statistical ecology. ii—a reassessment

STATISTICAL ECOLOGY. II.--A REASSESSMENT

ERIC ASHBY Botanical Department, The University, Manchester

INTRODUCTION

Eleven years ago the present writer published in this journal an, article on statistical ecology (1). The purpose of the present article is to review significant contributions to the subject since that time. This might be done in either of two ways : by making an an- notated survey of the literature on statistical ecology published since 1936, or by re-examining, in the light of present knowledge, the value of statistical methods in the study of plant communities. In this article the second of these two ways has been chosen.

The first aim of the ecologist, as of any other scientist, is to describe the objects of his study quantitatively and objectively. A further aim is to include as many of the objects as possible in generalisations, to deduce the "type" from individual examples of the "type". In pursuance of these aims, some ecologists have de- vised statistical methods to describe communities, and they have tried to use these methods as a basis for classifying communities as associations. Eleven years ago it seemed reasonable to conclude that these statistical methods could .not be used as a basis for classification, although they could be applied profitably in other ways. Is this conclusion still reasonable ? Is the classification of a moorland, like the classification of a flowering plant, still a matter for subjective judgment? And has statistical analysis disclosed important facts about plant distribution which could not have been disclosed otherwise ? The present article discusses, even if it does not clearly answer, these questions.

NATURE OF THE DATA IN STATISTICAL ECOLOGY

As soon as one proceeds from the mere inspection of a plant community to its analysis, it is necessary to focus attention upon one species at a time. The distribution of each species can be summarized by four characteristics:

(a). Fidelity: occurrence or non-occurrence of the species in the community;

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STATISTICAL ECOLOGY 223

(b). Abundance: the number of individuals of the species per unit area ;

(c). Cover: the proportion of area covered by the species ; (d). Frequency: the chance of finding the species and its mode

of dispersion. Each of these four characteristics can be measured objectively

and expressed quantitatively, and qualitative studies of them are, or ought to be, as out-of-date as qualitative studies of respiration. There is adequate proof that the older subjective methods are unreliable. West (27) has shown that estimates by different ob- servers of percentage cover in pastures are subject to wide dis- crepancies. Hope-Simpson (12) has shown that the familiar estimates of abundance ("dominant"; "abundant", "frequent", "oc- casional", "rare", etc.) are very unreliable, owing to heterogeneity among areas chosen as representative of the community, and to changes in the appearance of the community from season to season. Smith (25) has shown that a party of eight men, all familiar with the concept of density, and all with practical experience in the field, vary from 71.24~ to 139.81~ of the group average in their estimates of density of the same species over the same area. Over 40% of their estimates departed from the mean by more than 10%. Even after a week of intensive training, "high" estimators were finding range plants to have twice the density found by "low" estimators. There is no doubt, therefore, that statistical methods are required to give precise information about fidelity, abundance, cover and frequency.

ATTEMPTS TO ANALYSE PLANT COMMUNITIES OBJECTIVELY

Quantitative methods have not been of much help in the central problem of plant sociology: the extrapolation from the concrete community to the abstract association. When the ecologist stops his car and decides he has reached a "suitable place" for throwing quadrats on a community, he has already performed the major act of classification and he has performed it subjectively. Any subse- quent quantitative analysis only elaborates, and possibly obscures, the original subjective decision. It is for this simple reason that the work of Scandinavian ecologists 25 years ago failed to reveal any- thing new about the status of associations: they had already settled the status before they laid a quadrat. It is still necessary, how-

224 THE BOTANICAL REVIEW

ever, to sample communities in order to secure precise information about their composition; and therefore it is desirable to know what constitutes an adequate sample of a community.

In recent papers Cain (6, 7) has attempted to redefine an adequate sample plot, especially for alpine vegetation in Wyoming (7). He points out that the location of a sample plot is "partially subjective . . . it should be placed well within the community in a position that seems to be as nearly typical or average as can be judged". And the size of the sample plot "must be at least as, large as the minimal area of the community type".

