Photosynthetically relevant foliar traits correlating better on a mass vs an area basis: of ecophysiological relevance or just a case of mathematical imperatives and statistical quicksand?

Authors

  • Jon Lloyd,

    Corresponding author
    1. School of Geography, University of Leeds, Leeds, UK
    • Centre for Tropical Environmental and Sustainability Science (TESS), School of Earth and Environmental Sciences, James Cook University, Cairns, Queensland, Australia
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  • Keith Bloomfield,

    1. School of Geography, University of Leeds, Leeds, UK
    2. Research School of Biology, Australian National University, Canberra, ACT, Australia
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  • Tomas F. Domingues,

    1. Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Univeridade de São Paulo, São Paulo, Brazil
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  • Graham D. Farquhar

    1. Research School of Biology, Australian National University, Canberra, ACT, Australia
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(Author for correspondence: tel +61 40 757 6864; email jon.lloyd@jcu.edu.au)

Plant biologists have long had a choice of the dimensions in which to express their studied traits and/or processes. For example, a plant physiologist working at the leaf level might typically measure and report photosynthetic rates as a flux per unit leaf area (e.g. μmol CO2 m−2 s−1) whereas a biochemist might typically express the same process per unit chlorophyll. An agronomist or forester would usually be more interested in dry matter accumulation rate per unit ground area (Mg ha−1 yr−1) or sometimes even as a relative growth rate (g DW g−1 DW d−1). Although expressing leaf-level photosynthesis on an area basis seems intuitive and was for decades the standard practice, starting with Field & Mooney (1986) and then Reich & Walters (1994), there has been an increasing tendency to express the photosynthetic characteristics of leaves on a dry-weight basis (typically nmol CO2 g−1 s−1). This trend has been due, at least in part, to stronger correlations for mass-based photosynthetic rates with foliar properties thought to be important in their modulation (Reich et al., 1998). Weaker associations between foliar properties when expressed on an area-basis have also provided one rationale for the inclusion of mass-based measures of photosynthesis, nitrogen and phosphorus into a so-called ‘leaf economics spectrum’ (Wright et al., 2004) and with mass-based measures of photosynthetic carbon exchange subsequently underlying further analyses (Shipley et al., 2006). Some modelling studies investigating the relative importance of nitrogen vs phosphorus as modulators of leaf photosynthetic capacity have likewise been parameterized on a mass rather than an area-basis because of the apparently superior model fit of the former (Domingues et al., 2010).

Area- and mass-based measures of any foliar trait are, of course, readily inter-convertible through the simple relationship

display math(Eqn 1)

where Θm is the value of trait Θ expressed on a per unit mass basis, Θa is its equivalent value on an area-basis and Ma is the leaf mass per unit leaf area (typically in units of g m−2). This means that any investigation of the nature of the variation in Θm in relation to Ma is also an investigation into how Θa/Ma varies in relation to Ma; and from which, as first pointed out by Pearson (1897), some interesting statistical properties emerge. Such ‘ratio correlations’ have been studied in some detail (Chayes, 1971) and, as we show later, the relevant theory can simply be applied to help understand the apparently different relationships between plant traits expressed on area vs mass bases. It turns out that when the coefficients of variation of any two traits are of a similar magnitude, then the area- vs mass-based correlations must, in all practical situations, be different. This inevitable difference can apply to both the sign and magnitude of the regression coefficient (r) and one must be very careful when interpreting the difference between the two metrics. In many cases this has nothing to do with plant function or with one measure being fundamentally better than the other. But rather to do with some less well known statistical properties of ‘ratio correlations’. Although the mathematical statistics involved are not overly complicated, we start with a simple simulation to help illustrate our points, then explaining observed results through the presentation of established statistical theory. This first analysis is presented for the simplest case: a single area-based trait not correlated with Ma. The effects of area-to-mass transformations for a single trait significantly correlated with Ma on an area basis is then examined: the results from which help to understand the final case investigated; viz., How do slopes and correlation coefficients change when two area-based traits are transformed and subsequently regressed on a mass basis? We then consider the implications of our results; especially in terms of whether species-level variations in leaf photosynthetic properties are, indeed, closely associated with intraspecific variability in other leaf traits such as Ma and leaf longevity (τ) as is now generally assumed to be the case (Westoby & Wright, 2006).

The simplest case: a single trait (Θ) expressed on an area basis not correlated with Ma

