1
Appendix S1 – Exploring the properties of the Cimp and Iimp indices
2
In this Appendix we set out in more detail some of our concerns with respect to Rees et al.’s
3
(2012) critique of the properties of the Cimp and Iimp indices, focussing in particular on certain
4
aspects that have not been covered in depth in our paper. Although this detail is not essential
5
for understanding the arguments we make in our paper, we present it here in case it is of
6
interest.
7
As noted in our paper, a critique of the indices Cimp and Iimp need not necessarily be a
8
critique of the concept of importance. However, given that an examination of these indices is
9
a substantial part of Rees et al.’s paper, it is clearly necessary to consider whether this
10
examination is reasonable. Our paper discusses the measurement of environmental severity
11
and the problems that arise from using proxies for it such as PNC or net primary productivity
12
that are more or less likely to be closely correlated with measurements of success (i.e. our
13
response variable; see “The use of PNC as a measure of “productivity””). In this Appendix we
14
consider in particular the data underlying the key figures provided by Rees et al. (e.g. Rees et
15
al.’s Fig. 1, referred to here as RF1). In so doing, it is essential to realise that “productivity”
16
in this case is PNC, the productivity of the target species. To help clarify this critical point,
17
where relevant in the discussion that follows we use “PNC” and not “productivity”.
18
In many cases the problems with Rees et al.’s criticism of the Cimp index are identical
19
to the problems associated with their criticism of Iimp. In the following, although we discuss
20
Cimp, it should be understood that unless explicitly stated these comments refer also to the
21
criticism of Iimp.
22
23
The complexity of the response of Cimp to “productivity”
24
Rees et al. illustrate much of their critique of the Cimp index in RF1. Close examination of
25
RF1 and associated text helps us to understand some of the assumptions lying behind this
26
critique, and in so doing leads us to question it.
27
First, RF1a presents some hypothetical relationships between C and PNC, which are
28
then used in RF1b to explore some of the characteristics of Cimp. Because C= PNC /PC, and we
29
know both PNC and C (from RF1a), we can then calculate PC (Fig. S1). Being able to
30
visualise PC as well as C and PNC allows us to assess whether the responses of Cimp shown in
31
RF1b are “more complex than the underlying patterns it seeks to summarize”. We would
32
suggest that the patterns in the raw data are not themselves simple, but that their summation
33
in RF1a using the derived ratio C make them appear simple, such that RF1b in contrast
34
appears complex, thereby lending weight to the argument that Cimp is somehow a difficult
35
index to understand.
36
In addition the ratio C is itself not intuitive: assuming a negative effect of neighbours,
37
competition will not be measured as a reduction in performance relative to plant performance
38
alone, as usually is the practice, but rather as a value larger than one. The difficulties with
39
interpreting C are even greater when considering facilitation. In extreme cases in which
40
plants depend on neighbours for their success (e.g. in arid ecosystems) and when PC>>PNC, C
41
then approaches 0. This measurement of biotic interactions therefore has an asymmetric
42
range of values for the two types of plant interactions: while it is highly restricted in the case
43
of facilitation (0<C<1), its range in the case of competition is unlimited (1<C<∞).
44
Examination of the raw data - as opposed to C - helps us to realise that the relationships
45
between Cimp and PNC shown in RF1b are understandable. If we consider first the dashed line
46
(C=2 in RF1a), this represents a case where PC increases linearly with PNC. In this case RF1b
47
usefully demonstrates the relationship between PNC and Cimp as PNC increases, and we see
48
that Cimp is constrained toward PNC when PNC either approaches Pmax, in which case Cimp
49
approaches 1, or as PNC approaches 0, in which case Cimp approaches 0 as well. To us such
50
constraint seems logical. When PNC is 0, how can Cimp not be 0 (and, indeed, how can C=2)?
51
When PNC = Pmax, it is clear that the only factor constraining plant success (P) is competition,
52
and so how can Cimp not be 1 (i.e. how can competition not be 100% of the impact of the
53
environment on plant growth)? However, although there is clearly constraint in the
54
relationship between Cimp and PNC, it is essential to remember that this does not demonstrate
55
that Cimp shows an inevitable relationship with environmental severity: this conclusion
56
depends on how environmental severity is expressed (e.g. as PNC, as NPP, or as a measure of
57
the abiotic environment such as water availability).
