Comparing mixture estimates by parametric bootstrapping likelihood ratios

Editor’s Invited Article


Wildlife managers and researchers often need to estimate the relative contributions of distinct populations in a miture of organisms. Increasingly, there is interest in comparing these mixture contributions across space or time. Comparisons usually just check for overlap in the interval estimates for each population contribution from each mixture. This method inflates Type I error rates, has limited power due to its focus on marginal comparisons, and employs a fundamentally inappropriate measure of mixture difference. Given the difficulty of defining an appropriate measure of mixture difference, a powerful alternative is to compare mixtures using a likelihood ratio test. In applications where the standard asymptotic theory does not hold, the null reference distribution can be obtained through parametric bootstrapping. In addition to testing simple hypotheses, a likelihood ratio framework encourages modeling the change in mixture contributions as a function of covariates. The method is demonstrated with an analysis of potential sampling bias in the estimation of population contributions to the commercial sockeye, salmon (Oncorhynchus nerka) fishery in Upper Cook Inlet, Alaska.

Key Words

Compositional data Compositional difference discrete mixture analysis Genetic stock identification Mixed stock analysis Mixture homogeneity 


  1. ADF&G (2000), “SPAM version 3.2 User’s Guide: Statistics Program for Analyzing Mixtures,” Special Publication No. 15, Alska Department of Fish and Game. Division of Commercial Fisheries. Anchorage, AK, http: // Page.html.Google Scholar
  2. Aitchison, J. (1982), “The Statistical Analysis of Compositional Data” (with discussion). Journal of the Royal Statistical Society, Ser. B, 44, 139–177.MATHMathSciNetGoogle Scholar
  3. — (1986) The Statistical Analysis of Compositional Data, New York: Chapman and Hall.MATHGoogle Scholar
  4. — (1992), “On Criteria for Measures of Compositional Difference,” Mathematical Geology, 24, 365–379.MATHCrossRefMathSciNetGoogle Scholar
  5. Allendorf, F. W., and Phelps, S. R. (1981), “Use of Allelic Frequencies to Describe Population Structure” Canadian Journal of Fisheries and Aquatic Sciences, 38, 1507–1514.CrossRefGoogle Scholar
  6. Begg, G. A., Friedland, K. D., and Pearce, J. B. (1999), “Stock Identification—Its Role in Stock Assessment and Fisheries Management: A Selection of Papers Presented at a Symposium of the 128th Annual Meeting of the American Fisheries Society in Hartford, Connecticut, USA, 23–27 August 1998,” Fisheries Research, 43, 1–3.CrossRefGoogle Scholar
  7. Billheimer, D., Guttorp, P., and Fagan, W. F. (2001), “Statistical Interpretation of Species Composition,” Journal of the American Statistical Association, 96, 1205–1214.MATHCrossRefMathSciNetGoogle Scholar
  8. Blischke, W. R. (1963), “Mixtures of Discrete Distributions,” in Proceedings of the International Symposium on Classical and Contagions Discrete Distributions, New York: Pergamon Press, pp. 351–372.Google Scholar
  9. Burgner, R. L. (1991) “Life History of Sockeye Salmon (Oncorhynchusnerka),” Pacific Salmon Life Histories, eds. C. Groot and L. Margolis, Vancouver, BC: University of British Columbia Press, pp. 3–117.Google Scholar
  10. Cassie, R. M. (1954), “Some Uses of Probability Paper in the Analysis of Size Frequency Distributions,” Australian Journal of Marine and Freshwater Research, 5, 513–523.Google Scholar
