A D‐vine copula‐based model for repeated measurements extending linear mixed models with homogeneous correlation structure

We propose a model for unbalanced longitudinal data, where the univariate margins can be selected arbitrarily and the dependence structure is described with the help of a D-vine copula. We show that our approach is an extremely flexible extension of the widely used linear mixed model if the correlation is homogeneous over the considered individuals. As an alternative to joint maximum-likelihood a sequential estimation approach for the D-vine copula is provided and validated in a simulation study. The model can handle missing values without being forced to discard data. Since conditional distributions are known analytically, we easily make predictions for future events. For model selection, we adjust the Bayesian information criterion to our situation. In an application to heart surgery data our model performs clearly better than competing linear mixed models.

[1]  C. Czado,et al.  Modeling Longitudinal Data Using a Pair-Copula Decomposition of Serial Dependence , 2010 .

[2]  H. Joe.  Generating random correlation matrices based on partial correlations , 2006 .

[3]  Marta Nai Ruscone,et al.  Modelling the Dependence in Multivariate Longitudinal Data by Pair Copula Decomposition , 2016, SMPS.

[4]  Thibault Vatter,et al.  Generalized Additive Models for Pair-Copula Constructions , 2016, Journal of Computational and Graphical Statistics.

[5]  S. Müller,et al.  Model Selection in Linear Mixed Models , 2013, 1306.2427.

[6]  Claudia Czado,et al.  Growing simplified vine copula trees: improving Di{\ss}mann's algorithm , 2017, 1703.05203.

[7]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[8]  J. Cavanaugh,et al.  Generalizing the derivation of the schwarz information criterion , 1999 .

[9]  Hutan Ashrafian,et al.  Longitudinal study of the profile and predictors of left ventricular mass regression after stentless aortic valve replacement. , 2008, The Annals of thoracic surgery.

[10]  Joseph G. Ibrahim,et al.  Missing data methods in longitudinal studies: a review , 2009 .

[11]  Ingrid Hobæk Haff,et al.  Parameter estimation for pair-copula constructions , 2013, 1303.4890.

[12]  Peter J. Diggle,et al.  joineR: Joint modelling of repeated measurements and time-to-event data , 2012 .

[13]  P. Embrechts,et al.  Dependence modeling with copulas , 2007 .

[14]  George Biddell Airy,et al.  On the Algebraical and Numerical Theory of Errors of Observations and the Combination of Observations , 2007 .

[15]  Claudia Czado,et al.  Selecting and estimating regular vine copulae and application to financial returns , 2012, Comput. Stat. Data Anal..

[16]  Philip H. Ramsey.  Nonparametric Statistical Methods , 1974, Technometrics.

[17]  Claudia Czado,et al.  D-vine copula based quantile regression , 2015, Comput. Stat. Data Anal..

[18]  Claudia Czado,et al.  Evading the curse of dimensionality in nonparametric density estimation with simplified vine copulas , 2015, J. Multivar. Anal..

[19]  Shaojun Li,et al.  Sequential Dependence Modeling Using Bayesian Theory and D-Vine Copula and Its Application on Chemical Process Risk Prediction , 2014 .

[20]  Ludwig Fahrmeir,et al.  Regression: Models, Methods and Applications , 2013 .

[21]  Claudia Czado,et al.  Analysis of Australian electricity loads using joint Bayesian inference of D-Vines with autoregressive margins , 2011 .

[22]  Donald Hedeker,et al.  Longitudinal Data Analysis , 2006 .

[23]  Peng Shi,et al.  Multilevel modeling of insurance claims using copulas , 2016 .

[24]  Ulf Schepsmeier,et al.  Estimating standard errors in regular vine copula models , 2013, Comput. Stat..

[25]  Changyu Shen,et al.  A copula model for repeated measurements with non‐ignorable non‐monotone missing outcome , 2006, Statistics in medicine.

[26]  S. R. Searle,et al.  Generalized, Linear, and Mixed Models , 2005 .

[27]  C. Genest,et al.  Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask , 2007 .

[28]  Ana Ivelisse Avilés,et al.  Linear Mixed Models for Longitudinal Data , 2001, Technometrics.

[29]  Claudia Czado,et al.  Examination and visualisation of the simplifying assumption for vine copulas in three dimensions , 2016, 1602.05795.

