Ecological Indicators 4 (2004) 29?37
Soil total phosphorus threshold in the Everglades: a Bayesian
changepoint analysis for multinomial response data
Song S. Qian a,?, Yangdong Pan b, Ryan S. King c
a The Cadmus Group Inc., 6330 Quadrangle Drive, Suite 180, Chapel Hill, NC 27517, USA
b Environmental Sciences and Resources, Portland State University, Portland, OR 97207, USA
c Smithsonian Environmental Research Center, 647 Contees Wharf Road, Box 28, Edgewater, MD 21037, USA
Abstract
The Everglades is a historically phosphorus limited freshwater wetland ecosystem. Increasing the supply of phosphorus will
lead to changes in the ecosystem. As a cumulative indicator of phosphorus loading to the wetland system, the top 10 cm soil
TP concentration is often used to indicate eutrophication. The current study used multivariate species composition data to find
the TP threshold, above which species compositional changes are expected. A Bayesian changepoint model was developed
in this study. The statistical model can directly use species assemblage data as a measure of structural change. Using diatom
species and macroinvertebrate species assemblage data collected from the field as well as from a mesocosm experiment, the
soil TP threshold was estimated to be around 400 mg/kg.
? 2003 Elsevier Ltd. All rights reserved.
Keywords: Everglades; Species composition data; Soil
1. Introduction
Because shifts in species composition are often
caused by changing environmental conditions, ecol-
ogists have long considered relative abundance of
different species an important measure for evaluating
ecosystems (Odum, 1985; Schindler, 1990; Stoemer
and Smol, 1999). For example, an anthropogenic dis-
turbance, such as a slight to moderate increase in nu-
trient supply, can alter the relative abundance of peri-
phyton (Pan et al., 2000). Under intense human-related
stress, significant compositional and functional
changes in biotic communities may occur, which may
result in the interruption or loss of some or all eco-
logical functions in the ecosystem. Odum (1985) sug-
? Corresponding author. Tel.: +1-919-403-5105;
fax: +1-919-401-5867.
E-mail address: sqian@cadmusgroup.com (S.S. Qian).
gested and Schindler (1990) experimentally demon-
strated that many measures of ecological function such
as nutrient cycling, energy flow, and succession can
be severely disrupted as anthropogenic stress intensi-
fies. For example, excessive nutrient enrichments may
result in complete replacement of sensitive and native
algal species (Schindler, 1974; Pan et al., 2000), and
algal species assemblages, dominated by exotic and
non-palatable nuisance species (e.g., toxic blue?green
algae), may alter the energy transfer path from pri-
mary producers to consumers at higher trophic levels
and reduce ?food-chain? length in aquatic ecosystems
(Schindler, 1990). As a result, shifts in species com-
position are often used as indicators or early warning
signs of worsening environmental conditions.
Studies of how an ecosystem responds to human
disturbances can have important management impli-
cations, such as the establishment of water-quality or
emission standards for particular geographic regions
1470-160X/$ ? see front matter ? 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ecolind.2003.11.005
30 S.S. Qian et al. / Ecological Indicators 4 (2004) 29?37
(Adams and Greeley, 2000). One common approach
to estimating environmental criteria is to examine the
spatial changes of selected population, community, or
ecosystem attributes along a gradient of environmen-
tal conditions (e.g., Karr and Chu, 1997). Using this
approach, ecologists can quantify the environmental
threshold, which is an important environmental vari-
able given that ecological attributes often show little
change until a critical environmental value, or thresh-
old, is reached (Fore et al., 1996; Richardson and
Qian, 1999). In this way, quantitative descriptions of
exposure-response relationships can be very useful to
support the development of numerical environmental
criteria (Suter, 1993; USEPA, 1998a).
Statistical analysis of compositional data is a dif-
ficult subject. Aitchison (1982, 1986) developed the
logit normal distribution as a framework for compo-
sitional data analysis. The logit normal distribution
is, however, difficult to interpret in ecological terms.
The Bayesian hierarchical extension by Billheimer
et al. (2001) used a conditional multinomial distri-
bution on top of the logit normal distribution, such
that internal parameter structure changes can be trans-
mitted to changes in relative abundance of species.
