×

Faster Monte Carlo estimation of joint models for time-to-event and multivariate longitudinal data. (English) Zbl 07345929

Summary: Quasi-Monte Carlo (QMC) methods using quasi-random sequences, as opposed to pseudo-random samples, are proposed for use in the joint modelling of time-to-event and multivariate longitudinal data. The QMC integration framework extends the Monte Carlo Expectation Maximisation approaches that are commonly adopted, namely using ordinary and antithetic variates. The motivation of QMC integration is to increase the convergence speed by using nodes that are scattered more uniformly. Through simulation, estimates and computational times are compared and this is followed with an application to a clinical dataset. There is a distinct speed advantage in using QMC methods for small sample sizes and QMC is comparable to the antithetic MC method for moderate sample sizes. The new method is available in an updated version of the package .

MSC:

62-XX Statistics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Albert, P. S.; Shih, J. H., An approach for jointly modeling multivariate longitudinal measurements and discrete time-to-event data, Ann. Appl. Stat., 4, 3, 1517-1532 (2010) · Zbl 1202.62144
[2] Andrinopoulou, E.-R.; Rizopoulos, D., Bayesian shrinkage approach for a joint model of longitudinal and survival outcomes assuming different association structures, Stat. Med., 35, 26, 4813-4823 (2016)
[3] Antonov, I. A.; Saleev, V., An economic method of computing LP \(\tau \)-sequences, USSR Comput. Math. Math. Phys., 19, 1, 252-256 (1979) · Zbl 0432.65006
[4] Asar, O.; Ritchie, J.; Kalra, P.; Diggle, P. J., Joint modelling of repeated measurement and time-to-event data: an introductory tutorial, Int. J. Epidemiol., 1-11 (2015)
[5] Atanassov, E.; Karaivanova, A.; Ivanovska, S., Tuning the generation of Sobol sequence with Owen scrambling, (Lirkov, I.; Margenov, S.; Waśniewski, J., Large-Scale Scientific Computing (2010), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 459-466 · Zbl 1280.65005
[6] Caflisch, R. E., Monte Carlo and quasi-Monte Carlo methods, Acta Numer., 7, 1-49 (1998) · Zbl 0949.65003
[7] Chi, H.; Beerli, P.; Evans, D. W.; Mascagni, M., On the scrambled Sobol sequence, (International Conference on Computational Science (2005), Springer: Springer Atlanta, GA, USA), 775-782, URL http://dx.doi.org/10.1007/11428862_105 · Zbl 1120.65303
[8] Cools, R., Advances in multidimensional integration, J. Comput. Appl. Math., 149, 1, 1-12 (2002) · Zbl 1013.65019
[9] Crowther, M. J., Merlin-a unified modelling framework for data analysis and methods development in stata (2018), arXiv preprint arXiv:1806.01615
[10] Crowther, M. J.; Abrams, K. R.; Lambert, P. C., Joint modeling of longitudinal and survival data, Stata J., 13, 1, 165-184 (2013)
[11] Dempster, A.; Laird, N.; Rubin, D. B., Maximum likelihood from incomplete data via the EM algorithm, J. R. Stat. Soc. Ser. B Stat. Methodol., 39, 1, 1-38 (1977) · Zbl 0364.62022
[12] Dutang, C.; Savicky, P., Randtoolbox: Generating and testing random numbers (2018), R package version 1.17.1
[13] Emura, T.; Nakatochi, M.; Murotani, K.; Rondeau, V., A joint frailty-copula model between tumour progression and death for meta-analysis, Stat. Methods Med. Res., 26, 6, 2649-2666 (2017)
[14] Gould, A. L.; Boye, M. E.; Crowther, M. J.; Ibrahim, J. G.; Quartey, G.; Micallef, S.; Bois, F. Y., Joint modeling of survival and longitudinal non-survival data: current methods and issues. Report of the DIA Bayesian joint modeling working group, Stat. Med., 34, 2181-2195 (2015)
[15] Halton, J. H., On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math., 2, 1, 84-90 (1960) · Zbl 0090.34505
[16] Hardy, G. H., On double Fourier series and especially those which represent the double zeta-function with real and incommensurable parameters, Quart. J. Math., 37, 1, 53-79 (1906) · JFM 36.0501.02
[17] Henderson, R.; Diggle, P. J.; Dobson, A., Joint modelling of longitudinal measurements and event time data, Biostatistics, 1, 4, 465-480 (2000) · Zbl 1089.62519
[18] Hickey, G. L.; Philipson, P.; Jorgensen, A.; Kolamunnage-Dona, R., Joint modelling of time-to-event and multivariate longitudinal outcomes: recent developments and issues, BMC Med. Res. Methodol., 16, 1, 1-15 (2016)
[19] Hickey, G. L.; Philipson, P.; Jorgensen, A.; Kolamunnage-Dona, R., JoineRML: a joint model and software package for time-to-event and multivariate longitudinal outcomes, BMC Med. Res. Methodol., 18, 1, 50 (2018)
