U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigaitons Report 2012–5006

Prepared in cooperation with the City of Houston

Regression Model Development and Computational 

Procedures to Support Estimation of Real-Time 

Concentrations and Loads of Selected Constituents in Two 

Tributaries to Lake Houston near Houston, Texas, 2005–9

Regression Model Development and 

Computational Procedures to Support 

Estimation of Real-Time Concentrations

and Loads of Selected Constituents in

Two Tributaries to Lake Houston near

Houston, Texas, 2005–9

By Michael T. Lee, William H. Asquith, and Timothy D. Oden

Prepared in cooperation with the City of Houston

Scientific Investigaitons Report 2012–5006

U.S. Department of the Interior

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

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2012

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

reproduce any copyrighted materials contained within this report.

Suggested citation:

Lee, M.T., Asquith, W.H., and Oden, T.D., 2012, Regression model development and computational procedures to 

support estimation of real-time concentrations and loads of selected constituents in two tributaries to Lake Houston 

near Houston, Texas, 2005–9: U.S. Geological Survey Scientific Investigations Report 2012–5006, 40 p.
iii

Contents

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

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

Purpose and Scope ..............................................................................................................................3

Description of Study Area ...................................................................................................................3

Methods..................................................................................................................................................4

Streamflow Measurements ................................................................................................................4

Continuous Water-Quality Monitoring...............................................................................................4

Discrete Water-Quality Sample Collection, Analysis, and Results ..............................................5

Sample Design and Collection...................................................................................................5

Sample Analysis...........................................................................................................................5

Quality Control..............................................................................................................................8

Regression Model Development.................................................................................................................8

Retransformation Bias Correction ............................................................................................9

Analysis of Censored Data.......................................................................................................10

Censored Data at Spring Creek and East Fork San Jacinto River Site ............................10

Regression Analysis Summaries and Presentation of Equations......................................10

Computational Procedures of Concentration and Loads......................................................................23

Summary........................................................................................................................................................23

References....................................................................................................................................................24

Appendixes ...................................................................................................................................................27

Figures

1. Lake Houston watershed and tributary subwatersheds

and location of U.S. Geological Survey streamflow-gaging

stations 08068500 Spring Creek near Spring, Texas, and 08070200

East Fork San Jacinto River near New Caney, Tex ...........................................................2

2. Flow duration curve and corresponding discrete water-quality

samples, (A) Spring Creek near Spring, Texas, and (B) East Fork

San Jacinto River near New Caney, Tex.............................................................................6

3. Abbreviations of mathematical functions and statistical terms

related to regression analysis of water-quality constituents

shown in figures 4–14...........................................................................................................11

4. Summary of regression analysis for nitrite plus nitrate nitrogen

for U.S. Geological Survey streamflow-gaging station 08068500

Spring Creek near Spring, Texas, 2005–9..........................................................................12

5. Summary of regression analysis for total phosphorus for

U.S. Geological Survey streamflow-gaging station 08068500

Spring Creek near Spring, Texas, 2005–9..........................................................................13

6. Summary of regression analysis for total organic carbon for

U.S. Geological Survey streamflow-gaging station 08068500

Spring Creek near Spring, Texas, 2005–9..........................................................................14

7. Summary of regression analysis for Escherichia coli for

U.S. Geological Survey streamflow-gaging station 08068500

Spring Creek near Spring, Texas, 2005–9..........................................................................15
iv

8. Summary of regression analysis for atrazine for

U.S. Geological Survey streamflow-gaging station 08068500

Spring Creek near Spring, Texas, 2005–9..........................................................................16

9. Summary of regression analysis for suspended sediment for

U.S. Geological Survey streamflow-gaging station 08068500

Spring Creek near Spring, Texas, 2005–9..........................................................................17

10. Summary of regression analysis for nitrite plus nitrate nitrogen

for U.S. Geological Survey streamflow-gaging station 08070200

East Fork San Jacinto River near New Caney, Texas, 2005–9.......................................18

11. Summary of regression analysis for total phosphorus for

U.S. Geological Survey streamflow-gaging station 08070200

East Fork San Jacinto River near New Caney, Texas, 2005–9.......................................19

12. Summary of regression analysis for total organic carbon for

U.S. Geological Survey streamflow-gaging station 08070200

East Fork San Jacinto River near New Caney, Texas, 2005–9.......................................20

13. Summary of regression analysis for Escherichia coli for

U.S. Geological Survey streamflow-gaging station 08070200

East Fork San Jacinto River near New Caney, Texas, 2005–9.......................................21

14. Summary of regression analysis for suspended sediment for

U.S. Geological Survey streamflow-gaging station 08070200

East Fork San Jacinto River near New Caney, Texas, 2005–9.......................................22

Tables

1. Subwatershed drainage areas for tributaries to Lake Houston

near Houston, Texas...............................................................................................................4

2. Summary statistics for samples collected at two tributaries

to Lake Houston near Houston, Texas, 2005–9...................................................................6
v

Conversion Factors, Datums, and Water-Quality Units

Inch/Pound to SI

Multiply By To obtain

Length

inch (in.) 25.4 millimeter (mm)

mile (mi) 1.609 kilometer (km)

Area

square mile (mi2) 2.590 square kilometer (km2) 

Flow rate

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

Mass

pound, avoirdupois (lb) 0.4536 kilogram (kg)

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

°F=(1.8×°C)+32

Datums

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 

(NAVD 88).

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

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

Water-Quality Units

Specific conductance is given in microsiemens per centimeter at 25 degrees Celsius (μS/cm at 

25 °C).

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

or micrograms per liter (μg/L).

Bacteria are given in most-probable number per 100 milliliters (MPN/100 mL).

Turbidity is given in Formazine Nephelometric Units (FNU).

Regression Model Development and Computational 

Procedures to Support Estimation of Real-Time 

Concentrations and Loads of Selected Constituents in Two 

Tributaries to Lake Houston near Houston, Texas, 2005–9 

By Michael T. Lee, William H. Asquith, and Timothy D. Oden 

Abstract

In December 2005, the U.S. Geological Survey (USGS), 

in cooperation with the City of Houston, Texas, began 

collecting discrete water-quality samples for nutrients, total 

organic carbon, bacteria (Escherichia coli and total coliform), 

atrazine, and suspended sediment at two USGS streamflowgaging stations that represent watersheds contributing to 

Lake Houston (08068500 Spring Creek near Spring, Tex., 

and 08070200 East Fork San Jacinto River near New Caney, 

Tex.). Data from the discrete water-quality samples collected 

during 2005–9, in conjunction with continuously monitored 

real-time data that included streamflow and other physical 

water-quality properties (specific conductance, pH, water 

temperature, turbidity, and dissolved oxygen), were used to 

develop regression models for the estimation of concentrations 

of water-quality constituents of substantial source watersheds 

to Lake Houston. The potential explanatory variables 

included discharge (streamflow), specific conductance, pH, 

water temperature, turbidity, dissolved oxygen, and time (to 

account for seasonal variations inherent in some water-quality 

data). The response variables (the selected constituents) at 

each site were nitrite plus nitrate nitrogen, total phosphorus, 

total organic carbon, E. coli, atrazine, and suspended 

sediment. The explanatory variables provide easily measured 

quantities to serve as potential surrogate variables to estimate 

concentrations of the selected constituents through statistical 

regression. Statistical regression also facilitates accompanying 

estimates of uncertainty in the form of prediction intervals. 

Each regression model potentially can be used to estimate 

concentrations of a given constituent in real time. Among 

other regression diagnostics, the diagnostics used as indicators 

of general model reliability and reported herein include the 

adjusted R-squared, the residual standard error, residual plots, 

and p-values. Adjusted R-squared values for the Spring Creek 

models ranged from .582–.922 (dimensionless). The residual 

standard errors ranged from .073–.447 (base-10 logarithm). 

Adjusted R-squared values for the East Fork San Jacinto River 

models ranged from .253–.853 (dimensionless). The residual 

standard errors ranged from .076–.388 (base-10 logarithm). In 

conjunction with estimated concentrations, constituent loads 

can be estimated by multiplying the estimated concentration 

by the corresponding streamflow and by applying the 

appropriate conversion factor. The regression models 

presented in this report are site specific, that is, they are 

specific to the Spring Creek and East Fork San Jacinto River 

streamflow-gaging stations; however, the general methods 

that were developed and documented could be applied to most 

perennial streams for the purpose of estimating real-time water 

quality data.

Introduction

Houston, Texas, is the fourth largest city in the United 

States, with an estimated population of about 5.9 million 

people in 2009 (Texas State Data Center, 2011). Historically, 

groundwater has been the major source of supply for the 

City of Houston; however, development of groundwater 

resources has contributed to deleterious water-level declines 

and land-surface subsidence (Kasmarek and Strom, 2002; 

Kasmarek and others, 2010). Lake Houston is a surface-watersupply reservoir for the city of Houston and currently (2011) 

supplies between 10 and 20 percent of the total source-water 

supply (City of Houston, 2011). Furthermore, as a result 

of regulations restricting groundwater withdrawals for the 

purpose of mitigating or arresting land-surface subsidence, 

Lake Houston is expected to become the primary source of 

water for the city in the future; the overall goal is to increase 

the use of surface water to no less than 80 percent of the 

total demand by 2030 (Harris-Galveston Subsidence District, 

1999). Because Lake Houston is a major source of potable 

water and also a recreation resource for the Houston area, 

the possible effects of urbanization on the water quality of 

tributaries to Lake Houston are of interest to water managers 

and planners. Two of the seven tributaries to Lake Houston, 

Spring Creek and East Fork San Jacinto River (fig. 1), 
2 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

Lake

Houston

Cypress Creek

Spring Creek

West Fork

San Jacinto River

Caney

Creek

Peach

Creek

East Fork

San Jacinto

River 

Luce

Bayou

TEXAS

Lake Houston

watershed

 Lake

Houston

Lake

 Conroe

WALLER

GRIMES

WALKER

MONTGOMERY

HARRIS

SAN JACINTO

LIBERTY

EXPLANATION

Subbasin boundary

Subwatershed boundary

U.S. Geological Survey

streamflow-gaging station

and number

08068500

Base from Houston-Galveston Area Council, 2004

Lake Houston watershed, 1:602,933

Universal Transverse Mercator projection, Zone 15

95°30'

95°00'

30°00'

30°30'

08068500

08070200

Houston

Location of study area and Houston, Texas

Study area

WESTERN

SUBBASIN

EASTERN

SUBBASIN

08068500

20 MILES

20 KILOMETERS

0

0

10

10

Figure 1. Lake Houston watershed and tributary subwatersheds and location of U.S. Geological Survey streamflow-gaging stations 

08068500 Spring Creek near Spring, Texas, and 08070200 East Fork San Jacinto River near New Caney, Tex.
Introduction 3

representing approximately 31 percent of the drainage area for 

tributaries to Lake Houston, are the focus of this report.

In compliance with the Federal Clean Water Act, the 

Texas Commission on Environmental Quality compiles and 

maintains an inventory, commonly known as the “303(d) 

list,” of water bodies that are either impaired (do not meet 

applicable State water-quality standards) or threatened 

(are not expected to meet standards in the future) (Texas 

Commission on Environmental Quality, 2011). Lake Houston 

(segment 1002) first appeared in 2006 and again in 2008 on 

the inventory for bacteria. All of Spring Creek (segment 1008) 

has been listed on the 303(d) list for bacteria since 1996, and 

one portion of Spring Creek has been listed for depressed 

dissolved oxygen concentrations that are not conducive to 

healthy ecosystems since 1996. In addition, the East Fork San 

Jacinto River (segment 1003) first appeared on the 303(d) list 

in 2006 for bacteria and was still listed in the 2008 303(d) list.

The U.S. Geological Survey (USGS) and the City of 

Houston maintain a cooperative program to monitor water 

quality in Lake Houston and its contributing watersheds. 

Watershed water-quality monitoring began in December 

2005 and is currently (2012) ongoing. Continuous, realtime monitoring of streamflow and water-quality properties 

(specific conductance, pH, water temperature, turbidity, and 

dissolved oxygen) in Spring Creek and East Fork San Jacinto 

River are collected to alert drinking-water managers to 

potential changes in quality of water entering Lake Houston. 

The continuously monitored streamflow and water-quality 

properties, in conjunction with regression-equation modeling 

using those data as surrogates for selected constituents (nitrite 

plus nitrate nitrogen, total phosphorus, total organic carbon, 

Escherichia coli, atrazine, and suspended sediment) can 

be used to estimate concentrations for constituents lacking 

continuous record. The estimated concentrations can be used 

to compute estimated constituent loads (a value proportional to 

the product of streamflow and concentration). Oden and others 

(2009) developed regression models to estimate constituent 

concentrations on Spring Creek and East Fork San Jacinto 

River based on data collected from 2005–7. 

With near real-time water-quality data for the tributaries 

(every 15 minutes), water managers and planners will be able 

to identify potential effects of tributary inflows on the water 

quality of Lake Houston with sufficient alert time and adjust 

drinking-water plant operations accordingly. In addition, 

over time the results of tributary water-quality monitoring 

will contribute to the broader understanding of watershed 

influences on Lake Houston and the effects of those influences 

on Lake Houston as a drinking water and recreational 

resource. 

Purpose and Scope

This report documents updates that were made to 

previously published (Oden and others, 2009) regression 

models developed using data collected during 2005–7 to 

estimate real-time concentrations of nitrite plus nitrate, 

total phosphorus, total organic carbon, E. coli, atrazine, and 

suspended sediment in two tributaries to Lake Houston: 

Spring Creek and East Fork San Jacinto River. The regression 

models were updated using data collected during 2005–9. 

Real-time (every 15-minutes), continuously measured 

streamflow and water-quality properties (specific conductance, 

pH, water temperature, turbidity, and dissolved oxygen); 

discrete water-quality samples analyzed for nitrite plus nitrate, 

total phosphorus, total organic carbon, E. coli, atrazine, and 

suspended sediment; and time as an additional explanatory 

variable for seasonality were used in the models. The data 

were collected at two USGS streamflow-gaging stations, 

08068500 Spring Creek near Spring, Tex. (hereinafter the 

Spring Creek site), and 08070200 East Fork San Jacinto 

River near New Caney, Tex. (hereinafter the East Fork 

San Jacinto River site). The regression models for each 

constituent are presented for each site. Lastly, examples of 

detailed computational analysis are provided to give the 

reader a step-by-step algorithmic or procedural guideline to 

independently use the regression models herein to calculate 

constituent concentrations, 90-percent prediction intervals, and 

instantaneous loads.

Results from these regression models can be used to 

better understand fluctuations of concentration and loads 

during changing seasons and flow conditions and to assess 

water-quality conditions relative to total maximum daily load 

goals and water-quality standards. The information also is 

useful for evaluating loading characteristics, such as range 

and variability, and for determining effectiveness of best 

management practices (Rasmussen and others, 2008). 

While this report serves primarily as an update to Oden 

and others (2009), the 2 years of additional data used to 

update the models also extends the scientific and statistical 

understanding between continuously measured streamflow 

and water-quality properties and constituent concentrations 

measured in the laboratory. The regression models presented 

in this report are therefore considered to represent actual 

conditions more accurately than the earlier report and should 

serve as a replacement to previous models and the primary 

source for constituent concentration and load calculations.

Description of Study Area

Lake Houston is about 25 miles northeast of Houston, 

Tex. The watershed of Lake Houston comprises the 

subwatersheds of seven tributaries and the area immediately 

adjacent to the lake in parts of seven counties (fig. 1), 

including large areas of densely populated Harris and 

Montgomery Counties. Sneck-Fahrer and others (2005) 

divided the Lake Houston watershed into eastern and western 

subbasins, primarily on the basis of relative amounts of 

development, with the eastern subbasin being less developed 

than the western subbasin. The western subbasin encompasses 

three tributary subwatersheds, and the eastern subbasin 

encompasses four tributary subwatersheds (table 1). The study 

area of this report consists of subwatersheds from the western 
4 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

and eastern subbasins— Spring Creek in the western subbasin 

and East Fork San Jacinto River in the eastern subbasin. 