58
The dotted line in RF1a shows a rapid decrease in C with increasing PNC (note that the
59
apparent rapidity of this decrease is to some extent dampened by the use of a log scale). For
60
this to occur, PC must initially be incredibly small whilst PNC increases rapidly. This line
61
therefore represents a situation where, when there is a huge reduction in plant performance
62
due to the abiotic environment (i.e. PNC <<Pmax), competition is then also impacting much
63
more severely on the plant. However, despite C being very large when PNC is low, the
64
reduction in performance due to competition is still relatively small in comparison to the
65
absolute reduction in performance due to the total impact of the environment (Pmax-PNC).
66
Consequently Cimp is small. As PNC increases, PC then increases, eventually approaching PNC
67
when PNC approaches Pmax. Whilst PNC is < Pmax, competition still represents a smaller
68
proportion of the total impact of the environment, but as PNC approaches Pmax the only
69
limiting constraint on plant performance is competition, and consequently Cimp approaches 1.
70
The solid line shows a situation where C is initially very low, but then increases as
71
PNC increases, and approaches Pmax. If competition is most influential at low levels of
72
severity (where we assume PNC ≈ Pmax) then the relationship shown is again understandable:
73
at lower levels of PNC, competition is initially unimportant (C is small). Competition becomes
74
increasingly important and hence Cimp approaches PNC as PNC increases, but as PNC is still
75
less than Pmax competition is clearly not the only factor influencing plant success, and so Cimp
76
only equates to Pmax when Pmax = PNC.
77
In all these cases, we suggest that the responses of Cimp to variation in PNC as shown
78
in RF1b make logical sense if we view the data for PC as well as C, and we also suggest that
79
the apparent simplicity in RF1a, and its contrast to the “complexity” of RF1b, belies
80
complexity in the raw data used to calculate the apparently simple patterns of RF1a.
81
However, this discussion does help to clarify the differences between absolute (C) and
82
relative (Cimp) measures of competition. Whether these measures then make biological sense
83
can only be tested by exploring their relationship with measures of plant success and
84
community structure.
85
Finally, we return to the statement by Rees et al. concerning RF1b, especially the
86
expression of severity as productivity, and more specifically as PNC. From RF1a and RF1b,
87
Rees et al. conclude “Therefore, in general, we expect Cimp to increase with productivity
88
regardless of whether effects of competition, C, increase, decrease or are independent of
89
productivity.” It is worth stating again that what we are exploring in these figures (RF1a and
90
RF1b) is not the relationship between Cimp and severity (assumed to equate to productivity) in
91
general, but between Cimp and PNC. Clearly PNC and Cimp have a strong relationship, but as
92
PNC is one of the variables used to calculate Cimp it is not surprising that they are related.
93
If we accept the argument that for many studies and research questions PNC is not
94
used as the measure of severity, we cannot then use the relationship between PNC and Cimp to
95
assess the relationship between Cimp and severity. This is critical: much of the criticism of
96
Cimp, for example it’s inevitable decline in response to changes in severity, or the possibility
97
of approximating it using PNC /Pmax, is based on the assumption that PNC = productivity =
98
severity. We suggest this is potentially one of the most confusing points throughout this
99
debate.
100
101
Selection of random data
102
RF1c illustrates the randomly generated data used by Rees et al. to assess the relationship
103
between Cimp and “productivity” (i.e. environmental severity) in RF1d, from which Rees et
104
al. conclude that “we again find a strong positive relationship between Cimp and productivity
105
(Fig. 1 c, d)”. First, and as noted above, “productivity” is actually PNC, and productivity may
106
itself not be a good proxy for the ideal explanatory variable of environmental severity, so
107
conclusions about the relationship between Cimp and “productivity” must be seen in this
108
context.
109
Furthermore, and independent from this first point, the approach to selecting data for
110
this assessment is biased such that it suggests even random data produce a direct, simple
111
linear relationship between Cimp and PNC. The reasoning behind this second point is set out in
112
some detail in our paper (see The use of C as an index of competition). Briefly, for any given
113
value of PNC, as C increases away from 0 any increase in C has a much smaller impact on PC
114
(Fig. S2a). Consequently, generating a uniform distribution of C between 1 and 100 results in
115
a non-uniform distribution of PC such that in general PC << PNC. If we repeat the production
116
of RF1c, but use the resulting data to plot PC and C against PNC, we can see that the resulting
117
distribution of PC is strongly biased, such that PC tends to be very small irrespective of PNC
118
(Fig. S2b).
