Introduction

Metopium brownei (Jacq.) Urb. (Anacardiaceae) or chechén negro is a tree up to 25 m in total height (at, m) and 60 cm normal diameter (d, cm). It is distributed on the slope of the Gulf of Mexico, from the south of the states of Veracruz, Tabasco, the Yucatán Peninsula and north of Chiapas. It is abundant in the middle and upper strata of the semi-evergreen and sub-deciduous forests, with less abundance in the low forests; both in shallow soils with good surface drainage, and in deep soils that withstand periodic flooding (^{Pennington and Sarukhán, 2005}). From the attractive grain of its wood given by its anatomical structure and the combination of colors of its heartwood and sapwood, it is commercially used for the manufacture of veneer, staves, floors, wood paneling or wainscot, furniture and handicrafts with a high market value (^{Tamarit and López, 2007}). Thus, it is an interesting taxon for the forest industry (^{Gallegos et al., 2012}; ^{Silva et al., 2011}).

The state of Quintana Roo stands in the second place in the country in timber production of tropical species from forests, with a use of 47 139 m³ of roundwood of tropical hard and soft species (12.10 % of national production) and 4 807 m³ of precious taxa (15.18 % of the national production), with an important economic benefit (^{Semarnat, 2016}).

In carrying out timber forest inventories, d and at are the two most important variables (^{Liu et al., 2017}), but measuring at of all trees is very costly in terms of time and resources. In addition, it is the variable in which the greatest errors occur during its indirect measurement (^{Hunter et al., 2013}). In contrast, the measurement of d is relatively simple, accurate and less expensive (^{Larjavaara and Muller-Landau, 2013}).

The application of forest management requires knowledge of d and at at the tree and stand level to estimate the existing volumes in the forest masses (^{Pompa-García et al., 2011}). The foregoing is achieved with the use of growth or allometric models that estimate the volume from them (^{Liu et al., 2017}). Therefore, the construction and use of models that describe functional relationships and growth dynamics over time are very useful (^{Regalado et al., 2008}). An allometric model is defined as an equation that quantitatively describes a relationship that allows one tree variable (at, volume or biomass) to be predicted as a function of another, for example, d (^{Picard et al., 2012}).

The relationships between at and d, or some other variable such as stump diameter (dt), obey a proportionality rule that is the same for trees growing under similar conditions (^{Bohlman and O'Brien, 2006}). This is a basic principle of allometry that makes it possible to predict variables that are difficult to measure based on others that are easy to measure (^{Huxley, 1924}).

The generation of reliable equations for some variables of the tree is complex due to the variability they present and the different conditions in which they grow (^{Quiñonez et al., 2012}).

Since some time ago, the adjustment of mixed effects models (MEM, for its acronym in Spanish) has become a viable option for this type of study (^{Corral-Rivas et al., 2014}), which replaces the classic regression techniques. The inclusion of random and specific parameters per sampling unit makes it possible to model the variability of a given phenomenon such as growth in height, between the different locations or grouping factors of the variable of interest, and with this, the variance of the error is notably reduced (^{Rijal et al., 2012}; ^{Seoane, 2014}). The MEM technique includes fixed parameters common to the entire population and other random parameters specific to grouping levels (^{Chenge, 2021}; ^{Raptis et al., 2021}). By using MEM, more efficient, precise and reliable estimators are obtained than the fixed parameters of the model adjusted by LCNM; therefore, the source of variation of the variable of interest can be reduced (^{Carrero et al., 2008}).

From the ecological and economic importance of M. brownei, it is essential that this taxon be studied in detail to generate biometric tools derived from the use of modern modeling techniques that guarantee the estimation of variables of interest such as the overall height, with the highest possible precision. Therefore, under the hypothesis that by including the MEM technique and grouping covariates within an at-d model, it will contribute to improving the quality of statistical adjustment, and so, the predictive capacity. The objectives of this study were: 1) To assess the fit of allometric and growth models under the MCNL technique to predict at as a function of d in M. brownei trees to select the one with the best fit quality, and 2) To adjust the base model selected by MEM, under the combination of two grouping factors to increase the predictive capacity of the at-d relationship for trees of the taxon in Quintana Roo, Mexico.