The minimal area, which Cain defines as "the least area upon: which a community can develop its typical composition and structure", has played a notorious part in statistical ecology. Its attrac- tion lies in the almost horizontal flattening of the species-area curve at a surprisingly low area: e.g., 50 square inches in an English pasture; 40 square metres in an Australian desert; 32 square metres in an American alpine fell. The assumption commonly made is that the number of species is related to the area sampled by a curve of the form

n ~ f n m ( 1 - e -h)

where na = the number of species on an area a, and nm= the maximum number of species in the community. Cain draws attention to the fact that the point of apparent flattening of the curve, which is the point selected as the minimal area, depends on the scales used for ordinates and abscissae in the graph; he proposes to overcome this source of error by selecting for the minimal area that point on the abscissa where "the increment in number of species added per unit: increment of area is equal to the maximum number of species found, divided by the total area sampled". This procedure adds to the complexity of the analysis without increasing its precision, for it assumes a knowledge of the shape of the extrapolated species-area curve, and there is no justification for such an assumption. The differential coefficient of the species-area curve, one value of which Cain proposes as a measure for minimal area, is dependent on the asymptotic value of the curve, which can only be guessed. Cain's method for selecting minimal area is not, therefore, really objective. It seems fair to conclude that Cain's studies confirm those of earlier workers in demonstrating the obstacles which lie in the way of using quantitative methods as a basis for the description of plant corn-


munities. Even the minimal area should still be selected subjectively on the basis of the observer's field experience.

Cain's attempt to define an adequate sample area brings into relief the limitations of the statistical method in plant sociology. The first limitation is that samples must be taken at random (even the method of composing larger quadrats from smaller ones, which Cain uses, is not free from objection). The second limitation is that the objects sampled must be approximately randomly distributed themselves, or in randomly distributed groups. In the study of plant communities the ecologist is obliged to transgress both these limitations. Even the micro-climate and soil of a plant community are not (except by a coincidence) randomly distributed. Having regard to thepresent state of our technique it is safe to predict that the classification of communities, like the dassification of species, will be based on subjective criteria for a long time to come.

ANALYSIS OF THE DISTRIBUTION OF INDIVIDUAL SPECIES

Although attempts to classify communities objectively are dis- appointing, attempts to analyse the distribution of individual species are promising. During the last 11 years methods of analysis have been improved and good progress has been made toward putting this branch of plant sociology on a firm basis.

Advances in technique. The situation in 1936 was that estimations of the frequency of a species, although the most convenient way to sample its distribution, were unreliable unless the species was randomly distributed; and it had been shown that many species are in fact not randomly distributed. However, Blackman (5) has shown for some Swiss and English pastures that even where there is a notable departure from randomness the percentage frequency may provide quite a good measure of density. By using small enough quadrats Blackman obtained linear regressions between the logarithm of percentage absence for a species and its density. The slopes of the regressions did not as a rule coincide with the slopes calculated on the assumption of random distribution: there were as a rule too many empty quadrats at higher densities. This indicates, of course, that individuals are over-dispersed (contagiously distributed, as P61ya (20) has called over-dispersion). Despite this over-dispersion the l~ercentage absence, suitably cor-


rected, can be used as a measure of density. Blackman has ac- cordingly put into the hands of ecologists a useful tool for following the changes in composition of small homogeneous plots. Blackman worked on areas of a few hundred square yards, and unpublished work indicates that the linear regressions he obtained can not be obtained when data are collected from larger areas. He found two kinds of divergence from Poisson distributions among the species he studied: (i) skewness without evidence of non-random dispersion: the Chi square values being high, but the relative variance re- maining about unity ; and (ii) some skewness accompanied by non- random distribution: the relative variance being significantly higher than unity, even though Chi square values are not very high. The first kind of divergence may indicate that individuals are randomly distributed within groups which are not randomly distributed; the second kind of divergence may indicate that individual plants are over-dispersed, though they may be in groups which are randomly distributed. Further studies are needed of these two kinds of divergence.

Clearly the study of over-dispersion is of major importance in statistical ecology, for recent work (8, 19, 22, 23) indicates that it is the rule rather than the exception in the distribution of plants, even over small areas. Recently a notable contribution to this study has been made by Cole (9). He points out that contagious distribution, i.e., over-dispersion, is the rule rather than the exception among animals, for individuals are commonly associated in groups: in nests, or in holes, or as mating couples. It may be as- sumed, however, that over restricted areas the groups are randomly distributed. Making this assumption, Cole shows how a population of individuals may be analysed into n s groups, consisting of nl single individuals, n2 pairs, n3 groups of three, and so on. Then:

ng --- nl + n2 + n3 + . . .

