Environmental and Ecological Statistics

, Volume 1, Issue 4, pp 287–302

Small sample estimation for Taylor's power law

  • S. J. Clark
  • J. N. Perry
Papers

Abstract

An analysis of counts of sample size N=2 arising from a survey of the grass Bromus commutatus identified several factors which might seriously affect the estimation of parameters of Taylor's power law for such small sample sizes. The small sample estimation of Taylor's power law was studied by simulation. For each of five small sample sizes, N=2, 3, 5, 15 and 30, samples were simulated from populations for which the underlying known relationship between variance and mean was given by σ2 = cμd. One thousand samples generated from the negative binomial distribution were simulated for each of the six combinations of c=1,2 and 11, and d=1, 2, at each of four mean densities, μ=0.5, 1, 10 and 100, giving 4000 samples for each combination. Estimates of Taylor's power law parameters were obtained for each combination by regressing log10s2 on log10m, where s2 and m are the sample variance and mean, respectively. Bias in the parameter estimates, b and log10a, reduced as N increased and increased with c for both values of d and these relationships were described well by quadratic response surfaces. The factors which affect small-sample estimation are: (i) exclusion of samples for which m = s2 = 0; (ii) exclusion of samples for which s2 = 0, but m > 0; (iii) correlation between log10s2 and log10m; (iv) restriction on the maximum variance expressible in a sample; (v) restriction on the minimum variance expressible in a sample; (vi) underestimation of log10s2 for skew distributions; and (vii) the limited set of possible values of m and s2. These factors and their effect on the parameter estimates are discussed in relation to the simulated samples. The effects of maximum variance restriction and underestimation of log10s2 were found to be the most severe. We conclude that Taylor's power law should be used with caution if the majority of samples from which s2 and m are calculated have size, N, less than 15. An example is given of the estimated effect of bias when Taylor's power law is used to derive an efficient sampling scheme.

Keywords

bias negative binomial distribution parameter estimation response surface sample scheme simulation small samples Taylor's power law variance-mean relationship 

