U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2008–5028 

Prepared in cooperation with the Michigan Department of Environmental Quality 

Environmental Factors and Flow Paths Related 

to Escherichia coli Concentrations at Two Beaches 

on Lake St. Clair, Michigan, 2002–2005 
Cover. Metropolitan Beach near Mt. Clemens, Michigan. (Photograph by Lisa R. Fogarty, U.S. Geological Survey) 
Environmental Factors and Flow Paths 

Related to Escherichia coli Concentrations 

at Two Beaches on Lake St. Clair, 

Michigan, 2002–2005 

By David J. Holtschlag, Dawn Shively, Richard L. Whitman, Sheridan K. Haack, 

and Lisa R. Fogarty 

Prepared in cooperation with the Michigan Department of Environmental Quality 

Scientific Investigations Report 2008–5028

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

DIRK KEMPTHORNE, Secretary

U.S. Geological Survey

Mark D. Myers, Director

U.S. Geological Survey, Reston, Virginia: 2008

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Although this report is in the public domain, permission must be secured from the individual copyright owners to 

reproduce any copyrighted materials contained within this report.

Suggested citation: 

Holtschlag, D.J., Shively, Dawn, Whitman, R.L., Haack, S.K., and Fogarty, L.R., 2008, Environmental factors and flow 

paths related to Escherichia coli concentrations at two beaches on Lake St. Clair, Michigan, 2002–2005: U.S. Geological Survey Scientific Investigations Report 2008–5028, 38 p. 
iii

Contents

Abstract ...........................................................................................................................................................1

Introduction.....................................................................................................................................................1

Purpose and Scope ..............................................................................................................................2

Site Description .....................................................................................................................................2

Data Used in the Study ........................................................................................................................2

Escherichia coli Concentrations ...............................................................................................2

Ancillary Environmental Data ....................................................................................................5

Analytical Approach and Methods .............................................................................................................5

Regression Models ...............................................................................................................................6

Linear Regression ........................................................................................................................6

Regression Trees .........................................................................................................................7

Logistic Regression .....................................................................................................................8

Hydrodynamic Modeling......................................................................................................................8

RMA2 Hydrodynamic Code ........................................................................................................8

Two-Dimensional Hydrodynamic Model of Lake St. Clair .....................................................9

Particle Tracking ..........................................................................................................................9

Relation of Environmental Factors to Escherichia coli Concentrations .............................................10

Statistical Distributions of Escherichia coli Concentration and Effects of 

Wind Direction .......................................................................................................................10

Linear Regression Results ........................................................................................................14

Memorial Beach ...............................................................................................................14

Metropolitan Beach .........................................................................................................15

Regression-Tree Results ...........................................................................................................16

Memorial Beach ...............................................................................................................16

Metropolitan Beach .........................................................................................................18

Logistic Regression Results .....................................................................................................18

Memorial Beach ...............................................................................................................18

Metropolitan Beach .........................................................................................................19

Relation of Flow Paths to Escherichia coli Concentrations .........................................................20

Flow Simulation ..........................................................................................................................20

Particle Tracking ........................................................................................................................21

Steady-State Simulations ................................................................................................21

Transient Simulations .......................................................................................................24

Additional Analysis of Period Covered by Transient 

Hydrodynamic Simulation .........................................................................27

Suggestions for Future Studies .................................................................................................................30

Summary and Conclusions .........................................................................................................................31

Acknowledgments .......................................................................................................................................33

References Cited..........................................................................................................................................33

Appendix 1. Equations and Detail Used in the Linear and Logistic Regression Models ..............37

Linear Regression ...............................................................................................................................37

Logistic Regression ............................................................................................................................37
iv

Figures

 1. Map showing Lake St. Clair study area ....................................................................................3

 2–10. Graphs showing: 

2. Time-series plot of the daily geometric mean E. coli concentrations at 

Memorial Beach on Lake St. Clair, Mich., 1998–2006 ....................................................4

3. Time-series plot of the daily geometric mean of E. coli concentrations at 

Metropolitan Beach on Lake St. Clair, Mich., 1998–2006 ..............................................4

 4. Statistical distributions of E. coli concentrations at (A) Memorial and 

(B) Metropolitan Beaches on Lake St. Clair, Mich., 2002–05 ......................................11

 5. Relation between wind conditions and E. coli concentrations at 

Memorial Beach on Lake St. Clair, Mich., 2002–05 ......................................................12

 6. Distributions of Macomb County Health Department E. coli concentrations 

at Memorial Beach by wind direction on Lake St. Clair, Mich., 2002–05 ..................12

 7. Relation between wind conditions and E. coli bacteria concentrations at 

Metropolitan Beach on Lake St. Clair, Mich., 2002–05 ................................................13

 8. Distributions of Macomb County Health Department E. coli data, by 

wind direction for Metropolitan Beach on Lake St. Clair, Mich., 2002–05 ...............13

 9. Relation between measured and main-effect model estimates of E. coli

concentrations at Memorial Beach on Lake St. Clair, Mich., 2002–05 .....................15

 10. Relation between measured and linear regression estimates of E. coli 

concentrations at Metropolitan Beach on Lake St. Clair, Mich., 2002–05 ...............16

 11. Schematic showing the regression tree for estimating log10 E. coli 

concentrations for Memorial Beach on Lake St. Clair, Mich., 2002–05 .............................17

 12–15. Graphs showing: 

 12. Regression tree root mean square error for estimation and prediction 

for Memorial Beach on Lake St. Clair, Mich. ................................................................17

 13. Receiver operating characteristics (ROC) curves for logistic equations 

predicting the probability of E. coli concentrations exceeding 

300 MPN/100 mL at Memorial Beach on Lake St. Clair, Mich., 2002–05 ...................19

 14. Receiver operating characteristics (ROC) curve for logistic equations 

predicting the probability of E. coli concentrations exceeding 

300 MPN/100 mL at Metropolitan Beach on Lake St. Clair, Mich., 2002–05 .............20

 15. Wind characteristics, by sector, on Lake St. Clair based on data from 

Lake St. Clair Monitoring Station LSCM4 in Michigan ...........................................21

 16–19. Maps showing: 

 16. Simulated reverse particle paths from Memorial Beach for selected 

wind sectors on Lake St. Clair, Mich. .............................................................................22

 17. Simulated reverse particle paths from Metropolitan Beach for selected 

wind sectors on Lake St. Clair, Mich. .............................................................................23

 18. Reverse particle tracks for selected sampling events in 2004 starting 

from Memorial Beach on Lake St. Clair, Mich. .............................................................25

 19. Reverse particle tracks for selected sampling events in 2004 starting 

from Metropolitan Beach on Lake St. Clair, Mich. .......................................................26
v

 20–23. Graphs showing: 

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

foot (ft) 0.3048 meter (m)

meter (m) 3.281 foot (ft)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2

) 259.0 hectare (ha)

square mile (mi2

) 2.590 square kilometer (km2) 

Volume

milliliter (mL) 0.06102 cubic inch (in3

)

Velocity and flow rate

mile per hour (mi/h) 1.609 kilometer per hour (km/h)

cubic foot per second (ft3

/s) 0.02832 cubic meter per second (m3/s)
vi

Temperature in degrees Celsius (°C) may be converted to degrees Fahrenheit (°F) as follows:

°F=(1.8×°C)+32

Temperature in degrees Fahrenheit (°F) may be converted to degrees Celsius (°C) as follows:

°C=(°F-32)/1.8

Vertical coordinate information is referenced to the International Great Lakes Datum of 1985 

(IGLD 85). This datum is a dynamic height system for measuring elevation, which varies with 

the local gravitational force, rather than an orthometric system, which provides an absolute 

distance above a fixed point. The primary reason for adopting a dynamic height system within 

the Great Lakes is to provide an accurate measurement of potential hydraulic head. The 

reference zero for IGLD 85 is a tide gage at Rimouski, Quebec, which is located near the outlet 

of the Great Lakes-St. Lawrence River system. The mean water level at the Rimouski, Quebec 

gage approximates mean sea level. 

Elevation, as used in this report, refers to distance above the vertical datum. 

Horizontal coordinate information is referenced to the Michigan South (2113) State Plane 

Coordinate System of 1983 (SPCS 83).

Concentrations of chemical constituents in water are given in milligrams per liter (mg/L). 

Concentrations of bacteria are given in colony-forming units per 100 milliliters (CFU/100 mL) or 

most probable number per 100 milliliters (MPN/100 mL). 

Turbidity is given in Nephelometric Turbidity Units (NTU).

Abbreviations used in this report

ANGB Air National Guard Base

AUROC Area under the receiver operating characteristic

MDEQ Michigan Department of Environmental Quality

MPN Most probable number

NOAA National Oceanic and Atmospheric Administration

NDBC National Data Buoy Center

RMSE-E Root-mean square error

RMSE-P Root-mean square error prediction

ROC Receiver operating characteristics

SMS Surface-water modeling system

TMDL Total maximum daily load

USGS U.S. Geological Survey 
Abstract

Regression analyses and hydrodynamic modeling were 

used to identify environmental factors and flow paths associated with Escherichia coli (E. coli) concentrations at Memorial and Metropolitan Beaches on Lake St. Clair in Macomb 

County, Mich. Lake St. Clair is part of the binational waterway 

between the United States and Canada that connects Lake 

Huron with Lake Erie in the Great Lakes Basin. Linear regression, regression-tree, and logistic regression models were 

developed from E. coli concentration and ancillary environmental data. 

Linear regression models on log10 E. coli concentrations 

indicated that rainfall prior to sampling, water temperature, 

and turbidity were positively associated with bacteria concentrations at both beaches. Flow from Clinton River, changes in 

water levels, wind conditions, and log10 E. coli concentrations 

2 days before or after the target bacteria concentrations were 

statistically significant at one or both beaches. In addition, 

various interaction terms were significant at Memorial Beach. 

Linear regression models for both beaches explained only 

about 30 percent of the variability in log10 E. coli concentrations. 

Regression-tree models were developed from data from 

both Memorial and Metropolitan Beaches but were found to 

have limited predictive capability in this study. The results 

indicate that too few observations were available to develop 

reliable regression-tree models. 

Linear logistic models were developed to estimate the 

probability of E. coli concentrations exceeding 300 most 

probable number (MPN) per 100 milliliters (mL). Rainfall 

amounts before bacteria sampling were positively associated 

with exceedance probabilities at both beaches. Flow of Clinton 

River, turbidity, and log10 E. coli concentrations measured 

before or after the target E. coli measurements were related to 

exceedances at one or both beaches. The linear logistic models 

were effective in estimating bacteria exceedances at both 

beaches. A receiver operating characteristic (ROC) analysis 

was used to determine cut points for maximizing the true positive rate prediction while minimizing the false positive rate. 

A two-dimensional hydrodynamic model was developed 

to simulate horizontal current patterns on Lake St. Clair in 

response to wind, flow, and water-level conditions at model 

boundaries. Simulated velocity fields were used to track 

hypothetical massless particles backward in time from the 

beaches along flow paths toward source areas. Reverse particle 

tracking for idealized steady-state conditions shows changes 

in expected flow paths and traveltimes with wind speeds and 

directions from 24 sectors. The results indicate that three to 

four sets of contiguous wind sectors have similar effects on 

flow paths in the vicinity of the beaches. In addition, reverse 

particle tracking was used for transient conditions to identify 

expected flow paths for 10 E. coli sampling events in 2004. 

These results demonstrate the ability to track hypothetical 

particles from the beaches, backward in time, to likely source 

areas. This ability, coupled with a greater frequency of bacteria sampling, may provide insight into changes in bacteria 

concentrations between source and sink areas. 

Introduction

The Macomb County Health Department (MCHD) 

began monitoring bathing beaches for bacteria in 1948 to help 

ensure a healthy recreational experience for people who visit 

area beaches (Macomb County Health Department, 2006). 

Michigan’s water-quality standard (R323.1062 Rule 62) for 

total body contact recreation states that no site should have an 

Escherichia coli (E. coli) concentration that exceeds 130 colony forming units per 100 milliliters (CFU/100 mL) of water 

as a geometric mean of all samples collected over a 30-day 

period. According to this standard, the daily geometric mean 

of three or more samples should not exceed 300 CFU/100 mL 

of water. If beach waters exceed these standards, an advisory 

or closure is implemented by the MCHD until standards are 

met. 

Environmental Factors and Flow Paths Related to 

Escherichia coli Concentrations at Two Beaches on 

Lake St. Clair, Michigan, 2002–2005

David J. Holtschlag, Dawn Shively, Richard L. Whitman, Sheridan K. Haack, and Lisa R. Fogarty
2 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

Public beaches along Lake St. Clair, in Macomb County 

(fig. 1), are intermittently closed when E. coli concentrations exceed the single-day Michigan recreational waterquality standard (referred to as the “recreational water-quality 

standard” hereafter). According to MCHD data from 1998 to 

2005, this standard was exceeded at two beaches along the 

lake, Memorial Beach and Metropolitan Beach (fig. 1). It was 

exceeded 32 times out of 360 daily geometric mean samples 

(8.9 percent) at Memorial Beach and 23 times out of 367 samples (6.3 percent) at Metropolitan Beach. Because there is 

typically a 24-hour delay in obtaining test results, beach 

closings based on E. coli tests are not completely effective in 

preventing exposures of concern or allowing contact to resume 

when conditions are considered safe. 

Uncertainties in source areas of microbial pathogens 

and indicator bacteria contribute to the difficulty of managing beach access to maximize public safety and to minimize 

the public inconvenience and economic losses associated with 

unnecessary or prolonged beach closings. The U.S. Geological 

Survey (USGS), in cooperation with the Michigan Department 

of Environmental Quality (MDEQ), began a study in 2006 

to identify environmental factors and flow paths related to 

E. coli concentrations at Memorial and Metropolitan Beaches. 

Purpose and Scope

This report presents environmental factors that are related 

to concentrations of E. coli at Memorial and Metropolitan 

Beaches on Lake St. Clair. Linear regression, regression-tree, 

and logistic regression models were used to identify these 

environmental factors. In addition, this report presents flow 

paths on Lake St. Clair that were developed using a hydrodynamic and particle-tracking model. These flow paths can be 

used to help identify potential source areas of E. coli to the 

two beaches. The statistical and hydrodynamic models can be 

used in combination by regulatory agencies for the development of Total Maximum Daily Loads (TMDLs) for E. coli at 

both beaches. 

Site Description

Memorial and Metropolitan Beaches are on the west 

shore of Lake St. Clair in Macomb County, Mich. (fig. 1). 

Memorial Beach is about 2.5 mi north of St. Clair Shores, 

Mich., and about 2.6 mi south of the mouth of the Clinton 

River Cutoff Canal on Lake St. Clair. Metropolitan Beach is 

on the northern edge of L’anse Creuse Bay, along a peninsula 

that carries water from the Clinton River to Lake St. Clair. 

Metropolitan Beach is about 2.5 mi east of the Cutoff Canal 

and about 3.0 mi south of the mouth of Clinton River. The 

beaches provide recreational opportunities to Detroit-area residents and stimulate considerable economic activity. Changing 

current patterns on Lake St. Clair bring water and associated 

materials and organisms to the beaches from various source 

areas contributing to the lake. 

