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?

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). 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 where is the sample mean. Assuming that 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, ℓAa ~ 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 ℓMa ~ 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 ℓAa and ℓMa (r = 0.03; P = 0.58). It is, however, informative to see what happens when we divide our randomly generated estimates of ℓAa by the randomly generated estimates of ℓMa; 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., ℓAa and ℓMa) being randomly generated, there emerges a ‘highly significant’ relationship between ℓAm and ℓMa (r = −0.74, P < 0.0001). 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 ℓAa to be relatively constant and ℓMa to be highly variable, then ρ → −1. However, were ℓMa to be constant and ℓAa highly variable, then ρ → 0. Had the two variances been exactly equal, then Eqn 2 predicts ρ = = −0.71. Even if 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 ℓAa and ℓMa within the dataset. For the earlier simulations we have used artificially generated data based on the means and variance of ℓAa and ℓMa taken from Wright et al. (2004). But also noting that the bivariate ℓAm↔ℓMa 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 ℓAm and ℓMa 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 ℓAa values within Wright et al. (2004) 1000 times, in each case dividing the ‘swapped’ ℓAa of each leaf (denoted ) by its true ℓMa to give a permutated ℓAm; this in turn being regressed against the true ℓMa. Estimates of r from these 1000 simulations range from −0.70 to −0.78 with the distributions of t-statistics from the simulations shown in Fig. 3 with (as shown by an arrow) the true data regression value for the actual ℓAm and ℓMa 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 |t | statistics confirming that the statistically strong relationship between ℓAm and ℓMa is no more than a consequence of ℓAa varying independently of ℓMa. That is, of course, totally consistent with the actual lack of correlation between ℓAa and ℓMa (r = 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 ℓAm and ℓMa with r ˜ −0.75 would still have been found: a concept we hope that some of our readers find at least a little disconcerting. For the ℓAm↔ℓMa 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 ℓAm and ℓMa (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 ℓAa; ℓMa 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 ℓAa the covariance term is minimal, whilst for the ℓAm case this is ‘inflated’ because, as is shown by Eqn A5.2, the covariance in the ‘null’ Am↔Ma 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 (t = 23.0). Similar to before, randomizing ℓMa 1000 times to give the required 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 ℓAm and ℓAa is exactly (just) what would be expected for ℓAa and ℓMa varying independently of each other. 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 we simply this the the word or considered to be in we also note that the apparently of an area- to could in some cases be considered to It is no that was also the of a with a of the same and the subsequently That our could in all be as the Wright et al. (2004) also in foliar nitrogen in to Ma on both a leaf-area basis and however, in to a significant association of with ℓMa was observed of Wright et al. as Fig. r = This us to the magnitude of the when some correlation in the parent for we have used only foliar nitrogen and Ma data for which the dataset also had estimates of Aa of in estimating σ2 = for and through its substitution for ℓAa in Eqn 2 calculating ρ = we find a stronger than that observed for CO2 This is as expected because the variance of is than that for ℓMa (see Eqn the actual Wright et al. (2004) data for which r = we also find that of the correlation is stronger than that The significance of this can be using a permutation approach and for the 1000 so generated, r from −0.78 to with the frequency distribution of the t estimates shown in Fig. even there is a strong and correlation between and it is also the case that this correlation would have been even stronger had there been no correlation at all between and ℓMa. This is further in Fig. where a randomly selected data and the actual data are area-based can the nature of both the Am↔Ma and in our this into one of the ‘leaf economics spectrum’ of leaf N on a mass vs an area basis different because of their different of with (Wright et al., This is because, as be from the earlier the two are not only but also And with the nature of their different through simple for which only a of the variance and of two area-based are In as Ma so but with the magnitude of this than that required to as is in the Fig. A1 because for the dataset the variances for both ℓAa and are than for it is in this for their mass-based slopes to be A1 also that negative mass-based correlation for both these are to the of the area-based especially for nitrogen where is only c. This means that the same mass-based associations would have been for both photosynthetic rate and foliar nitrogen even had Aa and/or shown different with We also note that Ma to leaf area to the as there is an For example, Eqn 1 as is the leaf simply to a of Θa with for the of Fig. A1 with the variance ratios expressed in terms of and we logarithmically transformed data and the between the area variances and mass variances (e.g. Eqn As is the then then to it is the area- or mass-based representation of the trait that is (as to in a physiological The so has shown that, even leaf CO2 assimilation rates can independently of Ma, a strong effect is observed when expressed on a mass For nitrogen there was correlation of with Ma resulting in the negative correlation of with Ma and with a slope than that expected on the basis of the because the variances of both Aa and are relative to Ma, the nature of the negative mass-based correlations is to the (or even of the or As for the correlations and as shown in Notes S1 (Eqns we could have estimated the correlation just from a of the relative variances and of and ℓMa. for the slope illustrates that, a higher correlation to rather than This that the slope of the relationship is to in to the between leaf nitrogen and Ma resulting in between leaf nitrogen and traits when expressed on a mass vs area basis (Wright et al., there is an and more important to this That is the of the large ℓMa variance term in both the and of the covariance for the mass-based regression. This can be seen by = for which we then find b1 = and r = both terms still being greater than the case = and r = Although Wright et al. (2004) were interested in the regression the same general as for the OLS earlier also We also note that b1 for the mass as to area-based case estimates of ± vs ± as for the correlations earlier, the standard errors of the OLS slope estimates are the greater significance of the mass-based relationship is simply due to the from an area to a mass basis giving through the of the relatively large Ma variance term in both the and Eqn we note that, Eqn might be considered relatively it provide a for vs area-based in a relatively And as by with its relatively being a true of