## Services on Demand

## Journal

## Article

## Indicators

- Cited by SciELO
- Access statistics

## Related links

- Cited by Google
- Similars in SciELO
- Similars in Google
- uBio

## Share

## Biota Neotropica

##
*On-line version* ISSN 1676-0611

### Biota Neotrop. vol.8 no.4 Campinas Oct./Dec. 2008

#### http://dx.doi.org/10.1590/S1676-06032008000400012

**ARTICLES**

**Allometric models for estimating the phytomass of a secondary Atlantic Forest area of southeastern Brazil**

**Modelos alométricos para estimativa de fitomassa de Floresta Atlântica secundária no sudeste do Brasil**

**Déborah Moreira Burger ^{1}; Welington Braz Carvalho Delitti**

Departamento de Ecologia, Instituto de Biociências Universidade de São Paulo – USP Rua do Matão, Travessa 14, n° 321, Cidade Universitária Armando de Salles Oliveira, CEP 05508-090 São Paulo, SP, Brazil

**ABSTRACT**

The purpose of this study was to develop and validate equations to estimate the aboveground phytomass of a 30 years old plot of Atlantic Forest. In two plots of 100 m^{2}, a total of 82 trees were cut down at ground level. For each tree, height and diameter were measured. Leaves and woody material were separated in order to determine their fresh weights in field conditions. Samples of each fraction were oven dried at 80 °C to constant weight to determine their dry weight. Tree data were divided into two random samples. One sample was used for the development of the regression equations, and the other for validation. The models were developed using single linear regression analysis, where the dependent variable was the dry mass, and the independent variables were height (h), diameter (d) and d^{2}h. The validation was carried out using Pearson correlation coefficient, paired *t*-Student test and standard error of estimation. The best equations to estimate aboveground phytomass were: lnDW = –3.068+2.522lnd (r^{2 }= 0.91; s_{y/x} = 0.67) and lnDW = –3.676+0.951ln d^{2}h (r^{2} = 0.94; s_{y/x} = 0.56).

**Keywords: **biomass, allometry, carbon, secondary Atlantic Forest, forest biomass estimation.

**RESUMO**

O objetivo deste estudo foi desenvolver e validar modelos preditores para a fitomassa epigéa de uma área de Floresta Atlântica secundária. Em duas parcelas de 100m^{2}, 82 árvores foram cortadas, ao nível do solo, e anotadas suas medidas de altura e diâmetro. As folhas foram separadas dos ramos para determinação do massa fresca da porção foliar e lenhosa. Amostras de cada fração foram secas em estufa a 80 °C, até massa constante, para determinação do massa seca. As árvores foram distribuídas em duas amostras aleatórias, sendo uma utilizada para o desenvolvimento das equações de regressão, e a outra para validá-las. Os modelos foram desenvolvidos através da análise de regressão linear simples, tendo como variável dependente a massa seca (DW) e, como variáveis independentes a altura (h), o diâmetro (d) e o quadrado do diâmetro multiplicado pela altura (d^{2}h). A validação foi analisada através da comparação entre os valores obtidos em campo e os estimados pelas equações, através da análise de correlação intraclasse de Pearson e teste *t*-Sudent pareado. As melhores equações para estimar o massa seca das árvores foram: lnDW = –3,068+2,522lnd (r^{2} = 0,91; s_{y/x} = 0,67) e lnDW = –3,676+0,951ln d^{2}h (r^{2} = 0,94; s_{y/x} = 0,56).

**Palavras-chave: **biomassa, alometria, regressão, Mata Atlântica, floresta secundária.

**Introduction**

The Brazilian Atlantic Forests known as Mata Atlântica is the biome that recovered the Brazilian coast before the European colonization. The Atlantic Forest is a mosaic of ecosystems with a complex structure and hight biodiversity and endemism. The Atlantic Dominion includes dense tropical rain forest and areas of coastal flooded forest, lowland, submontane and montane forests. It taked up about 1.1million km^{2} but today it is one of the most threaten biomes due to human activities as forest clearance for agriculture and pasture, construction of highways, mining and charcoal production. The forest remnants have survived in areas of difficult access (Joly et al. 1999). The restoration and preservation of these ecosystems can have a relevant role in the global carbon cycle. Knowledge about the biomass is necessary for the estimation of carbon reserves, which is currently very important and demanded for climate change issues.

