1. science
2. mathematics
3. statistics

Wetlands Ecol Manage (2007) 15:335–345
DOI 10.1007/s11273-007-9034-x
ORIGINAL PAPER
Power analysis to determine sample size for monitoring
vegetation change in salt marsh habitats
Mary-Jane James-Pirri Æ Charles T. Roman Æ
James F. Heltshe
Received: 21 February 2006 / Accepted: 12 February 2007 / Published online: 21 March 2007
Springer Science+Business Media B.V. 2007
Abstract Numerous initiatives are underway
throughout New England and elsewhere to quantify salt marsh vegetation change, mostly in
response to habitat restoration, sea level rise, and
nutrient enrichment. To detect temporal changes
in vegetation at a marsh or to compare vegetation
among different marshes with a degree of statistical certainty an adequate sample size is required.
Based on sampling 1 m2 vegetation plots from 11
New England salt marsh data sets, we conducted a
power analysis to determine the minimum number
of samples that were necessary to detect change
between vegetation communities. Statistical
power was determined for sample sizes of 5, 10,
15, and 20 vegetation plots at an alpha level of 0.05.
Detection of subtle differences between vegetation data sets (e.g., comparing vegetation in the
same marsh over two consecutive years) can be
accomplished using a sample size of 20 plots with a
M.-J. James-Pirri (&)
Graduate School of Oceanography, University of
Rhode Island, Box 8, Narragansett, RI 02882, USA
e-mail: [email protected]
C. T. Roman
National Park Service, Graduate School of
Oceanography, University of Rhode Island,
Narragansett, RI 02882, USA
J. F. Heltshe
Department of Computer Science and Statistics,
University of Rhode Island, Kingston, RI 02881, USA
reasonable probability of detecting a difference
when one truly exists. With a lower sample size,
and thus lower power, there is an increased
probability of not detecting a difference when
one exists (e.g., Type II error). However, if
investigators expect to detect major changes in
vegetation (e.g., such as those between an unimpacted and a highly impacted marsh) then a
sample size of 5, 10, or 15 plots may be appropriate
while still maintaining adequate power. Due to the
relative ease of collecting vegetation data, we
suggest a minimum sample size of 20 randomly
located 1 m2 plots when developing monitoring
designs to detect vegetation community change of
salt marshes. The sample size of 20 plots per New
England salt marsh is appropriate regardless of
marsh size or permanency (permanent or nonpermanent) of the plots.
Keywords Power analysis Salt marsh Sample
size Vegetation
Introduction
Salt marsh vegetation changes in response to
climate change, sea-level rise, hydrologic alteration from ditching and tide-restricting structures
(e.g., bridges, causeways), nutrient enrichment,
and other natural and human-induced factors
(e.g., Wigand et al. 2003; Bertness et al. 2002;
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Wetlands Ecol Manage (2007) 15:335–345
Donnelly and Bertness 2001; Wolfe 1996; Rozsa
1995; Roman et al. 1984; Niering and Warren
1980). In addition, numerous studies have documented the response of salt marsh vegetation to
restoration activities, such as re-introduction of
tidal flow to hydrologically altered systems (e.g.,
Burdick et al. 1997; Roman et al. 2002; Warren
et al. 2002). Currently many agencies and conservation organizations are sponsoring research
and monitoring programs aimed at better understanding the factors contributing to salt marsh
vegetation change and documenting the effects of
restoration efforts (e.g., Merkey et al. 2005).
Plot sampling is commonly used to evaluate
changes in salt marsh vegetation species composition and abundance; however, to detect change with
a degree of statistical certainty an adequate sample
size is required. In salt marsh vegetation studies the
number of sample plots is often arbitrarily determined. Researchers may sample too few plots or
expend critical resources and funds collecting more
data than necessary. Insufficient sample size leads to
a high probability of not detecting differences
between two vegetation communities when a
significant difference is actually present.
The purpose of this paper is to present the
findings of a power analysis with the objective of
guiding investigators in the selection of an appropriate sample size when attempting to detect
changes in salt marsh vegetation communities
using 1 m2 plots. Power is defined as 1-b (beta), b
being the probability of wrongly failing to reject
the null hypothesis of no difference when the null
hypothesis is actually false (Type II error). Power
is the desirable likelihood of correctly rejecting
the null hypothesis and is a function of the
differences between two populations, the sample
size, the alpha level of the test (the probability of
detecting a difference between two populations
when no difference exists, i.e., Type I Error,
commonly set at 0.05 in ecological analyses), and
the variability of the measured response.