119
The obvious consequence of using data where PC << PNC, is that Cimp will inevitably
120
be found to approximate PNC (as in RF1d). A better approach is to randomly generate values
121
of PC. PC should be free to vary between 0 and PNC at any given point along the gradient of
122
PNC. If we redraw RF1c and RF1d, but using randomly generated values of PC selected from
123
a uniform distribution between 0 and PNC, first we see that C tends to be small (Fig. S3a). We
124
suggest this is reasonable as in many cases, particularly when PNC << Pmax, the impact of
125
competition is likely to be limited. Second we see that there is variability in Cimp away from
126
PNC. From these data (Fig. S3a, b) we cannot conclude that “the expected relationship
127
between Cimp and productivity is… PNC /Pmax, which is a linear relationship with slope
128
1/Pmax”.
129
130
An exploration of relationships using field data
131
The next obvious question is whether in field data values of Cimp occur away from the
132
upper boundary of the potential values. Rees et al. use data from Kadmon (1995) to test
133
whether their conclusions based on synthetic data hold true also for data from field studies.
134
While the work of Kadmon (1995) is highly elegant, for the reasons explained in our paper
135
(see The use of only the data from Kadmon (1995) to explore the Cimp-PNC relationship in
136
field studies) we reject it as a proof that Cimp can be simply calculated from PNC /Pmax. We
137
suggest that it is better instead to consider a number of works simultaneously. As for the data
138
from Kadmon (Fig. S4b1) the data from Belcher et al. (1995) show a strong relationship
139
between PNC and Cimp (Fig. S4a). However, in the data from Reader et al. (1994; Fig. S4c)
140
and Ariza & Tielbörger (2011; Fig. S4d), although PNC and Cimp are clearly related it is not
141
possible to predict Cimp using PNC/Pmax. Although it might be argued that, even in the data of
142
Reader et al. and Ariza & Tielbörger, PNC/Pmax is a reasonable approximation of Cimp, the use
1
The analyses by Brooker & Kikvidze (2008) used the data from Kadmon (1995) provided at
http://esapubs.org/archive/ecol/E080/006/. In these data, from which Fig. S4b has been produced, there are
12 separate points along the productivity gradient. However, in the figures of Rees et al. there appear to be
only 8 points along the productivity gradient, with points missing from the least productive end of the severity
gradient. Although this does not alter the relationship between Cimp and PNC, which remains strongly linear
with the addition of the extra points, it does alter that between PNC and C as shown in RF4c. With the addition
of these missing points the relationship becomes far more curve-linear, trending toward the hyperbolic.
143
of PNC/Pmax to approximate Cimp hides some important information. Perhaps Rees et al.’s
144
analysis is helpful in highlighting that in some data Cimp is closely related to PNC (i.e. the data
145
of Kadmon and Belcher et al.), whereas in others it is not (i.e. the data of Reader et al.), and
146
we might be able to use this distinction to separate out those situations where competition and
147
strain are or are not interacting.
148
The examination of these data again shows the problems of using PNC as a measure of
149
productivity (and proxy for severity). Data from the study by Kadmon show a strong linear
150
relationship between productivity and Cimp, even when productivity is expressed as g m-2
151
(Brooker & Kikvidze 2008, Fig. 1, Table 1). In addition there is relatively little variation in
152
Cimp away from this line. However, data from two of the other studies show no relationship,
153
(Belcher et al. 1995), or a positive relationship (Reader et al. 1994), but in both cases there is
154
substantial variation in Cimp along the gradients of productivity recorded in the study
155
(Brooker & Kikvidze 2008). To put it another way, there is not an inevitable strong positive
156
linear relationship between Cimp and stand-level productivity in these other two studies, in
157
contrast to the conclusion that we would draw if we used PNC as the measure of productivity.
158
159
Specific inaccuracies in the criticism of Iimp
160
As noted above, the problems associated with Rees et al.’s critique of Cimp (the use of PNC as
161
a measure of productivity; the use of C as a measure of competition, thereby hiding
162
underlying complexity within the raw data) apply also to their criticism of Iimp as
163
demonstrated in RF2 and the associated discussion. Therefore we highlight here only one
164
inaccuracy which is specifically relevant to Rees et al.’s critique of Iimp. Rees et al. state that:
165
“Seifan et al. 2010 explore the properties of their index using plant performance with
166
competitors as a measure of productivity (see their Fig. 1). This is unusual as
167
productivity is usually defined with respect to performance in the absence of
168
competition”.
169
This is an error in the interpretation of Seifan et al. 2010, which might itself results from
170
Rees et al.’s definition of productivity as a measurement of target plant performance. Seifan
171
et al.’s Fig. 1 is presented only to demonstrate a change in the common biotic indices’ values
172
in relation to a systematic change in the effect of neighbours. In theory, this change may be
173
correlated with a productivity gradient. The potential effect of such correlations between
174
neighbour effect and productivity (or other environmental gradients) on the behaviour of Iimp
175
and the other common indices are then explored in Seifan et al.’s Fig. 2.
