Two Tests of Significance for Preferred Direction in Tree Radial Growth Under a Linear-Circular Regression Model with Correlated Random Errors

Abstract

To analyze tree growth statistically through annual ring widths measured in 2-D horizontal trunk sections, we propose two tests of significance defined under a linear-circular regression model with fixed trigonometric effects and normal random errors with a variance-covariance structure from the symmetric circulant family. The associated von Mises distribution has a preferred direction parameter. Accordingly, the first test aims to assess the presence of a preferred direction in the radial growth of a tree from the center of its trunk in a given year. Assuming there is a preferred direction of radial growth for the tree in two years, the second test extends the first one by assessing the equality of tree radial growth in the two preferred directions. Both tests of significance are modified F-tests with the denominator df adjusted for the presence of autocorrelation. Their validity is analyzed for two autoregressive symmetric circulant correlation structures, as a function of the number (n) of angular data and the autocorrelation parameter value. Effects of the inter-year correlation coefficient value are also studied in the two-year case. The performance of REstricted Maximum Likelihood as estimation method is scrutinized in an extensive Monte Carlo study, and the power of the tests is analyzed when valid. The new testing procedures are applied with \(n = 32, 64\) ring widths per year for a white spruce tree during 18 years of growth until its harvest. R codes are available. Conclusions and perspectives for future research are given. Supplementary materials accompanying this paper appear on-line.