Methods
Study sites
Vegetation plot data from seven individual salt
marshes, two sampled in three different years
resulting in a total of 11 unique data sets, were
used to determine statistical power curves for
different sample sizes (Table 1). The salt
marshes, located along the New England coast
Table 1 Location, number of plots, and sampling year for vegetation data used in the power analyses
Salt marsh
Location (latitude & longitude)
Study area
(ha)
Sampling
year
Total plots
sampled
Hatches Harbor reference
Provincetown, MA (N424¢,
W7014¢)
Wells, ME (N4316¢, W7036¢)
Biddeford, ME (N4325¢,
W7023¢)
Cape Porpoise, ME (N4323¢,
W7026¢)
Cape Porpoise, ME (N4323¢,
W7026¢)
Middletown, RI (N4129¢,
W7115¢)
Middletown, RI (N4129¢,
W7115¢)
Middletown, RI (N4129¢,
W7115¢)
Middletown, RI (N4129¢,
W7115¢)
Middletown, RI (N4129¢,
W7115¢)
Middletown, RI (N4129¢,
W7115¢)
90
1997
61
3.2
10.1
1999
1999
77
68
0.9
1999
28
1.5
1999
40
6.3
1996
22
3.7
1996
29
6.3
1998
22
3.7
1998
38
6.3
1999
22
3.7
1999
38
Moody marsh
Granite Point marsh
Marshall Point impacted
Marshall Point reference
Sachuest Point reference
Sachuest Point impacted (before tidal
restoration)
Sachuest Point reference
Sachuest Point impacted (after tidal
restoration)
Sachuest Point reference
Sachuest Point impacted (after tidal
restoration)
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from Maine to Rhode Island, varied from
relatively un-impacted (or reference) marshes to
severely impacted sites, due to tidal restrictions or
other hydrologic alterations, and also included
marshes that had recently undergone restoration
via tidal re-introduction. All the marshes,
including the reference sites, had been grid
ditched for mosquito control, a common practice
throughout New England, during the early- to
mid-twentieth century. Vegetation data used in
the power analyses were from Granite Point
marsh (ME), Marshall Point reference and
impacted marshes (ME), Moody marsh (ME),
Hatches Harbor reference marsh (MA), and
Sachuest Point reference and impacted marshes
(RI). Vegetation data from Granite Point and
Moody marshes, collected in 1999, were prior to
hydrologic alterations on portions of these
marshes and therefore represent un-impacted
marshes. Hatches Harbor was an un-impacted
reference marsh. Marshall Point impacted marsh
had been hydrologically altered by the placement
of sediment plugs in grid ditches, a method used
to increase open water on grid ditched marshes.
The plugs were installed one year (1998) prior to
the collection of the vegetation data. At Sachuest
Point the impacted marsh had been tiderestricted for several decades due to an undersize
culvert that impounded water and severely
reduced tidal range. In the early spring of 1998,
two larger culverts were installed and tidal flow
was restored to the impacted marsh. Vegetation
data were collected at both the Sachuest Point
reference and impacted marsh both before (1996)
and after (1998 and 1999) the impacted marsh
underwent tidal restoration.
The vegetation communities of the reference
and un-impacted marshes (Granite Point marsh,
Moody marsh, Hatches Harbor reference, and
Sachuest Point reference 1996, 1998, 1999) were
dominated by vegetation typical of New England
salt marshes (e.g., Spartina alterniflora Loisel., salt
marsh cordgrass; S. patens (Ait.) Muhl., salt
meadow cordgrass; Juncus gerardii Loisel., blackgrass; Iva frutescens L., marsh elder; Portnoy
et al. 2003 for Hatches Harbor; Niering and
Warren 1980). Marshall Point reference and
impacted marshes were dominated by J. gerardii;
Distichlis spicata (L.) Greene, spike grass;
337
Triglochin maritimum L., seaside arrow-grass;
Glaux maritima L., sea milkwort; and S. patens
(Adamowicz et al. 2004). The community of the
impacted Sachuest Point marsh prior to restoration (1996) was dominated by Phragmites australis (Cav.) Trin. Ex Steud., common reed. After
tidal restoration (1998, 1999) this marsh was in
transition from a P. australis dominated community to a community with increasing abundance of
S. alterniflora and S. patens (Roman et al. 2002).
Vegetation data were collected at the end of the
growing season (August–early October) at all sites.
