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Many biological investigations seek to understand variation in a trait of interest by studying underlying components. In some cases these components are related to the trait of interest in a multiplicative manner. One example is relative growth rate, which is the product of one or more ‘physiological’ and ‘morphological’ components (Evans, 1972; Poorter, 2002). Another is the leaf mass per area (LMA), which is the product of leaf thickness and leaf density (Witkowski & Lamont, 1991). One method that can be used to investigate the relative importance of multiplicative subcomponents for variation in a trait of interest is known as log–log scaling slope analysis. For example, in their meta-analysis of the causes and consequences of variation in LMA, Poorter et al. (2009) use this method to determine the relative contribution of variations in density (D) and thickness (T) to observed variability in LMA (henceforth abbreviated as L for equations) across various data sets. The relation of interest is thus L = D × T. The method involves calculating the slope of the log-transformed D or T when regressed linearly against log-transformed LMA (Poorter & van der Werf, 1998), and then using the value of this slope to generalize about the importance of density and thickness in causing LMA differences across a range of species or experiments. More specifically, if log(D) is modelled as a linear function of log(L), that is
and log(T) is similarly modelled as a linear function of log(L), that is
where aD, bD, aT and bT are model parameters determined by least-squares or maximum-likelihood fitting, then bD +bT will always be equal to 1. (Note that L is the dependent variable in the log–log scaling analysis in Eqns 1 and 2, which is a common source of confusion because it is more usually written as the independent variable in the equation L = D × T.) The interesting feature of this approach is that the values of bD and bT indicate the relative contribution of D and T to the variability of LMA; for example, if bD is close to 1 and bT is close to 0, then D is largely responsible for variability in LMA, whereas if bD is close to 0 and bT is close to 1, then T is largely responsible for variability in LMA. If bD and bT are both close to 0.5, then T and D are similarly responsible. Therefore, this is a powerful approach to summarize a wealth of data and to weigh the importance of variability in two or more ‘underlying’ variables.
Poorter & van der Werf (1998) derived this method analytically for use with two observations, but not for the case where the number of observations is more than two. Here, we explain formally why the log–log scaling method works for more than two data points, describe particular situations where it may give results that are less clearly interpretable and propose an extension that may make the relationships more transparent in certain conditions. Recently, a clear and complementary discussion of these approaches has been presented in the context of an analysis of the contributions of net assimilation rate, leaf mass ratio and specific leaf area to relative growth rate (Rees et al., 2010). This letter provides additional insights into why the log–log scaling method works, when care should be taken in interpreting its results and how it can usefully be extended, using a range of simple, yet deliberately contrasting, examples in the context of partitioning variability in LMA.
First, it must be noted that the log–log scaling slope analysis method aims to determine the relative responsibility of density and thickness for the observed variability in LMA, rather than for the mean value of LMA. The question is analogous to considering a collection of rectangles, and asking whether the variability in the area of the rectangles is caused more by the variability in their height than in their width. Although the rectangles may all be much taller than they are wide, their width may still contribute more to their variability in area than their height – for example, if all of them have exactly the same height. Whenever a variable of interest (e.g. LMA, rectangle area or relative growth rate) can be decomposed into the product of two or more other ‘component’ variables (e.g. thickness and density, or height and width, or net assimilation rate, leaf mass ratio and specific leaf area), a log–log scaling slope analysis is supposed to show the extent to which each of the component variables is responsible for the variance in the variable of interest. It must also be noted that this problem of finding the extent to which each of two variables contribute to the variability of their product is a different problem to fitting a line to a log-transformed bivariate data set to estimate and/or test allometric relationships through methods such as linear regression or major axis or standardized major axis procedures (Warton et al., 2006), even though the methods used to address these problems may superficially appear to be similar and both problems have previously been addressed in the context of LMA (e.g. Wright et al., 2004 and Poorter et al., 2009). One problem is estimating a power–law relationship between two variables, while the other is simply partitioning variation in a variable between two component variables. The latter uses linear regression on log-transformed variables as part of the method but is not aiming to estimate a functional relationship.
The reason that the log–log scaling method works for more than two points can be explained as follows, with formal details in the Supporting Information Notes S1 and S2. In Notes S1 it is shown that the variance of log(L) is equal to the variance of log(D) plus the variance of log(T) plus two times the covariance of log(D) with log(T). Thus, the variability of log(L) can be partitioned into one fraction that is a direct result of variability in D (D fraction), a second fraction that is a direct result of variability in T (T fraction) and a third fraction that is a result of the covariance between the two (covariance fraction). In Notes S2 it is shown that the log–log scaling slope analysis method is similar, but results in two fractions rather than three by effectively splitting the fraction resulting from covariance in half and adding it to the other two fractions; therefore, one slope is the D fraction plus half of the covariance fraction, and the other slope is the T fraction plus half of the covariance fraction. This can, in turn, be used to show why the slopes must always sum to 1.
