The predictive ability of seven sigmoid curves used in modelling forestry growth : a thesis submitted to the Institute of Information Sciences and Technology in partial fulfilment of the requirements for the degree of Master of Applied Statistics at Massey University, February, 2000
In this thesis we study seven sigmoid growth curve families to determine which best fit pinus radiata basal area against age data. We fit the sigmoid models to the data of distinct plots rather than the pooled data of sets of plots. The seven growth models are the three-parameter Chapman-Richards, Hossfeld, Schumacher, Weibull and Gompertz models and the four-parameter Levakovic and Sloboda models. This investigation was inspired by Dr. Richard Woollons' observation that sigmoid curves vary consistently in their estimation of the asymptote, and those functions giving bigger asymptotes have better goodness-of-fit properties. It is shown that models with better goodness-of-fit properties are better predictors of basal area. This indicates that the three-parameter Chapman-Richards model and the four-parameter Levakovic and Sloboda models are superior to the Hossfeld, Schumacher, Weibull and Gompertz models. We do however recommend caution in the use of the Levakovic model where convergence of the nonlinear least squares algorithm is often difficult. Also, parameter-effects curvature of the four-parameter models was on occasion seen to be unacceptably large and we recommend careful examination of curvature in the selection of a candidate function. We demonstrate that the Schumacher model predicts larger asymptotes than the other models but do not conclude that this model has better goodness-of-fit properties, contrary to Dr. Woollons' observation. We do however conclude that models with better goodness-of-fit do have better predictive power. The study of the growth curves is divided into six parts. Firstly, we investigate fundamental properties of the growth curves with particular attention paid to the point of inflection and the asymptote. Secondly, the models are fitted to pinus radiata data and goodness-of-fit properties are investigated. Additionally, we discuss practical considerations required when fitting these models using nonlinear least squares; in particular the location of starting values and reparameterization of the growth models. We note the ease of fitting the Chapman-Richards model and the relative difficulty in fitting the Levakovic and Sloboda models. Thirdly, we empirically demonstrate that the Schumacher model has the largest asymptote. Fourthly, we investigate the robustness of the models in predicting basal area using both pinus radiata data and simulated data. Next, Padé rational approximations of the growth curves are investigated to begin a theoretical study of the observed behaviour of the fitted growth curves in an attempt to explain the goodness-of-fit and predictive power of the curves by providing a common basis of comparison. A possible additional area of research is indicated by this analysis - the estimation of properties at the point of inflection and the fitting of the associated rational function to the data. Finally, the results in preceding chapters are summarised to rank the growth curves according to their effectiveness in modelling pinus radiata growth data. We do not conclude that one model is optimal in all respects but do give a hierarchy of suitability, with the Chapman-Richards model at the top of this hierarchy.