Two-dimensional penalized splines via Gibbs sampling to account for spatial variability in forest genetic trials with small amount of information available

E. P. Cappa 1 , M. Lstiburek 2 , A. D. Yanchuk 3 , 4 ,  und Y. A. El-Kassaby 4
  • 1 Instituto Nacional de Tecnología Agropecuaria (INTA), Instituto de Recursos Biológicos, 1686, Buenos Aires, Argentina
  • 2 Department of Dendrology and Forest Tree Breeding, Faculty of Forestry and Wood Sciences, Czech University of Life Sciences Prague, 165 21, Praha
  • 3 British Columbia Forest Service, Tree Improvement Branch, V8W 9C2, Victoria, Canada
  • 4 Department of Forest Sciences, Faculty of Forestry, University of British Columbia, V6T 1Z4, Canada


Spatial environmental heterogeneity are well known characteristics of field forest genetic trials, even in small experiments (<1ha) established under seemingly uniform conditions and intensive site management. In such trials, it is commonly assumed that any simple type of experimental field design based on randomization theory, as a completely randomized design (CRD), should account for any of the minor site variability. However, most published results indicate that in these types of trials harbor a large component of the spatial variation which commonly resides in the error term. Here we applied a two-dimensional smoothed surface in an individual-tree mixed model, using tensor product of linear, quadratic and cubic B-spline bases with different and equal number of knots for rows and columns, to account for the environmental spatial variability in two relatively small (i.e., 576 m2 and 5,705 m2) forest genetic trials, with large multiple-tree contiguous plot configurations. In general, models accounting for site variability with a two-dimensional surface displayed a lower value of the deviance information criterion than the classical RCD. Linear B-spline bases may yield a reasonable description of the environmental variability, when a relatively small amount of information available. The mixed models fitting a smoothed surface resulted in a reduction in the posterior means of the error variance (σ2e), an increase in the posterior means of the additive genetic variance (σ2a) and heritability (h2HT), and an increase of 16.05% and 46.03% (for parents) or 11.86% and 44.68% (for offspring) in the accuracy of breeding values, respectively in the two experiments.

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  • Anekonda, T. S. and W. J. Libby (1996): Effectiveness of nearestneighbor data adjustment in a clonal test of Redwood. Silvae Genet. 45(1): 46–51.

  • Bohmanova, J., I. Misztal and J. K. Bertrand (2005): Studies on multiple trait and random regression models for genetic evaluation of beef cattle for growth. J Anim Sci 83: 62–67.

  • Cantet, R. J. C., A. N. Birchmeier, A. W. Canaza Cayo and C. Fiorett (2005): Semiparametric animal models via penalized splines as alternatives to models with contemporary groups. J Anim Sci 83: 2482–2494.

  • Cappa, E. P. and R. J. C. Cantet (2006): Bayesian inference for normal multiple-trait individual-tree models with missing records via full conjugate Gibbs. Can J For Res 36: 1276–1285.

  • Cappa, E. P. and R. J. C. Cantet (2007): Bayesian estimation of a surface to account for a spatial trend using penalized splines in an individual-tree mixed model. Can J For Res 37: 2677–2688.

  • Costa e Silva, J., G. W. Dutkowski and A. R. Gilmour (2001): Analysis of early tree height in forest genetic trials is enhanced by including a spatially correlated residual. Can J For Res 31: 1887–1893.

  • Cornillon, P. A., L. Saint-Andre, J. M. Bouvet, P. Vigneron, A. Saya and R. Gouma (2003): Using B-splines for growth curve classification: applications to selection of eucalypt clones. Forest Ecology and Management 176: 75–85.

  • De Boor, C. (1993): B(asic)-spline basics. Fundamental Developments of Computer-Aided Geometric Modeling. Edited by L. Piegl, Academic Press, San Diego, CA.

  • Durban, M., I. Currie and R. Kempton (2001): Adjusting for fertility and competition in variety trials. J Agric Sci (Camb) 136: 129–140.

