Tree height (<i>H</i>) in a natural stand or forest plantation is a fundamental variable in management, and the use of mathematical expressions that estimate <i>H</i> as a function of diameter at breast height (<i>d</i>) or variables at the stand level is a valuable support tool in forest inventories. The objective was to fit and propose a generalized <i>H-d</i> model for <i>Pinus montezumae</i> and <i>Pinus pseudostr</i><i>o</i><i>bus</i> established in forest plantations of Nuevo San Juan Parangaricutiro, Michoacan, Mexico. Using nonlinear least squares (NLS), 10 generalized <i>H-d </i>models were fitted to 883 and 1226 pairs of <i>H-d</i> data from <i>Pinus montezumae</i> and <i>Pinus pseudostrobus</i>, respectively. The best model was refitted with the maximum likelihood mixed effects model (MEM) approach by including the site as a classification variable and a known variance structure. The Wang and Tang equation was selected as the best model with NLS; the MEM with an additive effect on two of its parameters and an exponential variance function improved the fit statistics for <i>Pinus montezumae</i> and <i>Pinus pseudostrobus</i>, respectively. The model validation showed equality of means among the estimates for both species and an independent subsample. The calibration of the MEM at the plot level was efficient and might increase the applicability of these results. The inclusion of dominant height in the MEM approach helped to reduce bias in the estimates and also to better explain the variability among plots.
The height of trees (H; m) in forest stands or plantations (FP) is an attribute that is useful to calculate timber estimates, to evaluate site productivity at the stand or plot level, to project growth and yield scenarios, and to create distribution tables of usable products (Crecente-Campo et al., 2010; Santiago-García et al., 2017; Santiago-García et al., 2020) . In a forest inventory, measuring the H of all the trees within a sampling unit involves an important demand of resources, time, and effort; therefore, a subsample of tree heights is usually obtained (Mehtätalo, 2005; Bronisz & Mehtätalo, 2020) .
A reliable and low-cost alternative for error reduction in tree height estimation and to reduce the costs of measuring this variable in the field, is using local or generalized height—diameter (d; cm) models (Zambrano, Suarez, & Jerez, 2001; Barrio, Álvarez, & Díaz-Maroto, 2004; Diéguez-Aranda, Barrio, Castedo-Dorado, & Álvarez, 2005) . Besides, acquiring the data necessary to make calculations at the population level does not imply additional costs, since these are mostly office work to further estimate the dimensions of the remaining trees in each plot, and the generalized expressions consider characteristics at the stand level (Diéguez-Aranda et al., 2005; Crecente-Campo et al., 2010; Gómez-García et al., 2015) .
The generalized H-d models also enable the evaluation of tree height response to different growth conditions, forest management, or forest stand characteristics (Sharma & Parton, 2007; Crecente-Campo et al., 2010; Bronisz & Mehtätalo, 2020) . Statistical fit of these models has been performed with nonlinear least squares (NLS) (Misir, 2010; García-Cuevas et al., 2013) , mixed effects models (MEM) (Vargas-Larreta, Castedo-Dorado, Álvarez-González, Barrio-Anta, & Cruz-Cobos, 2009; Corral-Rivas, Álvarez-González, Crecente-Campo, & Corral-Rivas, 2014; Corral, Silva, & Quiñonez, 2019) , and neural networks (Correia et al., 2018) .
The mathematical structure of MEMs is characterized by fixed and random parameters; the former is related to an average response of the data used, and the second represents the difference of the average plot with respect to each unit or classification variable (Sharma & Parton, 2007; Gómez-García et al., 2015) . The application of MEM is a statistically improved alternative to NLS, which allows the inclusion of covariates that decrease the error and reduce the specific variability by classification level (Baty et al., 2015; Corral et al., 2019; Flores-Ayala et al., 2023) . Consequently, the variability of the response (i.e. height) to different growth conditions can be efficiently explained and estimated with greater certainty (Corral et al., 2019; Bronisz & Mehtätalo, 2020) . In addition, they have been an effective alternative solution to the lack of independence of measurements within a sampling unit (Calama & Montero, 2004; Corral et al., 2019) and for the reduction of the estimation error through the use of NLS in the optimization of parameter estimates (Bronisz & Mehtätalo, 2020) .
In Mexico, owing to the practical utility of generalized H-d models in forest management, this type of structure has been fitted to different coniferous and broadleaf species in natural temperate forests in Durango (Vargas-Larreta et al., 2009; Corral-Rivas et al., 2014; Corral, Silva, & Quiñonez, 2019) , Michoacán (García-Cuevas et al., 2013; Hernández et al., 2018) , and Hidalgo (Hernández et al., 2015) , as well as to even-aged forest stands in Ixtlán de Juarez, Oaxaca (Santiago-García et al., 2020) and in Ixhuacan de los Reyes, Veracruz (Hernández et al., 2020) . However, to our knowledge, there is no research work that had been performed in pine plantations, in particular for the two species that are analyzed in this research, and it is anticipated that these results would be better than those reported for natural forests owing to the homogeneity of forest plantations.
