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1 The Negative Binomial as a Dynamic Ecological Model for Aggregation, and the Density Dependence of k Author(s): L. R. Taylor, I. P. Woiwod and J. N. Perry Source: Journal of Animal Ecology, Vol. 48, No. 1 (Feb., 1979), pp Published by: British Ecological Society Stable URL: Accessed: 08/11/ :03 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. British Ecological Society is collaborating with JSTOR to digitize, preserve and extend access to Journal of Animal Ecology.

2 Journal of Animal Ecology (1979), 48, THE NEGATIVE BINOMIAL AS A DYNAMIC ECOLOGICAL MODEL FOR AGGREGATION, AND THE DENSITY DEPENDENCE OF k BY L. R. TAYLOR,* I. P. WOIWOD* AND J. N. PERRYt Departments of Entomology* and Statistics,t Rothamsted Experimental Station, Harpenden, Hertfordshire SUMMARY (1) The negative binomial model for aggregation was found to have severe ecological, as distinct from statistical, limitations. (2) The behaviour of the index 1/k is complex in theory and an inconsistent measure of aggregation in practice, whether the negative binomial fits the data or not. (3) It lacks a common value (l/kq) for all densities in most species. (4) It can yield positive or negative linear regressions on density over short ranges of the mean, but is nonlinear over longer ranges and implies randomness where none exists. (5) It is not related to mean density by any simple functional form. (6) It is double-valued in all those species for which the power relating variance to the mean is between 1 and 2. This includes most of the material in the literature. (7) It has no independent behavioural criterion for aggregation other than variance itself. (8) The negative binomial cannot model regularity. (9) The moment estimate of 1/k has high variance and behaves too erratically at low densities for the underlying form to be detected easily. INTRODUCTION The word 'model' means different things in different disciplines. Skellam (1973) distinguished 'the scientists model, in all its conceptual richness' from 'those highly abstract systems which form part of its logical structure.' These 'abstract systems' are often probabilistic or purely numerical models such as statistical distributions. We do not wish to labour the familiar risk of creating ecological concepts from the statistical models that chance to describe sample counts from populations. But there is another risk that may be less obvious, especially to the mathematical ecologist. It comes from using statistics as parameters in population dynamics without due regard for the different meanings attached to a 'population model' in statistics and ecology. We have recently examined the fit of the negative binomial model having a 'common k' for all densities (Taylor, Woiwod & Perry 1978) and we now extend this study to the behaviour of the model when k is not constrained to be a constant. The negative binomial is widely used as a statistical model for the instantaneous distributions describing ad hoc ecological samples. The many combinations of sub-models, of individuals within groups and groups within larger units, that lead to this distribution (Boswell& Patil 1970; Patil & Stiteler 1974) mean that it is versatile yet without too deep /79/ $0.200?1979 Blackwell Scientific Publications