If the mean number of groups per sample is nag, this is the sum of the mean number of single individuals (ml), plus themean number of pairs (m2), plus the mean number of groups of three (ms), etc.: and:

ng m s = m x + m 2 + m s + . . . = ~

where N = the total number of samples collected. If each group is randomly distributed, then the distribution of each group will fol-


low its own Poisson series. The chance (No) of finding a sample without any groups at all in it will be the product of the zero terms ef the several Poisson series, i .e . :

N0=N (e-re" e-~" e-re ' . . . ) =N e -m' From this foundation it is easy to express the chances of finding:

1 individual in a sample: N1 = N (role -m'' e -~' ' e -~ ' . . . ) = m l N o

2 individuals" " " : N2 = No (m2 + mJ /2 !) 3 . . . . . . . . : Ns= No (ms+m~m2+m~3/3!)

and so on. Cole applied this technique to the analysis of various over-

dispersed populations: centipedes, beetles, yeast, isopods and fleas; and he obtained very much closer fits than are given on the assumption of simple random distribution. His analysis disclosed, for instance, that only about 87~5 of the centipedes he sampled were present as individuals, and about 6.3~5 were present in pairs. Cole's method may not be easily applicable to plant populations where aggregation is due often to vegetative propagation, which gives many individuals in each group, and where the data are more likely to fit binomial than Poisson distributions. But as a first attempt to separate contagious distributions of organisms into com- ponents, Cole's work is important for plant ecologists, and it ~teserves serious attention.

An entirely different approach to the quantitative description of the distribution of species in a community has been made by Ashby and Pidgeon (2). By plotting lines representing equal fre- quencies of a particular species in a community, called " i s o n o m e s " ,

these authors have shown that the distribution of a species may be resolved into centres of high frequency surrounded by a pattern of contours of lower frequency (Fig. 1). The technique has proved useful in reconstructing the history of eolonisation of species in sandstone scrub in eastern Australia.

A few other recent contributions to the technique of statistical .ecology are summarized in the following paragraphs.

Fisher (11) recently published a theoretical paper on the relation between the number of species and the number of individuals ~btained in samples of organisms. The analysis has been applied successfully to the number of species in the samples of insects


caught in a light-trap at Rothamsted (11), and it could be applied to the analysis of species-area curves in plant sociology.

Ramensky (21) published in 1938 an important monograph on

FIG. 1. Distribution of Baeckea diomnifolia. The numbers on the isonomes refer to percentage frequency. ecological methods with special reference to pastures. He has de- vised an ingenious method of estimating percentage cover in closely cropped pastures: it consists of laying a lattice at random on the pasture, and comparing the pattern of vegetation under the lattice


with a series of photographed patterns representing 20, 40, 60, etc., per cent cover. The productivity of each component of the pasture is estimated by multiplying the percentage cover of each species by a factor which is derived from the regression of cover on total bulk of the species.

Pechanec and Stewart (17) have demonstrated a simple way of improving the efficiency of random sampling in pasture research. Instead of sampling at random over the whole area to be investi- gated, these authors divide the area into a certain number of blocks, and sample at random within each block. They are then able to separate out in their analysis the variance due to blocks and the variance due to error within blocks. The benefit of this simple procedure is evident from data published in their paper.

Vestal (26) has attempted to improve the precision of subjective estimates of density and cover, such as the familiar scales used by Hult, Sernander, Norrlin, Braun-Blanquet, and others. The "classes" of these scales are, of course, distorted measures of density. To minimise this distortion, Vestal proposes to substitute classes arranged in arithmetical or geometrical progressions. Such a proposal would indeed minimise one of the errors of these subjective methods, but it would still teave the methods open to the other objections which have been raised against them.

Penfound (18) has studied the way in which frequency is af- fected by quadrat size, cover and number of species. He used coloured cards, but his methods could be extended to field studies. He concludes that cover is the most reliable measure of plant distribution, since estimates of cover do not depend on the size of the quadrat used for sampling; but this conclusion does not meet the objection that estimates of cover are inevitably in some degree subjective. This objection can be met in part, at any rate, by working out a regression between percentage cover and some suitable measure of cover, e.g., crown weight of leaves, as Kittredge (13) has done.