Due to the large extension of the Brazilian territory, Brazil plays a significant role in the global carbon balance, accounting for 4 to 5% of total carbon emissions to the atmosphere, related to land use changes (Schroeder & Winjum 1995). However, the precision of estimates of C stores and of carbon fixation depends on the adequate estimates of the biomass for each type of ecosystem. Biomass is unusually measured directly in forest ecosystems due to the difficulties inherent in this type of study, such as its high cost and the need to cut the vegetation. Biomass estimations are based on predictive models that were previously developed on the basis of direct measurement using destructive methods. Specific models for the various ecosystems are necessary to minimize errorr in biomass estimates (Rochow 1974, McWillian et al. 1993).

Predictive models are elaborated based on regression analysis between tree mass (generally given in dry weight) and their dimensional data, such as height and diameter. Various types of regression models and different combinations of variables have been used in the development of predictive models of phytomass for tropical ecosystems (Folster et al. 1976, Golley et al. 1978, Saldarriaga et al. 1988, Brown et al. 1989, Overman et al. 1994, Moreira-Burger & Delitti 1999, Chave et al. 2001, Chave et al. 2005, Delitti et al. 2006. These models have been applied to quantify nutrient stores, accumulated organic material and potential carbon reserves, to determine vegetation growth rate after perturbation, and to identify determinants of variation in biomass.

The biomass of Atlantic forests has only been evaluated through indirect methods, using equations developed for other tropical forests. Delitti & Burger (1998) estimated the phytomass in various areas of the Atlantic Forest, testing 25 models developed for tropical forests using data from phyto-sociological surveys. Based on these studies, it was concluded that the Atlantic forest has a maximum accumulation of above-ground biomass of 350 Mg.ha^{–1}. Secondary and disturbed forests have less than 200 Mg.ha^{–1}, and in extreme cases of perturbation, the forests accumulate less than 100 Mg.ha^{–1}. However, the precision of the estimates of Atlantic Forest biomass can only be evaluated with the development of specific models, or the validation of models published for other tropical forests, through comparison with directly obtained data. This type of study is becoming increasingly difficult, as the Atlantic Forest, and its various physiognomies, are being reduced to areas protected by legislation. In this study, it was possible to take advantage of a rare opportunity to conduct a study of a destructive nature in this type of vegetative formation, due to the deforestation of portions of this vegetation that was undertaken to widen the Imigrantes highway, in the state of São Paulo. Due to their location inside the Serra do Mar State Park, and the difficult access, these portions were well-preserved forest a remnant was sampled for this study. Within this context, the objective of the study was to develop and validate predictive models for the above-ground phytomass of the Atlantic Forest.

**Material and Methods**

*1. Study area*

The study area is located at 23° 55' 13" S and 46° 31' 54" W in the Serra do Mar State Park, São Paulo, at an altitude of approximately 570 m. Field work was carried out in August 2000, in an area designated to be cleared for the construction of a highway descending from the Imigrantes Highway. Sampling was carried out in an area made available by the Imigrantes Consortium and closely followed the recommendations and limits established by the highway construction project.

The study area lies within the Atlantic Forest Domain, defined by Decree 750, February 10, 1993 (Brasil 1993). The vegetation is classified as Dense Ombrophylus or Tropical Pluvial, according to the classification of Veloso et al. (1991). Although it is a secondary forest area, vegetation was dense, with trees as tall as 30 m and overlapping canopies that restricted the penetration of sunlight into the understory. With a complex vertical structure, vegetation was composed of many species of bushes and trees distributed in various strata, as well as lichens, mosses, pteridophytes, vines, and epiphytes covering the larger trees. In the 1970s, the area was affected by the disposal of soil material extracted from the road construction causing tree fall and the perturbation of the original soil. After this perturbation, vegetation naturally recovered, therefore it could be considered as a secondary formation, approximately 30 years old.

Topography is rolling hills. According to the Parque Estadual da Serra do Mar soil map (Rossi & Pffeifer 1991), dominant soils are Alic Cambisols and Alic Red-yellow Latosols, with loamy texture.

The climate in the region is hot and moist. According to information provided by ECOVIAS, the company responsible for the operation of the highway, the mean annual temperature in the study area during the period 2000 to 2005 was 18.7 °C. The mean monthly temperature during the coldest month (August) was 16.1 °C, and of the hottest month (February), 20.8 °C. Data collected by the Departamento de Águas e Energia Elétrica - DAEE (1952 to 1996) at the pluviometric station nearest the study area (station E3-153, Curva da Onça) indicate a mean annual precipitation of 3,400 mm, and seasonal variation, with the lowest precipitation in winter (June to August). Mean monthly precipitation in the driest month (June) was 116 mm and in the moistest month (January) 441 mm.