The Spring Creek subwatershed in the western subbasin 

is the second most densely populated of the seven Lake 

Houston subwatersheds, with a population density in 2000 of 

about 390 people per square mile (U.S. Census Bureau, 2000). 

Urban and agricultural land account for about 41 percent of 

the 453 square miles of the subwatershed and the predominant 

land-use classification is forest (31 percent) (Multi-Resolution 

Land Characteristics Consortium, 2003). 

The East Fork San Jacinto River subwatershed in the 

eastern subbasin is the least densely populated of the seven 

subwatersheds that drain to Lake Houston, with a population 

density in 2000 of about 80 people per square mile (U.S. 

Census Bureau, 2000). Urban and agricultural land together 

account for about 18 percent of the 404 square miles of the 

subwatershed and, as in the Spring Creek subwatershed, the 

predominant land-use classification in the subwatershed is 

forest (47 percent) (Multi-Resolution Land Characteristics 

Consortium, 2003). 

The climate in the study area is classified as humid 

subtropical (Texas State Climatologist, 2011), characterized 

by cool, temperate winters and long, hot summers with high 

humidity. During 2005–9, annual rainfall ranged from 41.2 to 

65.5 inches at George Bush Intercontinental Airport, Houston, 

Tex. (National Oceanic and Atmospheric Administration, 

2011). 

Methods

Streamflow Measurements

Streamflow is the volume of water passing an established 

reference point in a stream at a given time. Methods used to 

determine streamflow (discharge) are described in Buchanan 

and Somers (1969) and Turnipseed and Sauer (2010). 

Streamflow measurements during the course of the study 

(2005–9) were made about five times per year at the Spring 

Creek site and about five times per year at the East Fork San 

Jacinto River site. Stage, or gage height, was measured every 

15 minutes by using submersible pressure transducers (or 

other conventional stage-measurement technology as needed) 

to the nearest 0.01 foot at the Spring Creek and East Fork 

San Jacinto River sites. The data were electronically recorded 

and transmitted by satellite to a downlink site and then to the 

USGS Texas Water Science Center in Austin, Tex. Discharge 

measurements in this study were made to verify and modify a 

stage-discharge relation developed on the basis of streamflow 

measurements and the stage of the stream at the time of 

measurement (Kennedy, 1984). These unique relations were 

used to compute a continuous record of streamflow (Kennedy, 

1983) from the stage record at each site. Instantaneous stage 

and streamflow values are stored in the USGS National Water 

Information System (NWIS) database (U.S. Geological 

Survey, 2011).

Continuous Water-Quality Monitoring

Continuous monitoring of four physical properties 

(specific conductance, pH, water temperature, and dissolved 

oxygen) began at the Spring Creek site in November 1999 

by using a multiparameter monitor. In November 2005, 

a multiparameter monitor was installed at the Spring 

Creek site to include turbidity. Continuous monitoring of 

specific conductance, pH, water temperature, turbidity, and 

dissolved oxygen began at the East Fork San Jacinto River 

site in November 2005. Each of the five sensors on the 

multiparameter monitors were calibrated as described in the 

USGS “National Field Manual for the Collection of WaterQuality Data” (U.S. Geological Survey, variously dated); the 

continuous monitor and record were maintained as outlined in 

Wagner and others (2006).

The Spring Creek and East Fork San Jacinto River sites 

feature a swinging well design for monitoring real-time waterquality properties. Swinging wells respond to and swing in 

the direction of the force of the flowing water. The wells are 

constructed of schedule 80 polyvinyl-chloride pipe with holes 

in the lower 3 feet, allowing water to pass through wherever 

Table 1. Subwatershed drainage areas for tributaries to Lake 

Houston, near Houston, Texas (modified from Sneck-Fahrer and 

others, 2005).

Subwatershed

Drainage area

(square miles)

Western subbasin

West Fork San Jacinto River 998

Spring Creek1 453

Cypress Creek 305

Eastern Subbasin

East Fork San Jacinto River1 404

Peach Creek 151

Caney Creek 222

Luce Bayou 210

1

Subwatershed for which regression analysis was used to develop predictive 

equations in this report.
Introduction 5

a multiparameter monitor is located. Each multiparameter 

monitor is positioned near the centroid of base flow in each 

stream in a swinging well. The data from each multiparameter 

monitor were electronically recorded and transmitted by 

satellite to a downlink site and then to the USGS Texas Water 

Science Center in Austin, Tex. Specific conductance, pH, 

water temperature, turbidity, and dissolved oxygen data are 

stored in the USGS NWIS database in 15-minute intervals. 

The Spring Creek and East Fork San Jacinto River sites at 

the present (2012) remain operational for these five physical 

water-quality properties.

Discrete Water-Quality Sample Collection, 

Analysis, and Results

Discrete water-quality samples were manually collected 

at each sampling site. For this analysis, 58 samples were 

collected at the Spring Creek site, and 51 samples were 

collected at the East Fork San Jacinto River site. Samples were 

analyzed for nutrients, total organic carbon, bacteria, atrazine, 

and suspended sediment. The actual number of results 

reported varied by constituent because of a variety of reasons, 

for example, broken sample bottles, lost samples, or samples 

were determined to fail internal quality-assurance checks and 

therefore reviewed and rejected. 

Sample Design and Collection

Hydrologic conditions in the Spring Creek and East 

Fork San Jacinto River sites vary and might affect chemical 

constituent concentrations, so discrete water-quality samples 

were collected over a wide range of streamflow conditions 

(fig. 2). Discrete water-quality samples for the first year 

(December 2005–November 2006) of this study were 

collected about every 2 weeks to facilitate detection of 

seasonal patterns in water quality. Samples at these fixedfrequency sample times were collected as scheduled without 

regard to hydrologic condition, such as rising, falling, or 

stable streamflows. During storms or periods of high flow, 

unscheduled samples were also periodically collected during 

the first year of the study. During the second and third year of 

the study (December 2006–December 2008) discrete waterquality samples were collected approximately once a month 

at both the Spring Creek and East Fork San Jacinto River 

sites. During the fourth year of the study (December 2008–

December 2009), an approximate monthly sampling schedule 

was maintained for the Spring Creek site, whereas samples 

collected at East Fork San Jacinto River site were reduced 

to a quarterly schedule. Instead of fixed frequency sampling 

during the second through fourth years of the study, sampling 

focused on stormwater-runoff whenever possible. Changes to 

the sampling design during the course of the study (in terms 

of timing or responding to specific hydrologic events) added 

considerable complexity to the effort of assessing temporal 

trends in the selected water-quality constituents.

Discrete water-quality samples were collected either by 

wading, when flow conditions permitted, or by sampling from 

bridges during higher flows. All samples were collected and 

processed as described in the USGS “National Field Manual 

for the Collection of Water-Quality Data” (U.S. Geological 

Survey, variously dated). Depth-integrated samples were 

collected, by using a Teflon bottle and nozzle, either by 

multiple verticals when stream velocities were less than about 

1.5 feet per second or by the flow-weighted, equal-width 

increment method when stream velocities were greater than 

about 1.5 feet per second. Samples from each vertical were 

combined in a Teflon churn, dispensed into appropriate sample 

containers, and shipped at 4 degrees Celsius (°C) by overnight 

courier to appropriate laboratories. Samples for bacteria 

analysis were collected directly from the approximate centroid 

of flow in sterile, autoclaved bottles.

Sample Analysis

Samples collected and analyzed for nutrients and total 

organic carbon were analyzed by the USGS National Water 

Quality Laboratory, Denver, Colo., by using published 

methods. Methods for nutrient analysis are documented in 

Fishman (1993), U.S. Environmental Protection Agency 

(1993; method 365.1), and Patton and Truitt (2000). Total 

organic carbon analysis is documented in Wershaw and others 

(1987). Suspended-sediment samples were analyzed by the 

USGS Sediment Laboratory in Baton Rouge, La. (December 

2005 – September 2007), or Louisville, Ky. (October 2007–9), 

by using procedures described in Guy (1969) and Mathes 

and others (1992). Atrazine samples were analyzed by the 

USGS Organic Geochemistry Research Laboratory, Lawrence, 

Kansas, by using the Enzyme-Linked Immunosorbent Assay 

(ELISA) method documented in Aga and Thurman (1997). E. 

coli and total coliform bacteria were analyzed at the Houston 

Lab at Shenandoah, Tex., within the Gulf Coast Program 

Office of the USGS Texas Water Science Center by using the 

defined substrate method documented in American Public 

Health Association and others (2005) and were reported as 

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

confidence intervals. 

Summary statistics of the discrete water-quality samples 

are summarized in table 2. The data for the Spring Creek 

and East Fork San Jacinto River sites are stored in the USGS 

NWIS database and can be publicly accessed online (U.S. 

Geological Survey, 2011).
6 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9 Introduction 7

Nonexceedance probability

Streamflow, in cubic feet per second

0.1 1 2 5 10 20 30 50 70 80 90 95 98 99 99.9

10

20

30

50

70

100

200

300

500

700

1,000

2,000

3,000

5,000

7,000

10,000

Flow-duration curve of daily mean streamflow for October 1998–September 2010

Concurrent streamflow with water-quality sample for December 2005–October 2009

EXPLANATION

Nonexceedance probability

Streamflow, in cubic feet per second

0.1 1 2 5 10 20 30 50 70 80 90 95 98 99 99.9

10

20

30

50

70

100

200

300

500

700

1,000

2,000

3,000

5,000

7,000

10,000

Flow-duration curve of daily mean streamflow for October 1998–September 2010

Concurrent streamflow with water-quality sample for December 2005–October 2009

EXPLANATION

A

B

Figure 2. Flow duration curve and corresponding discrete water-quality samples, (A) Spring Creek near Spring, Tex., and 

(B) East Fork San Jacinto River near New Caney, Tex.

py

Flow-duration curve of daily mean streamflow for October 1998–September 2010

Concurrent streamflow with water-quality sample for December 2005–October 2009

EXPLANATION

Nonexceedance probability

Streamflow, in cubic feet per second

0.1 1 2 5 10 20 30 50 70 80 90 95 98 99 99.9

10

20

30

50

70

100

200

300

500

700

1,000

2,000

3,000

5,000

7,000

10,000

Flow-duration curve of daily mean streamflow for October 1998–September 2010

Concurrent streamflow with water-quality sample for December 2005–October 2009

EXPLANATION

B

Figure 2. Flow duration curve and corresponding discrete water-quality samples, (A) Spring Creek near Spring, Tex., and 

(B) East Fork San Jacinto River near New Caney, Tex.

Figure 2. Flow duration curve and corresponding discrete water-quality samples, (A) Spring Creek near Spring, Texas, and (B) East Fork 

San Jacinto River near New Caney, Tex. 

Table 2. Summary statistics for samples collected from two tributaries to Lake Houston near Houston, Texas, 2005–9.

[n, number of samples; <, less than1; E, estimated1; bold values indicate change in summary statistic from Oden and others (2009) report]

U.S. 

Geological

Survey

streamflow-gaging

station name

U.S.

Geological

Survey streamflow-gaging 

station number

Summary 

statistic

Ammonia plus

organic

nitrogen,

water, filtered

(milligrams

per liter

as nitrogen)

Ammonia plus

organic

nitrogen, water,

unfiltered

(milligrams

per liter

as nitrogen)

Ammonia, 

water,

filtered

(milligrams per

liter as

nitrogen)

Nitrite plus

nitrate,

water, filtered

(milligrams

per liter as

nitrogen)

Nitrite,

water,

filtered

(milligrams

per liter as

nitrogen)

Orthophosphate,

water,

filtered

(milligrams

per liter as

phosphorus)

Phosphorus,

water,

filtered

(milligrams 

per liter)

Phosphorus,

water,

unfiltered

(milligrams

per liter)

Escherichia

coli,

Colilert Quantitray 

method, water 

(most probable 

number per 100

milliliters)

Total

coliform,

Colilert

Quantitray 

method,

water (most 

probable

number per

100 milliliters)

Atrazine,

water, filtered, 

recoverable,

immunoassay,

unadjusted

(micrograms

per liter)

Organic

carbon, 

water,

unfiltered

(milligrams

per liter)

Suspended

sediment

(milligrams

per liter)

Spring Creek near 

Spring, Tex.

08068500 Minimum 0.48 0.64 <0.01 0.11 0.005 0.06 0.08 0.15 20.00 2,420 <0.1 7.51 15

Maximum 1.25 2.28 0.41 8.08 0.24 2.25 2.12 2.29 41,000 1,300,000 14 21.00 987

Median 0.82 1.21 0.06 2.12 0.037 0.67 0.67 0.86 300 38,800 0.80 11.53 44.5

Number of 

Samples

57 57 57 57 57 57 57 57 56 56 57 55 58

East Fork San

Jacinto River 

near New Caney, 

Tex.

08070200 Minimum 0.14 0.23 <0.01 <0.026 <0.00107 <0.00432 0.013 0.054 13.20 1,000 <0.1 3.97 6

Maximum 0.74 1.29 0.05 0.51 0.03 0.059 0.075 0.25 43,500 260,250 0.53 32.64 362

Median 0.37 0.43 E0.02 0.10 0.003 0.02 0.034 0.098 108 9,200 <0.1 7.77 25

Number of 

Samples

49 51 51 51 51 51 51 51 49 49 51 50 49

1

Concentrations measured as less than the long-term method detection level (LT-MDL) are reported as less than the laboratory reporting level (LRL).

Concentrations measured between the LT-MDL and LRL are reported but given an “E” remark code to indicate that they are semiquantitative

(Childress and others, 1999).
8 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

Quality Control

Quality-control (QC) samples were collected as described 

in the USGS “National Field Manual for the Collection 

of Water-Quality Data” (variously dated) and analyzed by 

the same laboratories and methods as the environmental 

samples. QC samples included equipment blanks, field blanks, 

and split replicate samples. QC samples were collected to 

evaluate any contamination as well as bias and variability 

of the water chemistry data that might have resulted from 

sample collection, processing, transportation, and laboratory 

analysis. Equipment blanks were collected about annually in a 

controlled environment to determine if the cleaning procedures 

for sample containers and the equipment for sample collection 

and sample processing were sufficient to produce contaminantfree samples. Field blanks were collected and processed at 

sampling sites prior to the collection of environmental samples 

Split replicate samples (referred to as replicate samples in this 

report) were collected and are prepared by dividing a single 

volume of water into multiple samples to provide a measure 

of the variability of sample processing and analysis. Replicate 

samples were compared to the associated environmental 

samples by calculating the relative percent difference (RPD) 

for each constituent. RPD was computed by using the equation 

 RPD = |C1 – C2| / ((C1 + C2)/2) x 100 (1)

where

C1 = concentration from environmental sample, and

C2 = concentration from replicate sample.

RPDs of 10 percent or less indicate good agreement 

between analytical results if the concentrations are sufficiently 

greater compared to the laboratory reporting level. The RPD 

exceeded 10 percent for 8 of the 80 sample pairs of nutrients, 

6 of 10 for total organic carbon, 3 of 11 for atrazine, and 6 of 

11 for suspended sediment. The RPD exceeded 10 percent 

primarily when constituent concentrations were at or near the 

LRL so that small variability in analysis caused large RPDs. 

Cases for which the analyte in both of the samples either 

was not detected or was detected at a concentration less than 

the LRL were defined as in agreement. Cases for which the 

analyte was detected at a concentration equal to or greater than 

the LRL in one of the samples and not detected or detected 

at a concentration less than the LRL were defined as being 

in nonagreement. This distinction was necessary because of 

a few instances in which an analyte was not detected or was 

detected at a concentration less than the LRL.

The 19 bacteriological replicate samples were analyzed 

in the same manner as the environmental samples. The 

acceptable RPD for bacteriological replicate samples was 

set at 30 percent. The Colilert method used for E. coli and 

total coliform allows the simultaneous detection of E. coli 

and total coliform and is reported as most probable number. 

