Constructing mortality models for natural forest state III in four provinces of the central region, Vietnam

Silviculture JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 3 CONSTRUCTING MORTALITY MODELS FOR NATURAL FOREST STATE III IN FOUR PROVINCES OF THE CENTRAL REGION, VIETNAM Cao Thi Thu Hien Vietnam National University of Forestry In this research, the dead trees were defined as the standing trees that died between the two occasions at which measurements were taken. The data on 300 subplots from 12 permanent sample plots were collected. The response variable was the number of dead trees per subplot. The results suggest that we successfully developed the mortality model by using both generalized linear and generalized linear mixed models for count data to address the problem of overdispersion. Arithmetic mean diameter of the subplot, plot basal area, and provinces as a categorical variable were found to be the most significant explanatory variable. With the generalized linear model, we found that the Negative Binomial GLM was the most appropriate model for predicting the number of recruitment for three groups. From using the provinces as a grouping variable, we realized that the mean numbers of dead trees was different in the four different locations, namely the number of dead trees for group 1 and group 2 in Thua Thien Hue were the highest, while in Ha Tinh these were the lowest. With the generalized linear mixed model, the Negative Binomial GLMM solves overdispersion by treating a plot as a random effect. The GLMM with random intercept was selected as the equation for the direct prediction of dead trees across each of the two species groups, and for N. melliferum. The GLMM with a random slope was chosen for S. wightianum. Keywords: Generalized linear model, generalized linear mixed model, mortality model, Negative Binomial GLM, tropical rainforests. 1. INTRODUCTION Natural mortality of trees is a crucial process that determines forest dynamics (Rüger et al., 2011). When a tree dies, the reduced competition benefits the trees near the dead tree, positively affecting their growth (Yang et al, 2003); in addition, gaps created by dead canopy trees are later filled by new trees (Oliver and Larson, 1996). McCarthy (2001) notes that these gap dynamics are crucial determinants of the structure and composition of a forest stand. For these reasons, the mortality process should be considered in stand simulation models. However, modeling mortality is difficult due to the stochastic nature of mortality events; standing death may be caused by intrinsic senescence (Carey et al., 1994) or extrinsic factors such as disease, insects, fungi, and wind. In previous studies, several statistical methods have been utilized to develop mortality models, including the logistic regression model (Monserud and Sterba, 1999), the two-step approach (Eid and Tuhus, 2001; Álvarez González et al., 2004; Diéguez-Aranda et al., 2005), the three-step approach (Fridman and Stahl, 2001; Meng et al., 2003), and neural networks (Hasenauer et al., 2001). Moreover, as recruitment data, mortality data consist of many zeros. Fortin and Deblois (2007), for instance, demonstrated that fitting a traditional Poisson distribution to this type of data can underestimate the occurrence of zeros or overestimate the occurrence of larger counts. One way to solve this issue is utilizing a method similar to conditional functions. Fortin and Deblois (2007) predicted tree recruitment with zero-inflated models, and Zhang et al. (2012) applied negative binomial mixture models (zero-inflated negative binomial, and Hurdle negative binomial models) to predict tree recruitments of Chinese pine trees (Pinus tabulaeformis). However, in the present study, we did not use zero-inflated models because of two reasons: (1) A zero-inflated model assumes that the zero observations