Imagine any trait (Θa) varying randomly across a plant population on a leaf-area basis – it could be, for example, trichome abundance, stomatal density or net CO2 assimilation rate. Let the variation of the area-based entity observed be simply quantified according to the variance (Va) of the sample, this being calculated as math formula where math formula is the sample mean. Assuming that math formula and Va provide unbiased estimators of the true population mean (μ) and variance (σ2), respectively, then if the frequency distribution of Θa within the population is known (or can be validly assumed) it is simple to generate an artificial population of any selected size using random number generation techniques (for the statistical method used see Supporting Information Notes S1). One of the best known plant physiological datasets is that of Wright et al. (2004) and so using this dataset as a basis for our random trait generation by taking the 786 measurements of area-based photosynthesis and calculating the mean rate (Aa) observed – but first log transforming to the base10 as proposed by the authors. We find that μ = 1.01 and σ2 = 0.054. Noting that the logarithmically transformed values can be regarded as dimensionless (being the base10 of the untransformed Aa and thus representing a value relative to the base10 of our chosen reference Aa = 10 μmol CO2 m−2 s−1) then by assuming a normal distribution, that is, Aa ~ N(μ, σ2) – and with the prefixed ℓ superscript to remind us that we are dealing with logarithmically transformed data – then the resulting artificial population of 1000 individual species so generated is shown in Fig. 1 (right-side histogram – note the logarithmic scale). A similar artificial randomly generated population may also be generated for leaf mass per unit leaf area (Ma) in units of g m−2 and for which Ma ~ N (2.02, 0.068) with the statistical distribution of the data so generated shown in the top histogram of Fig. 1. As these two artificial populations have been generated independently of each other, there is no significant correlation between Aa and Ma (= 0.03; = 0.58). It is, however, informative to see what happens when we divide our randomly generated estimates of Aa by the randomly generated estimates of Ma; this then giving a random population of Am (in units of nmol CO2 g−1 s−1) with Am so derived being plotted against the same Ma forming the basis of their calculation in Fig. 2 (again note the log–log scale). Despite both input variables (viz., Aa and Ma) being randomly generated, there emerges a ‘highly significant’ relationship between Am and Ma (r = −0.74, P < 0.0001).

Figure 1.

Relationship between 1000 randomly simulated data for net CO2 assimilation rate (area basis) and leaf mass per unit area (log–log scale). Frequency distribution histograms of the artificially generated population are shown on the upper and right side axis.

Figure 2.

The exact same 1000 data values as in Fig.1, but with net CO2 assimilation rate now expressed on a per unit mass basis.

At first sight this may seem surprising, but is exactly what is expected for a ‘part of whole correlation’ (Chayes, 1971); so termed because we are using log transformed data and because log(Aa/Ma) = AaMa = Am; we effectively have the whole (viz., AaMa) being examined as a function of the part (viz., Ma). Theory predicts that in such a situation a ‘spurious correlation’ should exist (i.e. even with Aa and Ma uncorrelated) with a ‘null regression coefficient’ (ρ) estimable as (Bartko & Pettigrew, 1968; Chayes, 1971)

display math(Eqn 2)

Exact agreement with the data is not to be expected due to uncertainties in our estimates of the true population means and variances, but Eqn 2 can be verified by inputting the variance estimates above which gives ρ = −0.75; very close to the r = −0.74 obtained from simple simulation. Equation 2 also illustrates that it is the ratio of the two variances that determines the magnitude of the observed effect – as can further be seen by investigating the likely limits. For example, were Aa to be relatively constant and Ma to be highly variable, then ρ → −1. However, were Ma to be constant and Aa highly variable, then ρ → 0. Had the two variances been exactly equal, then Eqn 2 predicts ρ = math formula = −0.71. Even if math formula were as large as 10, then ρ = −0.30 (see also Fig. A1 in the Appendix for a graphical representation of this). For the interested reader, a general derivation of Eqn 2 (based on Chayes, 1971, pp. 25–26) is given in Appendix A1 (Eqns A1–A6). Here it may be also worth noting that, because we are using logarithmically transformed data, our estimated variances are in both cases normalized and with their ratios therefore providing a measure of the relative variability of Aa and Ma within the dataset.

For the earlier simulations we have used artificially generated data based on the means and variance of Aa and Ma taken from Wright et al. (2004). But also noting that the bivariate AmMa relationship as reported by them (i.e. using the actual data) is r = −0.71, this being only marginally different to the ρ = −0.75 estimated earlier, then one reasonable conclusion might be that the statistically strong negative relationship between Am and Ma in the Wright et al. (2004) dataset is simply a consequence of Aa being in no way linked to Ma (functionally or otherwise). One way to test that hypothesis is through a statistical permutation approach (Manly, 2006), one variant of which is to simply randomly permute the dependent variable a large number of times (say > 1000), rerun the regression each time, and then see how the statistical significance of the population of t values obtained when the y variables have been shuffled compares with the true regression slope (as indicated by the t statistic). This is done by simply ranking the t statistics (or regression coefficients) from the permutation simulations and estimating the probability of significance of the original correlation as the proportion of the shuffled population that ends up with a higher (absolute) t statistic than observed for the original data. In applying this approach, we have therefore simply permuted the Aa values within Wright et al. (2004) 1000 times, in each case dividing the ‘swapped’ Aa of each leaf (denoted math formula) by its true Ma to give a permutated Am; this in turn being regressed against the true Ma. Estimates of r from these 1000 simulations range from −0.70 to −0.78 with the distributions of t-statistics from the math formula simulations shown in Fig. 3 with (as shown by an arrow) the true data regression value for the actual Am and Ma association in Wright et al. (2004) of t = 28.3 also indicated. Thus our actual t estimate lies around the upper 0.1 quantile of the distribution of permuted || statistics confirming that the statistically strong relationship between Am and Ma is no more than a consequence of Aa varying independently of Ma. That is, of course, totally consistent with the actual lack of correlation between Aa and Ma (= 0.05) in the underlying data as already demonstrated by Wright et al. (2004). But it also demonstrates that if these data had come from a single experiment with the leaf samples somehow all becoming mixed up after gas exchange measurements and their identifications confused before drying and leaf-area determinations, then it would not have mattered one little bit. A ‘strong’ negative correlation between Am and Ma with r ˜ −0.75 would still have been found: a concept we hope that some of our readers find at least a little disconcerting.