Transects were randomly located within each
marsh and placed perpendicular to the creek and
extended to the upland. Vegetation plots (1 m2)
were located along each transect, with the first plot
randomly located and subsequent plots spaced
systematically at a minimum of 10 m. Given the
random starts for setting both transects and plots,
and the minimum spacing between plots, it can
assumed that the individual vegetation plots were
independent (Elzinga et al. 2001). However, for
those pursuing any future vegetation plot sampling, it is suggested that plots be located in a pure
random manner along the transects instead of
systematic. The number of vegetation plots sampled per site ranged from 22 to 77 (Table 1).
Within each 1 m2 plot, the percent cover of live
species present was quantified using either the
point intercept method or ranked by visual inspection using a modified Braun-Blanquet scale (cover
classes: 0 = absent, 1 = 1–5%, 2 = 6–25%, 3 = 26–
50%, 4 = 51–75%, 5 = 76–100%, Kent and Coker
1992; Elzinga et al. 2001). Other cover-type categories recorded included bare, litter and wrack,
standing water, and standing dead vegetation.
Power analysis
Prior to analysis all point-intercept data were
converted to the modified Braun-Blanquet cover
classes. Converting the data to cover classes
served as a type of transformation that gave less
weight to dominant species and more weight to
rarer species and is typical of transformations
performed on multivariate species community
data (Clarke and Warwick 2001; Kent and Coker
1992). Statistical power was evaluated using the
cover class data and was determined as a function
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types for every marsh pair. A permutation testing
procedure (Appendix) was used to test the null
hypothesis of equivalent vegetation communities
between two marshes. For each pair of marshes a
decision to reject or fail to reject the null
hypothesis was made using an alpha level of
0.05. Empirical power was estimated by repeating
this process 200 times and was determined by the
number of times the null hypothesis was rejected
out of the 200 trials. The power associated with
the different sample sizes (n = 5, 10, 15, and 20)
were graphed for all 17 pairs of marshes.
of the Euclidean distance similarity index of
vegetation community data between pairs of
marshes using a permutation procedure (Clarke
and Green 1988; Smith et al. 1990). This procedure allowed statistical testing of equality
between the vegetation communities of the marsh
pairs by using the similarity index between the
two communities as the test statistic. While the
actual value of the test statistic does not indicate
statistical significance, it does provide a relative
measure of similarity (or dissimilarity) between
the vegetation communities of the marsh pairs.
Marsh pairs that were similar in vegetation
composition had low similarity indices and marsh
pairs with divergent vegetation composition had
larger similarity indices. Seventeen pairs of
marshes that exhibited a range of similarity
indices were chosen to construct the power curves
(Table 2). These pairs exhibited a range from
subtle differences in vegetation composition (e.g.,
Sachuest Point impacted sampled in 1998 versus
the same marsh sampled in 1999, similarity index
of 0.6) to dramatic differences (e.g., Rhode
Island’s Sachuest Point impacted marsh in 1996
versus the un-impacted Marshall Point reference
marsh in Maine, similarity index of 27.3).
Sample sizes of 5, 10, 15, and 20 plots were
randomly chosen in 200 trials from each marsh
data set for all 17 marsh pairs. In each trial the
average Braun-Blanquet value and corresponding
Euclidean distance were calculated for all cover
Independent confirmation of power curves
As an independent assessment of our power
curves, vegetation communities from data sets
not used in the power analyses were compared
using different sample sizes. Vegetation data were
from four marshes, Granite Point impacted marsh
(data from 2000 and 2001), Biddeford, ME;
Moody reference marsh (data from 2002), Wells,
ME; and Hatches Harbor impacted and Hatches
Harbor reference (data from 2000 to 2004),
Provincetown, MA. At Granite Point impacted
marsh (8.8 ha) mosquito ditches were plugged
with sediment in the spring 2000 to increase the
water table level and retain water on the marsh
surface upstream of the plugs. Vegetation sampling (n = 52 1 m2 plots) was conducted in 2000
shortly after the plugging and then again in 2001,
Table 2 Euclidean distance similarity indicesa for all possible combinations of marsh pairs
Marshb
HH_ MDY_99 GP_99 MP_I_99 MP_R_99 SP_R_99 SP_I_99 SP_R_98 SP_I_98 SP_R_96 SP_I_96
R_97
HH_R_97
MDY_99
GP_99