  • Dutkowski, G. W., J. Costa e Silva, A. R. Gilmour and G. A. Lopez (2002): Spatial analysis methods for forest genetic trials. Can J For Res 32: 2201–2214.

  • Dutkowski, G. W., J. Costa e Silva, A. R. Gilmour, H. Wellendorf and A. Aguiar (2006): Spatial analysis enhances modeling of a wide variety of traits in forest genetic trials. Can J For Res 36: 1851–1870.

  • Eilers, P. H. C. and B. D. Marx (1996): Flexible smoothing with B-splines and penalties (with comments and rejoinder). Stat Sci 11: 89–121.

  • Eilers, P. H. C. and B. D. Marx (2003): Multivariate calibration with temperature interaction using two-dimensional penalized signal regression. Chemometr. Intell Lab Syst 66: 159–174.

  • El-Kassaby, Y. A. and Y. S. Park (1993): Genetic variation and correlation in growth, biomass traits, and vegetative phenology of a 3-year-old Douglas-fir common garden at different spacings. Silvae Genet 42: 289–297.

  • Ericsson, T. (1997): Enhanced heritabilities and best linear unbiased predictors through appropriate blocking of progeny trials. Can J For Res 27: 2097–2101.

  • Federer, W. T. (1998): Recovery of interblock, intergradient, and intervarietal information in incomplete block and lattice rectangle designed experiments. Biometrics 54: 471–481.

  • Finley, A. O., S. Banerjee, P. Waldmann and T. Ericsson (2009): Hierarchical spatial modeling of additive and dominance genetic variance for large spatial trial data sets. Biometrics 65: 441–451.

  • Gilmour, A. R., B. R. Cullis and A. P. Verbyla (1997): Accounting for natural and extraneous variation in the analysis of field experiments. J Agric Biol Environ Stat 2: 269–293.

  • Gezan, S. A., D. A. Huber and T. L. White (2006): Post hoc blocking to improve heritability and precision of best linear unbiased genetic predictions. Can J For Res 36: 2141–2147.

  • Green, P. J. and B. W. Silverman (1994): Nonparametric Regression and Generalized Linear Model. Chapman & Hall, London, UK.

  • Grondona, M. O., J. Crossa, P. N. Fox and W. H. Pfeiffer (1996): Analysis of variety yield trials using two-dimensional separable ARIMA processes. Biometrics 52: 763–770.

  • Henderson, C. R. (1984): Applications of Linear Models in Animal Breeding. Canada, University of Guelph, Guelph, Ont.

  • Hamann, A., M. Koshy and G. Namkoong (2002): Improving precision of breeding values by removing spatially autocorrelated variation in forestry field experiments. Silvae Genet 51: 210–215.

  • Harville, D. A. (1997): Matrix algebra from a statistician’s perspective. Springer-Verlag. New York.

  • Iwaisaki, H., S. Tsuruta, I. Misztal and J. K. Bertrand (2005): Genetic parameters estimated with multi-trait and linear spline-random regression models using Gelbvieh early growth data. J Anim Sci 83: 499–506.

  • Joyce, D., R. Ford and Y. B. Fu (2002): Spatial patterns of tree height variations in a black spruce farm-field progeny test and neighbors-adjusted estimations of genetic parameters. Silvae Genet 51: 13–18.

  • Krakowski, J, Y. S. Park and Y. A. El-Kassaby (2005): Early testing of Douglas-fir: wood density and ring width. For Genet 12: 99–105.

  • Kroon, J., B. Andersson and T. J. Mullin (2008): Genetic variation in the diameter-height relationship in Scots pine (Pinus sylvestris). Can J For Res 38: 1493–1503.

  • Kusnandar, D. and N. Galwey (2000): A Proposed Method for Estimation of Genetic Parameters on Forest Trees Without Raising Progeny: Critical Evaluation and Refinement. Silvae Genet 49: 15–21.

