Advertisement

Bayesian Hierarchical/Multilevel Models for Inference and Prediction Using Cross-System Lake Data

  • Craig A. Stow
  • E. Conrad Lamon
  • Song S. Qian
  • Patricia A. Soranno
  • Kenneth H. Reckhow

Abstract

Cross-system data have been extensively used to estimate models for predicting lake responses to management actions. Using data from many lakes for model estimation is based on an implicit assumption that all lakes in the data set behave similarly. A common strategy to help meet this assumption is to group the data by common lake features, such as geography, landscape setting or morphometry, and estimate separate models for each lake group. Multilevel/hierarchical models offer a rigorous approach to combine data from many lakes and/or groups of lakes for inference at multiple levels of aggregation. We use data from 382 Michigan lakes and reservoirs to develop and evaluate several alternative multilevel models for predicting chlorophyll a concentration from total phosphorus concentration. Working in a Bayesian framework provides measures of uncertainty that can be used to evaluate probability that management objectives can be achieved under differing strategies.

Keywords

Posterior Distribution Markov Chain Monte Carlo Prior Distribution Secchi Depth Total Phosphorus Concentration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Abbreviations

AIC

Akaike’s Information Criterion

ANOVA

Analysis of Variance

BCART

Bayesian classification and regression tree

BIC

Bayesian Information Criterion

CART

Classification and Regression Tree

DIC

Deviance Information Criterion

LIL

Log Integrated likelihood

MCMC

Markov Chain Monte Carlo

MLE

Maximum Likelihood Estimator

SBC

Schwarz's Bayesian criterion

TP

Total phosphorus

References

  1. Akaike, H., 1973. Information theory and an extension of the maximum likelihood principle, In Proc. 2nd Int. Syp. Information Theory (eds. B.N. Petrov and F. Csáki), pp. 267–281. Budapest: Akadémiai Kiadó.Google Scholar
  2. Breiman, L., J. H. Friedman, R. A. Olshen, and C. J. Stone. 1984. Classification and Regression Trees. Chapman & Hall. New York, NY.Google Scholar
  3. Burnham, K. P., and D. R. Anderson. 1998. Model Selection and Inference, New York: Springer.Google Scholar
  4. Canfield, D. E., and R. W. Bachmann. 1981. Prediction of total phosphorus concentrations, chlorophyll a, and secchi depths in natural and artificial lakes. Canadian Journal of Fisheries and Aquatic Sciences 38: 414–423.CrossRefGoogle Scholar
  5. Chambers, J. M., A. E. Freeny, and R. M. Heiberger 1992. Analysis of Variance: designed experiments, in /Statistical Models in S/, J. M. Chambers and T. J. Hastie (eds.), Wadsworth and Brooks/Cole Advanced Books and Software, Pacific Grove, California.Google Scholar
  6. Chipman, H. A., E. I. George, and R. E. McCulloch. 1998. Bayesian CART model search. Journal of the American Statistical Association 93: 935–948.CrossRefGoogle Scholar
  7. Chipman, H. A., E. I. George, and R. E. McCulloch. 2002. Bayesian treed models. Machine Learning 48: 299–320.CrossRefGoogle Scholar
  8. Clark, L. A., and D. Pregibon, 1992. Tree Based Models, in Statistical Models in S, J. M. Chambers and T. J. Hastie (eds.), Wadsworth and Brooks/Cole Advanced Books and Software, Pacific Grove, California.Google Scholar
  9. Cole, J., G. Lovett, and S. Findlay (eds.). 1991. Comparative analyses of ecosystems: patterns, mechanisms and theories. Springer-Verlag. 375 pp.Google Scholar
  10. Dennis, B. 1996. Discussion: Should ecologists become Bayesians? Ecological Applications 6: 1095–1103.CrossRefGoogle Scholar
  11. Ellison, A. M. 2004. Bayesian inference in ecology. Ecology Letters 7: 509–520.CrossRefGoogle Scholar
  12. Freeman, A. M., E. C. Lamon, and C. A. Stow. 2008. Regional nutrient and chlorophyll a relationships in lakes and reservoirs: A Bayesian TREED model approach. Ecological Modelling. in press.Google Scholar
  13. Gelman, A. 2006. Prior distributions for variance parameters in hierarchical models. Bayesian Analysis 1: 515–533.CrossRefGoogle Scholar
