Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Biometrical Letters

The Journal of Polish Biometric Society

2 Issues per year

Open Access
See all formats and pricing
More options …

Finite mixture models with fixed weights applied to growth data

Marek Molas / Emmanuel Lesaffre
  • Department of Biostatistics, Erasmus MC, P.O. Box 2040, 3000 CA Rotterdam, Netherlands
  • Catholic University of Leuven, L-Biostat, U.Z. St. Rafael, Kapucijnenvoer 35, 3000 Leuven, Belgium
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2013-08-17 | DOI: https://doi.org/10.2478/bile-2013-0008


To model cross-sectional growth data the LMS method is widely applied. In this method the distribution is summarized by three parameters: the Box-Cox power that converts outcome to normality (L); the median (M); and the coeficient of variation (S).

Here, we propose an alternative approach based on fitting finite mixture models with several components which may perform better than the LMS method in case the data show an unusual distribution. Further, we explore fixing the weights of the mixture components in contrast to the standard approach where weights are estimated. Having fixed weights improves the speed of computation and the stability of the solution. In addition, fixing the weights provides almost as good a fit as when the weights are estimated. Our methodology combines Gaussian mixture modelling and spline smoothing. The estimation of the parameters is based on the joint modelling of mean and dispersion.

We illustrate the methodology on the Fourth Dutch Growth Study, which is a cross-sectional study that contains information on the growth of 7303 boys as a function of age. This information is used to construct centile curves, so-called growth curves, which describe the distribution of height as a smooth function of age. Further, we analyse simulated data showing a bimodal structure at some time point.

In its full generality, this approach permits the replacement of the Gaussian components by any parametric density. Further, different components of the mixture can have a diferent probabilistic (multivariate) structure, allowing for censoring and truncation.

Keywords: mixture models; growth curves; splines; IWLS algorithm; exible distributions

  • Cole T.J., Green P.J. (1992): Smoothing reference centile curves: The LMS method and penalized likelihood. Statistics in Medicine 11: 1305-1319.CrossrefGoogle Scholar

  • Dempster A.P., Laird N.M., Rubin D.B. (1977): Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society Series B 39: 1-38.Google Scholar

  • Eilers P., Marx B. (1996): Flexible smoothing with b-splines and penalties. Statistical Science 11: 89-121.CrossrefGoogle Scholar

  • Ghidey W., Lesa re E., Eilers P. (2004): Smooth random e ects distribution in a linear mixed model. Biometrics 60: 945-953.CrossrefGoogle Scholar

  • Harrell F.E. (2001): Regression Modelling Strategies. Springer-Verlag, New York.Google Scholar

  • Komarek A., Lesa re E., Hilton J.F. (2005): Accelerated failure time model for arbitrarily censored data with smoothed error distribution. Journal of Computational and Graphical Statistics 14: 726-745.CrossrefGoogle Scholar

  • Lee Y., Nelder J.A., Pawitan Y. (2006): Generalized Linear Models with Random E ects. Chapman & Hall / CRC: Boca Raton.Google Scholar

  • McLachlan G.J., Peel D. (2000): Finite Mixture Models. John Wiley and Sons, New York.Google Scholar

  • Muthen B., Brown H.C. (2009): Estimating drug e ects in the presence of placebo response: Casual inference using growth mixture modelling. Statistics in Medicine 28: 3363-3385.Web of ScienceCrossrefGoogle Scholar

  • Nelder J.A., Pregibon D. (1987): An extended quasi-likelihood function. Biometrika 74: 221-232.CrossrefGoogle Scholar

  • Nelder J.A., Wedderburn R.W.M. (1972): Generalized linear models. Journal of Royal Statistical Society A 135: 370-384.Google Scholar

  • Ramsay J.O. (1988): Monotone regression splines in action. Statistical Science 3: 425-461.CrossrefGoogle Scholar

  • van Buuren S., Fredriks M. (2001): Worm plot: a simple diagnostic device for modelling growth reference curves. Statistics in Medicine 20: 1259-1277. CrossrefGoogle Scholar

About the article

Published Online: 2013-08-17

Published in Print: 2012-12-01

Citation Information: Biometrical Letters, Volume 49, Issue 2, Pages 103–119, ISSN (Print) 1896-3811, DOI: https://doi.org/10.2478/bile-2013-0008.

Export Citation

This content is open access.

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Namhee Kim, Moonseong Heo, Roman Fleysher, Craig A. Branch, and Michael L. Lipton
Frontiers in Public Health, 2014, Volume 2

Comments (0)

Please log in or register to comment.
Log in