iii

CONTENTS

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

Introduction ............................................................................................................................................................................2

Purpose and scope .........................................................................................................................................................2

Previous investigations ..................................................................................................................................................2

Acknowledgments .........................................................................................................................................................2

Data description......................................................................................................................................................................2

Dependent variable ........................................................................................................................................................3

Explanatory (independent) variables .............................................................................................................................3

Logit model ............................................................................................................................................................................7

Significance testing of β coefficients ............................................................................................................................8

Model goodness-of-fit ...................................................................................................................................................9

Statistical model development..............................................................................................................................................10

Area of well influence .................................................................................................................................................10

Single variable models.................................................................................................................................................10

Multivariate model ......................................................................................................................................................10

Stepwise logistic regression.........................................................................................................................................10

Multicollinearity and interaction terms .......................................................................................................................12

Evaluation of model performance........................................................................................................................................14

Calibration ...................................................................................................................................................................14

Discrimination .............................................................................................................................................................14

Model validation..........................................................................................................................................................14

Confidence interval......................................................................................................................................................16

Analysis of variable effects .........................................................................................................................................16

Ground-water vulnerability to nitrate contamination in the Mid-Atlantic Region ..............................................................16

Summary ..............................................................................................................................................................................22

References cited ...................................................................................................................................................................23

Figures

1–3: Maps of the Mid-Atlantic Region showing:

1. Location of ground-water wells sampled for nitrate.........................................................................................4

2. Land cover statistically combined into three groups ........................................................................................5

3. Generalized geology classified into five categories..........................................................................................6

4. Graph showing Logit model for a single explanatory variable...................................................................................8

5. Graph showing transformation of measured values to ones and zeros based on a threshold value............................9

6. Map showing the extraction of land-cover variables within a statistical area of influence around

each ground-water well sampled in the Mid-Atlantic Region ............................................................................11
iv

Figures—Continued

7–10: Graphs showing:

7. Optimal radius for the best-fit model using land-cover variables for nitrate concentrations

in ground water exceeding a threshold of 3 milligrams per liter as nitrogen for the 

Mid-Atlantic Region ............................................................................................................................12

8. Example calibration of the best-fit model comparing the predicted (expected) probability

of nitrate concentrations in ground water to the percentage of the actual (observed)

nitrate concentrations exceeding a threshold of 3 milligrams per liter as nitrogen for 

the Mid-Atlantic Region ..................................................................................................................... 15

9. Frequencies of the confidence interval length and corresponding predicted probabilities of

nitrate concentrations in ground water from the Mid-Atlantic Region exceeding

management thresholds of 1 milligram per liter, 3 milligrams per liter, and

10 milligrams per liter as nitrogen .......................................................................................................17

10. Wald Type III analysis for relative importance of explanatory variables’ contribution

to the model at each threshold for ground water in the Mid-Atlantic Region .....................................18

11. Maps showing:

A. Probability of nitrate concentrations in ground water exceeding 1 milligram per liter as nitrogen

in the Mid-Atlantic Region ..................................................................................................................19

B. Upper limit of the confidence interval at 1 milligram per liter in the Mid-Atlantic Region........................19

C. Probability of nitrate concentrations in ground water exceeding 3 milligrams per liter as nitrogen

in the Mid-Atlantic Region ..................................................................................................................20

D. Upper limit of the confidence interval at 3 milligrams per liter in the Mid-Atlantic Region......................20

E. Probability of nitrate concentrations in ground water exceeding 10 milligrams per liter as nitrogen

in the Mid-Atlantic Region ..................................................................................................................21

F. Upper limit of the confidence interval at 10 milligrams per liter in the Mid-Atlantic Region.....................21

Tables

1. Explanatory variables and their descriptive statistics explored in the logistic-regression

models for the Mid-Atlantic Region......................................................................................................................7

2. Likelihood ratio tests for single variable logistic-regression models for the predicted

probability of nitrate concentrations in ground water from the Mid-Atlantic Region

exceeding management thresholds of 1 milligram per liter, 3 milligrams per liter, and

10 milligrams per liter as nitrogen.......................................................................................................................12

3. Final multivariate model significant variables, coefficients, and interaction terms for the

predicted probability of nitrate concentrations in ground water from the Mid-Atlantic Region

exceeding thresholds of 1 milligram per liter through 10 milligrams per liter as nitrogen.................................13

4. The c-statistic results for each multivariate logistic-regression model to predict the

probability of nitrate concentrations in ground water from the Mid-Atlantic Region

exceeding thresholds of 1 milligram per liter through 10 milligrams per liter as nitrogen.................................15

5. PRESS statistic results for each multivariate logistic-regression model to predict the

probability of nitrate concentrations in ground water from the Mid-Atlantic Region

exceeding thresholds of 1 milligram per liter through 10 milligrams per liter as nitrogen.................................15

6. Ordinal measure results for each multivariate logistic-regression model to predict the

probability of nitrate concentrations in ground water from the Mid-Atlantic Region

exceeding thresholds of 1 milligram per liter through 10 milligrams per liter as nitrogen.................................16
v

Conversion Factors, Vertical Datum, and Abbreviated Water-Quality Units

_____________________________________________________________________________________________________________________________________

Multiply By To obtain

_____________________________________________________________________________________________________________________________________

square meter (m 2) 0.0002471 acre

cubic meter (m3) 0.0008107 acre-foot (acre-ft)

cubic meter per second (m3/s) 35.31 cubic foot per second (ft 3/s)

meter (m) 3.281 foot (ft)

meter per kilometer (m/km) 5.27983 foot per mile (ft/mi)

liter per second (L/s) 15.85 gallon per minute (gal/min)

kilometer (km) 0.6214 mile (mi)

square kilometer (km2 ) 0.3861 square mile (mi 2 )

_____________________________________________________________________________________________________________________________________

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 (NAVD 88); horizontal 

coordinate information is referenced to the North American Datum of 1983 (NAD 83).

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

micrograms per liter (μg/L).

Application of fertilizer and atmospheric deposition are given in kilogram per square kilometer (kg/km2).
This Page Intentionally Left Blank
Abstract 1

Ground-Water Vulnerability to Nitrate Contamination at 

Multiple Thresholds in the Mid-Atlantic Region Using

Spatial Probability Models

By Earl A. Greene, Andrew E. LaMotte, and Kerri-Ann Cullinan

Abstract

The U.S. Geological Survey, in cooperation 

with the U.S. Environmental Protection Agency’s 

Regional Vulnerability Assessment Program, has 

developed a set of statistical tools to support 

regional-scale, ground-water quality and vulnerability assessments. The Regional Vulnerability 

Assessment Program’s goals are to develop and 

demonstrate approaches to comprehensive, 

regional-scale assessments that effectively inform 

managers and decision-makers as to the magnitude, extent, distribution, and uncertainty of current and anticipated environmental risks. The U.S. 