176
177
Putting the detail back into the broader context
178
Finally, it is necessary to place these detailed discussions back into the broader
179
context of understanding the relationship between the relative role of interactions (their
180
importance) and environmental severity. Here we have addressed what we think are some
181
inaccuracies or misleading aspects in the fine detail of the work by Rees et al.. For example,
182
we have demonstrated that the responses of Cimp to variation in PNC as shown in RF1b make
183
logical sense if we view the data for PC as well as C (see The complexity of the response of
184
Cimp to “productivity”, above). Similarly we have demonstrated that in some cases the
185
relationship between Cimp and PNC may not be related to the relationship between Cimp and
186
stand level productivity (see “An exploration of relationships using field data”, above).
187
But this focus on the fine detail should not distract us from the broader messages as
188
set out in our paper concerning analysis of the relationship between competition importance
189
and environmental severity. We need to take an additional step back and remember, for
190
example, that stand level productivity itself may not be a good indicator of environmental
191
severity for the types of analysis in which we are interested (see “The use of PNC as a
192
measure of “productivity”” in our paper).
193
we hope that the analyses that we have provided in this Appendix have given some
194
insight into why we have concerns regarding some of the detail of the analytical approach
195
used by Rees et al. for assessing the Cimp and Iimp indices, but we would caution against
196
excessive focus on this detail being a distraction from the bigger and perhaps more important
197
points concerning the general concept of importance and approaches to its measurement that
198
are made in our paper.
199
200
Acknowledgements
201
We would like to thank Clara Ariza for permission to use the data shown in Figure S4d.
202
203
References
204
Ariza, C. & Tielbörger, K. (2011) An evolutionary approach to studying the relative
205
importance of plant-plant interactions along environmental gradients. Functional
206
Ecology, 25, 932-942.
207
208
209
210
Belcher, J. W., Keddy, P. A. & Twolan-Strutt, L. (1995) Root and shoot competition intensity
along a soil depth gradient. Journal of Ecology, 83, 673-682.
Brooker, R. W. & Kikvidze, Z. (2008) Importance: an overlooked concept in plant interaction
research. Journal of Ecology, 96, 703-708.
211
Carlyle, C.N., Fraser, L.H. & Turkington, R. (2010) Using three pairs of competitive indices
212
to test for changes in plant competition under different resource and disturbance levels.
213
Journal of Vegetation Science, 21, 1025–1034.
214
Gaucherand, S., Liancourt, P. & Lavorel, S. (2006) Importance and intensity of competition
215
along a fertility gradient and across species. Journal of Vegetation Science, 17, 455-
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464.
217
Gross, N., Liancourt, P., Choler, P., Suding, K.N. & Lavorel, S. (2010) Strain and vegetation
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effects on limiting resources explain the outcomes of biotic interactions. Perspective in
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Plant Ecology, Evolution and Systematics, 12, 9-19.
220
221
222
223
224
225
Kadmon, R. (1995) Plant competition along soil moisture gradients: a field experiment with
the desert annual Stipa capensis. Journal of Ecology, 83, 253-262.
Lamb, E.G., Kembel, S.W. & Cahill, J.F. (2009) Shoot, but not root, competition reduces
community diversity in experimental mesocosms. Journal of Ecology, 97, 155–163.
le Roux, P.C. & McGeoch, M.A. (2010) Interaction intensity and importance along two stress
gradients: adding shape to the stress-gradient hypothesis. Oecologia, 162, 733–745.
226
Reader, R. J., Wilson, S. D., Belcher, J. W., Wisheu, I., Keddy, P. A., Tilman, D., Morris, E.
227
C., McGraw, J. B., Olff, H., Turkington, R., Klein, E., Leung, Y., Shipley, B., van
228
Hulst, R., Johansson, M. E., Nilsson, C., Gurevitch, J., Grigulis, K. & Beisner, B. E.
229
(1994) Plant competition in relation to neighbor biomass: an intercontinental study with
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Poa pratensis. Ecology, 75, 1753-1760.
231
Rees, M., Childs, D.A. & Freckleton, R.P. (2012) Assessing the role of competition and
232
stress: a critique of importance indices and the development of a new approach. Journal
233
of Ecology, 100, 577-585.
234
235
236
Seifan, M., Seifan, T., Ariza, C. & Tielborger, K. (2010) Facilitating an importance index.
Journal of Ecology, 98, 356–361.
237
Table S1. Summary information from studies used by Rees et al. to support their use of PNC
238
as an indicator of habitat productivity. For each study the table shows: its main aim; brief
239
details of the experimental design; environmental factors investigated; habitat type; definition
240
of productivity used.