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References

  1. Anderson, R.M., Gordon, D.M., Crawley, M.J. and Hassell, M.P. (1982) Variability in the abundance of animal and plant species. Nature 296, 245–8.Google Scholar
  2. Binns, M.R. (1986) Behavioural dynamics and the negative binomial distribution. Oikos 47, 315–318.Google Scholar
  3. Binns, M.R. and Nyrop, J.P. (1992) Sampling insect populations for the purpose of IPM decision making. Annual Review of Entomology 37, 427–53.Google Scholar
  4. Blackshaw, R.P. and Perry, J.N. (1994) Predicting leatherjacket population frequencies in Northern Ireland. Annals of Applied Biology, (in press).Google Scholar
  5. Bliss, C.I. (1941) Statistical problems in estimating populations of Japanese beetle larvae. Journal of Economic Entomology 34, 221–32.Google Scholar
  6. Boag, B. and Topham, P.B. (1984) Aggregation of plant parasitic nematodes and Taylor's power law. Nematologica 30, 348–57.Google Scholar
  7. Clark, S.J. and Perry, J.N. (1989) Estimation of the negative binomial parameter k by maximum quasilikelihood. Biometrics 45, 309–16.Google Scholar
  8. Downing, J.A. (1986) Spatial heterogeneity: evolved behaviour or mathematical artefact? Nature 323, 255–7.Google Scholar
  9. Duchateau, L., Ross, G.J.S. and Perry, J.N. (1994) Parameter estimation and hypothesis testing for Adès distributions applied to tse-tse fly data. Biométrie Praximétrie, (in press).Google Scholar
  10. Dye, C. (1983) Insect movement and fluctuations in insect population size. Antenna 7, 174–8.Google Scholar
  11. Dye, C. (1984) Reply to Dr Taylor. Antenna 8, 65–6.Google Scholar
  12. Finch, S., Skinner, G. and Freeman, G.H. (1975) The distribution and analysis of cabbage root fly egg populations. Annals of Applied Biology 79, 1–18.Google Scholar
  13. Finch, S., Skinner, G. and Freeman, G.H. (1978) Distribution and analysis of cabbage root fly pupal populations. Annals of Applied Biology 88, 351–6.Google Scholar
  14. Gaston, K.J. and McArdle, B.H. (1993) Measurement of variation in the size of populations in space and time: some points of clarification. Oikos 68, 357–60.Google Scholar
  15. Hanski, I. (1987) Cross-correlation in population dynamics and the slope of spatial variance-mean regressions. Oikos 50, 148–51.Google Scholar
  16. Holgate, P. (1988) Approximate moments of the Adès distribution. Biometrical Journal 31, 875–83.Google Scholar
  17. Kemp, A.W. (1987) Families of discrete distributions satisfying Taylor's power law. Biometrics 43, 693–9.Google Scholar
  18. Kemp, A.W. (1988) Response to Perry and Taylor. Biometrics 44, 888–9.Google Scholar
  19. Lepš, J. (1993) Taylor's power law and the measurement of variation in the size of populations in space and time. Oikos 68, 349–56.Google Scholar
  20. Marshall, E.J.P. (1985) Weed distributions associated with cereal field edges — some preliminary observations. Aspects of Applied Biology 9, 49–58.Google Scholar
  21. Marshall, E.J.P. (1989) Distribution patterns of plants associated with arable field edges. Journal of Applied Ecology 26, 247–257.Google Scholar
  22. May, R.M. and Southwood, T.R.E. (1990) Introduction. In Living in a Patchy Environment (B. Shorrocks and I.R. Swingland, eds), pp. 1–22. Oxford University Press.Google Scholar
  23. McArdle, B.H., Gaston, K.J. and Lawton, J.H. (1990) Variation in the size of animal populations: patterns, problems and artefacts. Journal of Animal Ecology 59, 439–54.Google Scholar
  24. Perry, J.N. (1981) Taylor's power law for dependence of variance on mean in animal populations. Applied Statistics 30, 254–63.Google Scholar
  25. Perry, J.N. (1984) Negative binomial model for mosquitos. Biometrics 40, 863–4.Google Scholar
  26. Perry, J.N. (1987) Iterative improvement of a power transformation to stabilise variance. Applied Statistics 36, 15–21.Google Scholar
  27. Perry, J.N. (1988) Some models for spatial variability of animal species. Oikos 51, 124–30.Google Scholar
  28. Perry, J.N. (1994a) Sampling and applied statistics for pests and diseases. Aspects of Applied Biology 37, 1–14.Google Scholar
  29. Perry, J.N. (1994b) Chaotic dynamics can generate Taylor's power law. Proceedings of the Royal Society of London Series B 257, 221–6.Google Scholar