The splitting of LMA variation into three variance components has been used as a complement to a log–log scaling slope analysis in a real data set by Hassiotou et al. (2010). Here we show four example data sets (Fig. 1; Table S1), purposely chosen to illustrate a few extreme contrasting cases. Table 1 shows the results of applying the log–log scaling method and three-way partitioning to these four example data sets. If T and D are uncorrelated (Example 1 in Table 1), and thus their covariance is equal to 0, then the log–log scaling slope analysis method works clearly, showing exactly what proportion of variability in LMA is caused by variability in each of the contributing variables. If T and D are slightly correlated, the method still works clearly because the fraction resulting from the covariance is relatively small and is split evenly between the two contributing variables, which is probably a ‘fair’ way to partition this small fraction. The variable that contributes more to variability in LMA will still show the higher slope. When D and T are highly positively correlated, but of different variability (Example 2 in Table 1) then the log–log method also works clearly, with the relatively large covariance fraction split evenly between the two slopes.
Table 1. Analysis results for four hypothetical examples, including the variance of each variable, the results of a log–log scaling slope analysis, the results of partitioning leaf mass per area (LMA) variability into three fractions and the correlation between log(D) and log(T)
Data for the four examples are shown in Fig. 1 and in the Supporting Information Table S1.
D, variation in density; T, thickness. * variance is of the log-transformed variables.
Correlation log(T) and log(D)
However, if T and D are highly positively correlated and similarly variable (Example 3 in Table 1) then the fraction of LMA variability caused by covariance (the covariance fraction) will be c. 0.5 and the T and D fractions will each be c. 0.25. When the log–log method splits the covariance fraction across T and D, the slopes for T and D will both be approximately equal to 0.5. Thus, the same result of both slopes being approximately equal to 0.5 will be found as when T and D are completely uncorrelated and similarly variable (Example 1 in Table 1). In both cases it will appear that T and D are contributing evenly to variability in LMA. In some senses, this may be a fair conclusion in both cases, but in one case T and D are contributing independently and in the other they are not. As the underlying biological relevance in these cases is quite different it may be preferable to also provide the data on the covariance because it provides better insight into the relationships between the three variables.
If T and D are negatively correlated, then the situation can be more complicated. If they are negatively correlated and similar in variability, then there will be little variability in LMA because the two will effectively cancel each other out. In such a case, it does not make much sense to try to attribute the variation in LMA to underlying factors at all; if it was applied in this case it would produce nonsensical results. If T and D are negatively correlated and differ in variability, then the log–log scaling slope analysis method still makes sense and can still be used with some care. Say, for example, that T and D are very negatively correlated and that log(D) is more variable than log(T), as in Example 4 in Table 1. The covariance betwen log(D) and log(T) will be negative and thus the covariance fraction will be negative. This means that the variance of log(LMA) will be less than the sum of the variance of log(D) and the variance of log(T), and thus the D fraction will be > 1 and the T fraction will be relatively small. When the log–log method evenly splits the negative covariance fraction across T and D, the resulting slope for D will still be > 1 and the slope for T will be negative (Example 4, Table 1). This can be reasonably interpreted as meaning that the relative increase in LMA occurs with a proportionally larger increase in D, whereas variability in T actually counteracts the increasing effect of D, thereby reducing the variability in LMA. However, the three-way partitioning method provides additional information, because the positive resulting values for T and D correspond to their actual variance, and the way that the less variable quantity (e.g. T) counteracts the more variable quantity (e.g. D) is shown separately by the negative covariance component.
We have shown here a formal justification for why the log–log scaling slope analysis is a useful method for determining the causes of variability in LMA and explained cases where care must be taken when interpreting the results of this method owing to positive or negative correlation between T and D. We have also discussed the possibility of partitioning variance of LMA into three components instead of two, in order to provide additional information on covariance not provided by the log–log scaling slope analysis. However, if the correlation between log(D) and log(T) is examined as part of a log–log scaling slope analysis then a similar depth of information is provided and the results may be clearer and simpler to interpret and present, particularly to a less mathematical reader. In particular, if the correlation is relatively weak, then the results of the log–log scaling slope analysis can be presented without further complication. We finally conclude that the log–log scaling slope analysis and its proposed extensions is a powerful method for analysing the contributions to variability in any biological trait that can be considered as the product of other variables.