U.S. Department of the Interior

U.S. Geological Survey

Prepared in cooperation with the

FEDERAL HIGHWAY ADMINISTRATION

Statistical Approaches to 

Interpretation of Local, Regional, and 

National Highway-Runoff and 

Urban-Stormwater Data

Open-File Report 00-491

A Contribution to the

NATIONAL HIGHWAY RUNOFF DATA AND METHODOLOGY SYNTHESIS
U.S. Department of the Interior

U.S. Geological Survey

Statistical Approaches to 

Interpretation of Local, Regional, and 

National Highway-Runoff and

Urban-Stormwater Data

By GARY D. TASKER and GREGORY E. GRANATO

Open-File Report 00-491

A Contribution to the

NATIONAL HIGHWAY RUNOFF DATA AND METHODOLOGY SYNTHESIS

Prepared in cooperation with the

FEDERAL HIGHWAY ADMINISTRATION

Northborough, Massachusetts

2000
U.S. DEPARTMENT OF THE INTERIOR

BRUCE BABBITT, Secretary

U.S. GEOLOGICAL SURVEY

Charles G. Groat, Director

The use of trade or product names in this report is for identification 

purposes only and does not constitute endorsement by the 

U.S. Government.

For additional information write to: Copies of this report can be purchased 

Chief, Massachusetts-Rhode Island District U.S. Geological Survey

U.S. Geological Survey Branch of Information Services

Water Resources Division Box 25286

10 Bearfoot Road Denver, CO 80225-0286

Northborough, MA 01532
PREFACE

Preface 

Knowledge of the characteristics of highway runoff (concentrations and loads of 

constituents and the physical and chemical processes which produce this runoff) is 

important for decision makers, planners, and highway engineers to assess and 

mitigate possible adverse impacts of highway runoff on the Nation’s receiving 

waters. In October 1996, the Federal Highway Administration and the U.S. 

Geological Survey began the National Highway Runoff Data and Methodology 

Synthesis to provide a catalog of the pertinent information available; to define 

the necessary documentation to determine if data are valid (useful for intended 

purposes), current, and technically supportable; and to evaluate available 

sources in terms of current and foreseeable information needs. This paper is 

one contribution to the National Highway Runoff Data and Methodology 

Synthesis and is being made available as a U.S. Geological Survey Open-File 

Report pending its inclusion in a volume or series to be published by the Federal 

Highway Administration. More information about this project is available on the 

World Wide Web at http://ma.water.usgs.gov/fhwa/

Fred G. Bank

Team Leader

Office of Natural Environment

Federal Highway Administration

Patricia A. Cazenas, P.E., L.S.

Highway Engineer

Office of Natural Environment

Federal Highway Administration

Gregory E. Granato

Hydrologist 

U.S. Geological Survey

Con

CONTENTS

Abstract .............................................................................................................................................................................

Introduction .......................................................................................................................................................................

Background .......................................................................................................................................................................

Basic Statistical Considerations ........................................................................................................................................

Classification of Variables .......................................................................................................................................

Characteristics of Water-Resources Data ................................................................................................................

Population Structure and Analysis ..........................................................................................................................

Transformations.......................................................................................................................................................

Regression Analysis ..........................................................................................................................................................

The Analytical Process ............................................................................................................................................

Linear Regression Methods.....................................................................................................................................

Nonlinear Regression Methods ...............................................................................................................................

Uncertainty, Quality Assurance, and Quality Control.......................................................................................................

Benchmarking of Analytical Tools..........................................................................................................................

Uncertainty in Modeling Efforts .............................................................................................................................

Uncertainty in Input Data........................................................................................................................................

Summary ...........................................................................................................................................................................

References .........................................................................................................................................................................

Appendix 1: Regression Tools...........................................................................................................................................

Appendix 2: Linear Regression Methods..........................................................................................................................

Appendix 3: Nonlinear Regression Methods ....................................................................................................................

Appendix 4: Uncertainty Analysis ....................................................................................................................................

Appendix 5: Region of Influence Method.........................................................................................................................

FIGURES

1–6. Examples of:

1. The generalized shape of the probability and cumulative distributions of unitless, exponential, 

gamma, and normal populations ...............................................................................................................

2. A scatterplot of total runoff volume and event mean sediment concentrations from a highway-runoff 

monitoring study along highway I-794 in Milwaukee, Wisconsin............................................................

3. A boxplot of total runoff volume and event mean sediment concentrations from a highway-runoff 

monitoring study along highway I-794 in Milwaukee, Wisconsin............................................................

4. Two histograms (in groups of 50 and 100 milligrams per liter) of event mean sediment concentrations 

from a highway-runoff monitoring study along highway I-794 in Milwaukee, Wisconsin ......................

5. A probability plot of event mean sediment concentrations from a highway-runoff monitoring study 

along highway I-794 in Milwaukee, Wisconsin ........................................................................................

6. Properties of the lognormal (natural logarithm) distribution as a function of the population coefficient 

of variation (COV).....................................................................................................................................

TABLES

1. Basic statistical techniques for parametric and nonparametric data analysis .....................................................

2. Documented metadata for selected reports that document highway-runoff regression analysis........................

3. General guide to regression methods..................................................................................................................
VI Contents
Ab

Statistical Approaches to 

Interpretation of Local, Regional, and 

National Highway-Runoff and 

Urban-Stormwater Data

By Gary D. Tasker and Gregory E. Granato

Abstract

Decision makers need viable methods for the 

interpretation of local, regional, and national-highway 

runoff and urban-stormwater data including flows, concentrations and loads of chemical constituents and sediment, potential effects on receiving waters, and the 

potential effectiveness of various best management 

practices (BMPs). Valid (useful for intended purposes), 

current, and technically defensible stormwater-runoff 

models are needed to interpret data collected in field 

studies, to support existing highway and urban-runoffplanning processes, to meet National Pollutant Discharge Elimination System (NPDES) requirements, 

and to provide methods for computation of Total 

Maximum Daily Loads (TMDLs) systematically and 

economically.

Historically, conceptual, simulation, empirical, 

and statistical models of varying levels of detail, complexity, and uncertainty have been used to meet various 

data-quality objectives in the decision-making processes necessary for the planning, design, construction, 

and maintenance of highways and for other land-use 

applications. Water-quality simulation models attempt 

a detailed representation of the physical processes and 

mechanisms at a given site. Empirical and statistical 

regional water-quality assessment models provide a 

more general picture of water quality or changes in 

water quality over a region. All these modeling techniques share one common aspect—their predictive 

ability is poor without suitable site-specific data for 

calibration. 

To properly apply the correct model, one 

understand the classification of variables, the un

characteristics of water-resources data, and the c

of population structure and analysis. Classifyin

ables being used to analyze data may determine 

statistical methods are appropriate for data analy

understanding of the characteristics of water-re

data is necessary to evaluate the applicability of 

ent statistical methods, to interpret the results o

techniques, and to use tools and techniques that 

account for the unique nature of water-resource

sets. Populations of data on stormwater-runoff q

and quality are often best modeled as logarithm

formations. Therefore, these factors need to be 

ered to form valid, current, and technically defe

stormwater-runoff models.

Regression analysis is an accepted metho

interpretation of water-resources data and for pr

tion of current or future conditions at sites that 

input data model. Regression analysis is design

provide an estimate of the average response of a 

as it relates to variation in one or more known v

ables. To produce valid models, however, regre

analysis should include visual analysis of scatte

an examination of the regression equation, eval

of the method design assumptions, and regressi

diagnostics. A number of statistical techniques 

described in the text and in the appendixes to pr

information necessary to interpret data by use o

appropriate methods.
2 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

Uncertainty is an important part of any decisionmaking process. In order to deal with uncertainty problems, the analyst needs to know the severity of the statistical uncertainty of the methods used to predict water 

quality. Statistical models need to be based on information that is meaningful, representative, complete, precise, accurate, and comparable to be deemed valid, up 

to date, and technically supportable. To assess uncertainty in the analytical tools, the modeling methods, 

and the underlying data set, all of these components 

need be documented and communicated in an accessible format within project publications. 

INTRODUCTION

Engineers, planners, economists, regulators, and 

others concerned with stormwater runoff from highways and urban areas often develop alternative plans to 

meet demands for desired quantity and quality of water 

at particular locations and times. Predictive procedures 

are needed for planning, permitting, and design of 

highway structures, as well as design of best management practice (BMP) structures (such as swales or 

retention ponds) and BMP maintenance procedures 

(such as street-sweeping programs). Predictive procedures include the development and application of conceptual models, simulation models, empirical models, 

and statistical models.

Any model, to be considered viable, must stem 

from an initial conceptualization of the system of 

concern. A conceptual model is a qualitative understanding of the sources and processes relevant to a 

problem. The model, which can commonly be represented by a diagram, is supported by common sense, 

scientific principles, data, and research to support the 

ideas. For example, expecting total pollutants in runoff 

to increase with increasing average daily traffic at a 

highway site—because vehicles are considered to be 

one of the primary sources of highway pollutants—is 

an example of a conceptual model. Analysis of available information, however, must substantiate the 

conceptual model to support development of more 

quantitative models through the application of 

empirical, deterministic, and (or) statistical techniques.

Stormwater-quality models have historically 

been used to characterize stormwater flow and quality, 

predict pollutant runoff loads, assess impacts on receiving waters, and determine the effectiveness of various 

BMPs to mitigate possible impairment of designated 

beneficial uses of receiving waters (Kobriger an

others, 1981; Shelley and Gaboury, 1986; Drisc

others 1990). Now, and in the future, valid, curre

technically defensible stormwater runoff model

needed to

• help in interpreting data collected by field stu

• support existing highway- and urban-runoff p

processes (Kobriger and others, 1981, Drisc

others 1990; Young and others, 1996; Gran

others, 1998),

• meet National Pollutant Discharge Eliminatio

System (NPDES) requirements (Young an

others, 1996), and 

• provide systematic and economical methods f

culation of Total Maximum Daily Loads (T

(Shoemaker and others, 1997). 

Sound statistical planning models that us

collected at monitoring stations representing the 

of interest can be used to assist policy makers a

public officials in the evaluations of alternative 

Statistical planning models represent a structure

ordered, and quantitative approach. These mode

provide information for debates over proper cho

water-management alternatives and to evaluate 

petitive alternatives. Statistical planning models 

needed for quantitative local, regional, or nation

water-quality assessments that can be demonstr

be valid (useful for intended purposes), current

scientifically/technically defensible when based 

readily available monitoring data, land use infor

and information about water-quality manageme

practices.

The general objective of this paper is to e

methods for water-quality modeling and to iden

and describe statistical methods and procedures 

may be used to predict concentrations and load

chemical constituents and sediment in highway 

urban runoff and the potential for effects in rece

waters in terms of the unique characteristics of 

resources data. Although the concentrations, loa

potential effects of runoff constituents are often 

ined on a site-specific basis, these issues are als

ined within the context of regional and (or) nati

interpretations. This report is designed to provid

eral (and sometimes very basic) information in t

the analysis of water-resources data for stormw

analysis, but it is not designed to be a textbook 

tistical methods. Appendixes are provided to exp

certain methods discussed in the text, and refere
Backg

suggested reading are provided where appropriate. 

Uncertainties in modeling are discussed, as well as the 

quality-assurance and quality-control measures that are 

necessary to address these uncertainties. 

BACKGROUND

Simulation models, empirical models, regression 

models, and other statistical models historically have 

been used as approaches for predicting the quantity, 

quality, and loads of constituents in highway and urban 

runoff. Simulation models are used with model parameters that have a direct physical definition in an attempt 

to provide a detailed description of the physical processes and mechanisms that affect water quality. These 

models therefore require a considerable degree of 

detail in the description of the physical system. In simulation models, parameter estimation is not as data 

dependent as in statistical water-quality assessment 

models. On the other hand, empirical and statistical 

water-quality-assessment models provide a more general picture of water quality or changes in water quality. This picture could be in the form of a map of a 

water-quality statistic, or tables, or simply an equation 

with error bands on the parameters and predictions. 

Statistical regional water-quality models may also be 

used to estimate nonpoint-source loadings as inputs for 

more detailed water-quality simulation models (Ichiki 

and others, 1996). All these modeling techniques share 

one common aspect—the predictive ability of almost 

any model will be poor without suitable site-specific 

data for calibration (Shelley and Gaboury, 1986).

Criteria for model selection depend on modeling 

logistics as much as on data quality objectives (DQOs) 

and other technical considerations (Shoemaker and 

others, 1997). Logistical criteria include

• the availability of hardware and software to implement the model of choice, 

• the availability of trained modelers to manipulate the 

model, to develop sound input parameters with an 

understanding of how they are used by the model, 

and to critically evaluate model results,

• organizational commitment to establish and support a 

model, to document the model, and to oversee 

subsequent applications of the model so that methods and results can be reviewed and accepted as 

valid, current, and technically defensible, 

• organizational expertise with the model, to ap

model and to review applications of the mo

maintain credibility of results, and 

• available financial resources to support mode

efforts. 

The resources needed to support a modeling eff

increase in direct proportion to the complexity o

model chosen for analysis. A political criterion 

involved in the selection and use of modeling m

To implement a successful modeling effort, the 

interest groups involved in a project must be wi

accept modeling results. Even the most success

sophisticated modeling effort will fail if the res

not understood and accepted by decision maker

(Shoemaker and others, 1997).

A detailed description of each available r

quality model is beyond the scope of this report

sary information, however, is readily available i

publications. Driscoll and others (1990) discuss 

application of different methods for national an

of highway-runoff quality and related environm

effects. Hall and Hamilton (1991) describe man

aspects of highway-runoff transport modeling. B

and Huber (1992) describe many of the hydrolo

simulation models in use today. Young and othe

(1996) provide examples of simulation, empiric

regression models, and the FHWA statistical mo

applied to individual highway sites. Shoemaker 

others (1997) provide a comprehensive guide d

ing most of the available predictive tools—inclu

most applicable simulation, empirical, regressio

statistical models—for water-quality assessmen

the TMDL process. A brief discussion of simul

empirical, regression, and other statistical mode

however, will provide information useful for sta

interpretation of local, regional, and national hi

and urban stormwater-quality data.

Simulation models—including SWMM (

and Dickinson, 1988), STORM (U.S. Army Co

of Engineers, 1977), HSPF (Bicknell and other

1993), and the Federal Highway Administration 

(FHWA) urban highway storm drainage model 

and others, 1983)—require detailed site-specifi

information and data to calibrate the model for 

conditions. By using simulation models, highw

engineers can evaluate the design of each highw

including the road surface, catch basins, and dr

structures—with respect to flow and water quali

and Hamilton, 1991). Simulation models are al
4 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

useful in estimating relations between variations in 

input parameters and resulting flows, constituent concentrations, and constituent loads in runoff. The validity of these estimates, however, depends upon a robust 

calibration of the model with site-specific data 

(Driscoll and others, 1990). For example, Zarrielo 

(1998) compared results of simulations made with nine 

uncalibrated runoff models to observed flows in two 

urban watersheds; although the modelers had very 

detailed information about site characteristics and precipitation, the resulting estimates of peak flow rates and 

total stormflow volumes differed from measured data 

by as much as 260 and 240 percent, respectively. Simulations are considered to be highly useful in the design 

phase or post construction-analysis phase of specific 

sections of a highway, but collection of the site-specific 

data and the high level of effort necessary to successfully implement these models are not practical for planning or management on a regional or national scale 

(Driscoll and others, 1990).

Empirical models include the U.S. Environmental 

Protection Agency (USEPA) Screening Procedures 

(Mills and others, 1985) and the Simple Method 

(Schueler, 1987). Empirical models involve the use of 

estimated concentrations (presumably derived from 

local studies and (or) the literature) multiplied by some 

fraction of local precipitation volumes (used to estimate 

runoff volumes) to estimate annual or storm loads of 

constituents of concern. Empirical models provide 

order-of-magnitude estimates of loads but do not indicate correlation among variables and do not provide 

estimates of the uncertainty in predictions without 

detailed site-specific data needed to estimate probability 

distributions of the precipitation/runoff volumes and 

concentrations used in the equation. 

