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Divide and conquer? Size adjustment with allometry and intermediate outcomes
© Nakagawa et al. 2017
- Published: 9 November 2017
Many trait measurements are size-dependent, and while we often divide these traits by size before fitting statistical models to control for the effect of size, this approach does not account for allometry and the intermediate outcome problem. We describe these problems and outline potential solutions.
Biologists measure traits of organisms, characterizing a range of features including morphology, physiology and behaviour. Many of these traits are size-dependent. For example, larger animals eat more and larger plants absorb more than smaller counterparts. However, when size is not our trait of interest, we often want to know values of focal traits after controlling for the effect of size. An intuitive way to account for organismal size is to divide a trait of interest by size (for example, the amount of food consumed or nutrient absorbed divided by mass or length). This method (hereafter called the “division” method) generates size-adjusted trait values, which can be used for statistical analyses. Indeed, the use of the division method is prevalent in the literature [1–4], but is it correct? The division method poses two major problems because, first, we assume a linear relationship between adjusted traits and size and, second, an experimental treatment often affects not only a trait of interest but also other variables, such as size. Here, we deliberate on these two problems and provide potential solutions.
The power law (meaning non-linear) relationship between body size and body parts is believed to have been first described by the evolutionary biologist Julian Huxley, (a grandson of “Darwin’s bulldog” Thomas Huxley) on the basis of his study of claw size and body size in fiddler crabs . This non-linear relationship was later termed “allometry” (meaning “different measure”) . Huxley’s original study was on ontogenetic allometry—the relationship between two traits while an organism is growing . This type of allometry is distinguished from the two other types: evolutionary allometry and static allometry . Evolutionary allometry concerns between-species variation in the relationship between traits (see  for an in-depth review), whereas static allometry concerns the relationship between two traits of mature individuals from the same species. Here, we focus mainly on static allometry.
where y [trait/size] is a variable derived from the focal trait divided by size, x [group] is a “dummy variable”, which takes the value 0 or 1 to indicate the presence or absence of a particular effect in order to sort data in mutually exclusive groups (for instance, 0 = control and 1 = experiment), b 0 is the intercept, b 1 is the slope (or in this case, the difference between the control and experimental group: experimental or treatment effect), e is the residual (error) term (which represents deviations from the regression line), and the subscript i indicates the i th value (i = 1…n, n = sample size; this linear model, Eq. 3, is equivalent to a t-test comparing the two groups). However, this model is not ideal because the division method (Eq. 3) creates a ratio variable (y [trait/size]), the distributional properties of which may not meet an important assumption of a linear model: the residuals are normally distributed . Furthermore, we are not able to estimate the allometric scaling exponent b.
where b 0 and b 2 correspond to lna and b in Eq. 2, respectively, and the other symbols are as above. It is also notable that Eqs. 4 and 5 could produce comparable results, depending on the exponent of the power law relationship, b, and the distribution of trait data (Fig. 1b–d). However, Eq. 4, like Eq. 3, could lead not only to a spurious treatment effect, b 1 , but also to spurious interactions (which means that the control and experimental groups have different slopes, b 2 as described in Fig. 1e, f); see the next section for modeling the interaction (that is, different slopes). However, it turns out that even Eq. 5 can provide an incorrect estimate of b 1.
where the symbols are as above. Then, at least we will get the total (direct and indirect) effect on y as the experimental effect, b 1 (for further discussion of intermediate outcomes and their problems see [14, 16–18]).
Another way to do such adjustment is through z-transformation instead of centering, which scales distributions of ln(size) for both the control and experimental group to have the mean of 0 and standard deviation of 1. These two methods (centering and z-transformation) are equivalent when the variances of the two groups are the same.
However, if slopes differ between the two groups after z-transformation, it should be checked whether the transformation caused the significant differences, which may happen if the variances for ln(size) to differ between the two groups. (Figure 3b; see below for how to detect differences in slopes). The choice of transformation to use (z-transformation or centering) should not affect the experimental effect, but z-transformation could lead to confusion if one is interested in obtaining allometric parameters (a and b in Eqs. 1 and 2). Centering size on the log scale may also be easier to interpret than z-transformation (the issue of different variances described above aside). However, z-transformation has some advantages over centering when applied normally to a predictor variable (meaning, not within-group transformations). For example, z-transformation can help another aspect of interpretation because regression coefficients of continuous variables become comparable (in that they are standardized beta coefficients) .
Notably, we still assume that the variances for the response variable are homogenous in this model (in fact, this is true for all the models above). When this assumption is not met (that is, the response variable, y, is heteroscedastic), either we can model different variances in the response between two groups, or we can use “robust” statistical estimators, which take heteroscedasticity into account (for details on how, see ). We suspect that when variance in body sizes between two groups is different, it is likely that a trait of interest is heteroscedastic.
where b 3 is the difference in slopes between the control and experimental groups, and x [group] x [wgc(ln(size))] is an interaction term between x [group] and x [wgc(ln(size))] (Figs. 1 and 3). An example of implementing the above procedures in the statistical environment R  is provided in the Additional file 1. In this supplement, we refer to the assumption that size is measured without error in linear models such as Eqs. 4, 5, 7 and 8 [26, 27] and also provide a solution to this problem when this assumption is not met.
We have described the two major shortcomings of the division method, so far focusing on scenarios when we use the ratio variable (for instance, dividing a focal trait by size) as a response variable (y). However, it is just as common to find the ratio variable being used as a predictor variable (x). Among the many issues of using ratios as predictors, there is one problem that is very general and straightforward to describe . When we have two variables (or traits), A and B, their ratio is A/B. The variable A/B fitted as a predictor can be considered an interaction term, because A/B can be re-expressed as AB−1 (compare it with x [group] x [wgc(ln(size))] in Eq. 8). Therefore, we should also fit A and B−1 as predictors (main effects), along with the interaction term (AB−1) . More generally, it is usually not advisable to create and fit a derived variable (in other words, a variable comprised of more than one variable, such as, A/B, AB2) to a model. For linear modelling, raw measurements or their direct transformations should be used to control for confounding effects. Finally, because the division method and other inappropriate modelling procedures could lead to spuriously significant results and/or biased effect size estimates [1, 2], correct modelling practice is essential to avoid exacerbating the current “reproducibility crisis” [28, 29].
The authors thank Dan Noble, Russell Bonduriansky, and Holger Schielzeth for comments and discussions. SN is supported by an ARC Future Fellowship (FT130100268) and JLP by a SNSF Early Postdoc.Mobility fellowship (P2ZHP3_164962).
SN and ML conceived the idea. All authors contributed to developing the idea further and writing of the paper. All authors read and approved the final manuscript.
The authors declare that they have no competing interests.
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