1
Appendix 1 in Laughlin: The intrinsic dimensionality of plant traits…
Supporting Information
2 Appendix S1. Methodological details
3 Daniel C Laughlin
4
5 Estimating the intrinsic dimensionality of plant traits using three large datasets
6 I expanded a previously published dataset from northern Arizona woodlands and
7
forests (Laughlin et al., 2010) to include 16 traits from 201 species. Traits in this dataset
8 include maximum height, seed mass, seed variance, flowering date, bark thickness, leaf area,
9 specific leaf area, leaf dry matter content, leaf C:N ratio, leaf N, leaf P, stem tissue density,
10
specific root length, root C:N ratio, root N, and leaf decomposition rate (Pérez-Harguindeguy
11
et al., 2013). The dataset was missing data for 11% (360 of 3216) of the cells in the matrix, so
12
I imputed missing data using the ‘mice’ function in the ‘mice’ library of R (R Development
13
Core Team, 2012, van Buuren and Groothuis-Oudshoorn, 2011). Imputing data may generate
14 redundancy among variables, thereby underestimating the true dimensionality of the dataset.
15 The sensitivity of this choice on the final outcome was explored with functions that can
16 handle missing data, and dimensionality was surprisingly underestimated using missing data.
17 Thus, for consistency and comparative purposes, I describe the analyses based on the filled
18 matrix. Variables were ln-transformed to improve the normality of their distributions, and
19 were scaled such that variance <10.
20 The other two datasets were extracted from the literature and represent some of the
21 most comprehensive list of traits included in any study to date. The trait dataset of California
22 chaparral species included 36 ecophysiological and morphological traits measured on 20
23
species (see Ackerly (2004) for which traits are included). Nineteen out of 720 (2.5%) data
24 points were missing, so these were imputed using the same methods as described above.
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25
Appendix 1 in Laughlin: The intrinsic dimensionality of plant traits…
The Sheffield trait dataset of Grime et al. (1997) included 67 traits on 43 species,
26 including whole-plant, leaf, seed, root, and flower traits. The pair-wise matrix of correlations
27
among the 67 traits in Figure 1 of Grime et al. (1997) are Spearman rank correlations. Ideally,
28 eigenanalysis is conducted on Pearson correlations, but given their similarity I used the
29 following procedure to analyses the Spearman ranks. The Spearman rank correlation matrix
30 was not positive definite, so I made minor adjustments to the matrix using the
31
‘make.positive.definite’ function of the ‘corpcor’ library of R (Schaefer et al., 2012, R
32
33
Development Core Team, 2012) to make it positive definite, and used the ‘cov2cor’ function
on the base ‘stats’ library or R to ensure that the diagonal elements were all ones. I used the
34
Procrustes test in the ‘vegan’ library of R (Oksanen et al., 2011) to examine the correlation
35 between the original and adjusted matrix, and they were statistically indistinguishable
36 (Procrustes r = 1, P < 0.001). Therefore, eigenanalysis was conducted on this adjusted matrix.
37 Furthermore, other functions required actual trait values on the subjects for analysis, so I
38 simulated data for 43 hypothetical species (to match Grime’s original number of species)
39 assuming a common mean of 0 and a covariance structure from above. Given these
40 assumptions, I conducted these adjustments and simulations 10 times to evaluate their
41 sensitivity, and the estimations of the intrinsic dimension were highly stable among the
42 simulations. I therefore used the median of the estimates of intrinsic dimension, which were
43 very similar to both the mean and mode.
44
45 Assessing trait importance in community assembly
46 To date, six studies have published results that show how explanatory power of a
47 trait-based model changes with increasing number of traits using the CATS model of
48
community assembly (Shipley, 2010). For each of the six studies I report results of models
49 that used observed community-weighted mean traits, because the focus of this comparison is
2

50
Appendix 1 in Laughlin: The intrinsic dimensionality of plant traits… to determine how well traits explain community composition. Different model fit indices
51 were used in these studies. Four papers reported the Pearson R 2 which measures the linear
52
relationship between observed and predicted relative abundances (Frenette-Dussault et al.,
53
2013, Laughlin et al., 2011, Shipley et al., 2011, Laliberté et al., 2012). Merow et al. (2011)
54 used a hellinger distance, but since the data was published I was able to calculate the change
55 in Pearson R
2
with additional traits for this dataset. Sonnier et al. (2012) used an index related
56 to Kullback Liebler divergence ( R 2
KL
), which, though different from Pearson R 2 , ranges
57 between 0 and 1, so is relatively comparable.