Most probable number analyses result in a statistical estimate 

of the original number of cells in a known volume of water; 

results are reported with a 95-percent confidence interval 

and upper and lower confidence intervals (Stoeckel and 

others, 2005). The RPD exceeded 30 percent for 8 of 38 

bacteriological sample pairs. Mean and median RPDs for all 

bacteriological samples, however, were 19 percent and 15 

percent, respectively. The confidence intervals for the eight 

replicate samples with RPDs exceeding 30 percent overlapped, 

indicating there were no statistically significant differences 

between replicate samples. Additionally, seven of the eight 

replicate samples with RPDs that exceeded 30 percent 

originated from the Spring Creek site. Environmental sample 

and associated QC sample pair results are listed in appendix 1.

Regression Model Development

The R environment for statistical computing (R 

Development Core Team, 2010) was used to develop 

algebraically representable, multiple-linear regression 

equations (Faraway, 2005; Helsel and Hirsch, 2002; 

Helsel, 2005; Maindonald and Braun, 2003) to estimate 

concentrations for selected water-quality constituents and 

estimate prediction intervals or quantification of uncertainty. 

The authors used an open-source computational environment 

(the R environment for statistical computing) because the code 

syntax in this environment accommodates the syntax required 

for mathematical operations, and because the R environment is 

available free of charge to readers and is available for different 

computer platforms. 

The regression models reported here are based on 

selected statistical relations between the constituent 

concentration acquired during discrete sampling and 

contemporaneous values of predictor variables that normally 

are measured continuously at a particular site (East Fork 

San Jacinto River or Spring Creek). Each equation provides 

estimates of the concentration of a single constituent as 

opposed to multivariate equations/methods that can provide 

for simultaneous estimation and inference of an ensemble of 

constituents; multivariate techniques are outside of the scope 

of this report. The relation between specific measurements 

and associated regression equations are site specific, and each 

equation is uniquely applicable for a particular site, but the 

methods could be applied to most perennial streams for the 

purpose of estimating real-time water quality data.

The potential predictor variables included streamflow, 

specific conductance, pH, water temperature, turbidity, 

dissolved oxygen, and various trigonometric functions of 

time (days into the year). Time was used when necessary 

to accommodate systematic seasonal variations of some 

constituents. The predictor variables are measured at the 

comparatively frequent sampling rate of every 15 minutes 

compared to the discrete sampling (intervals of every 2 weeks 

or longer) for the constituents. Lastly, the instantaneous 

daily load of a constituent can be estimated by multiplying 

the estimated constituent concentration (mass per volume) 

by contemporaneous streamflow (volume per time), 
Regression Model Development 9

accommodation of requisite unit-conversion factors, and 

applying an appropriate bias correction factor.

The application of multiple-linear regression equations 

to estimate constituent concentration and subsequent load 

estimation is well documented; for example, see Christensen 

and others (2000), Ryberg (2006), and Rasmussen and others 

(2009). Normally distributed response and explanatory 

variables with linear relations and constant variance 

are required for highly reliable equations. Logarithmic 

transformations on the response and explanatory variables 

are commonly used to improve linearity and to mitigate 

for nonnormality and heteroscedasticity (nonconstant 

variance about the regression line) in model residuals 

(Helsel and Hirsch, 2002). Consideration of nonlogarithmic 

transformations have been previously made (Oden and 

others, 2009) with a subset of the data considered in this 

report; however, additional analysis and implementation 

considerations resulted in a decision by the authors to 

exclusively use logarithmic transformation (base-10) 

trigonometric operations. 

The coefficient of determination, R-squared, describes 

the proportion of the total sample variability in the response 

explained by the regression model. The coefficient will only 

increase as additional explanatory variables are added to 

the model, thus it might not be an appropriate criterion for 

determining the usefulness of a model that has numerous 

explanatory variables. The adjusted R-squared statistic 

compensates for this by assessing a “penalty” for the number 

of explanatory variables in the model; adding additional 

explanatory variables increases the value of adjusted 

R-squared only when the predictive capability of the model 

increases. Choosing a model with the highest adjusted 

R-squared value is equivalent to choosing a model with the 

lowest mean standard error (Helsel and Hirsch, 2002). For 

the current (2012) investigation, reader attention is drawn 

to adjusted R-squared values provided with each regression 

figure (figs. 3-14) because the adjusted R-squared can provide 

a more realistic evaluation of the model fit or ability of the 

model to characterize uncertainty in this study (Helsel and 

Hirsch, 2002).

The residual standard errors (RSE) of the regression 

equations reported here are exclusively reported in logarithmic 

units, which is consistent with the application of the 

logarithmic transformation of the response (water-quality 

constituent concentration). The authors use base-10 logarithms 

because these logarithms are conventionally most accessible 

to water-resources managers and the supporting engineering 

community.

After the transformation or transformations for the 

response and explanatory variables are selected, the analysis 

continues with the selection of explanatory variables that 

produce reliable regression models. The preferred regression 

model contains the fewest explanatory variables for which 

model diagnostics (including adjusted R-squared, the residual 

standard error, residual plots, and p-values) are acceptable. 

The p-value represents the probability (ranging from zero to 

one), that the statistical test result could have occurred if the 

null hypothesis (the hypothesis representing no change or 

no difference) was true (Helsel and Hirsch, 2002). When the 

p-value is less than a specified significance level (.05 in our 

application), the null hypothesis is rejected. Preferable models 

are those judged to have an acceptable balance between model 

fit and the number of variables in the model. Variables with 

small statistical significance and (or) substantial variance 

inflation potential are excluded (Helsel and Hirsch, 2002).

Variance inflation factors (VIF) are used to check for 

high collinearity between explanatory variables (Stine, 1995). 

Explanatory variables carrying similar information about the 

response have a high collinearity, and when such variables 

are all included in the model, give rise to increased variance 

in the estimation of the regression coefficients and requisite 

expansion of prediction intervals. A VIF represents the 

increase in variance because of correlation between predictive 

variables, whereas a minimum value of 1 occurs when no 

correlation is present. Typically, VIFs numerically greater than 

10 are a cause of concern and indicate that a poor estimate of 

the associated regression coefficient has been produced by the 

model. Assessment of VIFs was made and no values greater 

than 10 were present for the variables shown in the regression 

equations reported here. 

Graphical analysis is a vital component of regression 

analysis and subsequent interpretation; it facilitates visual 

inspection and verification of data patterns such as linearity 

and constant variance underlying linear regression equation 

theory. The patterns seen in residual plots facilitate judgments 

in model reliability and are used to check if regression 

equations fit the observed data. 

Retransformation Bias Correction

When a water-quality constituent is transformed (that is, 

into logarithmic units) as part of the building of a regression 

equation, the constituent must be retransformed to obtain 

an estimate in the original units. Estimates of constituent 

concentration that are unbiased in the transformed scale 

will be biased upon retransformation to the original scale. 

Retransformation bias corrections are made to mitigate 

or remove bias; the form of the bias correction factor will 

depend on the transformation used in the regression analysis 

(Duan,1983; Helsel and Hirsch, 2002).

Because logarithmic transformation was used for the 

current (2012) investigation, a bias correction is necessary 

because retransformation yields a median estimate of a 

constituent; median estimates tend to underestimate the actual 

arithmetic mean for the data considered here. Simply inverting 

a log-transformed response will return a biased low and 

therefore inconsistent estimate of the arithmetic mean. This 

bias greatly influences how loadings (say in units of tons per 

unit time) are computed. Extensive research has been done by 

others to find estimators that return the expected value (mean) 

of the streamflow load of a constituent if the response was 
10 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

log-transformed, for example Duan (1983), Crawford (1991), 

and Cohn (2005).

A minimum variance unbiased estimator (MVUE) was 

derived by Finney (1941). This estimator adjusts for bias 

and returns an efficient estimate of the mean. The Finney 

estimator is a commonly used and a reliable choice when 

the log-normal model is correct and the residual errors are 

normally distributed. However, the requirement that the 

residuals are normally distributed is an assumption that is 

difficult to achieve, assess, or interpret for the water-quality 

data considered in this report. Nonparametric estimators can 

provide a useful alternative to the retransformation methods. 

Duan (1983) derived a “smearing” estimator that requires only 

the residuals to be independent and homoscedastic (constant 

variance about the regression line). In the case of a logtransformation, the correction factor involves re-expressing 

the residuals in the original units and computing their mean. 

This factor for a given regression equation is to be multiplied 

to the regression equation in circumstances involving the 

computation of loads (concentration multiplied by streamflow 

along with necessary unit conversions).

Analysis of Censored Data

To avoid false-positive quantification of a constituent, 

very low concentrations are censored and reported as a “less 

than” value by the laboratory (Childress and others, 1999). 

This kind of reporting results in what is referred to as leftcensored observations (Helsel, 2005). Another complicating 

feature of water-quality data is that the censored values can 

vary depending on the laboratory reporting level (LRL) at the 

time the analyses were done. Censored values are those less 

than the laboratory reporting level applicable at the time the 

analyses were done. The mathematical theory is thoroughly 

described by Helsel (2005).

For this report, the foundational computational scripts 

were constructed by the authors to auto-adapt to the presence 

of left-censored data values, identify these values, and 

subsequently apply a maximum-likelihood estimator (MLE) 

for regression in lieu of conventional ordinary least-squares 

(OLS) regression. Both regression estimation techniques are 

provided by the R environment for statistical computing and 

standard packages of R; the two specific functions for OLS 

and MLE are lm( ) (a linear regression modeling function) 

and survreg( ) (a survival regression modeling function), 

respectively (R Development Core Team, 2010). The end 

result for the current (2012) investigation is that the basic 

algebraic implementation of the equations, whether produced 

by OLS or MLE regression, will be familiar to water-resources 

managers and the engineering community.

To conclude the broader discussion of censored data, 

an additional remark concerning judgment exercised by the 

authors is needed. Along with censored data, the laboratory 

might report estimated values; these are identified in the 

National Water Information System database with an “E” 

remark code. A constituent concentration is considered 

estimated by the laboratory when results are greater than 

the long-term method detection level (LT–MDL) and less 

than the LRL; that is, a detection is considered likely, but 

numerical quantification is considered questionable. For this 

investigation, all occurrences of “E” were dropped and the 

remaining numerical values used in the regression analysis. 

Additionally, individual samples collected prior to 2008 that 

were deemed contaminated or determined to be substantial 

outliers by Oden and others (2009) also were not used in the 

development of the models reported herein. To further clarify, 

for the current (2012) investigation, the data files used by 

Oden and others (2009) were extended by using data collected 

after 2008. 

Censored Data at Spring Creek and East Fork 

San Jacinto River Site 

The data for the Spring Creek and East Fork San Jacinto 

River sites used in this study are stored in the USGS NWIS 

database and can be publicly accessed online (U.S. Geological 

Survey, 2011). Only one of the atrazine concentrations 

measured in environmental samples collected at the Spring 

Creek site was less than the LRL of 0.10 mg/L. In contrast, 

most of the atrazine concentrations measured from East 

Fork San Jacinto River site were less than the LRL. Because 

of the large amount of censored atrazine data in discrete 

environmental samples collected at the East Fork San Jacinto 

River site, a defensible multilinear regression equation to 

estimate atrazine concentrations could not be developed. 

None of the nitrite plus nitrate concentrations measured in 

environmental samples collected at the Spring Creek site was 

less than the LRL of 0.06 mg/L, whereas four of the nitrite 

plus nitrate concentration measured in samples collected from 

the East Fork San Jacinto River site were less than the LRL. 

Regression Analysis Summaries and 

Presentation of Equations

Summaries of the developed regression equations are 

provided in figures 3–14. Figure 3 shows abbreviations 

of terms used in figures 4-14. Each figure encapsulates an 

individual constituent per location and provides the significant 

explanatory variables used in the model, diagnostics factors 

used as indicators of general model reliability (including 

adjusted R-squared, the residual standard error, residual plots, 

and p-values), summary statistics for the explanatory variables 

and calculated constituent, the algebraic representation of the 

resultant model, and additional statistical parameters required 

for calculating prediction intervals, which were computed to 

display the uncertainty associated with the estimate (Helsel 

and Hirsch, 2002).

Regression equations for Spring Creek were developed 

for all constituents (table 2) analyzed for the study. A 

preliminary assessment (not reported here) of total ammonia 

plus organic nitrogen, dissolved ammonia plus organic 
Regression Model Development 11

nitrogen, ammonia nitrogen, nitrite nitrogen, orthophosphate 

phosphorus, dissolved phosphorus, and total coliform 

bacteria did not result in reliable equations because of large 

residual standard errors, and other unsatisfactory results from 

regression equation diagnostics. The regression equations 

developed for nitrite plus nitrate nitrogen, total phosphorus, 

total organic carbon, E. coli bacteria, atrazine, and suspended 

sediment are described in figures 4–9. Adjusted R-squared 

values for the Spring Creek models ranged from .582–.922 

(dimensionless). The residual standard errors ranged from 

.073–.447 (base-10 logarithm). 

Regression equations for East Fork San Jacinto River 

were developed for all constituents analyzed for the study 

(table 2), except atrazine. A preliminary assessment (not 

reported here) of total ammonia plus organic nitrogen, 

dissolved ammonia plus organic nitrogen, ammonia nitrogen, 

nitrite nitrogen, orthophosphate phosphorus, dissolved 

phosphorus, and total coliform bacteria did not result in 

reliable equations because of large residual standard errors, 

and other unsatisfactory results from regression equation 

diagnostics. Furthermore, an atrazine regression equation was 

not developed for the study because more than 50 percent 

of the data were below the LRL. The regression equations 

developed for nitrite plus nitrate nitrogen, total phosphorus, 

organic carbon, E. coli bacteria, and suspended sediment 

are described in figures 10–14. Adjusted R-squared values 

for the East Fork San Jacinto River models ranged from 

.253–.853(dimensionless). The residual standard errors ranged 

from .076–.388 (base-10 logarithm). 

ABBREVIATIONS OF MATHEMATICAL FUNCTIONS AND STATISTICAL TERMS SHOWN IN

FIGURES 4 THROUGH 14

Summary Statistics and Miscellaneous

Min. Minimum

1st Qu. First quartile

3rd Qu. Third quartile

Max. Maximum

log10() Base -10 logarithm

cos2piD The cosine of the cosine of the products 2 * pi * (Date)

sin2piD The sine of the sine of the products 2 * pi * (Date)

cos4piD The cosine of the cosine of the products 4 * pi * (Date)

sin4piD The sine of the sine of the products 4 * pi * (Date)

pi A mathematical constant approximately equal to 3.1415.

Date Julian days into year divided by 365.25

Regression Model , linear model (ordinary least squares for uncensored data)

lm() Linear regression modeling function

Std.Error Standard error

t-value T-statistic for the t-test

Pr(>|t|) Probability of absolute value of t-value

Signif.codes Default textual flags related to parameter significance

R-squared Coefficient of determination

F-statistic A statistic for the F-test

DF Degrees of freedom

p-value p-value (a standard computed probability)

Regression Model , survival model (for censored data)

survreg() Survival regression modeling function

dist = "guassian" The normal distribution is used for model estimation

Std.Error Standard error

z Standard normal z-statistic for survreg()

p-value p-value (a standard computed probability)

Scale Equivalent to residual standard error in transformed units

Loglik() Log -likelihood (a statistic for method of maximum likelihood)

Chisq A statistic for the Chi -squared distribution

Figure 3. Abbreviations of mathematical functions and statistical terms related to regression analysis of water-quality constituents shown in figures 4–14.

