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New Phytologist Supporting Information
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Article title: Decomposing leaf mass into photosynthetic and structural components explains divergent
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patterns of trait variation within and among plant species
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Authors: Masatoshi Katabuchi, Kaoru Kitajima, S. Joseph Wright, Sunshine A. Van Bael, Jeanne L. D.
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Osnas and Jeremy W. Lichstein
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Article acceptance date: Click here to enter a date.
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The following Supporting Information is available for this article:
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Notes S1 Defining and understanding leaf trait mass- vs. area-dependence.
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Notes S2 Implications of photosynthetic and structural leaf mass for trait mass- vs. area-dependence.
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Notes S3 Alternative leaf lifespan models and realized net photosynthesis.
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Notes S4 Model tests with simulated data.
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Fig. S1 Boxplots comparing posterior means of the latent variable f (the fraction of total LMA comprised
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by LMAp) across deciduous (D) and evergreen (E) leaves in GLOPNET and Panama, and across sites (wet
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and dry) and canopy strata (sun and shade) in Panama.
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Fig. S2 Boxplots comparing leaf mass per area (LMA), photosynthetic leaf mass per area (LMAp;
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posterior means), and structural leaf mass per area (LMAs; posterior means) across sites (wet and dry)
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and canopy strata (sun and shade) in all leaves in the Panama datasets.
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Fig. S3 Measured traits related to photosynthesis and metabolism (nitrogen and phosphorus per-unit
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leaf area; Narea and Parea) are positively correlated with LMA and with estimates (posterior means) of the
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photosynthetic and structural LMA components (LMAp and LMAs, respectively) in the GLOPNET dataset.
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Fig. S4 Boxplots comparing cellulose content (percent of total leaf mass) across sites (wet and dry) and
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canopy strata (sun and shade) in Panama.
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Fig. S5 Posterior means of LMAp vs LMAs in the (a) GLOPNET and (b) Panama datasets.
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Table S1 Summary of results for Potential LL Model (Eq. 5) and Optimal LL Models (Notes S3).
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Table S2 Correlations between LMAp or LMAs and other traits derived from the Potential LL model (Eq.
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5) fit to GLOPNET data.
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Table S3 Correlations between LMAp or LMAs and other traits derived from the three alternative LL
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models (Notes S3) fit to Panama data.
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Table S4 Correlations between observed and predicted LL for the Potential LL Model (Eq. 5) and the
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Optimal LL Model (Notes S3) fit to Panama data.
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Notes S1. Defining and understanding leaf trait mass- vs. area-dependence
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Let X represent the amount of a trait in an entire leaf. For example, X could be the photosynthetic
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capacity of the entire leaf (units = moles CO2 fixed per-unit time), the amount of nitrogen in the entire
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leaf (units = grams of nitrogen), etc. X could depend on leaf mass (in which case leaves of greater mass
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would have higher values of X than leaves of smaller mass), leaf area (in which case leaves of greater
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area would have higher values of X than leaves of smaller area), or both mass and area. We define a trait
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as being purely mass-dependent if X depends on leaf mass but not area, and we define a trait as being
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purely area-dependent if X depends on leaf area but not mass. We expect most traits to depend on both
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mass and area to some degree, but the extreme cases (pure mass- and area-dependence, which we
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explain in more detail below) are useful for illustrating and understanding the key concepts related to
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mass- and area-dependence.
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Rather than X representing the amount of trait in an entire leaf, we could instead define X as the
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amount of trait in a leaf sample of known mass and area. For example, we could define a trait as being
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purely mass-dependent if the amount of trait in a sample depended only on the mass of the sample. In
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what follows, we refer to ‘leaves,’ but the concepts are equally applicable to leaf samples, whether they
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be samples comprised of multiple leaves, or partial-leaf samples that are representative of entire leaves.
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First, consider a purely mass-dependent trait. In this case, if we were to compare leaves of equal
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mass that varied in area, there would be zero correlation between X and area across leaves. More
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generally, if we consider the case of pure mass-dependence with leaves of variable mass and area, and
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we assume that X is proportional to mass, we have:
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(Equation S1.1)
Xi = Cm × Massi × εi
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(Equation S1.2)
Xi/Areai = Cm × LMAi × εi
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(Equation S1.3)
Xi/Massi = Cm × εi
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where the subscript i refers to leaf i, Cm is a constant, and εi is a lognormally-distributed error term,
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which becomes an additive error term if we take the logarithm of both sides of the equations. The key
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point from Eqs. S1.1-S1.3 is that if X is purely mass-dependent, then the area-normalized trait (X/area)
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increases with LMA, and the mass-normalized trait (X/mass) is statistically independent of LMA (see
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Osnas et al. 2013 for further explanation and discussion). The assumption of mass-proportionality (i.e.,
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the Mass term in Eq. S1.1 has an exponent of one) leads to clean forms for Eqs. S1.2-S1.3, but our
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conclusions regarding the dependence of X/area and X/mass on LMA would be unchanged if this
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assumption were relaxed. That is, if Mass were raised to any positive exponent in Eq. S1.1, it would still
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be the case that X/area increases with LMA, and X/mass is independent of LMA.
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Next, consider a purely area-dependent trait. In this case, if we were to compare leaves of equal
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area that varied in mass, there would be zero correlation between X and mass across leaves. More
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generally, if we consider the case of pure area-dependence with leaves of variable mass and area, and
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we assume that X is proportional to area, we have:
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(Equation S1.4)
Xi = Ca × Areai × εi
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(Equation S1.5)
Xi/Areai = Ca × εi
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(Equation S1.6)
Xi/Massi = Ca × LMAi–1 × εi
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where Ca is a constant, and other details are as above. The key point from Eqs. S1.4-S1.6 is that if X is
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purely area-dependent, then the area-normalized trait (X/area) is statistically independent of LMA, and
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the mass-normalized trait (X/mass) decreases with LMA (Osnas et al., 2013). As explained above for
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mass-dependent traits, the qualitative conclusions do not depend on the proportionality assumption.