Highway-runoff regression models include 

those developed by the FHWA (Kobriger and others, 

1981; Driscoll and others, 1990); State departments 

of transportation in Washington (Chui and others, 

1982), California (Kerri and others, 1985), and Texas 

(Irish and others 1998); and the Ontario Ministry of 

Transportation (Thomson and others, 1996; 1997a). 

These models use information such as precipitation 

characteristics, highway-design features, traffic volumes, and interrelations between measured constituents to predict concentrations and (or) loads of 

highway-runoff constituents. The USGS regression 

method (Tasker and Driver, 1988; Driver and Tasker, 

1990) uses the percentage of impervious area, r

statistics, and in some cases the mean minimum 

January temperature to estimate concentrations 

(or) loads of urban-runoff constituents. This mo

may be used for estimating highway-runoff qua

urban areas (Young and others, 1996).

Other statistical techniques have also bee

as approaches for prediction of modeling urban

highway-runoff quantity, quality, and loads. Th

techniques use measures of the central tendency 

available data (such as the mean or median) and 

sures of the variability of data (the variance or c

cient of variation) to predict model outputs and 

uncertainty thereof. The model formulated by D

and others (1990), uses storm event statistics an

probability distribution of site event-mean conc

tions (EMCs) to estimate runoff volumes, conce

tions, and loads in runoff. Driscoll and others (1

then use these loads and the probability distribu

streamflow volume at a given site to estimate po

dilution in receiving waters. Statistical models t

readily available rainfall statistics and water-qu

data to produce a frequency distribution of conc

tions, loads, and potential for receiving-water e

are useful because assessments of risk and retur

periods can be calculated (Driscoll and others, 

The existing FHWA statistical pollutant l

and impact model approach described by Drisc

and others (1990) is, given appropriate input da

erally valid for intended purposes. This model u

site characteristics and other factors to calculate 

mates of runoff volumes, loads, and receiving-w

concentrations as a probability distribution. Thi

model, however, is not designed for scientific or 

cal interpretation of study-site data in terms of t

potential relations between constituents and (or

site characteristics. Therefore, other statistical 

models are necessary for scientific or technical 

pretation of local, regional, and national highwa

urban-stormwater data.

Statistical techniques are commonly best 

to highway-runoff modeling needs at any scale 

regional, or national). Population statistics and 

dard techniques for the analysis of error in pred

can be used to assess risk of decision error. Esti

of the error in predictions from empirical mode

uncalibrated simulation models is, in reality, im

ble, because inputs are complex and the effects 
Basic Statistical Consider

error of outputs can be multiplicative. Use of simulation models requires a high degree of institutional 

expertise and experience, as well as a substantial modeling effort, for each site of interest. Many simulation 

models are suitable for specific highway applications, 

but the detailed input requirements and extensive modeling efforts required may preclude use for planninglevel estimates. By comparison, use of existing statistical models minimizes the logistical burdens required 

for model application (Shoemaker and others, 1997). If 

the statistical model used is based on accepted statistical methods, is formulated from a valid, current, and 

technically defensible data set, and is documented and 

communicated in an accessible format, then these steps 

will ensure that model results will be accepted. The 

existence of a quantitative data set further lends credence to the acceptability of model results. The complexity of simulation models and the large range of 

reasonable input parameters inherent in the model calibration process can lead to differences in professional 

judgment, which can negatively affect acceptance of 

simulation-model results. Therefore, simple statistical 

models may be preferable to simulation models for 

many highway-runoff modeling needs.

Statistical analysis is also important to the design 

and implementation of highway- and urban-runoff 

data-collection programs that will yield results suitable 

for inclusion in local, regional, and national-synthesis 

efforts. Statistical analysis of available information is 

necessary to help determine the design of the sampling 

effort (random as opposed to systematic), the frequency of sampling, the number of and location of 

sites, and the quality of the resultant data (Averett and 

Schroder, 1994). For example, Thomson and others 

(1996; 1997b) used an extensive highway-runoff data 

set from Minnesota to determine that EMC samples 

from at least 15 to 20 storms are required to provide 

reasonable estimates of mean total suspended solids, 

total dissolved solids, total organic carbon, and zinc 

concentrations from each study site. Statistical methods can also be used to design and (or) optimize stormwater data-collection networks (Tasker and Raines, 

1995). Statistical-analysis techniques are also necessary for the design, implementation, and interpretation 

of quality-assurance and quality-control (QA/QC) programs necessary to demonstrate that data collected are 

valid (useful for the intended purposes), technically 

defensible, and complete (Jones, 1999).

BASIC STATISTICAL 

CONSIDERATIONS

Proper classification of variables and an u

standing of the characteristics of water-resource

are crucial for interpreting the results of individ

studies and for combining these results in a regi

national synthesis of stormwater-quality data. C

cation of the different types of variables being u

analyze data may determine which statistical m

are appropriate for data analysis. An understand

the unique characteristics of water-resources da

necessary to evaluate the applicability of variou

tical techniques, to interpret the results of these 

niques, and to use tools and techniques that acco

the nature of water-resources data sets. Underst

the methods and measures used to determine an

lyze the population structure is also important. W

necessary (as is common for water-resource and 

ularly for stormwater data sets), one must choo

appropriate methods of data transformation to e

use of statistical techniques without violating th

tical assumptions underlying the methods chose

analysis. Therefore, classification of variables o

est, an understanding of the statistical characteri

water-resources data, a familiarity with the pop

structure and basic methods of analysis, and (w

necessary) selection and proper use of populatio

transformation techniques are necessary to form 

current, and technically defensible stormwater-

models.

Classification of Variables

Classification of variables is useful in sev

ways because the proper method of data analys

depends on variable type. A variable can be cla

by inherent mathematical structure (discrete or 

uous), by statistical objectives of the study (resp

or predictor), and by level of measurement (nom

ordinal, interval, or ratio).

A variable is discrete if there is a gap bet

two successively observable values in which an 

observed value is not possible (for example, an 

scale). A variable is continuous if there is alway

another observable value between any two obse

values (for example, a real number scale). Exam
6 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

of a discrete variable are the number of floods above 

a threshold, the number of exceedences of a waterquality standard, or the identification of a group. It is 

not possible to have 3.2 floods above a threshold, 11.7 

exceedences of a standard, or to have a value between 

group A and group B. Examples of continuous variables are chloride concentration or the number of 

pounds of road salt applied per lane-mile of roadway 

per season. Discrete variables, however, can sometimes 

be treated as continuous if they cover a wide range with 

small gaps between observations. For example, the 

number of days per year with measurable rainfall could 

vary between 0 and 365. Furthermore, continuous variables may be grouped into categories and treated as 

discrete variables. For example, pH readings may be 

grouped into the categories of low, medium, and high.

A variable may also be classified according to 

the statistical objectives of the study without regard to 

the variable's mathematical structure. A variable that is 

described in terms of other variables in a regression 

model is called the response variable (or the dependent 

variable, or the predicted variable). Variables used to 

describe the response variable in a model are called 

predictors (or explanatory variables, carriers, or independent variables). For example, if one wishes to predict average nitrate concentration at a streamflow site 

on the basis of land-use characteristics of the upstream 

watershed, then average nitrate is the response variable, 

and land-use variables, such as percentage of urban or 

industrial land, would be predictors.

The level of measurement may also be used to 

classify variables. Nominal variables are variables 

whose observed values are labels for different unordered categories. For example, the "type" variable for a 

basin may be 1 for urban or 0 for nonurban. Ordinal 

variables are variables whose observed values are 

ordered with no implication of distance between different points on the scale. For example, Driscoll and 

others (1990) classified highway sites as either 

"urban"—average daily traffic (ADT) counts that are 

greater than or equal to 30,000 vehicles per day 

(VPD)—and "nonurban"—ADTs that are less than 

30,000 VPD. Another example of an ordinal variable 

would be the use of the previously mentioned pH categories of low, medium, and high. Interval variables 

have equal differences between successive points on 

the measurement scale, but the zero point is arbitrary. 

In this case, one can compare differences in observed 

values. An example of an interval variable would 

increase in population density since the last cen

Ratio variables have equal differences between 

sive points and a fixed zero. They allow one to c

differences in observed values and relative mag

For example, nitrate concentration measured at 

monitoring sites in a region could be treated as 

ratio-scale variable. 

Characteristics of WaterResources Data

The applicability of any given statistical p

dure depends directly on assumptions about the 

teristics of the data being explored. Results from 

statistical analysis may be meaningless or, even 

misleading if data do not conform to the design 

assumptions of the statistical method used. The

one must carefully consider which statistical m

are appropriate in terms of the data being evalu

The unique features of water-resources data aff

suitability of interpretations made by the applic

many classical statistical techniques. Pertinent c

teristics of water-resources data (Helsel and Hir

1992; Helsel, 1993; Hirsch and others, 1993), in

the following: 

• Nonrepeatability—Measurements are usually 

vational, not experimental. More specifical

tistical regularity cannot be demonstrated f

water-resources data by repeating a contro

experiment because exactly recreating the 

natural and anthropogenic influences that a

each measurement is impossible. 

• A lower limit of zero—Negative values are no

ble for many water-resources characteristic

example, negative values for measured pre

tion, flow, or constituent concentrations do 

have meaning.

• Censored data sets—Limits in methods for sa

collection and analysis cause data to be rep

either above or (more typically) below one 

more reporting limits, which produce a cen

population of data. 

• Meaningful outliers—Valid measurements th

are considerably higher or lower than most 

the measured population are common amo

hydrologic data sets.
Basic Statistical Consider

• Positive skewness—Data sets that are not symmetrical around mean or median values are typical for 

water-resources data sets because the combined 

effects of a lower bound of zero, censoring, and 

meaningful outliers tend to produce data sets in 

which the right tail of the distribution is extended 

and the left tail is truncated.

• Nonstandard distributions—Many statistical techniques are based on the assumption that the data 

(or in the case of regression models, the residuals) 

are normally distributed (the "bell-shaped" curve). 

The nonstandard distributions characteristic of 

water-resources data sets necessitate alternative 

tools for analysis. 

• Autocorrelation—Natural and anthropogenic effects 

tend to cause conditions in which consecutive 

measurements tend to be strongly correlated.

• Interdependence—Changes in one characteristic of 

interest (such as rainfall intensity) cause changes 

in other characteristics (such as measured flows 

and concentrations).

• Temporal variation—Measured water-resources 

characteristics vary cyclically at timescales 

of hours, days, weeks, seasons, years, and 

even decades because of both natural and 

anthropogenic influences. 

Therefore, data sets must be collected in a 

manner that will represent the underlying population 

distribution, and the statistical methods used to characterize the data must be appropriate for populations with 

these characteristics. Ancillary information and metadata (information about a given data set, including 

explanatory information and data-quality information) 

pertinent to the statistical characteristics of the sampled 

population need be documented and communicated in 

an accessible format with monitoring-study data to 

support interpretation methods. 

Ancillary data are important to quantify possibly 

confounding variables that may preclude meaningful 

interpretation of data because statistical regularity 

cannot be demonstrated through controlled experiments. For example, Driscoll and others (1990) noted 

the effect of local land use on regression analysis to 

predict the median EMC of zinc from measures of 

average daily traffic on a site-by-site basis. When all 

sites (with average daily traffic greater than 30,000 

vehicles per day) were included, regression analysis 

indicated a small negative slope with increasing traffic 

and an R2 of about 0.04, indicating that EMCs f

would decrease with increasing traffic but that t

tion was very weak. When one site (which was 

influenced by a local zinc-smelting operation) w

omitted, however, regression analysis indicated 

strong positive slope and relatively strong relati

between increasing traffic volume and increasin

EMCs for zinc (R2 was about 0.7). In this case, 

local land use is the ancillary information neces

meaningful interpretation of data based on the a

tion that traffic characteristics would affect (if n

control) zinc concentrations in highway runoff.

The effects of censored data can be espec

problematic for interpretation of water-quality d

Laboratory detection limits change with time, c

dramatically different from laboratory to labora

and may even be different from method to meth

within a laboratory. For example, Garbarino and 

Struzeski (1998) indicate that detection limits f

total recoverable copper in whole-water sample

0.4, 5., and 0.3 micrograms per liter for the grap

furnace-atomic absorption spectrophotometry 

(GF-AAS), inductively coupled plasma-optical 

emission spectrometry (ICP-OES), and inductiv

coupled plasma-mass spectrometry (ICP-MS) m

respectively. A number of methods may be use

compensate for the effects of one or more detec

limits while applying statistical analysis tools to 

set (Gilliom and Helsel, 1986; Helsel and Gillio

1986; Helsel and Cohn, 1988; Helsel, 1990; He

and Hirsch, 1992). These methods of statistical 

analysis require knowledge of the detection lim

truncating each data set. Detection-limit inform

however, may not be available. For example, in 

ing data to develop a pollutant-loading model, D

and others (1990), noted that "it was virtually im

ble to unequivocally determine the actual detec

limit associated with each pollutant concentrati

Driscoll and others (1990) could not determine 

detection limits because this information was n

explicitly documented in available ancillary dat

and because the data were produced by a numb

different analytical laboratories over the years 

between the mid 1970's and the mid 1980's. Tho

and others (1996) also noted the difficulty in id

ing detection-limit data in an extensive highway

runoff data set (416 storms from four sites mon

during 1976–83). Detection-limit artifacts also 
8 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

statistical properties of individual data sets. When a 

data set contains values reported as less than one or 

more detection limits an overestimation of central-tendency measures and an underestimation of dispersion 

measures will be caused by truncation of the lower tail 

of the true population (Driscoll and others, 1990). Also, 

because the relative uncertainty in the accuracy and 

precision of individual values tend to increase as 

reported concentrations approach the detection limit, 

the percent error expected for measurements near 

detection limits is much higher than for values well 

within the measurement range of the method of analysis (Granato and others, 1998).

The presence of meaningful high-end outliers 

(actual but extreme values) contributes to the positive 

skew and is a factor producing nonstandard distributions. High-end outliers represent times when, for 

example, regulatory criteria may be exceeded and the 

health of local ecosystems may be affected. These outliers, however, produce a host of potential problems for 

interpretation of data sets assembled for use in a 

regional or national synthesis. 

Certain assumptions and conditions need to be 

considered in terms of the decision to include or 

exclude individual data points or even entire data sets, 

among which are the following:

• It is often assumed that outliers are not meaningful. 

This assumption may be misleading unless documented QA/QC information indicates problems 

with sample collection, processing, and (or) analysis that would justify elimination of suspect data. 

• An outlier may be meaningful, but it may represent 

the effects of a process that should not be considered in a synthesis designed to characterize normal 

highway-runoff quality. For example, a chemical 

spill that occurs before or during a runoff-quality 

study may be representative of runoff quality and 

subsequent environmental effects at sites affected 

by spills, but the data may not be characteristic of 

"normal sites." In fact, the rate of substantial spills 

is about 0.0019 incident per lane kilometer per 

year (as estimated from the median of highway 

hazardous-materials incidents and public road 

mileage compiled by U.S. Environmental 

Protection Agency, 1999). Therefore, data from a 

spill site would incorrectly bias a national data set 

unless a sufficient number of sites were monitored 

to properly represent the probability of a spill at 

any given site. In another example, Driscoll and 

others (1990) eliminated data sets from site

affected by fallout from the eruption of Mt

Helens in Washington State. Although thes

"meaningful outliers" were representative o

effects of volcanic eruption on the quality 

way runoff, this effect was not deemed suit

estimation of typical highway-runoff qualit

United States.