have to come from two different sources, namely “structural” and Silviculture 4 JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 “sampling” (Hu et al., 2011). The sampling zeros are assumed to occur by chance, while structural zeros are observed due to some specific structure in the data. (2) Zero-inflated models are recommended if the overdispersion parameter is larger than 15 or 20 (Zuur et al., 2009), what was not the case with the data of the 12 sample plots under study. For this study, due to the low number of plots compared to the large number of species the purpose of this research is not to tackle the mortality of each single species, but rather, to concentrate on predicting tree mortality of particular species groups: across all tree species, all locally (province) important tree species, and important species spread over provinces. Two approaches were used here, generalized linear models (Poisson, Quasi- Poisson and Negative Binomial models), and generalized linear mixed effects models (Negative Binomial mixed model), the latter to take random plot effects into account. 2. RESEARCH METHODOLOGY 2.1. Study area Measurements were taken in a tropical rainforest, in four different provinces of Central region of Vietnam: Ha Tinh province, Thua Thien Hue province, Binh Dinh province and Khanh Hoa province. There were three plots in each of the four provinces. 2.2. Data collection In this research, 12 permanent sample plots (PSPs) in four provinces were selected from the network of PSPs which was established by the Forest Inventory and Planning Institute (FIPI) of Vietnam. Data from 2005 inherited, and re-measurement of these plots was done by the author in 2012, 2013. Each plot has a square shape (100 m x 100 m2) and is divided into twenty five 20 m x 20 m quadrats. It was aligned according to a magnetic-north direction and has four major corner posts made of concrete. All trees equal to or larger than 6 cm diameter at breast height (DBH ≥ 6 cm) were identified by species and permanently marked using a white metal tag. In 2005: On each plot, all trees in each plot with a diameter at breast height from 6 cm (DBH ≥ 6 cm) were marked and, identified by species; their diameter was measured at 1.3 m from the ground. The data within the plot were assigned to their 20 m x 20 m quadrat. In 2012 and 2013: Measurements were repeated on all 12 plots and standing dead trees were also recorded. 2.3. Data analysis 2.3.1. Species group There are a large number of tree species in natural tropical rainforests. Several species appear more frequently, some occur with only low frequency. For that reason, species might be aggregated into some groups to reduce the number of mortality models and to avoid the need for adding data for species with insufficient number of observations. For our study, simply the importance value index (IVI) was used to determine a group of most important species. Important tree species having IVI ≥ 5% in pooled data from three plots in each province were utilized to construct mortality model. 2.3.2. Mortality model In our case, the data on 300 subplots from 12 plots were used to fit the mortality model. The response variable was the dependent variable was the number of dead trees, which are the standing trees that died between the two occasions at which measurements were taken. Explanatory variables were measured at the beginning of the period, including arithmetic mean diameter of the subplot (DBH), subplot and plot basal area (BALsubplot, BALstand), subplot density (the number of trees on each subplot) (N), and provinces as a categorical variable (provincecode). Silviculture JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 5 - Generalized linear model (GLM) The GLM consists of two components, the response variable and the link function. The link function defines how the mean of the dependent