Geological Survey is developing and exploring the 

use of statistical probability models to characterize 

the relation between ground-water quality and 

geographic factors in the Mid-Atlantic Region.

Available water-quality data obtained from 

U.S. Geological Survey National Water-Quality 

Assessment Program studies conducted in the 

Mid-Atlantic Region were used in association with 

geographic data (land cover, geology, soils, and 

others) to develop logistic-regression equations 

that use explanatory variables to predict the 

presence of a selected water-quality parameter 

exceeding a specified management concentration 

threshold. The resulting logistic-regression equations were transformed to determine the probability, P(X), of a water-quality parameter exceeding 

a specified management threshold. Additional 

statistical procedures modified by the U.S. Geological Survey were used to compare the observed 

values to model-predicted values at each sample 

point. In addition, procedures to evaluate the confidence of the model predictions and estimate the 

uncertainty of the probability value were developed and applied. The resulting logistic-regression models were applied to the Mid-Atlantic 

Region to predict the spatial probability of nitrate 

concentrations exceeding specified management 

thresholds. These thresholds are usually set or 

established by regulators or managers at National 

or local levels.

At management thresholds of 1 milligram per 

liter and 3 milligrams per liter as nitrogen, the 

probability of nitrate concentrations exceeding 

these levels is greater than 50 percent (0.50) 

throughout much of the Mid-Atlantic Region. 

This includes extensive areas throughout central 

Maryland, southeastern Pennsylvania, northwestern Pennsylvania, and the Delmarva Peninsula. In 

addition, extensive areas in North Carolina and 

Virginia also have high probabilities of nitrate 

concentrations in ground water exceeding management thresholds of 1 milligram per liter and 

3 milligrams per liter. The mapped areas showing 

a high predicted probability of nitrate concentrations in ground water exceeding 1 milligram per 

liter and 3 milligrams per liter correspond to areas 

that are mapped as cultivated land cover and/or 

overlying carbonate rocks. At a management 

threshold of 10 milligrams per liter (corresponding 

to the U.S. Environmental Protection Agency standard for nitrate in drinking water of 10 milligrams 

per liter), the predicted probability of nitrate concentrations in ground water exceeding this level is 

low for most of the Mid-Atlantic Region, except 

for the Delmarva Peninsula, southeastern Pennsylvania, and areas mapped as carbonate rocks in 

Virginia, Maryland, and Pennsylvania.
2 Ground-Water Vulnerability to Nitrate Contamination at Multiple Thresholds in the Mid-Atlantic Region

Introduction

The U.S. Environmental Protection Agency (USEPA) 

has recognized that regional, State, and local water managers 

in the Mid-Atlantic Region need to know the condition of 

ground-water quality at a regional scale for environmental 

health and human health purposes (such as public drinkingwater consumption). Over the last several years, the USEPA 

and the U.S. Geological Survey (USGS) have been conducting National and regional assessments of water quality in 

ground water in the Mid-Atlantic region (Ator and 

Ferrari, 1997; Ferrari and others, 1997). Although these 

studies have addressed questions of overall water-quality 

conditions in the region, they were not designed to predict 

water quality in areas with little or no data. In order to determine ground-water vulnerability in these unknown areas, the 

USGS, in cooperation with the USEPA, began a study in 

2001 to address this issue.

Purpose and Scope

The USGS, in cooperation with the USEPA’s Regional 

Vulnerability Assessment (ReVA) Program, is developing a 

set of statistical tools to support regional-scale integrated 

ecological risk assessment studies. ReVA’s goal is to 

develop and demonstrate approaches to comprehensive, 

regional-scale assessments that effectively inform decisionmakers as to the magnitude, extent, distribution, and uncertainty of current and anticipated environmental risks. The 

USGS is developing and exploring the use of statistical probability models to characterize the relation between groundwater quality and geographic factors in the Mid-Atlantic 

Region.

This report describes a spatial statistical methodology 

that can be used to assess the risk of nonpoint-source 

contamination of ground water exceeding a management 

threshold in areas of the Mid-Atlantic Region where little 

data exits. Thresholds can be formulated and specified as 

management values such as 3 mg/L (milligrams per liter) of 

nitrate as nitrogen for environmental concerns or 10 mg/L of 

nitrate as nitrogen for regulating drinking water. In addition, 

the PRESS statistic (predictive ability) and the uncertainty 

(confidence interval) of the models were developed and 

applied to all spatial probability maps. Spatial probability 

maps showing the likelihood of elevated concentrations of 

nitrate above a management threshold were developed to 

identify areas that currently are vulnerable to nitrate contamination. These maps help identify areas where ground water 

has been affected by human activities, and can help regional 

and local water managers protect water supplies by targeting 

land-use planning solutions and implementing monitoring 

programs where ground water may be vulnerable.

Previous Investigations

Various investigations have shown there is a relation 

between ground-water quality and other variables, such as 

land cover and geology (Ator and Ferrari, 1997; Tesoriero 

and Voss, 1997). Cain and others (1989) evaluated regional 

ground-water quality using changes in land use. Barringer 

and others (1990) also related land use to ground-water quality, but discussed problems (misclassification, data closure, 

and spatial autocorrelations) with using land use as an 

explanatory variable. Some studies also have shown that 

when relating ground-water quality to land use, the size of 

the buffer around the well is an important factor. The maximum correlation between land use and nitrate concentration 

was reached when the land use around a well was quantified 

within a radius of 800 to 1,200 m (meters) (Hay and 

Battaglin, 1990). Tesoriero and Voss (1997) determined the 

best logistic-regression model relating land cover to nitrate 

concentration exceeding a 3-mg/L threshold was at a radius 

of 3,200 m.

Recent work relating regional ground-water quality to 

explanatory variables (land cover, geology, soils, and others) 

has used statistical models to explain these relations. These 

investigations usually use nitrate as a surrogate for groundwater contamination or for an estimate of vulnerability. A 

variety of statistical methods can be used to relate nitrate 

contamination to explanatory variables. The use of logisticregression techniques and their application to hydrology are 

well documented in the literature (Walker and Duncan, 1967; 

Harrell and others, 1980; Helsel and Hirsch, 1992; Eckhardt 

and Stackelberg, 1995; Tesoriero and Voss, 1997; Nolan, 

2001).

Acknowledgments

The authors thank Dr. Nagaraj Neerchal, Professor of 

Statistics at the University of Maryland, Baltimore County, 

for his statistical advice and Joseph Vrabel of the USGS for 

help in developing the data base for the project. The

authors would also like to thank Bernard T. Nolan and

A. Jim Tesoriero of the USGS for their technical reviews. 

Lastly, we thank Valerie Gaine, Jean Hyatt, Donna Knight, 

and Timothy Auer of the USGS Maryland-Delaware-D.C. 

