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ŠUMARSKI LIST 3-4/2012 str. 45     <-- 45 -->        PDF

 
extracted at dbh (1.3 m) to obtain the age. The tree rings were counted using a WILD M3Z binocular with polarized light source.
Data analyses – Obrada podataka
The following equations which have been widely used in modelling biological growth phenomena were used as candidate functions to model dominant height (H) of studied chestnut population: (i) Richards, Eq. 1 (Richards 1959; Pienaar and Turnbull 1973; Zeide 1993; Rennolls 1995; Rojo and Montero 1996; Amaro et al. 1998); (ii) Lundqvist-Korf Eq. 2 (Stage, 1963; Tome 1988; Zeide 1993); and (iii) Hossfeld, Eq. 3 (Pita 1964; Elfving and Kiviste 1997; Palahí et al. 2004)
                          H = a [1–exp(–kA)]c                     (1)
                             H = aexp(–kA–c)                        (2)
H = a 1+ k Ac
                                                                           (3)
where A is the age, a is the asymptote of H, k is a growth rate related parameter, and c is a shape parameter.
The procedure used to evaluate the models incorporated qualitative as well as quantitative examinations. The goodness of fit of all regression models was assessed through the coefficient of determination (R2), F-test for significance of the regression and t-tests for significance of the coefficients of the models. Plots of the predictor variables against the residuals and the predicted values against the residuals were examined to check for model deficiencies (Draper and Smith 1981). Cook’s distance, Leverage and DFFit residual statistics were employed to identify potential influence cases.
Models were further compared by the evaluation of the standard error of the model Sy (Eq. 4) and Akaike’s Information Criteria (AIC, Eq. 5)
                                                                           (4)
                         AIC = M ln (S/M)+2m                    (5)
where M is the sample size, S is the residual sum of squares, and m is the number of coefficients of the regression. AIC is measure of the relative goodness of fit of a statistical model. For chosen set of models for fitting the data, the one with minimum AIC value is the preferred one. Unlike R2, AIC is a relative measure, and cannot tell us how well a model fits the data in an absolute sense, but can be used as a mean of comparison among models (Stankova et al. 2006; Field 2009). At this stage, the residual plots of the models were further examined in order to check for violation of the assumptions of linearity and homoscedasticity. Biological realism and graphical appearance of the models were also considered.
The elaboration of Site Index Curves (SIC) for studied chestnut population followed the "guide curve method" procedure as suggested by Clutter at al. (1983).
Results
Rezultati
It is apparent from the model statistics shown in Table 1 that each growth function was well fitted to the tree age/height data. The overall fit of the models was significant at a < 0.01 and they accounted for at least 60 % of the total variation in dominant height, which according to Cohen (1986) is a large effect. Richards function performed marginally better compared to the others with R2 = 0.62. The Richards function had the smallest values of Standard error of the model (Sy) and AIC coefficients: Sy = 3.12 and AIC = 296.3. The Richards function was the only one with all model coefficients being statistically significant at a < 0.01. Although the three growth functions were fitted to the same data set, they resulted in different asymptote coefficients. The asymptote of the height was greatest for the Lundqvist-Korf function (28.91) and lowest for the Richards function (23.26) (Table 1). The three growth functions used in the current study similarly predicted the tree dominant heights for most age classes with exception for the older trees (Figure 1). The Lundqvist-Korf function predicted larger tree heights, followed by the Hossfeld and Richards functions. Furthermore, at older ages the Lundqvist-Korf and Hossfeld equations produced a less-asymptotic trend than the Richards one, hence the latter being considered as more biologically realistic. The residuals for the three models showed a random manner of distribution (Figure 2), which suggested that there were no violations of the assumptions about the errors. No potentially influential cases were detected.
In accordance with the evaluation tests, the Richards function (Eq. 1) was chosen as most adequate to express the age-dominant height relationship and further employed as a guide function to derive site index curves for studied chestnut population.