In Michoacán, Mexico, pine species have been established in 16,000 ha of FPs (CONAFOR, 2018) . Pinus pseudostrobus Lindl. and Pinus montezumae Lamb. are the most frequently used species as a result of their high growth rates, their potential to successfully expand the area already established, and consequently to increase Michoacán’s timber yield (Muñoz, Sáenz, García, Hernández, & Anguiano, 2011; Muñoz, Sáenz, García, Coria, & Muñoz, 2015) . These two Pinus species play also a major role in the resin, sawmill, furniture, and handicrafts industries of Nuevo San Juan Parangaricutiro (NSJP) (González, Gasca, & Heredia, 2014) . The objective of this research was to fit and propose a generalized H-d model for Pinus montezumae and Pinus pseudostrobus in the FPs of NSJP, Michoacán, Mexico with a mixed effects modelling approach.
The FPs evaluated are located in the forests of Nuevo San Juan Parangaricutiro, Michoacán, between 19˚17' and 19˚30'N and 102˚06' and 102˚17'W, at an altitude ranging from 2000 to 3100 m. The region has a rugged terrain topography, with an extrusive igneous geology and an Andosol-type dominant soil—typical of the physiographic sub-province known as sierra Neovolcánica Tarasca. The Cw-type climate has a 1,500 mm average annual precipitation and 15˚C average temperature (INEGI, 2017) . The study area is shown in
In this research, fourteen 7 - 32 years old Pinus montezumae and thirty-four 7 - 37 years old Pinus pseudostrobus plantations were analyzed. One-hundred
ninety-one 400 m2 (20 × 20 m) quadrangular sampling plots were systematically located in these plantations. The d and H of each tree within each plot were measured, and the mean height (mH; m) was further calculated. The average maximum dominant height of the five tallest trees (dH; m) was estimated with the information from each plot, and this was defined according to the concept of dominant height that uses the 100 thickest trees per hectare (Alder, 1980) ; also, root mean square diameter (Msd; cm), basal area per plot (Ba; m2∙plot−1) and density (N; tree∙plot−1) were calculated.
Ten generalized models used in similar studies were selected (
In a first approach, the equations in
| Statistics | d | H | md | mH | dH | Msd | Ba | N |
|---|---|---|---|---|---|---|---|---|
| Pinus montezumae: 57 plots and 1261 data pairs | ||||||||
| Mean | 17.43 | 9.17 | 17.44 | 9.17 | 12.46 | 18.02 | 0.62 | 24.70 |
| Minimum | 1.85 | 2.00 | 10.15 | 2.86 | 4.25 | 10.66 | 0.18 | 7.00 |
| Maximum | 46.40 | 29.00 | 37.59 | 25.86 | 29.00 | 37.82 | 1.45 | 40.00 |
| Standard deviation | 7.04 | 4.21 | 5.33 | 3.63 | 4.42 | 5.34 | 0.33 | 8.07 |
| Variance sample | 49.60 | 17.75 | 28.46 | 13.18 | 19.51 | 28.51 | 0.11 | 65.08 |
| Kurtosis | 0.1 | 1.0 | −0.1 | 2.0 | 0.7 | −0.3 | −0.1 | −0.8 |
| Asymmetry coefficients | 0.5 | 0.8 | 0.5 | 1.0 | 0.6 | 0.4 | 1.0 | −0.2 |
| Pinus pseudostrobus: 134 plots and 1751 data pairs | ||||||||
| Mean | 26.16 | 15.34 | 26.29 | 15.34 | 19.41 | 26.78 | 0.80 | 16.76 |
| Minimum | 2.50 | 2.50 | 6.26 | 3.98 | 6.60 | 7.16 | 0.10 | 5.00 |
| Maximum | 72.30 | 43.40 | 58.22 | 32.20 | 43.40 | 58.83 | 2.31 | 34.00 |
| Standard deviation | 0.29 | 0.17 | 0.27 | 0.16 | 0.19 | 0.26 | 0.01 | 0.20 |
| Variance sample | 12.48 | 7.38 | 11.36 | 6.90 | 7.93 | 11.08 | 0.40 | 8.32 |
| Kurtosis | 155.80 | 54.40 | 129.10 | 47.57 | 62.85 | 122.74 | 0.16 | 69.19 |
| Asymmetry coefficients | −0.5 | −0.7 | −0.6 | −1.0 | −0.6 | −0.7 | −0.3 | −0.9 |
d = diameter at breast height (cm). H = total height (m). md = mean diameter at breast height of the plot (cm). mH = mean height of the plot. dH = dominant height of the plot (m). Msd = mean square diameter (cm). Ba = basal area per plot (m2). N = number of trees per plot.
