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Separation of Shape and Rate Parameters in Statistical Growth Curve Modeling

Martin Ricker
Dietrich von Rosen

Wolfram Technology Conference 2012
Conference location

Champaign, Illinois, USA

Employing a Mathematica notebook, we explore the relationship between logarithmic relative growth and quantity "Q" with piecewise (multiple) linear regression. We exemplify our approach with tree growth data, with two linear segments. The linear relationship leads to an excellent growth curve function for predicting the relationship between time (or age) and quantity, which in Mathematica notation is: t2 = t1 + Exp[-a] (ExpIntegralEi[-b Subscript[Q, 2]] - ExpIntegralEi[-b Q1]), with "Exp" = exponential, "a" and "b" = regression coefficients, "t" = time or age, and "ExpIntegralEi" = the exponential integral Ei[x]. The Y-intercept "a" of the linear regression indicates the growth rate, while the slope "b" indicates the turning point quantity TPQ = -1/b, where a sigmoid curve turns from left to right-winged. The model is highly flexible: Each segment can represent sigmoid, exponential and almost linear, or over-exponential growth. Finally, the model can also be used in situations where time or age is unknown, but annual increment can be measured, such as in tropical trees without annual growth rings. The regression coefficients "t1" or "Q1", "a", and "b" can be determined with nonlinear regression without any problem, though with the unusual feature that time is determined as a function of quantity. After calculating the regression coefficients individually for 10 trees on two sites, we use the Generalized Multivariate Analysis of Variance (GMANOVA) model to compare the relative growth rate A = Exp[a] among tree growth curves on different sites, when the shape ("b") and position ("Q1") differ among growth curves. This is done with (t2 - t1) as the time variable and (ExpIntegralEi[-b Q2] - ExpIntegralEi[-b Q1]) as the quantity variable, while the only regression coefficient is "A". The statistical growth rate comparison can nicely be presented in a graph.

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