Lake St. Clair is a binational body of water within the 

United States and Canada that is part of the Great Lakes 

Basin. The St. Clair River, which has a drainage area of 

about 222,400 mi2, delivers water to Lake St. Clair at a rate 

of about 185,000 ft3/s from Lake Huron. In comparison, local 

tributaries—including the Clinton River and the Cutoff Canal 

in the United States, and the Thames and Sydenham Rivers in 

Canada—and precipitation contribute a small fraction of the 

flow of the St. Clair River directly to Lake St. Clair. Water in 

the lake is lost by evaporation and withdrawals for water supply or discharges through the Detroit River into Lake Erie. 

Lake St. Clair has a surface area of about 430 mi2 and an 

average depth of 10 ft (Great Lakes Informational Network, 

2006). The lake, which has a length of 26 mi and a breadth 

of 24 mi, contains about 1 mi3 of water. An 800-ft-wide by 

25-ft-deep channel is maintained through Lake St. Clair to 

facilitate navigation. Most of the navigational channel lies 

within the United States. Public and private beaches are 

located along the 130-mi-long shoreline of mainland areas. 

Data Used in the Study

This report was developed on the basis of existing 

E. coli concentration and ancillary environmental data 

collected and analyzed by the Macomb County Public Health 

Department (MCHD), in cooperation with the Michigan 

Department of Environmental Quality (MDEQ). Additional 

meteorological and water-level data were obtained from the 

stations at Selfridge Air National Guard Base (ANGB) and 

the Lake St. Clair, Mich., weather station (LSCM4). Station 

LSCM4 is operated by the National Data Buoy Center 

(NDBC). 

Escherichia coli Concentrations

The MCHD and MDEQ measured E. coli concentrations 

at Memorial and Metropolitan Beaches (figs. 2 and 3). E. coli

sampling generally occurred on selected weekday mornings 

from April through September. Beginning in 1998, MCHD 

samples were generally obtained twice per week on Mondays and Wednesdays, although supplemental samples were 

obtained occasionally. MCHD sampling has been ongoing but 

MDEQ sampling, which generally occurred once per week 

on Tuesdays or Thursdays, began in 2003 and ended in 2004. 

Each E. coli concentration reported herein represents the daily 

geometric mean of six samples collected at about the same 

time. 
Introduction 3

Figure 1. Lake St. Clair study area. 

13,500,000 13,550,000 13,600,000 13,650,000

300,000

350,000

400,000

450,000

Middle

Channel

South Channel

Clinton River

Clinton River 

Cutoff Canal

Thames River

Chenal

Ecarte

Bassett Channel

Walpole Island

Metropolitan Beach

Memorial

Beach

St. Clair Shores, Mich.

Algonac, Mich.

Fort Wayne, 

Mich.

North

Channel Sydenham

River

Beaches and canal

Finite-element mesh

EASTING, IN INTERNATIONAL FEET WITHIN THE MSPCS 83 (SOUTHERN ZONE)

NORTHING, IN INTERNATIONAL FEET WITHIN THE MSPCS 83 (SOUTHERN ZONE)

EXPLANATION

Navigational channel

United States

Canada

Lake St. Clair

Meteorological or 

water-level station

Selfridge

Air National 

Guard Base

LAKE

MICHIGAN

LAKE

HURON

LAKE SUPERIOR

MICHIGAN

CANADA

Lake

St. Clair

STUDY AREA

LAKE 

ERIE USA

MICHIGAN,

UNITED STATES

ONTARIO,

CANADA

Detroit

River

St. Clair

River

0 10,000 30,000 50,000 feet

Anchor Bay

L’anse

Creuse

Bay

LSCM4 Weather Station
4 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

Figure 2. Time-series plot of the daily geometric mean E. coli concentrations at Memorial Beach on Lake St. Clair, Mich., 1998–2006. 

(MCHD, Macomb County Health Department; MDEQ, Michigan Department of Environmental Quality).

Figure 3. Time-series plot of the daily geometric mean of E. coli concentrations at Metropolitan Beach on Lake St. Clair, Mich., 

1998–2006. (MCHD, Macomb County Health Department; MDEQ, Michigan Department of Environmental Quality).

1/1/98 1/1/99 1/1/00 1/1/01 1/1/02 1/1/03 1/1/04 1/1/05 1/1/06

DATE (MONTH/DAY/YEAR)

1

10

100

1,000

10,000

DAILY GEOMETRIC MEAN E. COLI, IN MOST PROBABLE NUMBER PER 100 MILLILITERS 

Michigan daily geometic mean total body contact water-quality standard

MCHD Samples

MDEQ Samples

1/1/98 1/1/99 1/1/00 1/1/01 1/1/02 1/1/03 1/1/04 1/1/05 1/1/06

DATE (MONTH/DAY/YEAR)

1

10

100

1,000

10,000

DAILY GEOMETRIC MEAN E. COLI, IN MOSTPROBABLE NUMBER PER 100 MILLILITERS 

Michigan daily geometic mean total body contact water-quality standard

MCHD Samples

MDEQ Samples
Analytical Approach and Methods 5

In general, E. coli concentrations reported by MDEQ 

were lower than those reported by MCHD during the same 

periods. The reasons for these apparent differences are not 

known. Although sample locations were approximately the 

same, depth of sampling or sample-collection techniques may 

have varied by agency and person collecting the sample. In 

addition, MCHD and MDEQ use somewhat different analytical protocols to determine E. coli concentrations. MCHD 

samples are analyzed by means of the Colilert reagent system 

developed by IDEXX Laboratories (Westbrook, Maine), which 

quantifies E. coli concentrations by use of the most probable number (MPN) per 100 mL of sample water (Macomb 

County Health Department, 2007). In contrast, the MDEQ 

used membrane filtration (Standard Method 9222G in 2003, 

and U.S. Environmental Protection Agency method 1103.1 

in 2004), a technique documented by the U.S. Environmental 

Protection Agency (2006), which quantifies E. coli concentrations by the number of colony-forming units (CFU) per 

100 mL of water. Eckner (1998) concludes that MPN concentrations are comparable to CFU concentrations; therefore, 

for the purposes of this report, 300 MPN/100 mL is used as a 

criterion equivalent to the daily Michigan recreational waterquality standard. 

Owing to concerns about possible systematic differences 

in E. coli concentrations measured by MCHD and MDEQ, 

the MCHD data were used in this report because they had 

been collected over a longer period and at more frequent 

intervals than the MDEQ data. Use of the MCHD dataset was 

restricted to April 2002 to September 2005 so that bacteria 

concentrations could be related to wind conditions on Lake 

St. Clair. In particular, the NOAA weather station LSCM4 on 

Lake St. Clair, which is near the middle of the lake just off 

the navigational channel at north latitude 42o27’54” and west 

longitude 82o45’18,” began reporting meteorological data in 

November 2001. 

Ancillary Environmental Data

Numerous environmental factors may be associated 

with variations in E. coli concentrations. Environmental data 

considered in this study included simultaneous water-quality 

measurements made by MCHD with E. coli concentrations at 

the two beaches. These measurements included water temperature (reported in degrees Fahrenheit), pH, specific conductance 

(reported in millisiemens per centimeter), turbidity (reported 

in nephelometric turbidity units, NTU), dissolved oxygen 

(reported in milligrams per liter (mg/L)), and salinity, which 

is computed as a function of specific conductance and water 

temperature and reported in parts per thousand (PPT). MCHD 

also furnished data on E. coli concentrations for the Clinton 

River (station identifier U–41–Clinton River) and the Clinton 

River Spillway (station identifier Z–42–Spillway). 

In addition to E. coli data and simultaneous water-quality 

measurements, continual measurements of meteorological, 

water-level, and flow data were available from nearby gaging 

stations operated by NOAA, Environment Canada, and the 

USGS. Hydrometeorological data were retrieved from Selfridge Air National Guard Base (ANGB) in Mount Clemens, 

Mich., which is operated by NOAA’s National Climatic Data 

Center as station 725377 (about 3 and 8 mi from Metropolitan 

and Memorial Beaches, respectively; fig. 1). Precipitation data 

were used to compute 24-, 48-, and 72-hour totals (in inches) 

preceding sampling for E. coli at the beaches. In addition, air 

temperature, wind velocity, atmospheric pressure, dewpoint, 

sky cover, and cloud ceiling data were obtained from NOAA. 

The National Data Buoy Center (NDBC) furnished wind 

speed and direction data for station LSCM4 on Lake St. Clair 

(approximately 7.5 mi offshore, fig. 1), which were used to 

compute the average 24-hour wind speeds and directions 

prior to E. coli sampling. In addition, sea-surface temperature 

(sometimes referred to in this report as SeaTemp), in degrees 

Celsius, barometric pressure, and dewpoint data were collected 

at station LSCM4. Environment Canada furnished data on 

lake-surface temperature, wave height, maximum wave height, 

and wave period from its buoy C45147 on Lake St. Clair. 

Missing values for the lake-surface temperature for NDBC 

station LSCM4 were estimated by linear regression, using 

lake-surface temperatures from Environment Canada Station 

C45147. The NOAA water-level gaging station at St. Clair 

Shores, Mich. (station 9034052) was the source of lake-level 

data. Water levels referenced to time t, WLt, are measured at 

6-minute intervals, Δ = 6 minutes, at the NOAA gaging station on Lake St. Clair at St. Clair Shores, Mich. A smoothed 

water-level series, SWLt was computed from the water-level 

data by use of a moving-average filter. The temporal averaging was used to dampen erratic fluctuations in water levels to 

help distinguish subtle changes in average water levels over 

short intervals. The smoothed water levels were computed 

as SWL WL t t i i i

= × å + =-f 4

4

 where the 9 f coefficients are 

symmetrical, such that f f -i i = . Coefficients fi= ±0 1 , ,...±4 »

{0.2042, 0.1802, 0.1238, 0.0663, 0.0276}. Changes in 

the smoothed water levels, DSWLt , were computed as

D D SWL SWL SWL t t = - t t - × 10 . The USGS provided streamflow (sometimes referred to in this report as ClintonFlow)

information at the streamgage on Clinton River at Mount Clemens, Mich. (station 04165500, about 10 river miles from the 

mouth of the river at Lake St. Clair), in cubic feet per second.

Analytical Approach and Methods 

An exploratory data analysis was done to evaluate the 

distributions of E. coli concentrations measured at Memorial 

and Metropolitan Beaches by MCHD, the relation between 

E. coli concentrations and wind velocity, and simple correlations between E. coli concentrations and other environmental 

factors. The results of this analysis were the basis for regression modeling, which attempted to identify environmental 

factors that are statistically associated with the E. coli concentrations or likelihoods of concentrations exceeding a specified 

threshold. Three types of regression models were developed 
6 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

and evaluated: linear regression, regression trees, and logistic 

regression models. The three types of models are based on different sets of statistical equations and constitute three unique 

ways to evaluate the data. Each is discussed individually in the 

following section. Briefly, linear regression models identify 

sets of environmental factors associated with different level of 

E. coli concentrations. With robust data sets, linear regression 

models have been used to statistically predict when E. coli

concentrations are likely to exceed water-quality standards, 

given a set an explanatory variables (U.S. Environmental 

Protection Agency, 1999; Olyphant and Whitman, 2004; Nevers and Whitman, 2005; Francy and others, 2006). However, 

a required assumption for linear regression models is that the 

data are linear, which may not always be the case. Therefore, 

regression-tree analysis, like linear regression, was used to 

evaluate the data based a series of if-then statements that help 

identify a series of environmental variables associated with 

higher E. coli concentrations without the assumption of linearity. Finally, logistic regression models were used to estimate 

the probability of E. coli concentrations exceeding the recreational water-quality standard on the basis of selected environmental variables. Estimating probabilities is an alternative to 

estimating E. coli concentrations and comparing the estimates 

to the standard; the logistic model describes the odds of 

exceeding the standard with a unit change in the corresponding environmental variable, given that the other environmental 

variables do not change.

An existing two-dimensional (2D) hydrodynamic model 

(Holtschlag and Koschik, 2004) was modified to simulate flow 

patterns on Lake St. Clair near Memorial and Metropolitan 

Beach areas. Steady-state simulations were used to identify 

idealized current patterns near the beaches for selected wind 

conditions. Transient simulations were used to identify current patterns likely associated with specific bacteria-sampling 

events. Simulation results were the basis for particle-tracking 

analyses that were used to identify expected flow paths of 

water prior to bacteria sampling. The framework for the 

regression and hydrodynamic models is discussed in the following section of this report and the results of the models are 

discussed in the subsequent sections. 

Regression Models

Regression models describe a statistical association 

between E. coli magnitudes and readily measurable environmental (explanatory) factors. These statistical associations 

do not ensure causal relations. Parameter values in regression 

equations, however, are commonly interpreted as indicators of 

possible physical relation between explanatory and response 

variables. In particular, positive parameters may indicate a 

direct relation between explanatory and response variables, 

whereas negative parameters may indicate an inverse relation. 

Any interpretation of parameter magnitudes is problematic 

because explanatory variables in these equations are often 

related or dependent on each other; therefore, distinguishing 

their individual contributions is difficult. 

Linear Regression

Linear regression provides a statistical model for estimating the continuous variation of E. coli concentrations as 

a function of environmental factors (explanatory variables). 

General linear regression equations are described by Draper 

and Smith (1998), and further explanations on the equations 

used for these models can be found in sections following. 

Statistics summarizing the regression help assess model 

performance. Key statistics include the following:

•	 Sum of squared errors, SSE = - (yyyy ˆ) (¢ - = ˆ) , e e ¢ × 

•	 Total sum of squares, SST = - ( )( ) yyyy ¢ - ,

where y is the mean of the response variable, and 

•	 Model sum of squares, SSM = SST-SSE. 

•	 Degrees of freedom for the error, dfe = n-p-1, and 

•	 Model degrees of freedom, dfm = p. 

•	 Mean square error, MSE = SSE/dfe ,

•	 Model mean square, MSM = SSM/dfm, and 

•	 Root mean square error, RMSE = MSE.

In addition, an F statistic, computed by dividing the 

MSM by the MSE, characterizes the statistical significance 

of the model. On the basis of the F probability distribution, 

a probability (p-value) is computed with the F statistic, and 

the degrees of freedom in the model and error components to 

assess the likelihood that the null hypothesis (that all model 

parameters are zero) is true. A small p-value, commonly less 

than 0.05, is used to reject the null hypothesis, thereby accepting the alternative hypothesis that the regression model is 

statistically significant. 

The fraction of the variability of response variables 

described by the predicted values is determined by the multiple coefficient of determination, R2

, which tends to decrease 

monotonically with increasing p whether or not there is an 

improvement in prediction accuracy. The adjusted coefficient 

of determination, Ra

2 is a better measure of the prediction 

accuracy and of comparing equations with different numbers 

of explanatory variables. 

Like the response estimates, estimated parameters associated with the individual explanatory variables are uncertain. 

As n becomes large, estimated parameters are assumed to be 

unbiased and normally distributed about their true values. A 

t statistic can be computed to assess the significance of individual parameters. On the basis of the t probability distribution 

with n-p-1 degrees of freedom, a p-value is computed with the 

t statistic to assess the likelihood that the null hypothesis of 

the model parameter being equal to zero is true. Again, a small 
Analytical Approach and Methods 7

p-value is used to reject the null hypothesis, thereby accepting 

that the parameter estimate as statistically different from zero. 

A stepwise procedure was used to automatically select 

explanatory variables for inclusion in preliminary regression 

equations. Preliminary equations only included main effects 

(simple variables), without consideration of interaction effects 

(explanatory variables formed as the products of other explanatory variables), or dynamic effects associated with past or 

future measurements of E. coli concentrations. In the stepwise 

procedure, variables are added one at a time to the model from 

a candidate list. In this report, the included variables were 

required to have a p-value less than 0.05. Once included, they 

were required to retain a p-value of less than 0.10 as new variables were added. Stepwise methods are efficient for evaluating a large candidate list of explanatory variables. Inferential 

measures, such as R2 and RMSE values, however, are biased 

towards indicating too much strength in the relation between 

response and explanatory variables. 

