In cooperation with the Texas Department of Transportation 

Alternative Regression Equations for Estimation of 

Annual Peak-Streamflow Frequency for Undeveloped 

Watersheds in Texas using PRESS Minimization 

Scientific Investigations Report 2008–5084
 

(Texas Department of Transportation Research Report 0–5521–2)
 

U.S. Department of the Interior 

U.S. Geological Survey 
— Technical Report Documentation Page — 

1. Report No. 2. Government Accession No. 3. Recipient’s Catalog No. 

FHWA/TX–08/0–5521–2 

4. Title and Subtitle 5. Report Date 

Alternative Regression Equations for Estimation of June 2008 

Annual Peak-Streamflow Frequency for Undeveloped 

Watersheds in Texas using PRESS Minimization 

7. Authors(s) 8. Performing Organization Report No. 

William H. Asquith and David B. Thompson USGS SIR 2008–5084 

9. Performing Organization Name and Address 10. Work Unit No. (TRAIS) 

U.S. Geological Survey 

8027 Exchange Drive 11. Contract or Grant No. 

Austin, Texas 78754 Project 0–5521 

12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered 

Texas Department of Transportation Technical report on research from 2007 

Research and Technology Implementation Office to 2008 

P.O. Box 5080 14. Sponsoring Agency Code 

Austin, Texas 78731 

15. Supplementary Notes 

Project conducted in cooperation with the Texas Department of Transportation and the Federal Highway Administration 

16. Abstract 

The U.S. Geological Survey, in cooperation with the Texas Department of Transportation and in partnership with Texas Tech 

University, investigated a refinement of the regional regression method and developed alternative equations for estimation of 

peak-streamflow frequency for undeveloped watersheds in Texas. A common model for estimation of peak-streamflow frequency 

is based on the regional regression method. The current (2008) regional regression equations for 11 regions of Texas are based 

on log10 transformations of all regression variables (drainage area, main-channel slope, and watershed shape). Exclusive use of 

log10-transformation does not fully linearize the relations between the variables. As a result, some systematic bias remains in the 

current equations. The bias results in overestimation of peak streamflow for both the smallest and largest watersheds. The bias 

increases with increasing recurrence interval. The primary source of the bias is the discernible curvilinear relation in log10 space 

between peak streamflow and drainage area. Bias is demonstrated by selected residual plots with superimposed LOWESS trend 

lines. To address the bias, a statistical framework based on minimization of the PRESS statistic through power transformation of 

drainage area is described and implemented, and the resulting regression equations are reported. Compared to log10-exclusive 

equations, the equations derived from PRESS minimization have PRESS statistics and residual standard errors less than the log10 ­

exclusive equations. Selected residual plots for the PRESS-minimized equations are presented to demonstrate that systematic bias 

in regional regression equations for peak-streamflow frequency estimation in Texas can be reduced. Because the overall error is 

similar to the error associated with previous equations and because the bias is reduced, the PRESS-minimized equations reported 

here provide alternative equations for peak-streamflow frequency estimation. 

17. Key Words 18. Distribution Statement 

Flood Frequency, Peak Streamflow, Texas No restrictions. 

This document is available to the public through the 

National Technical Information Service, Springfield 

Virginia, 22161, www.ntis.gov

19. Security Classif. (of report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price 

Unclassified Unclassified 40 

Form DOT F 1700.7 (8-72) Reproduction of complete page authorized 

Front cover: 

U.S. Geological Survey station 08134000 North Concho River near Carlsbad, Tex., May 12, 1953. Photographs by J.D. Yost.

Top: Panorama from left bank looking directly across channel showing gage shelter and control (dam); flow about 7 feet over dam at 

gage height of about 9.7 feet. 

Bottom: From left bank looking across channel showing extent of washout at the station. 

Back cover:

U.S. Geological Survey station 08171000 Blanco River near Wimberley, Tex., June 28, 1928. Photographer unknown. 

Left: Gage shelter and stilling well. 

