In longitudinal studies, it is often of great interest to cluster individual trajectories based on repeated measurements taken over time. Non-linear growth trajectories are often seen in practice, and the individual data can also be measured sparsely, and at irregular time points, which may complicate the modeling process. Motivated by a study of pregnant women hormone profiles, we proposed a shape invariant growth mixture model for clustering non-linear growth trajectories. Bayesian inference via Monte Carlo Markov Chain was employed to estimate the parameters of interest. We compared our model to the commonly used growth mixture model and functional clustering approach by simulation studies. Results from analyzing the real data and simulated data were presented and discussed.
HeckmanNEZamarRH. Comparing the shapes of regression functions. Biometrika2000; 87: 135–144.
2.
RaySMallickB. Functional clustering by Bayesian wavelet methods. J Royal Stat Soc: Ser B (Stat Methodol)2006; 68: 305–332.
3.
ChiouJMLiPL. Functional clustering and identifying substructures of longitudinal data. J Royal Stat Soc: Ser B (Stat Methodol)2007; 69: 679–699.
4.
JacquesJPredaC. Model-based clustering for multivariate functional data. Computat Stat Data Analys2014; 71: 92–106.
5.
JamesGMSugarCA. Clustering for sparsely sampled functional data. J Am Stat Assoc2003; 98: 397–408.
6.
YaoFMüllerHGWangJL. Functional data analysis for sparse longitudinal data. J Am Stat Assoc2005; 100: 577–590.
7.
DongJJWangLGillJ, et al.Functional principal component analysis of glomerular filtration rate curves after kidney transplant. Stat Meth Med Res2017; 27: 3785–3796, .
8.
McLachlan G and Peel D. Finite mixture models. Wiley Series in Probability and Statistics: Applied Probability and Statistics. Wiley-Interscience, New York, 2000. xxii+419 pp. ISBN: 0-471-00626-2.
MuthénBSheddenK. Finite mixture modeling with mixture outcomes using the EM algorithm. Biometrics1999; 55: 463–469.
11.
PaulerDKLairdNM. A mixture model for longitudinal data with application to assessment of noncompliance. Biometrics2000; 56: 464–472.
12.
ElliottMRGalloJJTen HaveTR, et al.Using a bayesian latent growth curve model to identify trajectories of positive affect and negative events following myocardial infarction. Biostatistics2005; 6: 119–143.
13.
De la Cruz-MesíaRQuintanaFAMarshallG. Model-based clustering for longitudinal data. Computat Stat Data Analys2008; 52: 1441–1457.
14.
DunsonDBHerringAHSiega-RizAM. Bayesian inference on changes in response densities over predictor clusters. J Am Stat Assoc2008; 103: 1508–1517.
15.
SlaughterJCHerringAHThorpJM. A Bayesian latent variable mixture model for longitudinal fetal growth. Biometrics2009; 65: 1233–1242.
16.
LeibyBESammelMDTen HaveTR, et al.Identification of multivariate responders and non-responders by using Bayesian growth curve latent class models. J Royal Stat Soc: Ser C (Appl Stat)2009; 58: 505–524.
17.
Frühwirth-SchnatterSPyneS. Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions. Biostatistics2010; 11: 317–336.
18.
NeelonBSwamyGKBurgetteLF, et al.A bayesian growth mixture model to examine maternal hypertension and birth outcomes. Stat Med2011; 30: 2721–2735.
19.
LuXHuangY. Bayesian analysis of nonlinear mixed-effects mixture models for longitudinal data with heterogeneity and skewness. Stat Med2014; 33: 2830–2849.
20.
BeathKJ. Infant growth modelling using a shape invariant model with random effects. Stat Med2007; 26: 2547–2564.
21.
ColeTJDonaldsonMDBen-ShlomoY. Sitar-a useful instrument for growth curve analysis. Int J Epidemiol2010; 39: 1558–1566.
De BoorCDe BoorCMathématicienEU, et al.A practical guide to splines1978; Vol 27New York, NY: Springer-Verlag.
24.
KassRERafteryAE. Bayes factors. J Am Stat Assoc1995; 90: 773–795.
25.
FraleyCRafteryAE. How many clusters? Which clustering method? Answers via model-based cluster analysis. Comput J1998; 41: 578–588.
26.
SpiegelhalterDJBestNGCarlinBP, et al.Bayesian measures of model complexity and fit. J Royal Stat Soc: Ser B (Stat Methodol)2002; 64: 583–639.
27.
HwangBSChenZLouisGMB, et al.A bayesian multi-dimensional couple-based latent risk model with an application to infertility. Biometrics2018. doi: 10.1111/biom.12972 .
28.
CeleuxGForbesFRobertCP, et al.Deviance information criteria for missing data models. Bayesian Analys2006; 1: 651–673.
29.
RousseauJMengersenK. Asymptotic behaviour of the posterior distribution in overfitted mixture models. J Royal Stat Soc: Ser B (Stat Methodol)2011; 73: 689–710.
30.
NasserinejadKvan RosmalenJde KortW, et al.Comparison of criteria for choosing the number of classes in bayesian finite mixture models. PloS One2017; 12: e0168838.
31.
Andrinopoulou ER, Nasserinejad K, Szczesniak R, et al. Integrating latent classes in the bayesian shared parameter joint model of longitudinal and survival outcomes. arXiv preprint arXiv:180210015 2018..
32.
GelmanARubinDB. Inference from iterative simulation using multiple sequences. Stat Sci1992; 7: 457–472. .
33.
DieboltJRobertCP. Estimation of finite mixture distributions through Bayesian sampling. J Royal Stat Soc Ser B (Methodological)1994; 56: 363–375. .
34.
CeleuxGHurnMRobertCP. Computational and inferential difficulties with mixture posterior distributions. J Am Stat Assoc2000; 95: 957–970.
35.
Frühwirth-SchnatterS. Markov chain Monte Carlo estimation of classical and dynamic switching and mixture models. J Am Stat Assoc2001; 96: 194–209.
36.
WangWL. Multivariate t linear mixed models for irregularly observed multiple repeated measures with missing outcomes. Biometric J2013; 55: 554–571.
37.
WangWLLinTI. Multivariate t nonlinear mixed-effects models for multi-outcome longitudinal data with missing values. Stat Med2014; 33: 3029–3046.
38.
Wu H and Jin-Ting Z. Nonparametric regression methods for longitudinal data analysis: mixed-effects modeling approaches. Vol. 515. John Wiley & Sons, 2006.
RandWM. Objective criteria for the evaluation of clustering methods. J Am Stat Assoc1971; 66: 846–850.
41.
RichardsonSGreenPJ. On Bayesian analysis of mixtures with an unknown number of components (with discussion). J Royal Stat Soc: Ser B (Stat Methodol)1997; 59: 731–792.
42.
EscobarMDWestM. Bayesian density estimation and inference using mixtures. J Am Stat Assoc1995; 90: 577–588.
43.
FahrmeirLKneibTKonrathS. Bayesian regularisation in structured additive regression: a unifying perspective on shrinkage, smoothing and predictor selection. Stat Comput2010; 20: 203–219.
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