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Notes S2 Implications of photosynthetic and structural leaf mass for trait mass- vs. area-dependence
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Here, we use simulated data to explore how variation in photosynthetic and structural LMA components
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(LMAp and LMAs, respectively) affect trait mass- and area-dependence (as defined in Notes S1). We
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show that variation among leaves in LMAp leads to mass-dependence of photosynthetic capacity (Amax)
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and related traits (e.g., Rdark and concentrations of N and P), whereas variation in LMAs leads to area-
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dependence of these same traits. We assume that traits associated with photosynthetic capacity, when
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expressed per-unit leaf area, increase with LMAp but are unaffected by LMAs. This is equivalent to
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assuming that the total amount of a photosynthetic trait in a leaf (represented by the symbol X in Notes
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S1) increases with the mass of photosynthetic tissue in a leaf but is unaffected by the mass of structural
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tissue. Our analysis of simulated data focuses on a single trait, photosynthetic capacity (Amax), since this
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trait is closely related to LMAp in our model (see main text).
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2.1 Simulated datasets
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We performed tests with simulated data to explore how the degree of trait mass-dependence is
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affected by the variances of LMAp and LMAs, and the covariance between LMAp and LMAs. In the
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simulated datasets, we used estimates from our analysis of the GLOPNET dataset (see Methods and
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Results in the main text) to assign the following: Aarea = 0.23  LMAp  exp(a), where a is normally-
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distributed with mean 0 and standard deviation 0.5; median value of LMAp = 70 g m–2; and median value
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of LMAs = 98 g m–2. To evaluate how variance and covariance of LMAp and LMAs affect estimates of
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mass-dependence, we considered all combinations of the following: standard deviations of log-scale
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LMAp and LMAs from 0.1 to 1.0 in increments of 0.1 (which determine how variable LMAp and LMAs are
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among leaves in the simulated sample), and correlation coefficients between log-scale LMAp and LMAs
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from –0.4 to 0.4 in increments of 0.1 (which determines how LMAp and LMAs covary among leaves in
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the simulated sample). In total, we estimated mass-dependence for 900 parameter combinations (10
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values of LMAp variance × 10 values of LMAs variance × 9 correlation coefficients). For each combination
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of parameters, we simulated 100 replicates. Each replicate had a sample size of 100 leaves, where the
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values of LMAp and LMAs for each leaf was a random draw from the distributions with the variances
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and covariances described above.
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Of the 90,000 simulated datasets, 7839 (~8.7%) showed negative covariance between LMA and
either LMAp or LMAs. We excluded these simulated datasets from our subsequent analysis because one
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goal of our analysis was to attribute total LMA variance due to variance in LMAp vs. LMAs (see main text
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Eq. 15), which is not straightforward when one of the covariances is negative. Furthermore, our
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empirical results based on the GLOPNET and Panama datasets suggested that covariances between LMA
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and both LMAp and LMAs are in reality positive, so excluding negative covariances from our analysis
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should not bias our inferences with respect to real datasets.
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2.2 Quantifying trait mass-dependence
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For each of the simulated datasets described above, we quantified mass-dependence of Amax using the
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following equation, equivalent to “Model-LN” in Osnas et al. (2013; see the “Models” section of their
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Supplementary Materials):
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(Equation S2.1)
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where Aareai is area-normalized Amax for leaf i, LMAi is total LMA for leaf i, εi is a normally-distributed
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error term, and the parameter b quantifies mass- vs. area-dependence. Parameter b equals one for a
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purely mass-proportional trait (Eq. S1.2) and zero for a purely area-proportional trait (Eq. S1.5). More
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generally, b is close to zero for traits that are mostly area-dependent, and b increases with increasing
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mass-dependence. Empirical estimates of b tend to range between 0 and 1 (Osnas et al. in review), and
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we interpret values of b > 0.5 as indicating greater mass- than area-dependence. We used ordinary least
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squares regression to fit Eq. S2.1 to each simulated dataset. Osnas et al. (2013) showed that this method
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yields similar results as other approaches to quantifying mass-dependence.
log(Aareai) = a + b × log(LMAi) + εi
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2.3 Results
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Variation among leaves in LMAp led to mass-dependence of Amax (b > 0.5), whereas variation in LMAs led
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to area-dependence of Amax (b near 0; Fig. S2.1a). That is, as total LMA variance became increasingly
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dominated by LMAs (i.e., moving to the right along the x-axis of Fig. S2.1a), mass-dependence of Amax
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decreased (or, equivalently, area-dependence increased). Intuitively, mass-dependence of Amax
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decreases (and area-dependence increases) with LMAs variance because LMAs has no effect on area-
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normalized Amax (Aarea). Thus, as LMAs variance increases, the photosynthetic capacity of entire leaves
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becomes increasingly dependent on leaf area (rather than leaf mass), and Aarea becomes increasingly
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independent of LMA. As expected, the percent variation of total LMA that is due to LMAs variance
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increased with LMAs variance (Fig. S2.1b). Thus, weak mass-dependence corresponds to cases where
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LMAs variance dominates total LMA variance (right side of x-axes in Figs. S2.1a and S2.1b), and strong
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mass-dependence corresponds to cases where LMAp variance dominates total LMA variance (left side of
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x-axes in Figs. S2.1a and S2.1b).
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Covariance between LMAp and LMAs (which was positive but weak in our analyses of GLOPNET
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and Panama data; see Fig. S6) also affected mass- vs. area-dependence. Empirical estimates of mass-
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dependence (b) tend to range from 0 to 1 (Osnas et al. in review). Within this typical range (0 < b < 1),
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positive covariance between LMAp and LMAs increases mass-dependence, whereas negative covariance
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between LMAp and LMAs (which is inconsistent with our results of GLOPNET and Panama data)
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increases area-dependence (Fig. S2.1a). Our simulations show that if LMAs has greater variance than
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LMAp, then b tends to be greater than 1, although such values are larger than most empirical estimates
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of b, which tend to range between 0 (pure area-proportionality) and 1 (pure mass-proportionality;
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Osnas et al. in review).