variable and the linear combination of the explanatory variables are connected (Faraway, 2006). In this study, a Poisson GLM (log link), Quasi-GLM and Negative Binomial were used, where the Poisson GLM (log link) was used to detect overdispersion. In the Poisson model, the variance equaled ϕ, with mean and dispersion parameter ϕ. The following formula (Zuur et al., 2009) was used in the calculation: ∅ = (1) Where: D is the residual deviance and n – p is degrees of freedom. n is the number of observations, and p is the number of regression parameters (intercept and slopes) in the model. If ϕ equals 1, there is no overdispersion and we have the Poisson GLM; if ϕ is larger than 1, this is evidence for the suggestion of overdispersion (Zuur et al., 2009), the Quasi- Poisson GLM and Negative Binomial models were used: log() = + + + + + + log () (2) = exp ( + + + + + + log()) (3) Where: x1i to x4i are independent variables of the i th subplot (DBH, BALsubplot, BALstand, N) and αk is the effect of province k (k = 1, 2, 3), αk = 0 for Ha Tinh; 0 to 4 and the αk are the parameters to be estimated; log(timei) is an offset factor. To estimate the regression parameters of the GLM, a maximum likelihood estimation was used (Zuur et al., 2013). The procedure for selecting Poisson, Quasi-Poisson, and Negative Binomial models followed Zuur et al. (2009). - Generalized linear mixed model (GLMM) Generalized linear mixed models are an extension of a GLM in which the linear predictor contains random effects in addition to the fixed effects. The random effects can account for the correlation between observations from the same plot in a province. In this study, a random plot effect was added to the intercept (equation 4), the slope (equation 5), or both intercept and slope (equation 6) of each model. The general equation is as follows: log = + + + 1. 1 + 2. 2 + 3. 3 + 4. 4 + log() (4) log = + + 1 + . 1 + (2 + ). 2 + (3 + ). 3 + (4 + ). 4 + log() (5) log = ( + ) + + 1 + . 1 + (2 + ). 2 + (3 + ). 3 + (4 + ). 4 + log() (6) To assess goodness of fit of the data in the GLM and GLMM models, Pearson’s χ2 was used. The parameter estimations for the GLMM in this chapter were fitted with glmmPQL in “MASS” package available from the open source statistical software R. All hypothesis testing was performed at the α = 0.05 significance level. 3. RESULTS 3.1. Results of the GLM A total of 1323 dead trees belonging to 189 species were counted at the four locations. The number of dead trees for all species and all important species counted per plot ranged from 21 to 221 and from 10 to 133 respectively. This was in correspondence with the number of species, which was respectively, from 17 to 58 and from 8 to 21 (Table 1). Silviculture 6 JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 Table 1. Descriptive statistics of the mortality data used for the model development Province Plot No. trees (cm) Stand basal area (m2/ha) All tree species All important tree species No. dead trees No. species No. dead trees No. species Ha Tinh 1 416 20.95 18.65 21 17 10 8 2 352 19.95 13.84 69 33 33 9 3 391 19.05 15.54 66 37 25 9 Total 156 66 68 14 Thua Thien Hue 4 932 17.35 33.04 154 44 109 21 5 855 18.20 33.50 77 41 45 19 6 1092 16.53 34.72 190 54 133 21 Total 421 73 287 21 Binh Dinh 7 1151 16.05 31.01 221 57 130 17 8 967 16.62 31.72 184 58 108 17 9 893 18.34 32.44 96 43 47 12 Total 501 86 285 17 Khanh Hoa 10 800 17.53 28.95 35 18 25 9 11 782 17.52 24.46 82 27 59 9 12 901 17.47 28.73 128 27 99 11 Toal 245 42 183 11 Total (4 prov.) 1323 189 823 49 The fitted Poisson GLM model for two species groups (all tree species and all important tree species) supported evidence for overdispersion through the ratio of deviance to degrees of freedom larger than 1 (2.79 and 2.38, respectively). Thus, we refitted the data to correct the standard errors using Quasi- Poisson and Negative