MP_I_99
MP_R_99
SP_R_99
SP_I_99
SP_R_98
SP_I_98
SP_R_96
SP_I_96
0.0
17.7
15.0
21.3
32.8
12.6
18.5
13.4
18.6
11.4
18.7
a
0.0
0.4
22.2
33.8
15.9
23.5
11.3
23.0
14.4
21.0
0.0
19.8
32.2
12.8
19.6
9.2
19.3
11.9
18.3
0.0
6.3
11.6
22.9
10.8
22.4
10.0
18.9
0.0
17.8
33.6
16.0
32.6
15.9
27.3
0.0
7.4
1.5
7.6
1.2
7.5
0.0
10.1
0.6
9.6
4.7
0.0
9.6
1.2
7.0
0.0
9.2
3.4
0.0
6.6
0.0
Bold values indicate marsh pairs that were used to construct the power curves (Fig. 1)
b
HH_R_97: Hatches Harbor reference (1997 data); MDY_99: Moody marsh (1999 data); GP_99: Granite Point marsh
(1999 data); MP_I_99: Marshall Point impacted (1999 data); MP_R_99: Marshall Point reference (1999 data); SP_R:
Sachuest Point reference 1999, 1998, and 1996 data; SP_I: Sachuest Point impacted 1996, 1999, and 1998 data
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339
1
0 .9
N=10
0 .8
N=5
N=15
0 .7
Power
0 .6
0 .5
0 .4
N=20
0 .3
n=5
n=10
0 .2
n=15
0 .1
n=20
0
0
Similar
marsh pairs
5
10
15
20
Euclidean Distance Similarity Index
25
30
Dissimilar
marsh pairs
Fig. 1 Estimation of power at alpha = 0.05 for 5, 10, 15,
and 20 sample plots per marsh. Distance was calculated by
the Euclidean distance similarity index between marsh
pairs. Plotted points are seventeen marsh pairs (Table 2,
bold values) that exhibited a range in similarity indices
from low (similar vegetation communities) to high (very
different vegetation communities). Curve fit lines are
shown to assist in the identification of the appropriate
sample size to achieve adequate power for a given
similarity distance
the second growing season after plugging. After
plugging there was an increase in open water and
decrease in S. patens (Adamowicz et al. 2004).
Moody reference marsh (0.8 ha) is an unimpacted marsh, with the exception of historic
mosquito grid ditches, and is located along a river
behind a barrier beach system. The dominant
vegetation of Moody reference marsh was
S. patens and D. spicata (n = 31 plots). The
Hatches Harbor marsh is bisected by a dike that
had severely reduced tidal flow to the upstream
portion of the marsh (impacted marsh, 80 ha) for
seven decades. The marsh downstream of the
dike (reference marsh, 90 ha) experienced normal tidal inundation. In 1998, the undersized
culvert in the dike was replaced with larger box
culverts and tidal flow was restored to the
impacted marsh (Portnoy et al. 2003). Both the
impacted and reference marshes were sampled
from 1997 through 2004 (impacted marsh n = 56
1 m2 plots, reference marsh n = 33 1 m2 plots).
Prior to installation of the new culvert, sampling
in 1997 revealed the impacted marsh was dominated by P. australis and a variety of upland
species such as Morella pensylvanica (Mirbel)
Kartesz, comb. nov. ined. (northern bayberry),
Rubus L. spp. (blackberry) and J. gerardii
(Portnoy et al. 2003). In 2004, seven growing
seasons after tidal restoration, S. alterniflora and
S. patens increased in cover on the impacted
marsh along with the amount of bare ground as
upland species continued to die due to tidal
inundation. The vegetation community of
Hatches Harbor reference marsh was typical of
un-impacted New England salt marshes, and was
dominated by S. alterniflora and S. patens.
We examined five pair-wise comparisons from
these four independent marsh data sets using
different sample sizes (n = 5, 10, 15, and 20 plots)
to determine the effect of sample size on the
observed P-value. For each sample size, plots
were randomly selected five times from the
complete data set and analyzed in five different
trials for each comparison. If the comparison was
for the same marsh but between different years
(i.e., Granite Point impacted 2000 vs. 2001) the
same plots were compared for each year. A oneway Analysis of Similarities (ANOSIM, Clarke
and Gorley 2006; Clarke and Warwick 2001) was
used to determine if there was a difference in salt
marsh vegetation communities for each comparison. ANOSIM is a non-parametric multivariate
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permutation procedure that analyses both species
composition and abundance and is considered a
non-parametric analog to multivariate analysis of
variance (MANOVA) without the often unattainable assumptions of MANOVA (Clarke and
Green 1988).
Results and discussion
The power graph (Fig. 1) shows the Euclidean
distance similarity index between pairs of salt
marsh vegetation communities on the X-axis and
the statistical power on the Y-axis. Marshes that
have similar or very subtle differences in vegetation communities have low similarity indices and
are located near the left side of the plot and
those that have different vegetation communities,
reflected by high similarity indices, are to the right.