Right: Gage shelter at far left (to left of automobiles). 
Alternative Regression Equations for Estimation of 

Annual Peak-Streamflow Frequency for Undeveloped 

Watersheds in Texas using PRESS Minimization 

By William H. Asquith and David B. Thompson 

In cooperation with the Texas Department of Transportation 

Scientific Investigations Report 2008–5084

(Texas Department of Transportation Research Report 0–5521–2)

U.S. Department of the Interior 

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

DIRK KEMPTHORNE, Secretary 

U.S. Geological Survey 

Mark D. Myers, Director 

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

For product and ordering information:

World Wide Web: http://www.usgs.gov/pubprod

Telephone: 1-888-ASK-USGS

For more information on the USGS—the Federal source for science about the Earth, its natural and living resources,

natural hazards, and the environment:

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Telephone: 1-888-ASK-USGS

Any use of trade, product, or firm names in this publication is for descriptive purposes only and does not imply

endorsement by the U.S. Government.

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:

Asquith, W.H., and Thompson, D.B., 2008, Alternative regression equations for estimation of annual peak-streamflow 

frequency for undeveloped watersheds in Texas using PRESS minimization: U.S. Geological Survey Scientific Investigations Report 2008–5084, 40 p. [ http://pubs.usgs.gov/sir/2008/5084 ] 

Typeset on July 9, 2008
iii 

Contents 

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

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

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

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

Current (2008) Regional Regression Equations for Peak-Streamflow Frequency Estimation in Texas . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Inconsistent Peak-Streamflow Frequency Curves by Regional Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Regional Regression Equation Applicability and Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Biased Peak-Streamflow Frequency Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas . . . . . . . . . . . . . . . . . . . . . . 4

Regression Equations Based on Logarithmic Transformation of Drainage Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Regression Equations Based on PRESS Minimization and Power Transformation of Drainage Area . . . . . . . . . . . . . . . . . . . . . . . 6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Appendix 1—Computational Script Using PRESS Minimization and Drainage Area, Mean Annual Precipitation, 

and Main-Channel Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Appendix 2—Computational Script Using PRESS Minimization and Drainage Area and Mean Annual Precipitation . . . . . . . . . . . . . . . 25

Appendix 3—Computational Script Using PRESS Minimization and Drainage Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Figures 

1.Residual plot of regression of 100-year peak streamflow using logarithmic transformation of drainage area using

three predictor variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.Residual plot of regression of 100-year peak streamflow using logarithmic transformation of drainage area using

drainage area as the only predictor variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.Conceptual display of the PRESS statistic minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4.Residual plot of regression of 100-year peak streamflow using power transformation of drainage area using three

predictor variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.Residual plot of regression of 100-year peak streamflow using power transformation of drainage area using

drainage area as the only predictor variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6.Graph showing comparison of PRESS statistics from regression analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7.Graph showing relation between bias ratio and drainage area by recurrence interval for the regressions using

drainage area as the only predictor variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Tables 

1.Asquith and Slade (1997) regression equations for region 11 (southeast Texas) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.Summary statistics of basin characteristics used in regression analysis described in this report . . . . . . . . . . . . . . . . . . . . . 4
iv 

3. U.S. Geological Survey streamflow-gaging stations identified as outliers and removed from analysis . . . . . . . . . . . . . . . . . . 5 

4. Regression equations based on logarithmic transformation of drainage area using three predictor variables . . . . . . . . . . . 12 

5. Regression equations based on logarithmic transformation of drainage area using two predictor variables . . . . . . . . . . . . 12 

6. Regression equations based on logarithmic transformation of drainage area using one predictor variable . . . . . . . . . . . . . 12 

7. Regression equations based on power transformation of drainage area using three predictor variables . . . . . . . . . . . . . . . 13

8. Regression equations based on power transformation of drainage area using two predictor variables . . . . . . . . . . . . . . . . 13