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Fig. S2.1 Effects of the LMAs:LMAp variance ratio in simulated datasets on (a) the mass-dependence of
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photosynthetic capacity (Amax) and (b) the percent of variance in leaf mass per area (LMA) explained by
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LMAs variance, where LMAs and LMAp are structural and photosynthetic components, respectively, of
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LMA (see Notes S2 for details). Mass-dependence (y-axis in the left panel) is estimated as parameter b in
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Eq. S2.1; thus mass-dependence of Amax decreases (and area-dependence increases) as the LMAs:LMAp
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variance ratio increases. Methods for generating simulated data and for estimating mass-dependence
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are described in Notes S2. Methods to quantify the percent variation of LMA due to LMAs (right panel)
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are described in the main-text (see Eq. 15). Note that empirical estimates of mass-dependence (y-axis in
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the left panel) tend to range between 0 (pure area-proportionality; Eq. S1.5) and 1 (pure mass-
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proportionality; Eq. S1.2). Thus, we interpret values less than 0.5 as indicating primarily area-dependent
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traits, and values greater than 0.5 as indicating primarily mass-dependent traits.
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Notes S3 Alternative leaf lifespan models and realized net photosynthesis
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Here, we describe three alternative LL models, and how we derive an approximation for realized rate of
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annualized (or time-averaged) net photosynthesis (which is required for the Optimal LL Models) based
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on measured Amax and Rdark.
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3.1 Alternative models of LL
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The fist model, which we refer to as “Potential LL Model”, assumes that LL is proportional to the amount
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of LMAs:
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E[LL𝑖 ] = 𝛽 LMAs𝑖
(Potential LL Model)
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where, E[] is the expected value of the variable in brackets, LLi is leaf lifespan of leaf i and  is a
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constant. We fitted the Potential LL Model to both GLOPNET and the Panama dataset.
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The second model assumes optimal LL theory (Kikuzawa, 1991). Details on the Kikuzawa model
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are described in the main text. According to optimal LL theory, LL should decrease with the realized rate
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of net photosynthesis, because frequently replacing old leaves to new leaves ensure high carbon gain
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per unit time for leaves that have high initial net photosynthesis rate (Kikuzawa, 1991). We call this
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second model “Optimal LL Model”. We scaled maximum net photosynthetic rate to use realized net
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photosynthetic rate to use realized:
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E[LL𝑖 ] = 𝛽√LMAs𝑖 LMA𝑖 /(𝜃L 𝐴area 𝑖 − 𝑅area 𝑖 )
(Optimal LL model)
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where, θ L (0 < θ L < 1) is the scaling parameter for shade leaves to describe realized net photosynthetic
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rate (Kikuzawa et al., 2004), with θ L = 1 for sun leaves, θ L < 1 for shade leaves, and Aarea i is the the net
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photosynthetic rate per unit area of leaf i, and Rarea i is the dark respiration rate per unit area of leaf i. In
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this model, parameter θ LAarea i – Rarea i is proportional to the mean daily net photosynthetic rate. We
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fitted the optimal LL model to the Panama dataset.
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The third model includes an additional parameter to account for site effects (). Because we
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combined data from wet and sites in a single analysis, it is possible that site differences in LL could
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confound our results.
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E[LL𝑖 ] = 𝛽𝜑√LMAs𝑖 LMA𝑖 /(𝜃L 𝐴area 𝑖 − 𝑅area 𝑖 )
(Optimal LL model with site effect)
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We report the optimal LL model in the main text as it produced the better WAIC (Watanabe, 2010;
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Gelman et al., 2014) value than the potential LL model. Although the site effect model had best WAIC
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value, we include this in supplementary materials, because this model includes non-mechanistic
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processes relevant to variance in LL (i.e., site effects). See Table S1 for the full result.
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3.2 Realized net photosynthesis
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The realized rate of net photosynthesis reflects multiple factors, including down-regulation of
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photosynthetic capacity and respiration of chronically shaded leaves (Chen et al., 2014), as well as the
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amount of light incident upon a leaf at a given time (Kikuzawa et al., 2004). The former (down-
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regulation) is already reflected in observed values of Amax (which is measured under saturating light
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conditions), but the latter (realized light availability) is not. Therefore, we introduced a scaling
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parameter ( L; with 0 <  L < 1) to account for shading effects when approximating realized net
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photosynthesis in the Optimal LL Models. We show that realized net photosynthesis per unit time is
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approximately proportional to  LAarea – Rarea, where Aarea and Rarea are net photosynthetic capacity (Amax)
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and dark respiration (Rdark) per-unit leaf area per hour respectively. For simplicity, we assume that (1)
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day and night are each 12 h long (which is approximately correct for the tropical forest sites in Panama
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where we applied the Optimal LL Models), (2) net photosynthetic rate is constant during the day, and (3)
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respiration rate is constant during the night. Then, we have:
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Net photosynthesis during 12-hour daytime = (12 hr) × LAarea
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Respiration during 12-hour nighttime = (12 hr) × Rarea
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The scaling parameter L accounts for the effects of light availability on net photosynthesis is, with L = 1
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for sun leaves, and L < 1 for shade leaves. Subtracting the two respiration terms from gross
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photosynthesis yields the approximation for 24-hour net photosynthesis is proportional to  LAarea – Rarea.
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Above, we express Aarea and Rarea per hour, but these are easily translated to other time periods.
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Although Rdark may not accurately measure nighttime respiration respiration because of
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complicated by processes such as nighttime phloem loading (Azcón-Bieto & Osmond, 1983; Azcón-Bieto
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et al., 1983), it is not easy to implement scaling parameters for nighttime respiration (RN). It is difficult
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to include both L and RN at the same time for convergence. Additionally, to implement RN into the
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models, RN should be same for both sun and shade leaves or RN = 1 for sun or shade leaves and RN > 0
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for shade or sun leaves, but there is no clear biological justification to set RN. In contrast, L has clear
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biological motivation as explained above and thus we only focus on  L in our models.
