Multilevel Nonlinear Mixed-effects Basal Area Models for Individual Trees of Quercus mongolica

FU Li-yong; TANG Shou-zheng; ZHANG Hui-ru; LEI Xiang-dong

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Multilevel Nonlinear Mixed-effects Basal Area Models for Individual Trees of Quercus mongolica

1.
Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091, China

Received Date: 2014-01-28

Abstract

Taking 10 111 Q. mongolica individuals obtained from 184 plots in Wangqing Forestry Bureau in Jilin Province as test materials, this study is to develop individual basal area increment model for Quercus mongolica Fisch by mixed effects model approach. The relationship between four dependent variables (the later diameter at breast height, the later basal area at breast height, the diameter increment and the basal area increment) and earlier stage diameter at breast height were analyzed using seven functions commonly used, i.e. linear function, Richards function, logistic function, exponential function etc. The best model was selected as the base model to develop mixed effects model. And then, the best combination form of formal parameters in the base model was determined with considering both the random effects of forest farms and the plot simultaneously. Forest variables contained in the formal parameters of the model were determined by stepwise regression method. Three kinds of residual variance functions (exponential function, power function and constant plus power function) that used to eliminate heteroskedasticity were analyzed and compared and the prediction efficiency of model was tested. The results are as follows. The Wykoff model that the dependent variable was later basal area at breast height had a better fit effect and used as the base model. In addition to the early diameter at breast height (D), the model included the stand variables, such as the tangent of slope (ST), the ratio of D of target tree to arithmetical mean diameter of plot (RAD), the total basal area of plots (TBA), the sum of basal area of trees with diameter larger than target tree (GSBA), the ratio of basal area of target tree to arithmetical mean basal area of plot (RABA) and the ratio of basal area of target to total basal area of plot (RBA), had a better prediction accuracy. For residual variance, the exponential function, power function and constant plus power function could eliminate the heteroskedasticity, but the power function was the best. The mixed effects model taking forest farm and plot effects into account has the highest prediction accuracy.
- Quercus mongolica stand,
- mixed effects model,
- individual basal area increment model