District for editorial review, report preparation, and report 

graphics.

Data Description

Data needed for model development were obtained from 

previously published sources and available USGS data 

stored in files for ground-water and water-quality data bases. 

These data bases contain information obtained from National 

Water-Quality Assessment (NAWQA) studies, statewide 

ground-water-quality networks, and individual projects. 

Geographic data sets, such as land cover, surficial geology, 

soil permeability, soil organic matter, depth of soil layer, 

depth to water table, soil texture, hydrologic groups, manure, 

fertilizer, atmospheric deposition, and population density, 

were obtained for the Mid-Atlantic Region from published 

sources.

Water-quality data collected from October 1985 through 

September 1996 as part of the NAWQA Program (Gilliom 

and others, 1995) and various other USGS projects were 

compiled in 1997 for a regional analysis of pesticides and 
Data Description 3

nitrate in surface and ground water of the Mid-Atlantic 

Region (Ator, 1998; Ferrari and others, 1997; Ator and 

Ferrari, 1997). Ground-water-quality data consisted of 

1,551 samples taken from 937 different shallow observation 

wells (95 percent less than 33 m), with multiple samples 

taken at various sites (Ator, 1998; Ator and Ferrari, 1997). 

When multiple samples were collected at a site, the most 

recent sampling date was used. Ten sites were eliminated 

because of either missing or incomplete data, including well 

depth, lack of nitrate testing, and site information, thereby 

reducing the data set to 927 sites (fig. 1). 

Dependent Variable

Many contaminants in ground water occur naturally; 

however, elevated nitrate concentrations in ground water 

typically are caused by anthropogenic (human-related) 

activities that include crop fertilization and domestic septic 

systems. Spalding and Exner (1993) suggested that nitrate 

may be the most widespread contaminant in ground water. 

Because of the extensive presence of this contaminant, 

nitrate concentrations in ground water may help identify 

environments that are susceptible to contamination (U.S. 

Environmental Protection Agency, 1996a).

Nitrate as nitrogen was chosen as a surrogate variable to 

develop spatial maps of ground-water vulnerability because 

it has been suggested that it can be used as an indicator of 

overall ground-water quality (U.S. Environmental Protection 

Agency, 1996a). In addition, nitrate is a USEPA-regulated 

contaminant and its presence in high concentrations is a 

potential health risk. The USEPA has set a standard for 

nitrate in drinking water of 10 mg/L (U.S. Environmental 

Protection Agency, 1996b).

Explanatory (Independent) Variables

Land cover and geology type have been related previously to the occurrence of elevated nitrate concentrations 

in ground water throughout the Mid-Atlantic Region (Ator 

and Ferrari, 1997; Nolan 2001). Land-cover data for the 

Mid-Atlantic Region were derived from the National Land 

Cover Dataset (NLCD). The NLCD was produced as a 

cooperative effort between the USGS and the USEPA to produce a consistent, land-cover dataset for the conterminous 

United States using early 1990s Landsat Thematic Mapper 

(TM) data purchased by the Multi-Resolution Land Characterization (MRLC) Consortium (U.S. Geological Survey, 

2000). There are 15 classes of land cover that represent the 

Mid-Atlantic area. These 15 classes were first aggregated 

into 6 similar classes (water, barren, wetland, developed, 

cultivated, and forested), then to 3 groups (developed, 

cultivated, and forested/wetland). Comparison of logisticregression analyses confirmed no loss in significance 

between models developed using six land-cover classes and 

models that were further aggregated into just three landcover groups representing developed, cultivated, and forested/wetland. Therefore, the logistic-regression models 

developed for this study included land-cover groups representing developed, cultivated, and forested/wetland. These 

land-cover groups were developed as continuous variables 

with units equal to percentage of land cover (table 1, fig. 2).

Geologic type was compiled from the digital version of 

the geology of the conterminous United States (King and 

Biekman, 1974; Schruben and others, 1994) and the MidAtlantic Coastal Plain geologic framework (Ator and others, 

in press). The geologic framework of the Coastal Plain (generalized into coarse unconsolidated and fine unconsolidated 

material), as developed by Ator and others (in press), 

replaced the Coastal Plain area described by King and 

Biekman (1974). Geologic units were included in the statistical model as nominal categorical variables identified as 

Siliciclastic, Carbonate, Crystalline, coarse Coastal Plain, 

and fine Coastal Plain (table 1, fig. 3).

The other variables considered by the statistical model 

were nutrient (nitrate) source inputs consisting of application 

of manure and fertilizer (kilogram per square kilometer) and 

atmospheric deposition (kilogram per square kilometer). 

Population density and soils data consisting of hydrologic 

groups, organic matter, depth of soil layer, depth to water 

table, percent sand, percent silt, and percent clay also were 

explored by the statistical model to determine if they were 

significant explanatory variables (table 1).

Manure input was derived from county-based estimates 

of nitrogen content of animal manure in the United States for 

1992 (Puckett and others, 1998). Fertilizer input was compiled from nitrogen-fertilizer sales for the conterminous 

United States in 1991 by county (Battaglin and Goolsby, 

1994). Atmospheric deposition of nitrate was derived from 

1992 data collected by the National Atmospheric Deposition 

Program (NADP) (National Atmospheric Deposition Program, 2000). Manure and fertilizer-input values were distributed evenly across agricultural areas of each county 

(cultivated land-cover variable of the model) (table 1).

Population density was derived from 1990 block group 

point data compiled by the USGS (Hitt, 1992). This variable 

was explored as a density value and as a natural log transformation. Hydrologically relevant soil characteristics were 

compiled by the USGS from the State Soil Geographic 

(STATSGO) Data Base produced by the U.S. Department of 

Agriculture’s Natural Resources Conservation Service 

(Schwartz and Alexander, 1995).

Well depth has been shown to be an important variable in 

explaining elevated nitrate levels in ground water (Hallberg 

and Keeney, 1993; Nolan, 2001; Tesoriero and Voss, 1997). 

For this study, however, models were developed to predict 

the probability of nitrate concentrations exceeding a specified management threshold in areas where there are little or 

no data on nitrate concentrations. Because well depth does 

not exist in these areas, this would not be a helpful explanatory variable and was therefore not included in the statistical 

models.
4 Ground-Water Vulnerability to Nitrate Contamination at Multiple Thresholds in the Mid-Atlantic Region
Data Description 5
6 Ground-Water Vulnerability to Nitrate Contamination at Multiple Thresholds in the Mid-Atlantic Region
Logit Model 7

 

Logit Model

General linear models (GLM) of the form

 

have their basis in classical statistics, and are used to predict 

the response of a dependent variable (y) from a single or 

group of independent explanatory (xi) variables. The intercept is β0 and the β i ’s are the slope coefficients. The unexplained variance is termed the error and is symbolized as 

(Ott, 1977). When the dependent variable is dichotomous 

(above/below, yes/no), linear regression is no longer an 

option and other multivariate statistical methods need to be 

considered. Rather than modeling the actual response with 

the explanatory variables, as in linear regression, the Logit 

model (logistic-regression model) is based on the assumption that the log-odds (logit) of the model are related to the 

independent variables (Brown, 1998).