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Notes S4 Model tests with simulated data
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We performed tests with simulated data to evaluate the performance of our modeling approach with
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different prior assumptions for the latent variables that determine LMAp and LMAs (fi in Eqs. 2-3). If the
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modeling approach (with a given set of priors) is approach is robust, then posterior parameter estimates
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(including those for latent variables) should closely match the assumed (“true”) parameter values used
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to create the simulate datasets. Below, we describe (i) the different parameter combinations used to
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generate the simulated datasets, (ii) alternative priors that we explored for the latent variables fi, and
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(iii) results of the tests with simulated data, including randomized versions of the simulated data
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(analogous to the LMA-randomization tests presented in the main text). In general, these tests with
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simulated datasets suggest that the results presented in the main text are robust. In particular, the
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simulation tests suggest that our main results depend primarily on underlying patterns in the GLOPNET
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and Panama datasets that we analyzed, as opposed to being pre-determined by our model assumptions.
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4.1 Methods
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4.1.1 Simulated datasets
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Four distinct combinations of parameter values were used to create simulated datasets with different
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properties (Fig. S4.1-S4.4, Tables S4.1-S4.2):
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 Simulated datasets GL1 and GL2, based on GLOPNET data. We used parameter values estimated
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from the GLOPNET analysis described in main text (Potential LL Model, Eq. 5) to create two simulated
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datasets (GL1 and GL2) with different LMA distributions for evergreen and deciduous leaves. The
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correlation between LL and LMA is strong while the correlation between LMA and Aarea is weak (Figs.
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S4.1-S4.2). GL1 has the same LMA distribution as the observed GLOPNET data (high LMA for
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evergreen leaves, and low LMA for deciduous leaves; Fig. S4.1 and Table S4.2), whereas GL2 assigns
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the observed evergreen LMA distribution to deciduous leaves, and vice versa (Fig. S4.2 and Table
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S4.2). We analyzed GL1 and GL2 using the same methods applied to the GLOPNET data in the main
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text, which allowed us to assess if inferred differences between evergreen and deciduous leaves (Fig.
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4) merely reflect model assumptions (in which case GL1 and GL2 should yield inferences similar to
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those shown in Fig. 4), or if inferred differences reflect meaningful patterns in the data (in which GL1,
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but not GL2, should yield results similar to Fig. 4).
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 Simulated dataset PA, based on Panama data. We used parameters values estimated from the
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Panama analysis with the lowest WAIC (Optimal LL Model with site effects, Notes S3) to create a
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simulated dataset with properties similar to the observed Panama data. But in the case of the
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simulated data, the true values of the parameters (including LMAp and LMAs for each leaf) are
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known, which allows us to test the modeling approach. In contrast to the GLOPNET dataset, the
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correlation between LMA and Aarea is strong while the correlation between LMA and LL is weak in this
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simulated dataset (Fig. S4.3), as in the observed Panama data (Figs. 2a and 2g).
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 Simulated dataset WC (“weak correlation”). We created one simulated dataset with weak
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correlations among LMAp, LMAs and other traits (Fig. S4.4). This dataset allows us to assess if our
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modeling approach is prone to inducing strong correlations (e.g., between LMAp and Aarea, or
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between LMAs and LL) when the true correlations are weak.
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For each of the above four parameter combinations, we analyzed a single simulated dataset, as well as
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nine additional datasets with randomized LMA values (see details below). A more rigorous approach
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would involve analyzing many replicate datasets (e.g., 1000), but this would be very computationally
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expensive because each of our analyses requires fitting a complex Bayesian model with many latent
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variables. Although our tests with simulated data are either non-replicated (in the case of the original,
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non-randomized, simulated data) or have limited replication (in the case of the LMA-randomized
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simulated data), we consider the test results presented below to be sufficiently clear to provide
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meaningful guidance on the interpretation of the results presented in the main text.
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4.1.2 Model forms applied to simulated data
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We fit each simulated dataset with the model form used to generate the data; i.e., we applied the
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Potential LL Model (Eq. 1) to the simulated GL1, GL2, and WC datasets, and the Optimal LL Model with
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site effects (Notes S3) to the simulated PA dataset.
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4.1.3 Priors
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For leaf i of group j (where j is evergreen or deciduous in the GL1 and GL2 datasets, and j is sun/dry-site,
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shade/dry-site, sun/wet-site, or shade/wet-site in the PA dataset), we considered two different sets of
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priors for the latent variables fi:
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(i)
fi ~ Uniform(0,1) (Non-hierarchical model)
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(ii)
logit–1(fi) ~ Normal(μ0 + rj, σ) (Hierarchical model)
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where 0 is constant across all leaves, rj is a group effect (a vector with constant value within each group
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j), and  is the variance of fi. In the hierarchical model, 0, rj, and  are free parameters. The first set of
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priors (uniform, non-hierarchical model) is used in the results reported in the main text and in Figs. S5.1-
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S5.6 because analyses of simulated data (see below) suggested that these priors yielded more robust
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inferences compared to the hierarchical model.
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4.1.4 Parameter estimation
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Models were fit and convergence of posterior distributions was assessed as described in the main text.
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4.1.5 Randomized LMA datasets
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As with the analyses described in the main text, we created randomized LMA datasets corresponding to
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each of the four simulated datasets described above, and we fit the models to both the original (non-
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randomized) simulated datasets, as well as the randomized datasets (see details below). The purpose of
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this analysis was to assess the validity of the randomization approach in the main text. For example,
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analyses of non-randomized data should yield correlations between LMAp and Aarea (or between LMAs
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and LL) that are similar to the true correlations in the original simulated data, whereas analyses of
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randomized data should yield weaker correlations than in the original simulated data.
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We generated 9 randomized LMA datasets for each of the four simulated datasets, and we fit
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the same models (Potential LL Model or Optimal LL Model with site effects) to the randomized and non-
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randomized data. For the simulated GL1, GL2, and WC datasets, we randomized LMA values across
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leaves, while maintaining the original (non-randomized) data for Aarea, Rarea and LL. For the simulated PA
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dataset, LMA values were shuffled within sites (wet and dry) and across canopy strata (sun and shade).