Both the logit normal distribution and its Bayesian
hierarchical extension can be used to describe differ-
ences in species composition at different locations.
The changepoint method we proposed directly links
the species composition changes to a dominant envi-
ronmental gradient, with an objective for estimating
the environmental threshold.
In a previous study, we presented a Bayesian
changepoint analysis for estimating environmental
thresholds using univariate response variables such as
percentage of pollution sensitive species (Qian et al.,
2003). Because the method in our previous study was
limited to univariate response variable, species com-
position data must be processed to produce an index
of some sort. By reducing the dimension of species
composition data, loss of information is inevitable.
In this paper, we present a Bayesian changepoint
analysis method that uses species composition as
the response variable. Four data sets were used to
illustrate the method: (1) macroinvertebrate species
compositions measured along a phosphorus gradient
created in a mesocosm experiment conducted in an
Everglades wetland; (2) macroinvertebrate species
compositions measured along a phosphorus gradient
in the northern Everglades; (3) periphytic algae (algae
attached on aquatic plants) species composition data
collected near the transition zone of the phosphorus
gradient in the northern Everglades; and (4) peri-
phytic algae species composition data with sampling
sites covering the entire phosphorus gradient in the
northern Everglades. The rest of the paper presents
the study area and data sets, the statistical methods,
the results, and a discussion.
2. Method
2.1. Study area and data
The Florida Everglades is a phosphorus limited
ecosystem (e.g., Richardson and Qian, 1999; Steward
and Ornes, 1975a,b; Swift and Nicholas, 1987; Flora
et al., 1988). Excessive inputs of phosphorus to the Ev-
erglades led to the 1994 Everglades Forever Act (EFA,
Florida Administrative Code, 17-302.530), which
mandated that a numerically defensible water-column
total phosphorus (TP) standard be established to
ensure a balance of flora and fauna.
Four data sets, collected from the Water Con-
servation Area-2A (WCA2A) in the northern Ever-
glades, are used in this paper. WCA2A is a 547 km2
diked marsh with a mosaic of sawgrass (Cladium
jamaicense) prairies and open water sloughs. It has
been receiving agricultural runoff from the north for
decades. As a result, a north to south phosphorus
gradient has developed (Craft and Richardson, 1993;
Reddy et al., 1993; Urban et al., 1993).
The first data set consists of macroinvertebrate
species composition data collected from a meso-
cosm dosing study conducted by the Duke Univer-
sity Wetland Center in the southern end of WCA2A
(Richardson et al., 2000; Pan et al., 2000). Qian et al.
(2003) estimated TP threshold for the Everglades us-
ing the same data set with a single response variable
model using a derived dissimilarity index and the
percentage of phosphorus tolerant species.
The second data set includes macroinvertebrate
species composition data collected along three north
to south transects directly south of the three major
water inlets in WCA2A. Details of data collection and
initial analysis can be found in King and Richardson
(2002).
S.S. Qian et al. / Ecological Indicators 4 (2004) 29?37 31
The third and fourth data sets are diatom species
composition data collected from WCA2A separately
by Pan et al. (2000) and Cooper and Goman (2002).
The Pan et al. (2000) data include algae and sedi-
ment total phosphorus from 32 slough sites near the
southern end of the north?south phosphorus gradient
in WCA2A sampled from 6 to 8 April 1995 (see Pan
et al., 2000 for details of data collection and anal-
ysis). A total of 45 diatom species were identified
and recorded in the Pan et al. (2000) data. The Pan
et al. (2000) data were collected in the transi-
tional zone of the phosphorus enriched and unen-
riched areas of WCA2A. The Cooper and Goman
(2002) data were collected in 1996 from 31 sites
covering the entire WCA2A. There were 90 diatom
species that can be positively identified in the data
set. The number of diatom species included in the
Cooper and Goman (2002) study is twice as many
as in the Pan et al. (2000) study. However, the site
locations were more carefully chosen in the Pan et al.
(2000) study to focus on the impact of phosphorus.