Figure 3.

Frequency distribution of t-statistics from ordinary least squares (OLS) regression of net CO2 assimilation rate (mass basis) as a function of leaf mass per unit area for 1000 artificially generated datasets for which area-based net CO2 assimilation rates had been permutated before the estimation of their mass-based values. The arrow shows the value of the t-statistic obtained through a regression of the original data which comes from Wright et al. (2004).

For the AmMa correlation, the OLS (ordinary least squares) regression slope (± standard error) observed, b1 ± s(b1), is −0.958 ± 0.034. This is very close to the value of −1.00 expected in the case of total independence between Am and Ma (easily seen through a substitution of Eqn A5.3 (shown in the Appendix A1) into Notes S1 Eqn S2.1 with x = Ma in the latter). This is also very different from the nonsignificant b1 estimate for the Aa; Ma correlation of 0.027 ± 0.034. But note identical standard errors for the slopes obtained from both area- and mass-based regressions: as indeed shown in Notes S1 (Eqns S2.1–S2.7) must be the case. It is just that for Aa the covariance term is minimal, whilst for the Am case this is ‘inflated’ because, as is shown by Eqn A5.2, the covariance in the ‘null’ AmMa case is the variance of Ma itself rather than being zero. With the same s(b1), this then causes the higher r and greater level of significance for the slope of the mass-based regression.

The relationship between Am and Aa is also of interest and so the data from Wright et al. (2004) are plotted in Fig. 4 for which r = 0.63 (= 23.0). Similar to before, randomizing Ma 1000 times to give the required math formula gives t ranging from 20.8 to 27.2 and with only c. 0.20 of the simulated observations yielding a higher level of statistical significance than that calculated from the unmanipulated data (histogram not shown). We can thus conclude, as for the relationship between Am and Ma, that the strong association observed between Am and Aa is exactly (just) what would be expected for Aa and Ma varying independently of each other.

Figure 4.

Relationship between net CO2 assimilation rates on an area vs a mass basis (log–log scale). Original data from Wright et al. (2004).

One question the persevering reader may be asking at this stage is to what extent this analysis has been influenced by the use of logarithmic transformations. In the case with no transformation we have a simple correlation of a ratio (Aa/Ma) with its denominator (Ma) for which the null correlation can be expressed as (Chayes, 1971, p. 14)

display math(Eqn 3)

which is similar to Eqn 2, but differs through the effects of the variances being weighted by the relative magnitudes of their mean. For the nonlogarithmically transformed data of Wright et al. (2004) we can calculate from the mean and variances that ρ = −0.65. This is somewhat less than the value of ρ = −0.75 found for the logarithmically transformed data earlier but also steeper than the r = −0.50 correlation for the original (nontransformed) data itself. Thus, the basic principles as outlined in this paper remain the same irrespective of whether the data have been log-transformed or not. This is despite the obvious algebraic and hence numerical differences associated with the logarithmic transformations

From the earlier analyses we conclude that significant correlations between Am and both Aa and Ma do indeed exist in the dataset of Wright et al. (2004), but that these correlations may be considered ‘spurious’: Yet only spurious in the sense that the parent variables required for the estimate of Am and against which it was subsequently regressed (viz., Aa and/or Ma) were themselves uncorrelated. That is to say, the correlations themselves are real. But also obviously requiring careful insights for their interpretation: especially in terms of their biological significance. And especially regarding inferences as to the nature of their underlying causes. Given these complexities and with the word ‘spurious’ perhaps having an unnecessarily derogatory connotation (Snedecor, 1956) we simply dub this the ‘lulu- effect’; the word ‘lulu’ meaning ‘a person or thing considered to be outstanding in size, appearance’ (Treffry, 2005). Less colloquially we also note that the apparently innocent action of undertaking an area- to mass-transformation could in some cases be considered equivalent to opening Pandora's box. It is no coincidence that Lulu was also the femme fatale of a radical early 20th century German play Die Büchse der Pandora (Wedekind, 1904) with a popular silent movie of the same name and the famous ‘Lulu’ opera subsequently emerging (Berg, 1937). That our anti-heroine could in all depictions be described as dangerously alluring makes the designation ‘lulu-effect’ seem remarkably apt.

The next simplest case: a single trait expressed on an area basis, correlated with Ma

Wright et al. (2004) also analysed variations in foliar nitrogen contents in relation to Ma on both a leaf-area (Na) basis and leaf-mass (Nm) basis. Here, however, in contrast to Aa, a significant association of Na with Ma was observed (Fig. 2d of Wright et al. redrawn here as Fig. 5a; r = 0.60). This allows us to probe the magnitude of the lulu-effect when some correlation exists in the parent (area based) variables. Here, for consistency, we have used only foliar nitrogen and Ma data for which the dataset also had corresponding estimates of Aa (764 out of 1968 in total). First, estimating σ2 = 0.037 for Na and through its substitution for Aa in Eqn 2 calculating ρ = −0.80 we find a lulu-correlation stronger than that observed for CO2 assimilation. This is as expected because the variance of Na is smaller than that for Ma (see Eqn 2). Applying the actual Wright et al. (2004) data for which r = −0.68, we also find that strength of the null correlation is actually stronger than that observed. The significance of this difference can again be tested using a permutation approach and for the 1000 math formula so generated, r varies from −0.78 to −0.83 with the frequency distribution of the t estimates shown in Fig. 5(b), Thus, even though there is a strong and convincing correlation between Nm and Ma, it is also the case that this correlation would have been even stronger had there been no correlation at all between Na and Ma. This is illustrated further in Fig. 5(c) where a randomly selected ‘permuted data regression’ and the actual data are compared.