Figure 3. Abbreviations of mathematical functions and statistical terms related to regression analysis of water-quality 

constituents shown in figures 4-14.
12 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

Inflow Statistics of Applicable Explanatory Variables: [Min., minimum, Qu., quartile, Max., maximum]

EXPLANATORY VARIABLE SUMMARY STATISTICS

log10(Q) log10(SC)

Min. :1.230 Min. :1.806

1st Qu.:1.580 1st Qu.:2.413

Median :1.914 Median :2.562

Mean :2.112 Mean :2.521

3rd Qu.:2.346 3rd Qu.:2.680

Max. :3.867 Max. :2.847

Summary of Regression Analysis for the Constituent of:

nitrite plus nitrate (NO2 NO3)

SUMMARY STATISTICS FOR NO2NO3 , IN LOG10() milligrams per liter

Min. 1st Qu. Median Mean 3rd Qu. Max.

-0.9586 0.1239 0.3263 0.2678 0.6031 0.9074

REGRESSION EQUATION

lm(formula = log10(NO2NO3) ~ log10(Q) + log10(SC), data = the.data)

Residuals:

Min 1st Qu. Median 3rd Qu. Max

-0.42257 -0.04910 0.02540 0.06092 0.32012

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) -3.38330 0.59894 -5.649 6.20e-07 ***

log10(Q) -0.10225 0.06142 -1.665 0.102

log10(SC) 1.53375 0.18900 8.115 6.44e-11 ***

---

Signif.codes: 0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

Residual standard error: 0.1241 on 54 degrees of freedom

Multiple R-squared: 0.9245, Adjusted R-squared: 0.9217

F-statistic: 330.7 on 2 and 54 DF , p-value: < 2.2e-16

Variance inflation factors:

log10(Q) log10(SC)

7.0611 7.0611

Duan (1983) smearing factor: 1.037 = mean(10^[ residuals(model)])

(Detransformed estimates are to be multiplied this factor.)

Nomenclature (all potential variables)

• Q is streamflow, in cubic feet per second;

• pH is pH, in standard units;

• SC is specific conductance, in microsiemens

per centimeter at 25° Celsius;

• Turb is turbidity in Formazine Nephelometric Units;

• Temp is water temperature, in ° Celsius;

• Date is Julian day DOY (days into year)

divided by 365.25; and

• log10(x) is base-10 logarithm of x.

Algebraic Equation

log10(NO2NO3) =−3.3833

−0.1023log10(Q)

+1.5338log10(SC)

Residual Plot for Regression

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0.1 0.2 0.5 1.0 2.0 5.0

−0.4 −0.2 0.0 0.2 0.4

Estimated nitrite plus nitrate,

in milligrams per liter

Residuals

CORRELATION OF COEFFICIENTS NO2NO3

(Intercept) log10(Q) log10(SC)

(Intercept) 1 -0.9537 -0.9963

log10(Q) -0.9537 1 0.9265

log10(SC) -0.9963 0.9265 1

EXPLANATION

● CONSTITUENT FOR STREAMFLOW-GAGING STATION

ORIGIN LINE

Comprehensive descriptions of abbreviations for the

computer output related to the regression analysis

documented in this figure are shown in figure 3.

Figure 4. Summary of regression analysis for nitrite plus nitrate (NO2 NO3) for 08068500 Spring Creek near Spring, Tex. based

on data as described in text for period 2005–09.

Figure 4. Summary of regression analysis for nitrite plus nitrate nitrogen for U.S. Geological Survey streamflow-gaging station 

08068500 Spring Creek near Spring, Texas, 2005–9.
Regression Model Development 13

Inflow Statistics of Applicable Explanatory Variables: [Min., minimum, Qu., quartile, Max., maximum]

EXPLANATORY VARIABLE SUMMARY STATISTICS

log10(SC) log10(Turb) sin2piD cos2piD

Min. :1.806 Min. :0.9956 Min. :-0.999761 Min. :-0.9976

1st Qu.:2.411 1st Qu.:1.2729 1st Qu.: -0.736086 1st Qu.:-0.6996

Median :2.565 Median :1.5796 Median :-0.175941 Median :-0.2322

Mean :2.522 Mean :1.6364 Mean : 0.005503 Mean :-0.1095

3rd Qu.:2.684 3rd Qu.:1.9303 3rd Qu.: 0.793275 3rd Qu.: 0.4928

Max. :2.847 Max. :2.4771 Max. : 0.999091 Max. : 0.9839

Summary of Regression Analysis for the Constituent of:

total phosphorus (Phos)

SUMMARY STATISTICS FOR PHOS , IN LOG10() milligrams per liter

Min. 1st Qu. Median Mean 3rd Qu. Max.

-0.82390 -0.30550 -0.06299 -0.12350 0.05011 0.35980

REGRESSION EQUATION

lm(formula = log10(PHOS) ~ log10(SC) + log10(Turb) + sin2piD +

cos2piD , data = the.data)

Residuals:

Min 1st Qu. Median 3rd Qu. Max

-0.24310 -0.05750 -0.00554 0.05387 0.22659

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) -3.58078 0.26045 -13.749 < 2e-16 ***

log10(SC) 1.25921 0.07907 15.926 < 2e-16 ***

log10(Turb) 0.16780 0.04699 3.571 0.000786 ***

sin2piD -0.02472 0.01722 -1.436 0.157185

cos2piD -0.06272 0.01973 -3.178 0.002519 **

---

Signif.codes: 0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

Residual standard error: 0.09536 on 51 degrees of freedom

Multiple R-squared: 0.8838, Adjusted R-squared: 0.8747

F-statistic: 96.96 on 4 and 51 DF , p-value: < 2.2e-16

Variance inflation factors:

log10(SC) log10(Turb) sin2piD cos2piD

2.0915 2.1556 1.0197 1.0297

Duan (1983) smearing factor: 1.0222 = mean(10^[ residuals(model)])

(Detransformed estimates are to be multiplied this factor.)

Nomenclature (all potential variables)

• Q is streamflow, in cubic feet per second;

• pH is pH, in standard units;

• SC is specific conductance, in microsiemens

per centimeter at 25° Celsius;

• Turb is turbidity in Formazine Nephelometric Units;

• Temp is water temperature, in ° Celsius;

• Date is Julian day DOY (days into year)

divided by 365.25; and

• log10(x) is base-10 logarithm of x.

Algebraic Equation

log10(Phos) =−3.5808

+1.2592log10(SC)

+0.1678log10(Turb)

−0.0247 sin[2π(Date)]

−0.0627 cos[2π(Date)]

Residual Plot for Regression

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−0.2 −0.1 0.0 0.1 0.2

Estimated total phosphorus,

in milligrams per liter

Residuals

CORRELATION OF COEFFICIENTS PHOS

(Intercept) log10(SC) log10(Turb) sin2piD cos2piD

(Intercept) 1 -0.9777 -0.8447 -0.0966 -0.0567

log10(SC) -0.9777 1 0.7190 0.0750 0.0324

log10(Turb) -0.8447 0.7190 1 0.1310 0.1361

sin2piD -0.0966 0.0750 0.1310 1 -0.0164

cos2piD -0.0567 0.0324 0.1361 -0.0164 1

EXPLANATION

● CONSTITUENT FOR STREAMFLOW-GAGING STATION

ORIGIN LINE

Comprehensive descriptions of abbreviations for the

computer output related to the regression analysis

documented in this figure are shown in figure 3.

Figure 5. Summary of regression analysis for total phosphorus (Phos) for 08068500 Spring Creek near Spring, Tex. based on

data as described in text for period 2005–09.

Figure 5. Summary of regression analysis for total phosphorus for U.S. Geological Survey streamflow-gaging station 08068500 Spring 

Creek near Spring, Texas, 2005–9.
14 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

Inflow Statistics of Applicable Explanatory Variables: [Min., minimum, Qu., quartile, Max., maximum]

EXPLANATORY VARIABLE SUMMARY STATISTICS

log10(Q) log10(Turb)

Min. :1.230 Min. :0.9956

1st Qu.:1.580 1st Qu.:1.2899

Median :1.914 Median :1.5911

Mean :2.115 Mean :1.6457

3rd Qu.:2.349 3rd Qu.:1.9414

Max. :3.867 Max. :2.4771

Summary of Regression Analysis for the Constituent of:

total organic carbon (OrgC)

SUMMARY STATISTICS FOR ORGC , IN LOG10() milligrams per liter

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.8751 0.9800 1.0610 1.0640 1.1480 1.3220

REGRESSION EQUATION

lm(formula = log10(ORGC) ~ log10(Q) + log10(Turb), data = the.data)

Residuals:

Min 1st Qu. Median 3rd Qu. Max

-0.11713 -0.05173 -0.01510 0.03307 0.18542

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) 0.71382 0.04244 16.819 < 2e-16 ***

log10(Q) 0.02522 0.02233 1.130 0.264

log10(Turb) 0.18017 0.04054 4.444 4.65e-05 ***

---

Signif.codes: 0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

Residual standard error: 0.07333 on 52 degrees of freedom

Multiple R-squared: 0.597, Adjusted R-squared: 0.5815

F-statistic: 38.51 on 2 and 52 DF , p-value: 5.479e-11

Variance inflation factors:

log10(Q) log10(Turb)

2.6598 2.6598

Duan (1983) smearing factor: 1.0142 = mean(10^[ residuals(model)])

(Detransformed estimates are to be multiplied this factor.)

Nomenclature (all potential variables)

• Q is streamflow, in cubic feet per second;

• pH is pH, in standard units;

• SC is specific conductance, in microsiemens

per centimeter at 25° Celsius;

• Turb is turbidity in Formazine Nephelometric Units;

• Temp is water temperature, in ° Celsius;

• Date is Julian day DOY (days into year)

divided by 365.25; and

• log10(x) is base-10 logarithm of x.

Algebraic Equation

log10(OrgC) = 0.7138

+0.0252log10(Q)

+0.1802log10(Turb)

Residual Plot for Regression

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−0.2 −0.1 0.0 0.1 0.2

Estimated total organic carbon,

in milligrams per liter

Residuals

CORRELATION OF COEFFICIENTS ORGC

(Intercept) log10(Q) log10(Turb)

(Intercept) 1 0.1291 -0.693

log10(Q) 0.1291 1 -0.790

log10(Turb) -0.6930 -0.7900 1.000

EXPLANATION

● CONSTITUENT FOR STREAMFLOW-GAGING STATION

ORIGIN LINE

Comprehensive descriptions of abbreviations for the

computer output related to the regression analysis

documented in this figure are shown in figure 3.

Figure 6. Summary of regression analysis for total organic carbon (OrgC) for 08068500 Spring Creek near Spring, Tex. based on

data as described in text for period 2005–09.

Figure 6. Summary of regression analysis for total organic carbon for U.S. Geological Survey streamflow-gaging station 08068500 

Spring Creek near Spring, Texas, 2005–9.
Regression Model Development 15

Inflow Statistics of Applicable Explanatory Variables: [Min., minimum, Qu., quartile, Max., maximum]

EXPLANATORY VARIABLE SUMMARY STATISTICS

log10(Q) log10(Turb)

Min. :1.230 Min. :0.9956

1st Qu.:1.564 1st Qu.:1.2553

Median :1.860 Median :1.5623

Mean :2.079 Mean :1.6230

3rd Qu.:2.283 3rd Qu.:1.9303

Max. :3.867 Max. :2.4771

Summary of Regression Analysis for the Constituent of:

Escherichia coli (ECB)

SUMMARY STATISTICS FOR ECB , IN LOG10() most probable number per 100

milliliters

Min. 1st Qu. Median Mean 3rd Qu. Max.

1.301 1.995 2.477 2.701 3.519 4.613

REGRESSION EQUATION

lm(formula = log10(ECB) ~ log10(Q) + log10(Turb), data = the.data)

Residuals:

Min 1st Qu. Median 3rd Qu. Max

-0.73223 -0.33382 -0.06325 0.22911 1.21151

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) -0.1913 0.2512 -0.762 0.449533

log10(Q) 0.4969 0.1394 3.564 0.000782 ***

log10(Turb) 1.1458 0.2468 4.642 2.31e-05 ***

---

Signif.codes: 0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

Residual standard error: 0.4468 on 53 degrees of freedom

Multiple R-squared: 0.7578, Adjusted R-squared: 0.7487

F-statistic: 82.92 on 2 and 53 DF , p-value: < 2.2e-16

Variance inflation factors:

log10(Q) log10(Turb)

2.7361 2.7361

Duan (1983) smearing factor: 1.8071 = mean(10^[ residuals(model)])

(Detransformed estimates are to be multiplied this factor.)

Nomenclature (all potential variables)

• Q is streamflow, in cubic feet per second;

• pH is pH, in standard units;

• SC is specific conductance, in microsiemens

per centimeter at 25° Celsius;

• Turb is turbidity in Formazine Nephelometric Units;

• Temp is water temperature, in ° Celsius;

• Date is Julian day DOY (days into year)

divided by 365.25; and

• log10(x) is base-10 logarithm of x.

Algebraic Equation

log10(ECB) =−0.1913

+0.4969log10(Q)

+1.1458log10(Turb)

Residual Plot for Regression

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−1.0 −0.5 0.0 0.5 1.0

Estimated Escherichia coli bacteria,

in most probable number per 100 milliliters

Residuals

CORRELATION OF COEFFICIENTS ECB

(Intercept) log10(Q) log10(Turb)

(Intercept) 1 0.1167 -0.6759

log10(Q) 0.1167 1 -0.7966

log10(Turb) -0.6759 -0.7966 1

EXPLANATION

● CONSTITUENT FOR STREAMFLOW-GAGING STATION

ORIGIN LINE

Comprehensive descriptions of abbreviations for the

computer output related to the regression analysis

documented in this figure are shown in figure 3.

Figure 7. Summary of regression analysis for Escherichia coli (ECB) for 08068500 Spring Creek near Spring, Tex. based on data

as described in text for period 2005–09.

Figure 7. Summary of regression analysis for Escherichia coli for U.S. Geological Survey streamflow-gaging station 08068500 Spring 

Creek near Spring, Texas, 2005–9.
16 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

Inflow Statistics of Applicable Explanatory Variables: [Min., minimum, Qu., quartile, Max., maximum]

EXPLANATORY VARIABLE SUMMARY STATISTICS

log10(Q) log10(Turb) sin2piD cos2piD sin4piD cos4piD

Min. :1.230 Min. :0.9956 Min. :-0.99976 Min. :-0.9976 Min. :-0.99901 Min. :-0.99904

1st Qu.:1.580 1st Qu.:1.2611 1st Qu.: -0.71865 1st Qu.:-0.7164 1st Qu.: -0.61393 1st Qu.: -0.71440

Median :1.894 Median :1.5796 Median :-0.05209 Median :-0.2322 Median :-0.02074 Median :-0.14635

Mean :2.101 Mean :1.6314 Mean : 0.01822 Mean :-0.1065 Mean : 0.03735 Mean :-0.08866

3rd Qu.:2.342 3rd Qu.:1.9191 3rd Qu.: 0.77246 3rd Qu.: 0.5273 3rd Qu.: 0.74837 3rd Qu.: 0.57287

Max. :3.867 Max. :2.4771 Max. : 0.99909 Max. : 0.9839 Max. : 0.99948 Max. : 0.99045

Summary of Regression Analysis for the Constituent of:

Atrazine (Atz)

SUMMARY STATISTICS FOR ATZ , IN LOG10() milligrams per liter

The variable as "survival time"

Min. 1st Qu. Median Mean 3rd Qu. Max.

-1.00000 -0.28830 -0.09971 -0.03906 0.24180 1.14600

The variable as "survival status"

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.0000 1.0000 1.0000 0.9828 1.0000 1.0000

REGRESSION EQUATION

survreg(formula = log10(ATZ) ~ log10(Q) + log10(Turb) + sin2piD +

cos2piD + sin4piD + cos4piD , data = the.data , dist = "gaussian")

Value Std.Error z p-value

(Intercept) -0.8096 0.1537 -5.266 1.40e-07

log10(Q) -0.2253 0.0760 -2.966 3.02e-03

log10(Turb) 0.7473 0.1359 5.498 3.84e-08

sin2piD 0.4027 0.0451 8.932 4.20e-19

cos2piD -0.0718 0.0489 -1.469 1.42e-01

sin4piD -0.0105 0.0497 -0.212 8.32e-01

cos4piD -0.2353 0.0486 -4.847 1.25e-06

Log(scale) -1.4200 0.0935 -15.181 4.70e-52

Scale= 0.242

Gaussian distribution

Loglik(model)= 0 Loglik(intercept only)= -35.6

Chisq= 71.35 on 6 degrees of freedom , p-value: 2.2e-13

Number of Newton -Raphson Iterations: 5

Sample size: 58

Variance inflation factors are not computable for a survival

regression using the DAAG:::vif() function..

