Partitioning variation in Douglas-fir xylem properties among multiple scales via a Bayesian hierarchical model

Abstract

Hierarchical biological scales permeate research in tree physiology and represent multiple sources of variation. We discuss the importance of matching the sampling and analysis scales to biological scales in the data. The advantages of statistical hierarchical modeling are demonstrated using the relationship between specific conductivity and tracheid diameter of secondary xylem as an example. The structure and results of three statistical models were compared within a Bayesian context: a simple linear regression (SLR); a repeated measures analysis (REP); and a hierarchical model (HM). The models share similar mean structures but differ in how variation is partitioned among scales: the SLR model assumes independence among observations (variation came from only a single scale); the REP allows multiple observations of each tree to be correlated; and the HM incorporates features of the REP with an additional variance structure that partitions variation across a broader scale. Our data included hierarchical scales of position on the tree, tree, fertilization treatment and species (Pseudotsuga menziesii (Mirb.) Franco). The HM gave more precise estimates for model parameters, was more robust to outliers, provided a more detailed description of covariances within the data at multiple scales compared with the SLR and REP and increased our ability to detect differences among positions on the tree. The proper statistical analyses increase the value of research by allowing the most exact interpretation.