Figure 5.

Relationships between mass- vs area-based leaf nitrogen measures and leaf mass per unit area (Ma). (a) Leaf nitrogen (area-basis) vs Ma. (b) Frequency distribution of t-statistics from ordinary least squares (OLS) regression of leaf nitrogen (mass basis) as a function of Ma for 1000 artificially generated datasets for which area-based nitrogen concentrations had been permutated before the estimation of their mass-based values. The arrow shows the value of the t-statistic obtained through a regression of the original data. (c) Leaf nitrogen (mass basis) vs Ma. Grey symbols, randomly permutated area-based nitrogen; black symbols, original data as in Wright et al. (2004).

The reason that the lulu-correlation between Nm and Ma is stronger than for the dataset itself is explained through the positive correlation between Na and Ma reducing the covariance between Nm and Ma below that which would be observed were Na and Ma to be unrelated. This can be formalized mathematically through a substitution of Na = Θa in Eqn A6.3 in the Appendix A1, viz.

display math(Eqn 4)

where r is the expected correlation between Nm and Ma and r* is the observed correlation between Na and Ma. Indeed, substitution of r* = 0.60 in Eqn 2 (along with the appropriate variances as detailed earlier), predicts a correlation coefficient of −0.69 for the NmMa association, very close to the value r = −0.68 observed. From Eqn S2.1 (Notes S1) and Eqn A5.2 (Appendix A1) the slope of the NmMa relationship can also be simply predicted from the area-based metrics:

display math(Eqn 5)

giving an estimated b1 for the NmMa relationship of −0.54, very close to that obtained via OLS of −0.55. Thus, the positive covariance between Na and Ma (as represented in Eqns 4 and 5 through the r* term) not only reduces the magnitude of the mass-based regression coefficient, but also the slope of the NmMa relationship below that expected for a zero correlation between Na and Ma.

Because area-based metrics can easily predict the nature of both the AmMa and NmMa relationships, in our view this brings into doubt one aspect of the ‘leaf economics spectrum’ viz., ‘gradients of leaf N on a mass vs an area basis hence represent fundamentally different multiple-trait gradients because of their different patterns of covariation with Ma’ (Wright et al., 2004). This is because, as should be clear from the earlier analysis, the two patterns are not only inevitably linked, but also inevitably different. And with the nature of their different relationships easily understood through simple equations for which only a knowledge of the variance and covariances of two area-based parameters are required. In short: as Ma increases so does Na, but with the magnitude of this increase less than that required to prevent Nm declining.

Indeed, as is illustrated in the Fig. A1 (Appendix A1), because for the dataset here the variances for both Aa and Na are less than for Ma, it is actually in this instance impossible for their associated mass-based slopes to be positive. Figure A1 also shows that negative mass-based correlation coefficients for both these entities are remarkably insensitive to the strength of the area-based correlation; especially for nitrogen where math formula is only c. 0.55. This means that approximately the same (relatively strong) mass-based associations would have been found for both photosynthetic rate and foliar nitrogen even had Aa and/or Na shown dramatically different relationships with Ma.

We also note that converting Ma to specific leaf area does nothing to solve the lulu-correlation problems as outlined above. Indeed there is an inevitable symmetry. For example, re-writing Eqn 1 as math formula (where S is the specific leaf area) simply leads to a swapping of Θa with Θm for the axes of Fig. A1 with the variance ratios now expressed in terms of Θm and S. Because we use logarithmically transformed data math formula, and the relationships between the area variances and mass variances (e.g. Eqn A4.5) inevitably remain unaltered. As is discussed later on, the question then comes down then to whether it is the area- or mass-based representation of the trait that is fundamental (as opposed to derived) in a physiological sense.

Two area-based traits compared on a mass basis

The analysis so far has shown that, even though leaf CO2 assimilation rates (area basis) can vary independently of Ma, a strong lulu-correlation effect is inevitably observed when expressed on a mass basis. For nitrogen there was positive correlation of Na with Ma resulting in the weaker negative correlation of Nm with Ma and with a slope shallower than that expected on the basis of the lulu-correlation alone. Most importantly, because the variances of both Aa and Na are small relative to Ma, the nature of the negative mass-based correlations is extremely insensitive to the strength (or even sign) of the AaMa or NaMa association.