Unlike linear regression, the logistic-regression procedure models a change in the logit or log-odds of the dependent variable rather than a change in the variable itself. 

Whereas ordinary least squares (OLS) regression attempts to 

minimize the distance squared between the regression line 

and the data points, logistic regression applies the “maximum likelihood estimation.” The maximum likelihood estimation attempts to maximize the log likelihood (LL), which 

reflects how likely it is (the odds) that the observed values of 

the dependent variable ( y) may be predicted from the 

observed values of the independent variables (xi ).

The response probability, pi , of the Logit model can be 

defined as

Table 1. Explanatory variables and their descriptive statistics explored in the logistic-regression models for the 

Mid-Atlantic Region

[NA, not applicable; %, percent; kg, kilogram; km2, square kilometer]

Variable name Data type Units or Categories Minimum Median Maximum

Quartiles

 25% 75%

Cultivated land cover Continuous Percentage 0 46 100 18 71

Developed land cover Continuous Percentage 0 9 84 0.1 5.4

Forested/Wetland land

cover

Continuous Percentage 0 33 100 16 59

Geology Categorical Siliciclastic NA NA NA NA NA

Carbonate NA NA NA NA NA

Crystalline NA NA NA NA NA

Coarse Coastal Plain NA NA NA NA NA

Fine Coastal Plain NA NA NA NA NA

Soil permeability Categorical Moderate NA NA NA NA NA

Moderately rapid NA NA NA NA NA

Rapid NA NA NA NA NA

Soil organic matter Continuous Percentage 0.1 0.6 52 .5 0.9

Depth of soil layer Continuous Meters .7 1.5 1.8 1.2 1.6

Depth to water table Continuous Meters .1 1.5 1.8 1.1 1.6

Soil texture Continuous Ratio (% silt and 

clay to % sand)

.1 .3 0.9 .2 .5

Hydrologic Group A Continuous Percentage 0 5 65 0 11

Population density Continuous People/km 2 1 40 3,022 17 98

Inorganic fertilizer 

application

Continuous kg/km 2 0 1,314 1,765 590 3,556

Manure application Continuous kg/km 2 0 968 15,864 282 2,367

Atmospheric deposition Continuous kg/km 2 213 505 887 429 554

εi
8 Ground-Water Vulnerability to Nitrate Contamination at Multiple Thresholds in the Mid-Atlantic Region

 

where, for each of the n observations, there is a response Yi

that can take on the values of 1 or 0, from corresponding 

independent explanatory variables Xi , i = 1, 2, ..., n, where 

Xi

 = (1, xi1, xi2, ..., xik ). βk represents the regression coefficient corresponding to the kth explanatory variable, with β0

representing the intercept. The vector β = (β0 , β1, β2, ..., 

βk ) is called the vector of slope parameters. Equation (2) 

can be simplified further by dividing both the numerator and 

denominator by the numerator itself to obtain a form of the 

logit equation (equation 3) with a desired property of anything that is substituted for Xi ,and βi

 , and the probability pi, 

will always be between 0 and 1.

 

If there is a single explanatory variable (x) with β0 = 0, 

and β1 = 1, the simplest logit model, the equation can be 

graphed to produce the classic S-shaped curve shown in 

figure 4. As x increases or decreases, the probability pi, 

always will be between 1 and 0. In addition, the graph of the 

S-shaped curve of the logit model shows that the probability 

response of a unit change in the x variable varies depending 

on the starting probability value. When the probability is 

near 0.50, a unit change in the x variable produces a large 

probability response; when the probability is near 0 or 1, a 

unit change in the x variable produces a small probability 

response (fig. 4).

Logistic regression was used in this investigation to identify a dichotomous response between dependent variables 

(for example, when the dependent variable ( y ) is in a binary 

format representing above or below) and independent variables (x ) that either are continuous or categorical. Specifically, logistic-regression equations where explanatory 

variables (x ) are used to predict the presence of nitrate concentrations ( y) above a specified management threshold 

value (such as Maximum Contaminant Levels, or MCLs) 

were developed. The resulting equations then were transformed to determine the probability, P( Y ) , of estimates of 

nitrate concentrations above a selected threshold value. The 

presence of elevated concentrations of nitrate in ground 

water was used as a surrogate for anthropogenic contamination of shallow ground water in the Mid-Atlantic Region.

Logistic regression allows nitrate concentrations to be 

evaluated at any threshold level, thereby transforming nitrate

from a censored continuous variable measured in milligrams 

per liter to a categorical variable with classes. Understanding thresholds is important when applying logistic-regression equations to water-quality data. After a threshold has 

been established, sampled values are classified as either 

“1’s” when they are above the threshold, or “0’s” when they 

are below the threshold (fig. 5). These threshold levels can 

be set up to parallel decision-making criteria, making this 

modeling approach a useful decision tool for resource managers. Therefore, it is possible to predict the probability of 

nitrate concentrations in ground water exceeding a certain 

pre-determined threshold level.

Significance Testing of β Coefficients

The use of the maximum likelihood estimation (MLE ) is 

the preferred way to test the significance of the logit coefficients (β ) of the logistic-regression model (Allison, 1999; 

Hosmer and Lemeshow, 1989). Essentially, this test considers the null hypothesis Ho : βi =0, i=1,2,...,k, indicating that 

the regression coefficients βi

 are not significantly different 

from zero, against the alternative Ha: β i ≠0 , that at least one 

coefficient is different from zero. The MLE compares the 

ratio of the maximum of the likelihood under Ho to the maximum of the likelihood under Ha (likelihood ratio test) and 

seeks to maximize the log likelihood (LL) , which reflects 

how likely (the odds) that the observed values of the dependent variable ( y ) are predicted from the observed values of 

the independent variables (x’s). 

The likelihood ratio test (G ) statistic for inclusion of an 

independent variable can be defined as
Logit Model 9

 

where the G statistic has a chi-square distribution with 

degrees of freedom depending on the number of parameters 

included under the null hypothesis that all β coefficients are 

zero. A significant G statistic means that adding the independent variables to the model, as compared to a model with 

only an intercept (β 0 ) with its corresponding βi

 coefficient, 

improves the model (for example, at least one of the β i

 coefficients is nonzero) (Hosmer and Lemeshow, 1989). To test 

the significance of each β i coefficient, the model is refitted 

multiple times, comparing each explanatory variable in turn, 

to the intercept-only model.