An example of a marsh comparison with a low
similarity index would be the comparison of
vegetation communities sampled at the same
marsh, but separated by just a few growing seasons,
such as the Sachuest Point reference marsh. These
data sets would reflect no difference or perhaps
only slight differences in vegetation communities.
An example of a marsh pair with a large similarity
index would be the comparison of vegetation
communities between a reference marsh and a site
that was severely impacted or between marshes
from different regions (i.e., Maine versus Rhode
Island). Marsh pairs with intermediate similarity
indices could be from a restoration site sampled
over time, with a similarity index reflecting the
gradual response of vegetation to the restoration
activity (e.g., re-introduction of tidal flow).
The statistical power of detecting a difference
between two vegetation data sets, using different
sample sizes, can be estimated from the power
curves (Fig. 1). With a sample size of five plots
there is low power, generally less than 0.5, to
detect most differences, except where the differences between the two data sets are great as
indicated by a large similarity index. Increasing
the sample size to 10, 15, or 20 vegetation plots
per marsh substantially increases the power to
detect a difference between marshes even if the
vegetation communities are relatively similar. A
power above 0.9 means there is a >90% chance of
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Wetlands Ecol Manage (2007) 15:335–345
detecting a difference between vegetation data
sets when a difference actually exists. With a low
power there is an increased probability of not
detecting a difference when the data sets are
actually different (i.e., Type II error). It becomes
clear that with a sample size of 15 or 20
vegetation plots there would be a high probability
of detecting actual differences between marsh
communities. If an investigator was interested in
detecting subtle changes in vegetation (e.g., evaluating change from one year to another in a
reference marsh), then it would be appropriate to
select a sample size of at least 20. If dramatic
changes were of interest and were expected, such
as comparing a tide-restricted marsh to a reference marsh then a smaller number of plots (n = 5)
could be justified (although not recommended
given the ease of collecting data from additional
plots), while still maintaining adequate power.
Vegetation cover class data from three pairs of
marshes used in the power analyses are presented
in Table 3 to illustrate how similarity indices
reflect a range from subtle to dramatic differences
in vegetation communities. For ease of comparison the data have been converted to the percent
mid-point for each Braun-Blanquet cover class
(e.g., cover class 1 = 1–5% = 3%; 2 = 6–25% =
15.5%, etc.). Subtle changes in vegetation were
observed between Sachuest Point impacted marsh
in 1996 and the same marsh in 1999, 3 years after
tidal flow was restored (similarity index = 4.7).
Seven of the 16 cover types for this marsh pair
were observed in both years, and they shared two
dominant species (P. australis and S. patens)
although these species differed in abundance
each year. On the other end of the spectrum,
dramatic differences in vegetation communities
were illustrated by comparing the Sachuest Point
impacted marsh in 1996 with the Marshall Point
reference marsh in 1999 (similarity index = 27.3).
This was a comparison between marshes from
different regions (Rhode Island and Maine) and
different hydrological conditions (tide-restricted
versus un-impacted). The vegetation communities
from these marshes were quite different, with
only three of the 19 species in common at both
marshes, and each marsh had different dominant
species (Sachuest Point impacted: P. australis and
S. patens; Marshall Point reference: J. gerardii and
Sachuest Point
impacted 1996
(SP_I_96)a
Similarity index = 4.7
b
a
–
–
–
4.2
3.4
–
1.7
–
–
–
2.0
–
–
–
6.9
–
–
25.9
43.8
2.9
–
–
–
–
–
–
–
1.2
–
–
–
19.6
4.4
–
2.7
3.7
6.6
19.9
–
–
–
1.4
–
–
Granite Point
marsh 1999
(GP_99)
6.9
2.4
22.8
–
5.2
–
–
–
–
–
–
3.5
29.4
2.7
–
2.6
4.4
3.0
–
2.5
–
–
6.5
–
–
Sachuest Point
impacted 1996
(SP_I_96)
–
5.3
6.4
6.6
–
–
–
3.7
3.7
–
–
–
–
–
7.2
–
–
37.2
3.1
–
3.9
–
–
14.5
–
Marshall Point
reference 1999
(MP_R_99)
Similarity index = 27.3
Data were converted to the mid-point percentage for each Braun-Blanquet cover class. Species not observed are indicated with a dash (-)
2.2
10.6
27.7
–
8.5
4.8
–
–
–
1.6
–
–
2.6
–
–
–
5.5
16.9
–
–
–
–
4.0
7.0
–
Sachuest Point
reference 1998
(SP_R_98)
Similarity index = 9.2
–
–
1.7
29.7
–
1.3
Sachuest Point
impacted 1999
(SP_I_99)
Identification codes used in Table 2 are indicated in parentheses
Agrostis stolonifera L.
