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Regression spline mixed models: A forestry example

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Abstract

In this article, regression splines are used inside linear mixed models to explore nonlinear longitudinal data. The regression spline bases are generated using a single knot chosen using biological information—a knot position supported by an automated knot selection procedure. A variety of inferential procedures are compared. The variance in the data was closely modeled using a flexible model-based covariance structure, a robust method and the nonparametric bootstrap, while the variance was underestimated when independent random effects were assumed.

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References

  • Akaike, H. (1974), “dA New Look at the Statistical Model Identification,” IEEE Transactions on Automatic Control, AC-19, 716–723.

    Article  MATH  MathSciNet  Google Scholar 

  • Altham, P. M. E. (1984), “Improving the Precision of Estimation by Fitting a Model,” Journal of the Royal Statistical Society, Series B, 46, 118–119.

    MathSciNet  Google Scholar 

  • Brown, H., and Prescott, R. (1999), Applied Mixed Models in Medicine, New York: Wiley.

    MATH  Google Scholar 

  • Brumback, B. A., and Rice, J. A. (1998), “Smoothing Spline Models for the Analysis of Nested and Crossed Samples of Curves,” Journal of the American Statistical Association, 93, 961–976.

    Article  MATH  MathSciNet  Google Scholar 

  • Burdon, R. D. (1994), “Annual Growth Stages for Height and Diameter in Pinus radiate,” New Zealand Journal of Forestry Science, 24, 11–17.

    Google Scholar 

  • Butler, S. M., and Louis, T. A. (1992), “Random Effects Models with Nonparametric Priors,” Statistics in Medicine, 11, 1981–2000.

    Article  Google Scholar 

  • Calama, R., and Montero, G. (2004), “Interregional Nonlinear Height-Diameter Model with Random Coefficients for Stone Pine in Spain,” Canadian Journal of Forest Research, 34, 150–163.

    Article  Google Scholar 

  • Candy, S. G. (1997), “Growth and Yield Models for Eucalyptus nitens Plantations in Tasmania and New Zealand,” Tasforests, 9, 167–198.

    Google Scholar 

  • Davidian, M., and Giltinan, D. M. (1998), Nonlinear Models for Repeated Measurement Data, London: Chapman & Hall.

    Google Scholar 

  • DeBoor, C. (1978), A Practical Guide to Splines, New York: Springer-Verlag.

    Google Scholar 

  • Durban, M., Harezlak, J., Wand, M., and Carroll, R. (2004), “Simple Fitting of Subject-Specific Curves for Longitudinal Data,” Statistics in Medicine, 00, 1–24.

    Google Scholar 

  • Eilers, P. H. C., and Marx, B. D. (1996), “Flexible Smoothing with B-Splines and Penalties,” Statistical Science, 11, 89–121.

    Article  MATH  MathSciNet  Google Scholar 

  • Fouladi, R., deMoor, C., and Sui, D. (2002), “Fixed Effects in General Linear Mixed Effect Modelling in Longitudinal Studies: Bootstrap and Modified Classical Procedures,” in American Statistical Association Proceedings of the Joint Statistical Meetings [CD-ROM], American Statistical Association, pp. 1020–1025.

  • Garcia, O. (1984), “New Class of Growth Models for Even-Aged Stands: Pinus radiate in Golden Downs Forest,” New Zealand Journal of Forestry Science, 14, 65–88.

    Google Scholar 

  • Garcia, O. (1999), “Height Growth of Pinus radiate in New Zealand,” New Zealand Journal of Forestry Science, 29.

  • Guo, W. (2002), “Functional Mixed Effects Models,” Biometrics, 58, 121–128.

    Article  MathSciNet  Google Scholar 

  • Hardin, J., and Hilbe, J. (2001), Generalized Linear Models and Extensions, Stata Press.

  • Hart, J. D., and Wehrly, T. E. (1986), “Kernel Regression Estimation using Repeated Measurements Data,” Journal of the American Statistical Association, 81, 1080–1088.

    Article  MATH  MathSciNet  Google Scholar 

  • Hartford, A., and Davidian, M. (2000), “Consequences of Misspecifying Assumptions in Nonlinear Mixed Effects Models,” Computational Statistics and Data Analysis, 34, 139–164.

    Article  MATH  Google Scholar 

  • Hastie, T. J., and Tibshirani, R. J. (1990), Generalized Additive Models, London: Chapman & Hall.

    MATH  Google Scholar 

  • Hastie, T., Tibshirani, R., and Friedman, J. (2001), The Elements of Statistical Learning, New York: Springer-Verlag.

    MATH  Google Scholar 

  • Insightful Corporation (2001), S-PLUS 6 for Windows User’s Guide, Seattle, WA: Author.

    Google Scholar 

  • Lin, X., and Zhang, D. (1999), “Inference in Generalized Additive Mixed Models by Using Smoothing Splines,” Journal of the Royal Statistical Society, Series B, 61, 381–400.

