Oecologia (Berl.) 13, 183--190 (1973) 9 by Springer-Verlag 1973

Some Problems of Testing for Density-Dependence in Animal Populations J. F. Benson Department of Forestry, Oxford Received May 10, 1973

Summary. A test for density-dependent mortality using the regression of log number of survivors of a mortality on the log initial number is discussed. Problems associated with the test, including those due to errors in the independent variable and certain problems of interpretation, are also discussed. Despite criticisms of this type of test in recent years, it is felt that the test is valuable as long as it is used carefully and critically, involving constant consideration of the biological relationships involved. Introduction Many methods have been used for detecting density-dependent mortality; however, almost ~ll of them have limitations and restrictions to their use, some statistical and some biological. It5 (1972) has recently discussed and reviewed these methods in some detail. The present paper extends It6's discussion of one method of analysis, a method which It6 recommends, but which involves several problems of interpretation which have not previously been adequately considered.

Morris' Method A widely used method of analysis is due primarily to Morris (1963). This involves the regression of the log numbers of a particular developmental stage in one generation on the log numbers of the same stage in the previous generation. Morris suggests t h a t density-dependent mortality is acting if the regression coefficient, b, is ~ 1. This method is subject to biological problems of interpretation (e.g. Southwood, 1967; Hassell and Huffaker, 1969a; Morris and Royama, 1969; Luck, 1971) and to statistical problems (e.g. Maelzer, 1970; St Amant, 1 9 7 0 ) a n d I agree with It6 (1972) t h a t the method cannot be recommended for the analysis of life table data. Despite these criticisms, the method m a y still be useful for simple prediction of populution chunge (us might be of interest to an economic entomologist) if the regression h~s low variance; however, a biologically realistic interpretation will probably be impossible. 13a

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It is generally agreed that realistic interpretations of the action of different factors causing mortality can only be gained from careful and critical analysis of detailed life table data, although the precise methods of analysis to be used are not fully developed or agreed. In order to investigate the density-relationship of each factor causing mortality, Varley and Gradwell (1968, 1970) first plot the k-value for each factor (log density of a particular stage, n -- log density of survivors of the mortality, s) against the log initial density, n. Density-dependent mortality is suggested when a regression analysis gives a slope significantly different from b = 0 . However, this procedure is statistically invalid (Varley and Gradwell, 1963, 1968, 1970; Watt, 1964) because the measurements used on the two axes are not independent of each other, the independent variable appearing in both. Varley and Gradwell (1963, 1968, 1970) (see also ItS, 1972) have therefore recommended additional tests for a suspected density-dependent relationship. If n is accurately known, proof is provided by plotting log s against log n and testing if the slope of the regression is significant in itself (i.e. a high correlation coefficient, r) and also differs significantly from b = 1. Although this test resembles Morris' test (see above), there is an extremely important distinction between the two; while Morris' test uses the log number of a particular stage in successive generations, Varley and Gradwell's test uses the log number of two successive stages within a generation. Morris' test is therefore confused by problems of serial correlation (each observation appearing successively in both axes) which do not affect Varley and Gradwell's test (Maelzer, 1970; ItS, 1972). In this latter test, each pair of observations must be from separate and preferably successive generations, since only density-dependent mortality factors which act between generations can, without further analysis, have definite implications for regulation of a population (Hassell, 1966). There has been some argument on the question of whether plotting k-values against log n gives a spurious suggestion of density-dependence due to the lack of independence of the axes (Luck, 1971; Williamson, 1972). Although statistically invalid, tho slope determined is exactly complementary to that determined in the log s/log n plot. Fig. l a shows an artificial k-value plot where the regression line (b-~0.88) explains 47 % of the variability in k by its relation with log n. The same data in a log s/log n plot (Fig. 1 b) gives a complementary slope (b'= 0.12= l--b), but only 2 % of the variability in log s is explained by its relation with log n. There therefore seems to be no reason for using the k-value/log n plot other than that it may be easier to interpret than the log s/log n plot, especially if one is familiar with thinking in terms of k-values. The

