A New Method to Compare Statistical Tree Growth Curves: The PL-GMANOVA Model and Its Application with Dendrochronological Data
Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius (r) over time (t) of...
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| Published in | PloS one Vol. 9; no. 11; p. e112396 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
United States
Public Library of Science
17.11.2014
Public Library of Science (PLoS) |
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| Abstract | Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius (r) over time (t) of trees on different sites, GMANOVA is combined here with the adapted PL regression model Q = A · T+E, where for b ≠ 0 : Q = Ei[-b · r]-Ei[-b · r1] and for b = 0 : Q = Ln[r/r1], A = initial relative growth to be estimated, T = t-t1, and E is an error term for each tree and time point. Furthermore, Ei[-b · r] = ∫(Exp[-b · r]/r)dr, b = -1/TPR, with TPR being the turning point radius in a sigmoid curve, and r1 at t1 is an estimated calibrating time-radius point. Advantages of the approach are that growth rates can be compared among growth curves with different turning point radiuses and different starting points, hidden outliers are easily detectable, the method is statistically robust, and heteroscedasticity of the residuals among time points is allowed. The model was implemented with dendrochronological data of 235 Pinus montezumae trees on ten Mexican volcano sites to calculate comparison intervals for the estimated initial relative growth A. One site (at the Popocatépetl volcano) stood out, with A being 3.9 times the value of the site with the slowest-growing trees. Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time. Without the Popocatépetl site, the numbers changed to 7%, 42%, and 51%. Further explanation of differences in growth would need to focus on factors that vary within sites and over time. |
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| AbstractList | Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius (r) over time (t) of trees on different sites, GMANOVA is combined here with the adapted PL regression model Q=A.T+E, where for b not equal 0 : Q=Ei[-b.r] - Ei[-b.r] and for b=0 : Q=Ln[r/r(1)], A = initial relative growth to be estimated, T=t-t(1), and E is an error term for each tree and time point. Furthermore, Ei[-b.r] = integral (Exp[b.r]/r)dr, b = -1/TPR, with TPR being the turning point radius in a sigmoid curve, and r(1) at t(1) is an estimated calibrating time-radius point. Advantages of the approach are that growth rates can be compared among growth curves with different turning point radiuses and different starting points, hidden outliers are easily detectable, the method is statistically robust, and heteroscedasticity of the residuals among time points is allowed. The model was implemented with dendrochronological data of 235 Pinus montezumae trees on ten Mexican volcano sites to calculate comparison intervals for the estimated initial relative growth (A) over cap. One site (at the Popocatepetl volcano) stood out, with (A) over cap being 3.9 times the value of the site with the slowest-growing trees. Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time. Without the Popocatepetl site, the numbers changed to 7%, 42%, and 51%. Further explanation of differences in growth would need to focus on factors that vary within sites and over time. Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius ( r ) over time ( t ) of trees on different sites, GMANOVA is combined here with the adapted PL regression model Q = A · T + E , where for and for , A = initial relative growth to be estimated, , and E is an error term for each tree and time point. Furthermore, Ei [– b · r ] = , , with TPR being the turning point radius in a sigmoid curve, and at is an estimated calibrating time-radius point. Advantages of the approach are that growth rates can be compared among growth curves with different turning point radiuses and different starting points, hidden outliers are easily detectable, the method is statistically robust, and heteroscedasticity of the residuals among time points is allowed. The model was implemented with dendrochronological data of 235 Pinus montezumae trees on ten Mexican volcano sites to calculate comparison intervals for the estimated initial relative growth . One site (at the Popocatépetl volcano) stood out, with being 3.9 times the value of the site with the slowest-growing trees. Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time. Without the Popocatépetl site, the numbers changed to 7%, 42%, and 51%. Further explanation of differences in growth would need to focus on factors that vary within sites and over time. Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius (r) over time (t) of trees on different sites, GMANOVA is combined here with the adapted PL regression model Q = A · T+E, where for b ≠ 0 : Q = Ei[-b · r]-Ei[-b · r1] and for b = 0 : Q = Ln[r/r1], A = initial relative growth to be estimated, T = t-t1, and E is an error term for each tree and time point. Furthermore, Ei[-b · r] = ∫(Exp[-b · r]/r)dr, b = -1/TPR, with TPR being the turning point radius in a sigmoid curve, and r1 at t1 is an estimated calibrating time-radius point. Advantages of the approach are that growth rates can be compared among growth curves with different turning point radiuses and different starting points, hidden outliers are easily detectable, the method is statistically robust, and heteroscedasticity of the residuals among time points is allowed. The model was implemented with dendrochronological data of 235 Pinus montezumae trees on ten Mexican volcano sites to calculate comparison intervals for the estimated initial relative growth A. One site (at the Popocatépetl volcano) stood out, with A being 3.9 times the value of the site with the slowest-growing trees. Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time. Without the Popocatépetl site, the numbers changed to 7%, 42%, and 51%. Further explanation of differences in growth would need to focus on factors that vary within sites and over time.Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius (r) over time (t) of trees on different sites, GMANOVA is combined here with the adapted PL regression model Q = A · T+E, where for b ≠ 0 : Q = Ei[-b · r]-Ei[-b · r1] and for b = 0 : Q = Ln[r/r1], A = initial relative growth to be estimated, T = t-t1, and E is an error term for each tree and time point. Furthermore, Ei[-b · r] = ∫(Exp[-b · r]/r)dr, b = -1/TPR, with TPR being the turning point radius in a sigmoid curve, and r1 at t1 is an estimated calibrating time-radius point. Advantages of the approach are that growth rates can be compared among growth curves with different turning point radiuses and different starting points, hidden outliers are easily detectable, the method is statistically robust, and heteroscedasticity of the residuals among time points is allowed. The model was implemented with dendrochronological data of 235 Pinus montezumae trees on ten Mexican volcano sites to calculate comparison intervals for the estimated initial relative growth A. One site (at the Popocatépetl volcano) stood out, with A being 3.9 times the value of the site with the slowest-growing trees. Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time. Without the Popocatépetl site, the numbers changed to 7%, 42%, and 51%. Further explanation of differences in growth would need to focus on factors that vary within sites and over time. Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius ( r ) over time ( t ) of trees on different sites, GMANOVA is combined here with the adapted PL regression model Q = A · T + E , where for and for , A = initial relative growth to be estimated, , and E is an error term for each tree and time point. Furthermore, Ei [– b · r ] = , , with TPR being the turning point radius in a sigmoid curve, and at is an estimated calibrating time-radius point. Advantages of the approach are that growth rates can be compared among growth curves with different turning point radiuses and different starting points, hidden outliers are easily detectable, the method is statistically robust, and heteroscedasticity of the residuals among time points is allowed. The model was implemented with dendrochronological data of 235 Pinus montezumae trees on ten Mexican volcano sites to calculate comparison intervals for the estimated initial relative growth . One site (at the Popocatépetl volcano) stood out, with being 3.9 times the value of the site with the slowest-growing trees. Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time. Without the Popocatépetl site, the numbers changed to 7%, 42%, and 51%. Further explanation of differences in growth would need to focus on factors that vary within sites and over time. Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius (r) over time (t) of trees on different sites, GMANOVA is combined here with the adapted PL regression model Q = A·T+E, where for and for, A = initial relative growth to be estimated,, and E is an error term for each tree and time point. Furthermore, Ei[-b·r] =,, with TPR being the turning point radius in a sigmoid curve, and at is an estimated calibrating time-radius point. Advantages of the approach are that growth rates can be compared among growth curves with different turning point radiuses and different starting points, hidden outliers are easily detectable, the method is statistically robust, and heteroscedasticity of the residuals among time points is allowed. The model was implemented with dendrochronological data of 235 Pinus montezumae trees on ten Mexican volcano sites to calculate comparison intervals for the estimated initial relative growth . One site (at the Popocatépetl volcano) stood out, with being 3.9 times the value of the site with the slowest-growing trees. Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time. Without the Popocatépetl site, the numbers changed to 7%, 42%, and 51%. Further explanation of differences in growth would need to focus on factors that vary within sites and over time. Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius (r) over time (t) of trees on different sites, GMANOVA is combined here with the adapted PL regression model Q = A · T+E, where for b ≠ 0 : Q = Ei[-b · r]-Ei[-b · r1] and for b = 0 : Q = Ln[r/r1], A = initial relative growth to be estimated, T = t-t1, and E is an error term for each tree and time point. Furthermore, Ei[-b · r] = ∫(Exp[-b · r]/r)dr, b = -1/TPR, with TPR being the turning point radius in a sigmoid curve, and r1 at t1 is an estimated calibrating time-radius point. Advantages of the approach are that growth rates can be compared among growth curves with different turning point radiuses and different starting points, hidden outliers are easily detectable, the method is statistically robust, and heteroscedasticity of the residuals among time points is allowed. The model was implemented with dendrochronological data of 235 Pinus montezumae trees on ten Mexican volcano sites to calculate comparison intervals for the estimated initial relative growth A. One site (at the Popocatépetl volcano) stood out, with A being 3.9 times the value of the site with the slowest-growing trees. Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time. Without the Popocatépetl site, the numbers changed to 7%, 42%, and 51%. Further explanation of differences in growth would need to focus on factors that vary within sites and over time. Growth curves are monotonically increasing functions that measure repeatedly the same subjects over time. The classical growth curve model in the statistical literature is the Generalized Multivariate Analysis of Variance (GMANOVA) model. In order to model the tree trunk radius (r) over time (t) of trees on different sites, GMANOVA is combined here with the adapted PL regression model Q= A?T+E, where for b=0 : Q~Ei½{b: r{Ei½{b: r1 and for b~0 : Q~Ln½r=r1, A = initial relative growth to be estimated, T~t{t1, and E is an error term for each tree and time point. Furthermore, Ei[–b?r] = Ð (Exp½{b: r=r)dr, b~{1=TPR, with TPR being the turning point radius in a sigmoid curve, and r1 at t1 is an estimated calibrating time-radius point. Advantages of the approach are that growth rates can be compared among growth curves with different turning point radiuses and different starting points, hidden outliers are easily detectable, the method is statistically robust, and heteroscedasticity of the residuals among time points is allowed. The model was implemented with dendrochronological data of 235 Pinus montezumae trees on ten Mexican volcano sites to calculate comparison intervals for the estimated initial relative growth A^. One site (at the Popocate´petl volcano) stood out, with A^ being 3.9 times the value of the site with the slowest-growing trees. Calculating variance components for the initial relative growth, 34% of the growth variation was found among sites, 31% among trees, and 35% over time. Without the Popocate´petl site, the numbers changed to 7%, 42%, and 51%. Further explanation of differences in growth would need to focus on factors that vary within sites and over time. |
| Audience | Academic |
| Author | von Rosen, Dietrich Ricker, Martin Peña Ramírez, Víctor M. |
| AuthorAffiliation | University of Cambridge, United Kingdom 3 Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden 2 Posgrado en Ciencias Biológicas, Universidad Nacional Autónoma de México (UNAM), México D.F., Mexico 1 Departamento de Botánica, Instituto de Biología, Universidad Nacional Autónoma de México (UNAM), México D.F., Mexico |
| AuthorAffiliation_xml | – name: University of Cambridge, United Kingdom – name: 3 Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden – name: 2 Posgrado en Ciencias Biológicas, Universidad Nacional Autónoma de México (UNAM), México D.F., Mexico – name: 1 Departamento de Botánica, Instituto de Biología, Universidad Nacional Autónoma de México (UNAM), México D.F., Mexico |
| Author_xml | – sequence: 1 givenname: Martin surname: Ricker fullname: Ricker, Martin – sequence: 2 givenname: Víctor M. surname: Peña Ramírez fullname: Peña Ramírez, Víctor M. – sequence: 3 givenname: Dietrich surname: von Rosen fullname: von Rosen, Dietrich |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/25402427$$D View this record in MEDLINE/PubMed https://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-152263$$DView record from Swedish Publication Index https://res.slu.se/id/publ/66179$$DView record from Swedish Publication Index |
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| CitedBy_id | crossref_primary_10_1016_j_jenvman_2019_02_084 crossref_primary_10_12729_jbtr_2017_18_2_038 crossref_primary_10_1002_ece3_2508 crossref_primary_10_3390_f10050428 crossref_primary_10_1111_rec_13977 |
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| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23 Competing Interests: The authors have declared that no competing interests exist. Conceived and designed the experiments: MR DVR. Analyzed the data: MR VMPR. Wrote the paper: MR DVR. Acquired the dendrochronological data: VMPR. |
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| SubjectTerms | Algorithms Biology and Life Sciences Data processing Dendrochronology Econometrics Growth curves Growth models Growth rate Mathematical analysis Mathematical models Methods Models, Statistical Multivariate analysis Normal distribution Outliers (statistics) Physical Sciences Pine trees Plant growth Probability Theory and Statistics Regression analysis Regression models Sannolikhetsteori och statistik Skewness Statistical analysis Tree growth Trees Trees - growth & development Trävetenskap Variance analysis Volcanoes Wood Science |
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