[30]  Claudia Czado,et al.  Simplified pair copula constructions - Limitations and extensions , 2013, J. Multivar. Anal..

[31]  Christian Genest,et al.  Beyond simplified pair-copula constructions , 2012, J. Multivar. Anal..

[32]  P. J. Lindsey,et al.  Multivariate distributions with correlation matrices for nonlinear repeated measurements , 2006, Comput. Stat. Data Anal..

[33]  Ethel M. Newbold,et al.  Practical Applications of the Statistics of Repeated Events' Particularly to Industrial Accidents , 1927 .

[34]  M. Smith,et al.  Copula Modelling of Dependence in Multivariate Time Series , 2013 .

[35]  H. Joe.  Multivariate Models and Multivariate Dependence Concepts , 1997 .

[36]  Claudia Czado,et al.  Pair-Copula Constructions of Multivariate Copulas , 2010 .

[37]  J. L. Myers,et al.  Regression analyses of repeated measures data in cognitive research. , 1990, Journal of experimental psychology. Learning, memory, and cognition.

[38]  A. Frigessi,et al.  Pair-copula constructions of multiple dependence , 2009 .

[39]  E. Frees,et al.  Heavy-tailed longitudinal data modeling using copulas , 2008 .

[40]  Richard H. Jones,et al.  Bayesian information criterion for longitudinal and clustered data , 2011, Statistics in medicine.

[41]  J. Silvester.  Determinants of block matrices , 2000, The Mathematical Gazette.

[42]  Philippe Lambert,et al.  A copula‐based model for multivariate non‐normal longitudinal data: analysis of a dose titration safety study on a new antidepressant , 2002, Statistics in medicine.

[43]  Antero Malin,et al.  Multilevel Modelling in Repeated Measures of the Quality of Finnish School Life , 2001 .

[44]  S. G. Meester,et al.  A parametric model for cluster correlated categorical data. , 1994, Biometrics.

[45]  Fabian Spanhel,et al.  The partial vine copula: A dependence measure and approximation based on the simplifying assumption , 2015, 1510.06971.

[46]  Maud Delattre,et al.  A note on BIC in mixed-effects models , 2014 .

[47]  Lu Yang,et al.  Pair Copula Constructions for Insurance Experience Rating , 2018 .

[48]  J. Cavanaugh,et al.  The Bayesian information criterion: background, derivation, and applications , 2012 .

[49]  Fabian Spanhel,et al.  Simplified vine copula models: Approximations based on the simplifying assumption , 2015, Electronic Journal of Statistics.

[50]  G. Molenberghs,et al.  Longitudinal data analysis , 2008 .

[51]  James R. Kenyon,et al.  Statistical Methods for the Analysis of Repeated Measurements , 2003, Technometrics.

[52]  J Ludbrook,et al.  Repeated measurements and multiple comparisons in cardiovascular research. , 1994, Cardiovascular research.

[53]  Jong-Min Kim,et al.  Mixture of D-vine copulas for modeling dependence , 2013, Comput. Stat. Data Anal..

[54]  T. Bedford,et al.  Vines: A new graphical model for dependent random variables , 2002 .

[55]  Farid Kianifard,et al.  Models for Repeated Measurements , 2001, Technometrics.

[56]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[57]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[58]  P. Diggle,et al.  Analysis of Longitudinal Data , 2003 .

[59]  H. Joe.  Families of $m$-variate distributions with given margins and $m(m-1)/2$ bivariate dependence parameters , 1996 .

[60]  Roger E Bumgarner,et al.  Clustering gene-expression data with repeated measurements , 2003, Genome Biology.

[61]  Catherine Potvin,et al.  THE STATISTICAL ANALYSIS OF ECOPHYSIOLOGICAL RESPONSE CURVES OBTAINED FROM EXPERIMENTS INVOLVING REPEATED MEASURES , 1990 .

[62]  Kjersti Aas,et al.  On the simplified pair-copula construction - Simply useful or too simplistic? , 2010, J. Multivar. Anal..

[63]  P. Diggle,et al.  A SELECTED BIBLIOGRAPHY ON THE ANALYSIS OF REPEATED MEASUREMENTS and RELATED AREAS , 1989 .