Page 1

An Assessment of Fecal Indicator

Bacteria-Based Water Quality

Standards

A N D R E W D . G R O N E W O L D , *, †

M A R K E . B O R S U K ,‡

R O B E R T L . W O L P E R T ,† , §A N D

K E N N E T H H . R E C K H O W†

Nicholas School of the Environment, Duke University,

Durham, North Carolina 27708, Thayer School of

Engineering, Dartmouth College, Hanover, New Hampshire

03755, and Department of Statistical Science, Duke

University, Durham, North Carolina, 27708

Received December 16, 2007. Revised manuscript received

April 1, 2008. Accepted April 4, 2008.

Fecal indicator bacteria (FIB) are commonly used to assess

thethreatofpathogencontaminationincoastalandinlandwaters.

Unlike most measures of pollutant levels however, FIB

concentration metrics, such as most probable number (MPN)

and colony-forming units (CFU), are not direct measures of

the true in situ concentration distribution. Therefore, there is

thepotentialforinconsistenciesamongmodelandsample-based

water quality assessments, such as those used in the Total

Maximum Daily Load (TMDL) program. To address this problem,

we present an innovative approach to assessing pathogen

contamination based on water quality standards that impose

limits on parameters of the actual underlying FIB concentration

distribution, rather than on MPN or CFU values. Such

concentration-based standards link more explicitly to human

healthconsiderations,areindependentoftheanalyticalprocedures

employed, and are consistent with the outcomes of most

predictivewaterqualitymodels.Wedemonstratehowcompliance

with concentration-based standards can be inferred from

traditional MPN values using a Bayesian inference procedure.

This methodology, applicable to a wide range of FIB-based

waterqualityassessments,isillustratedhereusingfecalcoliform

data from shellfish harvesting waters in the Newport River

Estuary, North Carolina. Results indicate that areas determined

to be compliant according to the current methods-based

standards may actually have an unacceptably high probability

of being in violation of concentration-based standards.

1. Introduction

Section 303(d) of the United States Clean Water Act requires

that states assess the condition of surface waters and report

those which fail to meet ambient water quality standards

(1, 2). These are added to the United States Environmental

Protection Agency (USEPA) list of impaired waters (3) and

can only be removed after the performance of a Total

Maximum Daily Load (TMDL) assessment (4, 5) followed by

sample-based verification that the standards are being met.

TheprimaryobjectiveofaTMDLassessmentistodetermine

themaximumallowablepollutantloadfrompoint,nonpoint,

and natural sources, including a margin of safety (MOS),

which can be discharged into a receiving water without

violating water quality standards (2, 4). Such predictive

assessmentsareusuallybasedonanempiricalormechanistic

waterqualitymodelrelatingpollutantloadinglevelstowater

body concentrations (6, 7).

The latest official assessment of U.S. water quality data

(8) indicates that pathogens are the leading cause of coastal

shoreline standard violations (275 total miles impaired) and

the second leading cause of violations in rivers and streams

(82,100 total miles impaired). Fecal indicator bacteria (FIB),

suchasfecalcoliform,arecommonlyusedtoassesspotential

pathogencontaminationincoastalwaters,andconcentration

estimatesarebasedonindirectmeasuresofmicrobialdensity.

These are typically reported as either the most probable

number(MPN)orthenumberofcolony-formingunits(CFU)

perunitvolume.Asaresult,MPN-andCFU-basedestimates

of FIB form the basis of numeric water quality standards for

several designated uses including drinking water supply (9),

recreational use (10), and shellfish harvesting (11).

Unlike most measures of pollutant levels, the probability

distributionsofMPNandCFUdonotdirectlycorrespondto

the true in situ concentration distribution. The true FIB

concentration at a fixed point over time is typically believed

tohaveacontinuousunimodal(oftenlog-normal)probability

distribution (12, 13). The probability distributions of cor-

responding MPN and CFU values for the same monitoring

location, however, are discrete and, in the case of the MPN,

often multimodal. In addition, MPN and CFU variability

depends on the number and volume of sample aliquots.

Accurately predicting MPN- or CFU-based standard

violations from FIB loading levels should therefore involve

explicitacknowledgmentofhowamodel-predictedreceiving

water concentration translates into an MPN or CFU value,

and then how frequently those MPN and CFU values violate

standards (5, 14). A review of current modeling practice,

however, indicates that such an approach is seldom, if ever,

implemented.

Wepresentathree-partapproachtoassessingwaterbody

impairment from model-based predictions of FIB concen-

tration,usingMPN-andCFU-basedfecalcoliformstandards

inshellfishharvestingwatersasanexample.First,wesimulate

the frequency with which a model-generated distribution of

thetrueFIBconcentrationcyieldsMPNandCFUvaluesthat

violate shellfish harvesting area water quality standards.

Second, we identify the distributional parameters of c

expectedtoleadtowaterqualitystandardviolationsnomore

thanaspecifiedpercentageofthetime(e.g.,10%).Wepropose

that these parameters be used as the basis for FIB-based

standards because they do not depend on the analytical

methodemployed,aremorecloselyrelatedtothetruewater

body concentration, and are consistent with the outcomes

of most water quality models. In the third part, we dem-

onstrate how compliance with such concentration-based

standards can be inferred from traditional MPN and CFU

values using a Bayesian approach. We demonstrate this

procedure using data from the most recent assessment of

shellfish harvesting waters in the Newport River Estuary of

Eastern North Carolina.