2.2. Statistical method
We discuss the discrete parameter changepoint
problem. Let y1, . . . , yn be the sequence of the eco-
logical response variables observed along the ordered
environmental gradient x1, . . . , xn. A changepoint
problem is to find r (1 ? r ? n) that separates the
response variable into two groups: y1, . . . , yr and
y
r+1, . . . , yn, each with distinct characteristics such
as the relative species abundance. The corresponding
value of the environmental variable x
r
is the threshold.
Another type of changepoint problem is discussed in
Qian and Richardson (1997).
Under a Bayesian framework, we make specific
probabilistic assumptions about the ecological re-
sponse variable. Specifically, we assume that the
response variable values, y1, . . . , yn, collected from
the n sites along the gradient of interest, are ran-
dom samples from the sequence of random variables
Y1, . . . , Yn. In other words, we define a random vari-
able for each site, and assume that these random
variables belong to the same family of distributions
with parameter ?.
The term ?site? is used to refer a sampling location
that has a distinct predictive variable value. Depending
on the scale of the study, a site can be a 1 m ? 1 m
sampling grid as in the Everglades example, or reaches
of streams miles apart.
The random variables Y1, . . . , Yn have a change
point at r (1 ? r ? n) if
Y1, . . . , Yr ? ?(Yi|?1)
Y
r+1, . . . , Yn ? ?(Yi|?2)
(1)
where ? represent a generic probability density func-
tion. The general setup of a Bayesian analysis can be
described as follow.
First, the likelihood function L(Y; r) under the
model of Eq. (1) is:
L(Y; r) =
r
?
i=1
?(Y
i
|?1)
n
?
i=r+1
?(Y
i
|?2) (2)
With the prior density of r:?(r), the posterior density
of r|Y becomes:
?(r|Y) =
L(Y; r)?(r)
?
n
r=1 L(Y; r)?(r)
(3)
Features of the posterior distribution (3) can be easily
estimated, and the posterior odds of no change (P(r =
n|Y)/{1 ? P(r = n|Y)}) is often used to determine
whether or not a change has occurred.
For the species composition data, we assume a
multinomial distribution. That is, y
i
= {y
i1, . . . , yik}
is a vector of observed counts of individual species
from site i. The parameter of interest is the relative
abundance. When the environmental variable is less
than the threshold (i ? r), the relative abundance vec-
tor is ?1 = {p11, . . . , p1k} and the relative abundance
vector is ?2 = {p21, . . . , p2k} for i > r. The number
of species included (the dimension of the multino-
mial distribution) is k. We further assume that ?1 is a
priori independent of ?2, and the prior distribution of
parameters ?1, ?2, r is:
?(?1, ?2, r) ? ?(?1)?(?2). (4)
The prior distribution on r is assumed to be a uniform
distribution over all possible values of i. Ideally, the
prior distribution of r should be chosen based on eco-
logical knowledge. In addition, TP threshold is a real
value, not an index to the ordered values. A continuous
prior distribution on the threshold would be closer to
the reality than the uniform distribution on the index.
We choose the the uniform distribution on the index
for two reasons. First, there is no consensus on what
32 S.S. Qian et al. / Ecological Indicators 4 (2004) 29?37
constitutes a change in the species assemblage. Dif-
ferent interpretation may lead to different prior belief.
Consequently, a uniform distribution would be a com-
promise. Second, a discrete distribution of r makes
computation convenient. Simulation studies we con-
ducted showed no discernable difference in the pos-
teriors of r estimated from models using continuous
and discrete priors.
We use the Dirichlet distribution to describe the
prior distribution of ?1 and ?2:
?(?1) ?
k
?
j=1
p
?1j?1
1j ,
and
?(?2) ?
k
?
j=1
p
?2j?1
2j .
The likelihood of observing data Y = {yij} is
L(Y, ?1, ?2, r) ?
r
?
i=1
?
?
k
?
j=1
p
yij
1j
?
?
?
n
?
i=r+1
?
?
k
?
j=1
p
yij
2j
?
?