*2. Sampling and methods of analysis*

Above-ground phytomass was measured using the destructive method (Whittaker et al. 1974, Chapman 1976, Golley et al. 1978), involving the cutting and weighing of all trees exceeding 1.5 m in height on two 10 x 10 m plots (100 m^{2}). In this area was harvest 82 trees whose height range from 1.9 to 27.9 m and diameter range between 1.6 and 47.8 cm.

Tree diameter was calculated from the measurement of perimeters breast height and at the base using a common measuring tape. Trees were cut at ground level using a chain saw, then heights were measured using a tape measure. Leaves were manually separated from the trunks and branches, and the fresh weight of each portion weighed with a manual scale to a precision of 200 g. From each tree, sub-samples were taken of a slice of trunk near the base, a portion of branches, and a portion of leaves, and were appropriately labeled.

All sub-samples collected in the field were taken to the laboratory, oven-dried (80 °C) until constant weight to determine water content and dry weight (kg) of each tree.

To develop and validate regression equations to estimate tree dry weight, the data on 82 trees collected from the two study plots were used. The trees were randomly separated into two independent samples composed of 41 trees each. One sample was used to develop the regression equations (sample 1), and the other to validate the equations (sample 2), as proposed by Snee (1977).

In this study, total dry weight (DW, kg) was used as the dependent variable and diameter (d, cm), height (h, m), and diameter squared times height (d^{2}h, cm^{2}m) as independent variables.

The variables were described using means, standard deviations, and maximum, minimum, and median values. Scatter diagrams were generated for all variables, using the original variables initially, then with the dependent variable transformed, next with the independent variables transformed, and finally, with all variables transformed. The scatter diagrams of the raw variables indicated that the model that adjusts well to these data is the power function. To obtain estimates of the parameters of the power function (Y = aX^{b}) natural logarithms were applied to the function (lnY=lna+blnX). The logarithmic transformation of the data stabilizes the effects of the increase in variance of the biomass with increasing tree size (heteroscedasticity) (Zar 1974), enabling the use of parametric statistical analyses, such as regression, in studies of this nature (Beauchamp & Olson 1973). The scatter diagrams of the transformed dependent variable and the transformed independent variables showed a linear relationship between them, indicating that a linear regression could be adjusted to the transformed values (Vieira 2004). Various models have been developed for tropical forests employing linear regression analysis with logarithmic transformation of the data (Jordan & Uhl 1978, Saldarriaga et al. 1988, Brown et al. 1989, Martinez-Yrizar et al. 1992, Scatena et al. 1993, Overman et al. 1994, Santos 1996, Higuchi et al. 1998). Thus, in the modeling process for this study, only transformed data was used for both variables.

Pearson's correlation analysis was conducted on the variables. To increase the precision of the phytomass estimates, different variables can be introduced into the regression equation as long as there is no co-linearity among them. The high correlation between the variables makes it impossible to separate the effects of the independent variables on the dependent variable in a multiple regression analysis. Several authors reported high correlation between tree diameter and height, and above-ground biomass (Jordan & Uhl 1978, Overman et al. 1994). Thus, the equations were obtained through simple linear regression analysis, after testing co-linearity between selected variables. Linear regression analysis was applied on the dependent and independent variables that presented a linear relationship, identified in the scatter diagrams. The models that showed bias in the analysis of the residuals were excluded.

In the second stage, validation analysis of the equations was carried out. Using the data from sample 2, Student's paired *t*-test was applied to compare the means of the real values obtained in the field survey to those estimated using the equations. Pearson's intra-class correlation coefficient (r_{icc}) was also calculated for comparison of the real values and those estimated by the equations.

Following these phases of modeling and validation, the most appropriate models were defined using the following selection criteria: highest coefficients of determination (r^{2}), lowest standard errors of the estimates (s_{y/x}), highest intra-class correlation coefficients (r_{icc}) found in the validation, greater similarity of means, and 95% confidence intervals between the real values and those estimated using the equations, according to visual analysis and the application of the model.

In the final stage, simple regression analysis was done using all the data from all the trees sampled (n = 82), with the independent variables being the predictor variables in the models developed, validated, and selected in the first phase of the study. The criteria for selecting the best model were the highest determination coefficient, standard deviation, and better residuals distribution.

