Small sample estimation for Taylor's power law

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 log10 s 2 on log10 m, where s 2 and m are the sample variance and mean, respectively. Bias in the parameter estimates, b and log10 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 small-sample 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 log10 s 2 and log10 m; (iv) restriction on the maximum variance expressible in a sample; (v) restriction on the minimum variance expressible in a sample; (vi) underestimation of log10 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 log10 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.