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k-value m e t h o d needs two equations in order to express change in the population k~a~- blogn (1) and log s=- log n - - k (2) whereas a single equation suffices from a log s/log n plot log s = p-f- q log n

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Further, the k-value plot m a y be subject to complex biases due to errors of m e a s u r e m e n t (see later). F u r t h e r problems m a y arise if care is not t a k e n to distinguish between constant or highly variable density-independent m o r t a l i t y (Varley and Gradwell, 1971). I n tests involving the log s/log n plot, the null hypothesis is t h a t b = 1. Fig. 2 a shows such a relationship where the slope passes through the origin; here, log s = log n and no m o r t a l i t y is acting. If the slope does not pass through the origin (Fig. 2b), then a constant percentage m o r t a l i t y acts on all values of log n. I f the m o r t a l i t y is variable, b u t still independent of log n, the picture m a y be quite different (Fig. 2c). The value of log s for a given value of log n can theoretically take a n y value between log n and - - c ~ . I n practice, the limits of observation m a y be reached as either log s or log n pass, for example, below 0 (1 animal per unit area) and the appearance of the graph m a y be m u c h as in Fig. 2 c, with perhaps some points at - - 0 o . These r e m a r k s a p p l y if the m o r t a l i t y is uniformly distributed; if it is 13 b Oecologia (Berl.), Vol. 13

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normally distributed, the variance may be less than that in Fig. 2 e, but still considerably greater than that in Fig. 2b. Such a relationship, and its interpretation, may therefore be biased, depending on the number of points in the graph (Maelzer, 1970) and on whether the variance of the dependent variable is constant or variable. I t should therefore be clear that, for a variety of reasons just outlined, low variance of the regression is a prime requirement in a test for density-dependence. The Problem of Errors

If there are errors attached to the measurement of log n, further biases may be introduced, and Varley and Gradwell (1963) have proposed that the following test be used. The regression of log n on log s is also calculated, and for significance, both regressions must be significantly different from, and both must lie on the same side of, b----1. The test assumes that the true functional or structural relationship between the variables lies somewhere between the two lines (Moran, 1971) and has been used by Hassell and Huffaker (1969b), Hassell and Varley (1969), Luck (1971), Whittaker (1973) and others. It6 (1972) criticizes this two-way regression test because a prediction of log n from log s is effectively a nonsense; however, the regression is not used for this purpose, but merely as a statistical test that a density-dependent relationship exists. The requirement that both regressions must lie on the same side of b = 1 is in fact asking that the variance about the regressions be small, since as the correlation coefficient increases, bsn and bus move closer together, and the lines are coincident when r = 4 - 1 and bns-=l/bs~. Hence a better measure of the coincidence of the two lines is again a high correlation coefficient. A further reason for using r is that it is difficult to decide whether the lines are on the same or opposite sides of b = l when considering overcompensating density-dependent mortality (see later). Errors in the independent variable reduce the regression coefficient for undereompensating density-dependent mortality (Southwood, 1966) in the log s/log n plot ( 0 < b < 1) but increase the coefficient for overcompensating density-dependent mortality (b
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t h a t they will tend to move the regression line towards b = l in this case (Kuno, 1971). If the error in log n is small relative to the range of log n, the bias in the log s/log n plot m a y be slight. I f the size of the errors is known, statistical methods have been developed and can be followed to take the errors into account (see Sprent, 1969; Moran, 1971). Kuno (1971) has derived some interesting estimates of the precision (sample standard error/mean) of population counts needed so t h a t the effect of any bias shall be minimal, and he shows t h a t as long as the precision is 0.1 or better (or an error of 10% of the mean), this m a y often be the case. He further suggests t h a t attempts should be made to standardize the precision of the estimate of each stage (at say 0.1), perhaps b y a method of sequential sampling (Southwood, 1966; Kuno, 1969). Problems of Interpretation I n many, if not the majority of population studies, some stages are estimated without a detailed knowledge of the errors involved, and it m a y be necessary to use a statistical test of the type discussed above, involving a two-way regression. A further possibility is to fit a regression line using grouping methods (Sprent, 1969; Moran, 1971), but assumptions must then be made about the magnitude of the errors involved. However, there are problems of interpretation even with the log s/log n plot which I wish to discuss. For example, on the basis of the test, the