[20] Hlawka, E., Funktionen von beschränkter variatiou in der theorie der gleichverteilung, Ann. Mat. Pura Appl., 54, 1, 325-333 (1961) · Zbl 0103.27604
[21] Hsieh, F.; Tseng, Y. K.; Wang, J. L., Joint modeling of survival and longitudinal data: Likelihood approach revisited, Biometrics, 62, 4, 1037-1043 (2006) · Zbl 1116.62105
[22] Ibrahim, J. G.; Chu, H.; Chen, L. M., Basic concepts and methods for joint models of longitudinal and survival data, J. Clin. Oncol., 28, 16, 2796-2801 (2010)
[23] Kim, S., Jointmodel: Semiparametric joint models for longitudinal and counting processes (2016), R package version 1.0. URL https://CRAN.R-project.org/package=JointModel
[24] Koksma, J., A general theorem from the theory of uniform distribution modulo 1, Math. B (Zutphen), 1, 7-11, 43 (1942)
[25] Lemieux, C., Monte Carlo and Quasi-Monte Carlo Sampling (2009), Springer: Springer New York, NY · Zbl 1269.65001
[26] Lin, H.; McCulloch, C. E.; Mayne, S. T., Maximum likelihood estimation in the joint analysis of time-to-event and multiple longitudinal variables, Stat. Med., 21, 2369-2382 (2002)
[27] Martin, E.; Gasparini, A.; Crowther, M., Merlin: Mixed effects regression for linear, non-linear and user-defined models (2020), R package version 0.0.2. URL https://CRAN.R-project.org/package=merlin
[28] McLachlan, G. J.; Krishnan, T., The EM Algorithm and Extensions (2008), Wiley-Interscience · Zbl 1165.62019
[29] Ökten, G.; Göncü, A., Generating low-discrepancy sequences from the normal distribution: Box-Muller or inverse transform?, Math. Comput. Modelling, 53, 5-6, 1268-1281 (2011) · Zbl 1217.65004
[30] Owen, A. B., Scrambling Sobol’and Niederreiter-Xing points, J. Complexity, 14, 4, 466-489 (1998) · Zbl 0916.65017
[31] Pan, J.; Thompson, R., Quasi-Monte Carlo estimation in generalized linear mixed models, Comput. Statist. Data Anal., 51, 12, 5765-5775 (2007) · Zbl 1445.62196
[32] Peng, M.; Xiang, L.; Wang, S., Semiparametric regression analysis of clustered survival data with semi-competing risks, Comput. Statist. Data Anal., 124, 53-70 (2018) · Zbl 1469.62128
[33] Philipson, P.; Sousa, I.; Diggle, P. J.; Williamson, P.; Kolamunnage-Dona, R.; Henderson, R.; Hickey, G. L., : Joint modelling of repeated measurements and time-to-event data (2017), R package version 1.2.4. URL https://github.com/petephilipson/joineR/
[34] Proust-Lima, C.; Sene, M.; Taylor, J. M.G.; Jacqmin-Gadda, H., Joint latent class models for longitudinal and time-to-event data: a review, Stat. Methods Med. Res., 23, 1, 74-90 (2012)
[35] Ripatti, S.; Larsen, K.; Palmgren, J., Maximum likelihood inference for multivariate frailty models using an automated Monte Carlo EM algorithm, Lifetime Data Anal., 8, 2002, 349-360 (2002) · Zbl 1116.62407
[36] Rizopoulos, D., : an R package for the joint modelling of longitudinal and time-to-event data, J. Stat. Softw., 35, 9, 1-33 (2010)
[37] Rizopoulos, D., Fast fitting of joint models for longitudinal and event time data using a pseudo-adaptive Gaussian quadrature rule, Comput. Statist. Data Anal., 56, 3, 491-501 (2012) · Zbl 1239.62122
[38] Rizopoulos, D.; Verbeke, G.; Lesaffre, E., Fully exponential Laplace approximations for the joint modelling of survival and longitudinal data, J. R. Stat. Soc. Ser. B Stat. Methodol., 71, 3, 637-654 (2009) · Zbl 1250.62049
[39] Sobol, I. M., On the distribution of points in a cube and the approximate evaluation of integrals, Zh. Vychisl. Mat. Mat. Fiz., 7, 4, 784-802 (1967)
[40] Song, X.; Davidian, M.; Tsiatis, A. A., An estimator for the proportional hazards model with multiple longitudinal covariates measured with error, Biostatistics, 3, 4, 511-528 (2002) · Zbl 1138.62360
[41] van der Corput, J. G., Verteilungsfunktionen. I. Mitt., Proc. Akad. Wet. Amst., 38, 813-821 (1935) · Zbl 0012.34705
[42] Wei, G. C.; Tanner, M. A., A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithms, J. Amer. Statist. Assoc., 85, 411, 699-704 (1990)
[43] Williamson, P. R.; Kolamunnage-Dona, R.; Philipson, P.; Marson, A. G., Joint modelling of longitudinal and competing risks data, Stat. Med., 27, 6426-6438 (2008)
[44] Wulfsohn, M.; Tsiatis, A. A., A joint model for survival and longitudinal data measured with error, Biometrics, 53, 1, 330-339 (1997) · Zbl 0874.62140
[45] Xu, C.; Hadjipantelis, P.; Wang, J.-L., Semi-parametric joint modeling of survival and longitudinal data: The R package JSM, J. Stat. Softw., 93, 2, 1-29 (2020)
[46] Xu, J.; Zeger, S. L., Joint analysis of longitudinal data comprising repeated measures and times to events, J. R. Stat. Soc. Ser. C. Appl. Stat., 50, 3, 375-387 (2001) · Zbl 1112.62312
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.