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

Ricker M, Peña Ramírez VM, von Rosen D - PLoS ONE (2014)

Bottom Line: 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.Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time.Further explanation of differences in growth would need to focus on factors that vary within sites and over time.

View Article: PubMed Central - PubMed

Affiliation: Departamento de Botánica, Instituto de Biología, Universidad Nacional Autónoma de México (UNAM), México D.F., Mexico.

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 b ≠ 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]  = ∫(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 A. One site (at the Popocatépetl volcano) stood out, with A 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.

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Growth of tree #52 from the Chichinautzin site.Above: Regression line  from nonlinear regression with (1) and fixed  cm−1 to find  = 1.24% per year and radius  = 15.98 cm at time  = 15 years (1989; R2 = 0.9998). Below: Initial relative growth (here  = 1.24% per year) represents a standardized measure for comparing relative growth (dr/dt)/r, even though the data points may be at different radiuses: The mean growth path is projected backward to a radius infinitesimally close to zero; when b<0, b determines the change of relative growth with increasing radius as a negative exponential function (Relative growth  = A·Exp[b·r]). Note that the graph does not show an extrapolation, but a standardized characterization of relative growth at the Y-intercept.
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pone-0112396-g001: Growth of tree #52 from the Chichinautzin site.Above: Regression line from nonlinear regression with (1) and fixed cm−1 to find  = 1.24% per year and radius  = 15.98 cm at time  = 15 years (1989; R2 = 0.9998). Below: Initial relative growth (here  = 1.24% per year) represents a standardized measure for comparing relative growth (dr/dt)/r, even though the data points may be at different radiuses: The mean growth path is projected backward to a radius infinitesimally close to zero; when b<0, b determines the change of relative growth with increasing radius as a negative exponential function (Relative growth  = A·Exp[b·r]). Note that the graph does not show an extrapolation, but a standardized characterization of relative growth at the Y-intercept.

Mentions: Here, is time, age, or date of data point i; at is the tree trunk radius at given time or age for positioning the overall growth curve (either or can be used as regression coefficient); Ei is the exponential integral Ei[–b·r]  = ; Exp refers to the exponential function; b is a regression coefficient, representing the negative inverse of the turning point radius (TPR) in a sigmoid curve; is the tree trunk radius at time of data point i; a is a regression coefficient, representing initial logarithmic relative growth, with Exp[a]  = A =  initial relative growth (see Figure 1 why it is called “initial”); is the error term for data point i; and Ln refers to the natural logarithm.


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

Ricker M, Peña Ramírez VM, von Rosen D - PLoS ONE (2014)

Growth of tree #52 from the Chichinautzin site.Above: Regression line  from nonlinear regression with (1) and fixed  cm−1 to find  = 1.24% per year and radius  = 15.98 cm at time  = 15 years (1989; R2 = 0.9998). Below: Initial relative growth (here  = 1.24% per year) represents a standardized measure for comparing relative growth (dr/dt)/r, even though the data points may be at different radiuses: The mean growth path is projected backward to a radius infinitesimally close to zero; when b<0, b determines the change of relative growth with increasing radius as a negative exponential function (Relative growth  = A·Exp[b·r]). Note that the graph does not show an extrapolation, but a standardized characterization of relative growth at the Y-intercept.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4234403&req=5

pone-0112396-g001: Growth of tree #52 from the Chichinautzin site.Above: Regression line from nonlinear regression with (1) and fixed cm−1 to find  = 1.24% per year and radius  = 15.98 cm at time  = 15 years (1989; R2 = 0.9998). Below: Initial relative growth (here  = 1.24% per year) represents a standardized measure for comparing relative growth (dr/dt)/r, even though the data points may be at different radiuses: The mean growth path is projected backward to a radius infinitesimally close to zero; when b<0, b determines the change of relative growth with increasing radius as a negative exponential function (Relative growth  = A·Exp[b·r]). Note that the graph does not show an extrapolation, but a standardized characterization of relative growth at the Y-intercept.
Mentions: Here, is time, age, or date of data point i; at is the tree trunk radius at given time or age for positioning the overall growth curve (either or can be used as regression coefficient); Ei is the exponential integral Ei[–b·r]  = ; Exp refers to the exponential function; b is a regression coefficient, representing the negative inverse of the turning point radius (TPR) in a sigmoid curve; is the tree trunk radius at time of data point i; a is a regression coefficient, representing initial logarithmic relative growth, with Exp[a]  = A =  initial relative growth (see Figure 1 why it is called “initial”); is the error term for data point i; and Ln refers to the natural logarithm.

Bottom Line: 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.Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time.Further explanation of differences in growth would need to focus on factors that vary within sites and over time.

View Article: PubMed Central - PubMed

Affiliation: Departamento de Botánica, Instituto de Biología, Universidad Nacional Autónoma de México (UNAM), México D.F., Mexico.

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 b ≠ 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]  = ∫(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 A. One site (at the Popocatépetl volcano) stood out, with A 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.

Show MeSH
Related in: MedlinePlus