Binomial GLMs. The estimated parameters, standard errors, and the p-values of Poisson, Quasi-Poisson, and Negative Binomial models are represented in table 2. The deviance across all species and all important species was the lowest when analyzed with a Negative Binomial GLM, leading us to conclude that the Negative Binomial model was preferrable over the Poisson and Quasi-Poisson models. The Negative Binomial GLM is given as follows: For all species: log() = + + + log()(7) and for all important species: log() = + + + + + log () (8) In the same way, the Negative Binomial GLM was the selected model for three important species (Syzygium wightianum, Diospyros sylvatica and Nephelium melliferum) spread over three or four provinces: S. wightianum: log() = + + + log () D. sylvatica: log() = + + log () (10) N. melliferum: log() = + log () (11) Estimated parameters and p-value values of the Poisson, Quasi-Poisson and Negative Binomial equations are shown in table 2. (9) Silviculture JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 7 Table 2. GLM (Poisson, Quasi-Poisson, Negative Binomial) results for standing dead trees (0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05) Objects Variables Paramete rs Poisson Quasi-Poisson Negative Binomial Parameter estimates Pr(>|z|) Parameter estimates Pr(>|z|) Parameter estimates Pr(>|z|) All tree species Intercept 0 0.9764 0.0152* 0.3103 0.614253 0.4251 0.4287 DBH 1 -0.0306 0.0019 ** - - - - BALstand 2 -0.0969 1.02e-05 *** -0.0999 0.010019 * -0.1082 0.0012 ** N 3 -0.0075 0.0046 ** - - - - Thua Thien Hue αk 2.8502 5.10e-12 *** 2.7961 0.000130 *** 2.9610 1.73e-06 *** Binh Dinh 2.8215 2.68e-14 *** 2.7672 2.39e-05 *** 2.9166 1.33e-07 *** Khanh Hoa 1.6269 3.19e-09 *** 1.6007 0.000901 *** 1.7165 2.99e-05 *** AIC 1684.3 NA 1457.9 Deviance 816.81 831.72 322.56 All important tree species Intercept 0 -0.1060 0.8359 -1.2507 0.0033 ** 0.1144 0.8782 DBH 1 -0.0433 0.0004 *** -0.0429 0.0280 * -0.0396 0.0287 * BALstand 2 -0.0718 0.0095 ** - - -0.0910 0.0245 * Thua Thien Hue αk 2.6382 3.24e-07 *** 1.3537 2.24e-09 *** 3.0239 5.53e-05 *** Binh Dinh 2.4632 1.05e-07 *** 1.3239 6.41e-09 *** 2.7783 3.51e-05 *** Khanh Hoa 1.7083 8.98e-07 *** 0.8861 0.0002 *** 1.9593 9.33e-05 *** AIC 1389.9 NA 1233.6 Deviance 700.32 706.97 319.22 S. wightian um Intercept 0 -2.3011 0.0512. -2.3011 0.0615. -2.2846 0.0566. BALstand 1 -0.1859 0.0059** -0.1859 0.0081** -0.1869 0.0063** Thua Thien Hue αk 5.4731 6.58e-05*** 5.4731 0.0002*** 5.4926 8.15e-05*** Binh Dinh 4.7208 0.0002*** 4.7208 0.0004*** 4.7376 0.0002*** Khanh Hoa 4.7277 8.96e-07*** 4.7277 3.48e-06*** 4.7409 1.21e-06*** AIC 375.95 NA 377.85 Deviance 216.95 216.95 210.84 D. sylvatica Intercept 0 -6.5974 6.72e-11*** -6.310 6.69e-08*** -6.3090 4.12e-10*** BALsubplot 1 0.4233 0.0384* - - - - Binh Dinh αk 2.8643 0.0054** 3.135 0.0071** 3.1370 0.0026** Khanh Hoa 2.9082 0.0047** 3.135 0.0071** 3.1280 0.0027** AIC 235.37 NA 231.26 Deviance 152.10 155.85 106.08 N. melliferu m Intercept 0 -4.1127 < 2e-16*** -4.1127 < 2e-16*** -4.1101 < 2e-16*** AIC 176.37 NA 176.36 Deviance 124.77 124.77 98.79 JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 Silviculture 8 JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 From the Negative Binomial model output, the coefficient of BALstand was always negative (all species, all important species, and S. wightianum) denoting that the number of dead trees declined as the stand basal area became larger, or 0 (i.e. nonsignificant) for the other two species (Table 2). Similarly, the number of dead trees of all important species decreased with an increasing DBH, indicating a higher number of dead trees among small, as opposed to larger trees. This number should rise in age- related senescence as the tree becomes older; however, the data in our model did not demonstate this relationship. The number of dead trees for the two groups and