McFadden (1974) R-squared: 1.001 and adjusted R-squared: 0.8046

Bias correction factor: 1.0696 = 10^[(Scale * Scale)/2]

(Detransformed estimates are to be multiplied this factor.)

Nomenclature (all potential variables)

• Q is streamflow, in cubic feet per second;

• pH is pH, in standard units;

• SC is specific conductance, in microsiemens

per centimeter at 25° Celsius;

• Turb is turbidity in Formazine Nephelometric Units;

• Temp is water temperature, in ° Celsius;

• Date is Julian day DOY (days into year)

divided by 365.25; and

• log10(x) is base-10 logarithm of x.

Algebraic Equation

log10(Atz) =−0.8096

−0.2253log10(Q)

+0.7473log10(Turb)

+0.4027 sin[2π(Date)]

−0.0718 cos[2π(Date)]

−0.0105 sin[4π(Date)]

−0.2353 cos[4π(Date)]

Residual Plot for Regression

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−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

Estimated Atrazine,

in milligrams per liter

Residuals

CORRELATION OF COEFFICIENTS ATZ

(Intercept) log10(Q) log10(Turb) sin2piD cos2piD sin4piD cos4piD

(Intercept) 1 0.0223 -0.6761 -0.2079 -0.1187 -0.4198 -0.0427

log10(Q) 0.0223 1 -0.7376 0.0807 -0.0864 0.1763 0.1573

log10(Turb) -0.6761 -0.7376 1 0.0791 0.1681 0.1588 -0.0654

sin2piD -0.2079 0.0807 0.0791 1 0.0031 0.1904 0.0994

cos2piD -0.1187 -0.0864 0.1681 0.0031 1 0.0582 0.1181

sin4piD -0.4198 0.1763 0.1588 0.1904 0.0582 1 0.1052

cos4piD -0.0427 0.1573 -0.0654 0.0994 0.1181 0.1052 1

EXPLANATION

● CONSTITUENT FOR STREAMFLOW-GAGING STATION

ORIGIN LINE

Comprehensive descriptions of abbreviations for the

computer output related to the regression analysis

documented in this figure are shown in figure 3.

Figure 8. Summary of regression analysis for Atrazine (Atz) for 08068500 Spring Creek near Spring, Tex. based on data as

described in text for period 2005–09. Figure 8. Summary of regression analysis for atrazine for U.S. Geological Survey streamflow-gaging station 08068500 Spring Creek 

near Spring, Texas, 2005–9.
Regression Model Development 17

Inflow Statistics of Applicable Explanatory Variables: [Min., minimum, Qu., quartile, Max., maximum]

EXPLANATORY VARIABLE SUMMARY STATISTICS

log10(Q) log10(Turb)

Min. :1.230 Min. :0.9956

1st Qu.:1.580 1st Qu.:1.2843

Median :1.919 Median :1.5966

Mean :2.135 Mean :1.6441

3rd Qu.:2.351 3rd Qu.:1.9191

Max. :3.867 Max. :2.4771

Summary of Regression Analysis for the Constituent of:

suspended sediment (SS)

SUMMARY STATISTICS FOR SS , IN LOG10() milligrams per liter

Min. 1st Qu. Median Mean 3rd Qu. Max.

1.176 1.535 1.648 1.867 2.128 2.994

REGRESSION EQUATION

lm(formula = log10(SS) ~ log10(Q) + log10(Turb), data = the.data)

Residuals:

Min 1st Qu. Median 3rd Qu. Max

-0.41044 -0.14107 -0.00053 0.15352 0.29726

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) 0.23531 0.10339 2.276 0.0268 *

log10(Q) 0.42817 0.05009 8.548 1.13e-11 ***

log10(Turb) 0.43664 0.09587 4.554 2.96e-05 ***

---

Signif.codes: 0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

Residual standard error: 0.179 on 55 degrees of freedom

Multiple R-squared: 0.8755, Adjusted R-squared: 0.871

F-statistic: 193.4 on 2 and 55 DF , p-value: < 2.2e-16

Variance inflation factors:

log10(Q) log10(Turb)

2.5089 2.5089

Duan (1983) smearing factor: 1.0801 = mean(10^[ residuals(model)])

(Detransformed estimates are to be multiplied this factor.)

Nomenclature (all potential variables)

• Q is streamflow, in cubic feet per second;

• pH is pH, in standard units;

• SC is specific conductance, in microsiemens

per centimeter at 25° Celsius;

• Turb is turbidity in Formazine Nephelometric Units;

• Temp is water temperature, in ° Celsius;

• Date is Julian day DOY (days into year)

divided by 365.25; and

• log10(x) is base-10 logarithm of x.

Algebraic Equation

log10(SS) = 0.2353

+0.4282log10(Q)

+0.4366log10(Turb)

Residual Plot for Regression

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Estimated suspended sediment,

in milligrams per liter

Residuals

CORRELATION OF COEFFICIENTS SS

(Intercept) log10(Q) log10(Turb)

(Intercept) 1 0.1478 -0.7223

log10(Q) 0.1478 1 -0.7755

log10(Turb) -0.7223 -0.7755 1

EXPLANATION

● CONSTITUENT FOR STREAMFLOW-GAGING STATION

ORIGIN LINE

Comprehensive descriptions of abbreviations for the

computer output related to the regression analysis

documented in this figure are shown in figure 3.

Figure 9. Summary of regression analysis for suspended sediment (SS) for 08068500 Spring Creek near Spring, Tex. based on

data as described in text for period 2005–09. Figure 9. Summary of regression analysis for suspended sediment for U.S. Geological Survey streamflow-gaging station 08068500 

Spring Creek near Spring, Texas, 2005–9.
18 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

Inflow Statistics of Applicable Explanatory Variables: [Min., minimum, Qu., quartile, Max., maximum]

EXPLANATORY VARIABLE SUMMARY STATISTICS

log10(Q) sin2piD cos2piD

Min. :1.041 Min. :-0.99894 Min. :-0.99777

1st Qu.:1.568 1st Qu.: -0.71442 1st Qu.: -0.64457

Median :1.903 Median : 0.11502 Median : 0.04604

Mean :2.086 Mean : 0.03078 Mean : 0.03567

3rd Qu.:2.474 3rd Qu.: 0.77167 3rd Qu.: 0.65656

Max. :3.899 Max. : 0.99998 Max. : 0.99336

Summary of Regression Analysis for the Constituent of:

nitrite plus nitrate (NO2 NO3)

SUMMARY STATISTICS FOR NO2NO3 , IN LOG10() milligrams per liter

The variable as "survival time"

Min. 1st Qu. Median Mean 3rd Qu. Max.

-1.5230 -1.2220 -1.0000 -0.9825 -0.7959 -0.2924

The variable as "survival status"

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.0000 1.0000 1.0000 0.9216 1.0000 1.0000

REGRESSION EQUATION

survreg(formula = log10(NO2NO3) ~ log10(Q) + sin2piD + cos2piD ,

data = the.data , dist = "gaussian")

Value Std.Error z p-value

(Intercept) -0.8938 0.1251 -7.144 9.05e-13

log10(Q) -0.0288 0.0596 -0.483 6.29e-01

sin2piD 0.0289 0.0504 0.575 5.66e-01

cos2piD -0.1911 0.0590 -3.241 1.19e-03

Log(scale) -1.3788 0.1030 -13.390 6.94e-41

Scale= 0.252

Gaussian distribution

Loglik(model)= -3.7 Loglik(intercept only)= -10.3

Chisq= 13.2 on 3 degrees of freedom , p-value: 0.0042

Number of Newton -Raphson Iterations: 4

Sample size: 51

Variance inflation factors are not computable for a survival

regression using the DAAG:::vif() function..

McFadden (1974) R-squared: 0.6412 and adjusted R-squared: 0.2527

Bias correction factor: 1.0758 = 10^[(Scale * Scale)/2]

(Detransformed estimates are to be multiplied this factor.)

Nomenclature (all potential variables)

• Q is streamflow, in cubic feet per second;

• pH is pH, in standard units;

• SC is specific conductance, in microsiemens

per centimeter at 25° Celsius;

• Turb is turbidity in Formazine Nephelometric Units;

• Temp is water temperature, in ° Celsius;

• Date is Julian day DOY (days into year)

divided by 365.25; and

• log10(x) is base-10 logarithm of x.

Algebraic Equation

log10(NO2NO3) =−0.8938

−0.0288log10(Q)

+0.0289 sin[2π(Date)]

−0.1911 cos[2π(Date)]

Residual Plot for Regression

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−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

Estimated nitrite plus nitrate,

in milligrams per liter

Residuals

CORRELATION OF COEFFICIENTS NO2NO3

(Intercept) log10(Q) sin2piD cos2piD

(Intercept) 1 -0.9572 -0.0923 0.4401

log10(Q) -0.9572 1 0.0703 -0.4678

sin2piD -0.0923 0.0703 1 -0.0730

cos2piD 0.4401 -0.4678 -0.0730 1

EXPLANATION

● CONSTITUENT FOR STREAMFLOW-GAGING STATION

ORIGIN LINE

Comprehensive descriptions of abbreviations for the

computer output related to the regression analysis

documented in this figure are shown in figure 3.

Figure 10. Summary of regression analysis for nitrite plus nitrate (NO2 NO3) for 08070200 East Fork San Jacinto near New

Caney, Tex. based on data as described in text for period 2005–09.

Figure 10. Summary of regression analysis for nitrite plus nitrate nitrogen for U.S. Geological Survey streamflow-gaging station 

08070200 East Fork San Jacinto River near New Caney, Texas, 2005–9.
Regression Model Development 19

Inflow Statistics of Applicable Explanatory Variables: [Min., minimum, Qu., quartile, Max., maximum]

EXPLANATORY VARIABLE SUMMARY STATISTICS

log10(Q) log10(Turb) sin2piD cos2piD sin4piD cos4piD

Min. :1.041 Min. :0.8633 Min. :-0.99894 Min. :-0.99777 Min. :-0.99919 Min. :-0.99993

1st Qu.:1.568 1st Qu.:1.0792 1st Qu.: -0.71442 1st Qu.: -0.64457 1st Qu.: -0.56101 1st Qu.: -0.74933

Median :1.903 Median :1.2041 Median : 0.11502 Median : 0.04604 Median :-0.08091 Median :-0.12020

Mean :2.086 Mean :1.3546 Mean : 0.03078 Mean : 0.03567 Mean : 0.04246 Mean :-0.05625

3rd Qu.:2.474 3rd Qu.:1.6127 3rd Qu.: 0.77167 3rd Qu.: 0.65656 3rd Qu.: 0.73417 3rd Qu.: 0.69467

Max. :3.899 Max. :2.2304 Max. : 0.99998 Max. : 0.99336 Max. : 0.99926 Max. : 0.99108

Summary of Regression Analysis for the Constituent of:

total phosphorus (Phos)

SUMMARY STATISTICS FOR PHOS , IN LOG10() milligrams per liter

Min. 1st Qu. Median Mean 3rd Qu. Max.

-1.2680 -1.1110 -1.0000 -1.0070 -0.9393 -0.5986

REGRESSION EQUATION

lm(formula = log10(PHOS) ~ log10(Q) + log10(Turb) + sin2piD +

cos2piD + sin4piD + cos4piD , data = the.data)

Residuals:

Min 1st Qu. Median 3rd Qu. Max

-0.207627 -0.060346 -0.001328 0.055656 0.236860

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) -1.458421 0.050022 -29.156 < 2e-16 ***

log10(Q) -0.081051 0.029817 -2.718 0.00935 **

log10(Turb) 0.457366 0.056263 8.129 2.63e-10 ***

sin2piD -0.044349 0.016831 -2.635 0.01158 *

cos2piD -0.019996 0.020621 -0.970 0.33750

sin4piD -0.003777 0.017487 -0.216 0.82998

cos4piD -0.052928 0.017576 -3.011 0.00430 **

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Signif.codes: 0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

Residual standard error: 0.08579 on 44 degrees of freedom

Multiple R-squared: 0.7314, Adjusted R-squared: 0.6948

F-statistic: 19.97 on 6 and 44 DF , p-value: 4.201e-11

Variance inflation factors:

log10(Q) log10(Turb) sin2piD cos2piD sin4piD

cos4piD

3.4164 2.6701 1.0348 1.3866 1.0624

1.0567

Duan (1983) smearing factor: 1.0172 = mean(10^[ residuals(model)])

(Detransformed estimates are to be multiplied this factor.)

Nomenclature (all potential variables)

• Q is streamflow, in cubic feet per second;

• pH is pH, in standard units;

• SC is specific conductance, in microsiemens

per centimeter at 25° Celsius;

• Turb is turbidity in Formazine Nephelometric Units;

• Temp is water temperature, in ° Celsius;

• Date is Julian day DOY (days into year)

divided by 365.25; and

• log10(x) is base-10 logarithm of x.

Algebraic Equation

log10(Phos) =−1.4584

−0.0811log10(Q)+

+0.4574log10(Turb)

−0.0443 sin[2π(Date)]

−0.0200 cos[2π(Date)]

−0.0038 sin[4π(Date)]

−0.0529 cos[4π(Date)]

Residual Plot for Regression

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0.05 0.10 0.15 0.20

−0.2 −0.1 0.0 0.1 0.2

Estimated total phosphorus,

in milligrams per liter

Residuals

CORRELATION OF COEFFICIENTS PHOS

(Intercept) log10(Q) log10(Turb) sin2piD cos2piD sin4piD cos4piD

(Intercept) 1 -0.0607 -0.5696 -0.1556 0.2165 -0.0813 -0.0310

log10(Q) -0.0607 1 -0.7713 0.0769 -0.4853 0.2306 0.2231

log10(Turb) -0.5696 -0.7713 1 0.0346 0.2444 -0.1423 -0.1484

sin2piD -0.1556 0.0769 0.0346 1 -0.1018 -0.0213 0.0660

cos2piD 0.2165 -0.4853 0.2444 -0.1018 1 -0.1299 -0.1356

sin4piD -0.0813 0.2306 -0.1423 -0.0213 -0.1299 1 0.0620

cos4piD -0.0310 0.2231 -0.1484 0.0660 -0.1356 0.0620 1

EXPLANATION

● CONSTITUENT FOR STREAMFLOW-GAGING STATION

ORIGIN LINE

Comprehensive descriptions of abbreviations for the

computer output related to the regression analysis

documented in this figure are shown in figure 3.

Figure 11. Summary of regression analysis for total phosphorus (Phos) for 08070200 East Fork San Jacinto near New Caney,

Tex. based on data as described in text for period 2005–09.

Figure 11. Summary of regression analysis for total phosphorus for U.S. Geological Survey streamflow-gaging station 08070200

East Fork San Jacinto River near New Caney, Texas, 2005–9.
20 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

Figure 12. Summary of regression analysis for total organic carbon for U.S. Geological Survey streamflow-gaging station 08070200 

East Fork San Jacinto River near New Caney, Texas, 2005–9.