• An extreme outlier may be meaningful and m

resent the effects of a process that should b

sidered in a synthesis designed to characte

normal highway-runoff quality, but this effe

obscure effects of other process-related var

The example of the effect of the highway s

under influence of a local zinc smelting op

indicates the necessity for detailed docume

to describe surrounding land use, but this e

precludes development of meaningful relat

between metal concentrations and average 

traffic for more normal highway conditions 

the United States (Driscoll and others, 199

Elimination of outliers is considered a dan

and unwarranted practice for the interpretation 

water-quality data, unless one has substantial ob

evidence demonstrating that the outliers are not 

sentative of the population under study (Helsel 

Hirsch, 1992). When statistical tests are used to 

outliers, these tests do not indicate that outliers 

sent errors; they do indicate that the population 

sured vales is not necessarily a normal distribut

Excessive numbers of extreme values may caus

nificance levels of tests requiring the normality 

assumption to be in error. Therefore, use of a te

requiring the normality assumption will produc

curate results when outliers affect population str

Outliers may have high leverage and thus a stro

potential for influencing the slope of a regressio

(Helsel and Hirsch, 1992). If an outlier is discov

have a strong influence on the slope of a regress

(the slope and (or) the correlation coefficient ch

significantly when the point is omitted), then on

determine whether the outlier represents extrem

values for a single process or if a secondary pro

is characterized by the outlier. Measurement an

mentation of explanatory variables such as prec

tion and flow (Church and others, 1999); real-ti

measures of water-quality characteristics such a

cific conductance, pH, temperature, and turbidi

(Spangberg and Niemczynowicz, 1992; Whitfie
Basic Statistical Consider

Wade, 1992; Granato and Smith, 1999); use of ratios 

between constituents of interest (Granato, 1996); and 

results from a comprehensive QA/QC program (Jones, 

1999) can be used to identify and explain outliers in 

terms of the potential effect of real physicochemical 

processes as opposed to the effects of sampling 

artifacts. 

The natural and anthropogenic processes controlling runoff quality and the methods for sampling, 

processing, and analysis often cause problems with 

autocorrelation (also referred to as serial correlation or 

correlation—the dependence of residuals in a time 

sequence because data reflect the effects of preceding 

conditions). One of the assumptions inherent in many 

regression techniques is that the residuals are independent (Helsel and Hirsch, 1992). Autocorrelation can be 

a problem within stormwater data sets because many of 

the variables used for analysis are pairs of data in a 

time series. For example, precipitation and flow, flow 

volume and concentration, and relations between measured constituents (total suspended solids and lead, for 

example) may be pairs of data in a time series. Timeseries effects may also occur between subsequent 

storms. For example, Irish and others (1998) indicate 

that the duration, the volume of runoff per unit area, 

and the intensity of runoff per unit area of the preceding storm are significant causal variables in a regression model developed for highway-runoff loads of 

suspended solids and metals in Texas. Autocorrelation 

can be important because it affects the optimization of 

regression coefficients, affects estimates of population 

variance (invalidating results of hypothesis tests), and 

produces confidence and prediction intervals that are 

too narrow for the real population being sampled. To 

address autocorrelation problems, one may group data 

into time periods and use summary statistics in an analysis, use methods that are robust with respect to autocorrelation (Helsel and Hirsch, 1992), incorporate 

explanatory variables into predictive models that will 

account for potential effects of autocorrelation (Irish 

and others, 1998), or subsample from large data sets to 

eliminate autocorrelation.

Temporal variation may also increase variability 

in data and affect the comparability of data between 

sites. Seasonality is an obvious factor that may affect a 

population of stormwater samples at any given site. For 

example Driscoll and others (1990) segregated "snow 

washoff events" from other events for analysis and 

found that, for many constituents, snow washoff events 

had substantially higher median site EMCs, much 

wider confidence intervals, and a relatively few 

of events than for the rest of the available data. 

subtle temporal variations may affect relations b

predictors and response variables and (or) contr

the variability in measured water quality in a po

tion of stormwater-quality samples. For exampl

Whitfield and Wade (1992) used results from au

water-quality monitoring stations to detect daily 

in receiving-water quality, as well as the effects 

storms. In addition, the magnitude of seasonal 

tion would be expected to be a function of local 

and may therefore partly obscure relations betw

predictors (such as ADT) and response variable

as constituent concentrations) if a statistical ana

includes sites with different patterns of tempora

tion. For example, Driver and Tasker (1990) us

mean minimum January temperature as a predic

variable to partly account for differences in the 

tude of seasonal variation among sites in an ana

National Urban Runoff Program (NURP) data. 

Population Structure and 

Analysis

Sampling theory—the concept that one c

itor a given number of events and with this infor

estimate the properties of the underlying distrib

is based upon the concept of a probability distri

The structure of a given population (the probabi

tribution) will determine which methods will be 

priate for statistical analysis and interpretation o

water-resources data. If, for example, measured 

a normal distribution, one measure of location (

tendency) and one measure of spread (variabilit

be used to define the entire population. Howeve

applicability, robustness, and relative power of 

ent measures of location, measures of spread, a

sures of skewness depend upon the structure of 

and the objectives of the analytical process. 

The structure of data is often described by 

of population-frequency distributions. If variabl

can be ascribed to a particular frequency distrib

then the known structure of the distribution has 

potential uses (Athayde and others, 1983; Drisc

others, 1990; Helsel and Hirsch, 1992), includin

following: 
10 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

• concisely reporting data in terms of population measures rather than total range (which may be misleading for data with outliers and multiple 

detection limits),

• examining the characteristics of the data (for example, the mean and standard deviation can be used 

to define the location and shape of a normal distribution),

• establishing the probability of any given value in the 

distribution (for example the probability of 

exceeding a water-quality standard), 

• comparing results from different sites on a common 

basis,

• providing a framework for examining the transferability of data quantitatively, and

• testing hypotheses (for example, establishing 

whether concentrations of metals at sites with an 

ADT of less than 30,000 vehicles per day are statistically different from concentrations of metals at 

sites with an ADT greater than 30,000 vehicles per 

day). 

The number of potential distributions is infinite (Ott, 

1993). For example, McLaughlin (1999) provides 

equations for the probability distribution and 

cumulative distribution functions for more than 50 

distributions in terms of the location, shape, and scale 

of each distribution. Figure 1 indicates generalized 

shapes of the probability-distribution and cumulativedistribution functions for the exponential, gamma, and 

normal distributions. Commonly, water-resources data 

can be characterized by relatively few distributions, 

and many of the available distributions are specialized 

variations of more general distributions (Helsel and 

Hirsch, 1992; Ott, 1993). For example, the lognormal 

frequency distribution is simply a normal distribution 

for data that have been transformed to logarithmic 

space so that the resulting distribution approximates 

that of the theoretical normal distribution.

Statistical-analysis methods can generally be 

classified as either parametric (methods in which a 

specified data distribution is necessary to support 

design assumptions) or nonparametric (methods that 

do not depend on a specified data distribution to establish their meaning). Nonparametric tests, because 

they are not as dependent on an assumed population 

distribution, may be more robust for data analysis. The 

power of parametric techniques is generally higher 

when the distributional assumptions are correct

many cases, however, the relative advantage of 

metric techniques decreases with increasing pop

size (Hirsch and others, 1993). 

Statistical measures of population locatio

spread, and skewness are summarized and defin

terms of the statistical basis and design assump

that define each method in table 1. Helsel and 

Hirsch (1992) and Hirsch and others (1993) use 

techniques to describe mathematical informatio

for statistical analysis of water-resources data. 

0

1.0

0.2

0.4

0.6

0.8

CUMULATIVE DISTRIBUTION

DATA VALUES

0

1.0

0.2

0.4

0.6

0.8

PROBABILITY DISTRIBUTION

EXPONENTIAL DISTRIBUTION

GAMMA DISTRIBUTION

NORMAL DISTRIBUTION

EXPLANATION

Figure 1. Example of the generalized shape of the 

probability and cumulative distributions of unitless, 

exponential, gamma, and normal populations.
Basic Statistical Considera

Table 1. Basic statistical techniques for parametric and nonparametric data analysis

[N, nonparametric; P, parametric]

Technique Measure Basis Definition Comments

Arithmetic mean

(Average)

Location P The sum of all data divided by the 

sample size.

Can be affected by the presence of and (or) changes 

magnitude of one or more outlying observations. 

Representativeness depends upon the assumption 

data are normal (or at least unimodal and symme

Median Location N The middle value when data are 

ordered from lowest to highest.

When there is an even number of data points the me

the average of the two central observations. The 

is also refered to as the 50th percentile.

Mode Location N The data value that occurs with the 

highest frequency.

A data set may have more than one modal value. 

Geometric mean Location P The mean of the logarithms of data 

that is transformed back into 

original units.

Representativeness depends on the assumption that 

are normal (or at least unimodal and symmetric) 

space. 

Trimmed

(or weighted) 

mean

Location P The mean of censored data divided 

by the sample size after 

censoring.

Trimmed means (or weighted) means are computed 

values judged as outliers have been eliminated (o

weighted with a value of zero). It is typical to trim 

percentage from the bottom and top of the data in 

attempt to apply systematic methods.

Range Spread N The difference between the largest 

and smallest measurements in a 

set.

Although nonparametric, the range is affected by th

presence of and (or) changes in the magnitude of 

more outlying observations. 

Variance Spread P The sum of the squared deviation 

of all measurements divided by 

one less than the total number of 

data points. 

The variance can be unduly affected by the value of 

more outlying observations.

Standard 

deviation

Spread P The positive square root of the 

variance.

The standard deviation can be unduly affected by th

of one or more outlying observations.

Coefficient of 

variation (COV)

Spread P The ratio of the standard deviation 

over the mean.

A measure of spread normalized to the magnitude o

mean.

Interquartile 

range (IQR)

Spread N The difference between the 75th 

and 25th percentile values (by 

number of measurements) when 

data are ordered from lowest to 

highest.

Typically used as a measure of central spread becau

25th, 50th (median), and 75th percentiles split th

into four equal-sized quarters (by number of 

measurements). Other percentile ranges may be u

well.

Median absolute 

deviation (MAD)

Spread N The median of the absolute values 

of the difference between each 

data point and the data-set 

median. 

The MAD, because it is the median of the populatio

absolute differences, is resistant to the effects of 

Coefficient of 

skewness

Skewness P The third central moment divided 

by the variance cubed.

A positive value indicates that the population is righ

skewed, and a negative value indicates left skew.

Quartile skew 

coefficient

Skewness N The difference between the range 

of each quartile (25th to 50th 

and the 50th to 75th) divided by 

the IQR

A positive value indicates that the population is righ

skewed, and a negative value indicates left skew. 

percentile ranges may be used as well.
12 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

An understanding of the assumptions inherent in statistical measures, the methods used to calculated these 

measures, and their sensitivity to changes in location, 

spread, and skewness are important for interpretation 

of results of more complex statistical analysis. For 

example, in the calculation of a simple regression, the 

means and variances of each population of interest 

determine the slope and intercept of the resultant line 

equation (Hirsch and others, 1993). Furthermore, the 

simple measures of population characteristics covary 

with differing population structure. If a data distribution fits the normal distribution, then the mean, median, 

mode, and trimmed means should be equivalent if not 

equal (Helsel and Hirsch, 1992; Ott, 1993). If, however, a population is lognormal, then the geometric 

mean and median should be equivalent if not equal 

(Helsel and Hirsch, 1992). As unimodal populations 

become increasingly skewed to the left or right, the 

mean and trimmed mean will fall increasingly to the 

left or right of the median, respectively, and the mode 

will fall increasingly to the right or left of the median, 

respectively (Ott, 1993). Indices of spread also will 

have unique relations when data are normally distributed. For example, if data are normally distributed, 

then dividing the range by 4 should produce a value 

approximately equal to the standard deviation because 

about 95 percent of values should lie in the range of 

plus or minus 2 times the standard deviation (Ott, 

1993).

For water-resources data, graphical analysis is an 

essential first step in the interpretation process. Use of 

graphical analysis can provide a visual summary of the 

data and can help reveal the most appropriate population structure and methods for analysis (Helsel and 

Hirsch, 1992). Graphical tools such as the scatterplot, 

the histogram, the boxplot, and the probability plot can 

be used to characterize data and find potential problems. For example, Driscoll and others (1990) present 

the event-mean suspended-solids concentrations and 

runoff volumes for the I-794 data set from Milwaukee, 

Wisconsin. A scatterplot (fig. 2), a boxplot (fig. 3), two 

histograms (fig. 4), and a probability plot (fig. 5) indicate that event mean suspended-solids concentrations 

are not normally distributed in linear space and that 

runoff volume does not control event mean suspendedsolids concentrations at this site. Driscoll and others 

(1990) used probability plots to establish that the lognormal distribution was a sufficiently close approximation for highway-runoff-quality data to be included in 

their national highway-runoff-quality model. In 

case, use of graphical analysis helped identify t

substantial number of below-detection-limit val

were incorporated (without special notation) int

of the historical data sets and indicated that the 

detection-limit data did not fit the lognormal pr

ity distribution and would affect population stat

(Driscoll and others, 1990). Helsel and Hirsch 

provide detailed descriptions about the use of gr

exploratory data analysis (EDA). The National I

of Science and Technology provides text and so

for mathematical and graphical techniques to ap

EDA (Croarkin and Tobias, 2000). A number of 

fication tests (methods to check that assumption

0 0.5 1.0 1.5 

RUNOFF, IN INCHES

0

500

50

100

150

200

250

300

350

400

450

TOTAL SUSPENDED SOLIDS, IN MILLIGRAMS PER LITER

Figure 2. Example of a scatterplot of total runoff volu

event mean sediment concentrations from a highway 

monitoring study along highway I-794 in Milwaukee, 

Wisconsin (data from Driscoll and others, 1990).
Basic Statistical Considera

estimation method are valid) have been develop

recent years and may also be useful for examin

runoff data before application of statistical-anal

techniques (Godfrey, 1988).

Within a discussion of population structu

statistical analysis, it should be noted the event 

concentration, or EMC, is not a statistical mean 

on any implied probability distribution; rather th

is an operational definition used to characterize 

vidual storm-event water quality. The EMC is d

as the total load of a stormwater-quality constit

divided by the total flow that occurs during the s

(13) (25) (12)

<0.5 ≥0.5 All

RUNOFF, IN INCHES

TOTAL SUSPENDED SOLIDS, IN MILLIGRAMS PER LITER

EXPLANATION

(13) NUMBER OF OBSERVATIONS

DATA VALUES OUTSIDE THE

10TH AND 90TH PERCENTILES

90TH PERCENTILE

75TH PERCENTILE

MEDIAN

25TH PERCENTILE

10TH PERCENTILE

0

500

50

100

150

200

250

300

350

400

450

Figure 3. Example of a boxplot of total runoff volume and 

event mean sediment concentrations from a highway 

runoff monitoring study along highway I-794 in Milwaukee, 

Wisconsin (data from Driscoll and others, 1990).

<100 100-200 201-300 301-400 

TOTAL SUSPENDED SOLIDS CONCENTRATION

IN MILLIGRAMS PER LITER

0

14

2

4

6

8

10

12

NUMBER OF OCCURRENCES

≤50 51-100

150

101-

200

151-

250

201- 

TOTAL SUSPENDED SOLIDS CONCENTRATION

IN MILLIGRAMS PER LITER

0

8

2

4

6

NUMBER OF OCCURRENCES

Figure 4. Example of two histograms (in groups of 

and 100 milligrams per liter) of event mean sedimen

concentrations from a highway-runoff monitoring stu

highway I-794 in Milwaukee, Wisconsin (data from 

and others, 1990).
14 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

interest (Huber, and others, 1979; Driscoll and others, 

1990; Huber, 1993). In theory, an EMC would be 

obtained by collecting and homogenizing all runoff 

from a given event and then sending a representative 

subsample for analysis. In practice, the EMC representing each storm is determined from analysis of one 

flow-weighted composite sample or from a number of 

discrete samples that are flow-weighted mathematically (Driscoll and others, 1990; Huber, 1993). In 

either case, the EMC is not a parametric avera

a time- and flow-integrated estimate of stormw

quality.