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Thus, when randomizing the PA dataset, we removed the effect of strata (sun vs. shade) on LMA, while
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maintaining site differences in LMA as well as the observed covariances among Aarea, Rarea and LL. To
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compare model fit between the non-randomized dataset and randomized LMA dataset, we calculated a
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standardized effect size (SES) for correlation between focal variables and translated them to P-value as
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described in the main text.
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4.2 Results
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Non-informative flat priors for latent variables (non-hierarchical models) performed better than the
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hierarchical models. True values of model structural parameters (i.e., all parameters except for latent
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variables) were usually located within the central part of their posterior distributions for models with
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either non-hierarchical or hierarchical priors on latent variables (Figs. S4.5-S4.7). However, the match
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between posterior estimates and true values tended to be better for the non-hierarchical models (Figs.
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S4.5-S4.6), and in one case (WC dataset), the hierarchical model did not converge.
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For latent variables (which are relevant to many results and conclusions presented in the main
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text), the non-hierarchical model yielded posterior means that were unbiased and strongly correlated
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with the true values for the GL1 and PA datasets, whereas the hierarchical models had weaker
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relationships with the true values and/or systematic bias (Fig. S4.8). For the simulated WC dataset, the
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non-hierarchical model performed better than the hierarchical model, although neither set of priors
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yielded posterior means that were well-correlated with the true values of latent variables (Fig. S4.8).
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LMA randomization is an effective way to identify meaningful correlations between LMAp and LMAs
341
and other traits. When the true correlations between LMAp and LMAs and other traits are strong (as in
342
the simulated GL and PA datasets), the estimated correlations (derived from posterior distributions of
343
the latent variables fi; Eqs. 2-3) are very similar to the true correlations and significantly different from
344
correlations derived from datasets in which LMA was randomized among leaves (see GL1 and PA in Fig.
345
S4.9). In contrast, when the true correlations between LMAp and LMAs and other traits are weak (as in
346
the simulated WC dataset), the estimated correlations and randomization-based correlations are very
347
similar to each other and significantly greater than the true correlations (see WC in Fig. S4.9). These
348
results suggest that (i) if the true correlations between LMAp and LMAs and other traits are weak, then
349
model assumptions (e.g., the degrees of freedom provided by latent variables) can lead to over-
350
estimates of these correlations; but (ii) such artefactual results can be diagnosed using the LMA-
351
randomization tests presented here and in the main text. Specifically, if results obtained from the
352
original data (non-randomized) are significantly different from results obtained from LMA-randomized
353
data, then the tests presented here (Fig. S4.9) suggest that the estimated correlations are similar to the
354
true correlations.
355
356
Inferred differences in LMAp and LMAs among groups (e.g, evergreen vs. deciduous leaves) and
357
inferred contributions of LMAp and LMAs to total LMA variance depend more on the data than on
358
model assumptions. Estimated distributions of LMAp and LMAs for evergreen and deciduous leaves
359
were very similar to the true distributions for simulated GL1 data (which has similar properties to the
360
original GLOPNET dataset), as well as for simulated GL2 data (where LMA distributions for evergreen
361
and deciduous leaves were swapped) (Figs. S4.10-S4.11). The estimated variance contributions of LMAp
15
362
and LMAs to total LMA variation were very similar to the true values for GL1 and roughly similar to the
363
true values for GL2 (Fig. S4.11). Despite the quantitative mismatch between estimated and true values
364
for GL2 variance partitioning (Fig. S4.12), the estimates are qualitatively correct in identifying LMAp as
365
the dominant variance component in GL2. These test results suggest that using our modeling approach,
366
estimated differences in LMAp and LMAs between groups of leaves and estimates of LMA variance
367
components are at least qualitatively robust, and depend more strongly on the properties of the dataset
368
than on model assumptions.
369
370
16
371
372
Fig. S4.1 Simulated GL1 dataset used to test the Potential LL Model analysis of GLOPNET data. The GL1
373
dataset has similar properties as the original GLOPNET dataset. But for GL1 (unlike GLOPNET), the values
374
of all model parameters (including the latent variables fi that determine LMAp and LMAs for each leaf)
375
are known.
376
377
378
379
17
380
381
382
Fig. S4.2 Simulated GL2 dataset used to test the Potential LL Model analysis of GLOPNET data. GL2 is
383
similar to GL1, except that in GL2, the LMA distributions for evergreen and deciduous leaves are
384
swapped (Table S4.2). These non-realistic LMA distributions were used to test the ability of our
385
modeling approach to correctly estimate differences in LMAp and LMAs among groups of leaves.
386
18
387
388
Fig. S4.3 Simulated PA dataset used to test the Optimal LL Model analysis of Panama data. The PA
389
dataset has similar properties as the original Panama dataset. But for PA (unlike Panama data), the
390
values of all model parameters (including the latent variables fi that determine LMAp and LMAs for each
391
leaf) are known.
392
393
394
19
395
396
Fig. S4.4 The weak correlation (WC) dataset used to test the Potential LL Model. Relationships among
397
traits are much weaker in the WC dataset compared to other simulated datasets (Figs. S4.1-S4.3). The
398
WC dataset was designed to evaluate the performance of our modeling approach in cases where the
399
hypothesized relationships between LMAp and LMAs and other traits were weak.
400
20
401
402
403
Fig. S4.5 Posterior distributions for model parameters in the simulated GL1 dataset. Vertical red lines
404
indicate the assumed (“true”) values used to create the simulated data. The non-hierarchical model
405
assumes flat (uniform) priors for each latent variable fi (Eq. 5), whereas the hierarchical model assumes
406
that each latent variable comes from a group-specific normal distribution, where groups are either
407
evergreen, deciduous, or unclassified leaves.