References

[1]	Cole D M, Stage A R. Estimating future diameter of lodgepole pine[M]. U S For Ser Res Pap,INT-131. 1972.
[2]	Dolph K L. Prediction of periodic basal area increment for young-growth mixed conifers in the Sierra Nevada U S[M]. For Ser Res Pap, 1988, RP-190.
[3]	Wykoff W R. A basal area increment model for individual conifers in the northern Rocky Mountains[J]. For Sci, 1990, 36(4): 1077-1104.
[4]	Calama R, Montero G. Interregional nonlinear height-diameter model with random coefficients for stone pine in Spane[J]. Can J For Res, 2004, 34: 150-163.
[5]	Calama R, Montero G. Multilevel linear mixed model for tree diameter increment in stone pine (Pinus pinea): a calibrating approach[J]. Sliva Fennica, 2005, 39(1): 37-54.
[6]	Uzoh F C C, Oliver W W. Individual tree diameter increment model for managed even-aged stands of ponderosa pine throughout the western United States using a multilevel linear mixed effects model . For Ecol Manage, 2008, 256: 438-445.
[7]	Gregoire T G, Schabenberger O, Barrett J P. Linear modeling of irregularly spaced, unbalanced, longitudinal data from permanent plot measurements[J]. Canadian Journal of Forest Research, 1995, 25: 137-156.
[8]	Biging G S. Improved estimates of site index curves using a varying-parameter model[J]. For Sci, 1985, 31(1): 248-259.
[9]	Keselman H J, Algina J, Kowalchuk R K, et al. A comparison of recent approaches to the analysis of repeated measurements[J]. Br J Math Stat Psychol, 1999, 52: 63-78.
[10]	Garrett M F, Laird N M, Ware J H. Applied Longitudinal Analysis[M]. Wiley-Interscience. John Wiley and Sons, Inc, Publication, New Jersey, 2004.
[11]	符利勇,李永慈,李春明,等.利用2种非线性混合效应模型(2水平)对杉木林胸径生长量的分析[J]. 林业科学, 2012, 48(5):36-43.
[12]	符利勇,唐守正.基于非线性混合模型的杉木优势木平均高[J].林业科学,2012, 48 (7):66-71.
[13]	陈新美,张会儒,武纪成,等.柞树林直径分布模拟研究[J]. 林业资源管理,2008(1):39-43.
[14]	洪玲霞,雷相东,李永慈.蒙古栎林全林整体生长模型及其应用[J]. 林业科学研究,2012, 25( 2): 201-206.
[15]	杜纪山,唐守正,王洪良.天然林区小班森林资源数据的更新模型[J].林业科学,2000, 36( 2):26-32.
[16]	亢新刚,崔相慧,王虹.冀北次生林3 个树种林分生长过程表的编制[J].北京林业大学学报,2001,23( 3):39-42.
[17]	王春霞,刘万成,刘瑰琦,等.大兴安岭林区蒙古栎生长过程研究[J].中国林副特产, 2005(5):12-14.
[18]	于洋,王海燕,雷相东,等.东北过伐林区蒙古栎天然林土壤有机碳研究[J].西北林学院学报,2011,26( 2):57-62.
[19]	程徐冰,韩士杰,张忠辉,等.蒙古栎不同冠层部位叶片养分动态 .应用生态学报,2011,22( 9):2272-2278.
[20]	沈琛琛,雷相东,王福有,等.金苍林场蒙古栎天然中龄林竞争关系研究[J].林业科学研究,2012,25( 3):339-345.
[21]	Pinherio J C, Bates D M. Mixed-Effects Models in S and S-PLUS[M]. Spring-Verlag, New York, NY, 2000.
[22]	Fang Z, Bailey R L. Nonlinear mixed effects modeling for slash pine dominant height growth following intensive silvicultural treatments[J]. Forest Science, 2001, 47: 287-300.
[23]	Yang Y, Huang S. Comparison of different methods for fitting nonlinear mixed forest models and for making predictions[J]. Canadian Journal of Forest Research, 2011, 41(8): 1671-1686.
[24]	Pinheiro J C, Bates D M. Approximations to the loglikelihood function in the nonlinear mixed effects model[J]. Journal of Computational and Graphical Statistics, 1995, 4: 12-35.
[25]	Pandhard X, Samson A. Extension of the SAEM algorithm for nonlinear mixed models with 2 levels of random effects[J]. Biostatistics, 2009, 10(1): 121-135.
[26]	Davidian M, Giltinan D M. Nonlinear Models for Repeated Measurement Data[M]. Chapman& Hall, New York, 1995.
[27]	Meng S X, Huang S. Improved calibration of nonlinear mixed-effects models demonstrated on a height growth function[J]. Forest Science, 2009, 55(3): 239-248.
[28]	Yang Y, Huang S, Meng S X, et al. A multilevel individual tree basal area increment model for aspen in boreal mixedwood stands[J]. Can J For Res, 2009, 39: 2203-2214.
[29]	Schrder J, Soalleiro R R, Alonso G V. An age-independent basal area increment model for maritime pine trees in northwestern Spain[J]. For Ecol Manage, 2002, 157(1-3): 55-64.
[30]	Monserud R A, Sterba H. A basal area increment model for individual trees growing in even-and uneven-aged forest stands in Austria[J]. For Ecol Manage, 1996, 80(1-3): 57-80.
[31]	唐守正,郎奎建,李海奎.统计和生物数学模型计算 [M].北京:科学出版社,2008.
[32]	Lindstrom M J, Bates D M. Nonlinear Mixed Effects Models for Repeated Measures Data[J]. Biometrics, 1990, 46: 673-687.
[33]	符利勇,孙华.基于混合效应模型的杉木单木冠幅预测模型[J].林业科学,2013, 49(8):65-74.