58 Another difference between these studies is how traits were chosen for sequential
59 entry into the model, which can influence the rate of change in R
2
with increasing traits. Four
60 studies entered traits into the model in the order of decreasing absolute value of the Lagrange
61
multiplier associated with each trait (Laliberté et al., 2012, Merow et al., 2011, Sonnier et al.,
62
2012, Shipley et al., 2011). Lagrange multipliers indicate the strength and sign of selection on
63 each trait relative to all others. The other two studies entered traits into the model in the order
64
of decreasing strength of relationship to the environmental gradient (Frenette-Dussault et al.,
65
2013, Laughlin et al., 2011). Though technically different, each method bases the choice of
66 traits on the strength of trait selection.
67
I used the scree plot criterion (Cattell, 1966) in combination with a minimum fit index
68 of 0.60 to determine the number of traits beyond which predictive power ceases to increase
69 substantially. The trait at which a bend or elbow in the curve is most prominent was chosen.
70 Beyond these ‘elbows’ in the curves, the relationship between the number of traits and
71 predictive power was linear and relatively flat.
72 I compared the consistency of the importance of each of the organ traits or whole-
73 plant properties by counting the number of times each was important and dividing this by the
74 number of times each was included across the six studies (Table S1). For example, leaf traits
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75
Appendix 1 in Laughlin: The intrinsic dimensionality of plant traits… were consistently useful because they were important in five out of the six studies that
76 included them. Seed traits were less consistently useful because they were important in two
77 out of the five studies that included them. Importance was determined by whether its
78 inclusion improved explanatory power of the trait-based model.
79
80 Table S1.
The importance of each of the seven organs and whole-plant properties for
81 predicting community structure in each of the six studies. The first answer addresses if it was
82 important, assessed by whether it increased predictive power (see Fig. 3 in the main text).
83 The second answer addresses if the trait was used in the study. Importance was assessed by
84 the number of times it was important divided by the number of time it was included.
85
Study
Pine forest understory
Montane and subalpine forest
Tussock grassland
Arid steppe
Upland rangelands
Fynbos
Important/Included
Importance frequency
Leaf Height Seed Root Stem yes/yes no/yes yes/yes yes/yes no/no
Flowers no/yes
Life history yes/yes no/yes yes/yes no/yes no/yes yes/yes yes/yes no/no yes/yes yes/yes no/yes yes/yes no/no yes/yes no/yes no/yes no/no yes/yes yes/yes yes/yes no/no yes/yes no/yes no/no
5/6 3/6 2/5
83% 50% 40% no/no
2/3
67% yes/yes no/no no/yes yes/yes yes/yes yes/yes no/yes no/yes yes/yes yes/yes no/yes
3/3
100%
4/6
67%
2/5
40%
86 References
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94
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89
90
95
96
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Ackerly, D. (2004) Functional strategies of chaparral shrubs in relation to seasonal water deficit and disturbance. Ecological Monographs, 74, 25-44.
Cattell, R. B. (1966) The scree test for the number of factors. Multivariate behavioral research, 1, 245-276.
Frenette-Dussault, C., Shipley, B., Meziane, D. & Hingrat, Y. (2013) Trait-based climate change predictions of plant community structure in arid steppes. Journal of Ecology,
101, 484-492.