Inflow Statistics of Applicable Explanatory Variables: [Min., minimum, Qu., quartile, Max., maximum]

EXPLANATORY VARIABLE SUMMARY STATISTICS

log10(Q) log10(SC) log10(Turb)

Min. :1.041 Min. :1.690 Min. :0.8633

1st Qu.:1.568 1st Qu.:2.161 1st Qu.:1.0792

Median :1.908 Median :2.248 Median :1.2553

Mean :2.111 Mean :2.185 Mean :1.3653

3rd Qu.:2.542 3rd Qu.:2.328 3rd Qu.:1.6232

Max. :3.899 Max. :2.442 Max. :2.2304

Summary of Regression Analysis for the Constituent of:

total organic carbon (OrgC)

SUMMARY STATISTICS FOR ORGC , IN LOG10() milligrams per liter

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.6021 0.7853 0.8865 0.9247 1.1140 1.2600

REGRESSION EQUATION

lm(formula = log10(ORGC) ~ log10(Q) + log10(SC) + log10(Turb),

data = the.data)

Residuals:

Min 1st Qu. Median 3rd Qu. Max

-0.16068 -0.03881 -0.01109 0.04640 0.20723

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) -0.53685 0.25485 -2.107 0.040767 *

log10(Q) 0.24611 0.02745 8.967 1.41e-11 ***

log10(SC) 0.32656 0.09239 3.535 0.000958 ***

log10(Turb) 0.16730 0.04918 3.402 0.001413 **

---

Signif.codes: 0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

Residual standard error: 0.07622 on 45 degrees of freedom

Multiple R-squared: 0.8619, Adjusted R-squared: 0.8527

F-statistic: 93.64 on 3 and 45 DF , p-value: < 2.2e-16

Variance inflation factors:

log10(Q) log10(SC) log10(Turb)

3.5481 2.9318 2.5231

Duan (1983) smearing factor: 1.0144 = mean(10^[ residuals(model)])

(Detransformed estimates are to be multiplied this factor.)

Nomenclature (all potential variables)

• Q is streamflow, in cubic feet per second;

• pH is pH, in standard units;

• SC is specific conductance, in microsiemens

per centimeter at 25° Celsius;

• Turb is turbidity in Formazine Nephelometric Units;

• Temp is water temperature, in ° Celsius;

• Date is Julian day DOY (days into year)

divided by 365.25; and

• log10(x) is base-10 logarithm of x.

Algebraic Equation

log10(OrgC) =−0.5369

+0.2461log10(Q)

+0.3266log10(SC)

+0.1673log10(Turb)

Residual Plot for Regression

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−0.2 −0.1 0.0 0.1 0.2

Estimated total organic carbon,

in milligrams per liter

Residuals

CORRELATION OF COEFFICIENTS ORGC

(Intercept) log10(Q) log10(SC) log10(Turb)

(Intercept) 1 -0.5581 -0.9852 -0.3450

log10(Q) -0.5581 1 0.5737 -0.4695

log10(SC) -0.9852 0.5737 1 0.2378

log10(Turb) -0.3450 -0.4695 0.2378 1

EXPLANATION

● CONSTITUENT FOR STREAMFLOW-GAGING STATION

ORIGIN LINE

Comprehensive descriptions of abbreviations for the

computer output related to the regression analysis

documented in this figure are shown in figure 3.

Figure 12. Summary of regression analysis for total organic carbon (OrgC) for 08070200 East Fork San Jacinto near New Caney,

Tex. based on data as described in text for period 2005–09.
Regression Model Development 21

Figure 13. Summary of regression analysis for Escherichia coli for U.S. Geological Survey streamflow-gaging station 08070200 

East Fork San Jacinto River near New Caney, Texas, 2005–9.

Inflow Statistics of Applicable Explanatory Variables: [Min., minimum, Qu., quartile, Max., maximum]

EXPLANATORY VARIABLE SUMMARY STATISTICS

log10(Turb) I(log10(Turb)^2) sin2piD cos2piD

Min. :0.8633 Min. :0.7453 Min. :-0.99894 Min. :-0.99777

1st Qu.:1.0792 1st Qu.:1.1646 1st Qu.: -0.59438 1st Qu.:-0.66325

Median :1.2041 Median :1.4499 Median : 0.13659 Median : 0.19970

Mean :1.3642 Mean :1.9853 Mean : 0.08057 Mean : 0.05389

3rd Qu.:1.6232 3rd Qu.:2.6349 3rd Qu.: 0.77665 3rd Qu.: 0.67111

Max. :2.2304 Max. :4.9749 Max. : 0.99998 Max. : 0.99336

Summary of Regression Analysis for the Constituent of:

Escherichia coli (ECB)

SUMMARY STATISTICS FOR ECB , IN LOG10() most probable number per 100

milliliters

Min. 1st Qu. Median Mean 3rd Qu. Max.

1.114 1.740 2.041 2.277 2.716 4.643

REGRESSION EQUATION

lm(formula = log10(ECB) ~ log10(Turb) + log10(Turb)^2 + sin2piD +

cos2piD , data = the.data)

Residuals:

Min 1st Qu. Median 3rd Qu. Max

-0.7373 -0.1958 -0.0143 0.2422 0.9307

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) 4.21852 1.02533 4.114 0.000168 ***

log10(Turb) -4.43709 1.45936 -3.040 0.003970 **

log10(Turb)^2 2.06978 0.49393 4.190 0.000132 ***

sin2piD -0.12086 0.07790 -1.552 0.127932

cos2piD 0.22499 0.08888 2.531 0.015009 *

---

Signif.codes: 0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

Residual standard error: 0.388 on 44 degrees of freedom

Multiple R-squared: 0.7968, Adjusted R-squared: 0.7783

F-statistic: 43.12 on 4 and 44 DF , p-value: 1.106e-14

Variance inflation factors:

log10(Turb) log10(Turb)^2 sin2piD cos2piD

86.0980 87.7590 1.0155 1.2253

Duan (1983) smearing factor: 1.4893 = mean(10^[ residuals(model)])

(Detransformed estimates are to be multiplied this factor.)

Nomenclature (all potential variables)

• Q is streamflow, in cubic feet per second;

• pH is pH, in standard units;

• SC is specific conductance, in microsiemens

per centimeter at 25° Celsius;

• Turb is turbidity in Formazine Nephelometric Units;

• Temp is water temperature, in ° Celsius;

• Date is Julian day DOY (days into year)

divided by 365.25; and

• log10(x) is base-10 logarithm of x.

Algebraic Equation

log10(ECB) = 4.2185

−4.4371log10(Turb)

+2.0700log10(Turb)

2

−0.1209 sin[2π(Date)]

+0.2250 cos[2π(Date)]

Residual Plot for Regression

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−1.0 −0.5 0.0 0.5 1.0

Estimated Escherichia coli bacteria,

in most probable number per 100 milliliters

Residuals

CORRELATION OF COEFFICIENTS ECB

(Intercept) log10(Turb) log10(Turb)^2 sin2piD cos2piD

(Intercept) 1 -0.9929 0.9752 -0.0179 -0.3501

log10(Turb) -0.9929 1 -0.9939 -0.0012 0.3715

log10(Turb)^2 0.9752 -0.9939 1 0.0149 -0.3929

sin2piD -0.0179 -0.0012 0.0149 1 -0.0248

cos2piD -0.3501 0.3715 -0.3929 -0.0248 1

EXPLANATION

● CONSTITUENT FOR STREAMFLOW-GAGING STATION

ORIGIN LINE

Comprehensive descriptions of abbreviations for the

computer output related to the regression analysis

documented in this figure are shown in figure 3.

Figure 13. Summary of regression analysis for Escherichia coli (ECB) for 08070200 East Fork San Jacinto near New Caney, Tex.

based on data as described in text for period 2005–09.
22 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

Inflow Statistics of Applicable Explanatory Variables: [Min., minimum, Qu., quartile, Max., maximum]

EXPLANATORY VARIABLE SUMMARY STATISTICS

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.8633 1.0790 1.2040 1.3610 1.6130 2.2300

Summary of Regression Analysis for the Constituent of:

suspended sediment (SS)

SUMMARY STATISTICS FOR SS , IN LOG10() milligrams per liter

Min. 1st Qu. Median Mean 3rd Qu. Max.

0.7782 1.1900 1.4310 1.5460 1.7310 3.0370

REGRESSION EQUATION

lm(formula = log10(SS) ~ log10(Turb), data = the.data)

Residuals:

Min 1st Qu. Median 3rd Qu. Max

-0.59759 -0.18961 -0.05024 0.08017 1.03328

Coefficients:

Estimate Std.Error t-value Pr(>|t|)

(Intercept) 0.06566 0.19021 0.345 0.731

log10(Turb) 1.08800 0.13538 8.037 1.68e-10 ***

---

Signif.codes: 0 "***" 0.001 "**" 0.01 "*" 0.05 "." 0.1 " " 1

Residual standard error: 0.3382 on 49 degrees of freedom

Multiple R-squared: 0.5686, Adjusted R-squared: 0.5598

F-statistic: 64.59 on 1 and 49 DF , p-value: 1.680e-10

Variance inflation factors:

log10(Turb)

1

Duan (1983) smearing factor: 1.4908 = mean(10^[ residuals(model)])

(Detransformed estimates are to be multiplied this factor.)

Nomenclature (all potential variables)

• Q is streamflow, in cubic feet per second;

• pH is pH, in standard units;

• SC is specific conductance, in microsiemens

per centimeter at 25° Celsius;

• Turb is turbidity in Formazine Nephelometric Units;

• Temp is water temperature, in ° Celsius;

• Date is Julian day DOY (days into year)

divided by 365.25; and

• log10(x) is base-10 logarithm of x.

Algebraic Equation

log10(SS) = 0.0657

+1.0880log10(Turb)

Residual Plot for Regression

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Estimated suspended sediment,

in milligrams per liter

Residuals

CORRELATION OF COEFFICIENTS SS

(Intercept) log10(Turb)

(Intercept) 1 -0.9685

log10(Turb) -0.9685 1

EXPLANATION

● CONSTITUENT FOR STREAMFLOW-GAGING STATION

ORIGIN LINE

Comprehensive descriptions of abbreviations for the

computer output related to the regression analysis

documented in this figure are shown in figure 3.

Figure 14. Summary of regression analysis for suspended sediment (SS) for 08070200 East Fork San Jacinto near New Caney,

Tex. based on data as described in text for period 2005–09.

Figure 14. Summary of regression analysis for suspended sediment for U.S. Geological Survey streamflow-gaging station 08070200

East Fork San Jacinto River near New Caney, Texas, 2005–9. 

Estimated suspended sediment, 

in milligrams per liter

20
Summary 23

Computational Procedures of 

Concentration and Loads

Implementation of the 11 equations shown in figures 

4-14 for estimation of constituent concentration is done in 

this report in the context of concurrent reporting of 90-percent 

prediction intervals. The prediction interval computation is 

important because the equations documented in this report 

are readily used for real-time estimation of the constituents. 

The regression equations documented in this report provide 

a computed constituent record available through the online 

USGS National Real-Time Water-Quality website (NRTWQ) 

(http://nrtwq.usgs.gov/ks).

A thorough description and presentation of numerical 

results for a selected equation for selected stream conditions 

has been provided so that interested parties can verify the 

basic algorithmic framework needed to implement the 

equations in this report. Example algorithmic frameworks 

are described in succession in appendix 2 (for constituents 

with noncensored data) and appendix 3 (for constituents 

containing censored data). By way of example, these 

appendixes describe the application of the equations described 

in figures 4 and 14 for computation of an instantaneous 

15-minute load (15 minutes is the smallest time interval for 

computing continuously measured water-quality values). The 

implementation does not involve a particularly high level of 

mathematics, but basic knowledge of logarithms, statistics 

associated with regression, and matrix algebra is required. 

Summary

In December 2005, the U.S. Geological Survey (USGS), 

in cooperation with the City of Houston, Texas, began 

collecting discrete water-quality samples for nutrients, total 

organic carbon, bacteria (Escherichia coli and total coliform), 

atrazine, and suspended sediment at two USGS streamflowgaging stations that represent watersheds contributing to 

Lake Houston (08068500 Spring Creek near Spring, Tex., 

and 08070200 East Fork San Jacinto River near New Caney, 

Tex.). These sampling sites are respectively referred to as the 

Spring Creek site and East Fork San Jacinto River site. Data 

from discrete water-quality samples collected during 2005–9, 

in conjunction with continuously monitored real-time data 

that included streamflow and other physical water-quality 

properties (specific conductance, pH, water temperature, 

turbidity, and dissolved oxygen), were used to develop 

regression models for estimation of concentrations of waterquality constituents of substantial source watersheds to Lake 

Houston.

The regression models presented in this report are site 

specific to streamflow-gaging stations on two tributaries to 

Lake Houston; however, the methods that were developed and 

documented could be applied to most perennial streams for the 

purpose of estimating real-time water quality data.

 The continuously monitored streamflow and other 

physical water-quality properties, in conjunction with 

regression models that use those data as surrogates for 

selected constituents (nitrite plus nitrate nitrogen, total 

phosphorus, total organic carbon, E. coli, atrazine, and 

suspended sediment) can be used to estimate concentrations 

for constituents lacking continuous record.

Streamflow, physical water-quality properties, and 

selected constituent concentrations were collected at the 

two sites. During 2005–9, discrete samples were collected 

at the Spring Creek site (58 samples) and at the East Fork 

San Jacinto River site (51 samples). Hydrologic conditions 

within the Spring Creek and East Fork San Jacinto River sites 

vary and might affect chemical constituent concentrations, 

so discrete water-quality samples were collected over a 

wide range of streamflow conditions. Discrete water-quality 

samples for the first year (December 2005–November 2006) 

of this study were collected about every 2 weeks to facilitate 

detection of seasonal patterns in water quality. Samples 

at these fixed-frequency sample times were collected as 

scheduled without regard to hydrologic condition, such as 

rising, falling, or stable streamflow. During storms or periods 

of high flow, unscheduled samples were also periodically 

collected during the first year of the study. During the second 

and third year of the study (December 2006–December 2008) 

discrete water-quality samples were collected approximately 

once a month at both the Spring Creek and East Fork San 

Jacinto River sites. During the fourth year of the study 

(December 2008–December 2009), an approximate monthly 

sampling schedule was maintained for the Spring Creek site, 

whereas samples collected at East Fork San Jacinto River site 

were reduced to a quarterly schedule. Instead of sampling on 

a fixed frequency during the second through fourth years of 

the study, stormwater-runoff samples were collected whenever 

possible.

Regression analyses were done by using streamflow, 

continuous water-quality, and discrete water-quality data 

collected during 2005–9 at the Spring Creek and East Fork 

San Jacinto River sites. The R environment for statistical 

computing was used to develop algebraically representable, 

multiple-linear regression equations to (1) estimate 

concentrations for selected water-quality constituents and (2) 

estimate prediction intervals or quantification of uncertainty. 

The potential explanatory or predictive variables included 

streamflow, specific conductance, pH, water temperature, 

turbidity, dissolved oxygen, and time (to account for 

seasonal variations inherent in some water-quality data). 

The response variables at each site were nitrite plus nitrate 

nitrogen, total phosphorus, organic carbon, E. coli, atrazine, 

and suspended sediment. Logarithmic transformations 

(base-10) on the response and explanatory variables were 

used to improve linearity and to mitigate for nonnormality 

and heteroscedasticity in model residuals. The explanatory 

variables provide easily measured quantities as a means to 
24 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

estimate concentrations of the various constituents under 

investigation, with accompanying estimates of measurement 

uncertainty. Each regression equation can be used to 

estimate concentrations of a given constituent in real time 

on the basis of explanatory variables also measured in real 

time. Corresponding 90-percent prediction intervals can 

be computed to display the uncertainty associated with 

the estimate. Factors used as indicators of general model 

reliability include the adjusted R-squared, the residual 

standard error, residual plots, and p-values.

Regression equations for the Spring Creek site were 

developed for nitrite plus nitrate nitrogen, total phosphorus, 

organic carbon, E. coli bacteria, atrazine, and suspended 

sediment. Adjusted R-squared values for the Spring Creek 

models ranged from .582–.922 (dimensionless). The residual 

standard errors ranged from .073–.447 (base-10 logarithm). 