=
k
?
j=1
p
?
r
i=1 yij
1j ?
k
?
j=1
p
?
n
i=r+1 yij
2j (5)
The joint distribution of ?1, ?2, r is the product of the
prior (Eq. (4)) and the likelihood (Eq. (5)):
?(?1, ?2, r|Y)
?
k
?
j=1
p
?
r
i=1 yij+?1j?1
1j ?
k
?
j=1
p
?
n
i=r+1 yij+?2j?1
2j (6)
The marginal posterior distribution of r is
?(r|Y) =
??
?(?1, ?2, r|Y) d?1 d?2 ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
k
j=1{?(?1j +
?
r
i=1 yij)}
? {
?
k
j=1(?1j +
?
r
i=1 yij)}
?
?
k
j=1{?(?2j +
?
n
i=r+1 yij)}
? {
?
k
j=1(?2j +
?
n
i=r+1 yij)}
, for r < n
?
k
j=1{?(?1j +
?
n
i=1 yij)}
? {
?
k
j=1(?1j +
?
n
i=1 yij)}
?
k
j=1{?(?2j)}
? {
?
k
j=1(?2j)}
, for r = n
This integral is evaluated using the fact that the two
parts in the posterior distribution resemble the density
function of a Dirichlet distribution.
The marginal posterior distributions of ?1 and ?2
can be estimated using the Gibbs sampler since their
conditional posterior distributions are easy to obtain:
?(?1|?2, r) ?
k
?
j=1
p
?
r
i=1 yij+?1j?1
1j ,
which is a Dirichlet distribution; and
?(?2|?1, r) ?
k
?
j=1
p
?
n
i=r+1 yij+?2j?1
2j ,
which is also a Dirichlet distribution.
Parameters of the Dirichlet prior distributions can
be selected based on our knowledge of the species
distribution or be selected to make the prior near
non-informative. There are three options for select-
ing a noninformative Dirichlet prior distribution. The
most obvious one is to set ?ij = 0, an improper prior
distribution. The posterior distributions of r, ?1, and
?2 will be proper distributions only when all species
appeared in the data at least once. The second option
is to set ?ij = 0.5 (the Jeffreys? prior, Jeffreys, 1961)
and the third option is to set ?ij = 1 (the ?natural?
uniform density used by Bayes and Laplace, Berger,
1985). Good arguments can be made for any of the
three options (Ripley, 1996). When the sample size is
large, the choice of priors will be less important. We
used the Jeffrey?s prior.
3. Results
The estimated changepoint distributions from the
data sets show a clear and distinct change point along
the TP gradient (Fig. 1). The mean change point val-
ues ranging from 367 to 452 mg/kg (Table 1). Com-
paring the 95% credible intervals for the four data sets
(Fig. 2), we note that the interval for the mesocosm
macoinvertibrate data is wider (more uncertain) than
the same for the gradient data. The credible interval
for the Pan et al. (2000) data is narrower than the
same for the Cooper and Goman (2002) data, even
though the latter had twice as many diatom species.
The field macroinvertebrate species data also produce
S.S. Qian et al. / Ecological Indicators 4 (2004) 29?37 33
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
Diatom (Pan)
100 250 500 1000
Diatom (Cooper)
100 250 500 1000
MacroInv. Mesocosm
(Sep. 96)
MacroInv. Field
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
Sediment TP (mg/kg)
Pr
ob
ab
ilit
y
Fig. 1. The probability distributions of soil TP thresholds. The heights of the lines represents the probability the corresponding soil TP
value being the threshold.
very narrow credible interval for the threshold, as
the samples were collected from a well established
phosphorus gradient shared by many research groups.
