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Bayesian modeling of the dependence in longitudinal data via partial autocorrelations and marginal variances. (English) Zbl 1277.62088
Summary: Many parameters and positive-definiteness are two major obstacles in estimating and modeling a correlation matrix for longitudinal data. In addition, when longitudinal data are incomplete, incorrectly modeling the correlation matrix often results in bias in estimating the mean regression parameters. We introduce a flexible and parsimonious class of regression models for a covariance matrix parameterized using marginal variances and partial autocorrelations. The partial autocorrelations can freely vary in the interval $$(-1, 1)$$ while maintaining positive definiteness of the correlation matrix, so the regression parameters in these models will have no constraints. We propose a class of priors for the regression coefficients and examine the importance of correctly modeling the correlation structure on estimation of longitudinal (mean) trajectories and the performance of the deviance information criterion (DIC) in choosing the correct correlation model via simulations. The regression approach is illustrated on data from a longitudinal clinical trial.

##### MSC:
 62F15 Bayesian inference 62J12 Generalized linear models (logistic models) 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62H20 Measures of association (correlation, canonical correlation, etc.) 62P10 Applications of statistics to biology and medical sciences; meta analysis 65C60 Computational problems in statistics (MSC2010)
covreg
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##### References:
 [1] Agresti, A.; Hitchcock, D. B., Bayesian inference for categorical data analysis, Statistical Methods and Applications, 14, 297-330, (2005) · Zbl 1124.62307 [2] Albert, J. H.; Chib, S., Bayesian analysis of binary and polychotomous response data, Journal of the American Statistical Association, 88, 669-679, (1993) · Zbl 0774.62031 [3] Barnard, J.; McCulloch, R.; Meng, X.l., Modeling covariance matrices in terms of standard deviations and correlations, Statistica Sinica, 10, 1281-1312, (2000) · Zbl 0980.62045 [4] Chiu, T. Y.M.; Leonard, T.; Tsui, K. W., The matrix-logarithmic covariance model, Journal of the American Statistical Association, 91, 198-210, (1996) · Zbl 0870.62043 [5] Clyde, M.; George, E. I., Flexible empirical Bayes estimation for wavelets, Journal of the Royal Statistical Society. Series B (Statistical Methodology), 62, 681-698, (2000) · Zbl 0957.62006 [6] Czado, C., Multivariate regression analysis of panel data with binary outcomes applied to unemployment data, Statistical Papers, 41, 281-304, (2000) · Zbl 1047.62509 [7] Daniels, M. J., Bayesian modeling of several covariance matrices and some results on propriety of the posterior for linear regression with correlated and/or heterogeneous errors, Journal of Multivariate Analysis, 97, 1185-1207, (2006) · Zbl 1089.62025 [8] Daniels, M. J.; Hogan, J. W., Missing data in longitudinal studies: strategies for Bayesian modeling and sensitivity analysis, (2008), Chapman and Hall/CRC · Zbl 1165.62023 [9] Daniels, M. J.; Kass, R. E., Nonconjugate Bayesian estimation of covariance matrices and its use in hierarchical models, Journal of the American Statistical Association, 94, 1254-1263, (1999) · Zbl 1069.62508 [10] Daniels, M. J.; Kass, R. E., Shrinkage estimators for covariance matrices, Biometrics, 57, 1173-1184, (2001) · Zbl 1209.62132 [11] Daniels, M. J.; Normand, S. L., Longitudinal profiling and health care units based on mixed multivariate patient outcomes, Biometrics, 7, 1-15, (2006) · Zbl 1169.62370 [12] Daniels, M. J.; Pourahmadi, M., Bayesian analysis of covariance matrices and dynamic models for longitudinal data, Biometrika, 89, 553-566, (2002) · Zbl 1036.62019 [13] Daniels, M. J.; Pourahmadi, M., Modeling