1.1. Background on Current FIB-Based Water Quality

Standards. Model Ordinance in the Guide for the Control of

Molluscan Shellfish, prepared by the National Shellfish

Sanitation Program (NSSP), includes recommended water

qualitycriteriaforshellfish-growingwatersbasedonnumeric

limits on MPN- and CFU-based measures of fecal coliform

* Corresponding author e-mail: adg12@duke.edu.

†Nicholas School of the Environment, Duke University.

‡Thayer School of Engineering, Dartmouth College.

§Department of Statistical Science, Duke University.

Environ. Sci. Technol. 2008, 42, 4676–4682

46769ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 42, NO. 13, 200810.1021/es703144k CCC: $40.75

2008 American Chemical Society

Published on Web 05/22/2008

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concentration (11). States which participate in the NSSP,

and which are also members of the Interstate Shellfish

Sanitation Conference, enforce the Model Ordinance as a

minimum requirement for sanitary control of shellfish (11).

WhilesimilarFIB-basedwaterqualitystandardsareenforced

in surface waters with other designated uses, such as

recreational use (10) and drinking water supply (9), we

illustrateourproposedmethodologyusingNSSPwaterquality

standards as an example.

While several MPN- and CFU-based FIB analysis proce-

dures are in common use (for a wide range of water body

designated uses), the three cited in NSSP-recommended

standards for shellfish growing waters (see ref (11), Chapter

2,SubsectionIV.02)arethe5-tubemultipletubefermentation

(MTF), the 3-tube MTF, and the membrane filtration (MF)

procedure for fecal coliform bacteria. Here, we explore only

the 5-tube MTF and the MF procedures because they are far

morecommonthanthe3-tubeMTFprocedure.Thestandard

5-tube MTF procedure involves diluting a water quality

sample into 3 sets of 5 tubes, with each set of 5 containing

either 10 mL, 1 mL, or 0.1 mL of the original sample

(alternativedilutionsmaybeuseddependingontheexpected

concentration of the sample (15)). The MPN is then the

maximumlikelihoodestimate(MLE)ofthetruefecalcoliform

concentration based on the number of positive tubes

observed in each dilution series after a period of incubation

(15–18). A positive tube is one containing visible gas, which

servesasanindicationofbacteriallactosefermentation.The

MF procedure involves filtering a water quality sample and

counting the number of bacteria colonies emerging from

the filter on a growth plate after a period of incubation. The

number of colonies is divided by the sample aliquot volume

and reported as the CFU (19–23).

MPNandCFUvaluesareintrinsicallyvariableandsubject

to additional uncertainty arising from minor variations in

experimental protocol. NSSP criteria address MPN and CFU

variability by recommending that a water body assessment

be based on a minimum number n of MPN or CFU values,

and by recommending separate numeric standards for the

median,geometricmean,and90thpercentileofthosevalues

(Table 1). For example, the NSSP criteria for MPN values

from a standard 5-tube MTF decimal dilution analysis state

thatagrowingareaisclassifiedas“Approved”ifa“minimum

of 30 of the most recent randomly collected samples” have

an MPN median and geometric mean no greater than 14

organismsper100mL,andanestimatedMPN90thpercentile

no greater than 43 organisms per 100 mL (11). Ninetieth

percentiles are calculated from a log-normal distribution

parametrized by the mean and variance of the natural

logarithm of the n MPN or CFU values (11).

A common goal of model-based TMDL assessments is to

evaluate pollution loading mitigation strategies which will

result in compliance with water quality standards. Despite

a growing body of research on the role of models and model

uncertainty in water resources management (7, 14, 24), we

know of no water quality models which explicitly acknowl-

edgehownaturalvariabilityinFIBconcentrations,combined

withintrinsicanalyticuncertainty,propagatesintoMPN-or

CFU-basedwaterqualitystandardviolationsandassociated

managementdecisions.Thefollowingsectiondescribesour

attempt to fill this gap.

2. Methods

Most water quality standards, including those based on FIB

concentrations, imply that long-term pollutant concentra-

tions at a monitoring station are adequately characterized

byanassumedstationarylog-normalprobabilitydistribution

(6, 13). The 30 fecal coliform samples required by NSSP

guidelines, however, are usually collected at bimonthly

intervals,implyingthatshellfishharvestingareawaterquality

assessments may take 4-5 years to complete. Changes in

fecal coliform sources and delivery routes during that time

are likely to translate into changes in the fecal coliform

concentrationdistribution.Here,weassumeaconstantfecal

coliform concentration probability distribution at a moni-

toring station throughout the assessment period. While we

recognize this is a potentially restrictive assumption, it is

consistentwithcurrentmanagementpracticeandallowsour

results to be easily compared with recommended water

quality criteria. Developing strategies for assessing a water

bodybasedonachangingpollutantconcentrationprobability

distribution is an area for additional research (25).

We also assume that water quality samples are either

simulated (as in the case of a model forecast) or collected

based on a systematic random sampling program. In other

words, we assume that the 30 samples (as required by NSSP

guidelines) are collected under environmental conditions

(such as rainfall and tide, for example) indicative of long-

term distributions. Furthermore, we presume that the NSSP

criteria for an “Approved” classification implies compliance

withshellfishharvestingareawaterqualitystandardsinNSSP

guidelines, Chapter 2, subsection 02F (“Standards for ap-

provedclassificationofgrowingareasaffectedbynon-point

sources”) (11).

