Study area

The study region is located in western Tlaxcala, Mexico. To the north, it borders with the region of Calpulalpan, and to the south, with the state of Puebla; the entire region of Nanacamilpa is included in the physiographic region of the Transverse Neovolcanic Belt, between the geographical coordinates 19°27'30" and 19°30'38" N, and 98°38'17" and 98°32'17” W (Figure 1). It is located at an altitude range of 2 600 to 3 300 m, and its climate is of the C(w₂) type, i.e. temperate subhumid, with summer rains, and a mean annual temperature of 12 °C to 18 °C (^{Inegi, 2009}).

Figure 1 Location of the forest region of Nanacamilpa, Tlaxcala. 

The most representative vegetation types correspond to the conifer and broadleaf forests, whose most prominent species are Pinus montezumae, P. teocote, P. pseudostrobus Lindl., Abies religiosa (Kunth) Schltdl. & Cham. and Quercus laurina Bonpl., which grow on humic and eutric cambisols, as well as on humic andosols with a clayey and clayey loamy texture (^{Inegi, 2009}). Initially, the forest management was developed under the regular system that includes regeneration fellings and thinnings during the entire rotation period (^{Guerra et al., 2008}).

Field data collection

The field information was collected by establishing sampling sites for inventory whose shape and distribution was based on the design used by the National Forest and Soils Inventory, which considers a cluster as a primary unit, made up of four 400 m² plots as secondary sampling units (^{Conafor, 2012}). 16 systematically distributed clusters were established in the study area; in these, the dasometric variables of interest of all the living, whole and healthy trees with a normal diameter (at 1.3 m above the ground) of 7.5 or more cm were measured. Furthermore, 186 P. teocote trees with diameters above 30 cm ―which were scarce in the sampling plots― were included.

The normal diameter was measured with a Richter^® 5 m tree diameter tape, while the total height was calculated using a Haglöf Vertex III digital hypsometer with a T3 ultrasonic transmitter. Although all the tree species in each secondary sampling unit were considered, only the data for P. montezumae and P. teocote were included in the study because they were the best represented taxa in the area, having an average density of 231 and 112 trees ha^-1, respectively. A database for the regression analysis was built with this information, which included 583 pairs of data of normal diameter and total height for P. montezumae and 468 pairs for P. teocote. Table 1 shows the descriptive statistics.

Statistical analysis

The height-diameter data for P. montezumae and P. teocote were adjusted with linear and non-linear regression analyses; 14 non-linear models widely used in the specialized literature on forest biometry were tested (Table 2). Because the databases only included trees with a normal diameter of ≥7.5 cm ―and therefore no diameters near or equal to zero were measured―, the constant of 1.3 was not explicitly integrated in any of the models; this constant conditions the estimated height to be 1.3 m when the normal diameter is equal to zero. Furthermore, in practical terms, only rarely is there a need to estimate heights of near-zero diameters, and therefore, their interest is merely theoretical (^{Trincado and Leal, 2006}).

Most of the studies on the h-d relation consider non-linear models because they best describe the growth; however, in this case, a model with a linear structure in the parameters (M10) was considered, since it has sometimes proven to attain good results in the adjustment of this type of relation (^{García et al., 2017}).

The regression analysis was performed with the SAS^® software, version 9.3 (^{SAS, 2011}). The MODEL procedure with ordinary least squares was used for non-linear models, while the REG procedure of the same statistical analysis software was utilized for the linear model. The goodness-of-fit of each model was assessed with the coefficient of determination corrected by the number of parameters (_adjR²) as an indicator of the proportion of the variance accounted for by the independent variable, the bias, the root of the mean squared error (RMSE), the sum of the squared error (SSE), the mean squared error (MSE), the significance of the estimators, and the Akaike information criterion (AIC). The latter parameter is a very reliable criterion, as it compares between models with different numbers of parameters (^{Sharma, 2009}).

In order to attain greater objectivity in the selection of the best model for each species, the goodness-of-fit criteria indicated were hierarchized according to the procedure described by ^{Tamarit-Urias et al. (2014)}, in order to generate a quality index whose total value facilitated the identification of the model with the best fit in each case. Although the statistical adjustment criteria constituted the main basis for the selection of the best model, its biological consistency was also analyzed, since it is a desirable characteristic in any growth model (^{Sharma, 2009}). Certain authors have pointed out the importance of considering the predictive capacity of the models ―particularly for the height of larger trees (^{Moore et al., 1996}; ^{Zhao et al., 2006})―, which entails the search for a reasonable behavior of the fitted curve through the entire data range, paying special attention to the upper limit.

Due to this, the asymptotic behavior of the curve of each species was judged in terms of the maximum values for growth in height cited in the literature and in the observed data. Thus, the graphic description of the model within and outside the range of the observed data was an additional criterion for the selection. Furthermore, the fulfillment of the basic assumptions of the regression (i.e. normality in the distribution of the errors) was verified with Shapiro-Wilk’s test, along with the graphic analysis of the residuals for observing the tendencies before making inferences about the estimators of the parameters in each model (^{Kutner et al., 2005}).

Pinus montezumae

Table 3 shows the adjustment of the analyzed models. In general, all the tested models for P. montezumae, regardless of their form and the number of their parameters, were observed to exhibit good fits, as the variance (_adjR²) accounted for by the independent variable was above 89 % in all cases.

