Analysis of Distribution of Root Length Density of Apple Trees on Different Dwarfing Rootstocks

Abstract

This paper considers statistical analyses for comparing the distribution of root length density (RLD) of apple trees under different rootstocks and tree spacing. The source data included RLD values (cm cm−3) measured by soil coring the root systems of eight trees in each of two seasons. We formulated a regression model which assumed the RLD dropped exponentially with soil depth, and this relationship varied with the radial distance from the tree. The model fitted to the log transformed mean data described the RLD distribution well. Young trees (5-year-old) of M.26 (semi-dwarf) and MM.106 (semi-vigourous) had a higher mean RLD and showed a more layered vertical distribution, compared with trees of the dwarf Mark rootstock. Differences among rootstocks were not evident in older (9-year-old) trees. In general, young root systems were more bowl shaped, whereas older trees had a higher RLD further away from the tree trunk. RLD is a positive and continuous variable except for the possibility of an excess of exact zeros. A generalized linear model with a Poisson-gamma type distribution allows modelling of individual RLD data with zeros contributing to parameter estimation. It does not, however, provide simplicity of biological interpretation. In this paper we present a model that assumes the realization of RLD data is due to a Bernoulli and an exponential process. The fitting of the Bernoulli-exponential model by maximum likelihood is illustrated, and further generalization suggested.