Study área

The research area is located within the Zona de Protección de Flora y Fauna Pico de Tancítaro (Tancítaro Peak Flora and Fauna Protection Area) (FFPA), in the state of Michoacán de Ocampo, specifically in Tancítaro, Peribán, Uruapan and Nuevo Parangaricutiro municipalities. It has an altitudinal range between 2 200 and 3 850 masl and a surface of 23405- 92-09.55 ha according to its creation decree. The extreme coordinates of the FFPA are 19°31’09.83’’ to 19°20’30.61’’ N and 102°13’14.34’’ to 102°24’07.42’’ W.

Specifically, the plot is located in Provincia Eje Neovolcánico (the Neovolcanic Axis Province), of the Subprovincia Neovolcánica Tarasca (Tarascan Subprovince) (X9) and the Estrato Volcán (Volcano Stratum) (S1) (^{Inegi, 1985}).

The prevailing climate is temperate sub-humid C(m)(w), with a sub-humid warm summer with abundant rains, a mean annual precipitation of 1000-1200 mm and a mean annual temperature of 10-12 °C (^{García, 1983}). The predominant soil unit is ochric andosol, with a coarse texture (To+1); this type of soils are found in areas where volcanic activity has recently occurred, and they hold primarily pine, spruce and holm oak forest vegetations (^{Inegi, 1985}).

Data

A sample of 1 083 normal diameter-stump diameter data pairs was collected in a surface area of 10 ha (Figure 1), from complete, totally healthy trees with a single stem as straight as possible, that are neither isolated nor located at the edges of the stand, in which trees with a normal diameter (ND) of 2.5 to 80 cm were measured; the measure was taken at a height of 1.30 m as well as the total height with a Suunto clinometer in 107 sampling sites of circular shape (500 m²) distributed in a systematic way, at a distance of 100 m between sites and of 100 m between sampling lines, which were marked with a slope compensated rope.

Figure 1 Location of the study area in the Zona de Protección de Flora y Fauna Pico de Tancítaro (Tancítaro Peak Flora and Fauna Protection Area) in Michoacán. 

Variables

The following variables were recorded: species, tree number, normal diameter (D), total height (H), condition, damage and predominance. The age and the passage of time were measured in three trees per site. 1083 pairs of data of D measured at a height of 1.3 m and SD measured at 0.3 m were obtained for different diametric categories and growth conditions through the direct measurement.

Utilized models

The models utilized were those proposed by ^{Quiñonez et al. (2012)} and ^{Pompa-García et al. (2011)} for the estimation of the SD from the ND, given that this section has the shape of a truncated neiloid (Table 1).

Table 1 Models used to predict the normal diameter (ND) from the stump diameter (SD). 

Statistical Analysis

The information was graphed in order to detect atypical data in the scatter plot, and a preliminary run of the PROG REG models and of the R-INFLUENCE option was carried out to obtain the studentized residuals; when the value of the latter was above 2 (absolute), they were considered to be atypical, and therefore the observation was deleted. The PROC MODEL procedure was used for the final adjustment (^{SAS, 2003}).

The selection of the best equation was based on the Mean Square Error (MSE), the Root of Square Means of Error (RMSE), the level of significance of the estimators (Pr>|t|) and the fitted coefficient of determination (R²_adj); the distribution of the residuals was also verified. Compliance with the regression assumptions was verified using the Shapiro-Wilk test (^{SAS, 2003}). On the other hand, the relative and accumulated frequencies of the residuals were graphed in order to determine any similarities with a straight line in relation to the probability of the normal distribution, as well as between their percentages and a Gauss bell curve (^{SAS, 2003}).

The capacity of fit was analyzed based on the residuals and on three statistics, namely the Bias (E¯), the Root of the Mean Square Error (RMSE) and the R²_adj the distribution of the percentage to which the response variable is explained and is taken into account in the total number of the estimated parameters (^{Barrio et al., 2004}; ^{Trincado and Leal, 2006}).