So, how is the lulu-correlation effect manifest when both Aa and Na are divided by Ma and with both traits then compared on a mass basis? This is termed a ‘common element’ correlation (Chayes, 1971) and, using a similar approach as in Eqns A3–A6 (Appendix A1), it is shown in Notes S1 (Eqns S3.1–S3.6) that if Aa and Na and Ma are all independent then

display math(Eqn 6.1)

which can be rewritten as

display math(Eqn 6.2)

Comparing 6.2 with Eqn 2 we see that the common-element lulu-correlation can be simply quantified as the product of the two part-whole correlations from which it is derived. Thus, applying 6.2 we obtain from our previous analyses for AmMa and AmNa an estimated ρ for the AmNm association of −0.75 × −0.80 = 0.60; a value confirmed by a simple randomizing of Aa and Na and Ma before the calculation of associated math formula and math formula; the results of which are shown along with a plot of the actual data in Fig. 6(c). Here, the actual data give r = 0.73, these also being shown in Fig. 6(c) together with the AaNa relationship (Fig. 6a; r = 0.37) and a histogram showing the t-values from 1000 random permutations of the actual data; with t = 28.1 for the unpermutated data also indicated by an arrow (Fig. 6b). Here, the randomized datasets (for which there is on average no correlation between any bivariate pairing of Aa, Na and Ma) gives rise to a median t = 19.9. This is a highly significant value (consistent with the lulu-correlation coefficient of −0.60 as calculated earlier). But all such randomized t values are much less than the 28.2 obtained from a regression of the actual data. This indicates that the mass-based regression is in any case still highly significant. But importantly, after correction for the lulu effect, no more significant than the Aa and Na regression itself for which t = 10.6 (P < 0.0001). Thus, for our theoretical case with all the sample labels mixed up somewhere before nitrogen and/or Ma determinations, a strong association (although not as good as the actual data) would still have been found with a correlation coefficient of c. 0.60 (grey dots in Fig. 6c); a concept we hope that at least some of our readers find at least a little disconcerting.

As for the part-whole correlations and as shown in Notes S1 (Eqns S3.1–S3.6), we could have easily estimated the AmNm correlation coefficient just from a knowledge of the relative variances and covariances of Aa, Na and Ma. Likewise for the slope equation whose complexity illustrates that, despite a generally higher correlation coefficient, lulu-effects serve to blur rather than enlighten:

Figure 6.

Relationships between mass- vs area-based photosynthesis and leaf nitrogen measures. (a) Net CO2 assimilation rate vs leaf nitrogen (area-basis). (b) Frequency distribution of t-statistics from ordinary least squares (OLS) regression of net CO2 assimilation (mass basis) as a function of leaf nitrogen (mass basis) for 1000 artificially generated datasets for which area-based net CO2 assimilation rates had been permutated before the estimation of their mass-based values. The arrow shows the value of the t-statistic obtained through a regression of the original data. (c) Net CO2 assimilation vs leaf nitrogen (mass basis). Grey symbols, with randomly permuted net CO2 assimilation rate (area basis); black symbols, original data as in Wright et al. (2004).

display math(Eqn 7.1)

where here C is the AmNm covariance, and each of the three r* refers to a separate regression coefficient (that between the two variables whose variances form part of the same product). 7.1 shows that, even if the variances stay reasonably constant (or constant in relation to each other), the slope of any AmNm correlation is also dependent on factors other than the nature of the AaNa correlation. For example, the strength of the NaMa association. As the intercept of any OLS relationship (b0) is simply the mean of the dependent variable less the mean of the independent variable by simple formula (i.e. in this case math formula) then any functional interpretation of a changes in the intercept must also be much more complicated for these mass-based regressions because of the complexity embedded within 7.1.

The potential magnitude of these confounding effects can readily be seen through the insertion of the relevant Wright et al. (2004) values into 7.1 (with all products multiplied by 1000 for ease of presentation) viz.

display math(Eqn 7.2)

This shows that the slope of the AmNm relationship is sensitive to many factors. Thus, in addition to the covariation between leaf nitrogen and Ma resulting in relationships between leaf nitrogen and other traits differing substantially when expressed on a mass vs area basis (Wright et al., 2004), there is an additional and potentially more important contributing factor to this difference: That is the existence of the large Ma variance term in both the numerator and denominator of 7.2 inflating the covariance for the mass-based regression. This can easily be seen by setting r*(NaMa) = 0 for which we then find b1 = 0.79 and = 0.74: both terms still being substantially greater than the AaNa case (b1 = 0.45 and = 0.37). Although Wright et al. (2004) were mostly interested in the (standard) major axis regression slope, the same general considerations as noted for the OLS earlier also apply.

We also note that although b1 varies substantially for the mass as opposed to area-based case (OLS estimates of 1.244 ± 0.042 vs 0.453 ± 0.042) as for the part-whole correlations earlier, the standard errors of the OLS slope estimates are identical. Again, the greater apparent significance of the mass-based relationship is simply due to the transformation from an area to a mass basis giving steeper slopes; mostly through the presence of the relatively large Ma variance term in both the numerator and denominator (Notes S1, Eqn S3.6).

Finally we note that, although Eqn 7.1 might be considered relatively complicated it does provide a quantitative explanation for mass- vs area-based differences in a relatively straightforward way. And as noted by Chayes (1971), with its relatively complex formulation being a true reflection of the not-so-simple hypotheses inadvertently being tested.

Consequences of the lulu-correlation effect

That ‘spurious correlations’ could exist when two variables were standardized in some way was first pointed out in a classic paper by Pearson (1897). Since then, occasional controversy has arisen in the biological literature regarding their importance, most recently in studies examining allometric/scaling relationships (e.g. Jackson & Somers, 1991; Hayes, 2001). Problems associated with the interpretation of correlations involving ratios are also long known to the geochemistry community. Here the relevant theory has been most extensively developed (Chayes 1971) as the problem is all but unavoidable. The composition of one element within a rock can only be expressed in relation to the composition of some part (or all) of the remaining elements leading Aitchison (1986) to develop a comprehensive theory of compositional analysis based on log-ratio normal distributions. For plant scientists not expressly interested in stoichiometric scaling (or so-called ‘nutrient use efficiencies’) the situation is different, but with the investigator often having to make a conscious choice regarding area- vs mass-based analysis of their data. But, as we hope the earlier examples illustrate, unless one is simply interested in finding the best correlation irrespective of any functional meaning, then basing one's interpretation on the strength of one relationship as opposed to the other may provide little practical guidance.