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relationships in Fig. 3a and Fig. 3b would be accepted as proof of density-dependence, but that in Fig. 3 c would not, because of the low correlation coefficient, even though the latter relationship is seen to be perfectly density-dependent. If the variance about the regressions is low, as in Fig. 3c, the expected coefficients approach b,n~-O , bn,=cx~ and r = 0 ; if the variance was in this example large, the coefficients approach bsn= 0, bns= 0 and r = 0, representing a highly variable mortality factor. The correlation coefficient cannot therefore be used as a measure of the variance in this case. Although it is possible to calculate confidence limits for the regression coefficients, or to test the ratio of the variances of log s and log n, the problem remains insoluble at intermediate situations when it is difficult to define a dividing line between acceptance or rejection of a significant relationship. I t seems reasonable to use r in most cases, because it becomes more difficult to obtain a high r as the slope moves towards b = 0 and also errors are in any case moving the slope towards b = 0. However, there may be occasions when a decision based on r will be difficult. The problem, of course, is that we attempt to read biologicM significance (perfect density-dependent mortality) into a relationship (e.g. Fig. 3c) that has no statistical significance (no correlation between log s and log n). Certainly any factor or process that gives the appearance of being perfectly density-dependent should be investigated experimentally; this would lead to more accurate measurements and a better certainty of the functional or structural relationship between the variables. Fig. 4 shows an example from a natural situation. This is the relationship between the log number of young Tawny owls (Strix aluco L.) in spring and the log number of young fledged in the previous year (data from Southern, 1970), a relationship that is highly significant biologically, the number of owls in spring being determined by the relatively stable number of territories available. In the k-value/log n plot (Southern, 1970, Fig. 23), this " m o r t a l i t y " (strictly a disappearance of young owls) appears to be almost perfectly density-dependent (b-~l.1, r ~ 0 . 9 8 , P ~ 0 . 0 0 1 ) yet density-dependence is not " p r o v e d " in Fig. 4 due to the low correlation coefficient. Note that it is difficult to decide from Fig. 4 whether the regression lines lie on the same or opposite sides of b : l . A further example is the almost perfectly density-dependent mortality of cabbage root fly pupae [Erioischia brassicae (Bouchd)] in England (see Benson, 1973). I t will be obvious that these tests may fail to " p r o v e " a densitydependent relationship, but do not necessarily "disprove" such a relationship. This is especially true when a mortality is a complex process made up of several components, for the apparent relationship will then depend on the density-relationships and sizes of the components involved. This is also a major criticism of Morris' method of analysis (Luck, 1971; ItS, 1972).

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Conclusions I t seems extremely naive t h a t ecologists should expect biologically realistic interpretations of complex population processes solely from crude statistical analyses of very limited and inaccurate data. Of course the practical problems involved in ecological work are immense, but this is no criticism of the methods of analysis which I have discussed; rather, it emphasizes t h a t the interpretation of data must involve a constant feedback to the biology of the animals concerned. The following conclusions and recommendations can therefore be made: (i) The use of It-values is felt b y m a n y to be a helpful and useful way of thinking about the effect of a mortality factor on a population, but must be regarded as a preliminary analysis of life table data and should on no account be regarded as evidence for any particular biological relationship. (ii) Firm conclusions about the action of a particular factor require careful consideration of the biological and statistical relationships involved. A high correlation coefficient is essential in any relationship, because of problems of interpreting variable mortalities which m a y include several different components. (iii) If measurement errors are unknown, the only possibility seems to be a two-way regression test (see Moran, 1971), remembering the difficulties when a factor appears to be causing almost perfect densitydependent mortality. Careful consideration of the biological relationships implied b y the analysis m a y shed ]ight on the reality of a suggested relationship; experimental evidence is in all cases highly desirable. (iv) If information is available on the errors involved, and of course such information should increasingly be regarded as essential in any population study, then other alternatives are available and have been discussed elsewhere (Sprent, 1969; Moran, 1971).