| ID - Name | Structure | Reference |
|---|---|---|
| 1—Gaffrey (1988) | H = 1.3 + ( d H − 1.3 ) e a 0 ( 1 − M s d d ) + a 1 ( 1 M s d − 1 d ) | Sánchez-González, Cañellas, & Montero (2007) and Hernández et al. (2015) . |
| 2—Nilson (1999) | H = d H 1 + a 0 ( 1 − ( m d d ) a 1 ) | Sánchez-González et al. (2007) and García-Cuevas et al. (2013) . |
| 3—Chapman & Richards modified (2007) | H = 1.3 + a 0 + m H a 1 ( 1 − e a 2 d ) a 3 ( N M s d ) 3 | Corral-Rivas et al. (2014) and Ercanli (2015) . |
| 4—Mirkovic (1958) | H = 1.3 + ( a 0 + a 1 d H + a 2 M s d ) e − a 3 d | García-Cuevas et al. (2013) and Santiago-García et al. (2020) . |
| 5—Hui y Gadow (1993) | H = 1.3 + a 0 d H a 1 d a 2 d H a 3 | Vargas-Larreta et al. (2009) and García-Cuevas et al. (2013) . |
| 6—Harrison et al. (1986) | H = d H ( 1 + a 0 e a 1 M s d ) ( 1 − e − q 2 d d H ) | García-Cuevas et al. (2013) and Hernández et al. (2015) . |
| 7—Temesgen and von Gadow modified (2004) | H = 1.3 + ( a 0 + a 1 B a ) M s d a 2 d a 3 d H a 4 | Vargas-Larreta et al. (2009) and Liu et al. (2017) . |
| 8—Sharma and Parton (2007) | H = 1.3 + a 0 d H a 1 ( 1 − e − a 2 M s d a 3 d ) a 4 | Vargas-Larreta et al. (2009) and Santiago-García et al. (2020) . |
| 9—Wang and Tang (1997) | H = 1.3 + a 0 d H ( a 1 e − a 2 d ) | García-Cuevas et al. (2013) and Hernández et al. (2015) . |
| 10—Schroeder y Álvarez II (2001) | H = 1.3 + ( a 0 + a 1 d H − a 2 M s d + a 3 B a ) e a 4 d | Vargas-Larreta et al. (2009) and Corral et al. (2019) . |
ID = model identifier; a0, a1 and a2 = parameters to be estimated; e = exponential function; d = diameter at breast height (cm); H = total height (m); md = mean diameter at breast height of the plot (cm); mH = mean height of the plot; dH = dominant height of the plot (m); Msd = mean square diameter (cm); Ba = basal area per plot (m2); and N = number of trees per plot.
R 2 = ∑ i = 1 n ( y ^ i − y ¯ i ) 2 ∑ i = 1 n ( y i − y ¯ i ) 2 (11)
R M S E = ∑ i = 1 n ( y i − y ^ l ) 2 n − 1 (12)
A I C = 2 ⋅ p + n ⋅ ln ( ∑ i = 1 n ( y i − y ^ l ) 2 n ) (13)
where y i , y ^ l and y ¯ i are the observed, estimated, and mean values, respectively; n is the total number of data used in model fitting (
Once the generalized H-d model was chosen from the models fitted, a classification variable corresponding to the plot was included additively in the model, using the MEM approach (for the MEM fit). This analysis would enable the inclusion of fixed effects and optional random effects; the former applies to the whole population and the latter are specific for each grouping level. In Addition, it reduces the random variability that NLS fails to explain, it groups the H response per classification level, and it reduces the estimation bias by eliminating the initial conditions that influence the growth dynamics within each plot (Calama & Montero, 2004; Mehtätalo, de-Miguel, & Gregoire, 2015; Ferraz et al., 2018) .
The structure of the nonlinear models was: Y i j = f ( X i j , θ i j ) + ε i j ; where f is the nonlinear function (selected from
The MEM was fitted using the “nlme” function of R® (R Core Team, 2021) and the marginal approximation of maximum likelihood of the empirical best linear unbiased prediction (EBLUP) (Mehtätalo & Lappi, 2020) . In addition, three variance functions were evaluated to correct for heteroscedasticity of the residuals and stabilize the variance of the inconsistent error in the estimates: 1) power function (varPower, Equation (14)); 2) exponential function (varExp, Equation (15)); and 3) constant and power function (varConstPower, Equation (6)) (Pinheiro & Bates, 2000; Zuur, Ieno, Walker, Saveliev, & Smith, 2009; Gałecki & Burzykowski, 2013; Mehtätalo & Lappi, 2020) .
var ( ε i j ) = v i 2 ⋅ δ 1 (14)
var ( ε i j ) = exp 2 ⋅ δ 1 ⋅ v i (15)
var ( ε i j ) = ( δ 1 + | v i | δ 2 ) 2 (16)
where var ( ε i j ) is the variance function evaluated in the variance covariate of the predictor residuals ( v i ), while δ 1 and δ 2 are the coefficients of the variance function, which will be specific to each level ( δ i ).