When comparing the relative proportion of each
species before and after the changepoint (i.e., the
soil TP less than or larger than the threshold), only
a handful of species are found to have substantial
changes (Fig. 3). For example, the relative abun-
dances of the three species on the right end of the
x-axis (the dominant species in P-limited Everglades
habitats or ?native species?) are much smaller when
the soil TP concentration is above the threshold level,
Table 1
Data sets used in the paper
Study ID Reference Date collected Species Location
1.1 King and Richardson (2002) January 1997 Macroinvertebrates Mesocosm
1.2 February 1998
1.3 September 1996
1.4 September 1998
2 Pan et al. (2000) 1995 Diatom Field
3 Cooper and Goman (2002) 1996 Diatom
4 King and Richardson (2002) 1998 Macroinvertebrates
while the relative abundance of Nitzschia frustulum
(a well known ?eutrophic? tolerant diatom species)
increased dramatically. As a result, it is conceivable
that a limited number of ?indicator? species may be
used for indicating eutrophication. However, as many
have pointed out (e.g., King and Richardson, 2002),
eliminating ?rare? species may result in loss of infor-
mation. We tested our method for its sensitivity to the
number of species by removing rare species (defined
as occurring less than 1, 5, and 10% in each observa-
tion). For all four data sets, removing species occur-
ring with less than 1% in each observation resulted in
34 S.S. Qian et al. / Ecological Indicators 4 (2004) 29?37
Se
di
m
en
t T
P 
(m
g/k
g)
Diatom (Pan)
Diatom (Cooper)
MacroInv. Field
30
0
40
0
50
0
60
0
(a)
Se
di
m
en
t T
P 
(m
g/k
g)
Jan. 97 Feb. 98
Sep. 96
Sep. 98
(b)
50
0
10
00
Fig. 2. Summary statistics plots of the soil TP threshold distributions: (a) the soil TP threshold distributions from the three field studies;
(b) the soil TP threshold distributions from the mesocosm study data collected in four dates.
insignificant or no changes. When removing species
that occurred with values less than 5% in each obser-
vation, the TP threshold for the Pan et al. (2000) data
changed significantly while the other three showed
slight changes. All show significant changes in the es-
timated thresholds when removing species occurring
less than 10% in each observation.
4. Discussion
In a separate study, Qian et al. (2002) reported the
historical soil TP in the Everglades to be 472 mg/kg
with a 95% credible interval of 250, 790 mg/kg, in-
ferred by using the fossil diatom species composition
collected by Cooper and Goman (2002) from both the
enriched and unenriched areas in WCA2A. The his-
torical soil TP concentrations were about the same in
both the phosphorus enriched and unenriched areas.
The TP thresholds estimated in the current study are
similar to the historical soil TP concentrations (see
Table 2). This similarity indicates that site-specific
fossil diatom species responses to historical phos-
phorus enrichment are similar to the responses of
diatom species along the current TP gradient in the
Everglades. Based on diatom species responses along
the current TP gradient, a diatom species transfer
function model can be developed and used for recon-
structing site-specific historical changes of TP in the
Everglades.
S.S. Qian et al. / Ecological Indicators 4 (2004) 29?37 35
Species
Pr
op
or
tio
n
0.0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
TP < Threshold
TP > Threshold
Fig. 3. The average relative abundances of the identified diatom species associated with soil TP concentrations below and above the
estimated TP threshold values. Results shown are derived from the Pan et al. (2000) data.
The statistical method presented here assumes a
multinomial distribution of the species counts. As a
result, the actual counts are needed for model fitting.
The conventional practice in many ecological studies
is to report only the relative abundance (percentage) of
each species. Although the percentage is an unbiased
estimate of a species relative abundance, the variance
of the relative abundance is determined by the total
number counted. Therefore, the reporting of only the
percentage could lead to loss of information, unless
the total numbers of species counted in all samples are
the same and reported.
Table 2
Estimated soil TP thresholds (mg/kg) from the four data sets
Study ID 2.5% Median Mean 97.5%
1.1 230.7 401.8 519.7 1210.4
1.2 345.0 649.7 664.6 1219.5
1.3 212.9 280.8 383.2 880.3
1.4 308.5 416.4 528.7 1240.2
2 392.0 447.2 460.9 481.5
3 305.0 337.5 366.8 506.0
4 430.9 441.0 444.4 461.1
The method developed in this paper assumes an
abrupt change in the species composition along an
environmental gradient. Ecological changes are not
always abrupt (Muradian, 2001). Because the change-
point is estimated as a probability distribution, our
method can be used to show a zone of change reflected
by, e.g., the 95% credible interval. Our experience
(from simulation studies) show that (1) when the un-
derlying pattern is an abrupt change, the model will
result in a posterior changepoint distribution that is
narrowly concentrated around the threshold; and (2)
a wide-spreading posterior changepoint distribution
may indicate a gradual change or other noise (e.g.,
secondary gradient). Care must be taken when making
inference about a wide-spreading posterior change-
point distribution. No statistics method is capable
of discerning whether the cause of a wide-spreading
posterior changepoint distribution is a gradual change
or other noise. In our study, we see highly concen-
trated posterior changepoint distributions from the
Pan et al. (2000) data and the field macoinvertibrate
data, which indicate that both diatom and macoinvert-
ibrate species compositions change abruptly as soil
36 S.S. Qian et al. / Ecological Indicators 4 (2004) 29?37
TP increases. We also see widely distributed posterior
changepoint distributions from the mesocosm ma-
coinvertibrate data and the Cooper and Goman (2002)
data (Fig. 1), which can only suggest additional source
of uncertainty in light of the other two data sets.