241
Study
Main goal
Main experimental
design
Neighbour removal
experiment in three
natural sub-alpine
grassland
communities
Main factors examined
Gaucherand
et al. 2006
Evaluating variations
in competitive
intensity and
importance along a
nutrient gradient
Lamb et al.
2009
Evaluating how
competitive intensity
and importance affect
plant community
structure
Experimental garden
communities
(mesocosms) in
which fertilization
level was
manipulated
Fertilization (which is
assumed to regulate
productivity levels) is
used to create
variation in root and
shoot competition
within communities
Productivity is not
measured
Carlyle et al.
2010
Evaluating the
response of
competition to
variation in resources
and disturbances
Greenhouse
experiments in which
water, fertilization,
clipping and neighbor
presence are
manipulated
Fertilizer levels
(high/low) are used
directly in the
comparison of
different competition
indices
Productivity is not
defined
Gross et al.
2010
Test whether the
outcome of plant
interactions is better
explained using
individual-based stress
measurements than by
overall productivity
Neighbor removal
experiment in four
natural communities
in sub-alpine
grasslands
Strain: an individual
level response to local
abiotic factors.
Measured as the
deviation of individual
performance without
neighbors from the
optimal performance
Above ground
standing biomass
(g/m2) is directly
connected to resource
availability and used
as a proxy for site
productivity
le Roux &
McGeoch
2010
Examine the response
of biotic interactions
to an environmental
gradient
Field sampling,
Marion Island
Altitude and wind
exposure are used as
proxies for abiotic
severity
Standing above
ground biomass per
year (g/m2), although
no analysis is
conducted with these
values
Fertility: nitrogen
dilution in grassland
canopies during
regrowth, based on
above ground biomass
and N content
Definition of
productivity
NI (Nitrogen Nutrition
Index) that correlates
with above ground
biomass
242
Figure S1. Our approximation of the change in PC associated with the values of C and PNC
243
shown in Fig. 1a of Rees et al. Note that the dash, dotted and solid lines in this figure equate
244
to the three relationships used (and similarly indicated) in Rees et al.
350
300
250
PC
200
150
100
50
0
0
50
100
150
200
PNC
245
246
250
300
350
247
Figure S2. Exploration of relationships between C, PC and PNC. a) Illustration of the impact
248
of variation of C on the associated values of PC. In this example PNC is held constant at 100,
249
and C varies between 1 and 100. As the figure shows, a given change in C results in a much
250
greater change in PC when C is small. When C is large, the response of PC is greatly reduced.
251
b) Relationship between PNC and both C and PC when C is randomly selected from a uniform
252
distribution between 1 and 100.
253
a.
PC
100
10
1
0
20
40
60
80
100
120
C
254
255
b.
160
140
120
C and PC
100
80
C
Pc
60
40
20
0
0
50
100
150
200
PNC
256
257
250
300
350
400
258
Figure S3. Values for a) C and b) Cimp, in relation to PNC, that result from selecting
259
randomly-generated values of PC (not shown) from a uniform distribution between 0 and PNC.
260
These can be contrasted directly with figures 1c and 1d in Rees et al..
261
a.
35
30
25
C
20
15
10
5
0
0
50
100
150
200
250
300
350
400
250
300
350
400
PNC
262
263
b.
1.2
1
Cimp
0.8
0.6
0.4
0.2
0
0
50
100
150
200
PNC
264
265
266
Figure S4. Plots of PNC vs. Cimp. Data shown are from three datasets used by Brooker &
267
Kikvidze (2008): a) Belcher et al. 1995, b) Kadmon 1995, c) Reader et al. 1994; as well as d)
268
from the study by Ariza & Tielbörger (2011). In the latter case, because of the large number
269
of negative values of Cimp (facilitation) only data for positive values of Cimp are shown, with
270
filled diamonds and open circles indicating values for Biscutella didyma and Hymenocarpos
271
circinnatis, respectively.
b.
1.2
1.2
1
1
0.8
0.8
0.6
0.6
Cimp
Cimp
a.
0.4
0.4
0.2
0.2
0
0
0
0.5
1
-0.2
1.5
2
200
400
600
800
1000
1200
1400
1600
PNC (seed no.)
d.
1.2
1.2
1
1
0.8
0.8
Cimp
Cimp
c.
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
-0.2
272
0
-0.2
PNC (Yiso biomass)
0.01
0.02
0.03
0.04
PNC (phytometer rgr)
0.05
0.06
0
0.5
1
PNC (total biomass, g)
1.5
2