  30. Perry, J.N. and Taylor, L.R. (1985) Adès: new ecological families of species-specific frequency distributions that describe repeated spatial samples with an intrinsic power-law variance-mean property. Journal of Animal Ecology 54, 931–53.Google Scholar
  31. Perry, J.N. and Taylor, L.R. (1986) Stability of real interacting populations in space and time: implications, alternatives and the negative binomial k c. Journal of Animal Ecology 55, 1053–68.Google Scholar
  32. Perry, J.N. and Taylor, L.R. (1988) Families of distributions for repeated samples of animal counts. Biometrics 44, 881–90.Google Scholar
  33. Perry, J.N. and Woiwod, I.P. (1992) Fitting Taylor's power law. Oikos 65, 538–42.Google Scholar
  34. Ross, G.J.S. (1990) Incomplete variance functions. Journal of Applied Statistics 17, 3–8.Google Scholar
  35. Routledge, R.D. and Swartz, T.B. (1991) Taylor's power law re-examined. Oikos 60, 107–12.Google Scholar
  36. Southwood, T.R.E. (1966) Ecological Methods. Methuen, London.Google Scholar
  37. Southwood, T.R.E. (1984) Insects as models. Antenna 8, 3–14.Google Scholar
  38. Taylor, L.R. (1961) Aggregation, variation and the mean. Nature 189, 732–5.Google Scholar
  39. Taylor, L.R. (1965) A natural law for the spatial disposition of insects. Proceedings of the 12th International Congress of Entomology, London, 1964, pp. 396-7.Google Scholar
  40. Taylor, L.R. (1970) Aggregation and the transformation of counts of Aphis fabae (Scop.) on beans. Annals of Applied Biology 65, 181–9.Google Scholar
  41. Taylor, L.R. (1971) Aggregation as a species characteristic. In Spatial Patterns and Statistical Distributions (G.P. Patil, E.C. Pielou and W.E. Waters, eds.), Pennsylvania State University Press, Pennsylvania, pp. 357–77.Google Scholar
  42. Taylor, L.R. (1984a) Assessing and interpreting the spatial distributions of insect populations. Annual Review of Entomology 29, 321–57.Google Scholar
  43. Taylor, L.R. (1984b) Anscombe's hypothesis and the changing distribution of insect populations. Antenna 8, 62–7.Google Scholar
  44. Taylor, L.R. (1986) Synoptic dynamics, migration and the Rothamsted Insect Survey. Journal of Animal Ecology 55, 1–38.Google Scholar
  45. Taylor, L. R. and Taylor, R.A.J. (1977) Aggregation, migration and population mechanics. Nature 265, 415–21.Google Scholar
  46. Taylor, L.R. and Woiwod, I.P. (1980) Temporal stability as a density-dependent species characteristic. Journal of Animal Ecology 49, 209–24.Google Scholar
  47. Taylor, L.R. and Woiwod, I.P. (1982) Comparative synoptic dynamics. I. Relationships between inter- and intra-specific spatial and temporal variance/mean parameters. Journal of Animal Ecology 51, 879–906.Google Scholar
  48. Taylor, L.R., Woiwod, I.P. and Perry, J.N. (1978) The density dependence of spatial behaviour and the rarity of randomness. Journal of Animal Ecology 47, 383–406.Google Scholar
  49. Taylor, L.R., Woiwod, I.P. and Perry, J.N. (1979) The negative binomial as a dynamic ecological model for aggregation and the density-dependence of k. Journal of Animal Ecology 48, 289–304.Google Scholar
  50. Taylor, L.R., Woiwod, I.P. and Perry, J.N. (1980) Variance and the large-scale spatial stability of aphids, moths and birds. Journal of Animal Ecology 49, 831–54.Google Scholar
  51. Taylor, L.R., Taylor, R.A.J., Woiwod, I.P. and Perry, J.N. (1983) Behavioral dynamics. Nature 303, 801–4.Google Scholar
  52. Taylor, L.R., Perry, I.N., Woiwod, I.P. and Taylor, R.A.J. (1988) Specificity of the spatial power-law exponent in ecology and agriculture. Nature 332, 721–2.Google Scholar
  53. Thórarinsson, K. (1986) Population density and movement: a critique of Δ-models. Oikos 46, 70–81.Google Scholar
  54. Woiwod, I.P. and Perry, J.N. (1989) Data reduction and analysis. Boletin de Sanidad Vegetal No. 17, Proceedings of PARASITIS 88, pp. 159-74.Google Scholar
  55. Yamamura, K. (1990) Sampling scale dependence of Taylor's power law. Oikos 59, 121–5.Google Scholar

Copyright information

© Chapman & Hall 1994

Authors and Affiliations

  • S. J. Clark
    • 1
  • J. N. Perry
    • 2
  1. 1.IACR, Statistics DepartmentsRothamsted Experimental StationHarpendenUK
  2. 2.Entomology and Nematology DepartmentsRothamsted Experimental StationHarpendenUK

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