3 290 The negative binomial a causative commitment. Its statistical value then lies in its flexibility and the simplicity of its parameters for the comparison of instantaneous data sets, for example from simultaneous plot samples in which efforts have been made to eliminate dynamic changes other than those caused by the experimental treatments. The development of other models, such as the Thomas, Neyman, Polya-Aeppli and Poisson-log-normal demonstrates that the negative binomial is not a universal fit to all non-random distributions but it often fits well enough for practical purposes. It then fulfills its function within the usual statistical constraint that the background abstract population is not extended beyond the bounds of the actual data. However, statistical concepts in ecology are often stretched far beyond the bounds of this convention and we have elsewhere (Taylor 1978; Kempton & Taylor 1979) indicated the danger of making quasi-theoretical justifications for using conventional statistics as ecological parameters when their behaviour has not been thoroughly investigated on real data for the special ecological purpose in hand. For example, we have shown that in certain circumstances the Poisson-log-normal can fit very well to species-abundance data but that the distribution's flexibility allows its parameters to over-respond to slight deviations in the data and so behave erratically. The log-series, though less flexible and fitting less well nevertheless yields more consistent, and therefore ecologically more useful parameter values (Kempton & Taylor 1974; Taylor, Kempton & Woiwod 1976). Models for the spatial distributions of single species in population dynamics are descriptive of samples taken from real populations of living, moving and reproducing organisms. In consequence we can neither expect a subsequent set of samples, however carefully collected at exactly the same place, to yield a replicate distribution nor even one with the same mean and variance. Indeed, this is improbable since one of the objectives of studies in population dynamics is to investigate individual behaviour and population processes, some of which will have taken effect in the meantime. Also, all environments change and species' behaviour is unique, and this is reflected in the variability of the samples by which they are investigated. For example, to describe even such a seemingly simple function as the density-distance distribution may require as many as four parameters although in a given instance, or for a given species, only one may be in use at a time (R. A. J. Taylor 1978) and this has given a false impression of the adequacy of a small number of parameters now incorporated into the literature on population 'diffusion' or migration (Taylor & Taylor 1977, 1979). In other words, in contrast to the statistical disadvantages of having too many parameters there can be an ecological hazard in having too few. The usefulness of a sampling statistic as an ecological population parameter depends on its behaving in a known and appropriate way with respect to changes in the 'dynamic population density surface' (Taylor 1965; Taylor & Taylor 1977). To be useful, the model it represents must be capable of extrapolation beyond the bounds of the data yielded by sampling in order that the model should cover the potential range of densities for the species in question. This extrapolation, which is a real part of the conceptual ecological model, will differ between species (Taylor 1971). It is not always necessary to know the detailed causative mechanism of the model, but the behaviour of the parameter must be predictable if its subsequent use in population dynamics theory is not to mislead. Thus the grounds for selecting a dynamic distribution model may be more demanding than, and are certainly different from, those for a conventional, instantaneous distribution model; it performs a different function.

4 L. R. TAYLOR et al. 291 THEORY In the following study of the behaviour of k, both theoretically and with real data, the sampling method and the sampling area is assumed to be identical at each sampled density for any given species. The ecological model being investigated consists of a dynamic surface representing real individuals over a given area. The density at a point on this conceptual surface is denoted by p, with mean p per unit area estimated by m. The variance of observed counts of real individuals, estimated by s2, consists of a systematic component a2, which is dependent upon some function of density, and a random component attributable to unknown causes such as sampling error. The negative binomial distribution has two defining parameters J(=_p), the mean, and the supposed aggregation parameter k, and these are related to variance V(= U2) by V-= + /2/k. (1) For the measured statistic, k, to be of use dynamically it must behave in a known way with respect to p. It cannot be claimed, a priori to be an 'index of aggregation'. It must be shown to behave in a biologically reasonable manner with respect to change in population density and should not, for example, be assumed to be linearly related to population density to obtain mathematical tractability. The problem is aggravated by the lack of an agreed behavioural definition for 'aggregation' (Taylor, Woiwod & Perry 1978); nevertheless several attempts have been made in the past to justify the use of 1/k. The statistical use of the negative binomial was already well established (Greenwood & Yule 1920; Beall 1942; Anscombe 1948, 1949, 1950; Bliss & Fisher 1953; Bliss & Owen 1958) before the ecological use of k was first proposed by Waters (1959) and this has not been restated more succinctly since. 'Aggregation' was assessed using as the criterion the variance of density,2(= V) estimated by s2; k was shown to be inversely correlated with s2, so was deemed to measure aggregation. Since 1/k is, in effect, V somewhat modified by the mean u, this would be expected, and, the status of k depends on the status of s2 as a measure of aggregation. We have discussed this elsewhere (Taylor, Woiwod & Perry 1978). Also, since the data were for four species of woodland insects only, with very little replication at the same or at other population densities, deficiencies in l/k were not likely to be exposed. Again, if 1/k depends on (--= p), it also depends on the relationship between,2 and p which had not then been discovered (Taylor 1961). Subsequent development of the Waters approach (Kuno 1968; Morisita 1971; Iwao 1977; May 1978) has not appreciably changed the original logic which still fails to incorporate the now established variance-mean relationship (Taylor, Woiwod & Perry 1978) and has added no further observational evidence for the ecological validation of the parameter k. To be effective, the parameter for aggregation should fit the observed behaviour over a range of taxa, population densities and scales of space and time, otherwise subsequent mathematical manipulation of equations using the parameter may obscure the initial premises and extrapolate arguments into unproven or forbidden regions. For example, we have recently shown (Taylor, Woiwod & Perry 1978) that Iwao's (1968) regression for Monte Lloyd's (1967) mean crowding parameter m on mean density, which is identical with Bartlett's earlier (1936) quadratic equation V = (a + 1)u/ + (b - 1)/2 (2) but which omits Lloyd's behavioural constraints, implies negative variances when fitted to