Acknowledgements. I wish to thank a number of colleagues, and especially G. R. Gradwcll, for their time and patience spent discussing these problems. This work was carried out while in receipt of financial support from the Natural Environment Research Council.

References Benson, J. F. : Population dynamics of cabbage root fly in Canada and England. J. appl. Ecol. 1O, 437-446 (1973). ttassell, 1~. P,: Evaluation of parasite or predator responses. J. Anita. Ecol. 3~, 65-75 (1966). Hassell, M.P., Huffaker, C.B.: The appraisal of delayed and direct densitydependence. Can. Ent. 1{)1, 353-361 (1969a).

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Hassell, M. P., Huffaker, C. B.: Regulatory processes and population cyclicity in laboratory populations of Anagazta kiihnlella (Zellcr) (Lepidoptera; Phycitidae). I I L The development of population models. Researches Popul. Ecol. Kyoto Univ. 11, 186-210 (1969b). Hassell, M. P., Varley, G. C.: New inductive population model for insect parasites and its bearing on biological control. Nature (Lond.) 223, 1133-1137 (1969). ItS, Y.: On the methods for determining density-dependence by means of regression. Oecologia (Berl.) 10, 347-372 (1972). Kuno, E. : A new method of sequential sampling to obtain the population estimates with a fixed level of precision. Researches Popul. Ecol. Kyoto Univ. 11, 127-136 (1969). Kuno, E.: Sampling error as a misleading artifact in "key factor analysis". Researches Popul. Ecol. Kyoto Univ. 18, 28-45 (1971). Luck, R. F.: An appraisal of two methods of analyzing insect life tables. Can. Ent. 103, 1261-1271 (1971). Maelzer, D. A.: The regression of log Nn+ 1 on log N n as a test of density dependence: an exercise with computer-constructed density-independent populations. Ecology 51, 810-822 (1970). Moran, P. A . P . : Estimating structural and functional relationships. J. Multivariate Analysis 1, 232-255 (1971). Morris, R. F.: Predictive population equations based on key-factors. Mem. ent. Soc. Can. 32, 16-21 (1963). Morris, R . F . , Royama, T.: Logarithmic regression as an index of responses to population density. Comment on a paper by M. P. Hassell and C. B. Huffaker. Can. Ent. 101, 361-364 (1969). Southern, H. N. : The natural control of a population of Tawny owls (Strix aluco). J. Zool. (Lond.) 162, 197-285 (1970). Southwood, T. R. E. : Ecological methods with particular reference to the study of insect populations. London: Methuen 1966. Southwood, T. R. E.: The interpretation of population change. J. Anita. Ecol. 36, 519-529 (1967). Sprent, P. : Models in regression and related topics. London: Methuen 1969. St Amant, J. L. S. : The detection of regulation in animal populations. Ecology 51, 823-828 (1970). Yarley, G. C., Gradwell, G. R.: Predatory insects as density dependent mortality factors. Proc. XVI Int. Congr. Zool., Washington 1, 240 (1963). Varley, G. C., Gradwell, G. R. : Population models for the winter moth. Syrup. R. ent. Soc. Lond. 4, 132-142 (1968). Varley, G.C., Gradwell, G. R.: Recent advances in insect population dynamics. A. Rev. Ent. 15, 1-24 (1970). Varley, G. C., Gradwell, G. R.: The use of models and life tables in assessing the role of natural enemies. In: Huffaker, C. B., Ed., Biological control, p. 93-112. New York: Plenum 1971. Watt, K. E . F . : Density dependence in population fluctuations. Can. Ent. 96, 1147-1148 (1964). Whittaker, J. B. : Density regulation in a population of Philaenus spumarius (L.) (Homoptera: Cercopidae). J. Anita. Ecol. 42, 163-172 (1973). Williamson, M.: The analysis of biological populations. London: Arnold 1972. Dr. J. F. Benson Department of Forestry South Parks Road Oxford, England