Results and Discussion

Table 2 presents a summary of the variance analysis for each of the fitted equations and shows the statistical estimators for the five models analyzed, as well as the indicators of goodness of fit.

Table 2 Values of the parameters and statistics of goodness of fit for the models used. 

The models with the least MSE were M2 and M5; however, the latter includes a non-significant parameter. Models M1 and M3 yielded equal values (Table 2). This is one of the most important indicators for deciding on the best model; its value shows models M2, M1 and M3 to be the best.

The values of the fitted R² coefficient explain the fit of the model to the data and take into account the number of coefficients used (^{Pompa-García et al., 2011}). Models M2 and M5 yielded the closest values to the unit, followed by models M1 and M3, both with an equal value. These results are higher than those reported by ^{Alder and Cailliez (1980)}, who state that the best features may have coefficients above 0.7 and 0.8, and by ^{Gujarati (2004)}, according to whom a model is satisfactory when the value of this coefficient is approximately 0.8. This is an indication that the models yield satisfactory estimates. In this case, model M2 remains the best, since M5 includes the non-significant parameter β₀. This model coincides with the one obtained by ^{Pompa-García et al. (2011)}, who estimated values of 0.96 for Pinus durangensis Martínez.

Coefficients β₀ and β₁ of models M1, M2 and M3 are significant (p<0.0001), but not so the coefficient β₀ of model M5 (p=0.1226), which shows the highest value for the standard error of the regression coefficient. The parameter β₀ is the one that characterizes the form of the ratio of the stump diameter to the normal diameter; for this reason, it is important to maximize the accuracy of its estimation, and therefore model M5 should not be used.

The RMSE makes it possible to evaluate the accuracy of the estimations carried out (^{Benítez-Naranjo et al., 2004}). Model M2 produced the lowest value of RMSE, and M1 and M3 both yielded the same index. In this case, the best model is M2.

These criteria lead to select the simplest models because they penalize the value according to the principle of parsimony (^{Gómez-Aparicio et al., 2013}).

According to the Shapiro-Wilk test, the assumption of normality is not violated in any case, since values close to 1 are obtained and the level of significance (Pr < W) is high (< 0.0001) (^{Velazco et al., 2006}). Given that the study sample was sufficiently large according to the central limit theorem, these samples tend to be closer to normality (^{Martínez-González, et al., 2006}), and consequently it may be assumed that the distribution of the residuals is close enough to normal (^{Augusto et al., 2009}).

Table 2 shows the deviation of models M1 and M3 with respect to the observed values versus the predicted values (E¯) as measures of accuracy of the estimations; such deviation is low, and therefore these models are followed in order by model M2.

These three models have good predictive capacity in accordance with the low values of the root of the mean square error and bias, and the higher value of R²_adj (Table 2). A positive bias indicates by how many units the prediction is underestimated at the individual tree level. This is an improvement over the results obtained by ^{Pompa-García et al. (2011)}, who obtained values above 5.7 for the RMSE, a bias of over 2.39, and an R²_adj of 0.96 for Pinus durangensis.

Figure 2 shows the fidelity of the predicted data in relation to those obtained with models that had the best fits; no significant bias was observed in the predictions. It is possible to obtain a reliable prediction of the normal diameter with any of the three equations, depending on the dimensions of the stump.

Figure 2 Estimates of the ND based on the SD using the models M1, M2 and M3 best adjusted for Abies religiosa (Kunth) Schltdl. et Cham. in Tancítaro, Michoacán . 

In order to illustrate the use of the equations, we will assume that we have a stump with a 50 cm diameter cut at a height of 30 cm. The following diameters were obtained with the three equations:

The estimated values of the normal diameter in cm are very similar in models M1 and M3-although, due to the parsimony or simplicity, model M1 is recommended by ^{Martínez-López and Acosta-Ramos (2014)}-and closely resemble the results reported by ^{Corral-Rivas et al. (2007)} for the Pinus species of northern Mexico. The estimation of the normal diameter based on the stump diameter makes it possible to quantify and assess clandestine loggings, natural disasters, the reconstruction of the structure of the forest and the exploitation practices applied.