S. wightianum was found to be the highest in Thua Thien Hue in comparison with Ha Tinh, Binh Dinh, and Khanh Hoa. Similarly, D. sylvatica, which appeared in three locations, had a much larger number of dead trees in Binh Dinh and Khanh Hoa than in Ha Tinh. As mentioned above, for dealing with different time intervals of each plot in each province, adding an offset variable (as log(time)) for the model. To avoid plotting two lines in each location, we calculate the annual number of recruits per province. Thus, the annual number of dead trees can be estimated using the Negative Binomial GLM for all species, for all important species per province and for four important species occurring on all plots in three/four locations are: For all species (with k = 2.487) Ha Tinh: log(/) = 0.4251 − 0.1082 (12) Thua Thien Hue: log(/) = 0.4251 − 0.1082 + 2.9610 (13) Binh Dinh: log(/) = 0.4251 − 0.1082 + 2.9166 (14) Khanh Hoa: log(/) = 0.4251 − 0.1082 + 1.7165 (15) For all important species (with k = 1.993): Ha Tinh: log(/) = 0.1144 − 0.0396 − 0.0910 (16) Thua Thien Hue: log(/) = 0.1144 − 0.0396 − 0.0910 + 3.0239 (17) Binh Dinh: log(/) = 0.1144 − 0.0396 − 0.0910 + 2.7783 (18) Khanh Hoa:log(/) = 0.1144 − 0.0396 − 0.0910 + 1.9593 (19) For S. wightianum (k = 13.786): Ha Tinh: log(/) = −2.2846 − 0.1869 (20) Thua Thien Hue: log(/) = −2.2846 − 0.1869 + 5.4926 (21) Binh Dinh: log(/) = −2.2846 − 0.1869 + 4.7376 (22) Khanh Hoa: log(/) = −2.2846 − 0.1869 + 4.7409 (23) For D. sylvatica (k = 0.667): Ha Tinh: log(/) = −6.3090 (24) Binh Dinh: log(/) = −6.3090 + 3.1370 (25) Khanh Hoa: log(/) = −6.3090 + 3.1280 (26) and for N. melliferum (k = 0.772): log(/) = −4.1101 (27) 3.2. Results of the GLMM The significant negative effects of the DBH (for the dead trees of all important species) and BALstand (for the dead trees of two species groups) as predicted by the Negative Binomial GLM became insignificant under the Negative Binomial GLMM, leading to removal of those variables from the model. Thus, for all species and all important species, the respective fixed effects models (7) and (8), were compared to the following mixed effects models: Silviculture JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 9 log = + + + log () (28) log = + + ( + ) + log () (29) log = + ( + ) + log () (30) For S. wightianum, the fixed effects model (3.3) was compared to four mixed effects models: log = ( + ) + + + log () (31) log = + ( + ) + + log () (32) log = ( + ) + + ( + ) + log () (33) log = + + ( + ) + log () (34) While for D. sylvatica, the fixed effects model (10), along with candidate mixed effects models (28), (29), and (30) were assessed, and for N. melliferum, the fixed effects model (11) was compared to only one mixed effects model (35): log = ( + ) + log () (35) Pearson’s 2 values of fixed and mixed models for predicting the number of dead trees across all species, all important species, one important species (S. wightianum) occurring in four, and two others (D. sylvatica, N. melliferum) found in three locations are presented in table 3. Table 3. A comparison of Pearson’s 2 values between the fixed effects model and the mixed effects models. Selected models are bolded Objects n Equation Model specification Pearson’s 2 Negative Binomial GLMM All tree species 300 3.1 Fixed effects model (GLM) 330.35 3.22 FM + plot intercept 289.50 3.23 FM + plot intercept + plot slope (provincecode) 289.55 3.24 FM + plot slope (provincecode) 289.55 All important tree species 300 3.2 Fixed effects model (GLM) 324.64 3.22 FM + plot intercept 289.59 3.23 FM + plot intercept + plot slope (provincecode) 289.68 3.24 FM + plot slope (provincecode) 289.68 S. wightianum 300 3.3 Fixed effects model (GLM) 312.14 3.25 FM + plot intercept 300.00 3.26 FM + plot slope (BALstand) 300.00 3.27 FM + plot intercept + plot slope (provincecode) 298.13 3.28 FM + plot slope (provincecode) 298.13 