  • Loo-Dinkins, J. A. and C. G. Tauer (1987): Statistical efficiency of six progeny test field designs on three loblolly pine (Pinus taeda L.) site types. Can J For Res 17: 1066–1070.

  • Loo-Dinkins, J. (1992): Field test design. In: Handbook of quantitative forest genetics. Edited by L. Fins, S. Friedman, and J.V. Brotschol. Kluwer Academic Publishers, Dordrecht, the Netherlands. pp. 96–139.

  • Lopez, G. A., B. M. Potts, G. W. Dutkowski, L. A. Apiolaza and P. Gelid (2002): Genetic variation and intertrait correlations in Eucalyptus globulus base population trials in Argentina. Forest Genetics 9: 223–237.

  • Magnussem, S. (1990): Application and comparison of spatial models in analyzing tree-genetics field trials. Can J For Res 20: 536–546.

  • Magnussen, S. (1993): Bias in genetic variance estimates due to spatial autocorrelation. Theor Appl Genet 86: 349–355.

  • Magnussen, S. and A. D. Yanchuk (1994): Time trends of predicted breeding values in selected crosses of coastal Douglas-fir in British Columbia: a methodological study. For Sci 40: 663–685.

  • Meyer, K. (2005): Random regression analyses using B-splines to model growth of Australian Angus cattle. Genet Sel Evol 37: 473–500.

  • Rehfeldt, G. E. (1995): Genetic variation, climate models and the ecological genetics of Larix occidentalis. For Ecol Manage 78: 21–37.

  • Ruppert, D. (2002): Selecting the number of knots for penalized splines. Journal of Computational and Graphical Statistics 11: 735–757.

  • Ruppert, D., M. P. Wand and R. J. Carroll (2003): Semiparametric Regression. Cambridge Univ Press, Cambridge, UK.

  • Saenz-Romero, C., E. V. Nordheim, R. P. Guries and P. M. Crump (2001): A case study of a provenance/progeny test using trend analysis with correlated errors and SAS PROC MIXED. Silvae Genet 50: 127–135.

  • Schabenberger, O. and C. A. Gotway (2005): Statistical Methods for Spatial Data Analysis. Boca Raton: Chapman & Hall.

  • Silverman, B. (1986): Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.

  • Smith, B. J. (2003): Bayesian Output Analysis Program (BOA) version 1.0 user’s manual. Cited 14 Aug 2008.

  • Spiegelhalter, D. J., N. G. Best, B. P. Carlin and A. Van der Linde (2002): Bayesian measures of model complexity and fit (with discussion). Journal of the Royal Statistical Society Series B 64: 583–639.

  • St. Clair, J. B. (2006): Genetic variation in fall cold hardiness in coastal Douglas-fir in western Oregon and Washington. Can J Bot 84: 1110–1121.

  • Thomson, A. J. and Y. A. El-Kassaby (1988): Trend surface analysis of provenance-progeny transfer data. Can J For Res 18: 515–520.

  • Verbyla, A. P., B. R. Cullis, M. G. Kenward and S. J. Welham (1999): The analysis of designed experiments and longitudinal data by using smoothing splines (with discussion). Applied Statistics 48: 69–311.

  • Wand, M. P. (2003): Smoothing and mixed models. Comput Stat 18: 223–249.

  • Woods, J. H., D. Kolotelo and A. D. Yanchuk (1995): Early selection of coastal Douglas-fir in a farm-field environment. Silvae Genet 44: 178–186.

  • White, I. M. S., R. Thompson and S. Brotherstone (1999): Genetic and environmental smoothing of lactation curves with cubic splines. J Dairy Sci 82: 632–638.

  • Ye, T. Z. and K. J. S. Jayawickrama (2008): Efficiency of using spatial analysis in firest-generation coastal Douglas-fir progeny tests in the US Pacific Northwest. Tree Genet Genomics 4: 677–692.

  • Zas, R. (2006): Iterative kriging for removing spatial autocorrelation in analysis of forest genetic trials. Tree Genet Genomics 2: 177–185.


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