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通讯作者: 陈斌, bchen63@163.com

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沈阳化工大学材料科学与工程学院沈阳 110142

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Multilevel Nonlinear Mixed-effects Basal Area Models for Individual Trees of Quercus mongolica

1. Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091, China

Received Date: 2014-01-28

Abstract: Taking 10 111 Q. mongolica individuals obtained from 184 plots in Wangqing Forestry Bureau in Jilin Province as test materials, this study is to develop individual basal area increment model for Quercus mongolica Fisch by mixed effects model approach. The relationship between four dependent variables (the later diameter at breast height, the later basal area at breast height, the diameter increment and the basal area increment) and earlier stage diameter at breast height were analyzed using seven functions commonly used, i.e. linear function, Richards function, logistic function, exponential function etc. The best model was selected as the base model to develop mixed effects model. And then, the best combination form of formal parameters in the base model was determined with considering both the random effects of forest farms and the plot simultaneously. Forest variables contained in the formal parameters of the model were determined by stepwise regression method. Three kinds of residual variance functions (exponential function, power function and constant plus power function) that used to eliminate heteroskedasticity were analyzed and compared and the prediction efficiency of model was tested. The results are as follows. The Wykoff model that the dependent variable was later basal area at breast height had a better fit effect and used as the base model. In addition to the early diameter at breast height (D), the model included the stand variables, such as the tangent of slope (ST), the ratio of D of target tree to arithmetical mean diameter of plot (RAD), the total basal area of plots (TBA), the sum of basal area of trees with diameter larger than target tree (GSBA), the ratio of basal area of target tree to arithmetical mean basal area of plot (RABA) and the ratio of basal area of target to total basal area of plot (RBA), had a better prediction accuracy. For residual variance, the exponential function, power function and constant plus power function could eliminate the heteroskedasticity, but the power function was the best. The mixed effects model taking forest farm and plot effects into account has the highest prediction accuracy.

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Reference (33)