Transformations

The general versatility, power, and math

cal elegance of parametric procedures designe

data that fit a normal distribution are usually ad

geous in the application of statistical analysis. 

matical transformations (used to redistribute 

population characteristics to approximate a no

distribution) are often employed to facilitate th

normal (parametric) statistical-analysis techniq

Transformations are usually made for one or m

the following reasons: to simplify the model; t

lize the variance; to normalize the data; or, for 

sion analysis, to define a transformed model wi

distributions that fit the assumptions of the mod

regression models, transformation of the respo

variable is often desirable if the response varia

nonnegative and the range of observed values i

more powers of 10. Log transformations are of

useful for minimizing the standard error of the 

mate (Driver and Tasker, 1990). If all the values 

from zero and the range of values is relatively 

however, transformation will have little effect. 

parametric methods are generally invariant to m

surement scale, so transformations do not alter 

with respect to these methods (Helsel and Hirs

1992). 

Transformations typically improve the sy

try of data by means of a mathematical functio

designed to alter the distance between observat

a line plot. The goal is to expand or contract th

tance between the median and extreme values i

population of interest. Power functions (in whic

transformed variable is raised to an exponent) a

rithms (natural and base 10) are usually used to 

form data (Helsel and Hirsch, 1992). To reduce 

negative (left) skewness, powers of greater than 

used. To address positive (right) skewness powe

than 1 (the square or cubed roots) or logarithms 

used. 

0 100 200 300 400 500

TOTAL SUSPENDED SOLIDS, IN MILLIGRAMS PER LITER

0

1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

CUMULATIVE FREQUENCY

Normal Distribution: mean = 170

standard deviation = 114

Highway I-794 data

(Driscoll and others, 1990)

Figure 5. Example of a probability plot of event mean sediment 

concentrations from a highway runoff monitoring study along 

highway I-794 in Milwaukee, Wisconsin (data from Driscoll and 

others, 1990).
Basic Statistical Considera

Logarithmic transformations are especially 

useful for normalizing values that vary by orders of 

magnitude, and they dampen the effect of very large 

outliers on statistical estimators. In addition, the standard error of the transformed population can be 

expressed as a percent error when transformed back 

into original space. Logarithmic transformations, however, increase the relative weight of small data values 

and can magnify uncertainties in the true (or assigned) 

value of concentrations near detection limits (Stedinger 

and others, 1993). 

Historically, populations of urban- and highwayrunoff data can be characterized by the log-normal 

distribution (Athayde and others, 1983; Driscoll and 

others, 1990). EMCs of highway-runoff constituents 

indeed vary by more than one and as much as four 

orders of magnitude at and between sites. For example, 

Smith and Lord (1990) report that total suspended 

solids concentrations range from 4 to 1,156 mg/L 

(milligrams per liter), total organic carbon concentrations range from 5 to 290 mg/L, zinc concentrations 

range from about 0.01 to 3.4 mg/L, and chloride concentrations range from 5 to 13,300 mg/L. Driscoll and 

others (1990) used the natural logarithmic (ln) transform to normalize data used to develop their national 

highway-runoff-quality model. Athayde and others 

(1983) also indicated that urban-runoff data collected 

for the National Urban Runoff Program (NURP) generally could be characterized by use of the lognormal distribution. Thompson and others (1996) examined the 

effects of logarithmic and exponential transformation 

on the R2 of various regression models designed to predict constituents from surrogate parameters; they found 

that these transformations demonstrate the best explanation (highest R2) for the water-quality predictor variables but that untransformed models worked best for 

the constituents predicted by traffic volume. Driscoll 

and others (1990) graphically compared the probability 

distribution defined by the mean and standard deviation 

of the transformed population with individual points in 

the data set and tested the fit by means of the probability plot correlation coefficient (PPCC) test (Vogel, 

1986). In that study, the lognormal distribution was 

found to be a satisfactory model for most runof

constituents at most runoff sites (Driscoll and o

1990). 

Athayde and others (1983) examined the 

matical properties of the lognormal distribution 

terms of the ratio of the mean and percentile va

relation to the median as a function of typical c

cients of variation (COV; the ratio of the standa

ation to the mean). For a population with a COV 

the mean would be expected to be about 1.4 tim

median, 90 percent of the data would fall in a ra

from about 0.25 to 3.9 times the median, and th

quartile range (about 50 percent of the data) wo

in a range from about 0.6 to 1.7 times the media

(fig. 6). Historically, COVs of highway- and urb

runoff-quality constituent populations have rang

from about 1/4 to about 3, with typical values th

about 1 (Athayde and others 1983; Driscoll and 

1990). 

Mathematical properties of transformatio

introduce error into the interpretation process if 

properties are not properly addressed. When us

arithmic and exponential transformations, one m

realize that statistical values such as the mean a

dard deviation need to be calculated within the 

formed data and then translated to original unit

(Helsel and Hirsch, 1992). Specifically, parame

tistics such as the mean and standard deviation 

be transformed and subsequently used to estima

ulation characteristics in linear space. For exam

lognormal suspended-solids EMC population fr

I-794 Milwaukee Wisconsin site described by D

and others (1990) has a geometric mean of abou

and a 95-percent confidence interval from abou

about 510. The standard deviation and COV (tr

formed from logarithmic space), however, are o

about 1.9 and 1.3, respectively. It is apparent, th

that confidence-interval values must be calculat

log space and then transformed. Nonparametric 

tics such as the median and quartiles, however, 

taken directly from the original or transformed 

because these statistics are associated with indi

values in the data set. 
16 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

0 1 2 3

UNITLESS COEFFICIENT OF VARIATION (COV)

0.02

20

0.05

0.1

0.2

0.5

1

2

5

10

RATIO OF THE EXPECTED VALUE OF GIVEN

PERCENTILES TO THE MEDIAN VALUE FOR 

LOGNORMAL DISTRIBUTIONS

0

4

1

2

3

RATIO OF THE MEAN TO THE MEDIAN VALUE

FOR LOG NORMAL DISTRIBUTIONS 

MEAN / MEDIAN = SQRT (1 + (COV2))

PERCENTILE VALUE / MEDIAN = EXP ( PROBABILITY * SQRT ( LN (1+(COV2))))

95th

90th

75th

Median

25th

10th

 5th

Figure 6. Properties of the lognormal (natural logarithm) distribution as a function of the population 

coefficient of variation (COV) modified from Athayde and others (1983).
Regression Ana

Mathematical artifacts introduced by use of log 

transformation in regression analysis may also affect 

the predictive ability of resultant models. Transforming 

an unadjusted log-regression equation back into linear 

space (the original units, for example, milligrams per 

liter of suspended sediment) provides a median (the 

geometric mean) estimate, not a mean estimate of concentrations and (or) loads (Helsel and Hirsch, 1992). 

Ferguson (1986) summarized the problem in the context of estimating river loads from flow and concentration data and proposed a method for a bias correction 

factor (BCF). The BCF is a multiplicative term 

included in regression models that are formulated in 

logarithmic space and then transformed into original 

units; it is designed to prevent underestimation of concentrations or loads as an artifact of transformation. 

Further research indicates that several methods may be 

used to estimate a BCF and that a nonparametric 

method developed by Duan (1983) generally provides 

reasonable estimates for a BCF that is not affected by 

the structure of the data (Driver and Tasker, 1990; 

Helsel and Hirsch, 1992). Driver and Tasker (1990) 

used the Duan (1983) method to estimate BCF for 

storm-runoff loads, volumes, and selected constituent 

concentrations from NURP data and calculated BCFs 

that ranged from about 1.1 to 2.8 for their runoff 

models. 

In the construction of regression models, the 

Box-Cox transformation (Box and Cox, 1964) can be 

used to stabilize variance and correct for nonnormality 

of a strictly positive response variable. John and Draper 

(1980) define a transformation that can be used when 

the response variable is not strictly positive. ArandaOrdaz (1981) and Guerrero and Johnson (1982) define 

transformations that can be used when the response is 

binary (0 or 1) or a proportion between 0 and 1. For 

multiple regression models, a graphical device that can 

be helpful in deciding on a transformation for a predictor is a partial residual plot (Larsen and McCleary, 

1972) in which the partial residuals (computed by 

taking the difference between the observed value and 

the components of the predicted value that exclude the 

variable of interest) are plotted against the predictor of 

interest. If the plot looks linear, no transformation is 

needed. If the plot shows some curvature, a tran

tion may be helpful. The practice of transforma

optimization (finding the perfect root or power 

transformation), however, is not encouraged be

is never known how well a sample represents th

underlying population, and a generalized transf

tion that works reasonably well for all data of in

better than multiple, slightly different transform

for each data set (Helsel and Hirsch, 1992). Alt

tively, one may use other methods such as Whit

Specification Test, which is done by regressing s

residuals on the predictor variables and cross-pr

of the predictors. In this test, a significant regre

implies that the specification is wrong or that he

eroscedasticity of the residuals is related to the 

predictors (White, 1980).

REGRESSION ANALYSIS

Regression analysis is an accepted metho

interpretation of water resources data and for pr

tion of current or future conditions at sites with 

teristics that fit the input data model. The FHWA

departments of transportation, and watershed m

need models to interpret data; predict runoff vo

concentrations, and loads; and predict potential 

of runoff on receiving waters at sites for which 

not exist. In the following discussions, it is assu

that the purpose of the regional regression is for 

pretation of data and for prediction of future res

including possible extrapolation (that is, predict

side the range of the sample data). If the regress

model is not a reasonable representation of reali

extrapolation could be erroneous. The purpose 

regression analysis will influence how predictor

selected for the regression model. Therefore, in 

ing predictors for a possible regression model, o

should choose variables that have a physical ba

explaining variations in the response, and the fi

regression coefficients should have a logical alg

sign. 

Historically, regression analysis was used 

interpretation and prediction in several studies t

resent the primary efforts for regional and (or) n
18 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

characterization of highway runoff in the United States 

(table 2). To date, five studies (Kobriger and others, 

1981; Chui and others, 1982; Kerri and others, 1985; 

Driscoll and others, 1990; Thompson and others, 1996) 

represent regional analysis of highway-runoff flow and 

quality based on data collected during the 1970's and 

early 1980's. Additionally, Young and others (1996) 

identified the urban-runoff regression equations developed by Driver and Tasker (1990)—identified as "the 

USGS method"—as applicable for estimating highway-runoff quantity and quality (table 2). The USGS 

method was developed from NURP data collected 

throughout the United States during the 1970's and 

early 1980s, but it does not include highway runoff as a 

specific land use. More recently, Irish and others (1996; 

1998) interpreted data collected in the 1990's using 

regression analysis, but they were clear that these equations and related interpretations may be valid only in 

the local Austin, Texas, area. Typically, for all the 

models featured in table 2, runoff data from a substantial number of storms at a number of sites were compared to predict runoff coefficients, stormflow volume, 

constituent EMCs, storm loads, and (or) annual loads 

from a number of explanatory variables. These 

response variables are predicted from runoff constituents, hydrologic variables, highway-design features, 

land-use characteristics, and climate (table 2). Tasker 

and Driver (1988) used both ordinary and generalized 

least-squares regression methods. The highway studies 

featured in table 2, however, were limited to ordinary 

least-squares regression analysis.

The Analytical Process

Regression analysis is designed to provide an 

estimate of the average response of a system as it 

relates to variation in one or more known variables. 

Regression equations are often derived by use of computer programs that calculate the regression parameters 

and provide an estimate of the correlation coefficient 

(R2— the proportion of the variability in the dependent 

variable explained by the predictor variable) without an 

investigation of the validity of the selected model. 

When this exercise yields a correlation coefficient that 

is close to 1, then it is often assumed that a good 

regression model has been selected. Many factors may 

produce high but invalid correlation coefficients. 

Regression analysis, therefore, should include visual 

analysis of scatterplots, examination of the regr

equation, evaluation of the method design assum

and regression diagnostics (Helsel and Hirsch, 

Examination of scatterplots is necessary t

examine the relations between predictor and res

variables and to assess response variability (He

and Hirsch, 1992). Scatterplots of interest inclu

graphs of the response as a function of each pre

variable and graphs of the residuals as a functio

predictor variables. For most regression method

one must ensure that a linear relation exists betw

predictor and response variables. If relations are 

linear, other, more linear predictors may be cho

as previously discussed, transformation method

be used to increase the linearity of relations bet

variables. For some regression methods, one m

determine whether the variability in the respons

ables is also a function of the magnitude of the 

tor. Transformations are also used to reduce or 

eliminate problems of nonconstant variance. Pa

residual plots (described in Appendix 1A) and t

White Specification Test (White, 1980) are often 

in determining the appropriate transformation o

basis of the structure of residuals. 

Often, it is also prudent to examine the ef

seasonality on highway- and urban-runoff data 

of scatterplots of each variable of interest and th

regression residuals during the monitoring perio

seasonality exists, then explaining and quantify

factor may increase the linearity in response to 

predictors by removing seasonality from the res

variable. Methods for quantifying seasonality a

described in Appendix 1B. 

Examination of the regression equation e

that the model is logical, useful, and quantitativ

(Helsel and Hirsch, 1992). In a logical regressio

model, the coefficients will have a sign and mag

that can be explained by a reasonable scientific 

esis. When explanatory variables in a multiple 

sion equation covary, this multicollinearity will 

some predictors to have illogical values and (or

(Helsel and Hirsch, 1992). Methods for analys

multicollinearity are described in Appendix 1C

Regression statistics also provide informa

about the suitability of the model. Examination 

regression equation includes interpretation of th

lation coefficient to determine whether the resu

equation explains much of the variance in the d
Regression Analysis 19

Reference Year of data 

collection Location/ environmental setting Seasonality Number of samples

Number

of sites Model Output

Chui and others, 1982 1979–81 Washingon: eastern (semi-arid) and 

western (wet)

Winter sanding events considered 500 9 OLS SL, AL

Driscoll and others, 1990 1976–84 National: 8 sites in Washington, 3 sites 

each in California and Wisconsin, 2 

sites each in Florida, Minnesota, 

Pennsylvania, and 1 site each in 

Arkansas, Colorado, North 

Carolina, and Tennessee

Separated storms into snowmelt and 

nonsnowmelt events

Used site 

median 

EMCs 

from 

about 900 

storms

24 OLS Q, RC,

EMC

Driver and Tasker, 1990 1977–83 30 urban areas nationwide in 3 regions 

designated by precipitation statistics

Developed equations for seasonal and annual 

loads

2,813 173 OLS

GLS

Q,EMC,

SL,AL

Irish and others, 1998 1993–95 Austin, Texas (semi-arid) Date of storm and temperature were not 

significant

58 1 OLS SL

Kerri and others, 1985 1975–81 California (arid to semi-arid) ND ND 3 OLS SL

Kobriger and others, 1981 1976–77 National: 3 sites in Wisconsin (humid), 

and 1 site each in Pennsylvania 

(humid), Tennessee (humid), and 

Colorado (arid)

Deicing mentioned but not quantified 159 6 OLS Q, QD, SL

Thompson and others, 1996 1976–83 Minnesota Classified storms as rainfall, snowmelt, or 

mixed

416 4 OLS Q, EMC

AL, annual loads; AP, annual precipitation; ATC, antecedent dry period traffic count; As, arsenic; 

B, boron; BODx, biochemical oxygen demand; Cd, cadmium; Cl, chloride; COD, chemical oxygen demand; Cr, chromium; Cu, copper; D, storm duration; DQ, discussed qualitatively; DP, dissolved phosphorus; DS, dissolved solids; EMC, event mean concentration; Fe, iron; FR, filterable 

residue (similar to TSS); GLS, generalized least squares regression; Hg, mercury; HT, highway 

type (indicating the degree of urbanization and (or) impervious area); I, storm intensity (flow 

divided by duration); IA, impervious area; OG, oil and grease; OLS, ordinary least squares regression; LR, TSS loading rate by climate; LU, land use; MJT, mean January temperature; N, total 

nitrogen; Na, sodium; ND, not documented; NFR, nonfilterable residue (dissolved solids); 

Pb, lead; PD, population density; PDUR, preceding storm duration; PI, preceding storm intensity; 

PQ, preceding stormflow; Q, stormflow; QD, stormflow duration; RC, runoff coefficient; 

SL, storm loads; SO4, sulfate; SS, suspendid solids; TDS, total dissolved solids; TKN, total 

kjeldahl nitrogen; TOC, total organic carbon; TP, total phosphate; TR, total residue (similar to total 

solids); TS, total solids; TSS, total suspended solids; TVS, total volatile solids; VDS, vehicles 

during storm; VSS, volatile suspended solids; Zn, zinc; *, indicates that different equations were 

developed for different categories or a unified equation was developed with this variable as a 

nominal variable]
stical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

Reference

p

Uncertainty of 

estimates Runoff constituents Hydrologic 

variables

Runoff

constituents

Hydrologic 

variables

Highway

characteristics

Land

use

Climate Other

Chui and others, 1982 TSS RC VDS LR* Mean: ~200%

Range: 0.3–

1,650%

COD, Cu, NOx, Pb, TKN, TP, TOC, 

VSS, Zn

TSS ADT ND

Driscoll and others,

1990

Q, RC P QD Yes

RC Percent 

impervious

Yes

BODx, Cd, Cl, COD, Cr, Cu, Fe, Hg, 

NOx, OG, Pb, TOC, TP, TKN, TS, 

TSS, VSS, Zn

ADT, HT*

Driver and Tasker, 1990 Q P IA R*, AP A Yes

Cd, Cu, COD, DP, DS, N, Pb, SS, TKN, 

TP, Zn

P, D, I A, IA,

LU

R*, MJT Yes

Irish and others, 1998 BODx, COD, Cu, Fe, NOx, OG, Pb, 

TP, TSS, VSS, Zn

ADP, D, 

I, PDUR, 

PI, PQ, Q

VDS, ATC Yes

Kerri and others, 1985 COD, FR, Pb, TKN, Zn VDS Yes

COD, NFR, Zn TR Yes

Kobriger and others, 

1981

Q ADP, P HT* DQ

QD D, ADP* HT* DQ

PA ADT DQ

TS I HT* DQ

BODx, Cd, Cl, Cr, Cu, COD, Fe, Hg, 

Pb, TKN, TOC, TP, TSS, TVS

TS HT* DQ

Thompson and others, 

1996

Q P

Al, As, BODx, Cd, Cl, COD, Cr, Cu, 

Fe, Hg, N, Na, Ni, NOx, Pb, SO4, 

TKN, TP, Zn

TDS, TOC,

TSS, TVS

ADT Examine 

potential 

leverage of 

outliers

TDS, TOC, TSS, TVS ADP, D, 

I, P, Q

ADT ND
Regression Analysis 21

Chui and others, 1982 The regression equations provided adequate estimates of the central tendency of all storm loads, but were not accurate for predicting individual storm 

loads.