Once the preliminary regression equation identified maineffect variables, first-order interaction variables were computed as the product of all main-effect variable pairs. Finally, 

both causal and noncausal dynamic components were evaluated. In this report, E. coli concentrations measured 2 days 

before to the time of the response E. coli concentration were 

treated as an explanatory variable in a causal dynamic regression formulation. The relation is considered causal because 

past E. coli concentrations might be used to predict current 

E. coli concentrations. E. coli concentrations, measured 2 days 

after the response E. coli concentration was measured, also 

were evaluated as a noncausal dynamic component. Noncausal 

implies that the future E. coli concentrations do not affect 

current concentration, but may be helpful in estimating them 

offline. Including both causal and noncausal components in 

the same equation provides a mechanism to investigate the 

symmetry of the dynamic component. 

Regression Trees

Regression trees are an alternative to linear regression 

for estimating a response variable on the basis of one or more 

explanatory variables. Regression trees subdivide responses 

into a hierarchical system of nodes by a sequence of logical 

if-then statements based on selected threshold values of the 

explanatory variables. Threshold values are computed as part 

of the regression-tree analysis. In this study, E. coli measurements were grouped, divided, and subdivided on the basis of 

one of the environmental factors. Graphically, regression trees 

are depicted with the single root node at the top, which contains all observations (or E. coli measurements), and branches 

forming below. Starting at the root (parent) node, two terminal or internal (child) nodes are placed below the root. If the 

response to the logical if-then statement is true, the response 

is included with the observations in the child node below and 

to the left of the parent; otherwise the response is included 

in a child node to the lower right. For example, the if-then 

statement might be, if water temperature on day of sampling 

were less than 20oC, then all E. coli measurements on those 

days would be grouped as “true” and be represented on the 

left side below the parent node; all other measurements would 

be grouped together on the right side below the parent node. 

As the level of the branching increases, child nodes become 

parent nodes. In the above example, the two groups of E. coli 

measurements based on water temperature (new parent nodes) 

could be subdivided in four groups (two additional child nodes 

under each new parent). The if-then statements might be different for each of the new parent groups. In the group defined 

by water temperatures less than 20oC, the if-then statement 

could be if rainfall 24 hours prior to day of sampling were 

less than 0.1 in., then all E. coli samples that were previously 

collected on days with water temperature less than 20oC and 

those also collected on days with less than 0.1 in. of rain 

would be “true” and subdivide a new child node under that 

parent node. Those that haven’t met this criterion would be 

“false” and grouped to the right as a new child node under that 

parent. The branching process continues until all child nodes 

are terminal nodes. In this report, branching was constrained 

so that every internal node had to have at least 10 observations 

for a split to occur. Terminal nodes can have a minimum of 

one observation, which is why regression trees are robust to 

the presence of outliers (extreme values of responses). 

Regression trees are not based on a probability model, 

so there is no statistical criterion, analogous to a p-value, for 

determining the appropriate level of branching. Therefore, 

statistical resampling techniques were used to assess possible 

overfitting of regression trees to E. coli data and resulting 

effects on the degradation of prediction accuracy. In overfitting or parameterizing, the sample is fitted rather than the 

population. In particular, if we drew another random sample 

from the population, the model we would develop might not 

be similar to the one that was estimated from the sample we 

have. Consequently, we may not be able to infer how well we 

can predict new observations based on the overfitted model 

from the sample data. Resampling is based on the distinction 

between estimation errors—differences between measured and 

predicted values for observations used in the model building 

process—and prediction errors—differences between measured and predicted values for observations not used in the 

model building process. 

To implement the resampling analysis, the Memorial and 

Metropolitan E. coli data sets (including the explanatory variables) were repetitively subdivided into two random subsets 

for model development and testing. The model development 

dataset contained about 75 percent of the measurements, and 

the testing dataset contained the remaining 25 percent. In each 

repetition, a new regression-tree model was created with the 

model development dataset. The root-mean-square error of 

estimation (RMSE-E) was computed with the development 

data, in which the number of terminal nodes was used to estimate the number of model parameters. Then the model-testing 

data set was applied to the model structure identified with the 

model-development data, and a root-mean-square error for 

prediction (RMSE-P) was computed without adjustment for 
8 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

the number of model parameters. After 500 repetitions, the 

paired distributions of RMSE for estimation and prediction 

were displayed as boxplots as a function of the number of 

terminal nodes. 

Logistic Regression

As an alternative to estimating E. coli concentrations 

and comparing the estimates to thresholds of interest, one 

can directly estimate the probability that a specified threshold 

will be exceeded using classification methods. In this report, 

logistic linear regression models were developed to estimate 

probabilities of exceedance of the recreational water-quality 

standard. Application of the logistic model involves a transformation of E. coli concentrations into two classes: those that 

exceed the standard and those that do not. In this report, binary 

logistic models were developed to estimate the probability of 

E. coli concentrations exceeding 300 MPN/100 mL. Equations used to develop this model are further explained in a later 

section. 

Logistic regression equations describe probabilities that 

E. coli concentrations will exceed a specified magnitude. 

A cut point is a fixed probability value for classifying these 

computed probabilities as exceedances or nonexceedances. 

Although a cut point of 0.5 might be taken as an arbitrary 

choice, a receiver operating characteristic (ROC) analysis 

can be used for exploring the sensitivity and specificity of 

the logistic equation for alternative cut points. Here, sensitivity refers to the probability that the model will indicate 

an exceedance when an E. coli concentration greater than 

300 MPN/100 mL is measured. Sensitivity also is referred to 

as the true positive rate. Specificity is the probability that the 

model will indicate nonexceedance when the E. coli measurement is below the threshold concentration. One minus the 

specificity is the false positive rate. 

An ROC analysis is depicted by use of an ROC curve, 

which depicts the tradeoff between sensitivity and specificity. 

An ROC curve for a given logistic regression equation and 

measured concentrations, typically plots the false positive rate 

on the x-axis and the true positive rate on the y-axis for a set of 

cut points ranging from 0 to 1. A diagonal line, which is drawn 

from the lower left-hand corner of the plot at [0,0] to the upper 

right hand corner at [1,1], is a reference line for evaluating 

alternative models. The area under the diagonal line is 0.5; 

the area under the receiver operating characteristic (AUROC) 

curve is a measure of the utility of the logistic model. An 

AUROC of 1 indicates a perfect model, and a value of 0.5 

indicates a complete model failure. A rough guide classifies 

AUROC value from 0.9 to 1.0 as excellent and from 0.8 to 0.9 

as good (Tape, 2007). 

Hydrodynamic Modeling

Hydrodynamic simulations were used to describe the 

horizontal (vertically averaged) velocity components that 

are expected to occur for specified flow, water level, and 

wind boundary conditions, given the geometry and hydraulic 

characteristics of the waterway. Hydrodynamic simulations 

were developed by means of the generalized hydrodynamic 

code RMA2-WES version 4.5 (RMA2). The site-specific 

geometric and hydraulic characteristics within the Lake 

St. Clair area were based on the hydrodynamic model of the 

St. Clair-Detroit River Waterway developed for the MDEQ 

Source Water Assessment Program by Holtschlag and Koschik 

(2002). 

RMA2 Hydrodynamic Code 

RMA2 is a generalized computer code for two-dimensional hydrodynamic simulation of surface-water bodies. 

RMA2 implements a finite-element solution of the Reynolds 

form of the Navier-Stokes equations for turbulent flows. 

Donnell and others (2005) provide detailed documentation of 

the governing equations, their solution by the finite-element 

method, and the recommended uses and limitations of the 

RMA2 code. RMA2 is maintained and frequently updated 

with enhanced features by the U.S. Army Corps of Engineers 

Waterways Experiment Station in Vicksburg, Miss. 

Following is a brief overview of those modeling aspects 

needed to help understand this application of the hydrodynamic model. To numerically solve the partial differential 

equations describing flow in two horizontal dimensions, the 

waterway area is discretized into a mesh of quadrilateral and 

triangular elements defined at points by nodes (fig. 1). The 

finite-element discretization provides a flexible representation 

of an irregularly shaped waterway. Corner nodes are placed 

at the vertices of elements, and midside nodes are placed 

approximately midway between the arcs (edges) connecting 

the corner nodes. Thus, eight nodes are defined per quadrilateral element, and six nodes are defined per triangular element. Contiguous elements forming branches or subreaches 

of the waterway are grouped into material zones to facilitate 

characterization of the hydraulic properties of the waterway. 

Channel-roughness coefficients and eddy viscosity characteristics are assigned to material zones, which in turn affect the 

velocities and water-surface elevations simulated at nodes. 

Hydrodynamic simulations compute water levels and 

flow velocities at interior nodes on the basis of boundary 

conditions specified at exterior nodes, hydraulic characteristics 

of the waterway, and flow equations. Boundary conditions 

include flow (discharge across one or more elements), water 

level, and wind conditions. Quadratic interpolation may be 

used to determine water levels and velocities anywhere within 

the elemental areas on the basis of nodal values. Therefore, 

greater densities of nodes result in improved spatial resolution 

of velocity fields but also require additional computational 

resources. 

RMA2 supports steady-state and transient simulations. 

When boundary conditions remain unchanged for a prolonged 

period, flow is considered at steady state (not changing 

with time). Although there is always some variability in 
Analytical Approach and Methods 9

the boundary conditions with time, these changes may 

be sufficiently minor that steady-state simulations are 

useful approximations of actual flow conditions. In other 

situations, steady-state simulations are idealized scenarios 

that facilitate generalizing the effect of various forces on flow 

and circulation patterns. When changes at the boundaries 

occur before flow has approximately equilibrated to previous 

changes, flow is considered transient. Although transient 

simulations require greater computational and storage 

resources, they can result in useful flow information for 

historical events of specific interest. In this report, transient 

flow was simulated at 1.0-hour time steps. 

The horizontal water velocity field is characterized by 2D 

hydrodynamic simulation. Water velocities may decrease in 

shallow areas because aquatic vegetation effectively increases 

channel roughness, and adjacent points may have more or 

less similar velocities based on the effectiveness of turbulent 

exchange in mixing the water. Neither channel roughness nor 

turbulent exchange, which is the fluid momentum transfer 

due to chaotic motions of fluid particles (Donnell and others, 

2005), can be measured directly in the field, but they can be 

inferred from measurements of flow, water level, and velocity within the waterway. The hydrodynamic model of the St. 

Clair-Detroit River developed for the MDEQ’s Surface Water 

Assessment Program (SWAP) was calibrated with extensive 

sets of velocity, water level, and discharge information by use 

of a nonlinear, parameter estimation technique developed by 

Poeter and others (2005). 

The Surface-Water Modeling System (SMS) is a computer program for preprocessing and postprocessing RMA2 

datasets (Environmental Modeling Systems, Inc., 2005). SMS 

was used to facilitate the development of the hydrodynamic 

model of the St. Clair-Detroit River Waterway by helping to 

implement and quality-assure the mesh design and by helping to visualize simulated water velocities and elevations. To 

produce particle tracks, simulated velocities and locations of 

corner nodes were output from SMS into an ASCII file format 

and read into a plotting program (Tecplot by Amtec Engineering, Inc., 2006), which was then used to track hypothetical, 

massless particles by use of linear interpolation across elements at a time step of 0.1 hour through both steady-state and 

transient flow fields. 

Two-Dimensional Hydrodynamic Model of Lake 

St. Clair 

The St. Clair-Detroit River Model is a structured finiteelement data set in the RMA2 input format that describes 

the geometry, bathymetry, and hydraulic characteristics of 

the St. Clair-Detroit River Waterway. In particular, the input 

includes a connectivity table that indicates which nodes are 

associated with each element and which elements are associated with each material zone. In addition, the easting and 

northing specification of each node is referenced to International feet within the southern zone (2113) of the Michigan 

State Plane Coordinate System of 1983 (MSPCS 83), and the 

bottom elevation at each node is referenced to the International 

Great Lakes Datum of 1985 (IGLD 85). 

The St. Clair-Detroit River hydrodynamic model extends 

from the head of the St. Clair River at the water-level gage 

operated by NOAA near Fort Gratiot, Mich. (Station ID: 

9014098) to the Bar Point, Ontario, gage (station number 

12005) at the mouth of the Detroit River, which is operated by 

the Canadian Hydrographic Survey. Inflows are specified to 

the St. Clair River from Lake Huron, and the Black, Pine, and 

Belle Rivers; to Lake St. Clair from the Clinton, Sydenham, 

and Thames Rivers; and to the Detroit River from the River 

Rouge. Water levels are specified only at the Bar Point gage. 

Wind conditions are specified for the entire mesh on the basis 

of data at the NOAA weather station LSCM4 on Lake St. Clair 

(fig. 1). 

Several versions of the model were developed in response 

to changing data availability and application emphasis. The 

initial version discretized the waterway into 13,783 quadratic 

elements defined by 42,936 nodes and was used to estimate 

channel-roughness characteristics in 25 material zones. Data 

from velocity surveys (Holtschlag and Koschik, 2003a,b) were 

used to enhance the model calibration so as to describe the 

variation in channel roughness with depth of flow by material 

zones and to estimate the turbulent exchange coefficient or 

eddy viscosity. This version is referred to as the “standard version”. Following this, the “public water intake version” refined 

the mesh geometry in the vicinity of 13 U.S. public water 

intakes, thereby increasing the number of elements to 30,306 

and the number of nodes to 90,386. 

Particle Tracking

Particle tracking uses a spatial-temporal interpolation 

technique to move hypothetical massless particles across the 

lake with time and is based on simulated velocities of water 

movement at model nodes. Particle tracking is an inherently 

transient procedure even in steady-state velocity fields. This 

transient component indicates traveltimes along a flow path. 

In steady-state flow fields, the paths of particles started at different initial positions do not cross. In contrast, particle paths 

in unsteady flow fields may cross because of different arrival 

times at various points in the time-varying flow field. 

Particles may be tracked forward or backward in time 

through simulated steady or transient flow fields. In steady 

flow, particles are tracked forward in time by specifying an 

initial position, the length of the time step, the duration of the 

tracking period, and a temporally constant but spatially varying flow-velocity field. Some experimentation is required to 

determine a tracking time step that is sufficiently short so that 

further decreases have little impact on ending locations. Similarly, particles may be tracked backward in time by reversing 

the sign of the simulated velocity components at each node 

and specifying ending, rather than starting, locations. Particles 

may be forward tracked through a transient flow field by spec-
10 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

ifying a particle-tracking time step and a series of transient 

flow solutions computed at specified time intervals that may 

differ from the particle-tracking time step. Finally, particles 

may be backtracked through a transient flow field by reversing 

the order of the transient-flow solutions and reversing the sign 

of simulated velocities.

Relation of Environmental Factors to 

Escherichia coli Concentrations

Statistical Distributions of Escherichia coli 

Concentration and Effects of Wind Direction

During 2002–05, the 139 selected E. coli concentrations 

at Memorial Beach ranged from 1 to 2,020 MPN/100 mL, 

with mean, standard deviation, and skewness characteristics of 

137-, 268-, and 4.26 MPN/100 mL, respectively. In this report, 

the coefficient of skewness was computed as 

Skewness n

x x

n

x x

i i

n

i i

n

=

å -

å - æ

è

ç

ç

çç

ö

ø

÷

÷

÷

÷

=

=

1

1

3

1

2

1

3 2

( )

( )

/

(1)

where 

xi are the individual observations, 

x is the average of the xi values,

and

 n is the number of observations.