2.1. TranslationofModel-PredictedConcentrationsto

MPN- and CFU-Based Standard Compliance. As a starting

point, we assume that uncertainty in a water quality model

or variability in the environment leads to a probability

distribution on the fecal coliform concentration c (in organ-

isms per 100 mL) that is log-normal, LN(µc, σc), where µcand

σcare the mean and standard deviation, respectively, of the

natural logarithm of c . Therefore, we began our analysis by

simulating 300,000 concentrations drawn at random from

this distribution. We then randomly grouped the 300,000

simulations into 10,000 “samples” of size 30. We repeated

the simulation for values of µcevenly spaced between 0 and

3.5 in log(organisms per 100 mL) (at intervals of 0.05), and

valuesofσcevenlyspacedbetween0and3.0inlog(organisms

per100mL)(atintervalsof0.1),andallcombinationsthereof.

The ranges of values for µcand σcwere selected based on a

preliminary analysis of MPN values from the Newport River

Estuary.Foreachofthe300,000fecalcoliformconcentration

values, we randomly drew an MPN and a CFU value from

the conditional sampling distribution. For the MPN values,

weusedaPoisson/binomialprobabilitymodel(12,15,16,26–31)

written as

pi)1-e-cvi⁄100

c [∏

i)1

(1)

MPN)argmax

d

pi

xi(1-pi)mi-xi]

(2)

where pi) probability of a positive tube in dilution series

i, c ) fecal coliform concentration (in organisms per 100

mL), vi) volume of original sample in dilution series i (mL),

d)numberofdilutionseries(often3),xi)numberofpositive

tubes in dilution series i (i ? [1,. . .,d]), and mi) number of

tubes in dilution series i (often 5).

TABLE 1. NSSP Shellfish Harvesting Area Fecal Coliform Water

Quality Standards Based on a Minimum of 30 Randomly

Collected Samples

standard

µgeo

basis for standardq50

q90

n MPN observations from 5-tube

MTF procedure

n CFU observations from MF

procedure

141443

141431

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CFU values were generated using a Poisson Po(λ) prob-

ability model (32):

λ)cV⁄100

y|λ∼Po(y|λ))λye-λ⁄y!

CFU)100y⁄V

(3)

(4)

where y ) number of identifiable colony forming units, and

V ) sample aliquot volume.

Previous authors have suggested that the Poisson distri-

bution underestimates the variability of microorganisms in

both natural waters and sample aliquots and recommend

using the negative binomial distribution instead (32–37).

However, the Poisson probabilities in our eqs 1 and 3

representonlytheconditionalprobabilitiesofpositivetubes

and CFU values given the true fecal coliform concentration

c . Because we also assume a log-normal distribution on c,

the marginal distributions of positive tubes and CFU values

are very similar to the negative binomial. It is possible,

however, that the conditional distributions represented by

eqs 1 and 3 may also be more widely dispersed than Poisson

due to departures in laboratory procedure from standard

protocol,clumpingofbacteriacells,andsimilarissues(34,38).

We explored probability models more dispersed than the

Poisson and found that our analysis would only change

significantly if the methodological variability (e.g., measure-

menterror,celldamage,filterbypass)were33%greaterthan

that described by the Poisson distribution (see Supporting

Information for details). We believe that variability of this

magnitude is highly unlikely when standard laboratory

procedures are carefully followed.

For each (µc, σc) pair, we recorded the proportion of the

10,000 simulated size-30 samples which violated the geo-

metric mean standard, the median standard, the 90th

percentile standard, or any of the three standards for both

MPN and CFU values (see Table 1). Simulations and

calculationswereperformedusingthestatisticsandgraphics

software program R (39).

2.2. DeterminationofConcentration-BasedStandards.

Guidelines prepared by USEPA for assessing water quality

impairment suggest that a waterbody does not support its

designateduseif10%ofsamplesviolateassociateddesignated

use criteria (40, 41). NSSP guidelines for shellfish harvesting

waters address this provision by setting numeric limits on

statistics of a set of at least 30 consecutive water quality

samples. According to NSSP guidelines, any size-30 sample

which violates any of the standards (i.e., geometric mean,

median,or90thpercentile)indicatesimpairment.Givenour

probabilisticframework,however,itiseffectivelyimpossible

toobtainapredictedfecalcoliformconcentrationdistribution

thatwillviolatestandardswithzeroprobability.Wetherefore

identified(µc, σc)pairsthatwouldviolatestandardsnomore

than0.5%ofthetime.Wealsoidentifiedthosepairsviolating

standards no more than 10% of the time for purposes of

comparison.

FIGURE 1. Combinations of the mean µc and standard deviation σc of the log-transformed fecal coliform concentration distribution

which yielded MPN (solid lines) or CFU (dotted lines) samples in violation of the NSSP median standard (panel a), geometric mean

standard (panel b), 90th percentile standard (panel c), or any standard (panel d) with a frequency of either 0.005 or 0.1. The zone of

violations is in the upper right of each panel.

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Page 4

To formulate our results in the form of a more readily

implementableregulatorystandard,westatisticallyestimated

theequationofthelinein(µc, σc)spacethatdividesviolating

fromnonviolatingconcentrationdistributions.Thiswasdone

by fitting a regression model to points on the violation

boundary (see Supporting Information for details).