Driscoll and others, 1990 Regression analysis was used to examine factors that influence highway runoff characteristics. It was determined that there were not enough sites with 

consistent explanatory variables to quantitatively explore the effects of climate, atmospheric deposition, configuration, pavement (composition, 

quantity, or condition), design geometrics, right-of-way characteristics, drainage features, vehicle characteristics, maintenance practices, regulations, 

or surrounding land use characteristics.

Driver and Tasker, 1990 Flows and loads during individual storms, seasons and years were predicted for a number of constituents as a planning tool rather than for 

interpretation of cause and effect relations. Explanatory variables that would be readily available for planning purposes on a regional scale were 

used. Highways, as an individual land use, were not included in the analysis.

Irish and others, 1998 Regression analysis indicated that over 90 percent of the variation observed for most constituent loads may have been explained by in-storm-, 

antecedent dry period-, and preceding storm-variables. Date and time of storm, temperature, wind speed, and wind direction were not statistically 

significant predictors. Traffic mix, surrounding land use, curb height, guard-rail height, and maintenance activities could not be evaluated.

Kerri and others, 1985 ADP and ATC showed no statistical significance. B, Cd, NOx, OP, OG, TP, and TR cannot be estimated from explanatory variables

Kobriger and others, 1981 Sites were classified into 3 groups for regression: Type I—urban elevated bridge deck; Type II—curbed highways; and Type III—rural highways with 

flush shoulders. Equations were developed to estimate runoff volume, runoff duration, constituent accumulation during an ADP, pollution washoff, 

and constituent loadings (as a function of TS).

Thompson and others, 1996 Surrogate parameters were established to estimate most constituents. Also, models to predict surrogate parameters were developed. Additional data sets 

that were not used in the formulation of the model, were used to test and verify applicability of the models.
22 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

There is no general rule of thumb to determine a minimum acceptable correlation coefficient, but the risk of 

posing the wrong model must be considered in relation 

to the ability of the model to explain variance (Helsel 

and Hirsch, 1992). The t-ratio (the value of the estimate 

divided by the standard error of the estimate) also is a 

useful statistic. The t-ratio will usually indicate the significance of each coefficient so that the analyst can 

ensure that an apparent relation did not arise by chance 

when there is no real linear relationship. For OLS 

regression, t-ratio statistics that have an absolute value 

greater than 2 are generally considered to indicate a 

statistically significant non-zero relation between individual predictors and response variables (Helsel and 

Hirsch, 1992). 

Regression diagnostics include methods 

designed to determine whether the equation posed as a 

model is dominated by a few outliers in the data set. An 

analyst can use regression diagnostic methods to find 

influential observations and study their effects. Examination of a scatterplot of the residuals will often identify the effect of outliers when there is only one 

predictor variable, but influential outliers are more difficult to recognize in plots from multiple regression 

analysis. Typical problems that affect the validity of 

regression models are curvature, outliers, and highleverage points. Outliers, observations (or a subset of 

observations) that appear to be inconsistent with the 

remainder of that set of data, are fairly common in 

hydrologic data (Hirsch and others, 1993). Outliers 

should be checked for possible gross errors in measurement or mistakes in recording the observations, but 

rejecting them out of hand is not a prudent practice. 

Regression diagnostics are discussed further in 

Appendix 1D. 

Linear Regression Methods

As previously discussed, water-resources data—

and in particular, populations of urban- and highwaystormwater data—have statistical properties that 

require special treatment. Predictive modeling and 

interpretation of cause and effect relations of runoff 

quality and quantity may benefit from use of techniques applied in other water-resources studies. Some 

of the techniques and practical alternative methods for 

dealing with the realities of hydrologic data are discussed in the following sections. Applicable methods, 

classification of appropriate response and predictor 

variables, and general purpose and assumptions 

method are listed in table 3 with a reference to 

dix that describe some mathematical details for 

the methods discussed. This table, which distin

among several statistical methods discussed her

the basis of the type of data involved, can serve 

rough guide to help the researcher choose a met

study.

A method often used for predicting water 

tity and quality is ordinary-least-squares (OLS) 

sion (Haith, 1976; Lystrom and others, 1978; P

1984; Driver and Tasker, 1990; Irish and others

Jordan and others, 1997). The OLS model, how

requires several restrictive assumptions about th

parameters and errors in the model, which are o

valid for hydrologic data. To fully implement O

regression, one must demonstrate that the respo

variable (or the transformed values) is linearly r

to predictors (or the transformed values), the da

to fit the model are representative of the popula

interest, the variance of the residuals is constan

residuals are independent, and the residuals are 

mally distributed (Helsel and Hirsch, 1992). No

distributed errors are required, even with large s

for the determination of prediction intervals - al

certain empirical methods allow one to generate 

prediction intervals from OLS-determined resid

Appendix 2A more fully describes the OLS reg

model. 

In some cases, transformations either ma

adequately transform the error structure to fit th

required distributional assumptions or may be u

able because of the possible transformation arti

(Helsel and Hirsch, 1992). Nonparametric regre

provides a distribution-free alternative to OLS r

sion that does not require errors to be normally 

uted. Nonparametric regression can refer to mo

with a prespecified functional form but with an 

ified error distribution or models with neither a 

specified functional form nor error distribution. 

Appendix 2B contains more details.

Stormwater data often include outliers, w

may or may not be meaningful. Although it is o

imprudent to remove outliers, they often exert a 

age that will limit the accuracy and precision of 

sion results for the bulk of data included in the 

regression model. Robust regression procedures 

with outliers by reducing their influence withou

sarily rejecting them entirely from the analysis. 

sense, regression diagnostics and robust regress
Regression Ana

have the same goal of detecting outliers, but they 

approach the problem from different ends. Diagnostics 

use the classical fit of data to detect outliers and influential observations, whereas robust regression fits most 

of the data and detects outliers by their large residuals 

from the robust model. Robust regression techniques 

are discussed further in Appendix 2C. Some of the 

nonparametric methods discussed in Appendix 2B

(Kendall-Theil and LOWESS smooth) also can be 

resistant to outliers (Helsel and Hirsch, 1992).

In analysis of stormwater quality, observa

may not be independent in time and space. For 

ple, Irish and others (1998) noted the effect of t

volume, intensity, and duration of storm rainfal

acteristics on the water quality measured during 

next storm event. In using OLS regression, one 

assumes that observed values of the response va

are independent, resulting in independent residu

cases where this assumption is not approximate

estimated generalized least-squares regression (

can be used if the dependence of the residuals c

Table 3. General guide to regression methods

Method Response

classification

Predictor

classification General purpose and assumptions Ap

Ordinary leastsquares 

regression

Continuous Usually continuous but 

nominal can be used 

in addition

Describes the relation between response and predictors. 

Errors are independent and identically distributed with no 

outliers. Normality of errors is required for hypothesis 

testing.

Nonparametric 

regression

Continuous Usually continuous but 

nominal can be used 

in addition

Describes the relation between response and predictors. 

Error distribution unspecified. Functional form may or 

may not be specified. Useful when errors are not 

approximately normally distributed.

Robust 

regression

Continuous Usually continuous but 

nominal can be used 

in addition

Describes the relation between response and predictors. 

Useful for detecting outliers and highly influential 

observations. Fits main portion of data, giving outliers 

little or no weight.

Generalized 

least-squares 

regression

Continuous Usually continuous but 

nominal can be used 

in addition

Describes the relation between response and predictors. 

Errors can be correlated and variances of errors may be 

different. Useful when observations of response variable 

are not independent or not measured with equal accuracy.

Tobit regression Part continuous, 

part nominal

Usually continuous but 

nominal can be used 

in addition

Describes the relation between response and predictors. 

Useful when response variable is censored below a 

detection limit.

Logistic 

regression

Nominal Usually continuous but 

nominal can be used 

in addition

Predicts probability of response being in one category or 

another.

Contingency 

tables

Nominal Nominal or ordinal 

groups

Describes the relation between nominal response and 

nominal or ordinal predictors.

Ridge regression Continuous Usually continuous but 

nominal can be used 

in addition

Describes the relation between response and predictors. 

Useful when predictors exhibit high multicollinearity. 

Regression coefficients are biased.

SPARROW Continuous Continuous Nonlinear regression method to predict water quality for a 

stream reach based on spatially referenced predictors. The 

predictors are a function of the point and nonpoint sources 

and their location relative to the stream reach.

Artificial neural 

networks

Continuous or 

nominal

Continuous or nominal Flexible nonlinear nonparametric model for prediction. Any 

underlying model or functional relations may be 

impossible to extract. Data-in / predictions out black box.
24 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

estimated from the data. For example, Tasker and 

Driver (1988) and Tasker and Raines (1995) show that 

when the observed response in a regional regression to 

estimate mean annual loads is obtained from at-site 

rainfall-load models, responses may have nonconstant 

errors and correlated errors. Appendix 2D describes 

one estimated GLS regression approach.

As previously mentioned, concentrations of 

water-quality constituents below one or more detection 

limits in water-resources data sets are not uncommon. 

These values are considered as censored values for statistical analysis. Concentrations reported as less than a 

detection limit censor the data at the limit of detection 

and relegate all values below the limit to the nominal 

scale. When only a few observations are in the censored range, either fabricating values at or below the 

censoring threshold or ignoring the values in the censored range has been done but generally, these practices are not acceptable. When the response variable is 

moderately censored (below 20 percent censoring), 

Hirsch and others,(1993) recommend the KendallTheil robust regression method (Appendix 2C) or Tobit 

regression (Appendix 2E). When the level of censoring 

exceeds about 20 percent, logistic regression 

(Appendix 2F) or contingency tables (Appendix 2G) 

are recommended (Hirsch and others,1993).

As discussed, multicollinearity will affect the 

results of regression analysis, causing some predictors 

to have a regression coefficient with large standard 

errors as an artifact of regression-equation optimization. Large standard error may result in coefficient estimates having the wrong sign or unreasonable values, 

and it generally results in coefficients being insignificant. When using OLS or GLS regression, one must 

often eliminate some potentially valuable and logical 

predictors from a regression model. A method for dealing with multicollinearity without predictor elimination 

is ridge regression (Hoerl and Kennard, 1970). A successful ridge regression analysis (Appendix 2H) produces slightly biased regressor coefficients with 

smaller variance. Thus, one may trade absence of bias 

for a stable set of regression coefficients in the presence 

of multicollinearity.

Nonlinear Regression Methods

In the previous section, the structure of th

regression models was assumed to be linear in t

dictor coefficients. In some areas of hydrology, 

assumption of linearity may represent a distorti

the physical process being modeled. In such ca

more theoretically based nonlinear model may 

appropriate. In other cases, the analyst may wis

relax the linearity constraint for a more flexible 

without knowledge of the form of the model. 

Many regional regression studies relating 

quality to basin attributes treat contamination so

as homogeneously distributed throughout the w

shed in defining the predictors (basin attributes) 

(Lystrom, and others1978; Peters, 1984; Driver 

Tasker, 1990). This treatment limits the usefuln

the models because it fails to account for spatia

ences between sources and the water-quality m

ing points within a watershed. For example, con

the predictor "urban land use (in percent)" for t

watersheds both with 10 percent urban land use

urban land in one is in the lower part of the bas

immediately upstream from the water-quality m

ing point. In the other watershed, the urban land 

the upper part of the basin, far from the monito

point. The simple predictor of percent urban lan

fails to account for loss of contaminant mass du

instream and overland transport, a factor that m

substantially different between the two watersh

Smith and others (1997) deal with this pr

by developing water-quality predictors for poin

nonpoint sources as functions of both river reac

land-surface attributes, as well as by considerin

of material transport. Predictor formulas describ

transport of contaminant mass from source to th

end of a reach. This innovative technique, calle

SPARROW (SPAtially Referenced Regressions 

Watershed attributes), has the potential for grea

improving the usefulness of regional regression 

for water quality because it is able to include sp

sources of contaminants and their location relat

the stream reach. It includes substantial refinem
Uncertainty, Quality Assurance, and Quality Co

a prototype method described in Smith and others 

(1993). Appendix 3A provides a more complete 

description of the method.

Artificial neural networks (ANN) include a class 

of flexible nonlinear models that can be used, like 

regression models, to predict responses from a set of 

predictors. Lingras and Adamo (1996) use ANN to estimate average and peak traffic volumes on the basis of 

road classes and short-duration counts. Zhang and 

Stanley (1997) and Thirumalaiah and Deo (1998) use 

ANN to forecast water quality and river stage on the 

basis of previous time steps, respectively. Artificial 

neural networks attempt to simulate the manner in 

which humans think (Hertz, Krogh, and Palmer, 1991). 

An ANN is composed of simple processing units, 

called neurons, arranged in layers. Each unit receives 

input from other units and converts the input to a single 

output, which it sends to other units. The conversion 

takes place in two stages: first, a net input is computed 

as a weighted sum of inputs, then an activity function 

transforms the net input into an output. The flexibility 

of ANN comes from one's being able to specify multiple layers of neurons with nonlinear activity functions 

and alternative methods for computing the net input. 

Observed values of predictors (inputs) and responses 

(targets) are used to "train" the ANN by iteratively 

adjusting the weights used by the neurons to produce 

output so that the sum of squared differences between 

output and target data is small. 

ANN can over train (fit the observed data well, 

but not predict well for new data). For this reason, one 

should always set aside a portion of the observed data 

as a test data set that is not involved in any way with 

training the ANN, so that the ANN can be tested for 

predictive ability on new data. Just as in linear regression analysis, omission of important variables or inclusion of unimportant variables can be a problem. 

Artificial neural networks are data-in/predictions-out 

black boxes. Any underlying model or functional relation may be impossible to extract from the network. 

Appendix 3B gives more details about ANNs. 