The best structure was selected through the evaluation of the statistical fit with R2, RMSE, and AIC values, as well as the significance of the parameters (p > 0.05) (Corral et al., 2019; Bronisz & Mehtätalo, 2020) . Similarly, the likelihood ratio of each fit was verified through the ANOVA test included in the nlme function (Pinheiro & Bates, 2000) . In the event of a tie or equality, priority will be given to the analysis that presents the smallest deviation (RMSE) and that provides a greater explanation of the sample variability (R2), since an integral evaluation between both mentioned criteria as a system of adjustment evaluation statistical is the best alternative (Mayer & Butler, 1993; Sakici et al., 2008) . The estimation bias (Bias) Equation (17) for the best structure was calculated in order to verify the quantitative deviations resulting from the model application.
B i a s = ∑ i = 1 n ( y i − y ^ l n ) (17)
In addition, a means comparison (as independent populations) was carried out with a t test at a = 0.05 (Infante & Zarate, 2012) . For this purpose, 30 % of the randomly selected sample (which was not used for model calibration) was used (i.e., 378 and 525 pairs of H-d data for Pinus montezumae and Pinus pseudostrobus, respectively). The null hypothesis (H0) tested was that there is no difference between heights of both independent populations ( μ 1 = μ 2 ) , while the alternative hypothesis (H1) was that the real value of the population mean (H) is different from the value established by H0 ( μ 1 ≠ μ 2 ) .
Further, a subsample ( m i ) of the H-d data was used to calibrate the MEM and to estimate the value of random parameters of the vector b i , (Vonesh & Chinchilli, 1997; Corral-Rivas et al., 2014; Corral et al., 2019) ; data from Equation 9 and Equation 12 independent sampling plots were used for Pinus montezumae and Pinus pseudostrobus, respectively. The equation used was the following:
b ^ l = D ^ Z ^ l T ( R ^ l D ^ Z ^ l T ) − 1 ε ^ l (18)
where D ^ common variance-covariance matrix of the random parameters for which q × q refers to the number of parameters with the mixed effect, Z i : m × q matrix of the values of the partial derivatives, corresponding to the dimension of the trees selected for the calibration according to the chosen criterion, and evaluated as b ^ l = 0 , R ^ l : variance-covariance matrix of the global error of the fitted model m j × m j ; ε ^ l : error vector of the fixed model, which represents the residuals between the observed values minus the predicted ones m × 1 , and Z T ; refers to the transpose matrix of each expression.
Four calibration options (C) or localization of the MEM were analyzed to obtain the best estimation strategy of the random parameters for each sampling unit (plot), and to adjust the response of the model fixed component to each specific growth condition in the FPs (Sharma et al., 2016) . The criteria to calibrate the specific response per sampling unit of the MEM and obtain a value with which the random parameter ( + φ i ) was fit in an additive way to estimate the height of the trees based on the normal diameter, were the following:
C1: Measure the heights of the two thickest trees in normal diameter (d) within the plot.
C2: Measure the heights of the two thinnest trees in d within each plot.
C3: Measure the height of the thickest and thinnest trees in d per plot.
C4: Measure the heights of the three thickest trees in d within each plot.
Once the calibration was completed, tree heights on each plot was estimated by strategy (C1, C2, C3 and C4); the estimates were evaluated through the RMSE (Equation 12), Bias (Equation 17) and the mean absolute percentage error values (E%AM: Equation19) (Mayer & Butler, 1993) .
E % A M = 100 ⋅ [ ∑ ( y i − y ^ l / y i ) ] / n (19)
The two-parameter models (Equation 1 and Equation 2) had the lowest precision and provided the less accurate explanation. The Wang and Tang model (Equation 9) was chosen as the best owing to it goodness-of-fit and parsimony, being significant all of its parameters, which is not the case with Equations 3 and 10. Therefore, this expression was used for the analysis with MEM (
The random effect at plot level ( φ i ) was additively included within the generalized H-d structure in the Wang and Tang model (Equation 9.1, 9.2, and 9.3); however, the fit resulted in non-significant parameters when including ( φ i ) in the ( a o ) parameter or jointly with ( a 1 ) and ( a 2 ); therefore, those parameters are not shown.