Similarly, the 143 selected E. coli concentrations at 

Metropolitan Beach ranged from 1 to 1,270 MPN/100 mL, 

with mean, standard deviation, and skewness characteristics of 

93.2, 189, and 3.72 MPN/100 mL, respectively. Thus, the statistical distributions of bacteria at both beaches were positively 

skewed (fig. 4). This skewness is a problem in regression analyses because it effectively weights the larger concentrations at 

both beaches too highly with respect to their frequency in the 

sample. In addition, direct use of the E. coli concentrations in 

a regression analysis would not result in a residual distribution 

that was approximately normally distributed, thereby violating a fundamental assumption of least-squares regression. To 

overcome this problem, a common logarithmic (log10) transformation was applied to E. coli concentrations. For Memorial Beach, the mean, standard deviation, and skewness of the 

transformed concentrations are 1.69, 0.63, and 

-0.017 log10 MPN/100 mL, respectively. Likewise, for Metropolitan Beach, the mean, standard deviation, and skewness of 

the transformed concentrations are 1.46, 0.67, and 

0.10 log10 MPN/100 mL, respectively. Thus, the transformed 

concentrations show little skewness, and their distributions are 

closely approximated by normal distributions (fig. 4). All subsequent analyses used the transformed concentration values. 

For simplicity in discussing the statistical analyses, the log10

transformed E. coli concentration values are referred to as 

E. coli concentrations, unless ambiguity between the original 

and transformed metrics would result.

Wind conditions are thought to be the primary determinant of current patterns and source areas of water to beaches 

on Lake St. Clair. Thus, differences in 24-hour average wind 

conditions before a sampling event may be associated with 

different source areas. If the release of bacteria is constant and 

differs among sources, then a polar plot of bacteria concentrations with wind speed and direction may indicate possible 

source-area effects. In a polar plot, angular and radial distances used to plot bacteria concentrations are determined by 

use of 24-hour vector averages of wind speeds and directions 

from LSCM4, and the posted location of the sample is color 

coded by concentration. For the bacteria data from Memorial 

Beach, the polar plot indicates little apparent differences in 

bacteria concentrations with wind speed or direction (fig. 5). 

Possible variations in median bacteria concentrations 

with wind direction were evaluated by subdividing the bacteria 

data into corresponding equiangular sectors of 45 degrees 

each. Analysis by use of a notched boxplot indicates no statistically significant differences in median E. coli concentrations 

at Memorial Beach among sectors (fig. 6). In a notched boxplot, a notch is formed in each boxplot that corresponds to the 

95-percent confidence interval about the median. If the range 

in concentrations formed by notches from all sectors overlaps 

in some area, the null hypothesis of no differences in median 

concentrations among sectors is not rejected. A Kruskal-Wallis 

test (Conover, 1980), which is a nonparametric form of an 

analysis of variance, confirms the absence of statistical differences in E. coli concentrations among wind directions (p-value 

equals 0.3335). 

In contrast to Memorial Beach, median concentrations 

at Metropolitan Beach differed significantly among wind sectors. In particular, the polar plot shows a higher percentage of 

large -magnitude E. coli concentrations when winds are from 

the east-northeast (ENE.) than from some of the other wind 

sectors (fig. 7). And unlike Memorial Beach, exceedance of 

the water-quality criteria did not occur in all wind sectors. 

In the ENE., ESE., SSE., SSW., and WSW. sectors, three to 

five sample exceeded the Michigan recreational water-quality 

standard. In the NNE. sector, however, there was only one 

exceedance, and there were none in the NNW. and WNW. sectors. The corresponding boxplot (fig. 8) shows that the ENE. 

sector has the highest median value, but the fewest number 

of samples. Based on the notched boxplot (fig. 8), the median 

concentration in the ENE. sector is significantly greater than 

the median concentrations in the WSW., WNW., and NNW. 

sectors. In addition, the median concentration in the SSW. sector is significantly greater than median concentrations in the 

WNW. and NNW. sectors. The Kruskal-Wallis test also rejects 

the equality of median E. coli concentrations in all sectors 

(p-value equals 0.0009). 
Relation of Environmental Factors to Escherichia coli Concentrations 11

0 200 400 600 800

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0 200 400 600 800 1000 1200

E. COLI CONCENTRATIONS, 

IN MOST PROBABLE NUMBER 

PER 100 MILLILITERS

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

RELATIVE FREQUENCY

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

RELATIVE FREQUENCY

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

LOG10 OF E. COLI CONCENTRATIONS,

IN MOST PROBABLE NUMBER

PER 100 MILLILITERS

(A)

(B)

Normal Distribution

Normal 

Distribution

Figure 4. Statistical distributions of E. coli concentrations at (A) Memorial and (B) Metropolitan Beaches on 

Lake St. Clair, Mich., 2002–05.
12 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

EXPLANATION

E. coli geometric mean

most probable number 

per 100 mL (milliliter)

0 10 20 30 40

WIND SPEED, IN MILES PER HOUR

1 Wind data is from the Lake St. Clair, Mich. station (LSMC4), 

 operated by the National Data Buoy Center.

2

 Bacteria data is from the Macomb County Health Department, Mt. Clemens, Mich., 

 for 2002-05.

Wind direction indicates

the direction the wind 

is coming from. Both 

speed and direction

represent the average 24 hours

prior to bacteria sampling 

on Memorial Beach.

NNE.

ENE.

ESE.

SSW. SSE.

WSW.

WNW.

NNW.

1 to 10

11 to 30

31 to 70

71 to 150

151 to 300

300+

Figure 5. Relation 

between wind conditions and E. coli 

concentrations at 

Memorial Beach on 

Lake St. Clair, Mich., 

2002–05. 

Figure 6. Distributions of 

Macomb County 

Health Department E. coli

concentrations at 

Memorial Beach 

by wind direction 

on Lake St. Clair, 

Mich., 2002–05. 

E. COLI, IN MOST PROBABLE NUMBER PER 100 MILLILITERS OF WATER

1

10

100

1,000

7,000

AVERAGE DIRECTION THE WIND IS COMING FROM FOR THE 24-HOUR PERIOD BEFORE SAMPLING

(Number of samples)

NNE. ENE. ESE. SSE. SSW. WSW. WNW. NNW.

(27) (13) (16) (18) (50) (19) (15) (16) EXPLANATION

Median

Lower

quartile

Lower

hinge

Upper

quartile

Upper

hinge

Notch

Outlier

Michigan daily geometric mean total body contact water-quality standard

The “Notch” 

provides a 

95-percent 

confidence 

interval about 

the median.
Relation of Environmental Factors to Escherichia coli Concentrations 13

Figure 7. Relation 

between wind conditions and E. coli bacteria concentrations at 

Metropolitan Beach on 

Lake St. Clair, Mich., 

2002–05. 

EXPLANATION

E. coli geometric mean

most probable number 

per 100 mL (millimeters)

1 to 10

11 to 30

31 to 70

71 to 150

151 to 300

300+

0 10 20 30 40

WIND SPEED, IN MILES PER HOUR

1 Wind data is from the Lake St. Clair, Mich. station (LSMC4), 

 operated by the National Data Buoy Center.

2

 Bacteria data is from the Macomb County Health Department, Mt. Clemens, Mich., 

 for 2002-05.

Wind direction indicates

the direction the wind 

is coming from. Both 

speed and direction

represent the average 24 hours

prior to bacteria sampling 

on Memorial Beach.

NNE.

ENE.

ESE.

SSW. SSE.

WSW.

WNW.

NNW.

Figure 8. Distributions of 

Macomb County 

Health Department E. coli data, 

by wind direction 

for Metropolitan 

Beach on Lake 

St. Clair, Mich., 

2002–05. (Overlapping notched 

regions indicate 

that differences in 

median concentrations are not statistically significant. 

The two boxplots 

labeled WNW and 

NNW have median 

concentrations 

that differ statistically from the 

median concentraAVERAGE DIRECTION THE WIND IS COMING FROM FOR THE 24-HOUR PERIOD BEFORE SAMPLING tion at ENE.) 

(Number of samples)

NNE. ENE. ESE. SSE. SSW. WSW. WNW. NNW.

1

10

100

1,000

(23) (13) (17) (20) (53) (24) (15) (15) EXPLANATION

Median

Lower

quartile

Lower

hinge

Upper

quartile

Upper

hinge

Notch

Outlier

Michigan daily geometric 

mean total body contact 

water-quality standard

E. COLI, IN MOST PROBABLE NUMBER PER 100 MILLILITERS OF WATER

The “Notch”

provides a 95-

percent confidence interval

about the

median.
14 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

In summary, wind direction was not apparently associated with E. coli concentrations at Memorial Beach, but it 

did influence E. coli concentrations at Metropolitan Beach. 

Onshore winds are associated with higher waves, and higher 

waves are historically associated with higher E. coli concentrations. At Metropolitan Beach, onshore winds (from ESE. to 

SSW.) accounted for the largest number of exceedances of the 

recreational water-quality standard but not the largest median 

concentration of E. coli. During onshore winds, waves might 

resuspend or wash E. coli-laden sand, vegetative material 

(algae), or animal waste to nearshore beach water. Unfortunately, the wave data from Environment Canada proved to be 

problematic in characterizing this possibility; attempts to fit 

the wave data during regression analyses were not successful. 

Perhaps the Environment Canada buoy is too far offshore or 

the lake structure is too complex (shallow depth of 3-m with 

an 8-m navigation channel to the east of this buoy) to anticipate nearshore waves at the study beaches.

Linear Regression Results

Memorial Beach

In the linear regression analysis of log10 E. coli concentrations (LogEC) at Memorial Beach, the stepwise method 

identified total inches of rainfall at Selfridge ANGB 72 hours 

preceding the E. coli sample (Rainfall_72), sea-surface temperature of Lake St. Clair in degrees Celsius (SeaTemp), log10

of flow at Clinton River in cubic feet per second (ClintonFlow) 

1 day before the bacteria sample, and turbidity in Nephelometric Turbidity Units (Turbidity) as significant explanatory 

variables in the main-effects model (table 1). From this model, 

all first-order interaction terms were formed and evaluated, 

again by stepwise analysis. Although Rainfall_72-ClintonFlow 

and ClintonFlow-Turbidity interaction terms contributed to 

the main effects model and resulted in the lowest RMSE and a 

higher adjusted R2 value, neither interaction term was individually significant at the p = 0.05 level. Therefore, interactions 

were not included in the dynamic-effects model. Similarly, 

neither wind speed nor a north wind direction indicator variable contributed significantly to the regression model at the 

p = 0.05 level.

In a dynamic regression analysis of E. coli concentrations, 108 observations were formed from E. coli samples 

collected 2 days apart at Memorial Beach. Of these, 54 samples had explanatory E. coli concentrations measurements 

before the response E. coli concentrations and 54 samples 

had explanatory concentrations measurements afterwards. In 

addition to the presampling or postsampling concentrations 

(LagLeadLogEC), an indicator vector was formed to identify 

whether the explanatory E. coli concentrations were obtained 

before or after the response E. coli concentration. This type of 

model is used to help determine whether E. coli concentration 

in samples collected 2 days prior could be used as a predictor 

for E. coli concentrations on a particular day of interest. This 

would also help determine how frequently the beach should 

be sampled (Is sampling every other day for E. coli protective 

of human health?). Understanding the effects of E. coli 2 days 

after the response E. coli may help to understand the process 

that may transport bacteria to and from the beach area. 

Inclusion of the explanatory E. coli concentration data 

and the indicator vector in the main-effects regression indicated that the explanatory E. coli concentrations were statistically significant, but the timing indicator was not statistically significant (p = 0.4602). These results imply that the 

correlation between E. coli concentrations 2 days apart are 

statistically significant given the other main effects in the 

model. In addition, there is no statistical evidence that the 

correlation function is not symmetrical because the magnitude 

of the parameter was not significantly affected by whether 

the explanatory E. coli concentrations were measured 2 days 

before or after the response E. coli concentrations. In other 

words, the variability in E. coli concentrations for a specific 

date is somewhat explained by E. coli concentrations 2 days 

prior and (or) 2 days later. The final form of dynamic regression equation includes the main-effect variables, even though 

the statistical significance of Clinton River discharge was 

somewhat greater than 0.05 in the subset model, after inclusion of the lag and lead explanatory log10 E. coli concentrations. 

Although the preceding regression analyses identified 

explanatory variables that are statistically related to log10

E. coli concentrations at Memorial Beach, all the equations 

have limited explanatory ability as indicated by the adjusted 

R2 values and the match between the measured log10 E. coli

concentrations and those estimated by the main-effects model 

(fig. 9). In fact, 7 of the 11 measured concentrations that 

exceeded the Michigan recreational water-quality standard 

would not have resulted in exceedance based on the models 

estimation. Therefore, this model would not be useful as a 

predictive tool given the current set of data. 
Relation of Environmental Factors to Escherichia coli Concentrations 15

Table 1. Summary of linear regression models at Memorial 

Beach on Lake St. Clair, Mich., 2002–05. 

[<, less than; NA, not applicable; NI, not included because factor was 

insignificant in the interaction-effects model]

Environmental 

factor

Parameter estimate

(p-value)

Maineffects 

model

Interactioneffects 

model

Dynamiceffects 

model

Parameter summary

Intercept 0.88304

(<.001)

0.61808

(.0055)

0.32144

(.2431)

Rainfall_72 .20945

(.0004)

.31804

(.0030)

.21611

(.0101)

SeaTemp .02536

(.0038)

.03206

(.0006)

.03162

(.0101)

ClintonFlow .01200

(.0023)

.02209

(.0006)

.01620

(.0865)

Turbidity .00174

(.0014)

.00352

(.0029)

.00236

(.0039)

Rainfall_72-

ClintonFlow

NA -.00434

(.1470)

NI

ClintonFlowTurbidityNA -.00083

(.0866)

NI

LagLeadLogEC NA NA .21648

(.0123)

Model summary

Model F value 14.27

(<.0001)

10.48

(<.0001)

10.70

(<.0001)

Root mean 

square error

.53807 .53277 .54141

Adjusted R2 .2778 .2919 .3119

Number of 

observations

139 139 108

Figure 9. Relation between measured and main-effect model 

estimates of E. coli concentrations at Memorial Beach on 

Lake St. Clair, Mich., 2002–05. 

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Log10 OF MEASURED E. COLI CONCENTRATIONS, 

IN MOST PROBABLE NUMBER PER 100 MILLILITERS

Line of agreement

Log10 OF ESTIMATED E. COLI CONCENTRATIONS, 

IN MOST PROBABLE NUMBER PER 100 MILLILITERS

Metropolitan Beach

A preliminary regression model for estimating log10

concentrations of E. coli at Metropolitan Beach was identified 

by use of the stepwise procedure. The main effects for this 

model included total rainfall at Selfridge ANGB during the 

24 hours prior to bacteria sampling (Rainfall_24), sea-surface 

temperature (SeaTemp), turbidity, and the 1-hour change in 

smoothed water levels at the St. Clair Shores (DeltaWaterLevel) (table 2). In contrast to Memorial Beach, the Clinton 

River discharge was not significantly correlated with E. coli 

concentrations. 