UNCERTAINTY, QUALITY 

ASSURANCE, AND QUALITY 

CONTROL

Uncertainty is an important part of any de

making process. Success of a water-quality inter

model depends on uncertain future meteorologi

demographic, political, and technical condition

which may affect future costs and benefits. In o

deal with problems of external uncertainty, the 

first needs to know the severity of the statistical 

tainty inherent in the methods used to predict w

quality. Statistical models need to be based on i

tion that is meaningful, representative, complet

cise, accurate, and comparable to be deemed va

to date, and technically supportable. If sensitivi

yses reveal too much uncertainty in the predicti

new data and new methods may be needed, or s

factors based on prediction-interval estimates m

used. These criteria will also determine whethe

pretations based on the model will be admissibl

legal evidence.

Statisticians must deal with expected and 

pected uncertainties that can potentially affect i

tations made from a given data set. Expected 

uncertainty arises because many of the factors t

affect the process are unknown or cannot be kn

with certainty. Statisticians can often address ex

uncertainties, but these efforts are usually based 

assumption that data are collected and recorded 

rectly, thereby minimizing bias. Statistical meas

uncertainty are good for determining the level o

(variability) in a data set, but they cannot detect 

without a population of "true" values with which 

the hypothesis. Unexpected uncertainty can aris

faulty computer programs, faulty application of 

propriate statistical techniques, and faulty data 

Classical statistical measures cannot detect prob

caused by unexpected uncertainty because stati

analysis is done under the assumption that the c

tions are done correctly, that the correct model h

selected for the data, and that the data are repre

tive of the environmental system under study. T

ough documentation, quality assurance, and qu
26 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

control are necessary to ensure that bias is not introduced by the errors in the computer programs used for 

analysis, in the modeling effort, and in the data used for 

analysis. Prediction errors arise from natural heterogeneity, measurement errors, and structural differences 

between the real world and the methods used for predictions; therefore, quality-assurance and qualitycontrol programs must be designed to quantify these 

sources of uncertainty. Quality assurance and quality 

control are necessary throughout any study-from 

design through data entry and interpretation (Jones, 

1999). Increasingly legal and financial liability is 

driving modelers to implement rigorous qualityassurance and quality-control procedures at all stages 

of a modeling project (Van der Heijde, 1990).

Benchmarking of Analytical Tools

Results generated by the complex computer programs currently used for statistical analysis often are 

assumed to be correct because it is expected that the 

software companies have thoroughly tested their computer code under a number of different conditions 

(Landwehr and Tasker, 1999). If statistics that look 

reasonable but are, in fact, grossly incorrect are computed, this dangerous error is liable to remain undetected until applied to a real-world problem where the 

model noticeably fails. To prevent this situation, one 

can benchmark the statistical software to assess its 

reliability. 

Benchmarking consists of applying a suite of statistical analyses to various standard data sets for which 

the values of the statistics are known with great precision and assessing whether the resulting values are in 

conformance. A discussion and review of how to do 

such assessments can be found in Sawizki (1994a) 

and McCullough (1998), who use the Statistical 

Reference Datasets (StRDs) recently published by the 

National Institute of Standards and Technology (1998). 

Wilkinson (1985) also proposed a collection of simple 

tests designed to uncover common flaws in statistical 

programs, including an example in which the variables 

are collinear and the difference in magnitude between 

variables was extreme but the magnitude of the observations for each variable was not unlike that found 

among common statistical problems.

Benchmarking studies to provide quality assurance and quality control to verify the operation of statistical software packages sound esoteric; but when 

such studies have been done, real problems hav

discovered. For example, Sawitzki (1994b) repo

a joint effort by members of two working group

("Computational Statistics of the International B

rical Society" and "Statistical Analysis Systems

"GMDS," Deutsche Gesellschaft für Medizinisc

Informatik, Biometrie und Epidemiologie) to ap

Wilkinson tests to nine data-analysis systems, i

ing some running on multiple platforms. The gr

demonstrated performance difficulties with each 

tem, even between platform implementations of 

same package. More recently Landwehr and Ta

(1999) completed a benchmark study of comme

statistical-software packages commonly used b

USGS and found that several analysis packages 

more difficulty in providing computationally pr

and (or) correct values than did others, and som

cumbersome to use in obtaining specific statisti

Therefore, quality assurance and quality contro

interpretive efforts that make use of existing sof

should include documentation of the commerci

ware (including version and platform), docume

of benchmarking efforts, as well as scientific an

technical reviews of the software selection and 

benchmarking efforts.

In general, the analyst should be aware o

specific question or questions to be answered by 

analysis, as well as how a specific package is co

ing the answer it provides (Landwehr and Taske

1999). Quality assurance and quality control (u

tools such as the ANASTY data set) is especiall

important when an analyst will write computer 

implement a statistical process, even if the code 

ten within the interface of an existing commerc

software product that has been previously bench

marked. Quality assurance and quality control f

development includes verification of the structu

coding, model validation, record keeping, softw

documentation, and scientific and technical rev

Uncertainty in Modeling Efforts

Results generated by statistical models m

appear to be more certain, more precise, and mo

authoritative than they really are because design 

assumptions and results (even realistic ones) ca

stated with illusive precision and seeming accur

Models—like scientific theories—cannot truly 

dated; they can only be invalidated when an exc
Uncertainty, Quality Assurance, and Quality Co

or error is revealed by new data or circumstances. For 

example, Driscoll and others (1990) note that regression models have been criticized as poor predictors 

when applied beyond the original data set used to 

create the model. The analyst, therefore, has the 

responsibility to document efforts to 

• examine the representativeness of data used to 

construct statistical models,

• assess uncertainties in models, and 

• evaluate the potential predictive ability for sites not 

included in the construction of the model.

The application and documentation of these steps in the 

modeling procedure are essential for the transfer of a 

model to potential model users. Quality assurance and 

quality control in the processes used for data reduction, 

evaluation, and interpretation is as important as quality 

assurance and quality control for field and laboratory 

data-collection efforts (Brown and others, 1991). 

Quality assurance and quality control for modeling 

efforts include the procedural and operational 

framework used by an organization to ensure the 

technical and scientific adequacy of the tasks, and 

documentation thereof, to ensure that the results are 

fully reproducible and defensible. Modeling efforts are 

being used in the regulatory and legal domain, and the 

needs for model documentation describing the inherent 

uncertainty in predictions are increasing (Haan and 

others, 1990). Jones (1999) describes the quality 

assurance and technical review necessary to prepare 

data in a computerized database used to develop waterquality models.

A number of statistical tools are available to 

assess the uncertainties inherent in models during 

model development. The fit of a regression model is 

usually measured by means of the correlation coefficient, but a better measure of the model's predictive 

ability at unmonitored stations is the variance of prediction (Gilroy and others, 1990). This statistic is computed by estimating the mean or median variance of 

prediction for individual stations while using every station in the regression model. This computation is made 

on the basis of the assumption that the available stations are representative of the entire population of 

potential sites to be modeled. Irish and others (1996) 

note, however, that the standard error of forecasts (a 

measure of the spread of data points not used to formulate a regression model indicating the predictive ability 

of the coefficients in the model for data not included in 

the monitored population) is always larger than the 

standard error of the regression for modeled dat

When the errors in a regression model are appro

mately normally distributed, standard errors of 

tion and prediction intervals, which serve as a m

of the uncertainty in the predictions, can be com

on the basis of normal theory. Appendix 4A pro

some details. When using methods that do not i

the assumption that errors are normal or do not 

an error distribution, the analyst may be able to 

bootstrap method (Efron and Tibshirani, 1986). 

bootstrap is a simple, straightforward method fo

puting biases, standard deviations, and confiden

intervals for almost any nonparametric problem 

1982). More details are given in Appendix 4B. D

menting a sensitivity analysis—determining ho

parameters control model output—is necessary 

cate how uncertainty in input values will affect 

tainty in computed results. For example, Driver 

Tasker (1990) provided uncertainty estimates fo

regional regression equations, which had coeffi

of variation ranging from 0.2 to 0.65, standard e

of the estimate ranging from 79 to 145 percent, 

average prediction errors ranging from -67 to 2

percent.

When developing a regional hydrologic r

sion for a fairly large geographic area, such as a 

several states, it is sometimes advantageous to s

vide the region into several homogeneous subreg

which the basic regression assumptions are mor

to be true than for the whole region. These regio

or may not have geographic boundaries. Geogra

regions may be based on some general topograp

or geological feature of the region or may be ba

on ecoregions (Omernik, 1995). Regions can al

be defined on the basis of values of the predicto

Multivariate techniques of cluster and discrimin

analysis have been used to define regions based 

basin attributes (Tasker, 1982). Another possibl

method for dividing a large area into regions is r

to as the "region of influence method" (Tasker a

Slade, 1994; Tasker and others, 1996) a method 

which a unique regression equation is estimated 

each site where a prediction is to be made. The 

sion equation is based only on data observed fo

with basin characteristics similar to the site whe

prediction is to be made. Appendix 5 gives mor

details. Many of the characteristics that may be 

for regionalizing highway-runoff-quality data (i

ing maps of climatic characteristics, receiving-w

characteristics, and ecoregions) are provided as 
28 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

Geographic Information System (GIS) coverages for 

the conterminous United States by Smieszek and 

Granato (2000). The process of regionalization needs 

to be documented, as well as the regional characteristics, so that practitioners using the model may properly 

assign any given site within an appropriate region.

Proper model application as part of a planning, 

design, or assessment effort also requires substantial 

documentation. Information that should be documented includes the statistical characteristics and 

design assumptions supporting application of the 

model. Model-performance parameters as defined in 

the previous paragraph should be documented in terms 

of an uncertainty estimate in the predictions made by 

the model when the model is applied. For example, 

Young and others (1996) apply the USGS method 

(Driver and Tasker, 1990) to estimate an annual load of 

suspended solids in a hypothetical case study to illustrate use of the method. Young and others (1996) use 

watershed area (one significant figure), land use by 

percent (one significant figure), and the impervious 

area (two significant figures) to compute a storm load 

of 397.4 kg (four significant figures). This storm load 

is then multiplied by the average number of storms 

(two significant figures) to compute an annual load 

of 55,636 kg (five significant figures). Young and 

others (1996) provide the exact results of these hypothetical computations to illustrate the method unambiguously. In reality, however, model results are expected 

to have an uncertainty that is compounded by use 

of input values with only one significant figure. 

Computed storm-loads for this hypothetical site are 

about 400 kg per storm with a 90-percent confidence 

interval from about 40 to about 2,000 kg per storm. 

Computed annual-loads for this hypothetical site are 

about 60,000 kg per year with a 90-percent confidence 

interval that ranges from about 6,000 kg to about 

300,000 kg per year. Reporting the final rounded numbers and estimates of the uncertainty of the calculations 

provides the information necessary to evaluate the 

potential results from decisions based on these loads.

The analyst should also document an assessment 

of model suitability for site-specific conditions when 

models are applied. Regional regression models are 

designed to provide estimates for the average site with 

characteristics identified by predictor variables. One 

must determine that values of predictor variables for 

the site to be modeled fall within the range of da

to construct the models, and that no other distin

ing site characteristics would differentiate the si

the modeled population.

Uncertainty in Input Data

The quality of interpretations depends dir

upon the quality and representativeness of avail

data. Statistical models are empirical models ge

requiring large amounts of water-quality, land-u

and highway-related data for parameter estimat

Although a large amount of such data exist in th

country, the data are in disparate databases and 

comparability of this data is in doubt. De Vries 

Klavers (1994) demonstrate that the reliability o

eling estimates is determined primarily by the q

of the monitoring strategy and that computation 

ods can be much less important than the data in

rated into a given model. Models, at best, are on

good as the uncertainty in the input data (Montg

and Sanders, 1985). Furthermore, there is no gu

that water-quality data, no matter how carefully 

lected, will be transferable to other areas and ot

cumstances (Sonnen, 1983). Harrop (1983) obs

that the high uncertainty in highway-runoff-qua

models was caused by "too much analysis being 

applied to too little data." Driscoll and others (1

also noted that even with 31 sites and hundreds 

itored storms, many of the investigated factors—

theoretically should affect the quality of runoff—

not be quantitatively defined. Specifically, indiv

relations could not be defined because each site 

number of explanatory variables (including clim

traffic, highway-design features, and surroundin

use characteristics) that were not held constant 

site to site. In other words, it is difficult to deve

meaningful models to quantitatively predict wat

ity from physical or chemical differences betwe

unless the "noise" introduced by the sampling e

much smaller than the "signal" produced by diff

between sites. The NURP program recognized 

interpretation of data would be questionable un

field programs at different sites provided consist

sound data. Therefore, quality assurance and qu

control elements were adopted as integral parts 

site/project workplan, including elements to add
Uncertainty, Quality Assurance, and Quality Co

potential problems with the program, field monitoring 

and sample collection, laboratory analysis, data management, and data analysis (Athayde and others, 1983).

Standard least-squares methods for regression 

analysis depend on the assumption that the predictors 

are known without error. Nevertheless, measurement 

errors are inherent in most predictors used in regression 

analysis of environmental data. Measurement errors in 

the predictors may or may not be a problem in regional 

regression analyses. The effect of measurement errors 

can be ignored with little consequence when the variance of the measurement error is very small in relation 

to the variance in the predictor itself. Weisburg (1980), 

Seber (1977), and Davies and Hutton (1975) provide 

more details and methods for determining whether 

measurement errors are small enough to ignore.

When significant measurement errors are in the 

data set of predictors, use of the regional least-squares 

regression for prediction requires the predictions to be 

made by means of the same methods for measuring the 

predictors as used in the determination of the regression coefficients. For example, consider a regional 

regression in which the response, S, is annual sulfate 

load and predictors P and T are mean annual precipitation and average annual daily traffic flow, respectively. 

In addition, P is estimated from a contour map of the 

region based on 1930–60 data, and T is estimated 

from a regression on a sample short period count 

(Erhunmwunsee, 1991). As long as predictions from 

the regional model are made by use of the same methods for determining P and T, the measurement errors in 

P and T present little problem. However, if predictions 

from the regional model are made using P estimated 

from a nearby rainfall record or from a different contour map or if T is measured by means of some method 

different from the short-period-count regression, then 

the predictions from the regional least-squares model 

will not be appropriate. It follows that a problem exists 

when the observed predictors used to estimate the 

regression coefficients are measured using significantly 

different methods with different measurement errors. 

For example, consider a regional regression study covering several states in which each state uses a different 

method to estimate average annual daily traffic flow, T. 

In these cases, it is necessary to adjust the regression 

model for errors in the predictors. 

The problem with using standard least-sq

methods when measurement errors are in the pre

is that the observed predictors will correlate wit

regression errors, resulting in biased estimates o

regression coefficients. An alternative to standa

squares that deals with the measurement errors 

dictors is the method of instrumental variables 

(Johnston, 1972). Instrumental variables are var

that correlate with the predictor that contains m

ment error and but do not correlate with the reg

errors. Johnston (1972) and Fuller (1987) descr

eral methods for instrumental-variable estimatio

SAS/ETS procedure MODEL (SAS Institute, 1

can be used for the computations. 

Therefore, to assemble a regional or natio

data set, one must ensure that the methods used 

define both the predictor (explanatory) and resp

variables are the same, or that the different meth

will produce results that are neither substantiall

tistically different. Furthermore, it has been dem

strated that differences in monitoring objectives 

studies will affect the suitability of available da

inclusion in a regional or national synthesis beca

monitoring objectives of those studies will affec

representativeness of the selected sites when co

to the total population of existing sites (Norris a

others, 1990). This is because local studies are 

designed for addressing local problems rather th

national characterization (Norris and others, 19

Availability of reliable runoff-quality dat

an electronic format is necessary to facilitate fu

use of and interpretation of data collected. Dris

and others (1990) and Thompson and others (19

indicate the substantial difficulties involved in t

collection, examination, quality assurance, qual

control, and (when necessary) data entry of hist

runoff data in their efforts toward local, regiona

national interpretation of highway-runoff data. 

comparison, Driver and Tasker (1990) were abl

assemble a much larger National Urban Runoff 

Program (NURP) data from the USGS and the U

with less effort because these programs were su

by quality-assurance and quality-control measu

and the data were stored in readily available nat

water-quality databases. The Transportation Re

Board (1997) determined that development of c

tent electronic data structures is a primary resea
30 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

need to standardize results of hydraulics, hydrology, 

and water-quality research efforts for future use. Datacollection activities, therefore, need be conducted 

within a framework of an established quality-assurance 

and quality-control program to demonstrate that data 

collected meet data-quality objectives and will be 

meaningful for interpretation of the characteristics of 

runoff quantity and quality (Jones, 1999). Many of the 

criteria for basic information, acceptable uncertainty, 

and quality-assurance and quality-control documentation described by Granato and others (1998) are essential for reliable interpretation of local, regional, and 

national runoff-quality data sets.