2.5b
Argentina anserina L.
–
Atriplex patula L.
–
Bare ground
6.5
Distichlis spicata (L.) Greene
–
Euthamia tenuifolia var.
–
pycnocephala (Fern.) C.& J.
Taylor
Glaux maritima L.
–
Hibiscus moscheutos L.
2.6
Iva frutescens L.
4.4
Juncus gerardii Loisel.
3.0
Limonium carolinianum
–
(Walt.) Britt.
Morella pensylvanica (Mirbel) 2.7
Kartesz, comb. nov. ined.
Phragmites australis (Cav.)
29.4
Trin. Ex Steud.
Dead Phragmites australis
–
Plantago maritima L.
–
Toxicodendron radicans L.
3.5
Kuntze
Solidago sempervirens L.
6.9
Spartina alterniflora Loisel
2.4
Spartina patens (Ait) Muhl.
22.8
Triglochin maritimum L.
–
Typha angustifolia L.
5.2
Dead Typha angustifolia
–
Viburnum dentatum L.
–
Wrack
–
Unidentified grass
–
Species
Table 3 Percent cover for cover types that contributed approximately 90% of sampled vegetation for three marsh comparisons used in the power analyses (Fig. 1)
Wetlands Ecol Manage (2007) 15:335–345
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Wetlands Ecol Manage (2007) 15:335–345
D. spicata). A marsh comparison with an intermediate similarity index of 9.2 is represented by
Granite Point marsh in 1999 and Sachuest Point
reference marsh in 1998. These marshes were
both relatively undisturbed, but were from different regions (Rhode Island and Maine). Four of
the 15 vegetation covers were in common and
each marsh had the same dominant species
(Spartina patens), however there were 11 vegetation covers that were not in common reflecting
regional differences in vegetation communities.
Using data sets independent from those used in
the power analyses, ANOSIM results show that a
sample size of 20 was sufficient to detect
(P < 0.05) the subtle change in vegetation community that occurred at Granite Point impacted
marsh between 2000 and 2001, the first and
second growing season after the ditches were
plugged, but as suggested by the power curves this
change was not detected at lower sample sizes
(n £ 15; Table 4). The differences in vegetation
communities at Hatches Harbor between the
1997 impacted marsh and the same marsh 7 years
after tidal reintroduction (2004 impacted marsh)
could be detected with as few as 15 plots
(Table 4). A sample size of 10 plots detected
the fairly dramatic differences between the vegetation communities of Spartina-dominated
Hatches Harbor reference marsh and the Phragmites-dominated Hatches Harbor impacted
(although under a regime of tide restoration)
marsh in 2000. As few as five plots may be
sufficient to detect the divergent vegetation communities from geographically distant marshes,
such as the comparison between Hatches Harbor
reference (MA) and Moody reference (ME)
marshes (Table 4). Finally, as would be expected,
the vegetation community of a reference or
un-impacted marsh would be relatively stable
through time (e.g., Hatches Harbor reference),
and therefore no change would be detected at any
of the sample sizes.
The suggested minimum size of 20 plots per
marsh is the same regardless of the spatial area or
extent of the marsh that is being sampled. This is
because the proportion of a marsh that is actually
sampled (i.e., 20, 1 m2 plots = 20 m2) is often very
small compared to the entire area of the marsh
that could be potentially sampled. For example,
sampling a 10 ha marsh or a 1 ha marsh with 20
plots represents just 0.02% or 0.2% of the entire
marsh area, respectively. If the area sampled is
less than 5% of the total study area it is not
necessary to alter the sample size by applying the
finite population correction factor (Krebs 1999;
Elzinga et al. 2001). The suggested minimum
sample size of 20 plots also holds true if the plots
are permanent or non-permanent vegetation
plots. If permanent plots are re-sampled in
subsequent years, an investigator may be able to
sample less than the suggested minimum of 20
plots, as the variability would decrease since the
same plots were used in each sampling period.
