Annals of Botany 88 (Special Issue): 703-712, 2001
doi:10.1006/anbo.2001.1481, available online at http://www.idealibrary.com on
®
IDEA-
Developing Multisite Dynamic Models of Mixed Species Plant Communities
J. CONNOLLY* and M. WACHENDORFt
Department of Statistics, University College Dublin, Belfield, Dublin 4, Ireland
Received: 31 October 2000
Returned for revision: 4 January 2001
Accepted: 2 May 2001
Data on the development of two white clover cultivars (AberHerald and Huia) in mixed clover/ryegrass swards were
available at 12 sites in Europe from experiments conducted for several years under a common protocol. Swards were
measured up to seven times over winter and up to seven times over the growing season. In the overwintering period,
detailed morphological measurements were taken for clover at each sampling time and, during the growing season,
the clover contribution to total available biomass was recorded. Detailed meteorological data were available at all
sites. The development of these clover/ryegrass communities over time and sites was modelled. The modelling strategy
had three main elements: (a) division of the annual growth cycle of the clover/ryegrass community into a number of
functional periods; (b) development of relationships within each functional period using models incorporating plotlevel biotic variables characterizing each community at the start of the period and site-level climatic and management
variables measured during the period; and (c) introduction of a dynamic element by linking the models across
functional periods. The response variable(s) for a functional period was the biotic independent variable(s) of the
succeeding period. The object was to produce a dynamic series of models in which community development within
and across sites was described as a resultant of the initial state of the community and climatic and other forces acting
on it. The analysis used a mixed models technique in recognition of the complex error structure of the data. Various
statistical aspects of the modelling are discussed including the models and fitting strategy used, the complexity of the
error structure in an experiment that includes sites and years, and the desirability of transforming certain variables
before modelling. The issues in presenting the results of a series of complex models are discussed and a graphical/
tabular approach is outlined.
© 2001 Annals of Botany Company
Key words: Meta-analysis, multisite, community dynamics, competition, mixed model, white clover, Trifolium
repens, perennial ryegrass, Lolium perenne, AberHerald, Huia, dynamic model.
INTRODUCTION
In plant breeding where cultivars have to prove themselves
across a range of different environments multisite experiments with plants are quite standard (e.g. Talbot and
Verdooren, 1996), but they are otherwise relatively rare (e.g.
Reader et al., 1994), especially for mixed species commu-
nities. Although meta-analysis, the combination of information from many experiments, is of growing importance
in plant studies (e.g. Curtis and Wang, 1998; Xiong and
Nilsson, 1999; Poorter and Nagel, 2000) the experiments
being combined rarely share a common experimental
structure and philosophy, or a common measurement
framework. This inevitably leads to a somewhat simple
combined analysis. Few experiments with mixed plant
communities track the system with detailed measurements
through a number of annual cycles. Hence, the data from a
common experiment on clover/ryegrass mixtures performed
at 12 European sites over several years (Wachendorf et al.,
2001a, b) provide a rare opportunity for modelling dynamic
community relationships from a broad and complex body
of data. The results of the analysis of these data are
reported in Wachendorf et al. (2001a, b) and this paper
discusses the background to their analysis.
* For correspondence. E-mail john.connolly(a,uccl.ie
t Current address: Institute of Crop Science and Plant Breeding,
Department of Grass and Forage Science, University of Kiel,
Olshausenstr.40, 24098 Kiel, Germany.
0305-7364/01/100703 + 10 $35.00/00
Successive sections describe the experiments, consider the
principles underlying the modelling strategy, and discuss
various statistical modelling issues including methods for
presenting the results from a series of linked dynamic
models. The approach deals with difficulties in combining
data over many sites and also develops perspectives relevant
to evaluating data when information is available on several
phases of the annual cycle of mixed species plant
communities.
DESCRIPTION OF EXPERIMENTS
A detailed description of the experiments is given in
Wachendorf et al. (2001a) and is summarized here. A
common experiment was repeated at 12 sites in Europe
ranging from Iceland to Italy, and Ireland to Finland.
Generally, a common experimental protocol covering
design and measurement was followed. Experiments in
different sites began in different years and generally lasted
for 3-4 years. Usually measurement commenced after an
establishment phase of 1 year. At each site, two clover
cultivars were compared, each grown with a companion
ryegrass in a randomized complete block design with three
to four replicates of each clover/ryegrass mixture. The
clover/ryegrass mixtures were sown by broadcasting using
3 kg ha - l of one of two white clover cultivars (Trifolium
repens cultivars AberHerald and Grasslands Huia, both
medium-sized leaf) mixed with perennial ryegrass (in most
t
2001 Annals of Botany Company
704
Connolly and Wachendorf- Multisite Dynamic Models of Mixed Species Plant Communities
cases Lolium perenne Preference). Nitrogen fertilizer was
applied in equal amounts at the start of the growing season
and after each harvest. The total quantity of nitrogen used
varied among sites and, at some sites, among years.