    Article  MATH  MathSciNet  Google Scholar 

  • Littell, R. C., Milliken, G. A., Stroup, W. W., and Wolfinger, R. D. (1996), SAS System for Mixed Models, SAS Institute, Inc.

  • McCulloch, C., and Searle, S. R. (2001), Generalized, Linear, and Mixed Models New York: Wiley.

    MATH  Google Scholar 

  • McEwen, A. D. (1979), “Mensuration for Management Planning of Exotic Forest Plantations,” New Zealand Forest Service, Forest Research Institute Symposium No. 20.

  • Mehtätalo, L. (2004), “A Longitudinal Height-Diameter Model for Norway Spruce in Finland,” Canadian Journal of Forest Research, 34, 131–140.

    Article  Google Scholar 

  • Morris, J. S. (2002), “The BLUPs are not ‘Best’ When it Comes to Bootstrapping,” Statistics & Probability Letters, 56, 425–430.

    Article  MATH  MathSciNet  Google Scholar 

  • Ngo, L., and Wand, P. (2004), “Smoothing with Mixed Model Software,” Journal of Statistical Software, 9, 1–54. Available online at www.jstatsoft.org.

    Google Scholar 

  • R Development Core Team (2004), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. Available online at http://www.R-project.org

    Google Scholar 

  • Rice, J. A., and Silverman, B. W. (1991), “Estimating the Mean and Covariance Structure Nonparametrically When the Data are Curves,” Journal of the Royal Statistical Society, Series B, 53, 233–243.

    MATH  MathSciNet  Google Scholar 

  • Rice, J. A., and Wu, C. O. (2001), “Nonparametric Mixed Effects Models for Unequally Sampled Noisy Curves,” Biometrics, 57, 253–259.

    Article  MathSciNet  Google Scholar 

  • Roberts, J., and Fan, X. (2002), “Boostrapping Within the Multilevel/Hierarchical Linear Modelling Framework: A Primer for Use With SAS and SPLUS,” in American Educuational Research Association Annual Meeting, New Orleans, L.A.

  • Ruppert, D., Wand, M., and Carroll, R. (2003), Semiparametric Regression, Cambridge Series in Statistical and Probabilistic Mathematics.

  • SAS Institute (2000), SAS/STAT User’s Guide, Version 8, Cary, NC.

  • Schumaker, L. L. (1993), Spline Functions: Basic Theory, Melbourne, FL: Krieger Publishing Company.

    Google Scholar 

  • Verbeke, G., and Lesaffre, E. (1997), “The Effect of Misspecifying the Random-Effects Distribution in Linear Mixed Models for Longitudinal Data,” Computational Statistics and Data Analysis, 23, 541–556.

    Article  MATH  MathSciNet  Google Scholar 

  • Verbeke, G., and Molenberghs, G. (2000), Linear Mixed Models for Longitudinal Data, New York: Springer-Verlag.

    MATH  Google Scholar 

  • Verzinoa, G., Ingaramoa, P., Joseaua, J., Astinia, E., Rienzob, J. D., and Doradoa, M. (1999), “Basal Area Growth Curves for Pinus patula in Two Areas of the Calamuchita Valley, Córdoba, Argentina,” Forest Ecology and Management, 124, 185–192.

    Article  Google Scholar 

  • Vonesh, E. F., and Chinchilli, V. M. (1997), Linear and Nonlinear Models for the Analysis of Repeated Measurements, New York: Marcel Dekker.

    MATH  Google Scholar 

  • Wold, S. (1974), “Spline Functions in Data Analysis,” Technometrics, 16, 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  • Wood, S. N. (2000), “Modelling and Smoothing Parameter Estimation with Multiple Quadratic Penalties,” Journal of the Royal Statistical Society, Series B, 62, 413–428.

    Article  Google Scholar 

  • Zhang, D., Lin, X., Raz, J., and Sowers, M. (1998), “Semiparametric Stochastic Mixed Models for Longitudinal Data,” Journal of the American Statistical Association, 93, 710–719.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Centre for Research into into Ecological and Environmental Modelling, University of St. Andrews, KY 16 9LZ, St. Andrews, Fife, Scotland

    Monique L. Mackenzie (Lecturer) & C. R. Donovan (Lecturer)

  2. University of Auckland, Auckland, New Zealand

    B. H. McArdle (Associate Professor)

Authors
  1. Monique L. Mackenzie
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  2. C. R. Donovan
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  3. B. H. McArdle
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Correspondence to Monique L. Mackenzie.

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Mackenzie, M.L., Donovan, C.R. & McArdle, B.H. Regression spline mixed models: A forestry example. JABES 10, 394 (2005). https://doi.org/10.1198/108571105X80194

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  • DOI: https://doi.org/10.1198/108571105X80194

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