[1]	Cole D M, Stage A R. Estimating future diameter of lodgepole pine[M]. U S For Ser Res Pap,INT-131. 1972.
[2]	Dolph K L. Prediction of periodic basal area increment for young-growth mixed conifers in the Sierra Nevada U S[M]. For Ser Res Pap, 1988, RP-190.
[3]	Wykoff W R. A basal area increment model for individual conifers in the northern Rocky Mountains[J]. For Sci, 1990, 36(4): 1077-1104.
[4]	Calama R, Montero G. Interregional nonlinear height-diameter model with random coefficients for stone pine in Spane[J]. Can J For Res, 2004, 34: 150-163.
[5]	Calama R, Montero G. Multilevel linear mixed model for tree diameter increment in stone pine (Pinus pinea): a calibrating approach[J]. Sliva Fennica, 2005, 39(1): 37-54.
[6]	Uzoh F C C, Oliver W W. Individual tree diameter increment model for managed even-aged stands of ponderosa pine throughout the western United States using a multilevel linear mixed effects model . For Ecol Manage, 2008, 256: 438-445.
[7]	Gregoire T G, Schabenberger O, Barrett J P. Linear modeling of irregularly spaced, unbalanced, longitudinal data from permanent plot measurements[J]. Canadian Journal of Forest Research, 1995, 25: 137-156.
[8]	Biging G S. Improved estimates of site index curves using a varying-parameter model[J]. For Sci, 1985, 31(1): 248-259.
[9]	Keselman H J, Algina J, Kowalchuk R K, et al. A comparison of recent approaches to the analysis of repeated measurements[J]. Br J Math Stat Psychol, 1999, 52: 63-78.
[10]	Garrett M F, Laird N M, Ware J H. Applied Longitudinal Analysis[M]. Wiley-Interscience. John Wiley and Sons, Inc, Publication, New Jersey, 2004.
[11]	符利勇,李永慈,李春明,等.利用2种非线性混合效应模型(2水平)对杉木林胸径生长量的分析[J]. 林业科学, 2012, 48(5):36-43.
[12]	符利勇,唐守正.基于非线性混合模型的杉木优势木平均高[J].林业科学,2012, 48 (7):66-71.
[13]	陈新美,张会儒,武纪成,等.柞树林直径分布模拟研究[J]. 林业资源管理,2008(1):39-43.
[14]	洪玲霞,雷相东,李永慈.蒙古栎林全林整体生长模型及其应用[J]. 林业科学研究,2012, 25( 2): 201-206.
[15]	杜纪山,唐守正,王洪良.天然林区小班森林资源数据的更新模型[J].林业科学,2000, 36( 2):26-32.
[16]	亢新刚,崔相慧,王虹.冀北次生林3 个树种林分生长过程表的编制[J].北京林业大学学报,2001,23( 3):39-42.
[17]	王春霞,刘万成,刘瑰琦,等.大兴安岭林区蒙古栎生长过程研究[J].中国林副特产, 2005(5):12-14.
[18]	于洋,王海燕,雷相东,等.东北过伐林区蒙古栎天然林土壤有机碳研究[J].西北林学院学报,2011,26( 2):57-62.
[19]	程徐冰,韩士杰,张忠辉,等.蒙古栎不同冠层部位叶片养分动态 .应用生态学报,2011,22( 9):2272-2278.
[20]	沈琛琛,雷相东,王福有,等.金苍林场蒙古栎天然中龄林竞争关系研究[J].林业科学研究,2012,25( 3):339-345.
[21]	Pinherio J C, Bates D M. Mixed-Effects Models in S and S-PLUS[M]. Spring-Verlag, New York, NY, 2000.
[22]	Fang Z, Bailey R L. Nonlinear mixed effects modeling for slash pine dominant height growth following intensive silvicultural treatments[J]. Forest Science, 2001, 47: 287-300.
[23]	Yang Y, Huang S. Comparison of different methods for fitting nonlinear mixed forest models and for making predictions[J]. Canadian Journal of Forest Research, 2011, 41(8): 1671-1686.
[24]	Pinheiro J C, Bates D M. Approximations to the loglikelihood function in the nonlinear mixed effects model[J]. Journal of Computational and Graphical Statistics, 1995, 4: 12-35.
[25]	Pandhard X, Samson A. Extension of the SAEM algorithm for nonlinear mixed models with 2 levels of random effects[J]. Biostatistics, 2009, 10(1): 121-135.
[26]	Davidian M, Giltinan D M. Nonlinear Models for Repeated Measurement Data[M]. Chapman& Hall, New York, 1995.
[27]	Meng S X, Huang S. Improved calibration of nonlinear mixed-effects models demonstrated on a height growth function[J]. Forest Science, 2009, 55(3): 239-248.
[28]	Yang Y, Huang S, Meng S X, et al. A multilevel individual tree basal area increment model for aspen in boreal mixedwood stands[J]. Can J For Res, 2009, 39: 2203-2214.
[29]	Schrder J, Soalleiro R R, Alonso G V. An age-independent basal area increment model for maritime pine trees in northwestern Spain[J]. For Ecol Manage, 2002, 157(1-3): 55-64.
[30]	Monserud R A, Sterba H. A basal area increment model for individual trees growing in even-and uneven-aged forest stands in Austria[J]. For Ecol Manage, 1996, 80(1-3): 57-80.
[31]	唐守正,郎奎建,李海奎.统计和生物数学模型计算 [M].北京:科学出版社,2008.
[32]	Lindstrom M J, Bates D M. Nonlinear Mixed Effects Models for Repeated Measures Data[J]. Biometrics, 1990, 46: 673-687.
[33]	符利勇,孙华.基于混合效应模型的杉木单木冠幅预测模型[J].林业科学,2013, 49(8):65-74.

Multilevel Nonlinear Mixed-effects Basal Area Models for Individual Trees of Quercus mongolica

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通讯作者: 陈斌, bchen63@163.com

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Multilevel Nonlinear Mixed-effects Basal Area Models for Individual Trees of Quercus mongolica

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