Additional testing to evaluate the significance of each of 

the explanatory variables for each logistic-regression model 

was based on the Wald statistic. The Wald statistic was calculated for each of the β i coefficients of the model. This test 

statistic (equation 5) is computed as the coefficient divided 

by its standard error estimate of the coefficient, with the 

result squared.

 

This statistic, when compared to a chi-square random variable with 1 degree of freedom, was used to determine if each 

coefficient was significant in the model. Even though both 

tests were used to determine the significance of the β i coefficients, some researchers have suggested that the likelihood 

ratio tests are superior to the Wald test (Hauck and Donner, 

1977; Jennings, 1986; Collett, 2002).

Model Goodness-of-Fit

For this investigation, two statistical tests were used to 

assess the model’s goodness-of-fit. The first test used was 

the G statistic or model chi-square test and the second test 

was the “Hosmer and Lemeshow’s Goodness of Fit Test” 

(Hosmer and Lemeshow, 1989). The model chi-square test 

is a likelihood ratio test (equation 4 ), which reflects the difference between the error not including the explanatory variables (initial chi-square) and the error when the explanatory 

variables are included (deviance). The model chi-square follows a chi-square distribution with degrees of freedom equal 

to the difference in the number of explanatory variables 

included compared to the intercept-only model.

“Hosmer and Lemeshow’s Goodness of Fit Test” (H-L) 

(equation 6) tests the hypothesis that the data fit the logisticregression model. Based on the logistic-regression model, 

predicted probabilities associated with each observation are 

sorted and grouped into 10 intervals (deciles). Within each 

interval, the expected frequency is determined by adding the 

predicted probabilities. The expected frequencies are compared with the observed frequencies by the Pearson chisquare with 8 degrees of freedom (number of intervals minus 

2) at the α = .05 level of significance. If the H-L test statistic’s p-value is greater than 0.05, it is implied that the 

model’s estimates fit the data at an acceptable level. As with 

most model goodness-of-fit tests, as the sample size 

increases, the power to detect differences from the null 

hypothesis (no difference) based on the statistic improves 

(Hosmer and Lemeshow, 1989; Hosmer and others, 1997).

The H-L test statistic is:

 
10 Ground-Water Vulnerability to Nitrate Contamination at Multiple Thresholds in the Mid-Atlantic Region

where

Ni = the number of observations in the i

th decile,

Oi = the number of successes (Y =1) in the i

th decile,

π i

 = the average of the estimated probabilities, and

 

 = test statistic, approximated by the chi-square

 distribution with g-2 degrees of freedom.

Statistical Model Development

The statistical model was developed by first testing each 

single explanatory variable in a logistic-regression model to 

determine significant variables. Models that consisted of a 

set of single variable logistic-regression equations were analyzed. The explanatory variables that were significant individually then were tested in a multivariate logistic regression 

to determine how each contributed to the overall model 

when considered with other variables.

Area of Well Influence

Several studies have shown there is a significant relation 

between shallow ground-water quality (nitrate concentrations) and the types of land use near a sampled well (Cain 

and others, 1989; Hay and Battaglin, 1990; Barringer and 

others, 1990; Tesoriero and Voss, 1997). Using different 

types of statistical procedures (linear regression, rank correlations, logistic regression), these investigations all showed 

that the size of the land use/cover buffer around a sampled 

well is significant when evaluating ground-water quality in 

relation to land use/cover.

In developing the logistic-regression models for explanatory variables, the land cover within an optimal radius was 

used to establish an area of influence around each well

(fig. 6). The same area of well influence that was determined for land cover was applied to all explanatory variables 

that were a function of area. Explanatory variables that were 

dependent on the spatial area around the well (values varied 

in space) included land cover, soil permeability, soil organic 

matter, depth of soil layer, depth to water table, clay content 

of the soil, silt content of the soil, and hydrologic groups. 

The optimal land-cover radius was determined by examining how well the logistic-regression models at different 

radii fit the nitrate data sampled at each well. Using landcover data, logistic-regression models for 8 radii that ranged 

from 500 to 4,000 m in 500-m steps were tested. The best-fit 

model was determined by finding the radii that maximized 

the likelihood ratio test (G) and H-L test statistics. The bestfit model that maximized the test statistics for nitrate concentrations exceeding a threshold of 3 mg/L was at 1,500 m (fig. 

7). Repeating this analysis for thresholds of 1 mg/L through 

10 mg/L indicated that the test statistics also were maximized at a radius of 1,500 m. Results of this analysis indicate that the radius that contributes the most information to 

the model using land cover is at an optimal radius of 1,500 m 

regardless of the threshold level that is chosen.

Single Variable Models

To develop logistic-regression models that relate an 

event of an occurrence of nitrate concentrations above a predetermined threshold to an explanatory variable, various 

variables were tested individually to determine if they were 

significant predictors. Single-variable models that were considered and tested included land cover, surficial geology, soil 

permeability, soil organic matter, depth of soil layer, depth to 

water table, soil texture, hydrologic groups, manure, fertilizer, atmospheric deposition, and population density.

Regression coefficients and statistics for models 

developed using these variables are presented in table 2. 

All explanatory variables were found to be significant

(α = 0.05), except for Hydrologic Group A at the lower 

thresholds and atmospheric deposition for all thresholds. 

Even though Hydrologic Group A was not significant at the 

lower thresholds, this variable was included in the multivariate model because at higher thresholds, it was significant. 

Atmospheric deposition was dropped as an explanatory variable for the multivariate models at all thresholds.

Multivariate Model

The multivariate logistic-regression equations combined 

the results in direct relations between all tested explanatory 

variables and the probability of exceeding nitrate concentrations at a selected threshold value. The logistic-regression 

models were tested using goodness-of-fit statistical procedures (likelihood ratio test (G) and H-L test statistics) to 

define the final logistic-regression models at multiple thresholds.

Stepwise Logistic Regression

When building the logistic-regression model, the use of a 

stepwise selection procedure facilitated the development of 

models at multiple thresholds. The development of the 

models in a sequential fashion allowed for the examination 

of a collection of models that otherwise would not have been 

examined. A statistical algorithm that adds or removes 

variables from the model based on a pre-determined decision 

rule was used to develop the model. The decision rule 

usually is based on the statistical significance of the 

variable’s model coefficient. At any step in the procedure, 

the most important variable (in statistical terms) was the one 

that produced the greatest change in the log-likelihood 

relative to the model not containing the variable.

The stepwise selection procedure actually is a modification of the simpler forward selection procedure. With forward selection, the initial model begins with only the 

intercept and variables are added to the model one at a time. 

If the associated variable is considered significant at the 

α = 0.2 level of significance, it is entered into the model. 