Morphological measurements were made on four to
seven occasions during the winter. The first sampling took
place approx. 35 d after the last harvest (mean over sites)
before the onset of severe frost. The winter sampling period
ended before a considerable accumulation of biomass had
taken place. Up to seven harvests were taken during the
growing season. The first harvest was taken when the sward
height reached 20 cm and the last was taken when no more
biomass accumulation was detectable, but no later than the
end of October.
For each morphological sampling, three to four cores per
plot, 12 cm in diameter and 10 cm deep, were taken at a
distance of at least 30 cm from each other, avoiding
previously sampled locations. Samples were pooled and a
representative subsample was taken for processing. A range
of morphological characteristics was measured for each
plot. These included clover mass variables [clover dry mass
(DM), DM of stolons, petioles, buds and nodal roots],
clover morphological variables (number of terminal buds,
total buds, nodal roots and tap roots, leaf area index, stolon
length), clover biochemical variables [total non-structural
carbohydrates (TNC) and water soluble carbohydrates
(WSC) of stolons], grass variables (grass DM, tiller density)
and combined variables (clover DM as a percentage of total
DM). Swards were harvested two to seven times during the
growing season, cut to a stubble height of 5-7 cm. The
fresh weight of the harvested material was estimated from
sub-plots of 0.25 to 1 m2 within each plot generally cut to a
stubble height of 5-7 cm. Dry matter yield of the total
sward and of the grass, clover and unsown species
components was recorded. For all sites, climatic variables
(air temperature, radiation, precipitation) were typically
recorded daily.
PRINCIPLES OF THE MODELLING
STRATEGY
The object was to produce a series of models in which
community development within and across sites was
described as a resultant of the initial state of the community,
applied treatments and forces common to all plots at a site
(climate, management) acting on it. The modelling strategy
had three main elements: (a) the annual growth cycle of the
clover/ryegrass community was divided into a number of
functional periods; (b) relationships were developed for each
functional period, modelling some measure(s) of community performance, typically at the end of each period.
Variables in the models included biotic variables measured
on each plot at each site and characterizing community
status at the start of the period, applied treatments (clover
cultivar) and climatic and other variables measured at the
site level during the period; and (c) a dynamic element was
introduced by linking the models across functional periods.
This was done by taking as response variable(s) for a
functional period the biotic variable(s) that characterized
initial community status in the model for the succeeding
period. Some elements of this approach are similar to
methods for models of single crops described by France and
Thornley (1984) and of cloverjryegrass dynamics under
mixed and mono-grazing (Nolan et al., 2001).
(a) Division of annual cycle into functional periods
At each site, community development followed a
particular rhythm through its annual cycle, from overwintering through initiation of growth to growing season
growth. The timing of these phases varied widely across
sites and was reflected in the timing of morphological and
production measurements at each site, leading to a data set
uncoordinated in calendar time across sites. In consequence, a calendar-based approach to modelling (which
might suffice at any one site) was regarded as inappropriate.
Instead, the annual life history cycle of the community was
broken into four functional periods (Fig. 1):
Summer-from the first to the last growing season cut;
Spring- from the final morphological measurement to
the first growing season cut;
Winter- from the first to the last morphological
measurement; and
Autumn-from the last growing season cut to the first
morphological measurement.
The mean, maximum and minimum durations of each of
these periods are shown in Table I. Modelling plant
behaviour and community development within each of
these periods reduced the statistical difficulty and facilitated
interpretation by comparing a common phase of the life
cycle of the community across all sites.
(b) Modelling using plot and site independent variables
Each plot is a community whose development can be
affected by initial conditions that differ from other plots at
the same site, even those treated identically. Plant growth
and interaction in a community during a functional period
is largely a result of sward state at the start of the period,
clover cultivar and forces (e.g. climate, soil, management)
common to all plots at a site in a particular year during the
functional period. System responses (e.g. clover content,
clover leaf area index or grass tiller density at the end of, or
averaged over, a period) were modelled for each functional
period. Some independent variables for each model were
measured at the plot and some at the site level. Plot
variables are biotic descriptors characterizing the status of
the community at the start of the functional period (e.g.
clover content, grass tiller density, clover leaf area index).
Site variables (e.g. temperature, radiation, precipitation,
management, soil type) characterize growing conditions at
different sites and have a common value for all plots within
a site for a given functional period and year. Models also
included cultivar effects.
In this approach the prior history of swards is taken to be
largely integrated into the observed sward state at the start
of the period of interest, and other facets of the plots that
are not measured are taken to be largely allowed for by the
Connolly and Wachendorf-Multisite Dynamic Models of Mixed Species Plant Communities
705
FIG. 1. Organization of modelling of system over four functional periods.
TABLE 1.