2.3. Translation of MPN and CFU Values to Concentra-

tion-Based Standard Compliance. Assessing compliance

with concentration-based standards from traditional MPN

and CFU observations requires “reversing” the logic of the

probabilistic expressions given in eqs 2, 3, and 4 through a

process of Bayesian inference (42, 43). This involves using

these equations, together with the log-normal distribution

on c, as the likelihood functions for µc and σc and then

multiplyingtheselikelihoodsbypriordistributionsonµcand

σc. Following approaches of refs 44–47, we use diffuse prior

distributions for µc and σc in the model of J independent

observations of the concentration c at a given monitoring

station:

cj|µc,σc

µc

σc

iid∼

∼

∼

LN(cj|µc, σc),j)1,. . .,J

Normal(µ0)0,σc)1 000 000)

Uniform(lower)0,upper)100)

(5)

WethenimplementedaMarkov-chainMonteCarlo(MCMC)

proceduretogeneratesamplesfromthejoint(µc, σc)posterior

distribution using the Bayesian software package WinBUGS

(48). Posterior distributions used in our analysis are based

on 1000 MCMC samples collected after convergence, as

indicated by a potential scale reduction factor Rˆ) 1.0 (44).

Although we used very diffuse priors in this analysis,

the Bayesian approach is still preferable to a frequentist

method because it yields a full posterior distribution for

the distributional parameters, rather than mere point

estimates. This allows us to express our results in terms

of both “confidence of compliance” (6, 31) and in terms

of the posterior probability of a sample of size n revealing

a violation. We define confidence of compliance (CC) in

this case as the posterior probability that the actual values

of µcand σcare below our proposed violation boundaries.

We estimated CC by integrating the joint (µc, σc) posterior

probability density function over all possible µc and σc

values that fall below these boundaries. The posterior

probability of a sample revealing a violation is estimated

byusingMCMCsamplesfromthefulljoint(µc, σc)posterior

distribution to generate samples of fecal coliform con-

centration from the log-normal distribution. These con-

centrations are then propagated through eqs 1 and 2 to

generate size-30 sample sets of MPN values, from which

violation probabilities can be calculated.

2.4. Application to the Newport River Estuary. We

appliedourBayesianinferenceprocedureandconcentration-

based (µc, σc) standards to water quality monitoring data

collected from the Newport River Estuary, NC, during the

most recent assessment period (2000-2005). The NC De-

partment of Environment and Natural Resources, Shellfish

SanitationandRecreationalWaterQualitySection(NCDENR-

SSS)collectsambientfecalcoliformsamplesfromthisestuary

(and other shellfish harvesting waters throughout North

Carolina) at roughly bimonthly intervals and analyzes each

sampleusingthe5-tubeMTFprocedure.TheNewportRiver

Estuary is historically a productive shellfish harvesting area.

However, all of its segments and tributaries are either

permanently or conditionally closed to shellfishing based

on poor water quality or proximity to known or potential

sources of fecal contamination. As a result, the estuary and

its tributaries comprise 28 of the designated shellfish

harvesting areas in North Carolina which are currently

included in the USEPA 303(d) list and therefore require a

TMDL assessment. Our goal in analyzing these data is to

determinewhetherourconcentration-basedapproachwould

have yielded a different impairment decision than that

obtained using the MPN-based standard, and therefore

whether water quality data provide quantitative support for

includingtheNewportRiver(anditstributaries)onthe303(d)

list.

3. Results

3.1. Data Simulations. Simulation results indicate that (for

a given µc and σc) samples will violate NSSP water quality

standards at different frequencies depending on whether

theyareassessedusingtheMPN-basedstandardortheCFU-

basedstandard(Figure1).Additionally,therelativestringency

of the two standards will depend on the value of σc.

Specifically, application of the 5-tube MTF procedure (and

associatedMPNstandard)toaparticularsampleislesslikely

to lead to a violation than application of the MF procedure

(and CFU standard) when σc is greater than 0.65, and

more likely to lead to a violation when σcis less than 0.65

(Figure1d).Resultsalsoindicatethatsampleswithvaluesin

the range of those collected at the Newport River shellfish

harvesting area monitoring stations are more likely to be in

violation of the NSSP 90th percentile standards (Figure 1c)

thaneitherthemedianorgeometricmeanstandards(Figure

1a and b).

3.2. Concentration-Based Standards. Our regression-

based estimation of the lines that divide violating from

nonviolating values in the joint (µc, σc) space reveals that

“compliant”valuesofµcandσc(i.e.,thoseexpectedtoviolate

anyofthethreeMPNorCFU-basedstandardsnomorethan

0.5% of the time) are defined by the following inequalities

for the MPN procedure:

µce{

and by the following inequalities for the CFU procedure:

µce{

Thus, with point estimates of µc and σc determined for a

particular site from either a simulation model or sample-

basedinferenceprocedure,thefrequency-basedcompliance

status of that site can be established in a straightforward

way. Of course, when there is uncertainty about µcand σc,

then, rather than a binary compliance determination, a CC

is the result, as demonstrated in the next section.