The fit quality index, given by the total score of each model, evidence that the five best models are M10, M7, M2, M1 and M3 (all of them with three parameters), because they exhibited the best ―i.e. the lowest― values for all the goodness of fit criteria. Notably, the models with two parameters (M6, M11 and M14) exhibited poor values in the errors (SSE, MSE and RMSE), and therefore their total scores were generally less adequate than those of the models with three parameters. Besides, three of them failed to fulfill the assumption of normality in the residuals and therefore they were discarded in the selection.

Although the assessment with the index suggests which models had the best statistical behavior, the estimators of all the models were observed to be significant, except for the model with the best score (M10), which exhibited a non-significant parameter (b₀) (Table 4) and was therefore discarded.

The four next best models fulfilled the assumption of normality of residuals (H₀ is not rejected, p<0.05) (Figure 2). When studying the h-d relation for 25 species in northern Mexico, ^{Corral-Rivas et al. (2014)} stated that the local Bertalanffy-Richards model is the most consistent, as it exhibits values of R² of 0.6 to 0.8 in seven pine species. The results of this work coincide with the referred study, as the model (M1), together with Yang’s model (M2) ―which has a similar structure― exhibited very similar results, with _adjR² values of 0.81. Furthermore, ^{Castillo-Gallegos et al. (2018}) determined that Richards’ local model was the best for modeling the h-d relation for P. chiapensis (Martínez) Andresen in Veracruz, Mexico, although it had much lower R² values. Based on this, and having analyzed its graphic behavior, our final criterion for deciding which is the best model for P. montezumae was biological consistency.

Figure 2 Distribution of residuals of the four best height-diameter models adjusted for Pinus montezumae Lamb. 

In the observed data range, the four best models exhibited a very similar behavior, up to a diameter of approximately 60 m, after which the height projections differ (Figure 3), particularly with M3, which predicts higher values for height, without showing an asymptotic tendency even outside the data range. Conversely, Sharma’s model (M7) indicates lower values and a maximum height of 35 m for diameters slightly above 80 cm, after which it predicts lower heights, which may not be altogether acceptable for P. montezumae.

Figure 3 Trajectories of the curves generated with the best height-diameter models and data observed for Pinus montezumae Lamb. 

In this regard, ^{Zhao et al. (2006)} consider that two of the desirable attributes of the height-diameter equations are that they should attain an asymptote in larger diameters and that they should always exhibit a positive slope; neither attribute was observed with M7. On the other hand, the models of Richards (M1) and Yang (M2) have a similar behavior, even at the upper limit of the data range, and they exhibit an asymptotic tendency at a height of approximately 38 m, in diameters close to 100 cm. This is reasonable, from a biological point of view, for P. montezumae trees, particularly in regions with a good site quality (^{Rodríguez et al., 1988}). However, since Yang’s model (M2) is more conservative and has better adjustment values than that of Richards (M1), it is considered more adequate for estimating the total height in P. montezumae trees in the study region and is expressed by the following formula:

Where:

h = Total height

d = Normal diameter

Pinus teocote

In general, the adjustment values of the models tested for the species were also adequate, given that they are local models (Table 5). ^{Hernández et al. (2015)} estimated heights for P. teocote in the state of Hidalgo, using generalized equations, and they obtained _adjR² values of 0.93. According to the hierarchization of the criteria followed in this study, the quadratic model (M10) has the best fit to the data, since it has the lowest value for the index; this model’s adjustment coefficient of determination (_adjR²) is 0.87, which is above the value registered by ^{Corral-Rivas et al. (2014)} for the same species (R²<0.7) in northern Mexico.

According to the rating system, the best models after M10 turned out to be M3, M12, M9 and M7; however, in regard to the hypothesis tests on the estimators, neither the value of b ₀ with M9 nor that of b ₂ with M7 were significant (P<0.05) (Table 6); therefore, they were not considered for further analysis. In any case, M3 and M12 occupied the second and third place, respectively (Table 5), with the advantage that all their coefficients were significant (p<0.05), and in both cases the assumption of normality of residuals was fulfilled, according to the dispersion graphs (Figure 4) and the SW statistic (>0.97).

Figure 4 Distribution of residuals of the three best height-diameter models adjusted for Pinus teocote Schiede ex Schltdl. 

Graphically, models M3, M10 and M12 described similar trajectories until attaining a diameter of approximately 65 cm (Figure 5), after which only model M10 exhibited a descent in the value of the slope of the curve, while M3 and M12 maintained their rising trajectory, even outside the data range. It is evident that, with these two models, there is a tendency to overestimate the total height (>32 m) for trees with diameters close to 80 cm, and therefore they are not considered adequate for modeling this relation. The tendency shown by the curve generated by model M10 seems to be more reasonable and logical for P. teocote, whose maximum diameters and heights can hardly exceed 85 cm and 32 m, respectively. For example, ^{Hernández et al. (2015)} document maximum values of 72 cm for diameter and 31 m for height in the same species.

Figure 5 Trajectories of the curves generated with the best height-diameter models and data observed for Pinus teocote Schiede ex Schltdl. 

Therefore, the final equation (Model 10) for estimating the total height of P. teocote trees based on the normal diameter in the study region is expressed as follows:

Where:

h = Total height

d = Normal diameter

The local height-diameter models selected for Pinus montezumae and P. teocote are specific to the typical conditions of the forests of the western region of the state of Tlaxcala. Due to their appropriate statistical adjustment values and their biological consistency, they can be applied in the forest management of this area. Furthermore, they will be useful in future research ―e.g. for generating growth and timber yield systems of the two studied Pinus species―, but always restricted to the same geographical area.