Rather, the relevant question is which of the two is the ‘parent’ correlation: that is, the one justified on theoretical or logical grounds. Given that the main function of leaves is to intercept light for which the rate of arrival of photons into the plant canopy has a natural dimension of flux density (i.e. flux per unit area), then an area-based photosynthesis metric seems to us to be the more logical one. If one accepts that assertion, then because there is no systematic relationship between photosynthesis and leaf-mass-per unit area, as the relative variance of Aa is less than Ma (for reasons we discuss later), then this lack of correlation means that Aa/Ma must decline with Ma with an OLS slope close to −1.0 and with ≈ −0.75 for logarithmically transformed data.

Is there any functional meaning to the AmMa relationship that might serve to negate this contention of a mere tautological construct? For certain it demonstrates the inevitably lower instantaneous marginal rate of return in high Ma leaves on what is – if we accept the Ma/leaf lifetime (τ) trade-off – a longer-term investment. But as can be seen from Fig. A1, because math formula then, irrespective of how Aa varies with Ma, the existence of some sort of negative and likely significant relationship between Am and Ma is statistically inevitable. With the capital being lost at the end of the investment (i.e. a leaf needing to be replaced), the relative rate of return on the investment integrated over the effective duration of its depreciation should, however, still provide a legitimate metric by which to compare different ‘investment strategies’ (Feibel, 2003). But when looked at this way, the AmMa correlation (with Am now the total carbon return per unit invested carbon over a leaf's lifetime) all but ceases to exist (Kikuzawa & Lechowicz, 2006). That is to say, although the existence of a negative AmMa relationship across the terrestrial plant kingdom is by itself trivial, the intuitive and long-known Maτ trade-off (Chabot & Hicks, 1982) remains paramount for our understanding of the many different strategies employed in terms of foliar carbon investment and paybacks at the individual leaf level.

Choosing the ‘parent’ correlation

For many nutrients, the choice of ‘parent’ correlation will be less straightforward than for photosynthesis because of their multiple physiological roles within the leaf. For nitrogen, which typically displays a strong and well understood association with photosynthetic capacity (Evans, 1989), there is a clear rationale for any mechanistic modelling of photosynthesis–nutrient relationships to be done on a leaf-area basis. Especially as strong lulu-correlations can arise through the simple procedure of dividing two unrelated area-based entities by Ma (Fig. 6). In the past we too have chosen to analyse photosynthetic nutrient controls on a mass basis because of these stronger correlations (Domingues et al., 2010), but it is now clear that, despite the seductions of the lulu-effect, the true statistical strength of the photosynthesis–nutrient associations is actually no better under the Ma transformation. Unless it is somehow assumed that the photosynthesis–nutrient relationship under investigation cannot validly exist on an area basis. Further, when comparing different treatments, sites or plant functional groupings, Eqn 7 shows that AmNm relationships are potentially confounded by many factors not directly related to photosynthetic metabolism. This situation must become even more complicated as multivariate regression models are applied. As an indication, Eqns 4.15 and 4.16 of Edwards (1979) provide a means by which a group of correlation coefficient expressions of a similar form to Eqn S3.5 can be incorporated into a single algebraic expression allowing slope and significance estimates for multivariate Am↔Θm predictors to be expressed in terms of the associated area-based correlations. The resulting expression is mathematically tractable (not shown), but ends up of little practical use. This is because it contains a multitude of interacting terms underlying the almost incomprehensible complexity embodied in the (apparently simple) hypothesis being tested.

That is not to preclude, of course, cases where an analysis of photosynthetic properties in relation to mass-based traits might prove informative (Niinemets & Tenhunen, 1997; Niinemets, 1999) and in such cases we suggest that randomization techniques such as those illustrated here be employed as appropriate – especially as the background theory for applying such techniques in multivariate situations has now been developed (Pesarin, 2001) and with appropriate software freely available (Finos, 2012). The use of randomization tests to help solve ratio correlation problems was actually first suggested in Pearson's (1897) original article (see especially the note at the end by WFR Weldon) and Jackson & Somers (1991) have also suggested the use of permutation techniques to solve ratio-correlation problems inherent in many scaling ecology studies.

