Environmental Monitoring and Assessment

, Volume 105, Issue 1–3, pp 391–410 | Cite as

A Variance Estimator for Constrained Estimates of Change in Relative Categorical Frequencies

  • Steen MagnussenEmail author
  • Michael KÖhl


Consistent estimators of change and state becomes an issue when sample data come from a mix of permanent and temporary observation units. A joint maximum likelihood estimator of state and change creates estimates of state that depend on antecedent viz. posterior survey results and may differ from estimates of state derived from a single-date analysis of the sample data. A constrained estimator of change in relative categorical frequencies that eliminates this potential inconsistency is proposed and a model based estimator of their sampling variance is developed. The performance of the constrained estimator is quantified against six criteria and a joint maximum likelihood estimator in simulated sampling from 15 populations with three combinations of permanent and temporary samples, four to six categorical class attributes, and constant size between sampling dates. Bias of the constrained estimators was negligible but larger than for joint maximum likelihood estimators. Mean absolute deviations and variances of constrained estimators were generally at par with the joint estimators. Constrained estimators of root mean square errors and achieved coverage of nominal confidence intervals of constrained estimators were occasionally better. A generalized variance function for the constrained estimates of change is provided as a computational shortcut.


coverage rate generalized variance function iterative proportional fitting joint maximum likelihood estimation multinomial sampling root mean square error 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arthaud, G. J. and Rose, D. W.: 1996, ‘A methodology for estimating production possibility frontiers for wildlife habitat and timber value at the landscape level’, Can. J. For. Res. 26, 2191–2200.Google Scholar
  2. Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W.: 1975, Discrete Multivariate Analysis: Theory and Practice, MIT Press, Cambridge, MS.Google Scholar
  3. Casella, G. and Berger, R. L.: 2002, Statistical Inference, Duxbury, London.Google Scholar
  4. Cochran, W. G.: 1977, Sampling Techniques, Wiley, New York.Google Scholar
  5. Conover, W. J.: 1980, Practical Nonparametric Statistics, Wiley, New York.Google Scholar
  6. Cramer, E.: 1998, ‘Conditional iterative proportional fitting for Gaussian distributions’, J. Multivar. Anal. 65, 261–276.Google Scholar
  7. D’Agostino, R. B. and Rubin, D. B.: 2000, ‘Estimating and using propensity scores with partially missing data’, J. Am. Stat. Assoc. 95, 749–759.Google Scholar
  8. Diggle, P. J., Liang, K.-Y. and Zeger, S. L.: 1996, Analysis of Longitudinal Data, Clarendon, Oxford.Google Scholar
  9. Draper, N. R. and Smith, H.: 1981, Applied Regression Analysis, Wiley, New York.Google Scholar
  10. Efron, B. and Tibshirani, R. J.: 1993, An Introduction to the Bootstrap, Chapman & Hall, Boca Raton.Google Scholar
  11. Fahrmeir, L., Gieger, C. and Heumann, C.: 1999, ‘An application of isotonic longitudinal marginal regression to monitoring the healing process’, Biometrics 55, 951–956.PubMedGoogle Scholar
  12. Gaines, W. L., Harrod, R. J. and Lehmkuhul, J. F.: 1999, ‘Monitoring biodiversity: Quantification and interpretation’, PNW-GTR-443, USDA Forest Service, Portland, ORE, pp. 1–27.Google Scholar
  13. Gallant, A. R.: 1987, Nonlinear Statistical Methods, Wiley, New York.Google Scholar
  14. Green, E. J. and Strawderman, W. E.: 1986, ‘Reducing sample size through the use of a composite estimator: An application to timber volume estimation’, Can. J. For. Res. 16, 1116–1118.Google Scholar
  15. House, C. C., Goebel, J. J., Schreuder, H. T., Geissler, P. H., Williams, W. R. and Olsen, A. R.: 1998, ‘Prototyping a vision for inter-agency terrestrial inventory and monitoring: A statistical perspective’, Environ. Monit. Assess. 51, 451–463.Google Scholar
  16. Jovanovic, B. D. and Levy, P. S.: 1997, ‘A look at the rule of three’, Am. Stat. 51, 137–139.Google Scholar
  17. Kamaruzaman, J.: 1999, ‘Monitoring of forest area change using satellite data with special emphasis on Sungai Buloh forest reserve, Malaysia’, J. For. Plann. 5, 9–12.Google Scholar
  18. Katila, M. and Tomppo, E.: 2001, ‘Selecting estimation parameters for the Finnish multisource national forest inventory’, Rem. Sens. Environ. 76, 16–32.Google Scholar
  19. Lee, J. W. and DeMets, D. L.: 1991, ‘Sequential comparision of changes with repeated measurements data’, J. Am. Stat. Assoc. 86, 757–762.Google Scholar
  20. Li, H. G. and Schreuder, H. T.: 1985, ‘Adjusting estimates in large two-way tables in surveys’, For. Sci. 31, 366–372.Google Scholar