Response variables, plot and site independent variables in addition to clover cultivar, and mean, minimum and
maximum duration (d) for each functional period
Independent variables
Duration (d)
Functional
period
Model
Response variables
Plot
Site*
Mean
Minimum
Maximum
Summer
Spring
A
B
Clover content
Clover content
T
TS, PP, RRS
129
17
52
9
161
30
Winter
Winter
Autumn
Autumn
Summer
C1
C2
D1
D2
E
Clover leaf area index
Grass tiller density
Clover leaf area index
Grass tiller density
Clover content
Clover content
Clover leaf area index
Grass tiller density
Clover leaf area index
Grass tiller density
Clover content
Clover content
Clover content
T, RR
T, PP, D
PP, RR
T, PP, D
TS
172
172
37
34
130
81
81
13
15
48
239
239
119
55
204
*C, Clover cultivar; T, mean daily temperature; PP, mean daily precipitation; RR, mean daily radiation; TS, temperature sum; PPS,
precipitation sum; RRS, radiation sum; D, duration of period.
plot independent variables. Site variables and their
interaction with plot variables and clover cultivars are
likely to be major determinants of variation among sites
and over years. If variation in community behaviour across
plots, sites and years can be described by such variables it
could lead to a wider understanding of the role of the three
types of independent variable in the development of clover/
ryegrass communities and perhaps the degree to which
variation in community behaviour is beyond prediction,
being linked to unpredictable elements of the environment.
It might also provide a useful framework for hypothesis
generation and allow an understanding of the likely
response at a particular site to variation in the site variables
over a number of years. These insights would be difficult
to gain at a single site, even for an experiment repeated over
2 or 3 years, since climate would be unlikely to vary
sufficiently over years to allow good estimates of their
effects and some site variables (e.g. soil) would hardly
change at a site.
(c) Linking models across.functionalperiods
The models in different functional periods were linked to
produce a set of dynamic models describing sward development through the annual cycle. The independent plot
variable(s) for one functional period becomes the response
706
Connolly and Wachendorf-Multisite Dynamic Models of Mixed Species Plant Communities
variable(s) to be modelled in the preceding period (Fig. 1,
Table 1). For example, leaf area index at the end of winter
was an important determinant of clover content at the end
of spring, and, in turn, was modelled using sward variables
at the beginning of winter. A number of plot variables were
available at the start of each functional period as possible
candidates for inclusion as independent variables, particularly when the detailed morphological measurements were
available. The selection of the independent variables
actually used is described below. Also available for inclusion were climatic variables for each modelling period,
mean and cumulative temperature, radiation and precipitation. Since the duration of a functional period differed
across sites, mean and cumulative climatic variables are not
simply related and may describe different facets of the
response.
Modelling commenced by examining the determinants of
average clover content in swards during summer using
clover content at the end of spring as the plot variable.
Clover content at the end of spring was then modelled using
morphological variables at the end of winter as plot
variables, and so on. Finally, to complete the cycle, clover
content at the end of summer was modelled using end of
spring biotic variables. In all, seven models (Table 1) were
required for the four functional periods. A potential
difficulty with this approach is that several plot biological
variables could emerge as important independent variables
for some functional period(s), which could make the
modelling of the preceding functional period(s) quite
complex, with an expanding number of variables to be
modelled at each phase. This did not occur and the chain of
models was based on only a few biological variables
(Table 1; Wachendorf et al., 2001h).
STATISTICAL ISSUES
Statistical issues arising in modelling these data include (a)
the models and fitting strategy used, (b) the complexity of
the error structure in an experiment that includes plots,
blocks, sites and years, (c) the desirability of transforming
certain variables before modelling and (d) procedures for
presenting the results of the models. The modelling of mean
clover content during summer (Model A) is used to
illustrate various issues discussed below.
(a) Models and fitting strategy
Preliminary modelling was carried out to identify the
relative importance of site, year, clover cultivar and their
interactions, all as fixed effects, in determining morphological, chemical and productivity responses (Wachendorf
et al., 2001a). For the main modelling (Wachendorf et al.,
2001b) linear mixed models (Verbeke and Molenberghs,
1997) were used in each functional period to allow for the
estimation of the effects of cultivar and of plot and site
variables on responses and for the complex error structure.
The general form of the models was the standard mixed
model, with fixed and random effects:
y = Xfp+Zu+:
(1)
where y is the response, /i is the vector of fixed effects
parameters, X is the matrix of fixed independant variable
and factor level indicator values, u is the vector of random
effects for each response, other than the residual error, Z (a
matrix of zeros and ones) details which random effects are
associated with each plot and is a vector of residual error
terms which are assumed to have the same variance and to
be uncorrelated across plots and years. The form of the
random components is discussed below.
Modelling was carried out using the REML (restricted
maximum likelihood) option of the mixed procedure in
SAS. Determination of the significance of fixed terms was
through Wald tests with denominator degrees of freedom
estimated by Satterthwaite's approximation (SAS Institute,
1992). Selection among different transformations of independent variables was aided by comparison of full
maximum likelihood, with the random effects as defined
below.