SUMMARY

 Engineers, planners, economists, regulators, 

and other decision makers concerned with stormwater 

runoff need viable methods for the interpretation of 

local, regional, and national highway-runoff and urbanstormwater data. Stormwater-quality models have, historically, been used to characterize stormwater flow 

and quality, predict pollutant runoff loads, assess 

impacts on receiving waters, and determine the effectiveness of various best management practices to mitigate possible impairment of designated beneficial uses 

of receiving waters. Valid, current and technically 

defensible stormwater-runoff models are needed to 

interpret data collected by field studies; support existing highway and urban runoff planning processes, meet 

National Pollutant Discharge Elimination System 

requirements, and provide methods for calculation of 

Total Maximum Daily Loads, in a systematic and 

economical manner.

 Historically, conceptual models, simulation 

models, empirical models, and statistical models of 

varying levels of detail, complexity, and uncertainty 

have been used to meet various data-quality objectives 

in the decision-making processes necessary for the 

planning, design, construction, and maintenance of 

highways and for other land-use applications. Waterquality simulation models attempt a detailed description of the physical processes and mechanisms by 

means of model parameters with a direct physical definition, and require as input a considerable degree of 

detail in the description of the physical system. In simulation models, parameter estimation is not as data 

dependent as in statistical regional water-qualityassessment models. On the other hand, empirical and 

statistical regional water-quality-assessment mo

provide a more general picture of water quality 

changes in water quality over a region. Statistic

regional water-quality models may also be used 

mate nonpoint-source loadings as inputs for mo

detailed water-quality simulation models. All t

modeling techniques share one common aspect

predictive ability of almost any model will be p

without suitable site-specific data for calibration

 An understanding of the classification 

ables, the unique characteristics of water-resour

data, and the concept of population structure an

sis is necessary to properly interpret the results 

vidual studies and to combine these results to fo

meaningful interpretations as part of a regional 

national synthesis of stormwater quality data. C

cation of variables being used to analyze data m

determine which statistical methods are appropr

data analysis. An understanding of the fundam

characteristics of water-resources data is necess

evaluate the applicability of various statistical t

niques, to interpret the results of these techniqu

to use tools and techniques which account for th

unique nature of water-resources data sets. Und

standing the methods and measures used to dete

the population structure and analyze population 

teristics also is necessary to form valid, current

technically defensible stormwater runoff model

Regression analysis is an accepted met

interpretation of water-resources data and for pr

tion of current or future conditions at sites that 

input data model. The Federal Highway Admin

tion, state departments of transportation, have s

fully implemented regression models to interpre

identify quantitative relations between constitue

predict runoff volumes, concentrations, loads; a

dict potential effects of runoff on receiving wat

sites for which data do not exist. Regression ana

designed to provide an estimate of the average re

of a system as it relates to variation in one or m

known variables. To date, highway- and urban

studies have generally been limited to ordinary 

squares (OLS) and generalized least squares (G

regression techniques. There are, however, a num

linear and nonlinear regression methods that ma

appropriate for interpretation of local, regional, 

national highway-runoff and urban-stormwater 

when the classification of variables and the stru

the data violate the design assumptions of the O

(or) GLS methods.
Refere

 Uncertainty is an important part of any 

decision-making process. Success of a water-quality 

interpretive model depends on uncertain future meteorological, demographic, political, and technical conditions, all of which may affect future costs and benefits. 

In order to deal with uncertainty problems, the analyst 

needs to know the severity of the statistical uncertainty 

of the methods used to predict water quality. Statistical 

models need to be based on information that is meaningful, representative, complete, precise, accurate, and 

comparable to be deemed valid, up to date, and technically supportable. If sensitivity analyses reveal too 

much uncertainty in the predictions, new data and new 

methods may be needed, or safety factors based on prediction interval estimates may be used. To ensure that 

decision makers can assess uncertainty in the analytical 

tools, the modeling methods, and the underlying data 

set, the analyst must document and communicate each 

of these components in an accessible format within 

project publications. These criteria will also determine 

whether interpretations based on the models developed 

will be admissible in a regulatory framework and (or) 

as legal evidence when necessary.

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APPENDIX 1. Regression Tools

Appen

A. Partial Residual Plots

Transformation of predictors in a multiple regression model can be used to achieve linearity and simp

model. A graphical device that can be helpful in deciding on a transformation for a predictor is a partial res

plot (Larsen and McCleary, 1972). A partial residual plot for a predictor, xt, in a multiple linear regression i

plot of the partial residual, computed by subtracting the effects of all variables except xt, against xt. For exa

suppose we have the three-variable model

yi = b0 + b1x1,i + b2x2,i + b3x3,i + ei (i=1, n), 

and we wish a partial residual plot for variable x3. The partial residual, (yi-b0-b1x1,i-b2x2,i), is plotted again

If the plot appears to be linear, then no transformation is needed. If the plot shows some curvature, then a 

transformation may be helpful. This plot is sometimes called a component-plus-residual plot because the p

residual, (yi-b0-b1x1,i-b2x2,i), is equal to b3x3i+ei, which is easier to compute.

B. Seasonality

Data may exhibit seasonal patterns. One method for dealing with seasonal patterns is to develop diffe

regressions for the different seasons. Barbe and Francis (1995) use this method for coliform concentrations 

river in Louisiana. Driscoll and others (1990) classified snowmelt storms separately from rainfall-runoff eve

they did not develop seasonality as a quantitative variable in their highway-runoff-quality models. Effects o

sonality sometimes can be reduced by dealing with deviations from seasonal means or by use of periodic fun

When a simple periodic function such as a cosine function is used to describe the cyclic variation, the mod

, 

in which ti is a time unit and τ is the cycle length in the same time units. Cycle lengths may be known (suc

annual cycles, diurnal cycles, or tidal cycles) or may be estimated from the data. Tasker and Burns (1974) u

periodic function with estimated cycle lengths to model stream temperatures in New England. Other variab

may be included in the model. For example, Cohn and others (1992) use a periodic function to remove seas

from a model to estimate nutrient loading in Chesapeake Bay.

C. Collinearity

Correlation among the predictors in a regression results in near redundancies among the predictors, a

inferences based on the model can be misleading or erroneous. Multicollinearity is the problem of linear de

cies between predictors. Multicollinearity in a regression can cause prediction problems when predictions a

extrapolated beyond the sample space of the predictors. When the sample X-space is 3 or more dimensions

especially difficult to recognize a collinearity problem. 

One diagnostic for detecting possible collinearity problems is the variance inflation factor (VIF). It is 

computed as

, 

where Rj is the coefficient of determination from the regression of Xj on the other explanatory variables. A 

guideline for serious multicollinearity problems is VIF > 10. Belsley and others (1980) and Belsley (1991) d

other useful methods for detecting multicollinearity. Treatment of the multicollinearity problem depends som

on the source of the multicollinearity. Sources may be related to the sampling design, constraints on the mod

model specification. 

yi βo β1

2πti

τ

---------     cos β2

2πti

τ

---------     sin ε = + ++ i

VIF j

1

1 Rj

2 – = ---------------
40 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

It is sometimes possible to deal with multicollinearity created by the sampling design by collecting n

to fill in sparse areas on the predictor space. For example, consider a regression of sulfate load with two pre

drainage area (A) and percent of basin urbanized (U). If in designing the monitoring network, sites with lar

tended to have small U and sites with large U tended to have small A, a negative correlation in A and U wo

result. This could be fixed by adding new sites in the network with small A and small U and sites with large 

large U.

Multicollinearity problems created by constraints on the model or model specification can sometime

dealt with by redefining the predictors. For example, consider two regressors, drainage area (A) and stream 

(L). Suppose further that in this region all basins with a large A also have a large L and basins with small A 

small L, so that A and L are constrained by the population to have a positive correlation. It may be possible t

a new dimensionless shape variable, Sh=L2/A, which is not correlated with A or L, which can be substituted 

either A or L. 

It is also possible to deal with multicollinearity by means of predictor elimination. For example, if pre

x1, x2, and x3 exhibit strong multicollinearity, eliminating one of the predictors may reduce the problem gre

Predictor elimination, however, may not be an attractive alternative if the analyst wishes to extract informa

regarding the roles of individual predictors. 

D. Regression Diagnostics

Regression diagnostics are used to identify possible outliers. An analyst can use regression diagnosti

ods to find influential observations and study their effects. Regression diagnostics aid in the systematic loca

data points that are unusual or are highly influential in estimating regression parameters and standard error

nostics help to avoid misinterpretation of the regression model. Cook and Weisberg (1982; 1984) and Belsl

others (1980) give comprehensive treatments of regression diagnostics. Partial regression leverage plots are 

important part of regression diagnostics. Two commonly used diagnostics, leverage and Cook’s D are briefl

discussed below.

Observations that are far from the center of the X-variable space are considered high leverage points b

of their great potential to influence the regression results. The leverage of a point is defined as

hii = xi(X′X)-1x′i . 

The limits on hii are (1/n)<hii<1 and

. 

Therefore the average hii= , and a suggested value to identify a point with high leverage, is hii= . Leve

plays an important role in the calculation of influence statistics and standardized residuals.

Cook’s D is a measure of the shift in the vector of predicted values of y when the ith observation is no

It shows the influence of the observation on the regression estimates. Computationally, it is obtained as foll

. 

A suggested cutoff value to flag influential data points is Di>4/n.

hii

i = 1

n

∑ = p'

p'

n

---- 2 p'

n

----

Di

r2

i

p' ----- hii1 hii – --------------     =
APPENDIX 2. Linear Regression Methods

Appen

A. Linear Ordinary Least-Squares Regression

The linear model for relating a response variable, Y, to p predictors is

Yi = β0 + β1Xi,1 + β2Xi,2 + . . . +βpXi,p + εi. 

Subscript i denotes an observation at site i. There are p predictors and p′= (p+1) parameters to be estimated

denote the number of observations or sites.

 Denote the following matrices:

Y, a (n x 1) column vector of observed response,

X, a (n x p′) matrix of a column of ones followed by p columns of predictors,

β, is a (p′ x 1) vector of parameters to be estimated, and 

ε, is a (n x 1) column vector of random errors.

The linear model can be written in matrix notation as

Y = Xβ + ε , 

in which

The usual assumptions about random vector of errors, ε, is that all the elements, εi, have a common va

σ2,, and are statistically independent. These assumptions can be written in shorthand as

ε∼N(0, Iσ2). 

If the model is correct (another assumption), then

Y ~ N(Xβ, Iσ2) . 

The regression coefficients, β, are best estimated as

b = (X′X)-1(X′Y) . 

The predicted mean of the response variable at site k with basin characteristics xk =(1, xk,1, xk,2, . . ., 

=xkb . 

The variance of the prediction is Var( ) = σ2[1 + xk(X′X)-1x′k] .

Y

y1

y2

...

yn

 X =

1 x11 x21 ... xk1

1 x12 x22 ... xk2

...... .........

1 x1n x2n ... xkn

 β

β0

β1

...

βk

 ε

ε1

ε2

...

εn

= ==

yˆ k

yˆ k
44 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

B. Nonparametric Regression

The Kendall-Theil method (Helsel and Hirsch, 1992) is a nonparametric method for fitting a prespec

linear equation of the form: 

Y=B0+B1X. 

For each pair of points (xi,yi) and (xj,yj), such that i<j, a slope, Sij= , is computed and B1 is set equa

to the median of the computed slopes. Let Y.5 and X.5 denote the median of X and Y, respectively. The interc

B0 =Y.5 - B1X.5. Conover (1980) provides a method for estimating a confidence interval for B1. Instead of u

median of the pairwise slopes, median (Sij), one can use a weighted median of the slopes with weights prop

to the distance between the pairs of points. Birkes and Dodge (1993, p. 114) show that the nonparametric w

median estimator is the value of B1 that minimizes the sum

. 

Equation 12 may be generalized to more than one predictor for a multiple regression problem. Nonp

ric estimates of B1, B2, , Bp are found by minimizing the sum

. 

An estimate of B0 is obtained as the median of . Birkes and Dodge (1993, p. 123) 

iterative procedures to find the values of B1, B2, , Bp.

Regression smooths are nonparametric local averaging methods that require no prespecified model fu

tional form. Three major smoothing methods for problems with one predictor are kernel smoothing, k-near

neighbor (k-NN) smoothing, and splines (Härdle, 1990). Kernel smoothing uses local observations within a 

width to compute a weighted average defined by the kernel. The k nearest neighbors are used to estimate th

weighted average in k-NN smoothing. Splines are piecewise polynomials of order k that are smoothly joine

cubic spline (k=3) is usually good enough for most problems. Average smooths tend to follow outlying poi

are not particularly robust against outliers. However, one may choose median smooths (Helsel and Hirsch, 1

286) or LOWESS (Cleveland, 1979; Helsel and Hirsch, 1992, p. 288) when outliers are perceived to be a p

LOWESS is an iterative procedure that progressively downweights outliers in computing the local weights

The regression smooths described above are for problems with one predictor. Extending the local ave

smooths to problems with multiple predictors raises the problem of sparse data in local neighborhoods. Bre

and others (1984) proposed regression trees as a type of nonparametric, multiple-predictor smooth. Regres

trees define a piecewise constant regression surface based upon neighborhoods defined by hyper-rectangles 

sides parallel to coordinate axes. Friedman and Stuetzle (1981) describe projection pursuit regression, an ex

of regression trees, that uses smoothing methods on linear combinations of predictors to form the regressio

face. Research is needed to determine whether regression trees and projection pursuit regression would be u

regional water-quality-assessment models.

yi y j ( ) –

xi x j ( ) – -------------------

rank yi B1xi ( ) – ( ) n – 1

2 – ----------------     yi B1xi ( ) –

i = 1

n

∑

...

rank yi B1x1i B2x ( ) – ( ) + + 2i ... ( ) n – 1

2 – ----------------     yi B1x1i B2x ( ) – ( ) + + 2i ...

i = 1

n

∑

yi B1x1i B2x – ( ) + + 2i ...

...
Appen

C. Robust Regression

Robust regression methods are insensitive to the effects of outliers in the data; they are useful for det

outliers by accentuating observations with large residuals from the robust model. Observations that are dow

weighted in the robust regression model require close examination for the reasons for downweighting. The 

eter-estimation problem in regression analysis may be thought of as finding the estimates of β to minimize t

of some function, ρ,

, 

where z= and s is a scale factor. 

Robust procedures dampen the effects of outliers and tend to leave large residuals for the outliers. In 

estimation, the scale factor, s, must be a robust estimator of scale and not the standard deviation of the resid

because the standard deviation is relatively sensitive to outliers. Montgomery and Peck (1982, p. 367) sugg

robust estimator 

, 

where ei = yi-xiβ.

The least-squares estimator, corresponding to ρ(z)=z2, is sensitive to outliers because it gives relative

greater weight to large residuals. On the other hand, the least-absolute-deviation estimator, corresponding t

ρ(z)=|z|, is resistant to outliers but may give too much weight to small residuals. Huber (1973) proposed a c

mise estimator that is robust yet relatively efficient if data are normal. It weights small residuals as a least-s

estimator and large residuals as an absolute-deviation estimator. The ρ function for Huber’s estimator is 

, 

in which t is a constant usually equal to 2 or less. Parameter estimates are made from iteratively reweighted 

squares. For example, the weights, w, in a weighted least-squares regression minimizing the sum of the fun

equation 18 are

. 

