Introduction

Juniperus is the second most diverse genus of the conifer group in the world, second only to Pinus. Mexico is home to 16 species, of which Juniperus deppeana Steud. and Juniperus flaccida Schldtl. are the most widely distributed (^{Farjon, 2005}). J. deppeana forms copses in different geographic regions of Mexico. Particularly in the state of Tlaxcala, it covers approximately 12 711 ha; it is located in small and fragmented populations, to such extent that only patches of original vegetation can be observed (^{Islas et al., 2008}; ^{Herrerías and Nieto de Pascual, 2020}). In the central-western region (study area), only 24.6 % of the forested area is subject to forest management based on technical criteria, which, together with the lack of monitoring, increases forest degradation, mainly due to land clearing for agricultural and livestock purposes, as well as to the extraction of firewood for fuel (^{Conafor, 2003}). For this reason, it is necessary to know the dimensions of the subtracted trees (normal diameter, height, volume, biomass, carbon, among others), which is difficult if there are no models for estimating the dasometric variables of the trees such as normal diameter (nd, cm), total height (Th, m) or volume (V, m³) from the stump diameter (sd, cm) (residual variable). Based on this information, the losses can be quantified, and an appropriate forest management strategy can thus be designed (^{García-Cuevas et al., 2017}).

Within this context, the statistical modeling of the allometric relationships of trees is a useful resource for a correct estimation of their dimensions (^{Pompa-García et al., 2011}; ^{García-Cuevas et al., 2016}). These tools are reliable and reduce both the time and resources used for obtaining forest inventory information (^{Picard et al., 2012}).

In general, the dasometric variables have been statistically modeled by fitting regression techniques using the Ordinary Linear Squares (OLS) method (^{Hernández et al., 2016}; ^{García-Cuevas et al., 2017}; ^{Guerra-De la Cruz et al., 2019}). However, these have some limitations when used with correlated data from the same locality, site or sampling unit, or from repeated measurements of a variable over time (^{Calama and Montero, 2004}; ^{Corral et al., 2019}).

The application of Mixed Effects Models (MEM) is an alternative for statistical improvement, which allows the inclusion of covariates that reduce the error and the specific variability by classification level (^{Baty et al., 2015}; ^{Correa and Salazar, 2016}; ^{Corral et al., 2019}). Therefore, and because there are no models in the region of Tlaxcala for estimating the mensuration variables in Juniperus deppeana trees, the objective of this study was to represent in a quantitative way the allometric relationships between the variables of forest interest-sd, nd, cd, Th, and V- for Juniperus deppeana trees, through the incorporation of mixed-effects modeling.

Materials and Methods

Results

Descriptive statistics indicated that stump diameter ranged from 9.0 cm to 96.3 cm, the normal diameter of 7.5 cm to 88.0 cm with an average of 22.0 cm, and crown diameter from 0.9 m to 15.9 m. For total height the variation recorded was from 1.9 m to 17.0 m, and for the stem volume, from 0.0131 m³ to 4.0296 m³ per tree (Table 2).

OLS fits for the linear model and NLLS fits for the allometric expression were significant for all parameters (α<0.05), and the global deviations in the estimates resulted in less than two units (RMSE value), with the exception of the nd-sd ratio, which registered a value of 5.4 cm. Therefore, the selected equations correctly predict the nd-sd and cd-nd relationships with an R ² =0.7517 and 0.7741, respectively; on the other hand, for the prediction of volume with respect to the nd was the best ratio with an R ² =0.9442 (Table 3). The models for predicting Th from sd and nd had the lowest R ² (<0.50) in both cases (Table 3), this is due to the great variability in heights (Variance 5.173, Table 2) exhibited by the trees in the field.

When applying the MEM technique, better statistical fit values were observed for all relationships, since the random effect of site was included in an additive way, as a variable that groups individuals in one of the two locations (Españita, Atlangatepec), which ranged from 1.3 % in explaining the sample variability of V as a function of nd, to 42.3 % in the Th-sd relationship. Furthermore, the RMSE value decreased by 54.4 % for nd-sd, and by 7.4 % when V was estimated as a function of the stump dimensions (Table 4). The intervals of the estimated parameters, fixed and random effects fit value, and the variance-covariance matrix (vcov) of each allometric relationship are shown in Table 5.

No issues of non-compliance with the regression assumptions were observed for the fit in any of the proposed allometric relationships when the MEM was used, because the frequency distribution of the residuals is Gaussian (bell-shaped) for all variables (Figure 1), and the dispersion of the residuals showed a random trend (Figure 2).

A = nd-sd; B = Th-sd; C = cd-sd; D = V-sd; E = Th-nd; F = cd-nd; G = V-nd. sd = Stump diameter (cm); nd = Normal diameter (cm); cd = Crown diameter (m); Th = Total height (m); V = Stem volume (m³).