H = 1.3 + a 0 d H ( ( a 1 + φ i ) e − a 2 d ) (9.1)
H = 1.3 + a 0 d H ( a 1 e − ( a 2 + φ i ) d ) (9.2)
H = 1.3 + a 0 d H ( ( a 1 + φ i ) e − ( a 2 + φ i ) d ) (9.3)
The inclusion of an additive random effect ( + φ i ) in the parameter a 1 for the MEM within the Wang and Tang model, and in the combination of a 1 and a 2 provided good results. These results indicated that the inclusion with the best qualification is given in the combination of these parameters, and that a statistical gain in the fit is obtained with respect to NLS in R2, RMSE, and AIC: 4.3%, 10.9%, and 3.2%, for Pinus montezumae and 4.9%, 24.8%, and 6.5% for Pinus
| ID | Pinus montezumae | Pinus pseudostrobus | |||||
|---|---|---|---|---|---|---|---|
| R2 | RMSE | AIC | R2 | RMSE | AIC | ||
| 1 | 0.3075 | 3.480 | 4711.9 | 0.5518 | 4.788 | 7684.9 | |
| 2 | 0.3473 | 3.378 | 4659.7 | 0.5788 | 4.713 | 7605.2 | |
| 3 | 0.7627 | 2.039 | 3770.1 | 0.8769 | 2.550 | 6035.1 | |
| 4 | 0.8004 | 1.870 | 3617.4 | 0.8684 | 2.636 | 6119.9 | |
| 5 | 0.8025 | 1.860 | 3608.2 | 0.8656 | 2.665 | 6147.4 | |
| 6 | 0.8034 | 1.855 | 3601.9 | 0.8719 | 2.601 | 6084.2 | |
| 7 | 0.8037 | 1.856 | 3604.7 | 0.8705 | 2.617 | 6102.1 | |
| 8 | 0.8041 | 1.854 | 3602.8 | 0.8706 | 2.616 | 6100.9 | |
| 9 | 0.8051 | 1.847 | 3594.2 | 0.8726 | 2.593 | 6076.3 | |
| 10 | 0.8054 | 1.848 | 3597.0 | 0.8748 | 2.572 | 6058.0 | |
ID = model identifier; R2 = Coefficient of determination; RMSE = Root-mean-square error; and AIC = Akaike Information Criterion.
pseudostrobus, respectively. The inclusion of the varExp structure in the MEMs improved the results for both species since it regulated the unequal error variance (
According to the ANOVA, the Wang and Tang expression that included effects on the parameters a 1 and a 2 , and the varExp heteroscedasticity correction structure was the one that showed better results for Pinus montezumae (
A bell-shaped curve (i.e. Gaussian) of the residuals was obtained for both species (
An individual deviation of −0.1110 m for Pinus montezumae and 0.1560 m for Pinus pseudostrobus was observed for the H estimates calculated with the fixed parameters model, obtained with the MEMs approach (
| ID | Effects parameters | Correction structure | R2 | RMSE | AIC |
|---|---|---|---|---|---|
| Pinus montezumae | |||||
| 9 | a 1 | without structure | 0.8227 | 1.759 | 3578.2 |
| 10 | varPower | 0.8260 | 1.743 | 3504.1 | |
| 11 | varExp | 0.8258 | 1.744 | 3516.9 | |
| 12 | varConstPower | 0.8260 | 1.743 | 3506.1 | |
| 9 | a 2 | without structure | 0.8281 | 1.733 | 3572.1 |
| 10 | varPower | 0.8299 | 1.723 | 3487.0 | |
| 11 | varExp | 0.8299 | 1.724 | 3503.9 | |
| 12 | varConstPower | 0.8299 | 1.723 | 3489.0 | |
| 9 | a 1 , a 2 | without structure | 0.8349 | 1.698 | 3574.5 |
| 10 | varPower | - | - | - | |
| 11 | varExp | 0.8412 | 1.665 | 3484.1 | |
| 12 | varConstPower | 0.8299 | 1.723 | 3489.0 | |
| Pinus pseudostrobus | |||||
| 9 | a 1 | without structure | 0.9105 | 2.172 | 5884.3 |
| 10 | varPower | 0.9097 | 2.182 | 5653.1 | |
| 11 | varExp | 0.9101 | 2.177 | 5720.2 | |
| 12 | varConstPower | 0.9097 | 2.182 | 5655.1 | |
| 9 | a 2 | without structure | 0.9096 | 2.182 | 5901.1 |
| 10 | varPower | 0.9065 | 2.220 | 5681.4 | |
| 11 | varExp | 0.9076 | 2.207 | 5743.1 | |
| 12 | varConstPower | 0.9065 | 2.220 | 5683.4 | |
| 9 | a 1 , a 2 | without structure | 0.9104 | 2.050 | 5882.1 |
| 10 | varPower | 0.9174 | 2.086 | 5640.0 | |
| 11 | varExp | 0.9181 | 2.078 | 5707.7 | |
| 12 | varConstPower | 0.9174 | 2.086 | 5642.0 | |
where, ID = model variant identifier; R2 = coefficient of determination; RMSE = root-mean-square error; and AIC = Akaike Information Criterion.
