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
Oecologia (Berl.), u
18
184
J.F. Benson : Analysis of Life Table Data
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
Density-Dependence
185
o=
,,,
I log n
b=0.88 9 r-0,68 0 ~
~
-~,>~1"5 a5f /I."- (~
o~
,I/ll
~ ~~176
1.5
2.5
log n
Fig.1
1[)=0.12
I
~-n=0.12
2"5FI9.5.(b)|
I
] . . I ~ - -0' -0-"
I--
/!
9
1.5
0~
1
I
2.5
log n
~'
-
./~ e
(r *"0.oi. i , .
o
~
~
log n
log n
Fig.2
Fig. 1 a and b. Theoretical relationships to show a) a k-value/log initial number (n) plot, and b) the same data as a log survivors (s)/log initial number (n) plot Fig. 2a--c. Theoretical relationships between log s and log n to show a) no mortality, b) constant percentage mortality and e) variable percentage mortality
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
note t h a t
q = 1 -- b.
(3)
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
186
J.F. Benson:
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
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
Oecologia (Berl.), u
18
184
J.F. Benson : Analysis of Life Table Data
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
Density-Dependence
185
o=
,,,
I log n
b=0.88 9 r-0,68 0 ~
~
-~,>~1"5 a5f /I."- (~
o~
,I/ll
~ ~~176
1.5
2.5
log n
Fig.1
1[)=0.12
I
~-n=0.12
2"5FI9.5.(b)|
I
] . . I ~ - -0' -0-"
I--
/!
9
1.5
0~
1
I
2.5
log n
~'
-
./~ e
(r *"0.oi. i , .
o
~
~
log n
log n
Fig.2
Fig. 1 a and b. Theoretical relationships to show a) a k-value/log initial number (n) plot, and b) the same data as a log survivors (s)/log initial number (n) plot Fig. 2a--c. Theoretical relationships between log s and log n to show a) no mortality, b) constant percentage mortality and e) variable percentage mortality
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
note t h a t
q = 1 -- b.
(3)
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
186
J.F. Benson:
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