Among miscellaneous contributions to the technique of statistical ecology may be mentioned the work of Bauer (3) who reached, with the aid of 1,000 coloured discs, the not unexpected conclusion that transects can be used instead of quadrats for sampling communities; the work of Billings (4) who calculated an interesting regression between the organic matter of the soil and


the age of trees in a Pinus echinata forest, the work of Dice (10) who measured the degree of "ecologic association" between species by a technique similar to that described by the present writer (1) ; and the work of Lynch and Schumacher (14) who used the method of probits to sample regenerated forest.

Conclusions about plant distribution. It is evident from the foregoing paragraphs that the technique for studying plant distribution has improved in the last 11 years. What advances in ecology have followed from this improved technique ?

The most noteworthy advance is the knowledge that individuals are not, as a rule, normally dispersed, and that over-dispersion is much more common than under-dispersion. In large areas over- dispersion is to be expected, owing to heterogeneity of soil and micro-climate, which will have the effect of localising individuals in the more favourable habitats ; but Blackman (5) has shown that over-dispersion occurs on grassland plots as small as 18 by 15 feet, where one might expect uniformity; Singh and Chalam (22) and Singh and Das (23) have shown that many weeds on a fallow wheat field are over-dispersed ; and Pidgeon and Ashby (19) have shown that in a desert community, under uniform conditions of climate and soil, 68~b of the species show non-random distribution. It seems to be established that although the occurrence of any particular species in a habitat may be due to chance, as Palmgren (16) has ably shown, and although, as Clapham (8) has suggested, groups of plants may be randomly distributed, yet individual plants or shoots are usually over-dispersed.

This habitual over-dispersion is due to one or another of several causes, some still obscure. Obviously plants which spread by rhizomes or runners or heavy seeds will be over-dispersed, although the amount of over-dipersion for some rhizomatous plants is surprisingly small. However, over-dispersion is present in species where there is no obvious biological reason for it, and indeed both Blackman and Pidgeon and Ashby, found that the same species may be randomly distributed in one habitat and over-dispersed in another. Thus Campanula barbara grew randomly in three plots and non-randomly in 10 plots; and Senecio campestris grew randomly in 12 plots on chalk downland and non-randomly in three plots. This evidence points to a conclusion, which would be important if it were confirmed, that the mode of dispersion of a par-


ticular species is not entirely a property of the species, but is dependent also upon the habitat.

There is some evidence that one habitat factor, namely, density of plants, affects dispersion. Thus Clapham (8) records the following values for Senecio Jacobea:

Relative variance Mean density (value for random

dispersal = 1) Sparse 1.67 3.04 Intermediate 3.80 7.95 Dense 12.44 6.09

Pidgeon and Ashby (19) record a relation between dispersion and density of Australian desert plants as follows:

Density per Percentage Percentage distributed: 100 metre 2 of species Randomly Non-randomly

0 - 50 57.7 20.0 37.7 51 - 100 11.7 3.5 8.2

Over 100 30.6 1.2 29.4

and they find that, for instance, Eragrostis Dielsii is randomly distributed at low densities and over-dispersed at high densities.

Singh and Chalam (22) report that weeds on arable land are, on the whole, randomly distributed when their density is low and over-dispersed when their density is high.

What little evidence there is indicates that a species which has no biological predisposition toward over-dispersion is randomly distributed when it first occupies an area. As its density increases it loses its randomness and becomes over-dispersed; and if it dis- appears from the community, owing to competition or the course of succession, its distribution may become random again during its relict stages. This evidence requires a great deal of confirmation, but if it is confirmed it will be an interesting problem to discover the cause of this cycle of dispersion which may accompany the colonisa- tion, establishment and displacement of a species. One plausible speculation is that the over-dispersion of a species at high densities, when not caused by soil heterogeneity or by some obvious biological predisposition of the species, is caused by the presence at high densities of other species which are predisposed to aggregation. The fact that some species are over-dispersed may produce a secondary over-dispersion among the other species.