408
409
410
411
412
21
413
414
Fig. S4.6 Posterior distributions for model parameters in the simulated PA dataset. Vertical red lines
415
indicate the assumed (“true”) values used to create the simulated data. The non-hierarchical model
416
assumes flat (uniform) priors for each latent variable fi (Eq. 5), whereas the hierarchical model assumes
417
that each latent variable comes from a group-specific normal distribution, where groups are the four
418
combinations of two sites (wet and dry) and two canopy strata (sun and shade).
419
420
421
22
422
423
Fig. S4.7 Posterior distributions for model parameters in the simulated WC dataset. Vertical red lines
424
indicate the assumed (“true”) values used to create the simulated data. Posteriors for the model with
425
hierarchical priors did not converge, so only results for the non-hierarchical model (with uniform priors
426
for each latent variable) are shown.
427
428
429
23
430
431
432
Fig. S4.8 Posterior means for the latent variables fi (which determine LMAp and LMAs for each leaf; Eq.
433
5) plotted against their assumed (“true”) values in three simulated datasets. The dashed line indicates
434
the 1:1 relationship. The two rows correspond to results for two different sets of prior assumptions:
435
hierarchical priors (see legends for Figs. S4.5-S4.7 for details) and non-hierarchical uniform priors. The
436
hierarchical model for the simulated WC dataset did not converge, so no results are shown.
437
24
438
439
440
Fig. S4.9 Comparison of correlation coefficients estimated from non-randomized simulated data (‘Est’),
441
those obtained from analyses of LMA-randomized data (‘Rand’), and the assumed (‘True’) correlations
442
used to generate the simulated data. Correlations are shown between LMAp and Aarea, between LMAp
443
and Rarea, and between LMAs and LL. Red solid lines show the true correlations. Blue dashed lines show
444
the posterior means of the estimated (‘Est’) correlations obtained from non-randomized simulated data.
445
Black dashed lines show the posterior means of the correlations obtained from LMA-randomized
446
(‘Rand’) simulated data. Posterior distributions (estimated from 1000 random posterior samples) are
447
also shown for each correlation. P < 0.05 indicates a significantly stronger correlation in models fit to
448
non-randomized compared to models fit to randomized data (P-values based on the standardized effect
449
size, SES; see Methods in the main text for details).
450
451
25
452
453
454
455
456
Fig. S4.10 Boxplots of assumed (“true”) values of LMAp and LMAs in the simulated GL1 dataset, and the
457
corresponding posterior means for LMAp and LMAs estimated from the Potential LL Model applied to
458
the simulated data. The results show that for the GL1 dataset (which has similar properties to the
459
original GLOPNET data), the model accurately recovers the assumed distribution of LMAp and LMAs for
460
deciduous evergreen leaves.
461
462
463
26
464
465
Fig. S4.11 Boxplots of assumed (“true”) values of LMAp and LMAs in the simulated GL2 dataset, and the
466
corresponding posterior means for LMAp and LMAs estimated from the Potential LL Model applied to
467
the simulated data. The results show that for the GL2 dataset (in which LMA distributions for deciduous
468
and evergreen leaves were swapped relative to the original GLOPNET data), the model accurately
469
recovers the assumed distribution of LMAp and LMAs for deciduous and evergreen leaves.
470
27
471
472
473
Fig. S4.12 True and estimated proportions of total LMA variance contributed by variances in LMAp and
474
LMAs for the simulated GL1 and GL2 datasets. ‘True’ values refer to the assumed values in the simulated
475
datasets, whereas ‘Estimated’ values are posterior means from the Potential LL Model applied to the
476
simulated datasets. The results show that the modeling approach yields qualitatively correct inferences
477
(e.g., LMAs variance in dominant in GL1, whereas LMAp variance is dominant in GL2), although
478
quantitative estimates may not be robust (e.g., mismatch between True and Estimated values for GL2).
479
28
Table S4.1 Parameter values used to generate simulated datasets. The GL1 and GL2 parameter values were estimated from the Potential LL
Model (Eq. 5) applied to the GLOPNET global dataset, as described in the main text. GL1 and GL2 share the same parameter values, but differ in
the assumed LMA distributions (see Table S4.2). The PA parameter values were estimated from the Optimal LL Model with sites effects (Notes
S3) applied to the Panama data, as described in the main text. The WC (weak correlation) dataset was designed to have weak correlations
between LMAp and LMAs and other traits. Parameter values shown here were combined with LMA values (Table S4.2) to create the simulated
data are shown in Figs. S4.1-S4.4.
Parameter
Parameter value in each simulated dataset
Description
GL1, GL2
PA
WC

0.23
0.28
1
Slope coefficient relating LMAp to Aarea and Rarea

0.23
0.60
1
Slope coefficient relating LMAs to LL
rp
0.02
0.02
0.02
Slope coefficient relating LMAp to Rarea
rs
0.001
0.001
0.02
Slope coefficient relating LMAs to Rarea
L
NA
0.5
NA

NA
0.7
NA
12
0
0.8
0
13
0.7
0.03
0
23
0
0.1
0
1
0.5
0.2
1
2
0.5
0.5
1
3
0.5
0.7
1
Scaling parameter that accounts for shading and
thus affects the realized net photosynthetic rate
Parameter to account for site effects
Correlation coefficients in the covariance matrix in
Eq. 11 in the main text
Standard deviations in the covariance matrix in Eq.
11 in the main text
29
30
Table S4.2 LMA values in the simulated datasets. The latent variables fi determine LMAp and LMAs for
each leaf (Eq. 5).