 A stepwise evaluation of all first-order interaction 

terms from the main-effects model resulted in no additional 

variables at the 0.15 level of significance. Addition of wind 

speed (WindSpeed) and a north-wind indicator vector (NorthWindInd) marginally improved the accuracy of the model, 

although the individual terms were not statistically significant (p = 0.05). Finally, among the 143 E. coli measurements 

at Metropolitan Beach, 64 paired with explanatory E. coli

concentrations in samples collected 2 days before the response 

variable, and 50 paired with explanatory E. coli concentrations 

2 days after the response variables. In contrast to the Memorial 

Beach model, however, no dynamic components were significant at Metropolitan Beach when included with the main 

effects. Although the main-effects model was selected for the 

purpose of this report, it explains less than 30 percent of the 

variability in the log10 E. coli concentrations and so has limited 

utility for prediction of E. coli concentrations (fig. 10) with 

none of the 10 samples that exceeded the Michigan waterquality standard being predicted as exceedances by the model. 
16 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

Table 2. Summary of linear-regression models at Metropolitan 

Beach on Lake St. Clair, Mich., 2002–05. 

[<, less than; NA, not applicable; NI, not included because factor was insignificant in the interaction-effects model]

Environmental 

factor

Parameter estimate

(p-value)

Maineffects 

model

Windeffects 

model

Dynamiceffects 

model

Parameter summary

Intercept 0.29630

(.0842)

0.11153

(.6055)

0.23413

(.2430)

Rainfall_24 .26538

(.0270)

.23488

(.0531)

.18185

(.1987)

SeaTemp .05792

(<.0001)

.05810

(<.0001)

.05703

(<.0001)

Turbidity .00477

(.0014)

.00399

(.0088)

.00438

(.0087)

DeltaWaterLevel-114.14097

(.0101)

-102.80

(.0237)134

-97.187

(.0829)

NorthWindInd NA .20155

(.0406)

NI

WindSpeed NA .01608

(.4283)

NI

LagLeadLogEC NA NA .06563

(.4412)

Model summary

Model F value 15.78

(<.0001)

11.55

(<.0001)

9.30

(<.0001)

Root mean 

square error

.56647 .56045 .58927

Adjusted R2 .2935 .3084 .2685

Number of 

observations

143 143 114

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Line of agreement

Log10 OF MEASURED E. COLI CONCENTRATIONS, IN 

MOST PROBABLE NUMBER PER 100 MILLILITERS

Log10 OF ESTIMATED E. COLI CONCENTRATIONS, IN 

MOST PROBABLE NUMBER PER 100 MILLILITERS

Figure 10. Relation between measured and linear regression 

estimates of E. coli concentrations at Metropolitan Beach on 

Lake St. Clair, Mich., 2002–05. 

Regression-Tree Results

Regression trees were developed as an alternative to 

linear regression models for estimating E. coli concentrations. Like linear regression models, regression trees identify 

explanatory variables that are associated with different levels 

of E. coli concentrations. Regression trees are sometimes 

more effective than linear models when there is a nonlinear 

relation between environmental factors and E. coli concentrations or there are high-order interactions. The flexibility of the 

regression-tree structure, however, sometimes makes identification of relations in small data sets problematic. 

Memorial Beach

A regression tree was constructed for the Memorial 

Beach log10 E. coli concentrations by use of the same explanatory variables identified for the main effects linear regression 

model (fig. 11). The resulting full tree contained 28 terminal 

nodes, 27 parent nodes (where explanatory variables are used 

for branching), and 23 levels of branching before pruning. 

In the model, the explanatory variable Rainfall_72 was used 

3 times, SeaTemp 10 times, ClintonFlow 5 times, and Turbidity 9 times. E. coli concentrations were first partitioned by 

ClintonFlow. The second variable separating E. coli concentrations was Rainfall_72. A dynamic regression tree also was 

developed for the 108 observations at Memorial Beach where 

E. coli measurements 2 days apart could be paired. The RMSE 

of regression trees decrease monotonically with the number of 

terminal nodes. 
Relation of Environmental Factors to Escherichia coli Concentrations 17

 1.56

 2.30 27.1 95.1 0.552 2.95

 2.26 2.11 1.75

56.1 1.10 07.1 30.1 80.2 2.34 1.75 2.21

 1.10 0.728

 2.19 1.95 1.18

0.729

 1.28

 1.69

0.895

 1.43 1.00

 Disc < 16.0

651.0 < niaR 5360.0 < niaR 

1.21 < pmeT 70.6 < pmeT 

T 632 < bru

 

 Temp < 7.40

7.02 < pmeT 4.44 < csiD 

6.02 < csiD

2.29 < pmeT . 8.56 < bruT 

18.3 < csiD

 

5.11 < pmeT 3.91 < pmeT 6.01 < pmeT 51.3 < bruT Temp < 8.67

76.3 < csiD 7.34 < bruT 

 Temp < 25.2 Turb < 81.7

 Turb < 42.5

 Temp < 21.3

 Turb < 14.65

 Temp < 16.2

 Rain < 0.102

Rain: Rainfall_72, total rainfall 72-hours before sampling, in inches

Temp: SeaTemp, Water temperature, in degrees Celcius

Disc: ClintonFlow, Flow of Clinton River, in cubic feet per second

Turb: Turbidity, in nephelometric turbidity units (NTU).

The symbol “<” indicates “less than” the quanity to the right. 

Figure 11. Regression tree for estimating log10 E. coli concentrations for Memorial Beach on Lake St. Clair, Mich., 2002–05. 

E P E P E P E P E P E P E P E P E P E P 0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

AVERAGE ROOT MEAN SQUARE ERROR, IN LOG10 E. COLI CONCENTRATION UNITS

1 2 3 4 5 6 7 8 9 10

NUMBER OF TERMINAL NODES

E--Root-Mean-Square Error of Estimation

P--Root-Mean-Square Error of Prediction

As described in the approach, 

a statistical resampling analysis was 

used to evaluate the reproducibility 

of the regression trees. For data from 

Memorial Beach, the monotonic 

decrease in median RMSE of estimation 

(RMSE-E) with increasing number 

of terminal nodes (fig. 12) was not 

consistent with the nearly constant 

or slightly increasing median RMSE 

of prediction (RMSE-P). In addition, 

the interquartile ranges of RMSEs 

for prediction were nearly 3 times the 

RMSE for estimation in regression 

trees having 1 through 10 terminal 

nodes. This lack of reproducibility 

indicates that the regression trees may 

be overparameterized for the number of 

E. coli measurements available for model 

development and therefore not good 

predictors of E. coli concentrations.

Figure 12. Regression tree root mean square error for estimation and prediction for 

Memorial Beach on Lake St. Clair, Mich. 
18 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

Metropolitan Beach

A regression tree also was developed for the Metropolitan 

Beach log10 E. coli concentrations by use of the same three 

explanatory variables identified for the main-effects linear 

regression model. The resulting full tree for the variables in 

the main-effects model contained 33 terminal nodes, 32 parent 

nodes (where explanatory variables are used for branching), 

and 23 levels of branching. In this model, E. coli concentrations were first separated by SeaTemp, then by Turbidity. 

In a resampling analysis similar to the one for Memorial 

Beach data, the median RMSE of estimation (RMSE-E) for 

the regression tree decreased monotonically with the number 

of terminal nodes, whereas the median RMSE-P remained 

constant or increased slightly. Again, the inconsistency 

between the RMSE-E and the RMSE-P indicates an overparameterization of the regression tree. 

For Metropolitan Beach, where prior analyses had 

indicated a potential wind effect, regression trees were also 

constructed for two datasets partitioned by onshore or offshore 

winds. This analysis was not done for Memorial beach, where 

wind effects were not indicated. Concentrations of E. coli in 

the Clinton River Spillway, previously indicated to be correlated with E. coli concentrations at Metropolitan Beach 

(although not significant in the linear regression models) were 

also included in these regression-tree models. During onshore 

winds, E. coli concentrations were first separated by water 

temperature; if temperature was less than 17.1°C, the mean 

log E. coli was 1.2, and the branch was further subdivided 

by Clinton River Spillway discharge. If water temperature 

was greater than 17.1°C the mean log10 E. coli was 1.72, and 

this population was subdivided by turbidity. During offshore 

winds, E. coli concentrations were first split by Clinton River 

Spillway discharge, then by Clinton River Spillway E. coli

concentration. 

Logistic Regression Results

Logistic regression models were developed for Memorial 

and Metropolitan Beaches to provide a mechanism for estimating the probability of E. coli concentrations exceeding the 

water-quality standard as a function of selected environmental 

variables. Estimating probabilities is an alternative to estimating E. coli concentrations and comparing the estimates to the 

standard. Exponentiating parameters in the logistic model 

describe the odds of exceeding the standard with a unit change 

in the corresponding environmental variable, given that the 

other environmental variables do not change. 

Memorial Beach

Of the 139 samples from Memorial Beach analyzed 

for E. coli concentration, 127 had concentrations less than 

Michigan’s recreational water-quality standard, and 12 had 

concentrations that exceeded the standard. In a main-effects 

logistic model, Rainfall_72, ClintonFlow, and Turbidity were 

identified as significant explanatory variables (table 3). No 

interaction effects were significant. The parameters in the 

main-effects model indicate that all significant explanatory 

variables were positively associated with the odds of exceeding the beach-closure threshold. In particular, an increase in 

Rainfall_72 of 1 in. or more than doubles (2.36) the odds that 

E. coli concentrations will exceed the water-quality standard, assuming that the other explanatory variables remain 

unchanged. 

Table 3. Summary of logistic regression models for E. coli 

concentrations at Memorial and Metropolitan Beaches on 

Lake St. Clair, Mich., 2002–05. 

[<, less than; NA, not applicable]

Environmental 

factor

Memorial 

Beach 

(p-value)

Metropolitan 

Beach 

(p-value)

Maineffects 

parameter

Dynamiceffects 

parameter

Maineffects 

parameter

Parameter summary

Intercept -4.5845

(<.0001)

-8.0100

(.0002)

-1.6317

(.0080)

Rainfall_72

exp(Rainfall_72)

.8588

2.360

(.0052)

1.4930

4.451

(.0052)

NA

Rainfall_24

exp(Rainfall_24)

NA NA 3.3988

29.928

(.0013)

ClintonFlow

exp(ClintonFlow)

.0387

1.039

(.0350)

NA -.1918

.825

(.0205)

Turbidity

exp(Turbidity)

.00688

1.007

(.0291)

.0133

1.013

(.0031)

NA

LagLeadLogEC

exp(LagLeadLogEC)

NA 1.4343

4.197

(.0420)

NA

Intercept -4.5845

(<.0001)

-8.0100

(.0002)

-1.6317

(.0080)

Rainfall_72

exp(Rainfall_72)

.8588

2.360

(.0052)

1.4930

4.451

(.0052)

NA

Model summary

-2LogLa

- + ¢ 2LogLa b x

81.722

53.991

61.957

34.922

72.489

54.922

Chi-squared statistic 27.7313

(<.0001)

27.0343

(<.0001)

17.5667

(.0002)

Rank correlation c 

(area under the 

ROC) 

.891 .933 .807

Number of 

observations

139 139 108
Relation of Environmental Factors to Escherichia coli Concentrations 19

Similar to the linear regression model, a dynamic logistic 

regression also was developed for the 108 observations that 

could be formed by pairing an explanatory E. coli concentration (LagLeadLogEC) measured 2 days before or after the 

response E. coli concentration. Of these sample, 99 had concentrations less than 300 MPN/100 mL. An explanatory variable was initially included to indicate whether the explanatory 

E. coli concentration was sampled before or after the response. 

Results indicated that explanatory log10 E. coli concentrations 

were significant in estimating response E. coli exceedance 

probabilities, but the timing indicator was not (similar results 

to the linear regression model). Specifically, an increase of 

1 log10 unit in explanatory E. coli concentrations would quadruple (4.197) the odds that the response E. coli concentration 

would exceed the beach-closure threshold, assuming the other 

explanatory variables did not change (table 3). 

An ROC (receiver operating characteristics) analysis was 

done for the main-effects and the dynamic-effects logistic 

models for Memorial Beach (fig. 13). The area under the ROC 

for the main-effects model is estimated by c equal to 0.891, 

whereas the area under the ROC for the main and dynamiceffects model is slightly higher at c equal to 0.933. For both 

models, substantial gains in sensitivity are possible with modest loses in specificity above the arbitrary cut point of 0.5. For 

the main-effects logistic model of Memorial Beach, a cut point 

of 0.14 yields a sensitivity of about 0.83 and a specificity of 

0.89 (false positive rate of 0.11). For the main- and dynamiceffects logistic model, a cut point of 0.23 yields a sensitivity 

of about 0.77 and a specificity of 0.92 (false positive rate of 

0.08). 

Metropolitan Beach

Of the 143 samples from Metropolitan Beach analyzed 

for E. coli concentration, 133 had concentrations less than 

Michigan recreational water-quality standard, and 10 had 

concentration that exceeded the standard. In a main-effects 

logistic model, Rainfall_24 and ClintonFlow were identified 

as significant explanatory variables. No interaction or dynamic 

effects were found to be significant. In the main-effects model, 

E. coli exceedance odds increase nearly thirtyfold (29.9) 

for each inch of increase in total rainfall during the 24-hour 

period preceding sampling. In contrast, an increase of 1 log10

flow of Clinton River slightly decreases the odds of E. coli

exceedances. 

Figure 13. Receiver operating 

characteristics (ROC) curves 

for logistic equations predicting 

the probability of E. coli 

concentrations exceeding 

300 MPN/100 mL at Memorial 

Beach on Lake St. Clair, Mich., 

2002–05. 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FALSE POSITIVE RATE (1-SPECIFICITY)

TRUE POSITIVE RATE (SENSITIVITY)

0.23

0.40

0.44

0.50

0.14

0.073

0.073

0.50

Main effects curve

Probability cut points

Main and dynamic effects curve

Line of equal true- and false-positive rates
20 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

An ROC analysis was done for the main-effects logistic model for Metropolitan Beach (fig. 14). A cut point of 

0.11 yields a sensitivity of about 0.70 and a specificity of 

0.85 (false positive rate of 0.15). In contrast, a cut point of 

0.5 yields a sensitivity of only 0.2, with a specificity of 0.99. 

Relation of Flow Paths to Escherichia coli 

Concentrations

Because lakeshore beaches are part of a very dynamic 

system of hydrologic flow paths, it is important to understand 

these paths to identify potential source waters to the beach 

areas. Hydrodynamic modeling is a means of tracking the path 

particles may take under a given set of wind conditions. This 

study builds on the particle-tracking hydrodynamic model created specifically for Lake St. Clair. 

Flow Simulation 

The Lake St. Clair model referred to in this report is a 

subset of the 2D hydrodynamic model of the St. Clair-Detroit 

River Waterway model developed for the MDEQ SWAP. This 

version uses the same geometry, bathymetry, and hydraulic 

parameters as the standard model, except in areas excluded 

from the model domain. The Lake St. Clair version extends 

from a point near Algonac, Mich., on St. Clair River, through 

Lake St. Clair to the Fort Wayne water-level gaging station 

(station ID 9044036) operated by NOAA on the Detroit River. 

With respect to the standard model version, the mesh 

of the Lake St. Clair version has been refined northwest of 

the boundary between the United States and Canada (fig. 1). 

Refinement implies that arcs (lines) connecting opposing midside nodes are formed and used to subdivide each quadrilateral 

and triangular element into four subelements. The subelements 

have the same number of sides as the parent element. Nodes 

are added at the intersection of crossing arcs in quadrilateral 

elements and to the midsides of all refined elements to maintain their quadratic characteristic. The Lake St. Clair version 

contains 9,784 elements delimited by 28,749 nodes. 