The selection of terms for inclusion in the fixed part of
the model depended on preliminary analysis, standard
statistical model selection methods (Draper and Smith,
1998), the rules of hierarchy and marginality (Nelder. 1994;
Nelder and Lane, 1995) and biological insight. Fixed effect
terms were included if their significance exceeded the 5 ,
level. Models containing many fixed terms were initially
fitted and then reduced by a combination of statistical
testing and biological assessment. The rules of hierarchy
ensured that if an interaction was included in the fixed part
of the model then the variables (factors) involved in the
interaction were also included separately. The modelling
obeyed the marginality principle (Nelder and Lane, 1995).
which implies that if a term appears as part of a more
complex element in the model then, in general, the term is
not tested for significance. Thus, if there is an A x B
interaction in the model the estimates of A and B are not
tested. This is because the meaning of such terms is open to
misinterpretation as they depend on the particular way in
which SAS sets up the model with interaction and could
differ were another package used for analysis. Furthermore,
polynomial terms lower in order than the highest term fitted
(e.g. a linear term in a quadratic model) are also
particularly open to misinterpretation unless the independent variables are centred about their means prior to
analysis. As centring carries its own difficulties, uncentred
data were used in the analyses and the methods of
presentation of results were as described below. Thus,
t- and P-values for model coefficients where the term is
involved in a higher order term in the model were generally
omitted. There is an exception. Where a factor with only
two levels [as with the factor clover cultivar (C), where
levels 1 and 2 correspond to AberHerald and Huia]
interacts only with a variate x, and the model contains no
quadratic or higher order term in x, there is a useful
interpretation of the coefficient of x. This occurs for models
B, Cl and D2 (Wachendorf et al., 2001b). Let the relevant
terms in the model be 13lx + Bi3x C. Using the protocol in
SAS, the coefficient of x by itself (,) is the slope of the
response to .v for Huia and the slope for AberHerald is
(fl +3). When this occurs, the recommendation is to
present the t- and P-values for the linear coefficient fl and
Connolly and Wachendorf-Multisite Dynamic Models of Mixed Species Plant Communities 707
to interpret
cultivars.
3
as the change in slope between the two
(b) Error structure of models
The multisite and multiyear nature of the study suggests a
complex error structure and hence the necessity to use a
mixed model approach. Random components considered
were: site; year within site; block within site; and plot within
block within site, which leaves the (repeated measures) yearto-year variation associated with each plot as the residual
term. As data for 2 consecutive years only were available for
most plots, compound symmetry was assumed for the
repeated measures. The meaning of the variance components is as follows. Site, although not chosen randomly,
is considered random, conditional on having removed the
effects of the site variates included in the model, i.e. the
remainder of site differences, having removed the effects of
the site variables, are considered as random. The terms
involving blocks represent the structure imposed on the
experiment within each plot, recognizing that blocks have
no relationships across sites. It might be argued that blocks
are not random within a site, rather the contrary if an
effective blocking strategy has been employed. In the
current experiment, the use of biotic covariates seems to
have minimized block variation and so it generally appears
to contribute little or nothing to the variation (see Table 3,
Wachendorf et al., 2001b). The issue of its randomness is,
therefore, somewhat academic here. Plot within block
within site represents the variation among plots within a
block at a site averaged over years, and year within site
represents the way in which the site-to-site response (having
fitted the model) changes from year to year.
Estimates of these components for model A are: 0 for
site, 0.274 for year within site, 0.00253 for block within site
and 0.00189 for plot within block within site and 0111 for
residual. Estimates of these terms are shown for all models
in Table 3 of Wachendorf et al. (2001b). The year within
site component was large for all models.
The inclusion of non-random terms such as site and
block as random in mixed models needs to be treated with
caution, but sometimes this approach provides the only way
of assessing the influence of variables like climate in nonrandomized data sets. It is not always appropriate to
assume that such terms are random and normally
distributed. The issue is affected by considerations such as
there being a sufficient number of levels of the factor(s) and
the pattern of the effects for different levels e.g. would site
effects, having removed the influence of climate, look like a
sample from a normal distribution or are block effects
normally distributed, having removed the effects of plot
biotic variables? The decision as to whether effects are
random or otherwise is discussed in Searle et al. (1992) and
the non-normality of random effects in Verbeke and
Molenberghs (1997).
(c) Transformation of variables
Three variables were transformed before inclusion in the
models as response variables: clover content, leaf area index
and tiller density. Some of the independent variables were
also transformed. For more information on the rationale
behind transformations see e.g. Sokal and Rohlf (1995).
Mean summer clover content (g 100 g-l DM) ranged
from 08 to 95-2, and clover content at the end of spring
from 0.05 to 99-6. These values are close to the minimum
and maximum (0 and 100) achievable and suggest that
models in which the untransformed clover content was the
response could lead to predicted clover contents outside this
range, and might also include interactions induced by scale
rather than biology (Cox and Snell, 1989). Under conditions in which levels of clover content are low or close to
100 %, the absolute effects of treatments or other factors/
variables tend to be smaller than at intermediate values,
simply because there is less scope for response. For
example, a factor that doubles low clover content will
change 2 to 4 % at very low levels but will increase 10 to
20 %, a seeming interaction with clover level which
obscures the simplicity of the doubling effect of the factor.