| No | ID | Effect parameter | Correction structure | BIC | logLik | Test | L. ratio | p-value |
|---|---|---|---|---|---|---|---|---|
| Pinus montezumae | ||||||||
| 1 | 3.9.0 | a 1 | Without structure | 3602.1 | −1784.1 | |||
| 2 | a 2 | 3596.0 | −1781.1 | |||||
| 3 | a 1 y a 2 | 3608.0 | −1780.3 | 2 vs 3 | 1.55 | 0.4596 | ||
| 4 | 3.9.1 | a 1 | varPower | 3532.8 | −1746.1 | 2 vs 4 | 69.96 | <0.0001 |
| 5 | a 2 | 3515.7 | −1737.5 | |||||
| 6 | 3.9.2 | a 1 | varExp | 3545.6 | −1752.5 | |||
| 7 | a 2 | 3532.6 | −1745.9 | |||||
| 8 | a 1 y a 2 | 3522.4 | −1734.0 | 5 vs 8 | 6.88 | 0.032 | ||
| 9 | 3.9.3 | a 1 | varConstPower | 3539.6 | −1746.1 | 8 vs 9 | 24.06 | <0.0001 |
| 10 | a 2 | 3522.5 | −1737.5 | 8 vs 10 | 6.89 | 0.0087 | ||
| 11 | a 1 y a 2 | 3525.7 | −1735.8 | |||||
| Pinus pseuodstrobus | ||||||||
| 1 | 3.9.0 | a 1 | Without structure | 5910.1 | −2937.1 | |||
| 2 | a 2 | 5926.9 | −2945.6 | |||||
| 3 | a 1 y a 2 | 5918.2 | −2934.0 | 1 vs 3 | 6.206 | 0.0449 | ||
| 4 | 3.9.1 | a 1 | varPower | 5684.0 | −2820.6 | 3 vs 4 | 227 | <0.0001 |
| 5 | a 2 | 5712.4 | −2834.7 | |||||
| 6 | a 1 y a 2 | 5681.2 | −2812.0 | 4 vs 6 | 17.15 | <0.0001 | ||
| 7 | 3.9.2 | a 1 | varExp | 5751.1 | −2854.1 | 6 vs 7 | 84.21 | <0.0001 |
| 8 | a 2 | 5774.0 | −2865.5 | 6 vs 8 | 107.1 | <0.0001 | ||
| 9 | a 1 y a 2 | 5748.9 | −2845.8 | |||||
| 10 | 3.9.3 | a 1 | varConstPower | 5691.2 | −2820.6 | 9 vs 10 | 50.58 | <0.0001 |
| 11 | a 2 | 5719.5 | −2834.7 | 9 vs 11 | 22.26 | <0.0001 | ||
| 12 | a 1 y a 2 | 5688.4 | −2812.0 | 9 vs 12 | 67.73 | <0.0001 | ||
where ID = model variant identifier; BIC = Bayesian Information Criterion; logLik: log likelihood.
| Species | Parameters | Lower value | Value | Upper value | Standard Error | t-value | p-value |
|---|---|---|---|---|---|---|---|
| Pinus montezumae | a 0 | 0.5023 | 0.6559 | 0.8094 | 0.078 | 8.369 | <0.0001 |
| a 1 | 1.1280 | 1.2115 | 1.2951 | 0.043 | 28.414 | <0.0001 | |
| a 2 | 2.7927 | 3.5400 | 4.2873 | 0.381 | 9.282 | <0.0001 | |
| Pinus pseudostrobus | a 0 | 1.0223 | 1.2180 | 1.4137 | 0.100 | 12.194 | <0.0001 |
| a 1 | 0.9257 | 0.9719 | 1.0182 | 0.024 | 41.183 | <0.0001 | |
| a 2 | 3.4552 | 4.1490 | 4.8428 | 0.354 | 11.720 | <0.0001 |
The t test between the validation subsample data (i.e. observed) and the estimates indicated equality between populations for both species (Pinus montezumae: t = −0.8354 and p = 0.4038; Pinus pseudostrobus: t = −0.0792 and p = 0.9369). Therefore, Ha: ( μ 1 = μ 2 ) is accepted. The graphical representation confirms these results (
The evaluation of predictions with the calibration alternatives shows that the minor global deviations for Pinus montezumae are obtained by including the heights of two trees, one with the largest and another with the smallest d in the plot, whereas for Pinus pseudostrobus the heights of three trees with the largest d in the plot should be used to calibrate the value of the random parameter (
| Parameters | Pinus montezumae | Pinus pseudostrobus | ||||
|---|---|---|---|---|---|---|
| a 0 | a 1 | a 2 | a 0 | a 1 | a 2 | |
| a 0 | 0.00612 | −0.00305 | 0.02068 | 0.00995 | −0.00220 | 0.02662 |
| a 1 | −0.00305 | 0.00181 | −0.00600 | −0.00220 | 0.00056 | −0.00410 |
| a 2 | 0.02068 | −0.00600 | 0.14495 | 0.02662 | −0.00410 | 0.12504 |
| Variance function | 0.06510 | 0.08232 | 0.09953 | 0.04200 | 0.04886 | 0.05571 |
| Within-group standard error | 0.63651 | 0.75493 | 0.89537 | 0.86519 | 0.96558 | 1.07763 |
| Residual of random effects | 0.75493 | 0.29982 | ||||
| Calibration | RMSE | Bias | E%AM | RMSE | Bias | E%AM |
|---|---|---|---|---|---|---|
| Pinus montezumae | Pinus pseudostrobus | |||||
| C1 | 2.0931 | 0.4126 | −5.1874 | 2.5179 | −0.1964 | −2.7876 |
| C2 | 2.0730 | 0.3390 | −4.4834 | 2.5457 | −0.0266 | −3.3126 |