3.3. Newport River Estuary Assessment. Application of

ourconcentration-basedassessmenttotheMPNdataofthe

Newport River Estuary resulted in eight stations having an

assessedCClessthan1%,whetheremployingtheNSSPMPN

or CFU standard (Table 2). Samples collected between 2000

and 2005 actually revealed MPN standard violations at all

eight of these stations. While violations were not found at

any of the other stations during this period, our Bayesian

inference procedure indicates a low CC with the MPN

standard (less than 25%) at four other stations as well (Stns.

8, 8A, 25, and 84). This reflects the stringency of the NSSP

guideline that no violations, ever, will be tolerated (which

weoperationalizedasaviolationfrequencyoflessthan0.5%).

Thus, a station such as station 25, which did not actually

reveal a violation during the 2000-2005 assessment period,

still has a pattern of MPN values that indicate it would be

highlyunlikely(3%probability)tocomplywiththestandard

ifenoughsampleswerecollected.Ontheotherhand,wecan

2.65-σc

1.04

2.44-σc

1.05

1.39

for σc> 0.65

2.61

for σce 0.65

1.98-σc

0.66

1.65-σc

0.66

1.03

for σc> 0.65

2.35

for σce 0.65

VOL. 42, NO. 13, 2008 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 9 4679

Page 5

befairlyconfidentthatstationswithahighCCwillnotexhibit

a violation, even with the collection of many more samples.

The marginal probability that a water quality sample

of size 30 (the minimum required by NSSP) will reveal an

MPN violation at a particular location can be assessed by

applying(i)theposteriordistributionofµcandσcdescribing

knowledge of uncertainty about the concentration at that

location, (ii) the log-normal distribution of c capturing

environmental variability, and (iii) the Poisson/binomial

distributionrepresentingMPNsamplingvariability.Results

(Table2)indicatethat,indeed,thosestationswitharoughly

50%orgreaterassessed(posterior)probabilityofexhibiting

a violation actually did violate at least one of the three

NSSPstandards(Table1)duringthe2000-2005assessment

period. All of the remaining stations have a probability of

violation less than 33%, and samples from these stations

did not show a violation.

4. Discussion

Given the inherent differences between actual in situ FIB

concentrationsandresultsoflaboratoryanalyses,forecasting

water quality standard compliance in response to TMDL

levels should involve a two-step process. First, water body

FIB concentrations are predicted from changes in loading,

presumably using a water quality model. Then, these

concentrations are translated into laboratory results corre-

sponding to a particular method and compared to relevant

standards.Despitetherelativetransparencyofthisapproach,

nearlyallmodelspublishedasdecisionsupporttoolsforFIB

and pathogen TMDL assessment treat the predicted in situ

concentration as if it were equivalent to the corresponding

laboratory result (49–51). Thus, an important source of

uncertaintyandpotentialbiasinmodelpredictionsisignored.

In an effort to address the above problem, we have

outlined the probabilistic expressions required to translate

FIB concentrations into either MPN- or CFU-equivalent

measurements.Usingsimulations,wefoundthattheexisting

NSSP fecal coliform standards corresponding to these two

methodsareinconsistent,meaningthatagivendistribution

of actual water body concentrations may be found to be in

violationoftheCFU-basedstandard,butnottheMPN-based

standard. This is clearly an undesirable situation.

Rather than recommend a revision to the numeric limits

of the existing MPN- and CFU-based standards, we suggest

that FIB standards be based on the actual in situ concentra-

tion. The various laboratory analysis techniques would then

be explicitly understood to be imperfect measures of this

concentration,eachwiththeirownuncertainties.Thiswould

have a number of important benefits. First, and foremost,

such standards would have an explicit connection to the

protection of human health. This is because impairment

decisionswouldbebasedonourbestunderstandingofactual

conditions in the water body, rather than on the uncertain

outcome of a laboratory analysis.

Second, a focus on in situ concentrations would lead to

a single set of universal standards which are independent of

the analytical procedure used. We demonstrated how the

distribution of the actual fecal coliform concentration in

shellfishharvestingwaterscanbeinferredfromMPNorCFU

measurementsusingaBayesianapproach.Similarinference

could be performed on any laboratory method, for any

designated use category, by incorporating method-specific

measures of uncertainty and designated use-specific stan-

dards of quality. In fact, by estimating the “second-order”

uncertainty in the concentration distribution itself, in the

form of uncertainty in µcand σc(see Figure 2), the Bayesian

approach provides an incentive for states to employ or

developmorepreciselaboratoryanalysismethods.Methods

that have a lower measurement error will lead to a greater

“confidence of compliance” at a nonviolating site than

methods that are more uncertain.

Finally,aconcentration-basedstandardwouldencourage

a more consistent and accurate treatment of FIB concentra-

tion data in models forming the basis for TMDL determina-

tions. In addition to precluding the need for modelers to

probabilisticallytranslatetheirpredictionsintounitsofMPN

or CFU, a focus on concentration would facilitate proper

loading specification and model calibration. Currently, FIB

loading rates in models are often expressed in units of MPN

or CFU per unit volume (52). However, the discrepancy

between MPN- and CFU-based determinations of FIB

concentration coming from a particular loading source and

thetrueFIBcontributionofthatsourcesuggestthatsuchan

approachisinappropriate.Infact,usingMPNorCFUvalues

as FIB loading rates will lead to model predictions that

represent neither the true FIB concentration nor MPN- or

CFU-basedestimatesofthetrueconcentration(53).Similarly,

using MPN or CFU determinations to calibrate a FIB water

qualitymodeldoesnotimplythattheendpointsofthemodel

are correct MPN or CFU values. The concentration-based

perspective adopted here provides a potential solution to

these common, but flawed, FIB water quality modeling

practices.