the being That could exist when two variables were in some way was first in a by has in the biological regarding their in studies (e.g. & with the of correlations ratios are also long known to the Here the has been as the is all but The of one within a can only be expressed in to the of some (or of the (1986) to a of based on normal For plant not interested in (or so-called the is but with the having to a choice regarding area- vs mass-based of their data. as we hope the earlier one is simply interested in the best correlation of any then on the of one relationship as to the may provide little the is which of the two is the that is, the one on or Given that the of leaves is to for which the rate of of into the plant has a of flux density (i.e. flux per unit then an area-based photosynthesis seems to us to be the more one that then because there is no relationship between photosynthesis and unit as the relative variance of Aa is than Ma (for we then this lack of correlation means that must with Ma with an OLS slope close to and with r −0.75 for logarithmically transformed data. there any to the Am↔Ma relationship that might to this of a For it demonstrates the rate of in Ma leaves on what is – if we the – a But as can be seen from Fig. because of how Aa with Ma, the of some of negative and likely significant relationship between Am and Ma is statistically With the being at the of the (i.e. a leaf to be the relative rate of on the the of its however, still provide a by which to different But when at this the Am↔Ma correlation Am the total carbon per unit carbon a all but to exist & 2006). That is to say, the of a negative Am↔Ma relationship across the plant is by itself the intuitive and & for our of the different in terms of foliar carbon and at the individual leaf For the choice of correlation be than for photosynthesis because of their physiological within the For which typically a strong and association with photosynthetic capacity there is a rationale for any modelling of to be done on a leaf-area as strong can through the simple of dividing two area-based by Ma In the we have chosen to photosynthetic on a mass basis because of these stronger correlations (Domingues et al., but it is that, the of the the true statistical of the associations is no the Ma it is somehow that the relationship validly exist on an area when different or plant Eqn that are by not to photosynthetic This must even more as regression are As an and of provide a means by which a of correlation of a similar to Eqn can be into a single slope and significance estimates for to be expressed in terms of the area-based The resulting is but ends up of little This is because it a of terms underlying the in the hypothesis being That is not to of course, cases where an of photosynthetic properties in to mass-based traits might informative & and in cases we that techniques as be as – especially as the for applying techniques in has been and with The of to ratio correlation was first in original (see especially the note at the by and & have also the of permutation techniques to in foliar nitrogen has than photosynthetic carbon between Aa and are in any case This is especially the case for and/or For example, & estimated that, species also being highly variable within between and of the total foliar nitrogen in the leaves of species was as A also to foliar is the foliar of for which et al. report values as as c. g−1 – these to c. of the total leaf nitrogen This is perhaps an as the nitrogen of is as to but it to in to the of to et al., & and the (for which the are in the & that varying of nitrogen may be to the photosynthetic according to both species and This is in in to the of as phosphorus also photosynthetic capacity (Domingues et al., 2010). very strong correlations not be expected – especially in studies a range of different And dividing both measures by Ma to stronger statistical only to this Although area-based would more for of it is not that when some (for of leaf nitrogen on mass-based might be more This is likely to also be the for as given the one reasonable data might be to both area- and mass-based in with the Here we also note that for the of some in mass-based be observed if in density are the of variation in Ma and little be of this in an For example, it has been that in Ma leaves might be of a general of (Wright et al., 2004) also similar with increasing Ma et al., especially as a may to sometimes rather than the probability of to more of the leaf to be in for the same of to be the of some of significance to the with a higher Ma is is it – the is taken into – that photosynthetic characteristics are to a large from traits as Ma and mean leaf We this is because but trait dimensions – the concept of which is perhaps best by an a plant in the with one in the of have very and with little to be from the or so of (in one case due to in the due to an total of One might in as and of were that leaves of relatively (i.e. Ma) would be But the two are very different in terms of both and our plant the are usually but usually < 1000 μmol m−2 s−1). One might that the would be to the for leaf so as to the leaf area for – a in with the per unit leaf area Ma) due to the the lack of any for a photosynthetic capacity per unit However, the a for of the with to > 1000 μmol m−2 s−1). it is that the photosynthetic carbon be through the of but with higher et al., and with Wright et al. indeed a tendency for in with both and in their dataset. One of course, some in Ma for the higher leaves with the greater of nitrogen in the photosynthetic per unit leaf area et al., 2006). But the for foliar nitrogen for carbon in to can to a large be of the Here we note from some of our in a that for the species and even their Ma are Aa and are indeed very for both species at c. μmol CO2 m−2 and g (Domingues et al., with Fig. We further that there might be little in having a superior photosynthetic capacity because a higher would only be of the around the of the at would be to the in with in and leaf et al., et al., 2006), especially if strong stomatal with were to around these This then gives to the that there is for variability of a area-based photosynthetic capacity as to its (as by with the lack of any correlation between Aa and Ma, this relative variability in Aa as to Ma then to a of strong but In that we that the one not leaves with both a Ma and a is not that this is at with a because of or some (Wright et al., the that one not find and Ma leaves is because the could be for photosynthetic carbon even at the Although it seems likely that the long known & may be with foliar characteristics as and et al., et al., as as different of in our conclusion must be that photosynthetic characteristics and area-based in in the and/or of the photosynthetic not be with Ma or As a we therefore also that in the general area of plant trait and may have been times through a tendency for plant to the true variability in trait perhaps because of a to see and in the which are not be have been by the and physiological plant may be by the simple that the how a plant its photosynthetic leaf area vs more photosynthetic capacity per unit leaf are by a of of leaf and are not for the or of any by the authors. than be to the The is not for the or of any by the authors. than be to the for the