# A method for calculating the variance and confidence intervals for tree biomass estimates obtained from allometric equations

Keywords:
plant allometry, lognormal distribution, linear regression, savannah trees, confidence intervals, carbon sequestration

### Abstract

The need for accurate quantification of the amount of carbon stored in the environment has never been greater. Carbon sequestration has become a vital component of the battle against global climate change, and monitoring and quantifying this process are major challenges for policymakers. Plant allometric equations allow managers and scientists to quantify the biomass contained in a tree without cutting it down, and therefore can play a pivotal role in measuring carbon sequestration in forests and savannahs. These equations have been available since the beginning of the 20th century, but their usefulness depends on the ability to estimate the error associated with the equations – something which has received scant attention in the past. This paper provides a method based on the theory of linear regression and the lognormal distribution to derive confidence limits for estimates of biomass derived from plant allometric equations. Allometric equations for several southern African savannah species are provided, as well as the parameters and equations required to calculate the confidence intervals. This method was applied to data collected from a sampling campaign carried out in a savannah landscape at the Skukuza flux site, Kruger National Park, South Africa. Here the error was 10% of the total site biomass for the woody biomass and 2% for the leaf biomass. When the data were split into individual plots and used to estimate site biomass (as would occur in most sampling schemes) the error increased to 16% and 12% of the woody and leaf biomasses, respectively, as the sampling errors were added to the errors in the allometric equation. These methods can be used in any discipline that applies allometric equations, such as health sciences and animal physiology.### References

1. Brown S. Measuring carbon in forests: Current status and future challenges. Environ Pollut. 2002;116:363–372. doi:10.1016/S0269-7491(01)00212-3

2. Houghton RA, Lawrence KT, Hackler JL, Brown S. The spatial distribution of forest biomass in the Brazilian Amazon: A comparison of estimates. Glob Chang Biol. 2001;7:731–746. doi:10.1046/j.1365-2486.2001.00426.x

3. Ong JE, Gong WK, Wong CH. Allometry and partitioning of the mangrove, Rhizophora apiculata. For Ecol Manage. 2004;188:395–408.

4. Peichl M, Arain MA. Allometry and partitioning of above- and below ground tree biomass in an age-sequence of white pine forests. For Ecol Manage. 2007;253:68–80.

5. Zianis D. Predicting mean aboveground forest biomass and its associated variance. For Ecol Manage. 2008;256:1400–1407.

6. Cairns MA, Olmsted I, Grandos J, Argaez J. Composition and aboveground tree biomass of a dry semi-evergreen forest on Mexico’s Yucatan Peninsula. For Ecol Manage. 2003;186:125–132.

7. Niklas KJ. Plant allometry: Is there a grand unifying theory? Biol Rev. 2004;79:871–889. doi:10.1017/S1464793104006499, PMid:15682874

8. Zianis D, Mencuccini M. On simplifying allometric analyses of forest biomass. For Ecol Manage. 2004;187:311–332.

9. Duursma RA, Robinson AP. Bias in the mean tree model as a consequence of Jensen’s inequality. For Ecol Manage. 2003;186:373–380.

10. Duan N. A nonparametric retransformation method. J Am Statistical Assoc. 1983;78:605–610. doi:10.2307/2288126

11. Ai C, Norton, E. Standard errors for the retransformation problem with heteroscedasticity. J Health Econ. 2000;19:697–718.

12. Parresol BR. Modeling multiplicative error variance: An example predicting tree diameter from stump dimensions in Baldcypress. For Sci. 1993;39:670–679.

13. Carroll RJ, Ruppert D. Transformation and weighting in regression. New York: Chapman and Hall; 1988.

14. Seber GAF, Lee AJ. Linear regression analysis. Hoboken: Wiley; 2003.

15. García-Berthou E. On the misuse of residuals in ecology: Testing regression residuals vs. the analysis of covariance. J Anim Ecol. 2001;10:708–711. doi:10.1046/j.1365-2656.2001.00524.x

16. McArdle BH. The structural relationship: Regression in biology. Can J Zool. 1988;66:2329–2339. doi:10.1139/z88-348

17. Chave J, Andalo C, Brown S, et al. Tree allometry and improved estimation of carbon stocks and balance in tropical forests. Oecologia. 2005;145:87–99. doi:10.1007/s00442-005-0100-x,PMid:15971085