5 292 The negative binomial some real data. Equation (2) includes the negative binomial with common k (kc) as a special case, V= It + bu2; b= /k,; (3) and in that analysis (Taylor, Woiwod & Perry, 1978) of extensive data, we confirmed Bliss & Owen's (1958) finding that k is not constant for all species in all parts of their density range; in other words there is no general ecological property equivalent to k,. There remains the possibility that, whilst k is not constant with respect to p, it bears some other simple predetermined relationship that would permit its use as an index of aggregation. It may be noted that, unless this is a simple functional form, the index is useless for extrapolation because the whole density range must be investigated for each species separately before the parameter can be used, which would be an impossible task. In the same analysis, Taylor, Woiwod & Perry (1978) confirmed that Taylor's (1961) power function, V = aplb, (4) where V and,u are estimated by s2 and m respectively, fitted the majority of the data available from the literature better than the Bartlett-Iwao quadratic eqn (2) and the kc negative binomial (3). As a result an expected value for variance at any density, for any species, can simply be obtained from the linear form of eqn (4). We shall examine the expected behaviour of the parameter 1/k from eqns (1) and (4) to see if it varies with p in a useful, predictable manner, and then consider it in relation to the moment estimate of 1/k from real data. We therefore have four relevant issues to consider. (1) Unless the negative binomial distribution provides an adequate fit at all densities for the species concerned, and so has some ecological generality, there seems little reason for using k. We do not propose to examine this issue in detail because the parameter must be robust for it to be of any value at all, and we give it the benefit of this assumption. However, many authors have reported the fit to be inadequate in certain cases; Anscombe (1949, 1950); Bliss & Fisher (1953); McGuire, Brindley & Bancroft (1957) and others. Also, there is no published information, so far as we are aware, dealing with the robustness of k to departures in data from the stated distribution. (2) If the relationship of l/k (or k) with p has different forms which cannot easily be represented by a simple function, and especially if it may take double values, it should be discarded. (3) Although Waters (1959) correctly stated that for a random distribution l/k -> 0, it was untrue to claim, vice versa, that l/k - 0 implies a random distribution (see below). (4) Although regular spatial distributions are only rarely found, their occurrence cannot be accounted for by the negative binomial since k is constrained to be positive. We may only overcome this problem by defining k as in eqn (1) and allowing it to become negative, without specifying the distribution of individuals (see also the first issue above). We now investigate theoretically what forms 1/k may take in relation to /t. Since l/k should take the form i.e. with slope given by V = atub, (from 4) alb = M + u2/k, (from 1) 1/k = atb - -, (5) d(l/k) = a(b - d/-( 2)b (6)