(Of course such a factor could not work in this way at high
clover contents.) To avoid these difficulties, the logit of
clover content (CL), defined as log[CL/(100-CL)], was
used. The logit transformation tends to give a scale that
reduces the effect of interaction simply due to scale and also
ensures that the model will lead to predicted clover contents
lying within the range 0 to 100.
Clover leaf area index is a response variable in models C 1
and D1 (Table 1). In both cases the logarithm of leaf area
index was used, as residual plots indicated severe heteroscedasticity for models on the original scale. Residuals from
models on the logarithmic scale showed some evidence of
greater variance for smaller predicted values for model C1
but not for model Dl. Grass tiller density is a response
variable in models C2 and D2. Heteroscedasticity was a
problem for models on the natural scale and various
transformations (logarithm, inverse and square root) were
tested. By far the best of these in terms of eliminating
heteroscedasticity was the square root transformation, and
this was used in both models.
Clover leaf area index (LAI) was important as an
independent variable in two models (B and C1, Table 1).
It was tested untransformed and on the logarithmic scale.
The more appropriate form was checked by comparing the
full maximum likelihood under the two alternatives and the
maximum was considerably greater for the logarithmic
form. This suggests that a fixed proportional increase in
LAI, rather than a fixed absolute increase, produced a
constant effect on the response. This implies that additional
increments of LAI of equal size have less effect on the
response the greater the level of LAI. This agrees
qualitatively with the well-known phenomenon that the
ability of a canopy to capture additional light becomes
limited and eventually reaches a plateau as LAI increases,
setting a limit to many processes dependent on the products
of photosynthesis. Whether the response variable was
clover content at the end of spring or LAI at the end of
winter (Figs 3 and 4 of Wachendorf et al., 2001b), the
response to LAI was positive, but at a decreasing rate in
both cases. Other more complex response relationships
linked to the physiology of the growth relationships (e.g.
708
Connolly and Wachendorf-Multisite Dynamic Models of Mixed Species Plant Communities
France and Thornley, 1984) were not considered for several
reasons. The shape of the estimated response relationships
agrees with what is expected, and additional terms, testing
for a more subtle relationship, were not significant. In
addition, more complex relationships (e.g. a curve involving
an asymptote) could require inclusion of non-linear
elements in the model and this extra complexity was not
considered necessary at the level of sophistication attached
to these models. Finally, the physiological relationships in a
mixed canopy are considerably more complex than in a
monoculture, involving interactions between two competing canopies and two LAIs (e.g. Kropf and Spitters, 1991)
and it was considered that these data were not an
appropriate vehicle for exploring those relationships.
Tiller density was a useful independent variable in two
models (B and C2, Table 1). Four forms of this variable
were compared, untransformed, logarithm, square root and
inverse. The full maximum likelihood approach outlined
above was also used here. The inverse of tiller density was a
better independent variable than the three alternatives,
perhaps reflecting the common experience in competition
models that plant responses are inversely related to the
density of competitors (Suehiro and Ogawa, 1980; Wright,
1981; Spitters, 1983; Firbank and Watkinson, 1985;
Connolly et al., 1990; Menchaca and Connolly, 1990).
Underlying the analysis of all responses is the assumption
that they are normally, and hence symmetrically, distributed on the scale of the analysis and this is checked by
examination of residuals. When responses are transformed
prior to analysis, predictions from the model on the
transformed scale are taken to be symmetrically distributed
and the predicted value on the transformed scale is the
estimated mean of the population for the set of values of
the independent variables used. This has a number of
consequences for the reporting of results. All inference,
including tests of significance and computation of confidence intervals for predictions or predicted differences,
should be carried out on the transformed scale. It may be
desirable to back-transform to the scale of the original data
for reporting and interpretation. Direct back-transformation of predicted values gives values that are predicted
medians on the original scale and are generally not the
means of responses on that scale. If the mean is to be
predicted on the original scale a correction is required. In
the current analysis, back-transformation to the original
scale is used without correction and so the values presented
graphically for model A (Fig. 2) should be interpreted as
medians rather than means.
(d) Presentation of results
The final model(s) for a functional period may contain
many terms, sometimes interacting with each other. The
terms in the model may be variables (e.g. leaf area index,
average precipitation) or factors (e.g. clover cultivar). The
model, expressed as estimates of multiple regression
coefficients (e.g. Table 2), may be difficult to appreciate
and interpret, even for one familiar with multiple
regression, particularly if transformations of response
and/or independent variables have been used. Methods of
displaying the meaning of the models vary. The broad
principle underlying the presentation of many model terms
is that predictions are initially made from the model for the
term in question on the scale of the analysis [e.g. logit of
clover content predicted from log(clover leaf area index)].