One or more backward elimination steps follow the forward 

selection by examining the maximum likelihood estimates 

for each variable in the model. Those variables that are not 

significant at the α = 0.05 level of significance are removed 

from the model. The process continues until no more variables can offer a change in the log-odds, and therefore can 

not be added to or removed from the model.
Statistical Model Development 11
12 Ground-Water Vulnerability to Nitrate Contamination at Multiple Thresholds in the Mid-Atlantic Region

 

Multicollinearity and Interaction Terms

In developing general linear and logistic-regression 

models, multicollinearity and interaction among the explanatory variables must be examined. Multicollinearity occurs 

when there are strong dependencies among the explanatory 

variables. This interaction among variables will mean that 

the odds ratio for explanatory variables may vary with the 

value of another explanatory variable. It is possible to have 

an interaction in which no pairs of variables are highly correlated, but various variables together may be highly interdependent (Allison, 1999).

Interaction among explanatory variables was addressed 

early in the model-forming stage. All interaction terms were 

evaluated before any individual variable was eliminated. All 

explanatory variables that had significant interaction terms 

were included in the model. The final multivariate model 

variables, coefficients, and interaction terms for significant 

explanatory variables ( p ≤ 0.05) that were included in the 

model are presented in table 3. Cultivated land cover and 

surficial geology were the only significant variables for all 

thresholds. Other variables were significant at certain 

thresholds and were only included in the model when appropriate (table 3).

 

Table 2. Likelihood ratio tests for single variable logistic-regression models for the predicted probability 

of nitrate concentrations in ground water from the Mid-Atlantic Region exceeding 

management thresholds of 1 milligram per liter, 3 milligrams per liter, and

10 milligrams per liter as nitrogen

[mg/L, milligram per liter; <, less than]

Variable

Threshold 1 mg/L Threshold 3 mg/L Threshold 10 mg/L

Statistic p-value Statistic p-value Statistic p-value

Land cover

 (including developed, cultivated, 

and forested / wetland)

415.6207 <0.0001 412.2763 <0.0001 202.5157 <0.0001

Geology 262.3879 < .0001 210.7784 < .0001 94.5039 < .0001

Soil permeability 39.3112 < .0001 43.7395 < .0001 37.1519 < .0001

Soil organic matter 11.5722 < .0001 39.7203 < .0001 20.7725 < .0001

Depth of soil layer 180.5545 < .0001 193.2178 < .0001 89.6187 < .0001

Depth to water table 22.5801 < .0001 30.7165 < .0001 13.5250 < .0001

Soil texture, including

percent silt and clay

18.4296 < .0001 24.2340 < .0001 23.0355 < .0001

Hydrologic Group A 0.7826 .3764 2.8388 .0920 24.3645 < .0001

Population density 24.2443 < .0001 8.1261 .0044 10.0032 .0016

Inorganic fertilizer application 132.3953 < .0001 137.1383 < .0001 43.6472 < .0001

Manure application 76.1945 < .0001 96.3165 < .0001 100.2533 < .0001

Atmospheric deposition .0038 .9508 0.4737 0.4913 .0675 .7949
Statistical Model Development 13
14 Ground-Water Vulnerability to Nitrate Contamination at Multiple Thresholds in the Mid-Atlantic Region

Evaluation of Model Performance

Predictive statistical models need to be evaluated in order 

to establish that the model works satisfactorily for data that 

were not used in model development (Altman and Royston, 

2000). In developing logistic-regression models, various 

researchers have suggested that calibration, discrimination, 

and validation should be used to evaluate the predictive 

model (Altman and Royston, 2000; Hosmer and others, 

1997; Justice and others, 1999; Mittlbock and Schemper, 

1996; Schemper and Stare, 1996).

Calibration

During the calibration step, the models are assessed to 

determine whether the predicted probabilities agree with the 

observed probabilities. Two model calibrations were developed—the first looked at whether the predictions agreed on 

average with the observed probabilities, and the second 

investigated the calibration in more detail and the agreement 

between the predictions and observations over the entire 

range of predictions.

An example of the final calibration based on the method 

developed by Hosmer and Lemeshow (1980), Lemeshow 

and Hosmer (1982), and Hosmer, Lemeshow, and Klar 

(1988) is shown in figure 8. The actual observations 

(observed) compared to the percent predicted (expected) 

probabilities for nitrate concentrations exceeding management thresholds of 3 mg/L, for a model with only land cover, 

and for a final model with all significant variables, are presented in this figure. The 1:1 line in figure 8 illustrates the 

ideal calibration. A simple measure of fit (calibration) used 

in this investigation was the slope of the regression line for 

the observed and predicted values within a decile of risk. 

When the predictions were greater than the observations, 

then the slope was less than one. If the predictions were less 

than the observations, then the slope was greater than one. 

This form of the “mis-calibration” was compared statistically to a model with an intercept equal to 0 and a slope of 1 

(Harrell and others, 1980) and the best-fit model was developed for each threshold.

Discrimination

The ability of the predictive model to distinguish 

between sites with nitrate concentrations greater than the 

threshold and sites with nitrate concentrations less than the 

threshold was quantified using the measure of concordance 

(c statistic). The c statistic ranges between 0.5 and 1 for 

good models (Allison, 1999). The closer the c statistic is to 

1, the better the model is at distinguishing the correct binary 

outcome. The c statistic for the final predictive model at 

each of the thresholds is presented in table 4.

Model Validation

When developing general linear models, it is important 

to consider a class of statistical tools that are measures of 

how well the model can predict the response of a dependent 

variable (y) from a single or group of independent explanatory (xi ) variables. This type of assessment is often called 

model validation. This assessment is a different statistical 

measure than the model goodness-of-fit statistic. It is possible to have a model that predicts the dependent variable very 

well, but has a poor goodness-of-fit statistic (H-L test statistic). It is also possible that a model will have a very good 

goodness-of-fit statistic but a very poor prediction statistic. 

In developing general linear models, statistics such as R2 are 

used to estimate the proportion of the variance “explained” 

by the independent (explanatory) variables. In logistic 

regression, there is not a simple statistic that is equivalent to 

the R2 in linear regression. Although an R2 can be calculated, and is an estimate of how much variance is explained, 

it is specific to the particular data set and cannot be used to 

compare between different models. In logistic regression, 

three measures can be used in place of the R2, such as the 

D statistic, the Brier score, or the PRESS statistic, which was 

used for this investigation (Hosmer and Lemeshow, 1989; 

SAS Institute, Inc., 1990).

The PRESS statistic can be interpreted as the power of 

model predictability and can be used for model validation 

(SAS Institute, Inc., 1990). The PRESS statistic is defined 

as the prediction sum of squares and its purpose is to evaluate the regression model. A PRESS statistic for ordinary 

regression is available with SAS Institute, Inc. (1990) software, but there is not a similar algorithm for logistic regression. Therefore, as part of this investigation, a PRESS 

statistic was developed for use in logistic regression for 

model validation.