6 At very low densities, which are ecologically important in defining limits to real populations, regularity (i.e. 1/k < 0; V < u) may be impossible to detect, even if present, due to the counts approaching one individual in the complete sample of N units, i.e. when 1/k is also zero when s = m = 1/N, and lk = 0. (from 5) r 1]1(b- 1) =a Equations (5) and (6) give the following conditions: (i) If 3 < b then 1/k - oo as,t -- oo with ever-increasing positive slope. (ii) If 2 < b < 3 then 1/k -> oo as u -, oo with ever-decreasing positive slope. (iii) If b = 2 then 1/k -> a as / -* oo with slope tending to zero. (iv) If 1 < b < 2 then 1/k -> 0 as - oo and there is a turning point. The slope i.e. the turning point occurs when L. R. TAYLOR et al. 293 d( 1k) = 0 when a(2 - b)ub-a = 1/u, dpt = [l/(b- b)1 / = a(2 - b) n II 0,0 log /p FIG. 1. Four expected types of response curve for 1/k with respect to log mean density (log,) given that V = alb. Type I curve is for 1 < b < 2; type II for b -- 2; type III for 2 < b < 3; type IV for b > 3. The curves plotted are for the values a = 1.5; b = 3.1 (IV), b = 2.1 (III),b = 2 (II) and b = 1.1 (I). The turning point is at p = [1/(a(2 - b))]l(b-), the intercepts on the abcissa at,t = (l/a)ll(b-l) and all curves pass through the point log u = 0, 1/k = a - 1.

7 294 The negative binomial Interpretation is further confused since the forms of 1/k for 1 < b < 2, 2 < b < 3, and b > 3 are very different (Fig. 1), and clearly could not be fitted with a functional form more simple than eqn (4). Additionally for the range 1 < b < 2, which includes most of the data in the literature (Taylor, Woiwod & Perry 1978), 1/k has identical values for two different values of ut, i.e. is double-valued, although this seems an improbable ecological property for aggregation. Furthermore, as shown by Figs 2, 3, 8 and 9, although 1/k decreases and tends to zero, variance is still increasing with respect to mean density, i.e. the counts are not tending towards randomness as Waters wrongly inferred from a decrease in 1/k. Previous authors such as Bliss & Owen (1958) have sometimes been able to accept a constant for k, or more commonly a simple linear regression of 1/k on m, because their range of data has been insufficient to establish deviations from these models (Figs 2 and 9). This effect is even more marked due to the large variance of the moment estimate of I /k which is particularly evident when 1/k is fairly flat (Fig. 7). We have used the moment estimate of 1/k, (s2 - m)/m2, in our figures because, unless the complete data sets are published in the literature we cannot use the other common methods. We have no reason to expect the results to be appreciably affected by this. Anscombe (1950) gave the large sample efficiency of this method, and the 'method of zeros' as non-linear contours dependent on k and u. A maximum likelihood solution for k was proposed by Bliss & Fisher (1953) and efficient routines now exist for its calculation ( I, ^ o F *? (b00-10 c? 0.4 _(b) I O I I I - I 0*1 I Mean density (log scale) FIG. 2. Data for virus lesions (a) log variance x log mean regression. (b) linear negative regression (-) fitted to 1/k by Bliss & Owen (1958) and expected (-) functional response curve, type I, from regression in (a) projected over a longer range of mean density.

8 1-0 L. R. TAYLOR et al. 295 (a) 0 cr 10 - c.o a r t.1 0/ 0.l/ (b) II Y- C D I 0.l 10 Mean density (log scale) FIG. 3. The most detailed set of data in the literature, for wireworms in pasture. (a) log S2 x log m (b) double-valued (type I) 1/k response curve with sudden increase in 1 k at low densities and subsequent decline. TABLE 1. Parameter values of log s2 x log m regressions and other details of examples used in the figures Figure Organism Source of data 2 Virus lesions 3 Agriotes sp. in grass 4 Euttetix tenellus (Baker) 5 Hylaea fasciaria (L.) 6 Lycophotia porphyrea (D. & S.) 7 Mirid damage to cocoa 8 Earias clorana (L.) 9 Limonius sp. Kleckowski (1949) 1.32 Yates & Finney (1942) 1.18 Bowen (1947) 1.71 Rothamsted Insect 2.95 Survey (unpublished) Rothamsted Insect 3.04 Survey (unpublished) Johnson in Taylor 1.99 (1971) Rothamsted Insect 1-74 Survey (unpublished) Jones (1937) 1.35 Power law Fig. 1 b logio a Curve type 0.16* 0.13* -0.29* 0.43t 0.12t 0-16* 1.44t 0.06* * From Taylor, Woiwod & Perry (1978) Appendix B. t Light trap samples throughout Great Britain. Each point representing one year. I I I-II III IV II I I