Figure 1 Graphical normality test for the proposed models in each allometric relationship. 

A = nd-sd; B = Th-sd; C = cd-sd; D = V-sd; E = Th-nd; F = cd-nd; G = V-nd. sd = Stump diameter (cm); nd = Normal diameter (cm); cd = Crown diameter (m); Th = Total height (m); V = Stem volume (m³).

Figure 2 Graphical homoscedasticity test for the proposed models in each allometric relationship. 

The validation of the estimates made with the MEM-adjusted models for the various allometric relationships showed no significant differences (p<0.05) in any case. Therefore, H _o for equality of means is accepted, while H _a is rejected (Table 6).

In order to exemplify the application of the models, a clandestine logging area of one hectare with a density of 190 individuals was assumed, with J. deppeana trees whose sd averages 29 cm. Therefore, when applying model 1 (y=a0+a1x: linear) to the nd-sd relationship, we have: nd=2.5413+0.6778×29=22.2 cm.

Subsequently, the cd-nd, Th-nd, and V-nd ratios were calculated with the structure of model 2 (y=a0xa1: potential) cd=0.4513×22.20.7733=4.96 m; Th=1.3382×22.20.5039=6.38 m and V=0.0003×22.22.1005=0.2019 m3.

This procedure allows us to determine the dasometric characteristics of the trees within the affected stand. By multiplying the V by the density, it is possible to quantify the surface area subjected to clandestine felling, whose value would be 38.3609 m³. The same procedure can be performed by considering the sd.

Discussion

In general, the traditional approach provided acceptable results for the different allometric relationships, as in other related works (^{Quiñonez et al., 2012}; ^{Hernández et al., 2016}; ^{Guerra-De la Cruz et al., 2019}). However, the inclusion of the site within the MEM approach improved the average explanation of sampling variability by 16.63 % and reduced model deviations by 18.53 %. This situation is attributed to the fact that the variability of the analyzed information is grouped by site (^{Castedo et al., 2006}; ^{Corral et al., 2019}).

The proposed volume model (R ² =0.9563) does not explain the sample variability as well as the model utilized by ^{Buendía-Rodríguez et al. (2022}) (R ² =0.971), which is also potential; however, these authors only adjusted a homogeneous population of J. deppeana. The resulting gain was improved with the implementation of the MEM for the V-sd (7 %) and V-nd (1.3 %) ratios, with which higher increases were achieved than those quoted by ^{Guangyi et al. (2015}) of 1.3-2 %, when estimating the volume of Cunninghamia lanceolata (Lamb.) Hook trees.

The nd-sd, cd-sd and cd-nd models had conservative gains of 8.9, 18.7 and 5.9 %, respectively, with R ² values of 0.8187 (nd-sd), 0.7062 (cd-sd), and 0.8195 (cd-nd), which are lower than those recorded by ^{Pompa-García et al. (2011}) in Pinus durangensis Martínez (R ² =0.96) for nd-sd.

In a similar way, the models fitted by MEM (Th-sd, Th-nd) exhibited a better fit (0.5706 and 0.6281) than those documented by ^{Buendía-Rodríguez et al. (2022}), who used a traditional OLS fit (R ² =0.427).

^{Guerra-De la Cruz et al. (2023}) report that the incorporation of MEM in order to fit Th-nd models, as well as of the covariate sub-watershed as a grouping factor in Abies religiosa (Kunth) Schltdl. & Cham., results in gains of 12.19 % (R ² ), while, in this study, 32.3 % gain was achieved, and a better result was obtained with the grouping factor (site) for J. deppeana forests. In this regard, ^{Salas-Eljatib et al. (2019}) indicate that, by incorporating the effect of the covariate with the MEM approach, the variance is reduced and rendered homogeneous.

Juniperus deppeana requires forest management based on the use of models according to the mensuration characteristics of each region, since it is a species with great interspecific variation (^{Flores et al., 2018}), which in turn helps maintain biodiversity, as other taxa are associated with it (^{Maxted, 2013}), such as Quercus potosina Trel. and Pinus cembroides Zucc. (^{Díaz-Núñez et al., 2016}). Therefore, it is a conifer with high ecological and scientific potential.

Conclusions

Allometric models fitted using the mixed-effects approach satisfactorily explain the behavior of the analyzed mensuration variables

In addition, modeling using the MEM technique improves the fit with respect to the traditional linear Ordinary Least Squares (OLS) and nonlinear ordinary least squares (NLLS) techniques.

The results prove to be reliable for reconstructing the dasometric characteristics of trees within stands affected by illegal logging activities and for accurately quantifying the actual stocking of J. deppeana forests; therefore, they are a reliable alternative for the preparation of forest management plans in Tlaxcala and neighboring states.