Small sample estimation for Taylor's power law
 S. J. Clark,
 J. N. Perry
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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 σ^{2} = cμ^{d}. 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 log_{10} s ^{2} on log_{10} m, where s ^{2} and m are the sample variance and mean, respectively. Bias in the parameter estimates, b and log_{10} a, 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 smallsample estimation are: (i) exclusion of samples for which m = s ^{2} = 0; (ii) exclusion of samples for which s ^{2} = 0, but m > 0; (iii) correlation between log_{10} s ^{2} and log_{10} m; (iv) restriction on the maximum variance expressible in a sample; (v) restriction on the minimum variance expressible in a sample; (vi) underestimation of log_{10} s ^{2} for skew distributions; and (vii) the limited set of possible values of m and s ^{2}. 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 log_{10} s ^{2} 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 s ^{2} 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.
 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.
 Binns, M.R. (1986) Behavioural dynamics and the negative binomial distribution. Oikos 47, 315–318.
 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.
 Blackshaw, R.P. and Perry, J.N. (1994) Predicting leatherjacket population frequencies in Northern Ireland. Annals of Applied Biology, (in press).
 Bliss, C.I. (1941) Statistical problems in estimating populations of Japanese beetle larvae. Journal of Economic Entomology 34, 221–32.
 Boag, B. and Topham, P.B. (1984) Aggregation of plant parasitic nematodes and Taylor's power law. Nematologica 30, 348–57.
 Clark, S.J. and Perry, J.N. (1989) Estimation of the negative binomial parameter k by maximum quasilikelihood. Biometrics 45, 309–16.
 Downing, J.A. (1986) Spatial heterogeneity: evolved behaviour or mathematical artefact? Nature 323, 255–7.
 Duchateau, L., Ross, G.J.S. and Perry, J.N. (1994) Parameter estimation and hypothesis testing for Adès distributions applied to tsetse fly data. Biométrie Praximétrie, (in press).
 Dye, C. (1983) Insect movement and fluctuations in insect population size. Antenna 7, 174–8.
 Dye, C. (1984) Reply to Dr Taylor. Antenna 8, 65–6.
 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.
 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.
 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.
 Hanski, I. (1987) Crosscorrelation in population dynamics and the slope of spatial variancemean regressions. Oikos 50, 148–51.
 Holgate, P. (1988) Approximate moments of the Adès distribution. Biometrical Journal 31, 875–83.
 Kemp, A.W. (1987) Families of discrete distributions satisfying Taylor's power law. Biometrics 43, 693–9.
 Kemp, A.W. (1988) Response to Perry and Taylor. Biometrics 44, 888–9.
 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.
 Marshall, E.J.P. (1985) Weed distributions associated with cereal field edges — some preliminary observations. Aspects of Applied Biology 9, 49–58.
 Marshall, E.J.P. (1989) Distribution patterns of plants associated with arable field edges. Journal of Applied Ecology 26, 247–257.
 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.
 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.
 Perry, J.N. (1981) Taylor's power law for dependence of variance on mean in animal populations. Applied Statistics 30, 254–63.
 Perry, J.N. (1984) Negative binomial model for mosquitos. Biometrics 40, 863–4.
 Perry, J.N. (1987) Iterative improvement of a power transformation to stabilise variance. Applied Statistics 36, 15–21.
 Perry, J.N. (1988) Some models for spatial variability of animal species. Oikos 51, 124–30.
 Perry, J.N. (1994a) Sampling and applied statistics for pests and diseases. Aspects of Applied Biology 37, 1–14.
