Statistics and Its Interface

Volume 11 (2018)

Number 2

Time-varying copula models for longitudinal data

Pages: 203 – 221



Esra Kürüm (Department of Statistics, University of California at Riverside)

John Hughes (Department of Biostatistics and Informatics, Colorado School of Public Health, University of Colorado, Aurora, Co., U.S.A.)

Runze Li (Department of Statistics and The Methodology Center, Pennsylvania State University, University Park, Penn., U.S.A.)

Saul Shiffman (Department of Psychology, University of Pittsburgh, Pennsylvania, U.S.A.)


We propose a copula-based joint modeling framework for mixed longitudinal responses. Our approach permits all model parameters to vary with time, and thus will enable researchers to reveal dynamic response–predictor relationships and response–response associations. We call the new class of models TIMECOP because we model dependence using a time-varying copula. We develop a one-step estimation procedure for the TIMECOP parameter vector, and also describe how to estimate standard errors. We investigate the finite sample performance of our procedure via three simulation studies, one of which shows that our procedure performs well under ignorable missingness. We also illustrate the applicability of our approach by analyzing binary and continuous responses from the Women’s Interagency HIV Study and a smoking cessation program.


bimodal kernel, HIV, joint model, local regression, varying coefficient model

2010 Mathematics Subject Classification

Primary 62G08. Secondary 62H20.

Hughes’ research was supported by a grant from the Simons Foundation (#243657) and Li’s research was supported by National Science Foundation DMS 1512422, National Institutes of Health (NIH), National Institute on Drug Abuse (NIDA) grants P50 DA039838 and P50 DA036107, NIH, National Library of Medicine (NLM), T32 LM012415 and National Nature Science Foundation of China (NNSFC) grants 11690015. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NSF, the NIDA, the NLM, the NIH or the NNSFC.

Data in this manuscript were collected by the Women’s Interagency HIV Study (WIHS) Collaborative Study Group with centers (Principal Investigators) at New York City/Bronx Consortium (Kathryn Anastos); Brooklyn, NY (Howard Minkoff); Washington, DC, Metropolitan Consortium (Mary Young); The Connie Wofsy Study Consortium of Northern California (Ruth Greenblatt); Los Angeles County/Southern California Consortium (Alexandra Levine); Chicago Consortium (Mardge Cohen); Data Coordinating Center (Stephen Gange). The WIHS is funded by the National Institute of Allergy and Infectious Diseases (UO1-AI-35004, UO1-AI-31834, UO1-AI-34994, UO1-AI-34989, UO1-AI-34993, and UO1-AI-42590) and by the Eunice Kennedy Shriver National Institute of Child Health and Human Development (UO1-HD-32632). The study is co-funded by the National Cancer Institute, the National Institute on Drug Abuse, and the National Institute on Deafness and Other Communication Disorders. Funding is also provided by the National Center for Research Resources (UCSF-CTSI Grant Number UL1 RR024131).

Received 5 February 2017

Published 7 March 2018