D. sylvatica 225 3.4 Fixed effects model (GLM) 216.80 3.22 FM + plot intercept 225.00 3.23 FM + plot intercept + plot slope (provincecode) 224.99 3.24 FM + plot slope (provincecode) 224.99 N. melliferum 225 3.5 Fixed effects model (GLM) 235.47 3.29 FM + plot intercept 223.50 Silviculture 10 JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 Table 3 presents a comparison of selected models based on Pearson’s 2 values. Here, we see that the Pearson’s 2 statistic for the fixed model is significantly larger than that of mixed effects model, with the exception of one important species appearing in three places (D. sylvatica). The Negative Binomial GLMM performed better than the Negative Binomial GLM. In similar fashion to the recruitment model, the mixed models with random intercept/random slope (provincecode) was not different from the mixed models with random slope (provincecode) effects. Therefore, the mixed model with a random slope was chosen for S. wightianum for it was simpler, while the mixed model with random intercept was selected as the equation for the direct prediction of dead trees across each of the two species groups, and for N. melliferum, because it had the smallest Pearson’s 2 value (Table 3). For D. sylvatica, the fixed model using only the provincecode as a predictor (Equation 3.4) was used. The summary statistics for the parameter estimations, standard deviation errors, and the p-values for the Negative Binomial GLMM are reported in table 4. In general, for the two species groups, the number of dead trees in Thua Thien Hue and Binh Dinh was much greater than that in Ha Tinh, with the single exception of Khanh Hoa, where there was no significant difference when compared with Ha Tinh (p > 0.05). For S. wightianum, in Thua Thien Hue, Binh Dinh, and Khanh Hoa were significantly higher numbers of mortality in comparison with Ha Tinh. For N. melliferum, there was no difference in the number of both recruits and dead trees in three provinces Ha Tinh, Binh Dinh, and Khanh Hoa. The variance component of the random plot effects in Table 4 was rather small (from 7.22% to 25.77%); however, the random effect demonstrated evidence of unexplained variation at the plot level and provided a suitable adjustment for dispersion (the overdispersion parameter was more or less 1). 4. DISCUSSION The Negative Binomial regression for the mortality model expressed in this paper used variables BALstand and provinceocde for all species, and DBH, BALstand and provincecode for all important species as predictors in predicting mortality. The DBH had a negative sign, resulting in the high mortality of small diameter trees and suggesting that suppressed trees are more likely to be eliminated from stand level competition (Adame et al., 2010); the negative DBH coefficient also indicated that stand mortality is more likely in forest stands with many small trees as compared to those with larger trees (Juknys et al., 2006). This result was supported by Zhang et al. (2014) who likewise found that stand mortality was negatively associated with a stand arithmetic mean diameter among Chinese pines (Pinus tabulaeformis). The stand basal area was suggested as a measure of two-sided competition that can take into account both the vertical competition for light and the horizontal competition for rooting space, water, and nutrients (Yang et al., 2003). This indicator is a good measure of stand crowding because it accounts for both tree size and density. Trees in a stand with a larger basal area will experience more competition than those in another stand with a smaller stand basal area (Yang et al., 2003). The number of dead trees should grow along with the increase in the stand basal area as a result of competition pressure. In this study, however, the negative coefficient of the stand basal area demonstrated that with an increasing stand basal