Materials and Methods

The study was carried out in the state of Quintana Roo, Mexico, in medium sub-evergreen, medium sub-deciduous, low sub-evergreen and sub-deciduous forests on hills and rocky plains. The prevailing climate is warm subhumid (Aw), with 26 °C as average annual temperature and 1 300 mm as average rainfall (^{Pennington and Sarukhán, 2005}; ^{Conafor, 2014}).

A database made up of 2 794 at-d data pairs of M. brownei was integrated, which was obtained from 172 clusters of the National Forest and Soil Inventory 2004-2009 (^{Conafor, 2012}) (Figure 1).

Figure 1 Spatial distribution of the clusters by municipality with the presence of Metopium brownei (Jacq.) Urb. in Quintana Roo, Mexico. 

For the purposes of practical application of the at-d model to be generated, the information was grouped by combining the levels of classification of vegetation type (Veg): medium sub-deciduous tropical forest (SMS), medium sub-evergreen tropical forest (SMQ), low sub-evergreen tropical forest (SBQ) and low deciduous tropical forest (SBC), in addition to locality groups such as municipality (Mpio): Bacalar (Bac), Benito Juárez (BJ), Felipe Carrillo Puerto (FCP), Isla Mujeres (IM), José María Morelos (JMM), Lázaro Cárdenas (LC), Othón Pompeyo Blanco (OPB) and Solidaridad (Sol), as the use in Felipe Carrillo Puerto of this species for the elaboration of sleepers for several decades eliminated the best individuals and maybe affected the morphometry of the residual trees.

The d with bark was measured at 1.30 m from the ground, with a five-meter graduated diametric tape of the Forestry Suppliers Inc. Qualitäs-bandmaβ, graduated in centimeters and millimeters. The at was obtained from the base of the tree to the top of the crown with a Suunto® clinometer graduated in degrees and percentages.

The database was randomly divided into 60 %, which was used to perform the statistical adjustment of the at-d models and 40 % to validate the selected model. Six nonlinear models were fitted, as well as one potential-type allometric model (^{Burkhart and Tomé, 2012}; ^{Panik, 2014}; Thanh et al., 2019; ^{Hernández-Ramos et al., 2020}) (Table 1). In a first stage, in order to obtain a base equation, the fit was performed using the nonlinear least squares (NLS) approach in the R statistical package with the nls command (^{R Core Team, 2016}).

The evaluation of goodness of fit and selection of the model was carried out considering the significance of the parameters (α≤0.05) and the best values in the statistics determination coefficient (R ² ), Akaike and Bayesian information criteria (AIC and BIC) and log-likelihood (loglik) (^{Corral-Rivas et al., 2014}; ^{Guerra-De la Cruz et al., 2019}). After selecting the best model, the MEM was run under the combination of the Veg-Mpio clustering factors.

The mathematical structure in matrix form of the model to fit under MEM was based on ^{Bronisz and Mehtätalo (2020)} and ^{Chenge et al. (2021)}, which for a nonlinear model was expressed as:

Where:

f = Linear function

Yij and Xij = i ^-th dependent and independent observation, respectively from the i ^-th unit or classification level

θij = r×1 parameter vector. r = Number of parameters in the model and specific to the j ^-th classification group, this vector can be partitioned for the fixed and random parameters defined as:

Where:

Ai and Bi = Matrices of r×p and r×q size for the fixed and random effects, respectively, of each classification level γ and bi = p×1 and q×1 vector of the fixed and random parameter. p×1 = Number of fixed parameters. q×1 = Random parameters

The inclusion of the random effects (β _i ) was carried out individually in each of its model parameters separately, for instance:

In addition, a varPower type structure was included in the MEM adjustment, which represents a power type variance function structure in which the distribution of the residuals is corrected for each factor or level of grouping. In this case, the adjustments were made using maximum likelihood in the R statistical package with the nlme command (^{Pinheiro and Bates, 2000}; ^{R Core Team, 2016}).

The selection of the best model was made with the same criteria as the fixed effects models with LCNM. Likewise, the regression assumptions of the selected equation were verified by graphically verifying the assumptions of normality of the data with the distribution of frequencies and homoscedasticity of the residuals (^{R Core Team, 2016}). The global deviations of the selected model with MEM were verified with the root mean square error (RCME), while the prediction capacity was confirmed through the average bias of the residuals (E-, m) and the aggregate difference expressed in percent (DA %) (^{Lencinas and Mohr-Bell, 2007}; ^{Hernández-Ramos et al., 2020}).