Our results show that a new concentration-based FIB

standard can be conveniently expressed in terms of the

mean,µc,andstandarddeviation,σc,ofthelog-transformed

concentration distribution. We gave examples of such a

standard that are compatible with the stringency imposed

by the present MPN or CFU standards. Of course, rather

thanarbitrarilydecidingtomatchoneortheotherofthese

method-based standards, it would be more logical for the

NSSP to select an in situ concentration threshold that

should not, in theory, be exceeded more than a specified

TABLE 2. Estimated Confidence of Compliance (CC), Posterior

Probability of Violating any MPN Standard, and Observed

Violations for Monitoring Stations in the Newport River

Estuary during the 2000-2005 Assessment Period

posterior probability

of size-30 sample

violating any

MPN standard

CC (%)

stationMPN CFU

violated any MPN

standard during the

2000-2005 assessment

period?

3

4

4A

4B

5A

7

8

8A

9

10

11

14A

16A

18

24

25

27A

28

29

35

41

41A

55

56

83

84

85

86

52

44

<1

<1

<1

<1

14

15

93

100

53

51

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4680 9 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 42, NO. 13, 2008

Page 6

amountoftime.Anychosensamplingplanandlaboratory

analysis technique would then provide an estimate of this

exceedance frequency, together with the uncertainty in

thisestimateexpressedasthe“confidenceofcompliance.”

NSSP would therefore also need to specify the level of

confidence required to avoid shellfish harvesting area

closure.OurresultsfortheNewportRiverEstuaryindicate

that requiring a CC of 50% or greater would not be

unreasonable. All stations with a CC greater than or equal

50% according to the MPN standard were determined to

be in compliance according to samples collected during

the 2000-2005 assessment period. A CC threshold also

protects against possible inconsistencies in the current

standards under which a water body may have an

unacceptably high (e.g., greater than 10%) probability of

violating existing NSSP standards yet, because of the

inherent limitations in using only 30 samples, yield a

compliant water sample set (e.g., stations 8, 8A, 16A, 25,

and 84).

Water quality models used for TMDL assessment should

alsobeheldtoCCstandards,thoughperhapsnotasstringent

as those applied to monitoring programs. An approach to

makingCCdeterminationsbasedonmodel-basedparameter

uncertainty has been presented previously (6), and is

consistent with the approach taken here. This allows for

straightforward translation between concentration-based

standards, water quality analysis techniques, and model

predictions, accounting for the variability and uncertainty

characteristics of each.

Acknowledgments

ThisstudywassupportedwithfundsfromtheNorthCarolina

Division of Water Quality (Contract EW05049) through the

USEPA 319 program, and through grants from the National

Science Foundation (NSF Grants DMS-0112069 and DMS-

0422400). M.E.B. was partially supported by a grant from

USEPA Office of Research and Development’s Advanced

MonitoringInitiative(AMI)PilotProjectsFocusedonGEOSS

(Global Earth Observation System of Systems). In addition,

A.D.G.isgratefultotheWaterEnvironmentFederation(WEF),

the North Carolina Association of Environmental Profes-

sionals (NCAEP), and Quantitative Environmental Analysis

(QEA),LLCforscholarshipsupport.WethankNCDENR-SSS

(particularly Patti Fowler, Andy Haines, Shannon Jenkins,

and J.D. Potts) for making their water quality monitoring

data available, and for answering questions throughout the

preparationofthismanuscript.CraigStow,SongQian,Rachel

Noble,SeanMcMahon,andConradLamonprovidedvaluable

suggestions, as did two anonymous reviewers.

Supporting Information Available

Supporting Information is available for the analysis of

alternative probability models (Section 2.1), and regression

models (Section 3.2). This information is available free of

charge via the Internet at http://pubs.acs.org.

FIGURE 2. Joint posterior probability density contour lines (solid lines) for four monitoring stations in the Newport River Estuary.

Dashed and dotted lines indicate combinations of the mean µc and standard deviation σc of the log-transformed fecal coliform

concentration distribution which violate our concentration-based standards no more than 0.5% of the time using MPN or CFU

standards as the reference. Confidences of compliance (CC) are given in the lower left of each panel for both MPN- and CFU-based

standards.

VOL. 42, NO. 13, 2008 / ENVIRONMENTAL SCIENCE & TECHNOLOGY 9 4681

Page 7

Literature Cited

(1) Smith,E.P.;Ye,K.Y.;Hughes,C.;Shabman,L.A.Statisticalassessment

of violations of water quality standards under section 303(d) of the

Clean Water Act. Environ. Sci. Technol. 2001, 35, 606–612.

(2) Houck, O. A.; The Clean Water Act TMDL Program: Law, Policy,

and Implementation, 2nd ed.; Environmental Law Institute:

Washington, DC, 2002.

(3) U.S. Environmental Protection Agency. Guidance for 2006

Assessment, Listing and Reporting Requirements Pursuant to

Sections 303(d), 305(b) and 314 of the Clean Water Act; USEPA:

Washington, DC, 2005..

(4) National Research Council. Assessing the TMDL Approach to

Water Quality Management; NRC: Washington, DC, 2001.