| C3 | 2.0441 | 0.0641 | −1.9587 | 2.6539 | 0.1262 | −5.5435 |
| C4 | 2.0714 | 0.2988 | −4.2469 | 2.4907 | −0.1092 | −3.1220 |
where, C1: Measure the heights of the two thickest trees in normal diameter (d) in the plot. C2: Measure the heights of the two thinnest trees in d in the plot. C3: Measure the height of the thickest and thinnest tree in d in the plot. C4: Measure the heights of the three thickest trees in d in the plot. RMSE: root mean square error. E%AM: mean absolute percentage error.
Bronisz & Mehtätalo (2020) and Liu et al. (2017) pointed out that two-parameter models have some accuracy disadvantages with regard to three-parameters models, hence, in order to estimate tree height based on the diameter at breast height, choosing the latter is recommended. The Wang and Tang model proposed here differs from structures selected by other authors. For example, Misir (2010) proposed the Schnute model which includes dominant diameter (Dd) as predictor for Populus tremula L., whereas Bronisz & Mehtätalo (2020) projected a Schumacher-type expression with Msd and Ba as predictors for Eucalyptus globulus L.; in our case, the equations in which Msd was included (Equations 1, 6, 7, 8, and 10) showed non-significant parameters for both species. Conversely, our results concurred with the findings of Liu et al. (2017) for Metasequoia glyptostroboides Hu & Cheng; these authors concluded that including tree density as explanatory variables did not contribute significantly to models improvement after fitting 16 generalized models.
The inclusion of an additive effect ( + a i ) in the Wang and Tang model fit with MEM is similar to the procedure followed by Sharma & Parton (2007) , Vargas-Larreta et al. (2009) , Corral et al. (2014) , and Corral et al. (2019) . These authors also recorded a statistical gain in the model’s fit with respect to NLS in terms of the R2, RMSE and AIC values. Our results are also similar to those reported by Crecente-Campo et al. (2010) who used the Bertalanffy-Richards model to estimate H as a function of d and dH, obtaining in average 6.4% and 18.0% gains for the R2 and RMSE values, respectively. Therefore, the use of MEM with the inclusion of random effects makes it possible to model the random variability that results from different growth conditions of each site; in addition, more efficient and precise predictors are obtained, which is reflected in the goodness-of-fit statistics (Crecente-Campo et al., 2010; Ercanli, 2015) .
The R2 and RMSE values were higher than those reported by Santiago-García et al. (2020) who used a power function to weight the variance for Pinus patula Schiede ex Schltdl. et Cham. (R2 = 0.75, RMSE = 4.3), Pinus oaxacana Mirov (R2 = 0.82, RMSE = 4.3), Pinus ayacahuite Ehren. (R2 = 0.83, RMSE = 3.5), Pinus teocote Schlecht. & Cham. (R2 = 0.80, RMSE = 3.6), and Pinus leiophylla Schiede & Deppe (R2 = 0.83, RMSE = 3.4). They used the Sharma and Parton expression for the first three species and the Wang and Tang and Nilson for the rest. These differences might be due to the homogeneity of the plantations with respect to natural forests conditions; unfortunately, to our best knowledge, no work has been developed on forest plantations in Mexico in this subject.
In addition, our results also agree with Álvarez-González et al. (2007) findings, who concluded that including a structure to model the variance within the statistical fitting helps to correct heteroscedasticity problems in residuals distribution. Similarly, our findings are consistent with Ercanli (2015) and Corral et al. (2019) results, who reported that the inclusion of random effects in two parameters within the Schnute and Chapman-Richards generalized model improves the fit and precision indicators.