These predictions, perhaps back-transformed to the
original scale in which the biologist most readily thinks
(e.g. clover content), are displayed in graphical or tabular
form. Where an independent variable appears in the model
in transformed mode [e.g. log(clover leaf area index)], its
representation in graphs may again be on the original scale
(e.g. clover leaf area index). There are several types of model
terms: variables or factors not involved in interactions,
variables in higher polynomial (quadratic) forms, two
factor interactions of three types (factor by factor, factor
by variable and variable by variable) and perhaps more
complex terms such as three factor interactions. Presenting
the salient features of models may be done through
displaying these terms in graphical or tabular form.
There are two sides to the choice of scales for presenting
results. Graphs on the back-transformd scale may be
preferred as unit effects of independent variables on a
transformed scale may be difficult to understand, particularly if the independent variable is also transformed. On the
other hand, presenting graphical results on the transformed
scale(s) but with axes labelled on the original scale(s) is
sometimes acceptable to biologists and reduces the difficulty
of including inferential information. In some complex
models three-dimensional diagrams may display facets of
the model better.
The model for average summer clover content (Table 2
and Wachendorf et al., 2001b, Table 4) is used as an
example to illustrate some of the types of presentation
available. As it stands, interpretation of this model is very
difficult, since the response variable is transformed to the
logit scale and the model contains two interactions and a
quadratic term. The interaction between the clover content
at the end of spring (SCC) and clover cultivar (SCC x C) is
an interaction between a variable and a factor, where C is a
factor representing the two clover cultivars, with levels I
and 2 being AberHerald and Huia respectively. SCC x T is
an interaction between a variable and a variable, where T is
mean daily temperature(°C). The model also contains a
quadratic effect for clover content at the end of spring
(SCC x SCC). Since AberHerald is the first and Huia the
second level of C, Huia is used by SAS as the reference
cultivar. The coefficients for clover cultivar (C), clover
content at the end of spring (SCC) and temperature (T) are
not of great interest in themselves. The coefficient for C in
Table 2 is the predicted difference between AberHerald and
Huia at zero SCC and T; the coefficient for SCC is the slope
of the response to spring clover content for AberHerald at
zero SCC and the coefficient for T is the slope of the
response to temperature for AberHerald at zero SCC.
Quantitative interpretation of the interaction terms is also
not obvious.
A graphical presentation was chosen for model A and for
the models in Wachendorf et al. (2001b). Prediction of
values to set up the display of model terms and associated
tests of significance and measures of variability are derived
Connolly and Wachendorf-Multisite Dynamic Models of Mixed Species Plant Communities 709
80
.
60
rr
W
AberHerald
_
5)
U
0
Huia
To
.)O
_ 0
cUI
ao
E
EJ-
_
40
P-values for slopes
AberHerald <0.001
Huia
<0.001
20 _
0
Jv
0
20
40
60
80
100
0
20
Clover content at the end
of Spring (g 100 g-1 DM)
40
60
80
100
Clover content at the end
of Spring (g 100 g- 1 DM)
FIG. 2. Model A Summer: Predictions of mean clover content during summer (g 100 g I DM). A, The interaction of clover content at the end
of spring (g 100 g- DM) with clover cultivar, and B, the interaction of clover content at the end of spring with mean daily temperature during
summer (T) (C). Open horizontal bars indicate the range of values of clover content at the end of spring for which differences between the curves
are significant at the 5 % level. The significance of the response to clover content at the end of spring is indicated for each cultivar and each level of
temperature.
TABLE 2. Details of the modelfor mean clover content during summer (g 100 g-' DM)
Effect
INTERCEPT
Clover cultivar (C)t
Clover content at the end of spring (SCC) (g 100 g-' DM)
Mean daily temperature (T) (C)
SCC x C
SCC x T
SCC x SCC
Estimate
s.e.
-5.588
-0.225
0.128
0.242
0.0118
-0.00507
-0.00026
1.024
0.103
0.0266
0-0683
0-00268
0.00168
0.000057
d.f.
37
130
173
38
166
169
190
t
Pr > t
*
4.39
-302
-445
<0.001
0.003
<0.001
The response variable was transformed to the logit scale. The independent variables were the clover cultivar, clover content at the end of spring
and mean daily temperature over summer.
*t-values and probabilities of main effects were omitted when the effect was included in a significant interaction.
tCultivar is coded AberHerald and Huia as levels 1 and 2.
for the model in Table 2 and the other models in
Wachendorf et al. (2001b) using the LSMEANS/AT,
CONTRAST and ESTIMATE options of the SAS
MIXED procedure. In making predictions for a variable
or factor it is the usual practice to set all the other variables
in the model to their mean value and to attach equal weight
to each level of other factors in the model.
Presentation of the SCC x C interaction (variable x
factor). The model is used to predict mean summer clover
content (ACC) values for a range of clover contents at the
end of spring (SCC) for both AberHerald and Huia.