Available models in the literature were validated for the sudied vegetation, through the comparison between estimated biomass for sample 2 trees and the values obtained in the field.A significance level of 5% was used in all the analyses. All statistical analyses (descriptive analysis, correlation analysis, simple and multiple regression analyses) were carried out using *Statistica* for Windows^{®} (version 6.0) and *Statistical Package for the Social Sciences* (SPSS) for Windows^{®} (version 8).

**Results**

The equations from the first step in the work were developed using sample 1 data, which was composed of 41 trees with diameters, varying from 1.6 to 47.8 cm, with a mean of 8.5 cm. The height of trees varied between 2.3 to 27.9 m, with a mean of 7.0 m. Distribution parameters of the variables used in the study are presented in Table 1. Notice that the median values are generally close to the means, with the exception of dry weight (lnDW).

The matrix of the Pearson's linear correlations among all the variables (Table 2) shows a strong correlation between the independent variables, as reported in similar studies (Overman et al. 1994).

The models resulting from the simple linear regression analysis are presented in Table 3. All are statistically significant (p < 0.001), as can be observed by the confidence interval of the regression coefficient (CI _{95%} b) of each model. Analysis of residuals for the equations showed that the errors presented a normal distribution and no bias.

In the validation of the equations, the data from sample 2 were used. This was composed of 41 trees with heights varying between 1.9 m and 20.6 m, with a mean of 6.6 m. The mean diameter was 7.4 cm, varying from 1.6 to 24.5 cm.

Estimated biomass values from the models did not significantly differ from expected values (Figure 1, Table 4). All the models presented high intra-class correlation coefficient (r_{icc}). Student's paired t-test indicated no significant differences between the means of the estimated dry weights for the trees and those found in the field (Table 4). In spite all models were satisfactory, no sufficient data are available to to discriminate between them.

Considering that the variables used generated models appropriate for estimating phytomass of the Atlantic Forest, the final models were adjusted through simple linear regression analysis using all the information from all the trees collected (n = 82) in the field (Table 5). Despite being very similar to those developed using sample 1, they reflect all the variation in the dimensions of the trees represented in this study. Mean height of the 82 trees was 6.8 m, ranging from 1.9 to 27.9 m. Mean diameter was 8.0 cm, ranging between 1.6 and 47.8 cm. The equations were statistically significant (p < 0.001) and showed no tendencies in the analysis of residuals. The model with height as the predictor variable presented the lowest coefficient of determination and the largest standard error of the estimate. Considering the criteria established, the models whose predictor variables were diameter (lnDW = –3.068+2.522lnd) and diameter squared times height (lnDW = –3.676+0.951ln d^{2}h) were considered to be predictors of above-ground phytomass in the Atlantic Forest under conditions similar to those found in the local study.

Table 6 shows a validation of models selected from the literature. Estimated dry weight for trees of sample 2, when calculated using equations developed for secundary wet forests (Scatena et al. 1993; Brown et al. 1989) or primary wet forests (Chave et al. 2001, Overman et al. 1994), were not statistically different from the real mean. However, they significantly differed when the used equations were developed for moist forests.

**Discussion**

Two models were considered as good predictors for phytomass of *Mata Atlântica*: ln DW = –3.068+2.522lnd and lnDW = –3.676+0.951lnd^{2}h. The first one has the advantage of only requireing one variable, diameter that is easily measured in the field and less subjected to sampling errors. The second model adds the requirement of tree height which is not always available in forest samplings owing to the difficulty of getting accurate measurements. In spite of this limitation, this equation could be considered appropriate to estimate *Mata Atlântica* phytomass for similar conditions as those of the study area, owing to the fact of its showed the highest determination coefficient (r^{2} = 0.935) and the lowest standard error.

Further to tree size defined by its height and diameter, other factors should be considered in phytomass estimations. In this study, we validated models available in the literature that used height and diameter as predicting variables, similar to those developed in the sampled forest. Validation results indicate that model application should consider forest type (dry, moist or wet) for which the model was developed. In spite that forest type is not the major factor in determining the quantity of accumulated organic matter by trees, forest type has influence in allometric relationships (Brown et al. 1989, Chave et al. 2005). Developed models for moist life zones (Chave et al. 2001 and Higushi et al. 1998) produced higher estimates of tree dry weight for sample 2 trees, and therefore were not appropriated for the study area. According to Brown at al. (1989), for a given diameter, moist life zone trees show higher biomass than wet life zone because they tend to be taller. Therefore, moist life zone equations tend to over-estimate biomass in other forest types.