  21. Link, W. A. and Sauer, J. R.: 1998, ‘Estimating population change from count data: Application to the North American breeding bird survey’, Ecol. Appl. 8, 258–268.Google Scholar
  22. Liu, C. H. and Rubin, D. B.: 1998, ‘Ellipsoidally symmetric extensions of the general location model for mixed categorical and continuous data’, Biometrika 85, 673–688.Google Scholar
  23. Lloyd, C. J.: 1999, Analysis of Categorical Variables, Wiley, New York.Google Scholar
  24. Magnussen, S.: 2002, ‘Joint versus separate estimation of state and change in category frequencies from repeat stratified two-phase sampling with a fallible classifier’, Environ. Monit. Assess. 78, 63–87.PubMedGoogle Scholar
  25. Magnussen, S., Boudewyn, P., Wulder, M. A. and Seemann, D.: 2000, ‘Predictions of forest inventory cover type proportions using Landsat TM’, Silva Fenn. 34, 351–370.Google Scholar
  26. Mas, J. F.: 1999, ‘Monitoring land-cover changes: A comparison of change detection techniques’, Int. J. Rem. Sens. 20, 139–152.Google Scholar
  27. Matis, K. G., Hetherington, J. C. and Kassab, J. Y.: 1984, ‘Sampling with partial replacement – A literature review’, Commonw. For. Rev. 63, 193–206.Google Scholar
  28. Nakajima, N. Y., Yoshida, S. and Imanaga, M.: 1996, ‘Comparison of change estimation between four ground-survey methods for use in a continuous forest inventory system’, J. For. Plann. 2, 145–150.Google Scholar
  29. Niese, J. N. and Strong, T. F.: 1992, ‘Economic and tree diversity trade-offs in managed northern hardwoods’, Can. J. For. Res. 22, 1807–1813.Google Scholar
  30. Olsen, A. R., Sedransk, J., Edwards, D., Gotway, C. A., Liggett, W., Rathburn, S., Reckhow, K. H. and Young, L. J.: 1999, ‘Statistical issues for monitoring ecological and natural resources in the United States’, Environ. Monit. Assess. 54, 1–45.Google Scholar
  31. Piccioni, M.: 2000, ‘Independence structure of natural conjugate densities to exponential families and the Gibbs’ sampler’, Scand. J. Stat. 27, 111–127.Google Scholar
  32. Poso, S., Wang, G. and Tuominen, S.: 1999, ‘Weighting alternative estimates when using multi-source auxiliary data for forest inventory’, Silva Fenn. 33, 41–50.Google Scholar
  33. Purvis, A. and Hector, A.: 2000, ‘Getting the measure of biodiversity’, Nature 405, 212–219.CrossRefPubMedGoogle Scholar
  34. Rosenfield, G. H.: 1982, ‘Sample design for estimating change in land use and land cover’, Photogr. Eng. Rem. Sens. 48, 793–801.Google Scholar
  35. Sachs, D. L., Sollins, P. and Cohen, W. B.: 1998, ‘Detecting landscape changes in the interior of British Columbia from 1975 to 1992 using satellite imagery’, Can. J. For. Res. 28, 23–36.Google Scholar
  36. Schreuder, H. T., Lin, J.-M. S. and Teply, J.: 2000, ‘Estimating the Number of Tree Species in Forest Populations using Current Vegetation Survey and Forest Inventory and Analysis Approximation Plots and Grid Intensities’, RMRS-RN-8, USDA Forest Service, Rocky Mountain Research Station, pp. 1–7.Google Scholar
  37. Scott, C. T.: 1984, ‘A new look at sampling with partial replacement’, For. Sci. 30, 157–166.Google Scholar
  38. Scott, C. T. and Köhl, M.: 1993, ‘A method for comparing sampling design alternatives for extensive inventories’, 68, Eidgenössischen Forschungsanstalt für Wald, Schnee und Landschaft, Birmensdorf, Schweiz, p. 62.Google Scholar
  39. Searle, S. R.: 1982, Matrix Algebra Useful for Statistics, Wiley, New York.Google Scholar
  40. Turner, D. P., Koerper, G., Gucinski, H. and Peterson, C.: 1993, ‘Monitoring global change: Comparison of forest cover estimates using remote sensing and inventory approaches’, Environ. Monit. Assess. 26, 295–305.Google Scholar
  41. Valliant, R., Dorfman, A. H. and Royall, R. M.: 2000, Finite Population Sampling and Inference. A Prediction Approach, Wiley, New York.Google Scholar
  42. Van Deusen, P. C.: 1992, ‘Sampling moving strata on two occasions’, Can. J. For. Res. 23, 96–100.Google Scholar
  43. Van Deusen, P. C.: 1994, ‘Correcting bias in change estimation from thematic maps’, Rem. Sens. Environ. 50, 67–73.Google Scholar
  44. Woodcock, C. E., Macomber, S. A., Pax-Lenney, M. and Cohen, W. B.: 2001, ‘Monitoring large areas for forest change using Landsat: Generalization across space, time and Landsat sensors’, Rem. Sens. Environ. 78, 194–203.Google Scholar
  45. Zhang, L. C. and Chambers, R. L.: 2004, ‘Small area estimates for cross-classifications’, J. R. Stat. Soc. B Stat. Methodol. 66, 479–496.MathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Canadian Forest ServiceVictoriaCanada
  2. 2.University of Hamburg Federal Center for Forestry and Forest Products Institute for World Forestry Leuschnerstr.HamburgGermany

Personalised recommendations