Regression equations for the East Fork San Jacinto 

River site were developed for nitrite plus nitrate nitrogen, 

total phosphorus, organic carbon, E. coli bacteria, and 

suspended sediment. Adjusted R-squared values for the 

East Fork San Jacinto River models ranged from .253–.853 

(dimensionless). The residual standard errors ranged from 

.076–.388 (base-10 logarithm).

In conjunction with estimated concentrations, constituent 

loads can be estimated by multiplying the estimated 

concentration by the corresponding streamflow and by 

applying the appropriate conversion factor. By calculating 

loads from estimated constituent concentrations, a continuous 

record of estimated loads can be produced.

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Appendix 1 27

Appendix 1
28 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9 Appendix 1 29

Appendix 1. Results from environmental and quality-control sample pairs and equipment blanks collected for two tributaries

(Spring Creek and East Fork San Jacinto River) to Lake Houston near Houston, Texas, 2005–9.

[Environ., Environmental; —, not analyzed; <, less than laboratory reporting level; E, estimated; *, value reviewed and rejected; >, greater than; Equip., Equipment]

Sample

date

Sample

time

Sample

type

Ammonia

plus organic

nitrogen,

water ,filtered

(milligrams

per liter

as nitrogen)

Ammonia

plus organic

nitrogen,

water,

unfiltered

(milligrams

per liter

as nitrogen)

Ammonia,

water, filtered

(milligrams

per liter

as nitrogen)

Nitrite

plus nitrate,

water, filtered

(milligrams

per liter

as nitrogen)

Nitrite,

water, filtered

(milligrams

per liter

as nitrogen)

Orthophosphate,

water, filtered

(milligrams

per liter

as phosphorus)

Phosphorus,

water, filtered

(milligrams

per liter)

Phosphorus,

water,

unfiltered

(milligrams

per liter)

Escherichia coli,

Colilert Quantitray

method, water

(most probable

number per

100 milliliters)

Total coliform,

Colilert Quantitray

method, water

(most probable

number per

100 milliliters)

Atrazine,

water, filtered,

recoverable,

immunoassay,

unadjusted

(micrograms

per liter)

Organic carbon,

water, unfiltered

(milligrams

per liter)

Suspended

sediment

(milligrams

per liter)

Spring Creek near Spring, Tex. 08068500 Spring Creek near Spring, Tex. 08068500

12/1/2005 1330 Environ. 0.85 1.3 0.08 6.87 0.028 1.56 1.61 1.73 130 >2400 0.3 9.6 36

12/1/2005 1331 Replicate — — — — — — — — 90 >2400 — — —

2/7/2006 1400 Environ. 0.75 1.1 0.06 4.52 0.02 0.95 0.97 1.11 60 3400 0.95 9.8 33

2/7/2006 1401 Replicate — — — — — — — — 55 3300 — — —

2/21/2006 1400 Environ. 1.1 1.4 0.39 4.01 0.054 0.91 0.87 1.05 37 6100 1.51 9.6 24

2/21/2006 1401 Replicate 1.2 1.4 0.38 4.04 0.054 0.91 0.86 1.02 27 6500 1.34 9.3 23

5/16/2006 1115 Environ. 0.78 1.8 <.04 1.33 0.069 0.63 0.66 0.87 490 110000 2.56 13.4 60

5/16/2006 1116 Replicate 0.78 1.9 <.04 1.32 0.07 0.64 0.68 0.85 690 98000 2.68 15.3 61

6/20/2006 1000 Environ. 0.86 1.5 0.035 0.35 0.034 0.197 0.24 0.44 8000 410000 1.45 17 547

6/20/2006 1001 Replicate — — — — — — — — 7100 460000 — — —

6/28/2006 845 Blank 0.12 <.10 0.015 <.06 <.002 <.006 <.02 <.02 — — <.10 <.4 <1

6/28/2006 930 Environ. 1 1.2 0.088 4.66 0.028 1.01 1 1.25 43 >2400 0.62 8.9 31

7/12/2006 1300 Environ. 0.83 1.1 0.058 3.36 0.057 0.719 0.7 0.89 120 41000 0.68 9 38

7/12/2006 1346 Replicate — — — — — — — — 190 69000 — — —

8/23/2006 1320 Blank E.07 <.10 <.010 <.06 <.002 <.006 <.02 <.02 — — <.10 <.4 <1

8/23/2006 1345 Environ. 0.77 1.8 <.010 2.12 0.021 0.738 0.78 1 6800 440000 1.75 14.1 147

9/20/2006 1230 Environ. 1.1 1.5 0.166 2.01 0.101 0.891 0.94 1.1 270 82000 1.73 13.1 56

9/20/2006 1231 Replicate 1.1 1.5 0.17 2.02 0.101 0.901 0.95 1.09 360 49000 1.77 10.3 46

8/15/2007 1420 Environ. 0.64 0.73 0.055 3.56 0.053 0.735 0.75 0.92 310 17000 0.18 8.1 42

8/15/2007 1421 Replicate — — — — — — — — 190 17000 — — —

2/14/2008 1110 Environ. 0.7 1.1 0.011 1.47 0.022 0.252 0.27 0.41 2200 33000 1.95 11.6 43

2/14/2008 1110 Replicate — — — — — — — — 2100 37000 — — —

11/13/2008 1315 Environ. 0.5 0.9 <.020 0.14 0.006 0.069 0.09 0.19 3300 57000 0.43 11.4 587

11/13/2008 1316 Replicate 0.51 0.91 <.020 0.15 0.006 0.068 0.09 0.19 — — 0.45 11.3 755

3/10/2009 1027 Environ. 1 1.3 0.189 4.66 0.066 1.24 1.22 1.34 34 37000 0.64 8.3 32

3/10/2009 1028 Replicate 1 1.3 0.189 4.65 0.066 1.25 1.21 1.32 — — 1 8.3 37

7/29/2009 1120 Environ. 0.94 1.3 0.085 5.72 0.101 1.65 1.62 1.83 20 18000 0.3 9.9 46

7/29/2009 1121 Replicate 0.96 1.3 0.078 5.8 0.102 1.66 1.61 1.77 — — 0.39 7.8 139

East Fork San Jacinto River near New Caney, Tex. 08070200 East Fork San Jacinto River near New Caney, Tex. 08070200

12/1/2005 1020 Environ. 0.14 0.23 <.04 0.1 <.008 0.03 0.039 0.086 43 1700 <.10 4 11

12/1/2005 1021 Replicate — — — — — — — — 34 1700 — — —

12/21/2005 930 Blank E.06 <.10 <.04 <.06 <.008 <.02 E.002 <.004 <1 <1 <.10 <.4 1

12/21/2005 1030 Environ. 0.45 0.38 <.04 0.07 <.008 <.02 0.025 0.058 130 2000 0.1 7.7 16

3/7/2006 1130 Environ. 0.46 0.44 <.04 0.07 <.008 E.01 0.025 0.076 34 1600 <.10 7.3 17

3/7/2006 1131 Replicate — — — — — — — — 34 1700 — — —

4/4/2006 1100 Environ. 0.74 0.8 0.05 0.17 E.006 0.02 0.037 0.09 43 5200 0.13 14.2 19
30 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9 Appendix 1 31

Sample

date

Sample

time

Sample

type

Ammonia

plus organic

nitrogen,

water ,filtered

(milligrams

per liter

as nitrogen)

Ammonia

plus organic

nitrogen,

water,

unfiltered

(milligrams

per liter

as nitrogen)

Ammonia,

water, filtered

(milligrams

per liter

as nitrogen)

Nitrite

plus nitrate,

water, filtered

(milligrams

per liter

as nitrogen)

Nitrite,

water, filtered

(milligrams

per liter

as nitrogen)

Orthophosphate,

water, filtered

(milligrams

per liter

as phosphorus)

Phosphorus,

water, filtered

(milligrams

per liter)

Phosphorus,

water,

unfiltered

(milligrams

per liter)

Escherichia coli,

Colilert Quantitray

method, water

(most probable

number per

100 milliliters)

Total coliform,

Colilert Quantitray

method, water

(most probable

number per

100 milliliters)

Atrazine,

water, filtered,

recoverable,

immunoassay,

unadjusted

(micrograms

per liter)

Organic carbon,

water, unfiltered

(milligrams

per liter)

Suspended

sediment

(milligrams

per liter)

East Fork San Jacinto River near New Caney, Tex. 08070200—Continued East Fork San Jacinto River near New Caney, Tex. 08070200—Continued

4/4/2006 1101 Replicate 0.65 0.8 0.05 0.17 E.007 0.02 0.041 0.101 50 10000 <.10 12.7 14

7/25/2006 945 Environ. 0.35 0.47 0.027 0.51 0.007 0.059 0.071 0.158 93 9200 <.10 7 6

7/25/2006 946 Replicate — — — — — — — — 84 12000 — — —

8/8/2006 1000 Environ. 0.38 0.39 0.023 0.22 0.002 0.034 0.043 0.111 120 8100 <.10 5.5 12

8/8/2006 1001 Replicate 0.42 0.39 0.022 0.23 0.003 0.032 0.043 0.111 150 6900 <.10 6.1 13

9/6/2006 1100 Environ. 0.36 0.27 0.025 0.08 E.001 0.024 0.034 0.113 26 13000 <.10 4 6

9/6/2006 1130 Blank 0.14 <.10 E.009 <.06 <.002 <.006 <.004 <.004 — — <.10 <.4 <1

10/4/2006 945 Environ. 0.22 0.3 E.016 0.15 <.002 0.047 0.056 0.124 63 4900 <.10 4.2 15

10/4/2006 946 Replicate — — — — — — — — 52 3900 — — —

10/18/2006 1330 Environ. 0.65 0.98 <.020 E.05 0.003 0.013 0.036 0.134 610 44000 0.12 18.2 125

10/18/2006 1331 Replicate — — — — — — — — 690 39000 — — —

1/15/2007 1340 Environ. 0.5 0.76 0.038 0.09 0.003 0.021 0.029 0.122 2100 18000 <.10 16.1 110

1/15/2007 1341 Replicate 0.52 0.78 0.035 0.09 0.003 0.019 0.028 0.123 2000 17000 <.10 13.3 81

5/24/2007 1038 Environ. — — — — — — — — 250 10000 <.10 — 39

5/24/2007 1039 Replicate — — — — — — — — 240 9900 <.10 — 39

8/15/2007 1059 Blank 0.35 <.10 0.031 <.06 E.001 <.006 <.006 <.008 — — — — 1

8/15/2007 1132 Environ. — 0.33 <.020 0.07 0.004 0.015 0.017 0.081 71 15000 <.10 6.5 27

4/23/2008 1030 Environ. 0.2 0.32 <0.02 0.15 0.005 0.021 0.031 0.068 100 5200 <.10 5.77 45

4/23/2008 1030 Replicate — — — — — — — — 108 4600 — — —

1/7/2009 1050 Environ. 0.27 0.37 E.012 0.08 0.005 0.015 0.02 0.069 300 7600 <.10 7.8 168

1/7/2009 1051 Replicate 0.3 0.36 E.011 0.08 0.005 0.014 0.018 0.065 — — <.10 6.5 174

Houston Lab at Shenandoah, Tex. 301056095265000 Houston Lab at Shenandoah, Tex. 301056095265000

8/22/2006 1330 Equip. 

Blank

<.10 <.10 <.010 <.06 <.002 E.003 <.02 <.02 — — <.10 <.4 —

11/28/2007 1358 Equip. 

Blank

<.14 <.14 <.020 <.04 <.002 <.006 <.006 E.006 — — — — 1

11/28/2007 1359 Equip. 

Blank

— — — — — — — — — — — 0.9 —

12/3/2008 1035 Equip. 

Blank

E.09 <.10 <.020 <.04 <.002 <.008 <.02 <.02 — — — <.6 1

12/3/2008 1555 Equip. 

Blank

— <.10 <.020 <.016 — <.008 — <.008 — — — E.4 —

12/3/2008 1615 Equip. 

Blank

— E.07 <.020 <.016 — <.008 — <.008 — — — E.5 —

Appendix 1. Results from environmental and quality-control sample pairs and equipment blanks collected for two tributaries

(Spring Creek and East Fork San Jacinto River) to Lake Houston near Houston, Texas, 2005–9.

[Environ., Environmental; —, not analyzed; <, less than laboratory reporting level; E, estimated; *, value reviewed and rejected; >, greater than; Equip., Equipment]
32 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9
Appendix 2—Example computations 33

Appendix 2—Example computations

(definition of terms and quantities, 

concentration estimates, 90-percent prediction 

intervals, and loads) for nitrite plus nitrate (N02 

N03) at U.S. Geological Survey streamflowgaging station 08070200 East Fork San Jacinto 

River near New Caney, Texas.
34 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

Appendix 2—Example computations (definition of terms and quantities, concentration estimates, . . . 5

Example computations (definition of terms and quantities) for nitrite plus nitrate (NO2 NO3)

for 08070200 East Fork San Jacinto River near New Caney, Tex.

Given values for example problem

• Q = 7,930 cubic feet per second;

• DOY = 292.5243 day of year

(October 19, 2006 at 12:35 hours); and

• Co = <0.06 observed NO2NO3

concentration in milligrams per liter for

DOY.

Nomenclature (select variables and symbols)

• NO2NO3 is nitrite plus nitrate in milligrams per liter;

• Q is streamflow in cubic feet per second;

• S is residual scale in base-10 logarithms of milligrams per liter. The

survival regression provides the natural logarithm of the residual

scale as part of standard (conventional) output. A detransformation

by exp() is seen in these examples;

• Date is Julian day DOY (days into year) divided by 365.25;

• log10(x) is base-10 logarithm of x; and

• π or pi is the quantity “pi” of approximately 3.14159.

Note, text set in a monospaced font (often within brackets) reflects the precise nomenclature in computer output seen

in figure 10. This specific font change is used to promote parallelism between the regression figure and these example

computations.

Nitrite plus nitrate equation and diagnostics for 08070200 East Fork San Jacinto River near New Caney, Tex.

log10(NO2NO3) = −0.8938−0.0288log10(Q) +0.0289 sin[2π(Date)]−0.1911 cos[2π(Date)]

NO2NO3 = א ×10log10(NO2NO3) in milligrams per liter

S = 0.2519 = (exp(−1.3788)) [Log(scale)] base-10 logarithms of milligrams per liter

DF = 51 (samples)−4 (parameters) = 47 [DF] degrees of freedom

SEB = 0.1251 [Std.Error] Standard error of coefficient on intercept term

SEQ = 0.0596 [Std.Error] Standard error of coefficient on streamflow term

SEsin = 0.0504 [Std.Error] Standard error of coefficient on 2π sine term

SEcos = 0.0590 [Std.Error] Standard error of coefficient on 2π cosine term

א = 1.0758 [ALEPH] Duan (1983) smearing factor (dimensionless)

Correlation of coefficients matrix for given constituent

equation











1 −0.9572 −0.0923 0.4401

−0.9572 1 0.0703 −0.4678

−0.0923 0.0703 1 −0.0730

0.4401 −0.4678 −0.0730 1











The matrix on the left is used for prediction-limit

computation. The matrix will require additional operations

to convert it to the inverted X-prime X-transverse matrix.

This second matrix is critical for computation of leverage

for a given prediction. Once the leverage is known, the

subsequent computations to acquire the prediction limits

are relatively straightforward.

Figure 15. Example computations (definition of terms and quantities) for nitrite plus nitrate (NO2 NO3) for 08070200 East Fork

San Jacinto River near New Caney, Tex..
Appendix 2—Example computations 35

6

Example computations (computations of estimates and 90-percent prediction limits) for

nitrite plus nitrate (NO2 NO3) for 08070200 East Fork San Jacinto River near New Caney, Tex.