Median of LMA
SD of LMA
Mean of fi
Deciduous
80
0.3
0.6
Evergreen
170
0.5
0.4
Deciduous
170
0.3
0.6
Evergreen
80
0.5
0.4
Sun Dry
90
0.3
0.6
Shade Dry
70
0.3
0.7
Sun Wet
40
0.2
0.3
Shade Wet
30
0.4
0.6
120
0.3
0.4
Simulated GL1
Evergreen has higher LMA
Simulated GL2
Evergreen has lower LMA
Simulated PA
Weak Correlation (WC)
31
Fig. S1 Boxplots comparing posterior means of the latent variable f (the fraction of total LMA comprised
by LMAp) across deciduous (D) and evergreen (E) leaves in GLOPNET and Panama, and across sites (wet
and dry) and canopy strata (sun and shade) in Panama. Note that LMAp = f × LMA, and LMAs = (1 – f) ×
LMA. (a) Deciduous and evergreen leaves in the GLOPNET dataset; (b) deciduous and evergreen leaves
for Panama species for which both sun and shade leaves were available; (c) leaves for Panama species
for which both sun and shade leaves were available; and (d) all leaves for Panama. At the dry Panama
site, the increase in LMA from shade to sun (Fig. 5) is due to roughly equal proportional increases in
LMAp and LMAs (because f is similar between sun and shade), whereas at the wet Panama site, the
increase in LMA from shade to sun (Fig. 5) is due primarily to increased LMAp (because f is greater in sun
than shade). GLOPNET results are for the Potential LL Model (Eq. 5), and Panama results are for the
Optimal LL Model with site effects (Eq. 13 and Notes S3). The center line in each box indicates the
median, upper and lower box edges indicate the interquartile range, whiskers show 1.5 times the
interquartile range, and points are outliers. Groups sharing the same letters are not significantly
different (P > 0.05; t-tests).
32
Fig. S2 Boxplots comparing leaf mass per area (LMA), photosynthetic leaf mass per area (LMAp;
posterior means), and structural leaf mass per area (LMAs; posterior means) across sites (wet and dry)
and canopy strata (sun and shade) in Panama. The results shown here include all leaves in the Panama
dataset, whereas Fig. 5 in the main text only includes Panama species for which both sun and shade
leaves were available. Boxplot symbols as in Fig. S1. Groups sharing the same letters are not significantly
different (P > 0.05; t-tests).
33
Fig. S3 Measured traits related to photosynthesis and metabolism (nitrogen and phosphorus per-unit
leaf area; Narea and Parea) are positively correlated with LMA and with estimates (posterior means) of the
photosynthetic and structural LMA components (LMAp and LMAs, respectively) in the GLOPNET dataset.
LMAp yields stronger correlations and more consistent relationships compared to LMA and LMAs; e.g.,
evergreen and deciduous leaves align along a single relationship in panel b, but not in panels a or c. Gray
symbols show model results from one of ten randomized datasets in which LMA was randomized among
all leaves in GLOPNET. Pearson correlation coefficients are for observed LMA (left column) and posterior
means of LMAp (middle column) and LMAs (right column). P-values (*** P < 0.001) for LMA test the null
hypothesis of zero correlation, and for LMAp and LMAs test the null hypothesis of equal correlation in
observed and randomized datasets (see ‘Randomized LMA Datasets’ in Methods for details).
34
Fig. S4 Boxplots comparing cellulose content (percent of total leaf mass) across sites (wet and dry) and
canopy strata (sun and shade) in Panama. (a) All leaves in the Panama dataset; and (b) leaves for
Panama species for which both sun and shade leaves were available. Cellulose content (percent mass) is
similar for shade and sun at the dry site, but higher for shade than sun at the wet site. This pattern is
consistent with estimates of LMAp and LMAs fractions (Fig. S1c-d); i.e., at the wet site, LMAs comprises
a larger fraction of total LMA in shade than sun (because LMAp comprises a larger fraction in sun than
shade; Fig. S1c-d). Boxplot symbols as in Fig. S1. Groups sharing the same letters are not significantly
different (P > 0.05; t-tests).
35
Fig. S5 Posterior means of LMAp vs LMAs in the (a) GLOPNET and (b) Panama datasets. Correlations
between LMAp and LMAs are significantly positive, but the small r2 values indicate that a single axis
could not accurately represent the two-dimensional space. Symbols as in Main Text Figs. 1-2.
36
Table S1 Summary of results for Potential LL Model (Eq. 5) and Optimal LL Models (Eq. 13 Notes S3).
Column definitions:

Data = GLOPNET or Panama.

Model = Potential, Optimal or Optimal with site effects.

n.obs = number of samples analyzed.

n.par = number of free structural parameters (excluding latent variables) in models (see notes below table for details).

WAIC = Watanabe-Akaike information criterion; also known as widely applicable information criterion. WAIC is a predictive information
criterion for Bayesian models, with lower values indicating a more parsimonious model.

, , rp, rs, L and  = posterior mean and 95% credible interval for each parameter.
Data
LL Model
n.obs
n.par
WAIC


rs
L

0.026]
0.002 [0.001, 0.003]
NA [NA, NA]
NA [NA, NA]
0.021 [0.016,
0.002 [-0.001,
0.027]
0.007]
NA [NA, NA]
NA [NA, NA]
rp
0.023 [0.02,
GLOPNET
Panama
Potential
Potential
198
132
10
10
863
516
0.234 [0.208, 0.263]
0.298 [0.272, 0.329]
0.23 [0.203, 0.264]
0.499 [0.402, 0.636]
0.021 [0.018,
Panama
Optimal
132
11
378
0.282 [0.256, 0.316]
0.611 [0.542, 0.697]
0.279 [0.253,
0.026]
0 [-0.003, 0.003]
0.022 [0.018,
-0.001 [-0.004,
0.026]
0.002]
0.335]
NA [NA, NA]
Optimal
with site
Panama
effects
132
12
347
0.283 [0.254, 0.319]
0.691 [0.605, 0.799]
0.698 [0.564,
0.28 [0.253, 0.332]
0.863]
Note: Free structural parameters for the Potential LL Model: , , rp, rs, 12, 13, 23, 1, 2, and 3. Free structural parameters for the
Optimal LL Model: , , rp, rs, 12, 13, 23, 1, 2, 3 and L. The model with site effects includes one additional parameter (; see Notes
S3).