18. Henry HAL, Thomas SC. Interactive effects of lateral shade and wind on stem allometry, biomass allocation, and mechanical stability in Abutilon theophrasti (Malvaceae). Am J Bot. 2002;89:1609–1615. doi:10.3732/ajb.89.10.1609

19. Northup BK, Zitzer SF, Archer S, McMurtry CR, Boutton TW. Above-ground biomass and carbon and nitrogen content of woody species in a subtropical thornscrub parkland. J Arid Environ. 2005;62:23–43. doi:10.1016/j.jaridenv.2004.09.019

20. Son Y, Hwang JW, Kim ZS, Lee WK, Kim JS. Allometry and biomass of Korean pine (Pinus koraiensis) in central Korea. Bioresour Technol. 2001;78:251–255. doi:10.1016/S0960-8524(01)00012-8

21. Williams CJ, LePage BA, Vann DR, et al. Structure, allometry, and biomass of plantation Metasequoia glyptostroboides in Japan. For Ecol Manage. 2003;180:287–301.

22. Wainwright PC, Reilly SM. Ecological morphology: Integrative organismal biology. Chicago: University of Chicago Press; 1994.

23. Laclau J, Bouillet J, Gonçalves JLM, et al. Mixed-species plantations of Acacia mangium and Eucalyptus grandis in Brazil - 1: Growth dynamics and aboveground net primary production. For Ecol Manage. 2008;255:3905–3917.

24. Breusch TS, Pagan AR. A simple test for heteroscedasticity and random coefficient variation. Econometrica. 1979;47:1287–1294. doi:10.2307/1911963

25. Koenker R. A note on studentizing a test for heteroscedasticity. J Econometrics. 1981;17:107–112. doi:10.1016/0304-4076(81)90062-2

26. Crow EL, Shimizu K. Lognormal distributions: Theory and applications. New York: Dekker; 1988.

27. Stow CA, Reckhow KH, Qian SS. A Bayesian approach to retransformation bias in transformed regression. Ecology. 2006;87:1472–1477. doi:10.1890/0012-9658(2006)87[1472:ABATRB]2.0.CO;2

28. Zou GY, Huo CY, Taleban J. Simple confidence intervals for lognormal means and their differences with environmental applications. Environmetrics. 2009;20:172–180. doi:10.1002/env.919

29. Fenton LF. The sum of log-normal probability distributions in scatter transmission systems. IRE Trans Comm Syst. 1960;8:57–67. doi:10.1109/TCOM.1960.1097606

30. Pitard FF. Pierre Gy’s sampling theory and sampling practice: Heterogeneity, sampling correctness, and statistical process control. 2nd ed. Boca Raton: CRC Press; 1993.

31. Eckblad JW. How many samples should be taken? Bioscience. 1991;41:346–348. doi:10.2307/1311590

32. Kenkel NC, Podani J. Plot size and estimation efficiency in plant community studies. J Veg Sci. 1991;4:539–544. doi:10.2307/3236036

33. Ripley BD. Spatial statistics. Hoboken: Wiley; 2004.

34. Scholes RJ, Gureja N, Giannecchinni M, et al. The environment and vegetation of the flux measurement site near Skukuza, Kruger National Park. Koedoe. 2001;44:73–83.

35. Archibald SA, Kirton, A, Van der Merwe MR, Scholes RJ, Williams CA, Hanan N. Drivers of inter-annual variability in Net Ecosystem Exchange in a semi-arid savanna ecosystem, South Africa. Biogeosciences. 2009;6:231–266. doi:10.5194/bg-6-251-2009

36. Scholes RJ. Response of three semi-arid savannas on contrasting soils to the removal of the woody component. PhD thesis, Johannesburg, University of the Witwatersrand, 1988.

37. Goodman PS. Soil, vegetation and large herbivore relations in Mkuzi Game Reserve, Natal. PhD thesis, Johannesburg, University of the Witwatersrand, 1990.

38. Chidumayo EN. Above-ground woody biomass structure and productivity in a Zambezian woodland. For Ecol Manage. 1990;36:33–46.