Predictions are on the logit scale and are back-transformed
to the clover content scale (ACC as a percentage of total dry
matter) using
Predicted ACC
100 x exp[predicted logit(ACC)]
1 + exp[predicted logit(ACC)]
These values are plotted for AberHerald and Huia against
clover content at the end of spring (Fig. 2A). Measures of
the statistical importance of features observed in Fig. 2A
are presented in two ways. The cultivar effect is tested for
significance for several values of SCC. The range of values
of SCC for which the P-value for the cultivar comparison is
less than 0-05 is indicated by an open horizontal bar. Thus,
710
Connolly and Wachendorf-Multisite Dynamic Models of Mixed Species Plant Communities
there is a significant cultivar effect (Fig. 2A) for SCC
between about 30 and 80 %. Another feature of interest in
this diagram is the significance of the increase in ACC with
increasing SCC for both AberHerald and Huia. As there is
a quadratic effect of SCC in the model the slopes of these
lines are not constant, even on the logit scale. The increase
is tested for significance by testing the difference between
predictions made for a cultivar at a low (20 %) and high
(70 %) value of SCC. The results are shown in Fig. 2A as
'P-values for slopes' and for both cultivars the P-values are
very small. The significant SCC x C interaction in the
model shows that the lines do not have the same slope; they
are neither coincident nor parallel. The range of SCC values
chosen in Fig. 2A lies well within the range observed in the
data, as does the range of predicted values, and these two
criteria have been used in all graphical displays of models in
Wachendorf et al. (2001b).
Presentationof the SCC x T interaction (variable x variable). A low and high value of one of the variables is
chosen (e.g. mean T + s.d.) and the model is used to predict
ACC values for a range of SCC values at both levels of T.
These series are then plotted against the levels of SCC to
show the interaction (Fig. 2B), which can then be
interpreted directly. If either the response or independent
variables are transformed prior to analysis the predictions
are made on the transformed scales but the predicted
response and the independent variable may be backtransformed as above before plotting the graph. The salient
features of the graph are displayed as in the case of the
SCC x C interaction. The temperature effect was significant only for low SCC levels and the response to SCC was
significant for both low and high temperatures. The
significant interaction in the model means that the two
response lines in Fig. 2B do not have the same slope.
Other model terms and issues. In Model 2, there are no
independent variables or factors that are not involved in an
interaction, but in several of the other models this does
occur (e.g. mean daily radiation in Model C 1 and clover
cultivar in Model D of Wachendorf et al., 2001b). For such
a variable, predictions of the response and a 95 %
confidence interval (allowing for the random effects in the
model) are made on the transformed scale for a range of
values of the independent variable. These are backtransformed to the scale desired for presentation and
form the basis of a graph of predicted response and
confidence limits against the independent variable. To
explore the full interpretation of the effect of such a variable
in the model, predictions of the effect of the variable may be
made at values of the remaining independent variables
other than their means. When the factor clover cultivar (C)
is not involved in an interaction, predicted means are
derived for AberHerald and Huia together with a standard
error of difference between means (or other inferential
information such a P-value for the test of significance or a
confidence interval) and the means are presented on the
preferred scale. For factor x factor interactions [not in the
models of Wachendorf et al. (2001b), but included here for
completeness] a table of predicted means is produced for all
possible combinations of the two factors accompanied by
appropriate measures of variability or significance of
differences.
In Model A (Table 2), both interactions involve the same
variate SCC. In this situation the slopes of the responses to
SCC in Fig. 2A and B are average values. In Fig. 2A they
are the means of the slopes at 13.7 and 16.7 C respectively
for each cultivar. In Fig. 2B they are the means of the
slopes for the two cultivars at 13.7 and 16.7 C, respectively. It would be desirable to examine the four separate
slopes were the patterns of the average slopes very different
for the two panels in Fig. 2A and B, but this is not
necessary here.
DISCUSSION
The modelling strategy was successful in developing a series
of biologically meaningful linked models, which included
the effects of plot and site independent variables and the
effect of clover cultivar. The models give insight into the
annual development of the clover/ryegrass community
across a wide range of environmental, in this case climatic,
conditions. The use of functional periods greatly reduced
the complexity of the modelling and fitted well with a
biological view of the annual life history of the mixed plant
species community. The use of plot variables representing
initial conditions in the community at the start of each
functional period was very successful and led to models that
provided biological insights. The inclusion of site level
climatic variables to account for differences in community
behaviour across sites and years broadened the basis of
inference of the models and helped to quantify the impact
of forces that, though important, are uncontrolled. Other
variables describing site management or other site characteristics could readily be incorporated within the same
modelling framework.
The role of plot independent variables characterizing the
initial state of the system is taken largely to integrate the
effects of all plot history including past climate at the site.
The site variables help to describe the effects of changes in
the environment on the system from that initial state
forward. This is, of course, too simplistic, but at the level of
definition achievable in this study it is felt to be a useful
approximation. For functional periods of very short
duration, processes such as tiller production, which take a
certain minimum time to complete, could well be partly
influenced by environmental conditions at the end of the
previous period. Autumn is a period when this is likely to
occur. Although the minimum duration of autumn is 13 d,
its mean duration in this study is about 35 d (Table 1),
perhaps sufficient for the effect of the previous period to be
relatively small.