9 296 The negative binomial (Ross 1970) for sufficiently large sample sizes. The variance of the moment estimate of 1/k for small samples is investigated in the Appendix explaining the large scatter of this estimate especially at low means as shown empirically in the figures. PRACTICE In Figs 2, 3, 4, 5, 6, 7, 8, 9 sets of data illustrate the points made earlier. In each figure, the first part (a) shows a fitted log variance x log mean regression (Table 1) (Taylor, Woiwod & Perry 1978). The curve in the second part of each figure (b) is this regression transcribed to 1/k x log m as expected from the theoretical conclusions reached earlier, i.e. is an expected functional relation. The data points, 1/k, are moment estimates derived directly from the data and so are independent of any prior assumption associated with the expected curves. Nevertheless it is apparent that the curves describe the data points as well as such large variances permit. The figures are largely self-explanatory. They illustrate the different functional forms of 1/k with log m at different ranges of b in the power function V = al,b (see page 293). They show how negative (Figs 2 and 9), neutral (kc) (Figs 4 and 7), and positive (Figs 5 and 6) linear regressions of 1/k on log m can arise spuriously, due to inadequate data (a) a / c / ) / 10 / 0I- 0-(b) 0- Y * -0'1- / Mean density (log scale) FIG. 4. Leafhopper Eutettix tenellus (a) log s2 x log m (b) Type I - II response of 1/k x log m in which the data range covers the regular curve up to the slight maximum at m =

10 L. R. TAYLOR et al. 297 (a) a) / o ~I L I 2(b) / 10-0/ 0o- -10 I Mean density (log scale) FIG. 5. Geometrid moth Hylaeafasciaria (a) log s2 x log m (b) Type III response of l/k with positive linear regression over a short range becoming sigmoid when projected over longer range. ranges for mean density. They show how complex is the expected functional form of 1/k on m, having double values when the data range is long enough (Figs 3 and 4), and how poor is its definition because of its high variance at all densities (Fig. 7; 1/ = , 950o limits) even with good data and more so at low mean values. They show the increase in residuals on the 1/k plots over residuals from corresponding points on the variance-mean plots for these low densities. But especially they show how erratic 1/k is with real data, quite independently of any alternative model (Fig. 8). DISCUSSION Many theoretical models involving 'clumps' have been proposed for the ecological study of aggregation; these involve the compounding of distributions within and between clumps. These clump models often lead to the same theoretical distribution of individuals per unit area but unless the clumps are discrete the resulting spatial distribution cannot be interpreted (Skellam 1952; Anscombe 1950; Waters & Henson 1959). As Skellam pointed out, 'the limitations are inherent in the method itself.' Pielou (1964, 1977) notes that even for plants the clumps are not discrete and animals are more active and their aggregations more dynamic and ephemeral. Since behaviour is generally densitydependent, we have no a priori reason to assume that a particular distribution of clumps or of individuals will be the same at different population densities and we have evidence