  14. Gelman, A., and J. Hill. 2007. Data Analysis Using Regression and Multilevel/Hierarchical Models, Cambridge University Press, NY.Google Scholar
  15. Hession, W. C., D. E. Storm, S. L. Burks, M. D. Smolen, R. Lakshminarayanan, and C. T. Haan. 1995. Using Eutromod with a GIS for establishing total maximum daily loads to Wister Lake, Oklahoma. Pages 215–222 in K. Steele, editor. Animal Waste and the Land-Water Interface. Lewis, Boca Raton.Google Scholar
  16. Holling, C. S. ed. 1978. Adaptive Environmental Assessment and Management. John Wiley & Sons. NY.Google Scholar
  17. Hubbard, R., and M. J. Bayarri. 2003. Confusion over measures of evidence (p's) versus errors (α's) in classical statistical testing. American Statistician 57: 171–178.CrossRefGoogle Scholar
  18. Hubbard, R., and J. S. Armstrong. 2006. Why we don't really know what “statistical significance” means: A major educational failure. Journal of Marketing Education 28: 114–120.CrossRefGoogle Scholar
  19. Lamon E. C., O. Malve, and O-P. Pietiläinen. 2008. Lake classification to enhance prediction of eutrophication endpoints in Finnish lakes, Environmental Modeling and Software. 23: 947–948.Google Scholar
  20. Lamon, E. C., and C. A. Stow. 1999. Sources of variability in microcontaminant data for Lake Michigan salmonids: Statistical models and implications for trend detection. Canadian Journal of Fisheries and Aquatic Sciences 56, Supplement 1: 71–85.CrossRefGoogle Scholar
  21. Lamon, E. C., and C. A. Stow. 2004. Bayesian methods for regional-scale lake eutrophication models. Water Research 38: 2764–2774.CrossRefPubMedGoogle Scholar
  22. Lathrop, R. C., S. R. Carpenter, C. A. Stow, P. A. Soranno, and J. C. Panuska. 1998. Phosphorus loading reductions needed to control blue-green algal blooms in Lake Mendota. Canadian Journal of Fisheries and Aquatic Sciences 55: 1169–1178.CrossRefGoogle Scholar
  23. Lee, K. N. 1993. Compass and Gyroscope. Island Press. Washington, DC.Google Scholar
  24. Lunn, D. J., A. Thomas, N. Best, and D. Spiegelhalter. 2000. WinBUGS - A Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing 10: 325–337.CrossRefGoogle Scholar
  25. Magnuson J. J., W. M. Tonn, A. Banerjee, J. Toivonen, O. Sanchez, M. Rask. 1998. Isolation vs. extinction in the assembly of fishes in small northern lakes. Ecology 79: 2941–2956CrossRefGoogle Scholar
  26. Malve, O., and S. S. Qian. 2006. Estimating nutrients and chlorophyll a relationships in Finnish lakes. Environmental Science & Technology 40: 7848–7853.CrossRefGoogle Scholar
  27. Martin, S. L., and P. A. Soranno. 2006. Defining lake landscape position: Relationships to hydrologic connectivity and landscape features. Limnology and Oceanography 51: 801–814.CrossRefGoogle Scholar
  28. National Research Council. 2001. Assessing the TMDL Approach to Water Quality Management. National Academy Press, Washington, D.C.Google Scholar
  29. Pappenberger, F., and K. J. Beven. 2006. Ignorance is bliss: Or seven reasons not to use uncertainty analysis. Water Resources Research 42. Article number W05302.Google Scholar
  30. Qian, S. S., C. A. Stow, and M. E. Borsuk. 2003. On Monte Carlo methods for Bayesian inference. Ecological Modelling 159: 269–277.CrossRefGoogle Scholar
  31. Qian, S. S., and C. W. Anderson. 1999. Exploring factors controlling the variability of pesticide concentrations in the Willamette River Basin using tree-based models. Environmental Science & Technology 33: 3332–3340.CrossRefGoogle Scholar
  32. Reckhow, K. H. 1988. Empirical-models for trophic state in Southeastern United-States lakes and reservoirs. Water Resources Bulletin 24: 723–734.Google Scholar
  33. Reckhow, K. H. 1990. Bayesian-inference in non-replicated ecological studies. Ecology 71: 2053–2059.CrossRefGoogle Scholar
  34. Reckhow, K. H. 1993. Random coefficient model for chlorophyll nutrient relationships in lakes. Ecological Modelling 70: 35–50.CrossRefGoogle Scholar