Our results of the mixed effects models obtained for both species (Equation 9.2 + Equation 15: varExp) showed an efficient correction of heteroscedasticity for the Wang and Tang generalized model, in comparison with both NLS and MEM fit without a variance structure (varExp), and this helped to reduce the deviations in heights (H) estimation (better RMSE) and to maximize the explanation of the sample variability (R2) (Bronisz & Mehtätalo, 2020; Santiago-García et al., 2020) . Furthermore, a lower estimation error was achieved with the same field sampling effort by including the dominant height as inherent to the site (Corral et al., 2019) . Likewise, the biases calculated for both species are similar to those reported in similar works; for example, Corral et al. (2014, 2019) reported bias values of −0.013 m and −0.009 m, in seven pine species in Durango, Mexico for a generalized model that considered d, dH, density, and Msd as predictors of H.
The results also showed that by taking into consideration variables inherent to the population (mean square diameter: Dq, mH, A0, Ba, or age) to explain the growth trend of H in different stands is an alternative to improve the quality of the estimates, and increases the applicability of the expressions proposed on the H-d local equations (Misir, 2010; Liu et al., 2017) . The only differences were found in the explanatory variable(s) of the forest stand or FP, given the particular ecological conditions and requirements of each species studied.
Moreover, including dH as an explanatory variable helps to reduce bias in the estimates, and this is one of the variables with the greatest contribution to explain the sample variability and the dynamics of H in the stand or FP. The dH is directly related to site quality, it is sensitive to the environmental conditions in which the species grows, it remains constant after thinning (as long as it is not on the upper part), and it has shown better results than mH; besides, its registration does not entail a greater effort in the field (Crecente-Campo et al., 2010; Ferraz et al., 2018; Santiago-García et al., 2020) .
The model proposed can be a reliable quantitative tool for the development of forest management plans, owing to its statistical robustness and precision. The model is included in the growth and yield modeling, when it is combined with a site index equation or when it is used as a support in forest inventories (Ercanli, 2015; Santiago-García et al., 2020) . Furthermore, a dominant height by site index tag structure can be used within the structure proposed for the Wang and Tang model, which could help to expand its applicability and allow application of the results outside the research area or to other plantations of these two species, due to the direct relationship among site quality, tree age and dominant height within the site or plantation (Spurr & Barnes, 1982; Prodan et al., 1997; Torres & Magaña, 2001) .
The calibration of a MEM with a subsample of tree that does not generate additional sampling costs, allows the estimation of height of all the trees within a sampling plot, based on the d of an independent subsample of trees, and gives insight into it application in different areas. In addition, this is an alternative approach that allows for saving time and costs in the planning and execution of forest management programs (Corral et al., 2019) . Even though there are several calibration criteria for this type of fit (Calama & Montero, 2004; Crecente-Campo et al., 2010; Corral-Rivas et al., 2014) , several authors agree that the use of few trees (<4 individuals) yields specific results for each site or plot condition, reduces the deviations in the estimates and improves the predictive capacity of the equation proposed.
We obtained a significant statistical improvement with the use of mixed effects models (MEM) when fitting a generalized height-diameter (H-d) model, over nonlinear least squares (NLS). In addition, when a function to model the variance structure is included in the MEM, the residuals decreased and this helped to correct undesirable residuals trend.
The goodness of fit statistics, bias, means comparison test, and distribution of the estimates suggested that the Wang and Tang model that uses d and dominant height can accurately estimate H for Pinus montezumae and Pinus pseudostrobus forest plantations in Michoacán, Mexico.
The calibration or localization process of the MEM also allows the application of the expression proposed in plantations of these two species that were not included in the analysis, reduces the estimation error and improves the estimates of the H-d relationship. Finally, this does not represent an additional sampling effort or time invested in the inventory.
The first auhor would like to sincerely thank CONAHCYT for financially supporting this work. The authors gratefully acknowledge the COLPOS and INIFAP for their kind cooperation, and guidance during the study. We would also like to thank the anonymous reviewers for their edifying comments which have improved the quality of this manuscript.
The authors declare no conflicts of interest regarding the publication of this paper.
Hernández-Ramos, J., Reyes-Hernández, V. J., De los Santos-Posadas, H. M., Fierros-González, A. M., Buendía-Rodríguez, E., & Quiñonez-Barraza, G. (2024). Generalized Height-Diameter Models for Pinus montezumae Lamb. and Pinus pseudostrobus Lindl. Plantations in Michoacan, Mexico. Open Journal of Fores- try, 14, 214-232. https://doi.org/10.4236/ojf.2024.143014