Because foliar nitrogen has many roles other than photosynthetic carbon acquisition, clean ‘universal’ relationships between Aa and Na are in any case unlikely. This is especially the case for cross-species and/or cross-biome comparisons. For example, Miller & Woodrow (2008) estimated that, depending upon species (but also being highly variable within species), anywhere between 0% and 15% of the total foliar nitrogen in the leaves of Australian tropical forest species was present as cyanogenic glucosides. A second example also related to foliar defences is the foliar alkaloid concentration of Coffea arabica (coffee tree) for which Cavatte et al. (2012) report values as high as c. 40 mg g−1 – these amounting to c. 30% of the total leaf nitrogen (F. M. DaMatta, pers. comm.). This is perhaps an extreme example (especially as the nitrogen content of caffeine is unusually high as compared to most alkaloids), but it does serve to illustrate, in addition to the allocation of proteins to numerous nonphotosynthetic roles including cell wall proteins (Harrison et al., 2009; Hikosaka & Shigeno, 2009) and the apoplastic immune response (for which the proteins involved are initially located in the cytoplasm: Wang & Dong, 2011), that greatly varying proportions of available nitrogen may be allocated to the photosynthetic machinery according to both species and environment. This is in addition, in certain circumstances, to the likelihood of elements such as phosphorus also limiting photosynthetic capacity (Domingues et al., 2010). Thus, very strong AaNa correlations should not necessarily be expected – especially in survey studies employing a range of different genotypes. And dividing both measures by Ma to produce stronger statistical relationships mostly only serves to conceal this interesting variability.

Although area-based metrics would seem more logical for analysis of photosynthesis–nutrient relationships, it is not disputed that when examining some environmental–trait interactions (for example effects of leaf nitrogen concentration on herbivory rates) mass-based expressions might be more appropriate. This is likely to also be the situation for many other elements as well. So, given the issues raised here, one reasonable data strategy might be to analyse both area- and mass-based relationships in conjunction with the associated null-correlations. Here we also note that for the majority of physiologically relevant elements some decline in mass-based concentrations will be observed if changes in tissue density are the main source of variation in Ma and little should be imported of this in an ecological sense. For example, although it has been inferred that low Nm in high Ma leaves might be part of a general spectrum of low palatability/high τ; (Wright et al., 2004) numerous other elements also show similar declines with increasing Ma (Fyllas et al., 2009). Thus, especially as a low Nm may serve to sometimes increase rather than decrease the probability of herbivory (due to more of the leaf needing to be consumed in order for the same amount of protein to be obtained: Mattson, 1980) the assignation of some sort of ecological significance to the inevitably lower Nm associated with a higher Ma is tenuous.

Photosynthetic capacity and leaf structure/longevity as independent traits

So, why is it – once the lulu-effect is taken into account – that photosynthetic characteristics are decoupled to such a large extent from traits such as Ma and mean leaf turnover time? We believe this is because they represent distinct, but slightly overlapping trait dimensions – the concept of which is perhaps best illustrated by an example. Compare a plant growing in the Siberian tundra with one growing in the Sahel region of Africa. Both have very short growing seasons and with little to be gained from leaf-retention over the nine or so months of climate induced inactivity (in one case due to extremely cold weather, in the other due to an almost total absence of precipitation). One might predict in such circumstances (assuming other factors such as nutrient availability and risk of herbivore damage were similar) that leaves of relatively low construction cost (i.e. low Ma) would be equally favoured. But the two environments are very different in terms of both light and water. Taking our hypothetical tundra plant first: during the growing season, days are long, water usually abundant, but incoming photon irradiance (Q) levels usually low (< 1000 μmol m−2 s−1). One might reasonably anticipate that the most favourable strategy would be to invest the carbohydrate available for leaf development so as to maximize the leaf area for light interception – a strategy in harmony with the low construction cost per unit leaf area (low Ma) and, due to the generally low Q, the lack of any requirement for a high photosynthetic capacity per unit area. However, the Sahelian savanna tree experiences a limited water supply combined for much of the day with exposure to high light conditions (Q > 1000 μmol m−2 s−1). Under such conditions it is predicted that the optimal whole-plant photosynthetic carbon gain be attained through the construction of fewer leaves, but with higher Na (Farquhar et al., 2002) and with Wright et al. (2005) indeed reporting a tendency for increases in Na with both irradiance and decreased rainfall in their global dataset. One would, of course, expect some increase in Ma for the higher Na leaves associated with the greater packing of nitrogen in the photosynthetic apparatus per unit leaf area (Niinemets, 1999; Terashima et al., 2006). But the fundamental point remains: strategies for foliar nitrogen allocation for optimizing carbon gain in relation to water availability (and nutrient supply) can to a large degree be independent of those involving the Maτ trade-off.

Here we note from some of our own work in such a sub-Saharan environment that for the few dominant tree species present (Acacia senegal Wild. and Combretum glutinosum Perr. Ex DC.), even though their Ma are surprisingly different, Aa and Na are indeed very high for both species at c. 19–22 μmol CO2 m−2 s−1 and 2.6–3.0 g m−2, respectively (Domingues et al., 2010: compare with Fig. 6a). We further suggest that there might be little point in having a superior photosynthetic capacity (Amax), because a higher Amax would only be utilizable during periods of the highest Q around the middle of the day. Any marginal gains at high Q would be unlikely to offset the increased losses in nitrogen-use efficiency associated with increases in palisade and overall leaf thickness (Syvertsen et al., 1995; Terashima et al., 2006), especially if strong stomatal limitations associated with high leaf-to-air vapour pressure deficits were to simultaneously occur around these times.

This then gives rise to the idea that there is intrinsically less scope for variability of a leaf's area-based photosynthetic capacity as opposed to its structure (as approximated by Ma). Taken together with the lack of any obligatory correlation between Aa and Ma, this lower relative variability in Aa as opposed to Ma then leads to a family of strong but functionally irrelevant lulu-correlations. In that context we suggest that the main reason one does not observe leaves with both a high Ma and a high Nm is not that this combination is at odds with a high τ because of high herbivore susceptibility or some related trade-off (Wright et al., 2004). Rather, the reason that one does not find high Nm and high Ma leaves is because the associated high Na could never be efficiently utilized for photosynthetic carbon acquisition, even at the highest Q.