Because the weights depend on β, iteration is required until little or no changes in the parameters are observ

parameters for many other functions that have been suggested for the robust regression problem (Andrews 

others, 1972) also can be found by iteratively reweighted least squares.

One problem with robust methods is an apparent lack of agreement among authorities on how best to 

struct confidence intervals for the parameter estimates. Therefore, one important aspect of statistical region

models is not clearly determined for robust regression models.

ρ( )z

i = 1

n

∑

yi – xiβ

s ------------------

s

median ei median ei – ( )

0.6745 = -------------------------------------------------------------

ρ( )z

1

2

-- z2 z t ≤

z t

1

2

--t2 –     z t > 











=

w z( )

1.0 z t ≤

t

z

---- z t > 







=
46 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

D. Generalized Least-Squares Regression

In generalized least-squares (GLS) regression, β is estimated by

 , 

in which Λ is the covariance matrix of errors, E(eeT). The operational difficulty with this procedure is that Λ 

be estimated from the data at hand. Stedinger and Tasker (1985) show that Λ can be estimated as

Λ = γ2I + Σ , 

where γ2 is an estimate of the variance of the error inherent in the model, Σ is an estimate of the sampling-e

covariance matrix, and I is an (n by n) identity matrix. The model error variance, γ2, and regression coeffici

are found by iteratively searching for the best non-negative solution to the equation

 . 

A leverage statistic in GLS analogous to leverage in OLS regression is the ith diagonal element of

H* = X(X′Λ-1X)-1X′Λ-1 . 

The sum of the diagonal elements of H* is equal to the number of parameters in the model; and a high-lever

would be one in which the associated diagonal element is greater than 2 times the number of parameters div

the number of observations, as a rule of thumb. A GLS version of Cook’s D is 

 , 

where h′ii are diagonal elements of

H′ = X(X′Λ-1X)-1X′ . 

D′i is large if it exceeds about (4/n) (Tasker and Stedinger, 1989).

β

 ̃ = ( ) XTΛ–1X –1XTΛ–1Y

E ( ) Y X– β { } TΛ–1( ) Y X– β = n k – 1 –

D'

i

ei

2h'

ii

p' γ i h'

ii ( ) – 2 = -----------------------------
Appen

E. Tobit Regression

The tobit regression model for censored observations is 

 . 

If one arbitrarily assumes a value for those observations of y below the threshold and uses the entire sample

one uses the subsample of observations when yi is greater than the censoring threshold, the least-squares es

of β is biased and inconsistent. However, maximum-likelihood estimators for the tobit regression model are 

available and are described in Judge and others (1985). Liu and others (1996) use tobit regression to predic

atrazine concentrations in the Midwest. Hirsch and others (1993, p. 17.51) caution that application of tobit 

regression in hydrology is experimental.

F. Logistic Regression

In logistic regression, the response is a nominal variable with two possible values (0 and 1). For exam

is equal to zero if the value is below the detection limit and equal to 1 if the value is above the detection limi

model, the estimated response, E(y|x), is a proportion between 0 and 1 and is given by

 . 

The unknown β’s are estimated by maximizing the log likelihood function

 (Cox and Snell, 1989). 

Teso and others (1996) use logistic regression to estimate probability of pesticide contamination. Ma

tistics packages, such as SAS, Minitab, and STATIT, include procedures for fitting the logistic regression fu

Thus, the logistic regression procedure can be used to estimate the likelihood of a water-quality characteristi

above or below a censoring threshold at a site with basin characteristics equal to x.

yi xiβ e + i ( ) if yi ( ) > threshold 

 =

E y( ) x exp( ) xβ

1 + exp( ) xβ = ------------------------------

L( ) β yiln[ ] xβ 1 yi { } + ( ) – ln[ ] 1 – xβ

i = 1

n

= ∑

nominal otherwise
48 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

G. Contingency Tables

Contingency tables can be used when both response and predictors are nominal or censored. The num

observed values falling within a cell defined by the response and predictor groups divided by the total obser

in a group provides an estimate of the probability of a value being in the cell. If, for example, the effect of 

volume on the potential for detection of cadmium was of interest, then this effect could be examined by use 

following (hypothetical) contingency table:

In this hypothetical example, the probability of cadmium being greater than or equal to the detection 

5/20 = 0.25 for rural highways and 20/28 = 0.71 for urban highways. Helsel and Hirsch (1992) and other te

vide details for statistical analysis using contingency tables.

H. Ridge Regression

The ridge estimator of regression coefficients, β, is

 , 

in which constant κ is a biasing parameter to be determined, and Z and yo are standardized versions of X a

respectively. The choice of κ is the subject of several studies. Myers (1986) and Montgomery and Peck (19

describe several methods for choosing κ.

Computations can be made by augmenting the standardized data and using ordinary least-squares met

follows:

 , 

where is a p by p diagonal matrix with diagonal elements equal to the square root of κ and 0 is a p by 1 

of zeros (Montgomery and Peck, 1982). The estimate is then computed as

. 

The use of ridge regression requires thoughtful study of the data and careful analysis, but it can be an 

tive method for dealing with multicollinearity problems.

Contingency-table example using hypothetical cadmium concentrations in urban and rural highway runoff

[ADT, Average Daily Traffic; Cd, cadmium concentrations in micrograms per liter; VPD, vehicles per day]

Cadmium

concentrations

Rural highway

(ADT ≤30,000 VPD)

Urban highway 

(ADT>30,000 VPD) Total

Cd < detection limit ................... A=15 B=8 A+B=23

Cd >detection limit .................... C=5 D=20 C+D=25

Total: .......................................... A+C=20 B+D=28

β

ˆ R ( ) Z'Z + κI –1Z' yo =

Za

Z

κI

 ya

yo

0 = =

κI

β

ˆ R

βR

ˆ Za′Za ( )–1Za′ y = a
APPENDIX 3. Nonlinear Regression Methods

Appen

A. SPARROW

In the SPARROW (SPAtially Referenced Regression On Watershed attributes) model (Smith and oth

1997), the stream network in a region is divided into many stream reaches and the instream load, Li, of a co

ent in a stream reach indexed by i is equal to the sum of contributions to the load from all upstream sources, 

that

 , 

where N is the number of sources. Let J(i) represent the set of all stream reaches upstream from reach i and 

including reach i but downstream from all monitoring stations upstream from reach i. Let K(i) represent the 

all monitoring sites directly upstream from reach i. The source terms are determined by

 , 

in which βn is a coefficient for source n and is the predictor for source n and reach i associated w

upstream reaches J(i) or monitoring sites K(i). The values for Xn,i,j are given by

 , 

where sn,j is a measure of the contaminant mass from source n that is present in reach j or at monitoring sit

α’s are delivery coefficients associated with land-surface characteristics Zi,j, and the δ’s are decay coefficie

associated with flowpath characteristics Tn,i,j. For point sources or monitoring sites, the α’s are set equal to 

For example, sn,j might be the mass of chlorides placed on the roads within the drainage of reach j or meas

monitoring site k; the Z1,j might be average soil permeability for the drainage area of reach j; and T1,i,j migh

stream length between reach i and reach j. The predictors are spatially referenced because the contribution f

reaches above a given reach is tied to the reach by the flowpath characteristic, Ti,j.

The parameters (β’s, α’s, and δ’s) of this nonlinear regression technique are determined by means of 

SAS/ETS procedure MODEL (SAS Institute, 1988). Smith and others (1997) determine the standard errors 

parameters using bootstrap methods (Efron, 1982).

B. Artificial Neural Networks

An artificial neural network, or ANN (Hertz and others, 1991), is composed of simple processing un

called neurons, arranged in layers. Each unit receives input from other units and converts the input to a sing

put, which it sends to other units. The conversion takes place in two stages: first, a net input is computed as 

weighted sum of inputs, then an activation function transforms the net input into an output. The flexibility o

comes from the analyst’s being able to specify multiple layers of neurons with nonlinear activation function

alternative methods for computing the net input. A multilayer perceptron (MLP) with three nodes in the inpu

three nodes in the hidden layer, and one output node is shown in figure A. The hidden layer is so named be

has no direct connection to the outside world. 

Li Sn i ,

n = 1

N

= ∑

Sn i , βn Xnij , ,

j Ji ∈ ( )

∑ Xnik , ,

k Ki ∈ ( )

= + ∑

Xnij , ,

j ∈ J i( )

∑

Xnij , , sn j , α1Z1, j – α2Z – 2, j... δ1T1, ,i j – δ2T = exp( – 2, ,i j...)
52 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

Vj

HIDDEN LAYER 

OUTPUT LAYER  

yi

INPUT LAYER   

xk

Explanatory

 

 





Response, y1

Explanatory

Explanatory

variable, x2

 variable, x3

variable, x1 V1

V2

V3

Each explanatory variable has one input node in the model. Denote the kth input variable as xk and th

number of hidden nodes as nh. The net input to the jth hidden node is 

 , 

where aj and bjk are intercept and weights from input k to hidden node j. The activation function, g(u), for t

hidden nodes is usually a smooth nonlinear function with a single-valued first derivative, such as the sigmo

function, g(u)= , or the hyperbolic tangent function, g(u)=tanh(u). For example, the estimated respo

for the MLP in figure A with three input nodes, three hidden nodes with a hyperbolic tangent activation fun

and one output node is

 , 

where c1 and d1,j are intercept and weights from hidden node j to output 1. The activation function for the o

node in this example is the linear function, g(u)=u.

Observed values of predictors (inputs) and responses (targets) are used to train the ANN by iteratively 

ing the weights used by the neurons to produce output so that the sum of squared differences between outp

target data is small. The method used is called back-propagation with a conjugate gradient training algorith

(Hertz and others 1991). Neural Connection 2.0 (SPSS, Inc., 1997) software was used for the calculations. 

tively, one could estimate the intercepts and weights in equation 35 using nonlinear regression methods (Sa

1994). ANN can overtrain (fit the observed data well, but not predict well for new data). For this reason, a p

of the observed data is used as a validation data set that is not used in training the ANN. Artificial neural ne

are data-in/predictions-out black boxes. Any underlying hydrologic model or hydrologically significant fun

relation may be impossible to extract from the network.

a j b jk xk

k = 1

nh

+ ∑

1

1 e–u + ----------------

y1 c1 d1, j a j b jk xk

k = 1

3

+ ∑       tanh

j = 1

3

= + ∑

Figure A. Multilayer perceptron illustrating the function of an artificial neural network (ANN).
APPENDIX 4. Uncertainty Analysis

Appen

A. Uncertainty—Normally Distributed Errors

The predicted response at unmonitored site k with basin characteristics x0 =(1, x0,1, x0,2, .., x0,p) is

=x0b . 

The standard error of the prediction in OLS regression is

S( )= σ . 

In GLS regression the standard error of prediction is:

 . 

A 100(1-α) prediction interval would be

 , 

where

 , 

and where ta/2, n-p′ is the critical value from a t-distribution for n-p′ degrees of freedom. The use of the t-sta

requires the errors to be approximately normally distributed. If a log transformation had been made so that 

y0=log10(q0), then the prediction interval would be

 . 

When a log transformation has been made and the standard error in log units follows a normal distrib

the standard error may be expressed in percent of the predicted value in the original untransformed units. D

as the standard error in log (base 10) units, Sorg as the standard error in original units, and E(q|xk) as the pre

value of q, in original units, given xk, and xk =(1, xk,1, xk,2,..., xk,p) is a vector of basin characteristics at site 

standard error in percent, Spercent, is given by

 (Aitcheson and Brown, 1957). 

Sometimes it is said in OLS that two-thirds of the points lie within one standard error of estimate of 

regression function. This is true for the log unit standard error of estimate, σ, but it generally is not correct 

cent, because the errors in log space are symmetrically distributed under the assumption of normality of the 

errors, but the errors in original units are skewed. One can, however, calculate a +percent and -percent erro

the following formulas:

 and 

 . 

The three formulas (42, 43, and 44) above apply not only to the standard error of estimate for an OLS 

sion but also to the standard error of the model, , in GLS regression, and standard error of a prediction in 

OLS and GLS.

yˆ 0

yˆ 0 1 x0( ) X′X –1 x′ + 0

S yˆ 0 () γˆ 2 x0X′Λˆ –1X–1 x′ = + 0

yˆ 0 – T y0 yˆ 0 ≤ ≤ + T

T tα

2

---, ( ) n p – '

S yˆ 0 = ( )

10yˆ 0 – T

q0 10yˆ 0 + T ≤ ≤

Spercent 100

Sorg

Eqxk ( ) ------------------- 100 e5.302σ2 = = ( ) – 1

Splus 100 10σ = ( ) – 1

Sminus 100 10–σ = ( ) – 1

γˆ
56 Statistical Approaches to Interpretation of Local, Regional, and National Highway-Runoff and Urban-Stormwater Data

B. Uncertainty—Non-Normally Distributed Errors

Consider the general regression model

 , 

where g(xi, β) is a function of known form, xi is a vector of known predictors, and β is a vector of unknown 

coefficients. The εi are independent errors drawn from an unspecified distribution, F, centered at zero, that m

be normally distributed. Having observed yi for i=1, 2,..., n, β is estimated by minimizing the sum of some fu

ρ, of the errors

: min . 

Such a model may be too complicated for standard analysis, but a bootstrap method similar to that described 

can be used.

1. Compute the observed residuals: 

2. Draw a bootstrap sample by randomly selecting, with replacement, from the observed residuals, a boo

sample of residuals, , compute and calculate .

3. Repeat step 2 Z times to obtain bootstrap replications .

Let ; then, an estimate of β’s covariance matrix is

. 

A nonparametric (1-α) confidence interval for a prediction can be approximated by taking the (1-α) 

portion of Z predictions based on the bootstrap replications from step 3.

yi g xi ( )ε , β i = + i = 1, 2, 3, ..., n

β

ˆ ρ yi g xi ( ) – ( ) , β

i = 1

n

∑

εi

ˆ yi g xi β

ˆ = – ( ) ,

εˆ °1 εˆ °2 ... εˆ °n , ,, yi

° g xi β

ˆ ( )ε , i

ˆ = + ° β

ˆ °

β

ˆ ( ) ° 1 βˆ ( ) ° 2 ... βˆ , ,, ( ) ° Z

β

ˆ °mean

β

ˆ ( ) ° b

Z ------------

b = 1

B

= ∑

cov( ) β

ˆ 1

Z – 1 ------------ β

ˆ ( ) ° b β

ˆ ( )β – °mean ˆ ( ) ° b βˆ ( )′ – °mean

b = 1

B

= ∑
APPENDIX 5. Region of Influence Method

Appen

In this method of developing site-specific predictions with a data set spanning a large geographic are

regression equation for a site is computed using data from a unique region called the region of influence by 

(1990a, 1990b) and suggested by Acreman and Wiltshire (1987). The unique subset of monitoring sites tha

up the region of influence for each prediction site is made up of the Ns nearest neighbors. The method is an 

attractive alternative to the more traditional methods because it can be easily updated by simply updating th

quality data in a database file from which the method draws its basic data; furthermore, extrapolation errors 

be small because predictions by definition occur near the center of the space of the predictors. In this metho

nearness of two neighbors is not measured by the physical distance between the sites, but rather by a distan

defined in terms of the watershed characteristics. This distance between any two sites, indexed by i and j, is 

determined by the Euclidean distance metric;

 , 

in which, dij is the distance between the watershed characteristics at sites i and j, p is the number of watersh

characteristics needed to calculate dij, Xk represents the kth watershed characteristic, sd(Xk) is the sample s

deviation for Xk, and xik is the value of Xk at the ith site. The dij’s between the prediction site i and all moni

sites j = 1, 2,..., n in a region is determined, and the Ns monitoring sites with smallest dij make up the regio

influence for site i. For this method to work, the value of Ns should be large enough to have enough degree

freedom in the regression to estimate two or three parameters. The method is computer intensive and require

subjective judgement for selecting Ns and the attributes used in the distance metric.

dij

xik x jk –

sd Xk ( ) -------------------     2

k = 1

p

∑      1 2/

=