In the next step, this research would investigate the influence of wood density in the determination of biomass for Mata Atlântica trees. Chave et al. (2005) considered that tree diameter is the most relevant variable for predicting biomass in tropical forests, followed by wood density. The large species richness and the lack of knowledge of wood density for tropical forest trees would difficult wood density use. However, Chave et al. (2006) indicated that the identification of tress at the genus level would be enough for using wood density in predicting models. New studies in the various Mata Atlântica formations should include that variable, in the perspective of increasing biomass estimates accuracy.

**Conclusions**

The models developed and validated in this study explain about 90% of the variance in the dry weight (kg) of the trees, which generates a good estimate of this ecosystem descriptor. Information contained in different forest surveys, such as height and diameter of trees, can be used to estimate biomass. In spite of the limitations, and taking in account it is a secondary forest area, this information can contribute to estimates of carbon stores and exchanges, illustrating the role of the Atlantic Forests in the global carbon balance, as well as providing the basis for decisions regarding the management and recovery of these ecosystems.

**Acknowledgements**

We are grateful to Consórcio ECOVIAS for providing research facilities and to its staff members who considerably helped collecting the material in the field.

**References**

BEAUCHAMP, J.J. & OLSON, J.S. 1973. Corrections for bias in regression estimates after logarithmic transformation. Ecology 54(6): 1403-1407. [ Links ]

BRASIL. 1993 Decreto nº 750, de 11 de fevereiro de 1993. Dispõe sobre o corte, a exploração e a supressão de vegetação primária ou nos estágios avançado e médio de regeneração da Mata Atlântica, e dá outras providências. Diário Oficial da República Federativa do Brasil, Poder Executivo, Brasília, DF, Seção 1, p.1801. [ Links ]

BROWN, S., GILLESPIE, A.J.R. & LUGO, A.E. 1989. Biomass estimation methods for tropical forests with applications to forest inventory data. Forest. Sci. 35(4): 881-902. [ Links ]

CHAPMAN, S.B. 1976. Production ecology and nutrient budgets. In ______ Methods in plant ecology. Blackwell, Oxford, p.157-228. [ Links ]

CHAVE, J., RIÉRA, B., & DUBOIS, M. 2001. Estimation of biomass in a neotropical forest of French Guiana: spatial and temporal variability. J. Trop. Ecol. 17(1): 79-96. [ Links ]

CHAVE, J., ANDALO C., BROWN, S., CAIRS, M.A., CHAMBERS, J.Q., EAMUS, D., FOLSTER, H., FROMARD, F., HIGUCHI, N., KIRA, T., LESCURE, J.P., NELSON, B.W., OGAWA, H., PUIG, H., RIÉRA, B. & YAMAKURA, T. 2005. Tree allometry and improved estimation of carbon stoks and balance in tropical forests. Oecologia, 145(1): 87-99. [ Links ]

CHAVE, J., MULLER-LANDAU, H.C., BAKER, T.R., EASDALE, T.A., STEEGE, H.T. & WEBB, C.O. 2006. Regional and phylogenetic variation of wood density across 2456 neotropical tree species. Ecol. Appl. 16(6): 2356-2367. [ Links ]

DELITTI, W.B.C. & BURGER, D.M. 1998. Atlantic forests biomass estimates, Brazil. In The Earth's changing land GCTE-LUCC. Open Science Conference on Global Change, Barcelona, p. 341. [ Links ]

DELITTI, W.B.C., MEGURO, M. & PAUSAS, J. 2006. Biomass and mineralmass estimates in a "cerrado" ecosystem. Rev. Bras. Bot. 29(4): 531-540. [ Links ]

FOLSTER, H., DE LAS SALAS, G. & KHANNA, P. 1976. A tropical evergreen forest site with perched water table, Magdalena Valley, Columbia. Biomass and bioelement inventory of primary and secondary vegetation. Oecolog. Plantar. 11(4): 297-320. [ Links ]

GOLLEY, F.B., MCGINNIS, J.T., CLEMENTS, R.G., CHILD, G.I. & DUEVER, M.J. 1978. Ciclagem de minerais em um ecossistema de floresta tropical úmida. EDUSP, São Paulo. [ Links ]

HIGUCHI, N., SANTOS, J., RIBEIRO, R.J., MINETTE, L. & BIOT, Y. 1998. Biomassa da parte aérea da vegetação da Floresta Tropical Úmida de Terra-firme da Amazônia Brasileira. Acta Amazon. 28(2): 153-166. [ Links ]