Computation of an estimate and 90-percent prediction limits

The estimate of nitrite plus nitrate for the example problem is readily computed by

NO2NO3 = 1.0758×10−0.8938−0.0288log10(Q)+0.0289 sin[2π(292.5243/365.25)]−0.1911 cos[2π(292.5243/365.25)]

NO2NO3 = 0.0867 milligrams per liter

The lower (↓) and upper (↑) 90-percent prediction limits of NO2NO3 are respectively computed by

↓

NO2NO[α/2]

3 = 10log10(NO2NO3)−|t[α/2,DF]| S

√1+ho and ↑

NO2NO[α/2]

3 = 10log10(NO2NO3)+|t[α/2,DF]| S

√1+ho ,

where NO2NO3 is the estimate, t[α/2,DF] is the lower tail of the t-distribution for DF[NO2NO3] degrees of freedom at the α

significance level (α = [100−90 percent]/100), S is the residual scale of the maximum likelihood regression, and ho is the

leverage of the estimate. The lower and upper prediction limits of the estimated NO2NO3 = 0.0867 milligrams per liter

are respectively computed by

↓

NO2NO3 = 10log10(0.0867)−1.6779×0.2519√1+0.2256 = 0.0295 milligrams per liter and

↑

NO2NO3 = 10log10(0.0867)+1.6779×0.2519√1+0.2256 = 0.2546 milligrams per liter.

Thus, the 90-percent prediction interval is 0.030 ≤ NO2NO3 = 0.087 ≤ 0.255 milligrams per liter.

Computation of an estimate and 90-percent prediction limits from maximum likelihood regression by using

the R environment for statistical computing (R Development Core Team, 2010)

Q <- 7930 # streamflow in cubic feet per second

DOY <- 292.5243 # Days into year (Day of Year)

Date <- DOY/365.25; DF <- 47; ALEPH <- 1.0758; PERCENT <- 90

X <-c(1, log10(Q), sin(2*pi*Date), cos(2*pi*Date)) # array of predictor variables

CORMAT <- # Correlation of Coefficients Matrix

matrix(c( 1 , -0.9572, -0.0923, 0.4401,

-0.9572, 1 , 0.0703, -0.4678,

-0.0923, 0.0703, 1 , -0.0730,

0.4401, -0.4678, -0.0730, 1 ), ncol=4) # a 4x4 matrix

S <- exp(-1.3788) # Note the use of exp() for the "Log(scale)" of the regression

print(S) # in order to see at least four decimal points , compare to Scale= 0.252

[1] 0.2518806 # confirming , this value is in base -10 logarithmic units

SIGMAS <- c(0.1251, 0.0596, 0.0504, 0.0590) # Std. Error of the model coefficients

# The next two lines of code represent substantially complex mathematical operations for

# which the the use of a computing environment (not calculator or spreadsheet) such as R

# for this report is justified.

XPXI <- CORMAT * outer(SIGMAS , SIGMAS) / S^2 # inverted X-prime X-transverse matrix

ho <- t(X) %*% XPXI %*% X # leverage of the prediction

print(ho) # show the leverage of the prediction (ho used in hand computations shown above)

[1] 0.2256367

alpha <- 1 - PERCENT/100 # for the prediction interval

QT <- abs(qt(alpha/2, df= DF)) # t-distribution multiplier (sign change handled later)

print(QT) # show the lower -tail of t-distribution (QT used in hand computations shown above)

[1] 1.677927

Xbar <- -0.8938*X[1] - 0.0288*X[2] + 0.0289*X[3] - 0.1911*X[4] # the prediction

Xlo <- Xbar - QT*S*sqrt(1+ho); Xhi <- Xbar + QT*S*sqrt(1+ho) # lower/upper predict. limits

C <- ALEPH * 10^c(Xlo , Xbar , Xhi); names(C) <- c("lwr", "fit", "upr")

print(C) # show the estimated limits and value using numerical values of this report

lwr fit upr

0.02952812 0.08672235 0.25469844 # Lower limit , the prediction , and upper limit

# The predict.survreg() function does not provide direct capacity for prediction limits.

# If one has the model object (the.lm) that contains the NO2NO3 regression , then prediction is

print(ALEPH * 10^( predict(the.lm))[24]) # the 24th value in the underlying data

[1] 0.08719011 # numerical slight numerical difference because of rounding

Figure 16. Example computations (computations of estimates and 90-percent prediction limits) for nitrite plus nitrate (NO2 NO3)

for 08070200 East Fork San Jacinto River near New Caney, Tex..
36 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9 Appendix 2—Example computations (definition of terms and quantities, concentration estimates, . . . 7

Example computations (computation of load) for nitrite plus nitrate (NO2 NO3) for 08070200

East Fork San Jacinto River near New Caney, Tex.

Unit conversion to 15-minute load from streamflow and constituent concentration

The relation between a 15-minute load L, streamflow Q, and NO2NO3 concentration C is determined as follows

L

 kilogram

15 minutes

= Q



cubic feet

second 

×C



milligram

liter 

×



900 seconds

15 minutes 

×



28.317 liter

cubic feet 

×

 kilogram

1×106 milligram

  

K = 0.02549

Computation of 15-minute load of nitrite plus nitrate

The 15-minute load L in kilograms per 15 minutes for streamflow Q = 7,930 cubic feet per second of nitrite plus nitrate

with a concentration of C = 0.0867 milligrams per liter is computed by

L = Q×C ×K

L = 7,930×0.0867×0.02549

L = 17.53 kilograms per 15 minutes

Computation of 15-minute load of nitrite plus nitrate by using R environment for statistical computing (R Development

Core Team, 2010)

# Q is streamflow from code shown on previous page (7,930 cubic feet per second)

# C is the lower , estimate , and upper concentrations shown on previous page

# (0.02952812, 0.08672235, 0.25469844)

K <- 0.02549 # unit conversion factor shown in previous hand derivation shown

above

L <- Q * C * K # compute the 15-minute load prediction and prediction limits

print(L) # show the 15-minute load along with prediction limits

lwr fit upr

[1] 5.968687 17.529683 51.483647

Thus, the 15-minute load of nitrite plus nitrate with 90-percent prediction limits

is 5.97 ≤ L = 17.53 ≤ 51.48 kilograms per 15 minutes.

Figure 17. Example computations (computation of load) for nitrite plus nitrate (NO2 NO3) for 08070200 East Fork San Jacinto

River near New Caney, Tex..
Appendix 3—Example computations 37

(definition of terms and quantities, 

concentration estimates, 90-percent prediction 

intervals, and loads) for total phosphorus (Phos) 

at U.S. Geological Survey streamflow-gaging 

station 08068500 Spring Creek near

Spring, Texas.

Appendix 3—Example computations
38 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

Appendix 3—Example computations (definition of terms and quantities, concentration estimates, . . . 11

Example computations (definition of terms and quantities) for total phosphorus (Phos) for

08068500 Spring Creek near Spring, Tex.

Given values for example problem

• Q = 40 cubic feet per second;

• SC = 454 microsiemens per centimeter

at 25° Celsius;

• Turb = 18 Formazine Nephelometric

units;

• DOY = 52.5833 day of year (February

21, 2006 at 14:00 hours); and

• Co = 1.05 observed concentration, in

milligrams per liter for DOY.

Nomenclature (select variables and symbols)

• Phos is total phosphorus in milligrams per liter;

• Q is streamflow in cubic feet per second;

• RSE is residual standard error σ in base-10 logarithms of

milligrams per liter;

• SC is specific conductance, in microsiemens per centimeter at 25°

Celsius;

• Turb is turbidity in Formazine Nephelometric units;

• Date is Julian day DOY (days into year) divided by 365.25;

• log10(x) is base-10 logarithm of x; and

• π or pi is the quantity “pi” of approximately 3.14159.

Note, text set in a monospaced font (often within brackets) reflects the precise nomenclature in computer output seen

in figure 5. This specific font change is used to promote parallelism between the regression figure and these example

computations.

Total phosphorus equation and diagnostics for 08068500 Spring Creek near Spring, Tex.

log10(Phos) = −3.5808−1.2592log10(SC) +0.1678log10(Turb)−0.0247 sin[2π(Date)]−0.0627 cos[2π(Date)]

Phos = א ×10log10(Phos) in milligrams per liter

RSE = 0.0954 [Residual standard error] base-10 logarithms of milligrams per liter

DF = 56 (samples)−5 (parameters) = 51 [DF] degrees of freedom

SEB = 0.2605 [Std.Error] Standard error of coefficient on intercept term

SESC = 0.0791 [Std.Error] Standard error of coefficient on specific conductance term

SETurb = 0.0470 [Std.Error] Standard error of coefficient on turbidity term

SEsin = 0.0172 [Std.Error] Standard error of coefficient on 2π sine term

SEcos = 0.0197 [Std.Error] Standard error of coefficient on 2π cosine term

א = 1.0222 [ALEPH] Duan (1983) smearing factor (dimensionless)

Correlation of coefficients matrix for given constituent

equation

















1 −0.9777 −0.8447 −0.0966 −0.0567

−0.9777 1 0.7190 0.0750 0.0324

−0.8447 0.7190 1 0.1310 0.1361

−0.0966 0.0750 0.1310 1 −0.0164

−0.0567 0.0324 0.1361 −0.0164 1

















The matrix on the left is used for prediction-limit

computation. The matrix will require additional operations

to convert it to the inverted X-prime X-transverse matrix.

This second matrix is critical for computation of leverage

for a given prediction. Once the leverage is known, the

subsequent computations to acquire the prediction limits

are relatively straightforward.

Figure 18. Example computations (definition of terms and quantities) for total phosphorus (Phos) for 08068500 Spring Creek

near Spring, Tex..
Appendix 3—Example computations 39

12

Example computations (computations of estimates and 90-percent prediction limits) for total

phosphorus (Phos) for 08068500 Spring Creek near Spring, Tex.

Computation of an estimate and 90-percent prediction limits

The estimate of total phosphorus for the example problem is readily computed by

Phos = 1.0222×10−3.5808+1.2592log10(SC)+0.1678log10(Turb)−0.0247 sin[2π(52.8333/365.25)]−0.0627 cos[2π(52.8333/365.25)]

Phos = 0.8452 milligrams per liter

The lower (↓) and upper (↑) 90-percent prediction limits of Phos are respectively computed by

↓

Phos[α/2] = 10log10(Phos)−|t[α/2,DF]| RSE√1+ho and ↑Phos[α/2] = 10log10(Phos)+|t[α/2,DF]| RSE√1+ho ,

where Phos is the estimate, t[α/2,DF] is the lower tail of the t-distribution for DF[Phos] degrees of freedom at the α significance

level (α = [100−90 percent]/100), RSE is the residual standard error, and ho is the leverage of the estimate.

The lower and upper prediction limits of the estimated Phos = 0.8452 milligrams per liter are respectively computed by

↓

Phos = 10log10(0.8452)−1.6753×0.0954√1+0.0664 = 0.5780 milligrams per liter and

↑

Phos = 10log10(0.8452)+1.6753×0.0954√1+0.0664 = 1.2360 milligrams per liter.

Thus, the 90-percent prediction interval is 0.578 ≤ Phos = 0.845 ≤ 1.24 milligrams per liter.

Computation of an estimate and 90-percent prediction limits from least-squares regression by using

the R environment for statistical computing (R Development Core Team, 2010)

Q <- 40 # streamflow in cubic feet per second

SC <- 454 # specific conductance in microsiemens per centimeter at 25 degree Celsius

Turb <- 18 # turbidity in Formazine Nephelometric units

DOY <- 52.5833 # Days into year (Day of Year)

Date <- DOY/365.25; DF <- 51; ALEPH <- 1.0222; PERCENT <- 90

X <-c(1, log10(SC), log10(Turb), sin(2*pi*Date), cos(2*pi*Date)) # array of predictor variables

CORMAT <- # Correlation of Coefficients Matrix

matrix(c( 1 , -0.9777, -0.8447, -0.0966, -0.0567,

-0.9777, 1 , 0.7190, 0.0750, 0.0324,

-0.8447, 0.7190, 1 , 0.1310, 0.1361,

-0.0966, 0.0750, 0.1310, 1 , -0.0164,

-0.0567, 0.0324, 0.1361, -0.0164, 1 ), ncol=5) # a 5x5 matrix

RSE <- 0.0954 # residual standard error

SIGMAS <- c(0.2605, 0.0791, 0.0470, 0.0172, 0.0197) # Std. Error of the model coefficients

# The next two lines of code represent substantially complex mathematical operations for

# which the the use of a computing environment (not calculator or spreadsheet) such as R

# for this report is justified.

XPXI <- CORMAT * outer(SIGMAS , SIGMAS) / RSE^2 # inverted X-prime X-transverse matrix

ho <- t(X) %*% XPXI %*% X # leverage of the prediction

print(ho) # show the leverage of the prediction (ho used in hand computations shown above)

[1] 0.06637472

alpha <- 1 - PERCENT/100 # for the prediction interval

QT <- abs(qt(alpha/2, df= DF)) # t-distribution multiplier (sign change handled later)

print(QT) # show the lower -tail of t-distribution (QT used in hand computations shown above)

[1] 1.675285

Xbar <- -3.5808*X[1] + 1.2592*X[2] + 0.1678*X[3] - 0.0247*X[4] - 0.0627*X[5] # the prediction

Xlo <- Xbar - QT*RSE*sqrt(1+ho); Xhi <- Xbar + QT*RSE*sqrt(1+ho) # lower/upper predict. limits

C <- ALEPH * 10^c(Xlo , Xbar , Xhi); names(C) <- c("lwr", "fit", "upr")

print(C) # show the estimated limits and value using numerical values of this report

lwr fit upr

0.5779972 0.8452142 1.2359697 # Lower limit , the prediction , and upper limit

# If one has the model object (the.lm) that contains the PHOS regression , then

# the prediction limits are extracted using more significant figures than in this report.

print(sort(ALEPH * 10^( predict.lm(the.lm , interval =" prediction", level =0.90))[4,]))

lwr fit upr

0.5775619 0.8444275 1.2345999 # Values acceptably similar to previous , reliability is shown.

Figure 19. Example computations (computations of estimates and 90-percent prediction limits) for total phosphorus (Phos) for

08068500 Spring Creek near Spring, Tex..
40 Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9

Appendix 3—Example computations (definition of terms and quantities, concentration estimates, . . . 13

Example computations (computation of load) for total phosphorus (Phos) for 08068500 Spring

Creek near Spring, Tex.

Unit conversion to 15-minute load from streamflow and constituent concentration

The relation between an 15-minute load L, streamflow Q, and Phos concentration C is determined as follows

L



kilogram

day 

= Q



cubic feet

second 

×C



milligram

liter 

×



900 seconds

15 minutes 

×



28.317 liter

cubic feet 

×

 kilogram

1×106 milligram

  

K = 0.02549

Computation of 15-minute load of total phosphorus

The 15-minute load L in kilograms per 15 minutes for streamflow Q = 40 cubic feet per second of total phosphorus with a

concentration of C = 0.8452 milligrams per liter is computed by

L = Q×C ×K

L = 40×0.8452×0.02549

L = 0.862 kilograms per 15 minutes

Computation of 15-minute load of total phosphorus by using the R environment for statistical computing (R Development

Core Team, 2010)

# Q is streamflow from code shown on previous page (40 cubic feet per second)

# C is the lower , estimate , and upper concentrations shown on previous page

# (0.5779972, 0.8452142, 1.2359697)

K <- 0.02549 # unit conversion factor shown in previous hand derivation shown

above

L <- Q * C * K # compute the 15-minute prediction and prediction limits

print(L) # show the 15-minute load along with prediction limits

lwr fit upr

[1] 0.5893259 0.8617804 1.2601947

Thus, the 15-minute load of total phosphorus with 90-percent prediction limits is

0.589 ≤ L = 0.862 ≤ 1.26 kilograms per 15 minutes.

Figure 20. Example computations (computation of load) for total phosphorus (Phos) for 08068500 Spring Creek near Spring,

Tex..
Lee and others—Regression Model Development and Computational Procedures in Two Tributaries to Lake Houston, 2005–9—Scientific Investigations Report 2012–5006

ISBN 978-1 4113 3368-0

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