Figure 14. Receiver operating 

characteristics (ROC) 

curve for logistic equations 

predicting the probability 

of E. coli concentrations 

exceeding 300 MPN/100 mL 

at Metropolitan Beach on 

Lake St. Clair, Mich., 2002–05. 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FALSE POSITIVE RATE (1-SPECIFICITY)

TRUE POSITIVE RATE (SENSITIVITY)

.056

0.11

0.12

0.15

0.22

0.42

0.50

Main effects curve

.056 Probability cut points

Line of equal true- and false-positive rates
Relation of Environmental Factors to Escherichia coli Concentrations 21

Particle Tracking 

The paths that particles take may be important to the 

transport of E. coli to and from the beach. These paths may be 

highly variable throughout the year because they often depend 

on wind direction.

Steady-State Simulations

Steady-state hydrodynamic simulations were used to 

describe current patterns on Lake St. Clair for wind directions discretized into 24 sectors. Simulated current patterns on 

Lake St. Clair are primarily determined by wind conditions 

rather than flow magnitudes (Holtschlag and Koschik, 2004). 

Therefore, identification of steady-state flow paths to Memorial and Metropolitan Beaches were described as a function 

of wind conditions, while flow and water-level specifications 

were held constant. In particular, St. Clair River flow was 

specified as 185,000 ft3

/s, Clinton River flow was 637 ft3/s, 

Sydenham River flow was 1,860 ft3/s, and Thames River flow 

was 637 ft3/s. The water level specified for the boundary at the 

Fort Wayne, Mich. was 573.19 ft (IGLD 85). 

In this report, wind characteristics are summarized on 

the basis of LSCM4 data obtained from November 8, 2001 

(the first available data), to October 23, 2003, when this 

information was initially compiled. Vector averages of data at 

10-minute intervals were used to compute daily mean values, 

which were summarized in a wind chart (fig. 15). In this chart, 

wind directions were divided into 24 sectors, each having an 

angular width of 15 degrees. Sectors are numbered counterclockwise, starting at the sector from 90 to 75 degrees east of 

north. Within each sector, annuli describe the percentage of 

time that daily mean wind speeds were within specified 5-mi/h 

intervals. Average wind speeds and directions were computed 

for each sector. Average wind directions were transformed 

following the mathematical convention for input into RMA2, 

which describes the direction that winds are blowing toward 

and references angles counterclockwise from the positive 

x-axis (zero degrees east). Using the compass convention, a 

wind reported by NDBC from the southeast, at 130 degrees, 

for example, would be described in a mathematical convention 

as a wind blowing toward the northwest at 140 degrees for use 

in RMA2. 

In this analysis, particle tracking scenarios were used to 

identify flow paths and traveltimes for idealized steady-state 

conditions, where simulated current patterns on Lake St. Clair 

varied only as a function of wind sector. In the 24 scenarios, 

hypothetical particles were initially placed near Memorial and 

Metropolitan Beaches corresponding to their ending locations. 

Then, reverse particle tracking was used to map expected flow 

paths backward in time towards source areas. Expected flow 

paths do not reflect the increasing uncertainty of particle locations with increasing simulated traveltime. 

Subsets of contiguous wind sectors produced similar 

flow paths. For Memorial Beach, sector 1 simulation results 

(fig. 16) are similar to results from sectors 1–3, and 20–24, 

where particles generally maintain a southwestern flow direction along the northwestern shoreline of Lake St. Clair. In contrast, results from sector 9 and 14 simulations, which are representative of sectors 4–14, show particles that tend to initially 

flow in a southerly direction just east of L’anse Creuse Bay but 

revert to a northerly flow direction along the western shoreline 

of Lake St. Clair as they passed south of L’anse Creuse Bay. 

Sector 15 simulation resulted in a recirculation pattern within 

L’anse Creuse Bay. Finally, results from sector 17, which is 

representative of sectors 16–19 are similar to those for sector 1 

with the exception that these sectors circumvent a circulation 

pattern south of Metropolitan Beach area. Markers are placed 

along flow paths at 12-hour increments for 168 hours from 

their destination near the beaches or until they were simulated 

out of the model domain. 

For Metropolitan Beach, particle tracks resulting from 

simulations with sector 1 winds are similar to those for sectors 1–8 and 19–24 (fig. 17). In these simulations, flows are 

southerly past Clinton River and along the north shore of 

Anchor Bay from St. Clair River. Sector 10 is representative of 

sectors 9–11, where simulated particles flow are northerly past 

Cutoff Canal and Memorial Beach after flowing toward the 

south east of L’anse Creuse Bay. Finally, results for sector 15 

are representative of those for sectors 12–18, which result in a 

simulated recirculation pattern within L’anse Creuse Bay south 

of Metropolitan Beach. 

0o

45o

90o

135o

180o

225o

270o

315o

0% 2%

Sector 1

Sector 24

Sector 6

Wind speed, in miles per hour

Less than 5

5 to 9.9

10 to 14.9

15 to 19.9

Greater than 19.9

EXPLANATION

0 percent

2 percent

4 percent

6 percent

8 percent

Figure 15. Wind characteristics, by sector, on Lake St. Clair 

based on data from Lake St. Clair Monitoring Station LSCM4 in 

Michigan. (The location of station LSCM4 is shown on fig. 1.) 
22 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

Figure 16. Simulated reverse particle paths from Memorial Beach for selected wind sectors on Lake St. Clair, Mich. 

Grosse

Pointe

Shores

New Baltimore

MICHIGAN

+

++

++

+

+

+

13,500,000

EASTING, IN INTERNATIONAL FEET WITHIN THE MICHIGAN SPCS 83 SOUTHERN ZONE

13,550,000 13,600,000

440,000

420,000

400,000

380,000

360,000

340,000

LSCM4 Weather Station

Cutoff

Canal

Clinton River

Metropolitan Beach

Memorial Beach

North Channel

South Channel

Anchor Bay

Lake St. Clair

L’anse Creuse Bay

NORTHING, IN INTERNATIONAL FEET WITHIN THE MICHIGAN SPCS 83 SOUTHERN ZONE

Wind

sector

EXPLANATION

Particle

Path

Wind speed

(mi/hr)

Wind direction

(Counter clockwise

from 0 degrees east)

189

309

23.9

37.3

66.4

1 12.1

9 9.38

14 14.2

15 15.2

17 14.6

Distance between markers along

paths shows 12-hour traveltimes

0 10,000 20,000 30,000 feet
Relation of Environmental Factors to Escherichia coli Concentrations 23

Figure 17. Simulated reverse particle paths from Metropolitan Beach for selected wind sectors on Lake St. Clair, Mich. NORTHING, IN INTERNATIONAL FEET WITHIN THE MICHIGAN SPCS 83 SOUTHERN ZONE

Grosse

Pointe

Shores

New Baltimore

MICHIGAN

0 10,000 20,000 30,000 feet

Wind direction

(Counter clockwise

from 0 degrees east)

13,500,000 13,550,000 13,600,000

440,000

420,000

400,000

380,000

360,000

340,000

LSCM4 Weather Station

Cutoff

Canal

Clinton River

Metropolitan Beach

Memorial Beach

North Channel

South Channel

Anchor Bay

Lake St. Clair

L’anse Creuse Bay

Wind

sector

1

10

15

EXPLANATION

Particle

Path

Wind speed

(mi/hr)

189

324

37.2

12.1

8.99

15.2

Distance between markers along

paths shows 12-hour traveltimes

EASTING, IN INTERNATIONAL FEET WITHIN THE MICHIGAN SPCS 83 SOUTHERN ZONE
24 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

Transient Simulations

Transient flow simulation and reverse particle-tracking 

analyses were used to map flow paths from beach areas 

towards source areas for 10 selected E. coli sampling events. 

These events took place on Memorial and Metropolitan 

Beaches on Wednesday mornings (table 4). Data for both 

beaches included samples where E. coli concentrations were 

less than 20 and greater than 1,000 MPN/mL. There is no 

apparent serial correlation in the bacteria data, nor is there 

significant cross-correlation between E. coli concentrations 

sampled on the same day at the two beaches. 

Transient boundary conditions were used to simulate 

hourly flows during the selected E. coli sampling events. 

Hourly flows of St. Clair River were computed on the basis 

of the standard stage-fall-discharge rating, which was developed by the U.S. Army Corps of Engineers (John Koschik, 

U.S. Army Corps of Engineers, written commun., 2005) and 

supported by hourly water-level information from NOAA. 

Hourly flows for Clinton River were based on data from the 

USGS streamflow-gaging station on Clinton River at Mount 

Clemens, Mich. (station 04165500). Hourly water-level data 

were from the NOAA gage at Fort Wayne, and wind conditions were from the Lake St. Clair, Mich., weather station 

(LSCM4). Constant flows were specified for the Sydenham 

and Thames Rivers in Canada. 

Particle tracks at both Memorial and Metropolitan 

Beaches generally move laterally along the beach; but unlike 

steady-state simulations, particles can reverse directions along 

the same flow path and can cross flow paths from other events 

in response to transient wind and flow conditions. Visualizing the full results from a static display is problematic. On 

figures 18 and 19, simulated flow paths were color-coded by 

bacteria-concentration intervals. The complexity of the flow 

paths and the irregularity of the relation between flow path and 

bacteria concentrations confirms the need for model simulations and particle tracking, if mapping flow paths associated 

with individual sampling events is desired. 

Numerous potential sources of E. coli bacteria to Memorial Beach are within the probable paths indicated by hydrodynamic simulations. Sources include the Clinton River Cutoff 

Canal (Spillway), for which Macomb County Health Department data show bacteria concentrations > 1,000 MPN/100 mL 

for most samplings that follow rainfall events. However, 

16 drains, and 2 small rivers also deliver water to the shoreline 

within 3 mi north or south of Memorial Beach (Buzonik and 

Shoemaker, 2004). MCHD has evaluated nearshore bacteria 

concentrations (offshore of each of these drains) for several 

years. Annual geometric mean concentrations tend to be well 

below the recreational water-quality standard for the nearshore 

sampling sites (Buzonik and Shoemaker, 2004), but data for 

the water quality in the drains themselves is not available. 

Given that particle tracks indicate movement up and down this 

shoreline for most hydrodynamic conditions and that dynamic 

regression models indicate both E. coli concentrations 2 days 

presampling and postsampling are significant variables, it is 

likely that these numerous sources contribute to Memorial 

Beach water quality. 

Steady-state hydrodynamic simulations indicated that 

when winds were from the WNW. or NNW. (sector 1), particles generally originated near the mouth Clinton River within 

24 hours of their sampling at the beach. An exceedance of 

the recreational water-quality standard at Metropolitan Beach 

occurred on only 1 of the 10 dates for which transient hydrodynamic particle tracking was done August 25, 2004. On this 

date, the average wind direction for the preceding 24 hours 

was from the WNW., and there had been 0.38 in. of rain. Particle tracks for the August 25 event indicate particles originating from along the shoreline east of the beach, as well as from 

the Clinton River, within the preceding 24 hours. It appears 

likely that the source of contamination to Metropolitan Beach 

on this date was the Clinton River. 

Table 4. Selected E. coli sampling events on Memorial and Metropolitan Beaches on Lake St. Clair, Mich. 

[E. coli, Escherichia coli; CFU/100 mL, colony-forming units per 100 milliliters]

Date of 

E. coli

sample

Memorial Beach Metropolitan Beach Simulation 

start time (EST) 

on date of sample

Number of 

simulated 

hours

Time of sample 

(EST)

E. coli 

concentration 

(CFU/100 mL)

Time of sample 

(EST)

E. coli

concentration 

(CFU/100 mL)

May 26, 2004 8:34 a.m. 982 9:32 a.m. 141 9:00 a.m. 145

June 9, 2004 7:20 a.m. 39 7:55 a.m. 31 9:00 a.m. 141

June 21, 2004 7:16 a.m. 70 7:48 a.m. 170 9:00 a.m. 141

July 28, 2004 7:32 a.m. 16 8:22 a.m. 3 9:00 a.m. 141

Aug. 4, 2004 7:30 a.m. 1,150 8:24 a.m. 52 9:00 a.m. 141

Aug. 11, 2004 7:28 a.m. 18 8:18 a.m. 10 9:00 a.m. 141

Aug. 18, 2004 7:26 a.m. 17 8:18 a.m. 122 9:00 a.m. 141

Aug. 25, 2004 7:20 a.m. 37 8:02 a.m. 1,037 8:00 a.m. 137

Sept. 1, 2004 7:54 a.m. 172 8:42 a.m. 14 9:00 a.m. 141

Sept. 29, 2004 8:25 a.m. 340 9:16 a.m. 4 9:00 a.m. 141
Relation of Environmental Factors to Escherichia coli Concentrations 25

Particle tracks

EASTING, IN INTERNATIONAL FEET WITHIN THE MICHIGAN SPCS 83 SOUTHERN ZONE

Lake St. Clair

LSCM4 Weather Station

> 500

<= 500 to > 50

<= 50 to > 10

a

Clinton River

Cutoff Canal

Metropolitan Beach

Memorial

Beach

13,520,000 13,540,000 13,560,000 13,580,000

340,000

360,000

380,000

400,000

L’anse

Creuse Bay

Macomb County,

MICHIGAN

NORTHING, IN INTERNATIONAL FEET WITHIN THE MICHIGAN SPCS 83 SOUTHERN ZONE

EXPLANATION

Particle

track

E. coli concentration

(MPN / 100 mL)

0 10,000 20,000 feet

St. Clair Haven

Grosse

Pointe

Shores

Figure 18. Reverse particle tracks for selected sampling events in 2004 starting from Memorial Beach on Lake St. Clair, Mich. 
26 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

> 500

<= 500 to > 50

<= 50 to > 10

<= 10

EXPLANATION

Particle

track

E. coli concentration

(MPN / 100 mL)

0 10,000 20,000 feet

St. Clair Haven

Grosse

Pointe

Shores

a

Clinton River

Cutoff Canal

Metropolitan Beach

Memorial

Beach

LSCM4 Weather Station

13,520,000 13,540,000 13,560,000 13,580,000

340,000

360,000

380,000

400,000

Lake St. Clair

L’anse

Creuse Bay

Particle tracks

Macomb County,

MICHIGAN

EASTING, IN INTERNATIONAL FEET WITHIN THE MICHIGAN SPCS 83 SOUTHERN ZONE

NORTHING, IN INTERNATIONAL FEET WITHIN THE MICHIGAN SPCS 83 SOUTHERN ZONE

Figure 19. Reverse particle tracks for selected sampling events in 2004 starting from Metropolitan Beach on Lake St. Clair, Mich. 
Relation of Environmental Factors to Escherichia coli Concentrations 27

Additional Analysis of Period Covered by Transient 

Hydrodynamic Simulation

To further explore factors that may influence bacteria 

concentrations at Memorial and Metropolitan Beaches, the 

period from May 1 to September 30, 2004, was examined in 

more detail. This period encompasses the 10 transient particletracking simulations for the 2 beaches. 

Sky cover (cloudiness or solar radiation) can influence 

bacteria concentrations in water (Davies-Colley and others, 

1994). Between May 1 and September 30, 2004, 41 bacteria 

samplings were done at Memorial Beach and 46 at Metropolitan Beach. Table 5 lists the distribution of these samplings 

and the maximum E. coli concentrations for various sky-cover 

conditions. There were no exceedances of the recreational 

water-quality criterion when clear skies prevailed at the 

time of sampling. Preceding 24 and (or) 72 hour rainfall was 

statistically significant factor in several regression models. 