  11. Cavalli-Sforza, L. L., and Edwards, A. W. F. (1967), “Phylogenetic Analysis: Models and Estimation Procedures,” Evolution, 21, 550–570.CrossRefGoogle Scholar
  12. Do, K.-A., and McLachlan, G. J. (1984), “Estimation of Mixing Proportions: A Case Study,” Applied Statistics, 33, 134–140.CrossRefGoogle Scholar
  13. Davison, A. C., and Hinkley, D. V. (1997), Bpotstrap Methodsand Their Application. Cambridge, UK: Cambridge University Press.Google Scholar
  14. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977), “Maximum Likelihood from Incomplete Data via the EM Algorithm,” Journal of the Royal Statistical Society, Ser. B. 39, 1–38.MATHMathSciNetGoogle Scholar
  15. Fournier, D. A., Beacham, T. D., Ridell, B. E., and Busack, C. A. (1984), “Estimating Stock Composition in Mixed Stock Fisheries using Morphometric, Meristic, and Electrophoretic Characteristics,” Canadian Journal of Fisheries and Aquatic Sciences, 41, 400–408.CrossRefGoogle Scholar
  16. Grant, W. S., Milner, G. B., Krasnowski, P., and Utter, F. M. (1980), “Use of Biochemical Genetic Variants for Identification of Sockeye Salmon (Onchorhynchusnerka) Stocks in Cook Inlet, Alaska,” Canadian Journal of Fisheries and Aquatic Sciences, 37, 1236–1247.CrossRefGoogle Scholar
  17. Hsu, J. C. (1996) Multiple Comparisons: Theory and Methods, London: Chapman and Hall.MATHGoogle Scholar
  18. Ihssen, P. E., Booke, H. E., Casselman, J. M., McGlade, J. M., Payne, N. R., and Utter, F. M. (1981), “Stock Identification: Materials and Methods,” Canadian Journal of Fisheries and Aquatic Sciences, 38, 1838–1855.CrossRefGoogle Scholar
  19. Kiefer, J., and Wolfowitz, J. (1956) “Consistency of Maximum Likelihood Estimates in the Presence of Infinitely many Incidental Parameters,” Annals of Mathematical Statistics, 27, 887–907.MATHCrossRefMathSciNetGoogle Scholar
  20. Lindsay, B. G. (1995) Mixture Models: Theory, Geometry, and Applications, NSF-CBMS Regional Conference Series in Probability and Statistics (vol. 5), Alexandria, VA: Institute of Mathematical Statistics and the American Statistical Association.Google Scholar
  21. Marshall, S., Bernard, D., Conrad, R., Cross, B., McBride, D., McGregor, A., McPherson, S., Oliver, G., Sharr, S., and Van Allen, B. (1987), “Application of Scale Patterns Analysis to the Management of Alaska’s Sockeye Salmon (Onchorhynchus nerka) Fisheries,” in Sockeye Salmon (Oncorhynchus nerka) Population Biology and Future Management, eds. H. D. Smith, L. Margolis, and C. C. Wood, Canadian Special Publication on Fisheries and Aquatic Science 96, pp. 207–326.Google Scholar
  22. McLachlan, G. (1987) “On Bootstrapping the Likelihood Ratio Test Statistic for the Number of Components in a Normal Mixture,” Applied Statistics, 36, 318–324.CrossRefGoogle Scholar
  23. McLachlan, G., and Peel, D. (2000) “Finite Mixture Models, New York: Wiley.MATHCrossRefGoogle Scholar
  24. McParland, T. L., Ferguson, M. M., and Liskauskas, A. P. (1999), “Genetic Population Structure and Mixed-Stock Analysis of Walleyes in the Lake Eric-Lake Huron Corridor Using Allozy me and Mitochondrial DNA Markers,” Transactions of the American Fisheries Society, 128, 1055–1067.CrossRefGoogle Scholar
  25. Moles, A., and Jensen, K. (2000), “Prevalence of the Sockeye Salmon Brain Parasite Myxobolus arcticus in Selected Alaska Stream,” Alaska Fisheries Research Bulletin, 6, 85–93.Google Scholar
