×

Additive mixed models with Dirichlet process mixture and P-spline priors. (English) Zbl 1443.62098

Summary: Longitudinal data often require a combination of flexible time trends and individual-specific random effects. For example, our methodological developments are motivated by a study on longitudinal body mass index profiles of children collected with the aim to gain a better understanding of factors driving childhood obesity. The high amount of nonlinearity and heterogeneity in these data and the complexity of the data set with a large number of observations, long longitudinal profiles and clusters of observations with specific deviations from the population model make the application challenging and prevent the application of standard growth curve models. We propose a fully Bayesian approach based on Markov chain Monte Carlo simulation techniques that allows for the semiparametric specification of both the trend function and the random effects distribution. Bayesian penalized splines are considered for the former, while a Dirichlet process mixture (DPM) specification allows for an adaptive amount of deviations from normality for the latter. The advantages of such DPM prior structures for random effects are investigated in terms of a simulation study to improve the understanding of the model specification before analyzing the childhood obesity data.

MSC:

62G08 Nonparametric regression and quantile regression
62G05 Nonparametric estimation
62F15 Bayesian inference
62P10 Applications of statistics to biology and medical sciences; meta analysis
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Arenz, S.; Rückerl, R.; Koletzko, B.; Kries, R., Breast-feeding and childhood obesity—a systematic review, Int. J. Obes. Relat. Metab. Disord., 28, 1247-1256, (2004) · doi:10.1038/sj.ijo.0802758
[2] Beyerlein, A.; Toschke, A. M.; Kries, R., Breastfeeding and childhood obesity: Shift of the entire BMI distribution or only the upper parts?, Obesity, 16, 2730-2733, (2008) · doi:10.1038/oby.2008.432
[3] Brezger, A.; Kneib, T.; Lang, S., BayesX: Analysing Bayesian structured additive regression models, J. Stat. Softw., 14, 11, (2005)
[4] Brezger, A.; Lang, S., Generalized additive regression based on Bayesian P-splines, Comput. Stat. Data Anal., 50, 967-991, (2006) · Zbl 1431.62308 · doi:10.1016/j.csda.2004.10.011
[5] Dunson, D.; Yang, M.; Baird, D., Semiparametric Bayes hierarchical models with mean and variance constraints, Comput. Stat. Data Anal., 54, 2172-2186, (2007) · Zbl 1284.62187
[6] Durban, M.; Harezlak, J.; Wand, M. P.; Carroll, R. J., Simple fitting of subject-specific curves for longitudinal data, Stat. Med., 24, 1153-1167, (2005) · doi:10.1002/sim.1991
[7] Eilers, P. H.C.; Marx, B. D., Flexible smoothing with B-splines and penalties, Stat. Sci., 11, 89-121, (1996) · Zbl 0955.62562 · doi:10.1214/ss/1038425655
[8] Fahrmeir, L.; Kneib, T.; Lang, S., Penalized structured additive regression for space-time data: a Bayesian perspective, Stat. Sin., 14, 731-761, (2004) · Zbl 1073.62025
[9] Fahrmeir, L.; Lang, S., Bayesian semiparametric regression analysis of multicategorical time-space data, Ann. Inst. Stat. Math., 53, 11-30, (2001) · Zbl 0995.62098 · doi:10.1023/A:1017904118167
[10] Fenske, N., Fahrmeir, L., Rzehak, P., Höhle, M.: Detection of risk factors for obesity in early childhood with quantile regression methods for longitudinal data. Technical report 38. Department of Statistics, Ludwig-Maximilians-University Munich (2008)
[11] Ferguson, T. S., A Bayesian analysis of some nonparametric problems, Ann. Math. Stat., 1, 209-230, (1973) · Zbl 0255.62037
[12] Ferguson, T. S., Prior distributions on spaces of probability measures, Ann. Math. Stat., 2, 615-629, (1974) · Zbl 0286.62008
[13] Fritsch, A.; Ickstadt, K., Improved criteria for clustering based on the posterior similarity matrix, Int. Soc. Bayesian Anal., 4, 367-392, (2009) · Zbl 1330.62249 · doi:10.1214/09-BA414
[14] Gelman, A., Prior distributions for variance parameters in hierarchical models, Bayesian Anal., 1, 515-553, (2006) · Zbl 1331.62139 · doi:10.1214/06-BA117A
[15] Harder, T.; Bergmann, R.; Kallischnigg, G.; Plagemann, A., Duration of breastfeeding and risk of overweight: a meta-analysis, Am. J. Epidemiol., 162, 397-403, (2005) · doi:10.1093/aje/kwi222
[16] Ishwaran, H.; James, L. F., Gibbs sampling methods for stick-breaking priors, J. Am. Stat. Assoc., 96, 161-173, (2001) · Zbl 1014.62006 · doi:10.1198/016214501750332758
[17] Ishwaran, H.; James, L. F., Approximate Dirichlet process computing in finite normal mixtures: smoothing and prior information, J. Comput. Graph. Stat., 11, 508-532, (2002) · doi:10.1198/106186002411
[18] Jara, A., Applied Bayesian non- and semiparametric inference using DPpackage, R News, 3, 17-26, (2007)
[19] Jullion, A.; Lambert, P., Robust specification of the roughness penalty prior distribution in spatially adaptive Bayesian P-splines models, Comput. Stat. Data Anal., 51, 2542-2558, (2007) · Zbl 1161.62340 · doi:10.1016/j.csda.2006.09.027
[20] Kleinman, K.; Ibrahim, J., A semiparametric Bayesian approach to the random effects model, Biometrics, 54, 921-938, (1998) · Zbl 1058.62513 · doi:10.2307/2533846
[21] Lang, S.; Brezger, A., Bayesian P-splines, J. Comput. Graph. Stat., 13, 183-212, (2004) · doi:10.1198/1061860043010
[22] Li, Y.; Lin, X.; Müller, P., Bayesian inference in semiparametric mixed models for longitudinal data, Biometrics, 66, 70-78, (2009) · Zbl 1187.62057 · doi:10.1111/j.1541-0420.2009.01227.x
[23] Lin, X.; Zhang, D., Inference in generalized additive mixed models by using smoothing splines, J. R. Stat. Soc. B, 61, 381-400, (1999) · Zbl 0915.62062 · doi:10.1111/1467-9868.00183
[24] Liu, J. S., Nonparametric hierarchical Bayes via sequential imputations, Ann. Stat., 24, 911-930, (1996) · Zbl 0880.62038 · doi:10.1214/aos/1032526949
[25] Ohlssen, D. I.; Sharples, L. D.; Spiegelhalter, D. J., Flexible random-effects models using Bayesian semi-parametric models: Applications to institutional comparisons, Stat. Med., 26, 2088-2112, (2007) · doi:10.1002/sim.2666
[26] Panagiotelis, A.; Smith, M., Bayesian identification, selection and estimation of semiparametric functions in high-dimensional additive models, J. Econom., 143, 291-316, (2008) · Zbl 1418.62166 · doi:10.1016/j.jeconom.2007.10.003
[27] Ruppert, D., Wand, M.P., Carroll, R.J.: Semiparametric regression. Cambridge University Press, Cambridge (2003) · Zbl 1038.62042 · doi:10.1017/CBO9780511755453
[28] Sethuraman, J., A constructive definition of Dirichlet priors, Stat. Sin., 4, 639-650, (1994) · Zbl 0823.62007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.