Theoretically, the computation of the PRESS statistic 

begins by taking the i th observation out of the sample, 

building the model from the remaining n–1 observations, 

and then evaluating the model using the explanatory variables from the i th observation. The i th PRESS residual is 

calculated by subtracting this new predicted value from the 

actual value for the dependent variable. Calculating and 

summing these residuals over all n sample data points gives 

the PRESS statistic for the model. The best model can be 

considered the one with the smallest PRESS statistic value.

A logistic-regression model predicts the probability of an 

event occurring rather than the value of the dependent variable directly. To make the PRESS statistic procedure valid 

for logistic regression, each predicted probability was transformed into a 1 or 0 whether it was likely to have exceeded 

the threshold or not. When the predicted probability was 

greater than or equal to 0.5, the value was interpreted as a 1 

for that threshold, otherwise it was taken to be 0. These 

transformed values are used to estimate the probability of 

misclassification. After comparing these values to the actual 

values for that threshold, the PRESS statistic results can be 

calculated as a count of the number of “correctly predicted” 

observations. To measure the prediction power of the 

model, the “correctly predicted” observations are divided by 

the total number of observations to give a percentage of 

correctly predicted terms.

The PRESS statistic was used to determine the ability 

of each model to correctly predict the occurrence of nitrate 

concentrations above a predetermined threshold value. 

The PRESS statistic results for each threshold are listed in 
Evaluation of Model Performance 15

 

table 5. This statistic illustrates that even at the lower thresholds, the model is correctly predicting nitrate concentrations 

above the threshold about 81 percent of the time, and generally increases to about 89 percent at the higher thresholds 

(table 5).

Additional statistics based on the ordinal measures of 

association also can be used to measure the predictive power 

of the model. The ordinal measures (Somer’s D and 

Gamma) were considered and maximized when building the 

statistical model. Both measures range between 0 and 1, and 

the values closer to 1 correspond to stronger associations 

between the predicted and observed values (Allison, 1999; 

SAS Institute, Inc., 1990). The ordinal measures for the final 

model at each of the thresholds are listed in table 6.

 

 

Table 4. The c statistic results for each multivariate 

logistic-regression model to predict the 

probability of nitrate concentrations in 

ground water from the Mid-Atlantic 

Region exceeding thresholds of 

1 milligram per liter through 

10 milligrams per liter as nitrogen

 [mg/L, milligram per liter]

Thresholds c statistic

1 mg/L 0.902

2 mg/L .900

3 mg/L .895

4 mg/L .893

5 mg/L .893

6 mg/L .893

7 mg/L .903

8 mg/L .899

9 mg/L .893

10 mg/L .908

Table 5. PRESS statistic results for each multivariate 

logistic-regression model to predict the 

probability of nitrate concentrations in 

ground water from the Mid-Atlantic 

Region exceeding thresholds of 

1 milligram per liter through

10 milligrams per liter as nitrogen

 [mg/L, milligram per liter]

Thresholds

PRESS statistic

(decimal percent of 

correctly predicted 

observations)

1 mg/L 0.813

2 mg/L .814

3 mg/L .823

4 mg/L .822

5 mg/L .815

6 mg/L .829

7 mg/L .856

8 mg/L .861

9 mg/L .875

10 mg/L .891
16 Ground-Water Vulnerability to Nitrate Contamination at Multiple Thresholds in the Mid-Atlantic Region

 

Confidence Interval

Confidence intervals for the best-fit logistic-regression 

model for each threshold were evaluated in order to obtain 

the range in values with a specified probability (95 percent) 

containing the regression coefficients. Two types of confidence interval maps were developed as part of this investigation—one mapping the length of the 95-percent confidence 

interval, and one displaying the upper bound of the interval, 

or the worst-case probability within a 95-percent confidence 

interval. The frequency for the predicted probabilities compared to the length of the confidence intervals for these predictions for nitrate in ground water exceeding the 1 mg/L, 

3 mg/L, and 10 mg/L management thresholds is shown in 

figure 9. The lengths of the confidence intervals gradually 

became smaller (increase in confidence) as the thresholds 

increase. Thus, a smaller confidence-interval length indicates a greater confidence in the prediction, whereas a larger 

length depicts less confidence in the prediction. 

Analysis of Variable Effects

A Wald Type III analysis was performed on each model 

to determine the relative importance of each significant variable to the overall logistic-regression model at each threshold (Hosmer and Lemeshow, 1989). The histogram of the 

statistics associated with each significant variable at each 

threshold from 1 mg/L through 10 mg/L is shown in figure 

10. The Wald Type III “Analysis of Effects” includes variables that are categorical, continuous, and have interaction 

terms to determine the relative importance of each variable 

to the model. Each threshold analysis corresponds to 100 

percent of the Type III Wald Statistic with each variable representing the weighted proportion of the statistic.

The variables that have the most effect or that are most 

significant to the model are cultivated land cover and geology. For the lower thresholds, developed land cover and 

depth of soil layer are significant explanatory variables. At 

the upper thresholds, the variable manure application 

becomes a significant explanatory variable for the model.

Ground-Water Vulnerability to Nitrate 

Contamination in the Mid-Atlantic Region

Geographic information system (GIS) methods were 

used to generate an array of regularly spaced points on a 

1,500-m grid representing the Mid-Atlantic Region. These 

points represent the center of each grid cell and correspond 

to the center of the 1,500-m radius that was determined to be 

the best-fit model from the land-cover analysis. Logisticregression model coefficients that were developed from the 

explanatory variables were applied to each point on the grid, 

and maps were developed that illustrate the probability of 

nitrate concentrations exceeding thresholds of 1 mg/L 

through 10 mg/L. These maps show the vulnerability of 

ground water to nonpoint nitrate contamination at these 

thresholds (fig. 11).

Along with the probability maps of ground-water vulnerability in the Mid-Atlantic Region, the uncertainty of the 

probability estimates in the form of confidence maps was 

generated (fig. 11). The length of the confidence interval 

helps discern the error in the probability estimate. The upper 

bound of the confidence interval for each estimate can be 

thought of as the worst-case scenario, or highest possible 

probability, of nitrate concentrations in ground water exceeding a certain threshold within a 95-percent confidence 

interval. 

The predicted probability of nitrate concentrations 

exceeding management thresholds of 1 mg/L, 3 mg/L, and 

10 mg/L, along with the associated upper bound (worst-case 

scenario) of the 95-percent confidence interval of these predictions are shown in figures 11a–f. At management thresholds of 1 mg/L and 3 mg/L, the probability of nitrate concentrations exceeding these levels is low throughout much of 

the Mid-Atlantic Region. Near the northern part of the 

Chesapeake Bay, however, the probability of nitrate concentrations exceeding management thresholds of 1 mg/L and 

3 mg/L is high (greater than 0.5). This area includes much of 

central Maryland, southeastern and northwestern Pennsylvania, and the Delmarva Peninsula. In addition, extensive 

areas in North Carolina and Virginia also show high probabilities of nitrate concentrations in ground water exceeding 

management thresholds of 1 mg/L and 3 mg/L (figs. 11a–d). 