39. Chatterjee S, Hadi AS. Regression analysis by example. New York: Wiley; 2006. doi:10.1002/0470055464

40. Filgueira R, Labarta U, Fernández-Reiriz MJ. Effect of condition index on allometric relationships of clearance rate in Mytilus galloprovincialis Lamarck, 1819. Rev Biol Mar Ocean. 2008;43:391–398.

41. Gillooly JF, Brown JH, West GB, Savage VM, Charnov EL. Effects of size and temperature on metabolic rate. Science. 2001;293:2248–2251. doi:10.1126/science.1061967, PMid:2736382

42. Dial KP, Greene E, Irschick DJ. Allometry of behaviour. Trends Ecol Evol. 2008;23:394–401. doi:10.1016/j.tree.2008.03.005

2. Houghton RA, Lawrence KT, Hackler JL, Brown S. The spatial distribution of forest biomass in the Brazilian Amazon: A comparison of estimates. Glob Chang Biol. 2001;7:731–746. doi:10.1046/j.1365-2486.2001.00426.x

3. Ong JE, Gong WK, Wong CH. Allometry and partitioning of the mangrove, Rhizophora apiculata. For Ecol Manage. 2004;188:395–408.

4. Peichl M, Arain MA. Allometry and partitioning of above- and below ground tree biomass in an age-sequence of white pine forests. For Ecol Manage. 2007;253:68–80.

5. Zianis D. Predicting mean aboveground forest biomass and its associated variance. For Ecol Manage. 2008;256:1400–1407.

6. Cairns MA, Olmsted I, Grandos J, Argaez J. Composition and aboveground tree biomass of a dry semi-evergreen forest on Mexico’s Yucatan Peninsula. For Ecol Manage. 2003;186:125–132.

7. Niklas KJ. Plant allometry: Is there a grand unifying theory? Biol Rev. 2004;79:871–889. doi:10.1017/S1464793104006499, PMid:15682874

8. Zianis D, Mencuccini M. On simplifying allometric analyses of forest biomass. For Ecol Manage. 2004;187:311–332.

9. Duursma RA, Robinson AP. Bias in the mean tree model as a consequence of Jensen’s inequality. For Ecol Manage. 2003;186:373–380.

10. Duan N. A nonparametric retransformation method. J Am Statistical Assoc. 1983;78:605–610. doi:10.2307/2288126

11. Ai C, Norton, E. Standard errors for the retransformation problem with heteroscedasticity. J Health Econ. 2000;19:697–718.

12. Parresol BR. Modeling multiplicative error variance: An example predicting tree diameter from stump dimensions in Baldcypress. For Sci. 1993;39:670–679.

13. Carroll RJ, Ruppert D. Transformation and weighting in regression. New York: Chapman and Hall; 1988.

14. Seber GAF, Lee AJ. Linear regression analysis. Hoboken: Wiley; 2003.

15. García-Berthou E. On the misuse of residuals in ecology: Testing regression residuals vs. the analysis of covariance. J Anim Ecol. 2001;10:708–711. doi:10.1046/j.1365-2656.2001.00524.x

16. McArdle BH. The structural relationship: Regression in biology. Can J Zool. 1988;66:2329–2339. doi:10.1139/z88-348

17. Chave J, Andalo C, Brown S, et al. Tree allometry and improved estimation of carbon stocks and balance in tropical forests. Oecologia. 2005;145:87–99. doi:10.1007/s00442-005-0100-x,PMid:15971085

18. Henry HAL, Thomas SC. Interactive effects of lateral shade and wind on stem allometry, biomass allocation, and mechanical stability in Abutilon theophrasti (Malvaceae). Am J Bot. 2002;89:1609–1615. doi:10.3732/ajb.89.10.1609

19. Northup BK, Zitzer SF, Archer S, McMurtry CR, Boutton TW. Above-ground biomass and carbon and nitrogen content of woody species in a subtropical thornscrub parkland. J Arid Environ. 2005;62:23–43. doi:10.1016/j.jaridenv.2004.09.019

20. Son Y, Hwang JW, Kim ZS, Lee WK, Kim JS. Allometry and biomass of Korean pine (Pinus koraiensis) in central Korea. Bioresour Technol. 2001;78:251–255. doi:10.1016/S0960-8524(01)00012-8

21. Williams CJ, LePage BA, Vann DR, et al. Structure, allometry, and biomass of plantation Metasequoia glyptostroboides in Japan. For Ecol Manage. 2003;180:287–301.