  26. Millar, R. B. (1987), “Maximum Likelihood Estimation of Mixed Stock Fishery Composition,” Canadian Journal of Fisheries and Aquatic Sciences, 44, 583–590.CrossRefGoogle Scholar
  27. Milner, G. B., Teel, D. J., Utter F. M., and Burley, C. L. (1981) “Columbia River Stock Identification Study: Validation of Genetic Method,” unpublished manuscript (final report of research (FY80) financed by Bonneville Power Administration Contract DE-A179-80BP18488), National Marine Fisheries Service, Northwest and Alaska Fisheries Center, Seattle, WA.Google Scholar
  28. Pearce, J. M., Pierson, B. J., Talbot, S. L., Derksen, D. V., Kraege, D., and Scribner, K. T. (2000), “A Genetic Evaluation of Morphology used to Identify Harvested Canada Geese,” Journal of Wildlife Management, 64, 863–874.CrossRefGoogle Scholar
  29. Pella, J. J., and Masuda, M. (2001), “Bayesian Methods for Stock-Mixture Analysis from Genetic Characters,” Fishery Bulletin, 99, 151–167.Google Scholar
  30. Pella, J. J., Masuda, M., and Nelson, S. (1996) “Search Algorithms for Computing Stock Composition of a Mixture From Traits of Individuals by Maximum Likelihood,” NOAA/NMFS Technical Memo NMFS-AFSC-6L. National Marine Fisheries Service, Northwest and Alaska Fisheries Center, Seattle, WA.Google Scholar
  31. Pella, J. J., and Milner, G. B. (1987), “Use of Genetic Marks in Stock Composition Analysis,” in Population Genetics and Fishery Management, eds. N. Ryman and F. Utter, Seattle, WA: Washington Sea Grant Program, pp. 247–276.Google Scholar
  32. Planes, S., and Doherty, P. J. (1997), “Genetic and Color Interactions at a Contact Zone of Acanthochromis polyacanthus. A Marine Fish Lacking Pelagic Larvae,” Evolution, 51, 1232–1243.CrossRefGoogle Scholar
  33. Rannala, B., and Mountain, J. L. (1997), “Detecting Immigration by Using Multilocus Genotypes,” Proceedings of the National Academy of Sciences USA, 94, 9197–9201.CrossRefGoogle Scholar
  34. Redner, R. A., and Walker, H. F. (1984), “Mixture Densities, Maximum Likelihood and the EM Algorithm,” Society for Industrial and Applied Mathematics Review, 26, 195–239.MATHMathSciNetGoogle Scholar
  35. Reynolds, J. H. (2001) “SPAM (Statistics Program for Analyzing Mixtures) Version 3.5: User’s Guide Addendum,” Addendum to Special Publication No. 15, Alaska Dept. of Fish and Game, Division of Commercial Fisheries, Gene Conservation Laboratory, Anchorage, AK, Scholar
  36. Ruesch, P.H., and Fox, J., (1999), “Upper Cook Inlet Commercial Fisheries Annual Management Report, 1998,” Regional Information Report No. 2A99-21. Alaska Department of Fish and Game, Division of Commercial Fisheries, Anchorage, AK.Google Scholar
  37. Ruzzante, D. E., Taggart, C. T., Lang, S., and Cook, D. (2000), “Mixed-Stock Analysis of Atlantic Cod Near the Gulf of St. Lawrence Based on Microsatellite DNA,” Ecological Applications, 10, 1090–1109.CrossRefGoogle Scholar
  38. Saitou, N., and Nei, M. (1987) “The Neighbor-Joining Method: A New Method for Reconstructing Phylogenetic Trees,” Molecular Biology and Evolution, 4, 406–425.Google Scholar
  39. Seeb, L. W., and Crane, P. A. (1999), “Allozymes and Mitochondrial DNA Discriminate Asian and North American Populations of Chum Salmon in Mixed-Stock Fisheries Along the South Coast of the Alaska Peninsula,” Transactions of the American Fisheries Society, 128, 88–103.CrossRefGoogle Scholar
