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3542
dx.doi.org/10.1021/es102790d |
Environ. Sci. Technol.
2011, 45,
3539–3546
Environmental Science & Technology
ARTICLE
mean concentration of the entire BW discharge. More empirical
research is necessary to determine how the aggregation of organisms
in BW a
ff
ects sample estimates. In the examples that follow, we
assume that organisms are randomly distributed throughout BW or
that sampling protocols that eliminate or mitigate this problem are
used, and thus can be modeled using the Poisson distribution. Our
assumptions include the following:
1 The BW sample is time-integrated and proportional to the
discharge
fl
ow to control for any underlying spatial/
temporal structure of organism distribution.
2 The total BW sample volume is processed.
3 All live organisms
g
50
μ
m are captured and detected (i.e.,
recovery error is negligible; Table S1, Figure S1, Supporting
Information).
Equations.
Poisson Probability and Statistical Power
25 27
P
½
X
¼
x
¼
e
m
m
x
x
!
x
¼
0, 1, 2
:::
m >
0
ð
1a
Þ
P
½
X
e
c
¼
∑
c
x
¼
0
e
m
m
x
x
!
ð
1b
Þ
p
¼
P
½
X > c
¼
1
P
½
X
e
c
ð
1c
Þ
where
X
= a random variable taking values
x
where
x
= non-
negative integer (i.e., 0, 1, 2..., where
X
represents the count
observed in a sample taken from a population);
e
= base of natural
logarithms;
m
= mean of Poisson distribution (i.e., true concen-
tration of organisms in discharge);
c
= count of organisms at the
noncompliance threshold for a given
R
and sample volume
(Table 1);
p
= the probability of exceeding
c
. In this application,
p
is the false positive rate (
R
) when the BW is compliant and is
power when BW is noncompliant with the discharge standard, i.e.,
P
½
X > c
j
null hypothesis true
~
¼
R
ð
2a
Þ
P
½
X > c
j
alternate hypothesis true
¼
power
ð
2b
Þ
Applying Sampling Statistics.
Using the Poisson distribu-
tion,
25 27
we modeled the probability that a random sampling
unit of ballast discharge will contain a specific number of
organisms (eqs 1a and 1b). For example, if the true concentration
of ballast discharge is 5 zooplankton
3
m
3
, the probability that a
random sampling unit (1 m
3
) will contain 0 organisms is 0.0067
(eq 1a). Alternatively, the probability that a sampling unit will
contain
e
3 organisms is 0.265 (eq 1b). (See Supporting
Information 1 for example calculations.) The units for this
parameterization of the Poisson distribution equal the number
of organisms per sampling unit. To convert to concentration, the
total count is divided by the total sampling unit volume.
Stage 1 (Single Trial Analysis).
Inherent uncertainty around
sampling data is reduced by sampling larger volumes.
17
We
determined how increased sample volume improves the ability
to identify sample concentrations that exceed the IMO standard
of <10
3
m
3
. We compared the sampling distribution of a
zooplankton concentration of <10
3
m
3
to sampling distribu-
tions obtained from theoretical populations with concentrations
g
10 zooplankton
3
m
3
in order to calculate statistical power,
based on the Poisson distribution described in eq 1a.
In our framework, a sample of ballast discharge must be
statistically signi
fi
cantly
g
10 zooplankton
3
m
3
to be classi
fi
ed
as noncompliant. The noncompliance threshold represents the
maximum number of organisms that are likely to occur in a sam-
ple if the concentration does not exceed the standard (Figure 1)
given our predetermined
R
values. These noncompliance thresh-
old values (Table 1) were determined (eq 1b) by summing the
probabilities of obtaining counts from 0 to
x
, given a true
concentration of 10
3
m
3
, until the cumulative probability just
exceeded 0.95 (
R
= 0.05) or 0.80 (
R
= 0.20). Statistical power
was calculated for each
R
value to determine how reliably popu-
lation concentrations ranging from 10 to 20 zooplankton
3
m
3
could be discriminated from populations of <10 zooplank-
ton
3
m
3
(eq 1c) for sample volumes of 0.1, 1, 3, and 7 m
3
.
Single trial analyses may be the only tractable sampling approach
available on working ships, and best suited for detecting large
exceedances of the discharge standard.
Stage 2 (Multiple Trial Analysis).
An alternative approach for
gauging the efficacy of a treatment system is to pool the results
from multiple independent ballast trials and to examine them
simultaneously. The simplest, and arguably most powerful,
approach for evaluating multiple tests relies on the fact that
Poisson distributions are additive and generate a summed
Poisson distribution.
22,27
For example, the total number of
zooplankton from two 4-m
3
trials would be summed and
compared to a Poisson distribution where mean and variance =
80 (i.e., the expected count for a 10 zooplankton
3
m
3
discharge
standard and total sample volume of 8 m
3
). To determine how
summing the results from multiple trials affects statistical power,
we calculated the probability of identifying noncompliant con-
centrations of 11 14 zooplankton
3
m
3
for 1 15 independent
trials, using 7-m
3
sample volumes. For each total sample volume
(7 105 m
3
), we calculated a noncompliance threshold value,
based on the upper probable count expected in samples with
concentrations of 10 zooplankton
3
m
3
(
R
= 0.05 in this
scenario). Power was calculated by determining the predicted
proportion of samples with counts greater than noncompliance
threshold values (eqs 1a 1c). Multiple test trials may be most
feasible on land-based test beds, which have fewer logistical
constraints than ships, and allow for more controlled and
repeated sampling and analysis.
Application of Model to BW Treatment Test Results.
To
demonstrate the potential practical utility of this statistical
approach, we applied our analysis to discharge data from tests
Table 1. Noncompliance Threshold Values for
r
= 0.05
and 0.20; If Sample Counts or Concentrations Exceed the
“
Noncompliance Threshold
”
the Discharge Is Statistically
Unlikely To Be Compliant with the IMO Discharge Standard
(<10 zooplankton
3
m
3
)
noncompliance threshold
R
= 0.05
R
= 0.20
sample
volume (m
3
)
count
(
N
)
concentration
(zoo
3
m
3
)
count
(
N
)
concentration
(zoo
3
m
3
)
1
15
15.0
13
13.0
3
39
13.0
35
11.67
7
84
12.0
77
11.0
14
160
11.43
150
10.71
21
234
11.14
222
10.57
28
308
11
294
10.50
35
381
10.89
366
10.46