11 298 The negative binomial (a) / eo n / o ) * 10- / (b) Mean density (log scale) FIG. 6. Noctuid moth Lycophotia porphyrea (a) log s2 x log m (b) Type IV response of l/k with an ever increasing slope. that, although the same mean density has the same variance for a particular species, it does not necessarily have the same geographical distribution (Taylor 1977). The highdensity concentrations that are treated as clumps are peaks in a dynamic density surface that must be a continuum if there is movement, and movement is essential for survival (Taylor & Taylor 1979). Clump models are artifacts and they often lead to quadratic relationships between variance and mean (Anscombe 1950; Pielou 1964) even when the negative binomial is not specifically fitted, and this makes them unsuitable models for these monotonic processes (Taylor, Woiwod & Perry 1978). Pielou (1977) also emphasizes the constraint that ecological clump models for aggregation should be restricted to organisms where movement from one unit to another is negligible, although this is sometimes overlooked. Another justification given for using the negative binomial model is that random deaths will not change the degree of aggregation. The implication here is that an index such as 1/k is a reasonable measure of aggregation if it is not affected by random deaths; but this requires mortality to be densityindependent, another severely limiting constraint. In view of the limitation to discrete clumps and the constraint on movement, there are few organisms to which all these quasi-theoretical ecological arguments can apply, as Anscombe (1950) noted. Thus generally, models of aggregation for animals cannot satisfy the necessary stringent

12 L. R. TAYLOR et al. 299 (a), iooo- I0oo / 40 (b) * I O Mean density (log scale) FIG. 7. Mirid damage to cocoa (a) log s2 x log m (b) Expected response curve type II with almost constant 1/k but very large scatter, mean 1/I = (95% limits). ecological requirements whilst retaining the apparent simplicity of a frequency distribution like the negative binomial, or the Bartlett-Iwao equation. In this analysis we have not questioned the fit of the negative binomial to the data. We have assumed that it fits because we wish to investigate the proposition that the parameter 1/k is a general measure for aggregation and this proposition implies that the negative binomial distribution is a generally applicable model. The same assumption is also made implicitly when any theoretical treatment uses k as a parameter for aggregation in a dynamic model, or uses the negative binomial as part of a general argument. In the present instance, the assumption of a fit has some justification since many data-sets, although not all, do fit the negative binomial well (Bliss 1971; Iwao 1977). We are more concerned with k and we find beyond reasonable doubt, that 1/k behaves in practice as expected from Taylor's variance-mean model, the evidence for which is now very strong. The data used for illustration are typical of those in the literature. They show the expected strong curvature of 1 /k with respect to m, or log m and have turning points such that 1/k may have the same value at two different densities, a serious defect in a parameter which purports to define aggregation. The figures also show what was initially less obvious, that the large variance in the

13 300 The negative binomial 10- (a).2/ _I v 5 / 0,0011- L I (b) ' 60-? 40-20?f0 ~~O *J ~I I ~ I I Mean density (log scale) FIG. 8. Noctuid moth Earias clorana (a) log s2 x log m (b) Type I response of 1/k typical of much data in the literature where the curvature and double value of 1/k is hidden by the high variance. moment estimate of 1/k particularly at low densities makes it an unstable parameter, and this conforms with theoretical expectation (see Appendix). Although this 'noise' in the system may be partly attributed to the lack of fit of the negative binomial model, most of it is inherent in the estimate of the parameter and cannot be removed. It is one of the reasons why non-linearity in 1/k with respect to m has not previously been detected. All populations encompass lower densities if only at their edges and, although this may present problems of estimation (Taylor, Woiwod & Perry 1978 and Appendix), such problems are ecologically unavoidable and therefore cannot be ignored. This series of investigations into the spatial dynamics of single- and multi-species populations began in 1960 (Taylor 1961) with a search for dynamic parameters that would serve the ecologists' purpose, paying due regard to the behaviour of living organisms whilst attaining the rigour demanded statistically. The negative binomial is a particularly clear example of the difference in requirements of experimental statistics and population dynamics. In all, we find k an unstable parameter whose relationship with aggregation is doubtful and which, since it is non-linearly related to u depends ultimately on Vfor any aggregation