The validation process was carried out with two independent samples: (i) with 40 % of the sample separated from the original database, and (ii) with 1 861 pairs of at-d data obtained in forest plots in the study area. In both cases, a t-test was applied to compare independent means (α=0.01) (^{Infante and Zarate, 2012}).

Results

The average value for d was 17.88 cm, with minimum and maximum values of 7 and 80 cm, respectively; the coefficient of variation was 49.36 % and they showed a positive leptokurtic and asymmetric distribution. The average at was 10.88 m, with extreme values of 6 and 22 m, a coefficient of variation of 27.58 %, and a leptokurtic distribution with a tendency to be symmetric (Table 2).

In the adjustment of the local fixed effects equations, parameters significantly different from zero were obtained at a significance level of 5 % (p≤0.05) in all cases. The R ² values explained between 39 % and 45 % of the variability in the data; the values in the Akaike and Bayesian information criteria varied between 7 688 and 7 901, the average logarithm of likelihood was -3 857 (Table 3).

The best fits were obtained for model 4 which corresponds to the Chapman-Richards model (Table 3 and Figure 2). When analyzing the trends of the estimates, it was observed that this model reproduces an average trajectory similar to that of the observed data (Figure 2) and presented the lowest values of the AIC and BIC, and the highest of the loglik, therefore, it was selected to be adjusted using MEM and to analyze the effect of including classification variables. The potential model overestimates height dimensions from categories greater than 35 cm and the Strand, Hossfeld I Modified and Verhulst-logistic models underestimate height in the same categories, so they were rejected (Figure 2).

Figure 2 Distribution of data observed and estimated by the equations of fixed effects for the at-d models for Metopium brownei (Jacq.) Urb. in Quintana Roo, Mexico. 

When the random component was incorporated into the model by the level of grouping (Veg-Mpio), it was observed that in some of the parameters not all their values were significant. For instance:

Where:

βi= Inclusion of the random effect

In Table 4 only the effects that were better and significant are presented. It was determined that by including the mixed effect in the asymptotic parameter (β0+βi), the best graphic behavior of the residuals was obtained, indicating that the variance modeling is correct (Figure 3).

Figure 3 Distribution of standardized residuals with respect to the values predicted in the adjustments by the MCNL technique (a) and by the MEM technique (b) with the Chapman-Richards model for Metopium brownei (Jacq.) Urb. in Quintana Roo, Mexico. 

The inclusion of the combined covariates (Veg-Mpio) in the model, led to a statistical gain because the value of R ² improved 11.11 % with respect to the value of the model adjusted with MCNL. The likelihood indicators (AIC, BIC and logLik) also improved by an average of 2.5 % (Table 4). When contrasting the distribution of the residuals between the fit with MCNL (Figure 3a) and the MEM technique (Figure 3b), it was found that the latter provides a homoscedastic pattern with a distribution centered on zero. The Shapiro-Wilk normality test showed values higher than 0.91, which shows the fulfillment of this regression assumption.

When evaluating the adjustment capacity by MEM, it was determined that the value of the RCME was 2.36 m, the average bias was 0.003 and the DA was 0.0002 %. Validation by statistical comparison of means in independent samples at 99 % reliability (a=0.01) showed that there are no significant differences between them. For the first independent sample it was obtained t=0.582 and p-value=0.5604, while for the second validation sample, a value of t=1.623 and p-value=0.102 was produced, aspects that can be verified graphically in Figure 4 (a and b).

Figure 4 Comparison of means of the model validation process using independent samples versus estimated data: (a) sample composed of 40 % of the observations of the original base, and (b) independent sample of forest inventories. 

From the statistical robustness and improvement of the MEM with respect to the adjustment by MCNL (Tables 3 and 4), as well as its validation through independent samples (Figure 4), the specific random parameters were obtained for each combined level of grouping (Veg-Mpio) that can be used for each specific growth condition (Table 5).

In addition, when contrasting extreme trends of at-d, an average difference of 3.20 m in height was observed, and in the diameter categories of 35 and 40 cm (Figure 5).

Figure 5 Trend between at-d in some specific reference conditions for Metopium brownei (Jacq.) Urb. in Quintana Roo, Mexico. 