  40. Seeb, L. W., Habicht, C., Templin, W. D., Tarbox, K. E., Davis, R. Z., Brannian, L. K., and Seeb, J. E., (2000), “Genetic Diversity of Sockeye Salmon of Cook Inlet, Alaska, and its Application to Management of Populations Affected by the Exxon Valdez Oil Spill,” Transactions of the American Fisheries Society, 129, 1223–1249.CrossRefGoogle Scholar
  41. Self, S. G., and Liang, K. Y. (1987), “Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests Under Nonstandard Conditions,” Journal of The American Statistical Association, 82, 605–610.MATHCrossRefMathSciNetGoogle Scholar
  42. Shaklee, J. B., Beacham, T. D., Seeb, L., and White, B. A. (1999), “Managing Fisheries Using Genetic Data: Case Studies from Four Species of Pacific Salmon,” Fisheries Research, 43, 45–78.CrossRefGoogle Scholar
  43. Smouse, P. E., Waples, R. S., and Tworek, J. A. (1990), “A Genetic Mixture Analysis for Use With Incomplete Source Population Data,” Canadian Journal of Fisheries and Aquatic Sciences, 47, 620–634.CrossRefGoogle Scholar
  44. Stuart, A., Ord, J. K., and Arnold, S. (1999) Kendall’s Advanced Theory of Statistics, (vol. 2A): Classical Inference and the Linear Model (6th ed.), New York: Oxford University Press.Google Scholar
  45. Teicher, H. (1963), “Identifiability of Finite Mixtures,” Annals of Mathematical Statistics, 34, 1265–1269.MATHCrossRefMathSciNetGoogle Scholar
  46. Titterington, D. M., Smith, A. F. M., and Makov, U. E. (1985), Statistical Analysis of Finite Mixture Distributions, New York: Wiley.MATHGoogle Scholar
  47. Tobias, T., and Tarbox, K. E. (1999) “An Estimate of Total Return of Sockeye Salmon to Upper Cook Inlet, Alaska 1976–1998,” Regional Information Report 2A99-11, Alaska Department of Fish and Game, Division of Commercial Fisheries, Anchorage AK.Google Scholar
  48. Urawa, S., Nagasawa, K., Margolis, L., and Moles, A. (1998) “Stock Identification of Chinook Salmon (Onchorltynchus tshawytsha) in the North Pacific Ocean and Bering Sea by Parasite Tags,” North Pacific Anadromous Fish Commission Bulletin, 1, 199–204.Google Scholar
  49. Waltemeyer, D. L., Tarbox, K. E., and Brannian, L. K. (1993) “Presence of the Parasite Philonema oncorhynchic in Sockeye, Salmon Return to Upper Cook Inlet. Alaska in 1991,” Regional Information Report No. 2A93-24, Alaska Department of Fish and Game, Division of Commercial Fisheries, Anchorage, AK.Google Scholar
  50. Waples, R. S. (1990), “Temporal Changes of Allele Frequency in Pacific salmon—Implications for Mixed-Stock Fishery Analysis,” Canadian Journal of Fisheries and Aquatic Sciences, 47, 968–976.CrossRefGoogle Scholar
  51. Wilmot, R. L., Kondzela, C. M., Guthrie C. M., and Masuda, M. M. (1998), “Genetic Stock Identification of Chum Salmon Harvested Incidentally in the 1994 and 1995 Bering Sea Trawl Fishery,” North Pacific Anadronous Fish Commission Bulletin, 1, 285–299.Google Scholar
  52. Woody, C. A., Olsen, J., Reynolds, J., and Bentzen, P. (2000), “Temporal Variation in Phenotypic and Genotypic Traits in Two Sockeye Salmon Populations, Tustumena Lake, Alaska,” Transactions of the American Fisheries Society, 129, 1031–1043.CrossRefGoogle Scholar

Copyright information

© International Biometric Society 2004

Authors and Affiliations

  1. 1.U. S. Fish and Wildlife ServiceDivision of Natural ResourcesAnchorage
  2. 2.Gene Conservation Laboratory of the Commericial Fisheries DivisionAlaska Department of Fish and GameUSA

Personalised recommendations