Therefore, weather conditions appear to affect the concentrations of E. coli at these beaches, but whether this is due to 

increased sunlight or effects of rainfall or a combination of 

both is uncertain. 

Changes in water level induce dynamic flow patterns 

within beach sands and may wash material collected along the 

shoreline (algae, bird feces) into the nearshore waters. Several 

studies have shown that E. coli are retained in nearshore 

beach sand and may affect swimming waters (Alm and others, 

2003; Haack and others, 2003; McLellan and Salmore, 2003; 

Whitman and others, 2003; Francy and others 2006). Changes 

in water level, coupled with wind patterns, may induce flow 

reversals in the Clinton River, sometimes for significant distances upstream (W.F. Baird & Associates, 2005). Figure 20 

shows the relation between atmospheric pressure and water 

level at the St. Clair Shores recording station for May 2004. 

A shift from low water levels in winter/early spring to higher 

water levels in early summer is common in Lake St. Clair 

(Bolsenga and Herdendorf, 1993) and is evident in figure 20. 

The figure also shows a general pattern of decreasing atmospheric pressure corresponding to a general increase in water 

level towards the end of the month (May 2005), consistent 

with the passage of weather systems across the Great Lakes. 

Figure 21 shows rainfall and E. coli concentrations at 

Memorial and Metropolitan beaches, along with water levels 

for the 2004 during this same period. E. coli concentrations 

at both beaches increased generally as water levels increased 

in May. This relation may result from washing of materials 

accumulated on the shoreline during previously low water 

levels. Superimposed on this increase were several substantial rain events. E. coli concentrations at Memorial Beach 

increased rapidly in response to rain, but E. coli concentrations at Metropolitan Beach increased less dramatically. This 

difference might result from the greater density of sources 

(drains and rivers) surrounding Memorial Beach as opposed to 

Metropolitan Beach. It appears that other factors affect beach 

waters at Memorial and Metropolitan beaches, as is evident by 

exceedances at both beaches during periods of no precipitation 

(May 17 at Memorial Beach and August 25 at Metropolitan 

Beach; figs. 21 and 23). 

June and July water levels were more stable than in May 

(fig. 22). There was little rainfall in these months, and few 

there were exceedances of the recreational water-quality standard at either beach. In August and September, water levels 

typically start to decline as discharge from rivers begins to 

decrease (Bolsenga and Herdendorf, 1993). E. coli concentrations at both beaches generally followed this decline (fig. 23). 

The MCHD found that E. coli concentrations in foreshore 

sand at Metropolitan Beach were correlated with beach-water 

E. coli concentrations during sampling in 2004 (Buzonik 

and Shoemaker, 2004). No such correlation was reported for 

Memorial Beach. An increase in water level at this beach 

might have entrained E. coli in sand or materials (algae, gull 

feces) deposited on the beach during the preceding dry period 

(in this case, 11 days). Although particle tracking indicated 

the Clinton River as a probable source for the August 25 

exceedance, beach sources also may have contributed to this 

event.

Table 5. Number of sampling events and maximum E. coli concentrations associated with various sky-cover conditions, May 1 to 

September 30, 2004, on Lake St. Clair, Mich.

Beach

Sky cover

Clear Scattered 

(1/8–4/8 cover)

Broken 

(5/8–7/8 cover) Overcast

Memorial Beach

Number of samplings 8 15 12 6

Maximum E. coli

concentration

245 1,167 1,418 363

Median E. coli

concentration

73 71 57 90

Metropolitan Beach

Number of samplings 8 14 13 11

Maximum E. coli

concentration

172 6,881 982 1,150

Median E. coli

concentration

42.5 130 42.5 21
28 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

5/1/04 5/8/04 5/15/04 5/22/04 5/29/04

DATE (MM/DD/YR)

990

1,000

1,010

1,020

1,030

1,040

ATMOSPHERIC PRESSURE, IN MILLIBARS

573.0

573.5

574.0

574.5

575.0

575.5

WATER LEVEL, IN FEET ABOVE IGLD85

Atmospheric pressure

Water level

5/1/04 5/5/04 5/9/04 5/13/04 5/17/04 5/21/04 5/25/04 5/29/04 6/2/04

DATE (MM/DD/YR)

0.0

1.0

2.0

3.0

4.0

5.0

 RAINFALL, IN INCHESLOG 10 E. COLI CONCENTRATION

573.0

573.5

574.0

574.5

575.0

575.5

WATER LEVEL, IN FEET ABOVE IGLD85

Metropolitan Beach E. coli concentration

Memorial Beach E. coli concentration 

Rainfall

Michigan recreational water-quality standard

Water level

Figure 20. Atmospheric pressure and water levels in May 2004 at St. Clair Shores, Mich. 

Figure 21. Rainfall, E. coli concentrations, and water levels in May 2004 on Lake St. Clair, Mich.
Relation of Environmental Factors to Escherichia coli Concentrations 29

6/1/04 6/8/04 6/15/04 6/22/04 6/29/04 7/6/04 7/13/04 7/20/04 7/27/04

DATE (MM/DD/YR)

0.0

1.0

2.0

3.0

4.0

5.0

 RAINFALL, IN INCHESLOG 10 E. COLI CONCENTRATION

573.0

573.5

574.0

574.5

575.0

575.5

WATER LEVEL, IN FEET ABOVE IGLD85

Metropolitan Beach E. coli concentration 

Memorial Beach E. coli concentration

Rainfall

Michigan recreational water-quality standard

Water level

8/3/04 8/10/04 8/17/04 8/24/04 8/31/04 9/7/04 9/14/04 9/21/04 9/28/04

DATE (MM/DD/YR)

0.0

1.0

2.0

3.0

4.0

5.0

 RAINFALL, IN INCHESLOG 10 E. COLI CONCENTRATION

573.0

573.5

574.0

574.5

575.0

575.5

WATER LEVEL, IN FEET ABOVE IGLD85

Metropolitan Beach E. coli concentration 

Memorial Beach E. coli concentration

Rainfall

Michigan recreational water-quality standard

Water level

Figure 22. Rainfall and water levels in June and July 2004 on Lake St. Clair, Mich.

Figure 23. Rainfall and water levels in August and September 2004 on Lake St. Clair, Mich.
30 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

Suggestions for Future Studies

This study has demonstrated that circulation patterns and 

associated flow paths to Memorial and Metropolitan Beaches 

on Lake St. Clair are transient phenomena that are sensitive to 

variable wind conditions and uncertain dispersion characteristics. This sensitivity and uncertainty demonstrates the utility 

of hydrodynamic simulation for identifying flow paths, travel 

and exposure times, and source areas of bacteria to beaches. 

Changes in the flow paths alone may change the concentration or activation of E. coli bacteria during transit as a result of 

dieoff from solar radiation or other factors (Davies-Colley and 

others, 1993). 

The massless hypothetical particles used in this report 

have limitations for identifying source areas of bacteria. 

Bacteria may be preferentially associated with particles having 

mass rather than moving strictly with the advective component 

of flow. Settling of these actual particles may significantly 

reduce bacteria concentrations with time and travel distances. 

Similarly, resuspension of these particles may be associated 

with wind and wave conditions on Lake St. Clair, which may 

be unrelated to bacteria concentrations in source areas, unless 

source areas are defined as anywhere along historic flow paths. 

Despite these limitations, particle tracking yields information on the expected flow paths, traveltimes, and the 

integrative effects of transient hydrodynamic conditions 

on particle movement that is associated with the transport 

of bacteria. This information may be critical for identifying source areas, particularly if bacteria discharges in these 

areas are highly variable overtime. MCHD bacteria samples 

acquired at beaches and other selected areas at the rate of 

two per week have shown substantial variability and limited 

temporal correlation. Given the temporal variability of bacteria 

concentrations at the beaches, however, suspected source 

areas or beaches would need to be monitored more frequently 

than twice a week. Otherwise, the uncertainty in interpolating 

bacteria concentrations between samples would diminish the 

usefulness of particle tracking. Sampling only twice per week, 

therefore, it is difficult to relate specific samples of E. coli

concentrations at beaches with samples at suspected source 

areas. 

Future investigations of the temporal variation in bacteria 

concentrations would benefit from event sampling; that is, a 

series of frequent samplings (every few hours) at suspected 

source areas during two or more rainfall-runoff events when 

large variations in concentrations were expected over a short 

period of time. In addition, similar sampling would be needed 

at both beaches—in addition to suspected source areas—so 

that the dynamics of E. coli concentrations can be characterized. This type of data could help improve understanding of 

the uncertainty of the simulated flow paths. 

Loading rate is the product of flow and concentration 

data. For measurements of bacteria concentrations near the 

USGS streamflow-gaging station on Clinton River at Mount 

Clemens, Mich. (station 04165500), supporting streamflow 

data are available at 15-minute intervals. At other sites, such 

as the spillway on the Cutoff Canal, it may be possible to 

compute unit streamflow values on the basis of hydraulic 

information and continuous water-level data. Supporting measurements of E. coli at nearshore and offshore sites on Lake 

St. Clair would help verify dispersion and mixing characteristics. 

The hydrodynamic model of Lake St. Clair developed 

for this analysis could serve as a basis for future water-quality 

investigations. In particular, the velocity field simulated by an 

RMA2-based hydrodynamic model is the primary input to an 

RMA4-based water-quality model. RMA4 (Letter and others, 

2005) is a finite-element water-quality transport code that can 

simulate a depth-averaged advection-diffusion process in an 

aquatic environment. Within this framework, RMA4 could be 

used to model constituent concentrations that are conservative 

or that have dynamics that can be described by a first-order 

decay process. 

Future efforts to develop a statistical model that relates or 

predicts bacteria concentrations at beaches from environmental 

factors may benefit from these hydrodynamic simulations. To 

the extent that flow paths are associated with source areas, it 

may be helpful to develop separate statistical equations within 

defined sets of contiguous wind sectors. As in this report, the 

wind sector for a specific bacteria sampling event may be 

identified by use of wind data from the LSCM4 weather station for the 24-hour period preceding sample collection. As a 

starting point for model development at Memorial Beach, four 

distinct equations may be appropriate for winds for sectors 

1–3, 20–24, 4–14, 15, and 16–19. At Metropolitan Beach, 

three distinct equations may be appropriate for sectors 1–8, 

19–24, 9–11, and 12–18. Final determination of the number 

of equations and partitioning of sectors would be based on 

statistical considerations. 

Additional factors may influence both beaches and could 

be considered in future studies. Bathers at the beach are a 

source of E. coli bacteria, both to beach sands and to the water 

itself (Calderon and others, 1991). Warmer temperatures likely 

result in an increase in the number of bathers, which may 

increase water turbidity. No data are available on the number 

of bathers associated with various samples. Likewise, there 

is no information on the numbers of birds on the beach or on 

aggravating factors such as algal mats along the shore. These 

factors have been implicated at other beaches as potential 

sources of E. coli (Whitman and Nevers, 2003; Whitman and 

others, 2003; Olapade and others, 2006). Beach-grooming 

practices may influence beach-water bacteria concentrations 

(Kinzelman and others, 2004). Francy and others (2006) used 

microbial source-tracking techniques and modeling to show 

the influence of temperature, turbidity, and fecal sources on 

selected Lake Erie bathing beaches. Any of these factors may 

account for occasional high concentrations of E. coli bacteria, 

and may be correlated with factors that were significant in 

regression models. 
Summary and Conclusions 31

Summary and Conclusions 

In cooperation with the Michigan Department of Environmental Quality, the USGS analyzed environmental data 

and circulation patterns on Lake St. Clair, which is part of the 

connecting channel between Lake Huron and Lake Erie in the 

Great Lakes Basin. The analysis identified factors associated 

with E. coli concentrations at Memorial and Metropolitan 

Beaches, in Macomb County, Mich. Data from 2002–2005 

provided the basis for development of linear regression models, regression trees, and logistic regressions to help identify 

environmental factors that were statistically associated with 

E. coli concentrations. A hydrodynamic and particle-tracking 

model was used to identify flow paths in Lake St. Clair that 

may help identify potential source areas of E. coli to the study 

beaches. This study has demonstrated that circulation patterns and associated flow paths to Memorial and Metropolitan 

Beaches on Lake St. Clair are transient phenomena that are 

sensitive to variable wind conditions and uncertain dispersion 

characteristics. This sensitivity and uncertainty demonstrates 

the utility of hydrodynamic simulation for identifying flow 

paths, travel and exposure times, and source areas of bacteria 

to beaches. The hydrodynamic and statistical analyses can be 

used in combination by regulatory agencies for the development of Total Maximum Daily Loads (TMDLs) by helping 

to identify source areas and conditions that may influence 

exceedances of the daily Michigan recreational water-quality 

standard at these beaches. 

Exploratory analysis indicated that E. coli samples 

obtained by MDEQ in 2003 and 2004 tended to have lower 

concentrations than samples obtained by MCHD during the 

same period. Differences in exact sampling locations and 

procedures, and in analytical techniques between agencies 

may have contributed to the differences between concentration 

characteristics. MCHD data were used in this report because 

they had been collected over a longer period and at more frequent intervals than the MDEQ data. 

The statistical distributions of E. coli concentrations 

at both Memorial and Metropolitan Beaches are positively 

skewed, which is manifested by numerous low and medium 

concentrations interspersed with infrequent high values. For 

consistency with assumptions underlying statistical models 

used to associate environmental factors with concentration data, a log10 transformation was applied to the E. coli

data before analysis. The distribution of the log-transformed 

E. coli data closely approximated a normal distribution. 

Wind is a primary determinant of circulation patterns 

on Lake St. Clair, and may be associated with differences in 

E. coli concentrations because of changes in flow paths and 

potential source areas. To assess this potential effect, E. coli

concentration data were grouped into eight 45-degree winddirection categories. Results of statistical analyses indicated 

no differences in median E. coli concentrations among winddirection categories at Memorial Beach, whereas significant 

differences were detected from data at Metropolitan Beach. In 

particular, the median E. coli concentration in the east-northeast (ENE.) sector is significantly greater than the median 

concentrations in the west-southwest (WSW.), west-northwest 

(WNW.), and north-northwest (NNW.) sectors. In addition, the 

median concentration in the south-southwest (SSW.) sector is 

significantly greater than median concentrations in the WNW. 

and NNW. sectors. 

Results from linear regression analyses indicated that the 

amount of rainfall prior to E. coli sampling, the temperature 

of the water, and the turbidity of the water were positively 

associated with log10 E. coli concentrations at both beaches. 

Flow from Clinton River also was positively associated with 

concentrations at Memorial Beach and the hourly change in 

water level preceding sampling was negatively associated with 

concentrations at Metropolitan Beach. Some of the first-order 

interactions among “main-effects” model variables were 

marginally significant at Memorial Beach in the “interactioneffects” model. In addition, log10 E. coli concentrations 

measured 2 days before or after concentrations at Memorial 

Beach were statistically significant when added to the maineffects model. This result indicates that, for Memorial Beach, 

E. coli concentrations in conjunction with other environmental variables may help forecast E. coli concentrations at this 

beach. The fact that postsampling E. coli concentrations were 

also significant in the model suggests that complex processes 

may transport bacteria back and forth from the beach area. 