The interpretation of effects in this series of models should
take potential confounding into account. Some plot independent variables may contain effects of treatment (clover
cultivar) and this may result in some confounding of treatment effect with effects of independent variables. In model
A, for example, the coefficient(s) for initial clover content
shows the effect of variation in initial clover content across
plots with the same clover cultivar, i.e. the within treatment
Connolly and Wachendorf-Multisite Dynamic Models of Mixed Species Plant Communities 711
effect. The effect of treatment (clover cultivar) is the difference between species in their performance over the period
assuming that they have a common level of initial clover
content. Thus, any effect of treatment on the initial clover
content is not taken into account in the treatment effect for
the period, it only represents that portion of treatment
difference that arose during the period. The use of initial
state variables as covariates and then modelling these in turn
may also be subject to the charge that the relationship
between the performance of a species over successive periods
arises partly from intrinsic differences among the plots that
affect both variables systematically rather than reflecting a
causal association between them.
With these caveats in mind it is emphasized that the
models derived from this process are not claimed to be
explanatory. However, since many of the relationships are
very strong and biologically plausible, causality should be
entertained as accounting for at least part of the observed
relationships, particularly since the climatic variables
represent forces that would be universally agreed to be
important determinants of biological responses. Normally,
these variables are either excluded from use as field
experimental factors through reasons of cost or impracticality, or generally ineffectually dealt with through field
experiments at a single site lasting even 2 or 3 years.
Establishing insight into their importance was a real
strength of the modelling. Verification of the extent of
causality in the relationships requires controlled experimentation, some of which will be under laboratory rather than
field conditions. Although the dynamic models cover the
complete annual cycle they should not be used to predict
long-term effects in the system as they do not include any
regulatory mechanism to prevent elements of the population (e.g. leaf area index and tiller density) from
unbounded growth.
The use of initial biotic conditions in modelling growth in
mixed species communities is not widespread, with much of
the literature using the initial densities of species (Suehiro
and Ogawa, 1980; Spitters, 1983; Firbank and Watkinson,
1985; Connolly et al., 1990; Menchaca and Connolly, 1990;
Gibson et al., 1999). Connolly and Wayne (1996) used
initial seedling biomass of two species as independent
variables in a model of growth of two competing weed
species immediately following establishment. The ratio of
species leaf area indices early in the growth of a crop-weed
community was used in Kropf and Spitters (1991) in
modelling crop yield loss from weed competition. Connolly
et al. (2001) and Gibson et al. (1999) stressed the need to
allow for initial biotic conditions rather than species
densities in modelling competitive effects in community
development to avoid the possibility of bias due to
differences in the initial size of the competing species.
Indeed, for perennial species such as white clover and
perennial ryegrass, density may not be a meaningful
measure once they have spread by clonal reproduction,
whereas their contribution to swards can be determined by
sampling at any stage. Although the design of this
experiment does not fall within the usual range of design
for the study of interspecific interaction (Gibson et al.,
1999; Connolly et al., 2001), the variability among plots in
the development of the clover and ryegrass allowed
regression analysis on initial plot conditions, from which
some information on interspecific effects was estimable,
particularly during the winter period when more detailed
information was available. However, for a full competition
experiment it would be desirable to establish plots differing
widely in composition.
When an independent variable is measured imprecisely
the estimator of the regression coefficient will be biased
(Carroll et al., 1995), tending, in models like these, to be
somewhat reduced in absolute size. Morphological variables such as leaf area index, which are based on
subsamples within plots, may not measure the true plot
leaf area index precisely. If this is of concern, preliminary
detailed analysis of the variation among subsamples
compared with variation among plots will provide a basis
for deciding whether the variable is measured sufficiently
well to lead to negligible bias or to correct for it. As data
from subsamples were not available in the current study this
issue was not examined.
Presenting complex models is a necessary part of
modelling. This paper has outlined several aspects of a
graphical/tabular approach to this which allows the user to
engage directly with the biological issues raised by the
model and bypasses the difficulties of attempting to
understand a complex equation. These methods have been
extremely successful in communicating the results of seven
rather complex models to biologists.
CONCLUSIONS
The modelling strategy was successful in developing a series
of multisite, biologically meaningful, linked dynamic
models of white clover/ryegrass communities as affected
by clover cultivar and climatic independent variables. The
models gave insight into the annual development of the
clover/ryegrass community across a wide range of environmental conditions. The statistical issues were handled
within a linear mixed models framework. The methods of
presentation allowed a relatively simple appreciation of the
behaviour of community dynamics across four linked
functional periods.
ACKNOWLEDGEMENTS
Thanks are due to Eric Macklin and Michael O'Kelly for
helpful discussions on some of the issues in this paper. The
helpful comments of Adrian Dunne, David Elston, David
Kemp and David Williams have greatly improved this
work. We acknowledge the financial contribution of the
Commission of the European Community for support for
the COST (Cooperative Organisation of Science and
Technology) 814 Action.
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