(5) Cooter, W. S. Clean Water Act assessment processes in relation

tochangingU.S.EnvironmentalProtectionAgencymanagement

strategies. Environ. Sci. Technol. 2004, 38, 5265–5273.

(6) Borsuk, M. E.; Stow, C. A.; Reckhow, K. H. Predicting the

frequency of water quality standard violations: A probabilistic

approach for TMDL development. Environ. Sci. Technol. 2002,

36, 2109–2115.

(7) Benham, B. L.; Baffaut, C.; Zeckoski, R. W.; Mankin, K. R.;

Pachepsky, Y. A.; Sadeghi, A. M.; Brannan, K. M.; Soupir, M. L.;

Habersack, M. J. Modeling bacteria fate and transport in

watersheds to support TMDLs. T. ASABE 2006, 49, 987–1002.

(8) U.S.EnvironmentalProtectionAgency.NationalWaterQuality

Inventory: Report to Congress (2002 Reporting Cycle); EPA 841-

R-07-001; USEPA: Washington, DC, 2007.

(9) U.S. Environmental Protection Agency. Code of Federal Regula-

tions: Title 40, Chapter 1, Part 141, 2005.

(10) NCDepartmentofEnvironmentandNaturalResources.Coastal

Recreational Waters Monitoring, Evaluation, and Notification

Rules; 15A NCAC 18A 0.3400; 2004.

(11) National Shellfish Sanitation Program. Guide for the Control of

Molluscan Shellfish; 2005.

(12) Thomas,H.A.StatisticalAnalysisofColiformData.SewageInd.

Wastes 1955, 27, 212–222.

(13) Ott, W. Environmental Statistics and Data Analysis; Lewis

Publishers: Boca Raton, FL, 1995.

(14) Shirmohammadi, A.; Chaubey, I.; Harmel, R. D.; Bosch, D. D.;

Munoz-Carpena,R.;Dharmasri,C.;Sexton,A.;Arabi,M.;Wolfe,

M. L.; Frankenberger, J.; Graff, C.; Sohrabi, T. M. Uncertainty

in TMDL models. T. ASABE 2006, 49, 1033–1049.

(15) Cochran, W. G. Estimation of Bacterial Densities by Means of

the ‘Most Probable Number’. Biometrics 1950, 6, 105–116.

(16) McCrady,M.H.TheNumericalInterpretationofFermentation

Tube Results. J. Infect. Dis. 1915, 17, 183–212.

(17) deMan,J.C.MPNTablesForMoreThanOneTest.Eur.J.Appl.

Microbiol. 1977, 4, 307–316.

(18) Klee, A. J. A Computer Program For the Determination of Most

ProbableNumberandItsConfidenceLimits.J.Microbiol.Meth.

1993, 18, 91–98.

(19) Dufour, A. P.; Cabelli, V. J. Membrane-Filter Procedure for

Enumerating Component Genera of Coliform Group in Sea-

water. Appl. Microbiol. 1975, 29, 826–833.

(20) Dufour, A. P.; Strickland, E. R.; Cabelli, V. J. Membrane-Filter

Method for Enumerating Escherichia coli. Appl. Environ. Mi-

crobiol. 1981, 41, 1152–1158.

(21) Rippey, S. R.; Adams, W. N.; Watkins, W. D. Enumeration of

Fecal-Coliforms and Escherichia coli in Marine and Estuarine

Waters - an Alternative to the APHA-MPN Approach. J. Water

Pollut. Control Fed. 1987, 59, 795–798.

(22) Esham, E. C.; Sizemore, R. K. Evaluation of two techniques:

mFCandmTECfordeterminingdistributionsoffecalpollution

insmall,NorthCarolinatidalcreeks.WaterAirSoilPollut.1998,

106, 179–197.

(23) Rompre ´, A.; Servais, P.; Baudart, J.; de Roubin, M.-R.; Laurent,

P. Detection and enumeration of coliforms in drinking water:

currentmethodsandemergingapproaches.J.Microbiol.Meth.

2002, 49, 31–54.

(24) Refsgaard,J.C.;vanderSluijs,J.P.;Hojberg,A.L.;Vanrolleghem,P.A.

Uncertaintyintheenvironmentalmodellingprocess-Aframework

and guidance. Environ. Modell. Software 2007, 22, 1543–1556.

(25) Milly,P.C.;Betancourt,J.;Falkenmark,M.;Hirsch,R.M.;Kundzewicz,

Z.W.;Lettenmaier,D.P.;Stouffer,R.J.StationarityIsDead:Whither

Water Management. Science 2008, 319, 573–574.

(26) Greenwood, M.; Yule, G. U. On the Statistical Interpretation of

Some Bacteriological Methods Employed inWater Analysis. J.

Hyg. 1917, 16, 36–54.

(27) Thomas, H. A. On Averaging Results of Coliform Tests. Boston

Soc. Civil Engrs. J. 1952, 39, 253–270.

(28) Hurley, M. A.; Roscoe, M. E. Automated Statistical Analysis of

Microbial Enumeration by Dilution Series. Appl. Environ.

Microbiol. 1983, 55, 159–164.

(29) Haas,C.N.Estimationofmicrobialdensitiesfromdilutioncount

experiments. Appl. Environ. Microbiol. 1989, 55, 1934–1942.