covariance matrices via partial autocorrelations, Journal of Multivariate Analysis, 100, 2352-2363, (2009) · Zbl 1175.62090 [14] Daniels, M. J.; Zhao, Y. D., Modeling the random effects covariance matrix in longitudinal data, Statistics in Medicine, 22, 1631-1647, (2003) [15] Efron, B.; Morris, C., Multivariate empirical Bayes and estimation of covariance matrices, The Annals of Statistics, 4, 22-32, (1976) · Zbl 0322.62041 [16] George, E. I.; Foster, D. P., Calibration and empirical Bayes variable selection, Biometrika, 87, 731-747, (2000) · Zbl 1029.62008 [17] Hoff, P. D.; Niu, X., A covariance regression model, Statistica Sinica, 22, 729-753, (2012) · Zbl 1238.62065 [18] Joe, H., Generating random correlation matrices based on partial correlations, Journal of Multivariate Analysis, 97, 2177-2189, (2006) · Zbl 1112.62055 [19] Lapierre, Y. D.; Nair, N. P.V.; Chouinard, G.; Awad, A. G.; Saxena, B.; Jones, B.; McClure, D. J.; Bakish, D.; Max, P.; Manchanda, R.; Beaudry, P.; BIoom, D.; Rotstein, E.; Ancill, R.; Sandor, P.; Sladen-Dew, N.; Durand, C.; Chandrasena, R.; Horn, E.; Elliot, D.; Das, M.; Ravindran, A.; Matsos, G., A controlled dose-ranging study of remoxipride and haloperidol in schizophrenia — a Canadian multicentre trial, Acta Psychiatrica Scandinavica, 82, 72-77, (1990) [20] Leonard, T.; Hsu, J. S.J., Bayesian inference for a covariance matrix, The Annals of Statistics, 20, 1669-1696, (1992) · Zbl 0765.62031 [21] Liechty, J. C.; Liechty, M. W.; Muller, P., Bayesian correlation estimation, Biometrika, 91, 1-14, (2004) · Zbl 1132.62314 [22] Little, R. J.A.; Rubin, D. B., Statistical analysis with missing data, (2002), John Wiley New York · Zbl 1011.62004 [23] McCullagh, P.; Nelder, J. A., Generalized linear models, (1989), Chapman and Hall · Zbl 0744.62098 [24] Munoz, A.; Carey, V.; Schouten, J. P.; Segal, M.; Rosner, B., A parametric family of correlation structures for the analysis of longitudinal data, Biometrics, 48, 733-742, (1992) [25] Nelsen, R. B., An introduction to copulas, (1999), Springer · Zbl 0909.62052 [26] Pourahmadi, M., Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation, Biometrika, 86, 677-690, (1999) · Zbl 0949.62066 [27] Pourahmadi, M., Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix, Biometrika, 87, 425-435, (2000) · Zbl 0954.62091 [28] Pourahmadi, M.; Daniels, M. J., Dynamic conditionally linear mixed models for longitudinal data, Biometrics, 58, 225-231, (2002) · Zbl 1209.62152 [29] Smith, M.; Kohn, R., Parsimonious covariance matrix estimation for longitudinal data, Journal of the American Statistical Association, 97, 1141-1153, (2002) · Zbl 1041.62044 [30] Spiegelhalter, D. J.; Best, N. G.; Carlin, B. P.; Linde, A., Bayesian measures of model complexity and fit, Journal of the Royal Statistical Society. Series B (Statistical Methodology), 64, 583-639, (2002) · Zbl 1067.62010 [31] Verbyla, A. P., Modelling variance heterogeneity: residual maximum likelihood and diagnostics, Journal of the Royal Statistical Society. Series B, 55, 493-508, (1993) · Zbl 0783.62051 [32] Wang, C.; Daniels, M. J., A note on MAR, identifying restrictions, model comparison, and sensitivity analysis in pattern mixture models with and without covariates for incomplete data (correction, vol. 68, p. 994), Biometrics, 67, 810-818, (2011) · Zbl 1272.62096 [33] Wong, F.; Carter, C. K.; Kohn, R., Efficient estimation of covariance selection models, Biometrika, 90, 809-830, (2003) · Zbl 1436.62346 [34] Yang, R.; Berger, J. O., Estimation of a covariance matrix using the reference prior, The Annals of Statistics, 22, 1195-1211, (1994) · Zbl 0819.62013 [35] A. Zellner, On assessing prior distributions and Bayesian regression analysis with $$g$$-prior distributions, in: Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti, 1986, pp. 233-243.
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