14 100 - (a) L. R. TAYLOR et al _ :./ (b) o'l 1.0 si (o scale Mean density (log scale) FIG. 9. Wireworm Limonius sp. (a) log s2 x log m (b) Linear regression (-) fitted to 1/k by Bliss & Owen (1958) and expected (-) functional response curve, type I, projected over longer range. Note increased residual of the first point compared with (a) giving erroneous impression of linearity. property it may have. Unfortunately it also imposes an unnecessary distribution on the data. We conclude that the study of aggregation should, at this stage, be based solidly on real data unprejudiced by preconceptions from theoretical models having little correspondence with reality. REFERENCES Anscombe, F. J. (1948). The transformation of Poisson, binomial and negative-binomial data. Biometrika 35, Anscombe, F. J. (1949). The statistical analysis of insect counts based on the negative binomial distribution. Biometrics 5, Anscombe, F. J. (1950). Sampling theory of the negative binomial and logarithmic series distributions. Biometrika 37, Bartlett, M. S. (1936). Some notes on insecticide tests in the laboratory and in the field. Journal of the Royal Statistical Society, Supplement 3, Beall, Geoffrey (1942). The transformation of data from entomological field experiments so the analysis of variance becomes applicable. Biometrika 32, Bliss, C. I. (1971). The aggregation of species within spatial units. Statistical Ecology 1, (Ed. by G. P. Patil, E. C. Pielou & W. E. Waters), Pennsylvania State University Press, University Park. Bliss, C. I. & Fisher, R. A. (1953). Fitting the negative binomial distribution to biological data and note on the efficient fitting of the negative binomial. Biometrics 9,

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16 L. R. TAYLOR et al. 303 Waters, W. E. & Henson, W. R. (1959). Some sampling attributes of the negative binomial distribution with special reference to forest insects. Forest Science 5, Yates, F. & Finney, D. J. (1942). Statistical problems in field sampling for wireworms. Annals of Applied Biology 29, (Received 30 June 1978) APPENDIX We may calculate the small-sample variance of the moment estimate in terms of the higher moments of the distribution of counts (i.e. without necessarily assuming the negative binomial); a m2 J [E(m2)]2 [E(m2)]3 [E(m2)]4 where Var( ) denotes variance, E( ) denotes expected value and Cov( ) denotes covariance; by the A-method of Kendall & Stuart (1976). It can be shown after some algebra that; if the moments are represented by (7) and Lk = E[[X - E(X)]k], (k > 1), 1 = E(X), where X is the random variable denoting the counts, and sample size is N, then eqn (7) reduces to V S2-4- m\ ( N m2 (2 + N/2)2 2N,2 - D1P)[L4 + 2(N - 1)L N1,2 - (t2 + Np2)3 241] N(t-2-1x)2[tP4 + 4(N3-3N + 3)tlz/L3-4(4N2-12N + 9)L/2 + (3N - 4)1z + 4(N - 3)(N- 1)-4] 2 + N12)4 (8) We conclude from this that for large values of /, the variance should stabilize to a constant, although we cannot say how large this constant will be. For low means, however, the distribution of counts is often close to a Poisson for which Then t3 = t2 = t = A, say, and 4 =- 3A2 + A. (9) from (8) (2) (N- 1)(+ +A)

17 304 The negative binomial Thus the variance of 1/k will increase rapidly as the mean decreases and when only one individual is caught in the sample. - and Var S- m N In contrast eqn (4), fitted on logarithmic scales, has the variance of its y- variate for the Poisson case given by and if A = 1/N 2 1 Var(log s2)(- 1)+ N (10) Var(log s2) - (N + 1. Hence (10) unlike (9) is bounded above. The larger variance of the moment estimate of 1/k, predicted theoretically above for low means, can be seen empirically from the figures, especially by noting the large residuals on the. 1/k plots compared with the residuals from corresponding points on the variance-mean plots for these low densities.