JOLY, C.A., AIDAR, M.P.M., KLINK, C.A., MCGRATH, D.G., MOREIRA, A.G., MOUTINHO, P., NEPSTAD, D.C., OLIVEIRA, A.A., POTT, A., RODAL, M.J.N. & SAMPAIO, E.V.S.B. 1999. Evolution of the Brazilian phytogeography classification systems: implications for biodiversity conservation. Cienc. Cult. 51(5/6): 331-348. [ Links ]

JORDAN, C.F. & UHL, C. 1978. Biomass of a "tierra firme" forest of the Amazon Basin. Oecolog. Plantar. 13(4): 387-400. [ Links ]

MARTINEZ-YRIZAR, A., SARUKHAN, J., PEREZ-JIMENEZ, A., RINCON, E., MAASS, J.M., SOLIS-MAGALLANES, A. & CERVANTES, L. 1992. Above-ground phytomass of a tropical deciduous forest on the coast of Jalisco, México. J. Trop. Ecol. 8(1): 87-96. [ Links ]

McWILLIAN, A.L.C., ROBERTS, J.M., CABRAL, O.M.R., LEITÃO, M.V.B.R., COSTA, A.C.L., MAITELLI, G.T. & ZAMPARONI, C.A.G.P. 1993. Leaf area index above-ground biomass of terra firme rain forest and adjacent clearings in Amazonia. Funct. Ecol. 7(3): 310-317. [ Links ]

MOREIRA-BURGER, D. & DELITTI, W.B.C. 1999. Fitomassa epigéa da mata ciliar do Rio Mogi-Guaçu, Itapira - SP. Rev. Bras. Bot. 22(3): 429-435. [ Links ]

OVERMAN, J.P.M., WITTE, H.J.L. & SALDARRIAGA, J.G. 1994. Evaluation of regression models for above-ground biomass determination in Amazon rainforest. J. Trop. Ecol. 10(2): 207-218. [ Links ]

ROCHOW, J.J. 1974. Estimates of above-ground biomass and primary productivity in a Missouri Forest. J. Ecol. 62(2): 567-77. [ Links ]

ROSSI, M. & PFEIFER, R.M. 1991. Pedologia do Parque Estadual da Serra do Mar. I. Levantamento de reconhecimento de solos. Rev. Inst. Flor. 3(1): 1-44. [ Links ]

SALDARRIAGA, J.G., WEST, D.C., THARP, M.L. & UHL, C. 1988. Long-term chronosequence of forest succession in the upper Rio Negro of Colombia and Venezuela. J. Ecol. 76(4): 938-958. [ Links ]

SANTOS, J. 1996. Análise de modelos de regressão para estimar a fitomassa da floresta tropical úmida de terrra-firme da Amazônia brasileira. Tese de Doutorado, Universidade Federal de Viçosa, Viçosa. [ Links ]

SCATENA, F.N., SILVER, W., SICCAMA, T., JOHNSON, A. & SANCHEZ, M.J. 1993. Biomass and nutrient content of the Bisley Experimental Watersheds, Luquillo Experimental Forest, Puerto Rico, before and after Hurricane Hugo, 1989. Biotropica 25(1): 15-27. [ Links ]

SCHROEDER, P.E. & WINJUM, J.K. 1995. Assessing Brazil's carbon budget: II. Biotic fluxes and net carbon balance. Forest. Ecol. Manag. 75(1): 87-99. [ Links ]

SNEE, R.D. 1977. Validation of regression models: methods and examples. Technometrics 19(4): 415-428. [ Links ]

VELOSO, H.P., RANGEL FILHO, A.L.R. & LIMA, J.C.A. 1991. Classificação da vegetação brasileira adaptada a um sistema universal. IBGE, Rio de Janeiro. [ Links ]

VIEIRA, S. 2004. Bioestatística: tópicos avançados. 2.ed. Elselvier, Rio de Janeiro. [ Links ]

WHITTAKER, R.H., BORMANN, F.H., LIKENS, G.E & SICCAMA, T.G. 1974. The Hubbard Brook ecosystem study: forest biomass and production. Ecol. Monogr. 44(2): 233-252. [ Links ]

ZAR, J.H. 1974. Biosstatistical analysis. Prentice Hall, Englewood Cliffs. [ Links ]

Data Received 25/11/07

Revised 05/12/08

Accepted 18/12/08

1 Corresponding author: Déborah Moreira Burger, email: deborahmburger@yahoo.com.br