 Perry, J.N. (1994b) Chaotic dynamics can generate Taylor's power law. Proceedings of the Royal Society of London Series B 257, 221–6.
 Perry, J.N. and Taylor, L.R. (1985) Adès: new ecological families of speciesspecific frequency distributions that describe repeated spatial samples with an intrinsic powerlaw variancemean property. Journal of Animal Ecology 54, 931–53.
 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.
 Perry, J.N. and Taylor, L.R. (1988) Families of distributions for repeated samples of animal counts. Biometrics 44, 881–90.
 Perry, J.N. and Woiwod, I.P. (1992) Fitting Taylor's power law. Oikos 65, 538–42.
 Ross, G.J.S. (1990) Incomplete variance functions. Journal of Applied Statistics 17, 3–8.
 Routledge, R.D. and Swartz, T.B. (1991) Taylor's power law reexamined. Oikos 60, 107–12.
 Southwood, T.R.E. (1966) Ecological Methods. Methuen, London.
 Southwood, T.R.E. (1984) Insects as models. Antenna 8, 3–14.
 Taylor, L.R. (1961) Aggregation, variation and the mean. Nature 189, 732–5.
 Taylor, L.R. (1965) A natural law for the spatial disposition of insects. Proceedings of the 12th International Congress of Entomology, London, 1964, pp. 3967.
 Taylor, L.R. (1970) Aggregation and the transformation of counts of Aphis fabae (Scop.) on beans. Annals of Applied Biology 65, 181–9.
 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.
 Taylor, L.R. (1984a) Assessing and interpreting the spatial distributions of insect populations. Annual Review of Entomology 29, 321–57.
 Taylor, L.R. (1984b) Anscombe's hypothesis and the changing distribution of insect populations. Antenna 8, 62–7.
 Taylor, L.R. (1986) Synoptic dynamics, migration and the Rothamsted Insect Survey. Journal of Animal Ecology 55, 1–38.
 Taylor, L. R. and Taylor, R.A.J. (1977) Aggregation, migration and population mechanics. Nature 265, 415–21.
 Taylor, L.R. and Woiwod, I.P. (1980) Temporal stability as a densitydependent species characteristic. Journal of Animal Ecology 49, 209–24.
 Taylor, L.R. and Woiwod, I.P. (1982) Comparative synoptic dynamics. I. Relationships between inter and intraspecific spatial and temporal variance/mean parameters. Journal of Animal Ecology 51, 879–906.
 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.
 Taylor, L.R., Woiwod, I.P. and Perry, J.N. (1979) The negative binomial as a dynamic ecological model for aggregation and the densitydependence of k. Journal of Animal Ecology 48, 289–304.
 Taylor, L.R., Woiwod, I.P. and Perry, J.N. (1980) Variance and the largescale spatial stability of aphids, moths and birds. Journal of Animal Ecology 49, 831–54.
 Taylor, L.R., Taylor, R.A.J., Woiwod, I.P. and Perry, J.N. (1983) Behavioral dynamics. Nature 303, 801–4.
 Taylor, L.R., Perry, I.N., Woiwod, I.P. and Taylor, R.A.J. (1988) Specificity of the spatial powerlaw exponent in ecology and agriculture. Nature 332, 721–2.
 Thórarinsson, K. (1986) Population density and movement: a critique of Δmodels. Oikos 46, 70–81.
 Woiwod, I.P. and Perry, J.N. (1989) Data reduction and analysis. Boletin de Sanidad Vegetal No. 17, Proceedings of PARASITIS 88, pp. 15974.
 Yamamura, K. (1990) Sampling scale dependence of Taylor's power law. Oikos 59, 121–5.
 Title
 Small sample estimation for Taylor's power law
 Journal

Environmental and Ecological Statistics
Volume 1, Issue 4 , pp 287302
 Cover Date
 19941201
 DOI
 10.1007/BF00469426
 Print ISSN
 13528505
 Online ISSN
 15733009
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 bias
 negative binomial distribution
 parameter estimation
 response surface
 sample scheme
 simulation
 small samples
 Taylor's power law
 variancemean relationship
 Authors

 S. J. Clark ^{(1)}
 J. N. Perry ^{(2)}
 Author Affiliations

 1. IACR, Statistics Departments, Rothamsted Experimental Station, AL5 2JQ, Harpenden, Herts, UK
 2. Entomology and Nematology Departments, Rothamsted Experimental Station, AL5 2JQ, Harpenden, Herts, UK