To summarize, it is recommended that a minimum of 20 randomly located 1 m2 vegetation plots
be established when designing monitoring programs
to evaluate changes in vegetation communities of
salt marshes. To facilitate sampling, randomly
located transects that extend from creek bank to
upland with plots randomly located along each
transect can be used. The sample size of 20
vegetation plots is for the entire salt marsh encompassing all salt marsh zones (e.g., high marsh, low
marsh), as opposed to 20 plots within each salt
Table 4 An example of how sample size influenced P-values when comparing salt marsh vegetation community dataa
Comparison
All plots
Granite Point impacted (2000 versus 2001)
Hatches Harbor impacted (1997 versus 2004)
Hatches Harbor reference (2000) versus Hatches
Harbor impacted (2000)
Hatches Harbor reference (2002) versus Moody
reference (2002)
Hatches Harbor reference (2000 vs. 2004)
a
n = 20
n = 15
n = 10
n=5
<0.001 (n = 52, 52)
0.001 (n = 56, 56)
0.001 (n = 33, 56)
0.039
0.002
0.001
0.111
0.002
0.001
0.217
0.074
0.004
0.435
0.216
0.092
<0.001 (n = 33, 31)
<0.001
<0.001
0.013
0.016
0.582 (n = 33, 33)
0.628
0.807
0.687
0.603
For each sample size, plots were randomly selected from the entire data set and analyzed by ANOSIM. P-values are an
average obtained from five separate ANOSIM analyses for each sample size and comparison
123
Wetlands Ecol Manage (2007) 15:335–345
343
Euclidean distance similarity index (Krebs
1999):
marsh zone. However, if investigators are interested
in detecting changes between zones within a marsh
or within zones between years, 20 plots within each
zone would be an appropriate sampling design. It is
noted that a sample size of 15 vegetation plots would
probably be adequate (based on the power curves)
to detect the subtle vegetation changes that may be
of interest in long-term monitoring programs and as
few as 5 or 10 sample plots may be appropriate for
detecting more dramatic differences in vegetation
communities; however, given the relative ease of
collecting vegetation plot data, a minimum sample
size of 20 plots is recommended. The power analyses
presented in this paper are appropriate for New
England salt marshes. However, the approach can
be applied to other regions and habitat types to
assist with sample size selection for research and
monitoring programs designed to detect vegetation
community responses to natural and humaninduced factors.
D12 ¼
X
k (C1k C2k )2
where D12 is the Euclidean distance between
P
marsh 1 and marsh 2,
k is the sum of all
cover types or species (k) in marsh 1 and marsh 2,
C1k is the abundance of cover type or species k in
marsh 1, C2k is the abundance of cover type or
species k in marsh 2.
Synopsis of permutation procedure as outlined
in Smith et al. (1990):
The null hypothesis of the permutation test is
‘‘No difference between marsh 1 and marsh 2’’.
To illustrate the permutation procedure, the
original data matrix of marsh 1 and marsh 2 with
N samples and M variables is defined as:
Marsh 1(N M)
Marsh 2(N M)
X11 X12
X21 X22
... ...
XN1 XN2
= XðNþ1Þ1 XðNþ1Þ2
X
ðNþ2Þ1 XðNþ2Þ2
...
...
X
X
X =
Acknowledgements This research was supported, in part,
by the National Park Service in association with the design and
testing of monitoring protocols for the Long-term Coastal
Ecosystem Monitoring Program at Cape Cod National
Seashore. Support was also provided by the U.S. Fish and
Wildlife Service, U.S. Geological Survey, and the NOAANational Marine Fisheries Service, Restoration Center.
Appendix
ð2NÞ1
This appendix details the process used to conduct
the power analysis. It is included to clarify the
methodology used as well as to provide the stepby-step procedure so that others may perform this
same kind of analysis to evaluate the relationship
between power and sample sizes for other types
of multivariate community data.
Marsh /
Sample
No.
Marsh 1
1
2
3
...
N
Marsh 2
N+1
N+2
N+3
…
2N
1
0
2
Marsh 1
3
...
D12 D13 …
0 D22 …
0 …
0
ð2NÞ2
X23 . . . X2M
... ... ...
XN3 . . . XNM
XðNþ1Þ3 . . . XðNþ1ÞM XðNþ2Þ3 . . . XðNþ2ÞM ... ...
... Xð2NÞ3 . . . Xð2NÞM X13 . . . X1M
The distance (or similarity) between marsh 1 and
marsh 1 (i £ N and j £ N), marsh 2 and marsh 2
(i > N and j > N), and marsh 1 and marsh 2 (i £ N
and j > N) samples, respectively, can be calculated pair-wise. For the above data matrix, the
distance matrix is:
Marsh 2
N+3
N
N+1
N+2
...