(30) Beliaeff, B.; Mary, J.-Y. The Most Probable Number Estimate

and Its Confidence-Limits. Water Res. 1993, 27, 799–805.

(31) McBride, G. B.; McWhirter, J. L.; Dalgety, M. H. Uncertainty in

mostprobablenumbercalculationsformicrobiologicalassays.

J. AOAC Int. 2003, 86, 1084–1088.

(32) Eisenhart, C.; Wilson, P. W. Statistical Methods and Control in

Bacteriology. Bacteriol. Rev. 1943, 7, 57–137.

(33) Pipes, W. O.; Ward, P.; Ahn, S. H. Frequency Distributions for

Coliform Bacteria in Water. J. Am. Water Works Assoc. 1977, 69,

664–668.

(34) El-Shaarawi, A. H.; Esterby, S. R.; Dutka, B. J. Bacterial Density

in Water Determined by Poisson or Negative Binomial Distri-

butions. Appl. Environ. Microbiol. 1981, 41, 107–116.

(35) Christian, R. R.; Pipes, W. O. Frequency Distributions of

Coliforms in Water Distribution Systems. Appl. Environ. Mi-

crobiol. 1983, 45, 603–609.

(36) Haas, C. N.; Heller, B. Test of the validity of the Poisson

assumptionforanalysisofMost-Probable-Numberresults.Appl.

Environ. Microbiol. 1988, 54, 2996–3002.

(37) Teunis, P. F. M.; Medema, G. J.; Kruidenier, L.; Havelaar, A. H.

AssessmentoftheriskofinfectionbyCryptosporidiumorGiardia

in drinking water from a surface water source. Water Res. 1997,

31, 1333–1346.

(38) Russek, E.; Colwell, R. R. Computation of Most Probable

Numbers. Appl. Environ. Microbiol. 1983, 45, 1646–1650.

(39) Ihaka, R.; Gentleman, R. R: A Language for Data Analysis and

Graphics. J. Comput. Graph. Stat. 1996, 5, 299–314.

(40) Reckhow, K. H.; Arhonditsis, G. B.; Kenney, M. A.; Hauser, L.;

Tribo,J.;Wu,C.;Elcock,K.J.;Steinberg,L.J.;Stow,C.A.;McBride,

S. J. A predictive approach to nutrient criteria. Environ. Sci.

Technol. 2005, 39, 2913–2919.

(41) U.S. Environmental Protection Agency. Guidelines for the

Preparation of the Comprehensive State Water Quality Assess-

ments(305(b)Reports)andElectronicUpdates:ReportContents;

Technical Report; USEPA: Washington, DC, 1997.

(42) Berry, D. A. Statistics: a Bayesian Perspective; Duxbury Press:

Belmont, CA, 1996.

(43) Bolstad, W. M. Introduction to Bayesian Statistics; Wiley-

Interscience: Hoboken, NJ, 2004.

(44) Gelman, A. J.; Carlin, J. B.; Stern, H. S.; Rubin, D. B. Bayesian

Data Analysis; Chapman & Hall/CRC: Boca Raton, FL, 2004.

(45) Gelman, A. J. Prior distributions for variance parameters in

hierarchical models (Comment on Article by Browne and

Draper). Bayesian Anal. 2006, 1, 515–534.

(46) Van Dongen, S. Prior specification in Bayesian statistics: Three

cautionary tales. J. Theor. Biol. 2006, 242, 90–100.

(47) Qian, S. S.; Reckhow, K. H. Combining model results and

monitoring data for water quality assessment. Environ. Sci.

Technol. 2007, 41, 5008–5013.

(48) Lunn,D.J.;Thomas,A.;Best,N.G.;Spiegelhalter,D.J.WinBUGS

- A Bayesian modelling framework: Concepts, structure, and

extensibility. Stat. Comput. 2000, 10, 325–337.

(49) Shen,J.;Sun,S.-C.;Wang,T.-P.Developmentofthefecalcoliform

total maximum daily load using Loading Simulation Program

C++andtidalprismmodelinestuarineshellfishgrowingareas:

A case study in the Nassawadox coastal embayment, Virginia.

J. Environ. Sci. Health, A 2005, 40, 1791–1807.

(50) U.S.EnvironmentalProtectionAgency.DecisionRationaleLetter:

Total Maximum Daily Load for Shellfish Harvest and Primary

Contact Use Impairements in the Poquoson River, Back Creek

andBackRiverWatersheds,YorkCounty,Virginia;USEPA,2006.

(51) U.S.EnvironmentalProtectionAgency.DecisionRationaleLetter:

Total Maximum Daily Load for Shellfish Harvest and Primary

Contact Use Impairements in the Onancock Creek, Locklies and

Mill Creek Watersheds, Accomack and Middlesex Counties,

Virginia; USEPA, 2006.

(52) Alderisio, K. A.; DeLuca, N. Seasonal enumeration of fecal

coliform bacteria from the feces of ring-billed gulls (Larus

delawarensis) and Canada geese (Branta canadiensis). Appl.

Environ. Microbiol. 1999, 65, 5628–5630.

(53) Sanders, B. F.; Arega, F.; Sutula, M. Modeling the dry-weather

tidal cycling of fecal indicator bacteria in surface waters of an

intertidal wetland. Water Res. 2005, 39, 3394–3408.

ES703144K

4682 9 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 42, NO. 13, 2008