A new method to compare statistical tree growth curves: The PL-GMANOVA model and its application with dendrochronological data

Abstract

Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius (r) over time (t) of trees on different sites, GMANOVA is combined here with the adapted PL regression model Q = A.T+E, where for bne 0 : Q=Ei [-b.r]-Ei [-b.r1] and for b=0 : Q=Ln[r/r1], A = initial relative growth to be estimated, T = t - t1 , and E is an error term for each tree and time point. Furthermore, Ei[-b.r] = x0283 (Exp[-b.r]/r)dr, b= -1/TPR , with TPR being the turning point radius in a sigmoid curve, and r1 at t1 is an estimated calibrating time-radius point. Advantages of the approach are that growth rates can be compared among growth curves with different turning point radiuses and different starting points, hidden outliers are easily detectable, the method is statistically robust, and heteroscedasticity of the residuals among time points is allowed. The model was implemented with dendrochronological data of 235 Pinus montezumae trees on ten Mexican volcano sites to calculate comparison intervals for the estimated initial relative growth Â. One site (at the Popocaté petl volcano) stood out, with  being 3.9 times the value of the site with the slowest-growing trees. Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time. Without the Popocatépetl site, the numbers changed to 7%, 42%, and 51%. Further explanation of differences in growth would need to focus on factors that vary within sites and over time.