37
Table S2 Correlations between LMAp or LMAs and other traits (on logarithmic scales) derived from the
Potential LL model (Eq. 5) fit to GLOPNET data. Pearson correlations are given for models fit to the
observed datasets (robs) and randomized datasets (rrand). For randomized datasets, rrand is the mean
correlation for 10 datasets in which LMA was randomized across all leaves. P-values test the null
hypothesis of equal correlation in observed and randomized datasets (see ‘Randomized LMA Datasets’
in Methods for details).
LMA randomized across all
leaves
LL model
Potential
trait
LMA component
Aarea
LMAp
0.672
0.484
< 0.001
Aarea
LMAs
-0.234
-0.401
< 0.001
Rarea
LMAp
0.59
0.369
< 0.001
Rarea
LMAs
0.124
-0.237
< 0.001
LL
LMAp
0.108
-0.562
< 0.001
LL
LMAs
0.916
0.542
< 0.001
Narea
LMAp
0.645
0.074
< 0.001
Narea
LMAs
0.477
-0.027
< 0.001
Parea
LMAp
0.57
0.234
< 0.001
Parea
LMAs
0.304
-0.137
< 0.001
38
robs
rrand
P-value
Table S3 Correlations between LMAp or LMAs and other traits (on logarithmic scales) derived from the
three alternative LL models fit to Panama data: Potential LL Model (Eq. 5), Optimal LL Model without site
effect (Eq. 13), and Optimal LL Model with site effect (Notes S3). Pearson correlations are given for
models fit to the observed datasets (robs) and randomized datasets (rrand). For randomized datasets, rrand
is the mean correlation for 10 datasets in which LMA was randomized across all leaves, and 10 datasets
in which LMA was randomized within sites (wet and dry) across canopy strata (sun and shade). P-values
test the null hypothesis of equal correlation in observed and randomized datasets (see ‘Randomized
LMA Datasets’ in Methods for details).
LL model
Potential
Optimal
trait
LMA
component
robs
LMA randomized
LMA randomized within
across all leaves
sites across strata
rrand
rrand
P-value
P-value
Aarea
LMAp
0.970
0.646
< 0.001
0.668
< 0.001
Aarea
LMAs
0.057
-0.463
< 0.001
-0.428
< 0.001
Rarea
LMAp
0.618
0.528
0.157
0.496
0.063
Rarea
LMAs
0.167
-0.381
< 0.001
-0.355
< 0.001
LL
LMAp
-0.432
-0.613
0.254
-0.526
< 0.001
LL
LMAs
0.368
0.491
< 0.001
0.642
< 0.001
Narea
LMAp
0.854
0.575
< 0.001
0.592
< 0.001
Narea
LMAs
0.289
-0.376
< 0.001
-0.344
< 0.001
Parea
LMAp
0.768
0.571
< 0.001
0.550
< 0.001
Parea
LMAs
0.176
-0.383
< 0.001
-0.393
< 0.001
CLarea
LMAp
0.600
0.252
< 0.001
0.309
< 0.001
CLarea
LMAs
0.651
-0.141
< 0.001
-0.042
< 0.001
Aarea
LMAp
0.883
0.497
< 0.001
0.531
< 0.001
Aarea
LMAs
0.17
-0.366
< 0.001
-0.312
< 0.001
Rarea
LMAp
0.676
0.392
< 0.01
0.382
< 0.01
Rarea
LMAs
0.095
-0.308
< 0.001
-0.282
< 0.001
LL
LMAp
-0.549
-0.654
< 0.001
-0.573
0.307
LL
LMAs
0.556
0.600
0.317
0.726
< 0.01
39
Optimal
with site
effect
Narea
LMAp
0.823
0.427
< 0.001
0.459
< 0.001
Narea
LMAs
0.326
-0.284
< 0.001
-0.242
< 0.001
Parea
LMAp
0.766
0.448
< 0.001
0.441
< 0.001
Parea
LMAs
0.196
-0.31
< 0.001
-0.308
< 0.001
CLarea
LMAp
0.581
0.113
< 0.001
0.183
< 0.001
CLarea
LMAs
0.695
-0.021
< 0.001
0.080
< 0.001
Aarea
LMAp
0.846
0.454
< 0.001
0.502
< 0.001
Aarea
LMAs
0.208
-0.337
< 0.001
-0.327
< 0.001
Rarea
LMAp
0.690
0.439
< 0.001
0.400
< 0.01
Rarea
LMAs
0.063
-0.354
< 0.001
-0.331
< 0.001
LL
LMAp
-0.503
-0.502
< 0.01
-0.451
0.229
LL
LMAs
0.583
0.533
< 0.05
0.718
< 0.01
Narea
LMAp
0.808
0.409
< 0.001
0.444
< 0.001
Narea
LMAs
0.332
-0.300
< 0.001
-0.279
< 0.001
Parea
LMAp
0.740
0.384
< 0.001
0.391
< 0.001
Parea
LMAs
0.235
-0.249
< 0.001
-0.292
< 0.001
CLarea
LMAp
0.616
0.205
< 0.001
0.252
< 0.001
CLarea
LMAs
0.704
-0.132
< 0.001
0.001
< 0.001
40
Table S4 Correlations between observed and predicted LL (on logarithmic scales) for the Potential LL
Model (Eq. 5) and the Optimal LL Model (Eq. 13 and Notes S3) fit to Panama data. Pearson correlations
are given for models fit to the observed datasets (robs) and randomized datasets (rrand). For randomized
datasets, rrand is the mean correlation for 10 datasets in which LMA was randomized across all leaves,
and 10 datasets in which LMA was randomized within sites (wet and dry) across canopy strata (sun and
shade). P-values test the null hypothesis of equal correlation in observed and randomized datasets (see
‘Randomized LMA Datasets’ in Methods for details).
Model
robs
LMA randomized across
LMA randomized within
all leaves
sites across strata
rrand
P-value
rrand
P-value
Potential
0.382
0.502
< 0.001
0.652
< 0.001
Optimal
0.759
0.518
< 0.001
0.637
< 0.001
0.813
0.664
< 0.001
0.705
< 0.001
Optimal
with site
effect
41
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