The ranges of the estimated credible intervals re-
flect the level of uncertainty about the changepoint.
There are many sources of uncertainty, but the fol-
lowing two may be the most influential. First is the
source of phosphorus. In the mesocosm, orthophos-
phate is added as the source of phosphorus; while in
the gradient, the source of phosphorus is mainly from
the agriculture runoff. When the same mesocosm
macroinvertebrate species data were modelled against
the water column TP gradient, the estimated credible
interval (not shown) is very narrow (another evidence
of abrupt change). This shows that the mesocosm soil
TP may have not reached an equilibrium with TP
loading and thus may not be indicative of phosphorus
availability. Second, a secondary gradient (e.g., the
SO4 gradient near the southern end of WCA2A) may
affect the result as we move further into the southern
end of WCA2A, which may explain why the credible
interval for the Pan et al. (2000) data is narrower than
the same for the Cooper and Goman (2002) data, even
though the latter had twice as many diatom species.
Two additional factors further complicated our
study: (1) the difference between mesocosm and natu-
ral gradient; and (2) impact of species richness. Meso-
cosms are commonly used for hypotheses testing in
ecological and environmental studies. However, the
contribution of mesocosm studies to our understand-
ing of complex ecological systems remains unclear
(Daehler and Strong, 1996). Our data showed that
the credible intervals for the mesocosm macroinver-
tebrate data are wider than the same for the gradient
data. This may be because that duration of the meso-
cosm study is relatively short compared to the time
needed for the soil TP to reach equilibrium. Pan
et al. (2000) compared changes in periphyton assem-
blages along the experimental phosphorus gradient
in the mesocosm and observed phosphorus gradient
in the Everglades. They noted that several changes
in algal assemblages differed between the mesocosm
and Everglades phosphorus gradient. The discrepan-
cies were attributed largely to the difference between
observed and experimental phosphorus gradients in
habitat complexity, spatial and temporal scales of im-
portant ecological processes, and the frequency and
magnitude of environmental changes.
Species richness often increases as a function of
sampling area (Rosenzweig, 1996). Increased species
richness is typically contributed by so-called rare
species. Contribution of rare species to ecological
community analysis and bio-assessment remains un-
clear (see the review by Cao et al., 2001). Gauch
(1982) argued that rare species may only add noise
to the community analysis and thus should be ex-
cluded in the analysis. Other authors indicated that
rare species may contain significant information on
communities and their responses to environmental
disturbances (Faith and Norris, 1989). However, con-
ventional sampling and lab analytic procedures often
cannot quantify rare species and their population with
high certainty. On the one hand, we see significant
changes in our results when species occurring less
than 5% in each sample were removed. On the other
hand, the Pan et al. (2000) data with only half as
many species as the Cooper and Goman (2002) data
are most informative (much narrower credible inter-
val). A well-targeted sampling design may be most
important.
Acknowledgements
Qian and Pan?s work were supported by a US EPA
STAR grant (R827898). King?s work was supported
by the Duke University Wetland Center. E. Conrad
Lamon, Craig A. Stow, and Karen Wesson provided
constructive comments and suggestions on an earlier
version of the manuscript. Comments and suggestions
from two referees and the associate editor are greatly
appreciated. The data collected by Cooper and Goman
(2002) were provided by the Duke University Wetland
Center.
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