Grime, J. P., Thompson, K., Hunt, R., Hodgson, J. G., Cornelissen, J. H. C., Rorison, I. H.,
Hendry, G. A. F., Ashenden, T. W., Askew, A. P., Band, S. R., Booth, R. E., Bossard,
C. C., Campbell, B. D., Cooper, J. E. L., Davison, A. W., Gupta, P. L., Hall, W., Hand,
D. W., Hannah, M. A., Hillier, S. H., Hodkinson, D. J., Jalili, A., Liu, Z., Mackey, J.
4

122
123
124
125
126
127
128
116
117
118
119
120
121
129
130
131
132
133
134
135
136
137
138
106
107
108
109
110
111
112
113
114
115
98
99
100
101
102
103
104
105
Appendix 1 in Laughlin: The intrinsic dimensionality of plant traits…
M. L., Matthews, N., Mowforth, M. A., Neal, A. M., Reader, R. J., Reiling, K., Ross-
Fraser, W., Spencer, R. E., Sutton, F., Tasker, D. E., Thorpe, P. C. & Whitehouse, J.
(1997) Integrated Screening Validates Primary Axes of Specialisation in Plants. Oikos,
79, 259-281.
Laliberté, E., Shipley, B., Norton, D. A. & Scott, D. (2012) Which plant traits determine abundance under long-term shifts in soil resource availability and grazing intensity?
Journal of Ecology, 100, 662-677.
Laughlin, D. C., Fulé, P. Z., Huffman, D. W., Crouse, J. & Laliberté, E. (2011) Climatic constraints on trait-based forest assembly. Journal of Ecology, 99, 1489-1499.
Laughlin, D. C., Leppert, J. J., Moore, M. M. & Sieg, C. H. (2010) A multi-trait test of the leaf-height-seed plant strategy scheme with 133 species from a pine forest flora.
Functional Ecology, 24, 493-501.
Merow, C., Latimer, A. M. & Silander Jr, J. A. (2011) Can entropy maximization use functional traits to explain species abundances? A comprehensive evaluation. Ecology,
92, 1523-1537.
Oksanen, J., Blanchet, F. G., Kindt, R., Legendre, P., Minchin, P. R., O'Hara, R. B., Simpson,
G. L., Solymos, P., Stevens, M. H. H. & Wagner, H. (2011) Vegan: community ecology package. R package version 2.0-2. http:/CRAN.R-project.org/package=vegan.
Pérez-Harguindeguy, N., Díaz, S., Garnier, E., Lavorel, S., Poorter, H., Jaureguiberry, P.,
Bret-Harte, M. S., Cornwell, W. K., Craine, J. M., Gurvich, D. E., Urcelay, C.,
Veneklaas, E. J., Reich, P. B., Poorter, L., Wright, I. J., Ray, P., Enrico, L., Pausas, J.
G., de Vos, A. C., Buchmann, N., Funes, G., Quétier, F., Hodgson, J. G., Thompson,
K., Morgan, H. D., ter Steege, H., van der Heijden, M. G. A., Sack, L., Blonder, B.,
Poschlod, P., Vaieretti, M. V., Conti, G., Staver, A. C., Aquino, S. & Cornelissen, J.
H. C. (2013) New handbook for standardised measurement of plant functional traits worldwide. Australian Journal of Botany, 61, 167-234.
R Development Core Team (2012) R: A language and environment for statistical computing.
R Foundation for Statistical Computing, Vienna, Austria.
Schaefer, J., Opgen-Rhein, R., Zuber, V., Silva, A. P. D. & Strimmer, K. (2012) corpcor:
Efficient Estimation of Covariance and (Partial) Correlation. R package version 1.6.2.
Shipley, B. (2010) From Plant Traits to Vegetation Structure: Chance and Selection in the
Assembly of Ecological Communities. Cambridge University Press, Cambridge, U.K.
Shipley, B., Laughlin, D. C., Sonnier, G. & Otfinowski, R. (2011) A strong test of a maximum entropy model of trait-based community assembly. Ecology, 92, 507-517.
Sonnier, G., Navas, M. L., Fayolle, A. & Shipley, B. (2012) Quantifying trait selection driving community assembly: a test in herbaceous plant communities under contrasted land use regimes. Oikos, 121, 1103-1111. van Buuren, S. & Groothuis-Oudshoorn, K. (2011) mice: Multivariate Imputation by Chained
Equations in R. Journal of Statistical Software, 45, 1-67.
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