232 T H E BOTANICAL REVIEW

Some light is thrown on the interrelations of species in a community by the studies of Ashby and Pidgeon (2). From isonomes, whose construction is described along with Fig. 1, these authors have shown that the individual plants in sandstone scrub vegetation are not randomly distributed but are arranged in patterns of de- creasing frequency around centres of high frequency. If the ground is completely covered there is an interdigitafion between the patterns of the various species. Applied to some of the stages of succession on sand dune vegetation, the technique reveals that some species show a continuity of isonomes from one seral stage to another, while other species remain only in outliers as the succession proceeds. It would be a matter of some importance to know whether the centres Of distribution disclosed by this analysis are themselves randomly distributed. Work on these lines is proceeding. What is needed for the next step is some modification of Cole's method of analysis suitable for studying the peculiar type of contagious distribution which is so widespread in plant communities.

ADDENDUM

Through the courtesy of Dr. C. B. Williams I have been able to read four papers of his in typescript: (i) The logarithmic series and its application to biological problems, Journal of Ecology, (ii) The logarithmic series and the frequency of occurrence of plant species in quadrats, (iii) The generic relations of species in small ecological communities, Journal of Animal Ecology, (iv) The logarithmic series and the comparison of island floras, Journal of the Linnean Society of London. All these papers will shortly appear in scien- tific journals. Some of the work embodied in these papers concerns the relation between the number of species and the number of genera in plant and animal populations; this topic is beyond the scope of the present review. However, the second of the four unpublished papers contains an important contribution to statistical ecology, namely an analysis of species-area curves by the logarithmic series

n i x n i x 2 where nl = the number of species in the sample nl, ~-- , 3 - ' ' '

represented by only one individual, and where x is a constant, de- pending on quadrat size. Williams is able to give formulae for the theoretical relation between the area sampled and the number of species found in 1, 2, 3, . . . samples of the area. Using some of


Jaccard ' s data, Will iams obtains linear relationships between the number of species and the logari thm of the number of quadra ts ; and he measures the richness of the flora by an "index of diversity".

Will iams demonstrates neatly that Jaccard 's method of comparing two communities by a "coefficient of communi ty" (discussed in the present writer 's previous review, 1936), is unsatisfactory, for the level of the coefficient is determined by two quite different causes: on the one hand, the richness of the flora (which is relevant) , and on the other hand, the size of the area sampled (which is not relevant) .

BIBLIOGRAPHY

1. ASHBY, E. Statistical ecology. Bot. Rev. 2: 221. 1936. 2. and PIVGEON, I .M. A new quantitative method of analysis of

plant communities. Austral. Jour. Sci. 5: 19. 1942. 3. BAUER, H. L. The statistical analysis of chaparral and other plant

communities by means of transect samples. Ecology 24 : 45. 194,3. 4. BILLINGS, W.D. Quantitative correlations between vegetational changes

and soil development. Ecology 22 Suppl. : 448. 1941. 5. BL^CK~AN, G. E. Statistical and ecological studies on the distribution

of species in plant communities. (i) Dispersion as a factor in the study of changes in plant populations. Ann. Bot. 6: 351. 1942.

6. CAIN, S.A. The species area curve. Am. Mid. Nat. 19: 573. 1938. 7. . Sample plot technique applied to alpine vegetation in

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12. HOeE-SIMPSO~, J. F. On the errors in the ordinary use of subjective frequency estimations in grassland. Jour. Ecol. 28: 193. 1940.

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17. PECH^NEC, J. F. and STEWART, G. Sagebrush-grass range sampling studies: size and structure of sampling unit. Jour. Am. Soc. Agron. 32 : 669. 1940.

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21. RAMENSKY, L. G. Vvedenie v kompleksnoe pochvenno-geobotanicheskoe issledovanie zemel. [Russian] 620 pp. 1938. [Introduction to the complex soil-geobotanical research of land].

22. SINGH, B. N. and CHAL^M, G. V. A quantitative analysis of the weed flora on arable land. Jour. Ecol. 25: 213. 1937.

23. and DAS, K. Distribution of weed species on arable land. J0ur. Ecol. 26: 455. 1938.

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statistical ecology. ii—a reassessment

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