Bayesian Joint Semiparametric Mean–Covariance Modeling for Longitudinal Data

Abstract

Joint parsimonious modeling the mean and covariance is important for analyzing longitudinal data, because it accounts for the efficiency of parameter estimation and easy interpretation of variability. The main potential risk is that it may lead to inefficient or biased estimators of parameters while misspecification occurs. A good alternative is the semiparametric model. In this paper, a Bayesian approach is proposed for modeling the mean and covariance simultaneously by using semiparametric models and the modified Cholesky decomposition. We use a generalized prior to avoid the knots selection while using B-spline to approximate the nonlinear part and propose a Markov Chain Monte Carlo scheme based on Metropolis–Hastings algorithm for computations. Simulation studies and real data analysis show that the proposed approach yields highly efficient estimators for the parameters and nonparametric parts in the mean, meanwhile providing parsimonious estimation for the covariance structure.

Keywords

Cholesky decomposition / Longitudinal data / Bayesian semiparametric model / MCMC

References

[1] Cepeda EC, Gamerman D. Bayesian modeling of joint regressions for the mean and covariance matrix. Biome. J.. 2004, 46 4 430-440

[2] Diggle PJ, Heagerty PJ, Liang KY, Zeger SL. Analysis of Longitudinal Data. 2002 Oxford: Oxford University Press

[3] Diggle PJ, Verbyla AP. Nonparametric estimation of covariance structure in longitudinal data. Biometrics. 1998, 54 403-15

[4] Eilers PHC, Marx BD. Flexible smoothing with B-splines and penalties. Stat. Sci.. 1996, 11 89-121

[5] Gamerman D. Sampling from the posterior distribution in generalized linear mixed models. Stat. Comput.. 1997, 7 57-68

[6] Heckman NE. Spline smoothing in a partly linear model. J. R. Stat. Soc. Ser. B. 1986, 48 244-248

[7] Lang S, Brezger A. Bayesian P-splines. J. Comput. Graph. Stat.. 2004, 13 183-212

[8] Leng C, Zhang W, Pan J. Semiparametric mean–covariance regression analysis for longitudinal data. J. Am. Stat. Assoc.. 2010, 105 489 181-193

[9] Liang KY, Zeger SL. Longitudinal data analysis using generalized linear models. Biometrika. 1986, 73 13-22

[10] Lin X, Carroll RJ. Semiparametric regression for clustered data using generalized estimating equations. J. Am. Stat. Assoc.. 2001, 96 1045-1056

[11] Liu X, Zhang W. A moving average Cholesky factor model in joint mean-covariance modeling for longitudinal data. Sci. China Math.. 2013, 56 2367-2379

[12] Pourahmadi M. Joint mean–covariance models with applications to longitudinal data: unconstrained parameterisation. Biometrika. 1999, 86 677-90

[13] Pourahmadi M. Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix. Biometrika. 2000, 87 425-35

[14] Schumaker LL. Spline Functions: Basic Theory. 1981 New York: Wiley

[15] Stone C. Additive regression and other nonparametric models. Ann. Stat.. 1985, 13 689-705

[16] Wu H, Zhang JT. Nonparametric Regression Methods for Longitudinal Data Analysis: Mixed-Effects Modeling Approaches. 2006 New York: Wiley

[17] Wang Y, Carey V. Working correlation structure misspecification, estimation and covariate design: implications for generalised estimating equations performance. Biometrika. 2003, 90 29-41

[18] Welsh AH, Lin X, Carroll RJ. Marginal longitudinal nonparametric regression: locality and efficiency of spline and kernel methods. J. Am. Stat. Assoc.. 2002, 97 482-493

[19] Ye H, Pan J. Modelling of covariates structures in generalized estimating equations for longitudinal data. Biometrika. 2006, 93 927-941

[20] Zeger SL, Diggle PJ. Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters. Biometrics. 1994, 50 689-699

[21] Zhang W, Leng C. A moving average Cholesky factor model in covariance modeling for longitudinal data. Biometrika. 2012, 99 141-150

Funding

National Key Research and Development Plan(2016YFC0800100)

NSF of China(1671374)