The mapped areas showing a high predicted probability of 

Table 6. Ordinal measure results for each multivariate 

logistic-regression model to predict the 

probability of nitrate concentrations in 

ground water from the Mid-Atlantic 

Region exceeding thresholds of 

1 milligram per liter through 

10 milligrams per liter as nitrogen

[mg/L, milligram per liter]

Thresholds Somer’s D Gamma

1 mg/L 0.804 0.805

2 mg/L .800 .801

3 mg/L .789 .790

4 mg/L .786 .787

5 mg/L .786 .787

6 mg/L .786 .787

7 mg/L .806 .807

8 mg/L .777 .778

9 mg/L .786 .788

10 mg/L .817 .818
Ground-Water Vulnerability to Nitrate Contamination in the Mid-Atlantic Region 17
18 Ground-Water Vulnerability to Nitrate Contamination at Multiple Thresholds in the Mid-Atlantic Region
Ground-Water Vulnerability to Nitrate Contamination in the Mid-Atlantic Region 19
20 Ground-Water Vulnerability to Nitrate Contamination at Multiple Thresholds in the Mid-Atlantic Region
Ground-Water Vulnerability to Nitrate Contamination in the Mid-Atlantic Region 21
22 Ground-Water Vulnerability to Nitrate Contamination at Multiple Thresholds in the Mid-Atlantic Region

nitrate concentrations in ground water exceeding 1 mg/L and 

3 mg/L correspond to areas that are mapped as cultivated 

land cover (fig. 2) and/or overlying areas mapped as carbonate rocks (fig. 3).

Using a management threshold of 10 mg/L (corresponding to the USEPA standard for nitrate in drinking water of 

10 mg/L), the predicted probability of nitrate concentrations 

in ground water exceeding this level is low (less than 0.5) 

for most of the Mid-Atlantic Region. Areas within the 

Delmarva Peninsula and southeastern Pennsylvania, in addition to the mapped carbonate areas in Virginia, Maryland, 

and Pennsylvania, have high probabilities (greater than 0.5) 

of nitrate in ground water exceeding 10 mg/L (figs. 11e–f).

Summary

The U.S. Environmental Protection Agency has recognized the need for regional, State, and local water managers 

in the Mid-Atlantic Region to assess ground-water-quality 

conditions at a regional scale for environmental and human 

health purposes. The U.S. Geological Survey, in cooperation 

with the U.S. Environmental Protection Agency’s Regional 

Vulnerability Assessment Program, has developed a set of 

statistical probability models to characterize the relation 

between ground-water quality and geographic factors in the 

Mid-Atlantic Region to support regional-scale, integrated 

ecological risk-assessment studies. These models can be 

used to assess the risk of nonpoint-source contamination 

of ground water in areas of the Mid-Atlantic Region where 

little data are available at multiple management thresholds.

Logistic-regression models provide a statistical link 

between a dependent variable (nitrate concentration in 

ground water) and explanatory variables. Ground-water data 

for 927 sites sampled by the U.S. Geological Survey 

National Water-Quality Assessment Program and other 

projects were used to develop logistic-regression models to 

quantify the relation between geographic factors (land cover, 

geology, soils, and others) and the probability of nitrate in 

ground water exceeding a specified management concentration threshold. Thresholds can be formulated and specified 

as management values, such as 3 milligrams per liter for 

environmental concerns or 10 milligrams per liter for regulating drinking water.

A geographic information system in combination with 

logistic regression was used to establish the area of landcover influence around each ground-water sample site. The 

influence of land cover was analyzed by developing logisticregression equations at various radii (areas) to find the model 

that best described this influence. The radius (area) that best 

described the influence of land cover on nitrate concentrations was 1,500 meters.

Each of the other explanatory variables were all developed within this 1,500-meter radius and tested to determine 

the significance of including these variables in the logisticregression model. In some cases, the explanatory variable 

within this 1,500-meter radius was a continuous value (inorganic fertilizer application); in other instances, it was a 

weighted percentage (land cover) or a categorical variable 

(geology).

Individual explanatory variables that were significant 

were combined into an overall logistic-regression model 

using a stepwise selection process that developed an appropriate combination of variables to maximize the model’s performance, and by using a goodness-of-fit test incorporating 

Hosmer-Lemeshow, Wald, and PRESS statistics. In addition, multicollinearity and interaction terms were analyzed.

Geographic-information-system methods were used to 

generate an array of regularly spaced points to map the 

variables on a 1,500-meter grid representing the Mid-Atlantic Region. These points represent the center of each grid 

cell, and correspond to the center of the 1,500-meter radius 

that was determined to be the best-fit model from the landcover analysis. Logistic-regression model coefficients that 

were developed from the explanatory variables were applied 

to each point, and maps that determined the probability of 

nitrate concentrations exceeding management thresholds of 

1 milligram per liter through 10 milligrams per liter as nitrogen were developed.

In addition to the probability maps showing groundwater vulnerability to nitrate contamination in the MidAtlantic Region, uncertainty in the form of the confidence 

maps was generated and displayed. The upper bound of the 

confidence interval for each prediction of the probabilities of 

nitrate in ground water exceeding a threshold of 1 milligram 

per liter through 10 milligrams per liter was developed. 

Using these bounds, the worst-case scenario, or highest possible probability, of nitrate concentrations in ground water 

exceeding a certain management threshold was displayed.

The probability of nitrate concentrations exceeding 

management thresholds of 1 milligram per liter and 

3 milligrams per liter is extensive (greater than 0.5) throughout much of the Mid-Atlantic Region, especially near the 

northern part of the Chesapeake Bay. This area includes 

much of Maryland, southeastern and northwestern 

Pennsylvania, and the Delmarva Peninsula. In addition, 

extensive areas in North Carolina and Virginia also show 

high probabilities of nitrate concentrations in ground water 

exceeding these management thresholds. At a management 

threshold of 10 milligrams per liter (corresponding to the 

U.S. Environmental Protection Agency standard for nitrate 

in drinking water of 10 milligrams per liter), the predicted 

probability of nitrate concentrations in ground water exceeding this level is low (less than 0.5) for most of the MidAtlantic Region. Areas in the Delmarva Peninsula and 

southeastern Pennsylvania, along with the mapped carbonate 

areas in Virginia, Maryland, and Pennsylvania, show a 

higher predicted probability (greater than 0.5) of nitrate concentrations exceeding 10 milligrams per liter.
References Cited 23

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