Discussion

The gain in the quality of fit of model 6 obtained under the MEM technique, is due to the fact that according to ^{García and Rapelli (2011)}, grouping the information by some covariate, reduces the deviations with respect to the observed data and the estimation error, since the variance-covariance structure is corrected (^{Littell et al., 2006}) because with this analysis technique, the variance is specific for each grouping level (^{Rijal et al., 2012}; ^{Seoane, 2014}), thereby increasing R ² and improving the values of other goodness-of-fit statistics (AIC, BIC and loglik).

The inclusion of a single specific random effect in the cluster levels (Veg-Mpio) offered the highest predictive quality when associated with the asymptotic parameter, which is explained because it is the most variable and least linear attribute (^{Tamarit-Urias et al., 2014}). This situation led to obtain more efficient, precise and reliable estimators of the fixed parameters of the model and to predict random parameters of each experimental unit, which appropriately reflect the pattern of deviation, in relation to the mean (^{De los Santos-Posadas et al., 2006}; ^{García-Espinoza et al., 2019}).

Therefore, the Chapman-Richards expression adjusted under the MEM approach is reliable for estimating at as a function of d in M. brownei trees in the natural tropical forests of Quintana Roo, Mexico, as did ^{Arias (2004)} who obtained an R ² of 0.64 for six timber species from Costa Rica and ^{Castillo-Gallegos et al. (2018)} who cite values of R ² =0.49 for Pinus chiapensis (Martínez) Andresen in forest plantations in Tlapacoyan, Veracruz. In both studies the Chapman-Richards expression was used.

The Chapman-Richards model was also used successfully by ^{Saunders and Wagner (2008)} in an at-d relationship under the MEM technique and inclusion of covariates for nine species of trees from the northeast of the United States of America. These authors pointed out that sigmoid models are biologically more appropriate to study at-d relationships. ^{Rijal et al. (2012)} and ^{Raptis et al. (2021)} also indicate that with the inclusion of stand covariates as predictors, the accuracy of the estimates of the Chapman-Richards model is improved when adjusting it using MEM.

The favorable effect achieved when the MEM technique is applied in at-d relationships is remarkable. Thus, ^{Hernandez-Ramos et al. (2020)} obtained an R ² of 0.46 and an increase in the value referred to this statistic to 0.70 with the Hossfeld IV model under the fixed and random effects approaches, respectively for Lysiloma latisiliquum (L.) Benth. in Quintana Roo, while ^{Garcia et al. (2017)} determined local fixed effects models with R ² from 0.92 to 0.97 to predict at as a function of d in eight tropical species from Quintana Roo, Mexico.

^{Hernandez-Ramos et al. (2019)} adjusted at-d models for Bucida buceras L. (pukté) trees under the mixed effects approach. In this study, the control of variability, as for M. brownei, was carried out by groups when using the cluster, and they obtained a significant statistical improvement with the MEM approach compared to the adjustment of fixed parameters. However, the aforementioned authors recorded a bias of -0.46 m, which is higher than the estimate for M. brownei of 0.003 m, therefore, the proposed equation is reliable for estimating at, as a function of d for this species in the tropical forests of Quintana Roo.

When using the best model adjusted by MCNL, a constant growth trend is observed (Figure 5), while when applying the results obtained with the MEM adjustment this trend tends to an asymptote with a difference between the two of 4.16 m, so this approach contributes to reducing overestimation errors in the at used in the preparation of plans of forest management for M. brownie, species of commercial interest.

Conclusions

The selection of a base model and its subsequent adjustment through the combined use of the MEM technique with grouping factors given by covariates, is an efficient procedure to formulate expressions that allow estimating the total height as a function of the normal diameter in trees of Metopium brownei in Quintana Roo, Mexico, so it can be adopted as an excellent strategy to generate this type of biometric tools.

The Chapman-Richards model parameterized to an at-d relationship and adjusted by means of MEM with the combined factor between the types of vegetation and the municipality in the asymptotic parameter (β ₀ ), is the one with the best predictive quality because it increases the precision of the predictions, therefore, its use is recommended in carrying out timber inventories of M. brownei in the study region.

The application of the equations obtained will make it possible to complete the forest inventory databases, in which, due to time and economic logistics, it is not possible to measure the heights of all the trees in the sample.