22. Wainwright PC, Reilly SM. Ecological morphology: Integrative organismal biology. Chicago: University of Chicago Press; 1994.

23. Laclau J, Bouillet J, Gonçalves JLM, et al. Mixed-species plantations of Acacia mangium and Eucalyptus grandis in Brazil - 1: Growth dynamics and aboveground net primary production. For Ecol Manage. 2008;255:3905–3917.

24. Breusch TS, Pagan AR. A simple test for heteroscedasticity and random coefficient variation. Econometrica. 1979;47:1287–1294. doi:10.2307/1911963

25. Koenker R. A note on studentizing a test for heteroscedasticity. J Econometrics. 1981;17:107–112. doi:10.1016/0304-4076(81)90062-2

26. Crow EL, Shimizu K. Lognormal distributions: Theory and applications. New York: Dekker; 1988.

27. Stow CA, Reckhow KH, Qian SS. A Bayesian approach to retransformation bias in transformed regression. Ecology. 2006;87:1472–1477. doi:10.1890/0012-9658(2006)87[1472:ABATRB]2.0.CO;2

28. Zou GY, Huo CY, Taleban J. Simple confidence intervals for lognormal means and their differences with environmental applications. Environmetrics. 2009;20:172–180. doi:10.1002/env.919

29. Fenton LF. The sum of log-normal probability distributions in scatter transmission systems. IRE Trans Comm Syst. 1960;8:57–67. doi:10.1109/TCOM.1960.1097606

30. Pitard FF. Pierre Gy’s sampling theory and sampling practice: Heterogeneity, sampling correctness, and statistical process control. 2nd ed. Boca Raton: CRC Press; 1993.

31. Eckblad JW. How many samples should be taken? Bioscience. 1991;41:346–348. doi:10.2307/1311590

32. Kenkel NC, Podani J. Plot size and estimation efficiency in plant community studies. J Veg Sci. 1991;4:539–544. doi:10.2307/3236036

33. Ripley BD. Spatial statistics. Hoboken: Wiley; 2004.

34. Scholes RJ, Gureja N, Giannecchinni M, et al. The environment and vegetation of the flux measurement site near Skukuza, Kruger National Park. Koedoe. 2001;44:73–83.

35. Archibald SA, Kirton, A, Van der Merwe MR, Scholes RJ, Williams CA, Hanan N. Drivers of inter-annual variability in Net Ecosystem Exchange in a semi-arid savanna ecosystem, South Africa. Biogeosciences. 2009;6:231–266. doi:10.5194/bg-6-251-2009

36. Scholes RJ. Response of three semi-arid savannas on contrasting soils to the removal of the woody component. PhD thesis, Johannesburg, University of the Witwatersrand, 1988.

37. Goodman PS. Soil, vegetation and large herbivore relations in Mkuzi Game Reserve, Natal. PhD thesis, Johannesburg, University of the Witwatersrand, 1990.

38. Chidumayo EN. Above-ground woody biomass structure and productivity in a Zambezian woodland. For Ecol Manage. 1990;36:33–46.

39. Chatterjee S, Hadi AS. Regression analysis by example. New York: Wiley; 2006. doi:10.1002/0470055464

40. Filgueira R, Labarta U, Fernández-Reiriz MJ. Effect of condition index on allometric relationships of clearance rate in Mytilus galloprovincialis Lamarck, 1819. Rev Biol Mar Ocean. 2008;43:391–398.

41. Gillooly JF, Brown JH, West GB, Savage VM, Charnov EL. Effects of size and temperature on metabolic rate. Science. 2001;293:2248–2251. doi:10.1126/science.1061967, PMid:2736382

42. Dial KP, Greene E, Irschick DJ. Allometry of behaviour. Trends Ecol Evol. 2008;23:394–401. doi:10.1016/j.tree.2008.03.005