area, the number of dead trees decreased. This may imply that inter-specific competition does not cause tree mortality for these stands. Another study from Bravo et al. (2001) found that stand basal area was an insignificant predictor of Douglas-fir mortality across a range of stands. Plot level random effects on mortality models can address some of the unexplained variation in these processes due to unobserved plot level variables, which included topography, soil, microclimate, nutrients, and moisture (Ma et al., 2013). Silviculture JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 11 Table 4. Parameter estimates for the Negative Binomial GLMM across all species, all important species, and important species occurring in four or three provinces Objects Equation Fixed effects Variance components variation explained by the plot Overdispersion parameter Variables Parameters Parameter estimates Std- error Pr(>|z|) ran-in   ran-slop   res All species 5.56 Intercept 0 -1.3576 0.2586 0.0000 0.1661 - 0.8138 16.95 0.9814 Thua Thien Hue αk 1.0295 0.3600 0.0212 Binh Dinh 1.2125 0.3595 0.0097 Khanh Hoa 0.4374 0.3627 0.2623 All important species 5.56 Intercept 0 -2.1680 0.3048 0.0000 0.2200 - 0.7779 22.04 0.9817 Thua Thien Hue αk 1.4325 0.4192 0.0091 Binh Dinh 1.4350 0.4191 0.0090 Khanh Hoa 0.9419 0.4218 0.0560 S. wightianum 5.62 Intercept 0 -2.5911 1.2212 0.0347 - 0.3476 1.0013 25.77 1.0141 BALstand 1 -0.1670 0.0696 0.0475 Thua Thien Hue αk 5.1282 1.4038 0.0084 Binh Dinh 4.2865 1.3349 0.0152 Khanh Hoa 4.5158 0.9926 0.0026 N. melliferum 5.63 Intercept 0 -4.1223 0.2257 0.0000 0.0762 - 0.9793 7.22 1.0022 JOURNAL OF FOREST Y SCIENCE AND TECHNOLOGY NO. 5 - 2018 Silviculture 12 JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 5. 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Determinants of mortality across a tropical lowland rainforest community. Oikos 120, 1047-1056. 18. Vanclay, J.K. (1992). Modelling regeneration and recruitment in a tropical rain forest. Canadian Journal of Forest Research 22, 1235-1248. 19. Yang, Y., Titus, S.J., Huang, S., 2003. Modeling individual tree mortality for white spruce in Alberta. Ecological Modelling 163, 209-222. 20. Zhang, X., Lei, Y., Cai, D., Liu, X. (2014). Modeling stand mortality using Poisson mixture models with mixed-effects. iForest-Biogeosciences and Forestry. Silviculture JOURNAL OF FORESTRY SCIENCE AND TECHNOLOGY NO. 5 - 2018 13 XÂY DỰNG MÔ HÌNH CÂY CHẾT CHO RỪNG TỰ NHIÊN TRẠNG THÁI III Ở 4 TỈNH MIỀN TRUNG VIỆT NAM Cao Thị Thu Hiền Trường Đại học Lâm nghiệp SUMMARY Cây chết trong nghiên cứu này là những cây chết đứng giữa 2 chu kỳ đo. Số liệu từ 300 phân ô của 12 ô đo đếm (ODD) được dùng để xây dựng mô hình cây chết. Biến phụ thuộc là số cây chết/phân ô. Kết quả cho thấy có thể dùng mô hình tuyến tính tổng quát và mô hình tuyến tính tổng quát hỗn hợp để mô phỏng vì hai mô hình này giải quyết được vấn đề phân tán của số liệu. Với mô hình tuyến tính tổng quát (GLM), phương trình Negative Binomial GLM là phù hợp nhất để dựa đoán số cây chết cho 3 nhóm. Đường kính, tiết diện ngang lâm phần và biến phân nhóm “tỉnh” là các biến có ảnh hưởng rõ ràng nhất tới mô hình dự đoán cây chết. Từ biến phân nhóm cho thấy số cây chết ở các tỉnh là khác nhau, số cây chết của nhóm 1 và nhóm 2 nhiều nhất ở Thừa Thiên Huế và ít nhất là ở Hà Tĩnh. Với mô hình tuyến tính tổng quát hỗn hợp (GLMM), hàm Negative Binomial GLMM đã loại bỏ được những giá trị phân tán. GLMM với hiệu ứng ngẫu nhiên là ODD cho hệ số tự do được lựa chọn để dự đoán số cây chết cho nhóm 1, nhóm 2 và loài Trường vải. GLMM với hiệu ứng ngẫu nhiên là ODD cho hệ số hồi quy được lựa chọn để dự đoán số cây chết cho Trâm trắng. Từ khóa: Mô hình cây chết, Negative Binomial GLM, phương trình tuyến tính tổng quát, phương trình tuyến tính tổng quát hỗn hợp, rừng mưa nhiệt đới. Received : 14/5/2018 Revised : 10/9/2018 Accepted : 18/9/2018