  35. Reckhow, K. H. 1996. Improved estimation of ecological effects using an empirical Bayes method. Water Resources Bulletin 32: 929–935.Google Scholar
  36. Reckhow, K. H., and S. C. Chapra. 1983. Engineering approaches for lake management, Volume 1: Data Analysis and Empirical Modeling. Butterworth Publishers. Boston.Google Scholar
  37. Rohm C. M., J. M. Omernik, A. J. Woods, and J. L. Stoddard. 2002. Regional characteristics of nutrient concentrations in streams and their application to nutrient criteria development. Journal of the American Water Resources Association 38: 213–239.CrossRefGoogle Scholar
  38. Salsburg, D. 2001. The Lady Tasting Tea. Henry Holt and Company. NY.Google Scholar
  39. Schwarz, G., 1978. Estimating the dimension of a model, Annals of Statistics 6:461–464.CrossRefGoogle Scholar
  40. Sonquist, J. N., and J. N. Morgan. 1964. The Detection of Interaction Effects. Monograph 35, Survey Research Center, Institute for Social Research, University of Michigan, Ann Arbor, MI.Google Scholar
  41. Spiegelhalter, D. J., Best, N. G., Carlin, B. P., and van der Linde, A., 2002. Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society, Series B 64: 583–639.CrossRefGoogle Scholar
  42. Stow, C. A., S. R. Carpenter, and R. C. Lathrop. 1997. A Bayesian observation error model to predict cyanobacterial biovolume from spring total phosphorus in Lake Mendota, Wisconsin. Canadian Journal of Fisheries and Aquatic Sciences 54: 464–473.CrossRefGoogle Scholar
  43. Stow, C. A., S. R. Carpenter, K. E. Webster, and T. M. Frost. 1998. Long-term environmental monitoring: Some perspectives from lakes. Ecological Applications 8: 269–276.CrossRefGoogle Scholar
  44. Stow, C. A., K. H. Reckhow, and S. S. Qian. 2006. A Bayesian approach to retransformation bias in transformed regression. Ecology 87: 1472–1477.CrossRefPubMedGoogle Scholar
  45. Stow, C. A., and D. Scavia. 2008. Modeling Hypoxia in the Chesapeake Bay: Ensemble estimation using a Bayesian hierarchical model. Journal of Marine Systems. In press.Google Scholar
  46. U.S. Environmental Protection Agency. 2000. Nutrient Criteria Technical Guidance Manual, Lakes and Reservoirs. Office of water. EPA 822-B00-001.Google Scholar
  47. Vollenweider, R. A. 1968. The Scientific Basis of Lake and Stream Eutrophication with Particular Reference to Phosphorus and Nitrogen as Eutrophication Factors. Technical Report DAS/DSI/68.27. OECD, Paris.Google Scholar
  48. Vollenweider, R. A. 1969. Possibilities and limits of elementary models concerning the budget of substances in lakes. Archiv für Hydrobiologie 66: 1–36.Google Scholar
  49. Vollenweider, R. A. 1975. Input-output models with special reference to the phosphorus loading concept in limnology. Schweizerische Zeitschrift fur Hydrologie 37: 53–84.CrossRefGoogle Scholar
  50. Vollenweider, R. A. 1976. Advances in defining critical loading levels for phosphorus in lakes eutrophication. Mem. Ist. Ital. Idrobiol. 33:53–83.Google Scholar
  51. Webster, K. E., P. A. Soranno, K. S. Cheruvelil, M. T. Bremigan, J. A. Downing, P. Vaux, T. Asplund, L. C. Bacon, and J. Connor. 2008. An empirical evaluation of the nutrient color paradigm for lakes. Limnology and Oceanography. 53:1137–1148.Google Scholar
  52. Winkler, R. L., 2003. An Introduction to Bayesian Inference and Decision, Second Edition, Probabilistic Publishing, Gainesville, FL, USA.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Craig A. Stow
    • 1
  • E. Conrad Lamon
    • 2
  • Song S. Qian
    • 3
  • Patricia A. Soranno
    • 4
  • Kenneth H. Reckhow
    • 5
  1. 1.NOAA Great Lakes Environmental Research LaboratoryAnn ArborUSA
  2. 2.Levine Science Research CenterDuke University, Nicholas School of the Environment and Earth SciencesDurhamUSA
  3. 3.Duke University, Nicholas School of the Environment and Earth SciencesDurhamUSA
  4. 4.Department of Fisheries and wildlifeMichigan state UniversityEast LansingUSA
  5. 5.A317 Levine Science Research CenterDuke University, Nicholas School of the Environment and Earth SciencesDurhamUSA

Personalised recommendations