuous outcomes. Biometrics 56(4), 1047–1054 (2000)
Shah, A., Laird, N., Schoenfeld, D.: A random-effects model for multiple characteristics with possibly missing data. J. Am. Stat. Assoc. 92(438), 775–779 (1997)
Sklar, A.: Fonctions de repartition a n dimensions et leurs marges. Pub. Inst. Stat. Univ. Paris 8, 229–231 (1959)
Smith, M., Min, A., Almeida, C., Czado, C.: Modeling longitudinal data using a pair-copula decomposition of serial dependence. J. Am. Stat. Assoc. 105, 1467–1479 (2012)
Song, X.K.: Correlated Data Analysis: Modeling, Analytics, and Applications. Springer Science and Business Media, New York (2007)
Song, P.X.K., Li, M., Yuan, Y.: Joint regression analysis of correlated data using Gaussian copulas. Biometrics 65(1), 60–68 (2009)
Taylor, M.F., Brice, J., Buck, N., Prentice-Lane, E.: British Household Panel Survey User Manual. Volume A: Introduction, Technical Report and Appendices. Colchester: University of Essex (2005)
Teimourian, M., Baghfalaki, T., Ganjali, M., Berridge, D.: Joint modeling of mixed skewed continuous and ordinal longitudinal responses: a Bayesian approach. J. Appl. Stat. 42(10), 2233–2256 (2015)
Trivedi, P.K., Zimmer, D.M.: Copula modeling: an introduction for practitioners. Found. Trends Econom. 1(1), 1–111 (2007)
Verbeke, G., Molenberghs, G.: Linear Mixed Models for Longitudinal Data. Springer Science and Business Media, New York (2000)
Verbeke, G., Fieuws, S., Molenberghs, G., Davidian, M.: The analysis of multivariate longitudinal data: a review. Stat. Methods Med. Res. 23(1), 42–59 (2014)
Yau, K.K., Lee, A.H., Ng, A.S.: Theory & methods: a zero-augmented gamma mixed model for longitudinal data with many zeros. Aust. N. Z. J. Stat. 44(2), 177–183 (2002)
Zellner, A.: An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. J. Am. Stat. Assoc. 57(298), 348–368 (1962)
Zhu, C., Byrd, R.H., Lu, P., Nocedal, J.: Algorithm 778: L-BFGS-B: fortran subroutines for large-scale bound-constrained optimization. ACM Trans. Math. Softw. (TOMS) 23(4), 550–560 (1997)
Acknowledgements
The authors are grateful to School of Biological Sciences at IPM for their supports. This research is supported in part by a grant (BS-1396-02-01) from the Institute for Research in Fundamental Sciences (IPM), Tehran, Iran.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Baghfalaki, T., Ganjali, M. A transition model for analyzing multivariate longitudinal data using Gaussian copula approach. AStA Adv Stat Anal 104, 169–223 (2020). https://doi.org/10.1007/s10182-018-00346-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10182-018-00346-w