Although it seems likely that the long known Maτ trade-off (Chabot & Hicks, 1982) may be functionally correlated with other foliar characteristics such as cation concentrations and secondary chemistry (Poorter et al., 2009; Patiño et al., 2012) as well as different strategies of canopy development in competitive environments (Ackerly, 1996) our overall conclusion must be that photosynthetic characteristics and area-based associated variations in nutrients primarily involved in the structure and/or function of the photosynthetic machinery need not be associated closely with either Ma or τ. As a whole, we therefore also feel that progress in the general area of plant trait linkages and ‘trade-offs’ may have been hindered over recent times through a tendency for plant researchers (ourselves included) to underestimate the true (and meaningful) variability in trait relationships; perhaps because of a latent desire to see patterns and order in the natural world which are not (and probably should not) be there. Such ignes fatui have been reinforced by the lulu-correlation common denominator framework and physiological plant ecology may be better advanced by adopting the simple notion that the primary trade-offs dictating how a plant allocates its photosynthetic resources (more leaf area vs more photosynthetic capacity per unit leaf area) are driven by a set of environmental factors largely independent of those influencing leaf structure, longevity and habit.

Appendix A1

Take any physiologically relevant entity, Θ: expressible on either an area-or mass-based scale. Then

display math(Eqn A1.1)

where Θm is the mass-based measure, Θa is the area-based measure and Ma is the leaf mass per unit area. Expressed on a logarithmic basis

display math(Eqn A1.2)

The correlation of log(Θm) with log(Θa) can therefore be constituted to amount to a ‘part-whole correlation’; Ma vs (ΘaMa) where the superscript-prefixed reminds us we are working with logarithmically tranformed data. Clearly with Θm = (ΘaMa) then Ma and Θm cannot be independent and so our objective is to evaluate the correlation coefficient (r) expected to exist between Θm and Θa which should not be zero, even when Θ a and Ma are independent. Expressed in terms of the relevant variance/covariances:

display math(Eqn A2.1)

where C (Θm, Ma) is the sample covariance between Θm and Ma with math formula the associated standard deviations. In formal statistical terms we can define the covariance as

display math(Eqn A2.2)

where E denotes the expected value and with math formula = E(Θm) and math formula = E(Ma). Likewise, the two standard deviations can be formally defined through the population variances (V  ) as

display math(Eqn A2.3)

Noting the overlap in the expectation terms between Eqn A2.2 and Eqn A2.3, for ease of presentation later we define two deviations

display math(Eqn A2.4)

from which it then follows that

display math(Eqn A2.5)

Using standard, small sample statistical theoretical approaches (and also pointing out that what follows may be found expressed in general terms in Chayes (1971, p. 25), for Ma when expressed in terms of its mean (μ) and deviations (δ) from that mean then:

display math(Eqn A3.1)

and thus

display math(Eqn A3.2)

For Θm the situation is, however, less straight-forward, as from Eqn A1.2

display math(Eqn A4.1)

so that

display math(Eqn A4.2)

Then

display math(Eqn A4.3)

and so

display math(Eqn A4.4)

The expected value of the variance of Θm is therefore

display math(Eqn A4.5)

and which in the case of zero covariance between Θa and Ma reduces to

display math(Eqn A4.6)

The product of the deviances is

display math(Eqn A5.1)

and then converting to expectations

display math(Eqn A5.2)

which in the case of zero covariance between Θa and Ma reduces to

display math(Eqn A5.3)

Substituting Eqns A3.2, A4.4 and A5.3 into A2.1, we then obtain the expected null correlation between Θm and Ma (i.e. that expected in the absence of any correlation between Θa and Ma) as

display math(Eqn A6.1)

Equation A6.1 is a simpified version of a more complete equation allowing for some covariance between Θa and Ma, readily obtained through the substitution of Eqns A3.2, A4.5, A5.2 into Eqn 2.1 as

display math(Eqn A6.2)

from which we can remove the covariance term and replace it with the more readily intepretable ΘaMa correlation coefficient (denoted r*) through a rearrangement of Eqn A2.1 viz. math formula yielding

display math(Eqn A6.3)

In a manner similar to Bartko & Pettigrew (1968) Eqn A6.3 (right hand side) allows us to examine the effect of the strength of the parent (in our case area based) regression on the mass-based correlation coefficient as a function of math formula and this is shown in Fig. A1 for math formula = 0.1,1 and 10 as well as for the observed math formula and math formula. This shows, as is also evident from examination of Eqn A6.3 that, for math formula, a positive mass-based correlation coefficient (i.e. r > 0) is simply not possible and that even for math formula = 10, a significant lulu-correlation of ρ = –0.30 is obtained.

Figure A1.

Relationship between area- and mass-based trait; leaf mass per unit area (Ma) correlation coefficients as dependent upon the area-based trait (Θa) variance in relation to the Ma variance. Simulations are shown for three predefined Θa/ Ma variance ratios (0.1,1.0 and 10) as well as those observed in the dataset of Wright et al. (2004) for logarithmically transformed net CO2 assimilation rate (Aa) and area-based nitrogen concentration (Na).

Ancillary