A north wind direction was identified as a significant factor 

for estimating log10 E. coli concentrations at Metropolitan 

Beach when added to the main-effects model, although wind 

speed was not a significant factor. Linear-regression models 

developed for Memorial and Metropolitan Beaches explained 

only about 30 percent of the variability in log10 E. coli concentrations. From this result, it is evident that several factors 

or a combination of events that were not accounted for in the 

models contributed to bacteria concentrations at these beaches. 

Regarding the regression-tree models developed to 

estimate log10 E. coli concentrations from data collected at 

Memorial and Metropolitan Beaches, the apparent estimation 

error generally decreased with the number of terminal nodes 

in these models. A statistical resampling analysis indicated, 

however, that the apparent estimation error was not consistent 

with the prediction error, which tended to increase slightly 

for models with more than two terminal nodes. Models with 

two terminal nodes had little explanatory capability. These 

results are interpreted as indicating that too few observations of E. coli concentrations were available for developing 

regression-tree models in this study. Regression-tree analysis 

is commonly used to define variables associated with high 

E. coli concentrations. The results of this study show some 

limitations of this analysis and the need for robust datasets for 

the model to be useful. 

Results from the logistic regression models, which were 

developed to estimate the probability that E. coli concentrations would exceed 300 MPN/100 mL at Memorial and Metropolitan Beaches, showed that rainfall preceding the sampling 

events was positively associated with the probabilities of bac-
32 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

teria exceedances at both beaches. Flow in the Clinton River 

was positively associated with bacteria exceedances at Memorial Beach but negatively associated with bacteria exceedances 

at Metropolitan Beach. For Memorial Beach, turbidity and 

log10 E. coli concentrations measured two days before or after 

concentrations at Memorial Beach were positively associated 

with bacteria exceedances. The logistic models were effective 

in estimating bacteria exceedances at both beaches. A receiver 

operation characteristics (ROC) analysis identified cut points 

for maximizing the true positive prediction rate while minimizing the false positive rate. 

The hydrodynamic model of Lake St. Clair was developed from an existing 2D hydrodynamic model of the St. 

Clair-Detroit River Waterway. Steady-state simulations were 

used to depict velocity fields on Lake St. Clair as a function 

of wind direction. Reverse particle tracking was used with 

these fields to describe associated flow paths from the beaches 

towards source areas. Sets of contiguous wind sectors were 

associated with similar flow paths. Transient simulations were 

used to describe changing velocity fields on Lake St. Clair 

during periods preceding 10 selected bacteria sampling events 

in 2004. Reverse particle tracking was used with these velocity 

fields to identify expected flow paths of bacteria from beaches 

for specific E. coli sampling events. 

The main focus of this study was to determine what 

factors may influence or explain E. coli concentrations at the 

study beaches. Some significant explanatory variables from 

the linear regression analysis for the two beaches were different, but the common significant explanatory variables for 

both beaches were rainfall, water temperature, and turbidity. 

Several other studies have documented similar factors affecting beaches (Nevers and Whitman, 2005; Francy and others, 

2006). The study of beach watersheds is complex and needs to 

be associated with numerous water bodies (rivers and creeks) 

as well as an extensive and complex drainage system, such 

as outfalls that drain to Lake St. Clair directly or indirectly 

through the Clinton River. Rainfall as a significant variable 

may indicate that storm runoff or erosion is influencing the 

E. coli concentrations in nearshore beach water. Discharge 

was an explanatory variable for Memorial Beach; rainfall 

can increase stream discharge, which further corroborates 

this hypothesis. Turbidity also was a commonly included 

explanatory variable during model development for the study 

beaches; turbidity itself is influenced by rainfall, runoff, and 

high waves, and turbidity has historically been associated with 

increased E. coli concentrations. Water temperature may be 

an indication of seasonality; the highest E. coli concentrations 

were in August, with the highest water temperatures in July 

and August. Water temperature may also serve as a surrogate 

indicator for variables not addressed in the data set (such as 

algal accumulations, beach visitors, or waterfowl). 

At Memorial Beach, little systematic variation was 

observed in bacteria concentrations with respect to wind 

direction or magnitude. At least one exceedance of the 

300 MPN/100 mL criterion was observed for each 45o wind 

sector, and the median concentration for each wind sector was 

not statistically different. Hydrodynamic models indicated 

that, under most wind conditions, particles sampled at Memorial Beach would generally have traveled along the shoreline to 

the north and south of the beach or recirculated within L’anse 

Creuse Bay within the preceding 24 to 48 hours. 

Linear regression models indicated that rainfall in the 

preceding 72 hours, water temperature, turbidity, and discharge of the Clinton River were positively correlated with 

bacteria concentrations at Memorial Beach. Logistic regression models indicated that an increase of 1 in. of rain over the 

preceding 72-hour period more than doubles the probability 

that E. coli concentrations will exceed the 300 MPN/100 mL 

standard. E. coli concentrations 2 days before and 2 days after 

a sampling event were correlated at Memorial Beach. The 

limited spatial extent of particle tracks over 48- to 72-hour 

timeframes—and the several simulations indicating particle 

movement in first one direction, then in the opposite direction along the shoreline—may explain correlation between 

E. coli concentrations on successive dates at Memorial Beach. 

The tendency for water to move in different directions along 

the shoreline near Memorial Beach may result in a relatively 

constant and perhaps well-mixed source, regardless of wind 

direction.

Significant explanatory variables in the regressions 

may indicate a similar process influencing Memorial Beach. 

Clinton River discharge is likely a surrogate variable for the 

discharge from the drains and creeks in the area, and from 

the Clinton River Spillway after rainfall. Water draining from 

these sources after rainfall carries a relatively high load of suspended material. MCHD sampling data have shown a statistically significant relation between turbidity and E. coli concentrations nearshore to some of the drains and to the mouth of 

the Clinton River. 

In contrast to Memorial Beach, bacteria concentrations 

at Metropolitan Beach varied with respect to wind direction. Exceedances of the 300 MPN/100 mL standard were 

noticeably absent when wind was from the WNW. or NNW. 

Samplings during times when winds were from the SSW. and 

ESE. accounted for 7 of 14 exceedances. The median E. coli

concentration was greatest when winds were from the ENE. 

Steady-state hydrodynamic simulations indicated that when 

winds were from the WNW. or NNW. (sector 1), particles 

generally originated near the mouth of Clinton River within 

24 hours of when they were sampled at the beach. In contrast, 

when winds were from the ESE., particles originated from 

along the shoreline to the west or north of Memorial Beach 

within 24 hours of when they were sampled. When winds 

were from the northeast, a recirculation pattern within L’Anse 

Creuse Bay developed also might include particles from the 

western shoreline of Lake St. Clair. 

Linear regression models indicated that rainfall in the 

preceding 24 hours, water temperature and turbidity, but 

not Clinton River discharge, were positively correlated with 

bacteria concentrations at Metropolitan Beach. In addition, 

and in contrast to Memorial Beach, linear regression models 

indicated change in water level was inversely correlated with 
References Cited 33

bacteria concentrations at Metropolitan Beach. Logistic regression models indicated that an increase of 1 in. of rain over the 

preceding 24-hour period increases the probability that E. coli

concentrations will exceed the 300 MPN/100 mL standard 

more than 30 times. In contrast, an increase in discharge of 

the Clinton River was associated with a decreased probability 

that E. coli concentrations would exceed the standard. E. coli

concentrations 2 days before and 2 days after a sampling 

event were not correlated at Metropolitan Beach. Metropolitan 

Beach appears to be subject to short-term, transient events 

of poor water quality that are not necessarily associated with 

the Clinton River as a source. Spikes in E. coli concentrations 

occurred at the beach without always being associated with 

rainfall. These may be occasions when wind directions are 

appropriate for entrainment of bacteria trapped within beach 

sands or when contaminants may be drawn from sources to the 

west and southwest of the beach. The hydrodynamic simulations were based on a 2D model and depict the movement of 

massless particles. As noted previously, these models may not 

capture the full complexity of water circulation patterns or of 

bacterial transport in the nearshore. A 3D model might show 

additional complexities or water-shoreline interactions. 

The MCHD found that E. coli concentrations in foreshore sand at Metropolitan Beach were correlated with E. coli

concentrations in water near the beach during sampling in 

2004 (Buzonik and Shoemaker, 2004). No such correlation 

was reported for Memorial Beach. Winds from the south 

might force water onto the shoreline at Metropolitan Beach, 

perhaps entraining some of the bacteria from the foreshore 

sand. It is possible that this source might account for the association between winds from the southwest to southeast and 

exceedances of E. coli criteria. In addition, the median E. coli

concentration was highest at Metropolitan Beach for winds 

from the ENE., which were associated with a recirculation pattern in the bay. This recirculation might entrain bacteria from 

some of the same sources that influence Memorial Beach. 

Even though most particle tracks in the hydrodynamic models 

eventually move from Metropolitan Beach to the Clinton 

River within 24 to 48 hours, it appears that circulation patterns 

bringing particles from the shoreline to the west (or perhaps 

the sands themselves) may have greater influence on bacteria 

concentrations. 

Acknowledgments 

The authors gratefully acknowledge the assistance of 

Mr. James Buzonic, Manager, Macomb County Health Department, Environmental Health Services Division, for providing 

access to the division’s E. coli and supporting data. Donna S. 

Francy, USGS Ohio Water Science Center (WSC), Norman G. 

Grannemann, USGS Michigan WSC, and David L. Lorenz, 

USGS Minnesota WSC, provided technical reviews. Michael 

Eberle, USGS Columbus Publishing Service Center, provided 

editorial review. Dorothy H. Tepper, USGS Northeastern 

Region, recommended that the report be approved for publication by the Director of the USGS. 

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Appendix 1 37

Linear Regression

Linear regression provides a statistical model for estimating the continuous variation of E. coli concentrations as a 

function of environmental factors (explanatory variables). The 

general form of a linear regression model is 

y X = × b e + (1)

where 

y is a column n-vector of response variables, 

X is a n by (p+1) design matrix that is composed 

of p n-vectors, x1,x2,. . .,xp, of explanatory 

variables augmented with a leading column 

of ones,

	 β	 is a p-vector of coefficients, b b b 0 1 , ,..., p

with ordinary least-squares estimator 

bols = X X¢ × × X y ¢ × - ( ) 1 , 

and 

e is an n-vector of residuals that is assumed to 

be normally distributed and independent 

with mean zero and constant variance s2

,

which is commonly written e s  NI( , 0 ). 2

In addition, the regression model is based 

on the assumption that the covariance 

between e and X, Cov X ( , e ), equals zero. 

Finally, the design matrix X must be of full 

rank (p+1) < n, so that ( ) ¢ × -

X X 1

 exists. 

If these conditions are met, the predicted 

values, y X ˆ = × ols b , are the unbiased, 

minimum variance linear estimates of y. 

The multiple coefficient of determination, R2

, 

describes the fraction of the variability of y described 

by yˆ, where the adjusted value is computed as 

R SSE n p SST n a

2 = -1 1 [ (×- ×- )]/[ ( )].

Like the response estimates, estimated parameters bols

associated with the individual explanatory variables are uncertain. As n becomes large, estimated parameters are assumed 

to be unbiased and normally distributed about their true values 

with covariance Cov X ( ) b s ols = ( ) ¢ × X , 2 1 - which is commonly written b b b ols p N C ols ~ ( , ( ov )). +1 Diagonal elements 

of the covariance matrix describe the variance of the corresponding estimated parameters. Therefore, a t statistic can be 

computed to assess the significance of individual parameters 

as t Cov = ols i ols i i b b , , / ( ) . On the basis of the t probability 

distribution with n-p-1 degrees of freedom, a p-value is computed with the t statistic to assess the likelihood that the null 

hypothesis of the model parameter being equal to zero is true. 

Again, a small p-value is used to reject the null hypothesis, 

thereby accepting that the parameter estimate bols i, is statistically different from zero. 

Logistic Regression

As an alternative to estimating E. coli concentrations 

and comparing the estimates to thresholds of interest, one 

can directly estimate the probability that a specified threshold 

will be exceeded using classification methods. In this report, 

linear logistic models were developed to estimate probabilities 

of exceedance. In accordance with Michigan’s Water Quality 

Standard, a threshold of 300 CFU/100 mL was used. 

Application of the logistic model involves a transformation of E. coli concentrations into two classes. In this report, 

binary logistic models were developed to estimate the probability of E. coli concentrations exceeding 300 CFU/100 mL 

by coding response variables yi

 = 1 for samples exceeding this 

threshold and yi = 0 otherwise. Then, the probability p that 

y equals 1 given a set of explanatory variables x can be written 

p = Pr(y=1|x). 

The linear logistic model relates the logit transformation 

of probabilities p linearly to the explanatory variables. The 

logit transformation is the log of the odds, which is the ratio of 

the probability that an event will occur divided by the probability that it will not occur as 

logit( ˆ ˆ) log ˆ

ˆ p p

p o x -

é

ë

ê

ê

ù

û

ú

ú

= + ¢ 1 a b

.

As in linear regression, a is the intercept parameter and 

b¢ is a vector of parameters associated with explanatory 

(x) variables. Parameters are estimated by use of maximum 

likelihood. The inverse logit transformation is used to compute 

estimated probabilities as 

ˆ elogit( ˆ )

logit( ˆ ) p

e

p

= p +

é

ë

ê

ê

ù

û

ú

ú 1

Parameters in the logistic regression are estimated by the minimizing -2 times the log likelihood (-2 Log L) function, which 

is computed as 

- = 2 2 LogL - + åy p 1 1 - - y p i i i i iln(ˆ ) ( )ln( ˆ )

The difference between the -2 Log L function minimized 

under the intercept-only model -2LogLa and the linear model 

- + ¢ 2LogLa b x is used to compute a chi-square statistic, which 

Appendix 1. Equations and Detail Used in the Linear and Logistic Regression 

Models
38 Environmental Factors and Flow Paths Related to Escherichia coli at Two Beaches on Lake St. Clair, Michigan, 2002–2005

is then compared to the chi-squared statistical distribution to 

assess the overall statistical significance of the explanatory 

variables in the logistic equation. 

Logistic models provide a measure of rank correlation 

for assessing the predictive ability of the models. This measure is based on the number of concordant (nc) and discordant 

(nd) pairs in dataset. Specifically, a pair of observations with 

different response values is said to be concordant (discordant) 

if the larger response has a lower (higher) predicted probability than the smaller response. For example, if Pr(yi

=1|x) 

< Pr(yj=0|x), then observations {yi, yj} would be a concordant 

pair. If there are Ni measurements where E. coli concentrations 

are greater then 300 CFU/100 mL, and Nj measurements less 

than 300 in a set of size Ni + Nj, then there are t N = × i Nj

pairs. The selected measure of rank correlation was computed 

as c n = + ( . c t 0 5( ) - - nc nd )/ t .

The exponentiated values of the estimated parameters are 

commonly interpreted as the increase (or decrease) in the odds 

of an event associated with a unit increase in the explanatory 

variable, given that there are no changes in the other explanatory variables. For example, an estimated parameter value of 

0.2 would have an exponentiated value of 1.2214. Thus, one 

would expect that a unit increase in the associated explanatory 

variable (by itself) would be associated with a 22.14 percent 

increase in the odds of an event such as E. coli concentration 

exceeding 300 CFU/100 mL. Estimated parameters less than 

zero are associated with a decrease in the odds. The estimated 

standard deviation of the parameter can be used in a similar 

way to compute an approximate confidence interval about the 

increase in the odds. 

Holtschlag and others—Escherichia coli Concentrations at Two Beaches on Lake St. Clair, Michigan, 2002–2005—Scientific Investigations Report 2008–5028