2N
D1N
D2N
D3N
…
0
D1(N+1)
D2(N+1)
D3(N+1)
…
DN(N+1)
D1(N+2)
D2(N+2)
D3(N+2)
…
DN(N+2)
D1(N+3)
D2(N+3)
D3(N+3)
…
DN(N+3)
…
…
…
…
…
D1(2N)
D2(2N)
D3(2N)
…
DN(2N)
0
D(N+1)(N+2)
0
D(N+1)(N+3)
D(N+2)(N+3)
0
…
…
…
0
D(N+1)(2N)
D(N+2)(2N)
D(N+3)(2N)
…
0
123
344
Wetlands Ecol Manage (2007) 15:335–345
The distances in the rectangle are the distances
between marsh 1 and marsh 2 samples (the
Between Distances). The distances in the triangle
to the left of the rectangle are the distances
between pairs of marsh 1 samples (the Within
Distances for marsh 1), and the distances in the
triangle below the rectangle are the distances
between pairs of marsh 2 samples (the Within
Distances for marsh 2). From this matrix the
averages of the Between Distances ( B0) and the
Within Distances ( W0) are calculated, either one
of these can be used
as a test statistic. Alterna
tively, L0 ¼ B0 W0 can be used as the test
statistic. The test statistic L0 will become larger as
the difference between marsh 1 and marsh 2
samples increase. If the null hypothesis is true,
samples can be switched between marsh 1 and
marsh 2 groups. For example, after the mth
permutation, the data matrix may look like:
Group 1(N M)
Group 2(N M)
X32
X33
X31
XðNþ9Þ1 XðNþ9Þ2 XðNþ9Þ3
..
..
..
.
.
.
X22
X23
X21
= XN1
XN2
XN3
X12
X13
X11
..
..
..
.
.
.
X
X
X
L values from the permuted matrices at an alpha
level of 0.05. The P-value can be calculated by
comparing the rank order of the L0 value from
the original data with the rank ordered L values
from the 500 permutations. If the original L0 falls
out of the 95% range of the L values from the
permuted matrices, the null hypothesis of equality
of the two marshes would be rejected.
The basic procedure to determine empirical
power is summarized as follows:
1.
2.
3.
4.
Xm =
ðNþ2Þ1
ðNþ2Þ2
ðNþ3Þ3
...
..
.
...
...
...
..
.
...
XðNþ9ÞM ..
.
X2M XNM X1M ..
.
X
X3M
ðNþ2ÞM
The Between Distances ( B) and the Within
Distances ( W) are calculated for each permuted
matrix, and the test statistic L ( L ¼ B W) is
calculated. A series of L values are obtained after
many permutations (in this study we performed
500 permutations of each pair of selected data
sets). The L values are then arranged in descending order. For the Euclidean Distance metric, a
large L value indicates a large difference between
the two groups of data. As a result this test
statistic is one-sided.
If the null hypothesis of no difference between
marsh 1 and marsh 2 is true, L0 (from the original
data) should fall into the lower 95% range of the
123
5.
6.
7.
8.
9.
10.
11.
Select data sets from a pair of marshes (e.g.,
marsh 1 and marsh 2) for comparison.
Randomly select a sample size of 5, 10, 15,
or 20 samples from the data set for each
marsh pair.
Calculate the pair-wise Within Distances (or
similarities) within the marsh 1 and marsh 2
data sets (W0).
Calculate the pair-wise Between Distances
or similarities for the samples for marsh 1
and marsh 2 (B0).
Calculate the averages of all Within Distances and Between Distances.
Calculate the test statistic L0 from the
original data sets.
Randomly permute the marsh 1 and marsh 2
data in such a way that the identities of the
samples are randomly assigned.
Calculate the Within Distances and the
Between Distances (W and B) and their
averages for the permuted data sets.
Calculate the test statistic L for the permuted data sets.
Repeat (5) through (7) for 500 times to
obtain 500 L values.
Rank the L values in ascending order.
Compute the P-value by determining the
rank of L0 value among the rank order of
the L values. The level of the P-value is used
to determine whether there is a difference
between the two data sets at the 5% significance level. For example, if L0 value falls at
the 200th rank of the 500 rank ordered L
values, then the P-value is 200/500 or
P = 0.400; if L0 value falls at the 15th rank
of the 500 rank ordered L values, then the
P-value is 15/500 or P = 0.030.
Wetlands Ecol Manage (2007) 15:335–345
12.
13.
14.
Repeat steps (2) through (11) 200 times for
each sample size.
Determine empirical power as the number
of rejections (as indicated the permutation
procedure in step 11), at an alpha of 0.05 out
of the 200.
Repeat steps (1) through (13) for each
possible pair-wise comparison of marsh data
sets for each sample size.
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