9. Regression equations based on power transformation of drainage area using one predictor variable . . . . . . . . . . . . . . . . . 13

Conversion Factors 

Inch/Pound to SI 

Multiply By To obtain 

Length 

inch (in.) 25.4 millimeter (mm) 

foot (ft) .3048 meter (m) 

mile (mi) 1.609 kilometer (km) 

Area 

square mile (mi2) 2.590 square kilometer (km2) 

Flow 

cubic feet per second (ft3/s) .02832 cubic meters per second (m3/s) 

SI to Inch/Pound

Multiply By To obtain 

Length 

millimeter (mm) 0.03937 inch (in.) 

meter (m) 3.281 foot (ft) 

Area 

square kilometer (km2) .3861 square mile (mi2) 

Flow 

cubic meters per second (m3/s) 35.31 cubic feet per second (ft3/s) 
Alternative Regression Equations for Estimation of 

Annual Peak-Streamflow Frequency for Undeveloped 

Watersheds in Texas using PRESS Minimization 

By William H. Asquith and David B. Thompson1 

Abstract 

The U.S. Geological Survey, in cooperation with the 

Texas Department of Transportation and in partnership 

with Texas Tech University, investigated a refinement of 

the regional regression method and developed alternative 

equations for estimation of peak-streamflow frequency for 

undeveloped watersheds in Texas. A common model for 

estimation of peak-streamflow frequency is based on the 

regional regression method. The current (2008) regional 

regression equations for 11 regions of Texas are based on 

log10 transformations of all regression variables (drainage 

area, main-channel slope, and watershed shape). Exclusive use of log10-transformation does not fully linearize the 

relations between the variables. As a result, some systematic bias remains in the current equations. The bias results 

in overestimation of peak streamflow for both the smallest 

and largest watersheds. The bias increases with increasing recurrence interval. The primary source of the bias is 

the discernible curvilinear relation in log10 space between 

peak streamflow and drainage area. Bias is demonstrated 

by selected residual plots with superimposed LOWESS trend 

lines. To address the bias, a statistical framework based on 

minimization of the PRESS statistic through power transformation of drainage area is described and implemented, and 

the resulting regression equations are reported. Compared 

to log10-exclusive equations, the equations derived from 

PRESS minimization have PRESS statistics and residual standard errors less than the log10-exclusive equations. Selected residual plots for the PRESS-minimized equations are 

presented to demonstrate that systematic bias in regional 

regression equations for peak-streamflow frequency estimation in Texas can be reduced. Because the overall error is 

similar to the error associated with previous equations and

because the bias is reduced, the PRESS-minimized equa1Texas Tech University, Lubbock, Tex. 

tions reported here provide alternative equations for peakstreamflow frequency estimation. 

Introduction 

Peak-streamflow frequency estimates are needed for 

flood-plain management; for objective assessment of flood 

risk; for cost-effective design of dams, levees, and other 

flood-control structures; and for design of roads, bridges, 

and culverts. Peak-streamflow frequency represents the 

peak streamflow for recurrence intervals of 2, 5, 10, 25, 

50, and 100 years. 

Beginning in 2003, the U.S. Geological Survey 

(USGS), in cooperation with the Texas Department 

of Transportation (Research Project 0–4405) and in 

partnership with Texas Tech University, began a 3-year 

investigation of the influence of hydrologic scale (represented by drainage area for this report) on hydrologic 

model performance (Asquith and Thompson, 2005; 

Thompson, 2006). Hydrologic models for estimation of 

design floods are in widespread use by TXDOT engineers 

and the broader hydrologic engineering community. A 

common model for estimation of peak-streamflow frequency is based on the regional regression method. This 

method is the subject of this report. 

Bias exists in the regional regression equations for estimation of peak-streamflow frequency in Texas (Asquith 

and Slade, 1997), hereinafter referred to as AS1997. The 

source of the bias is the discernible curvilinear relation 

between peak streamflow and drainage area—the bias is 

graphically illustrated in this report. The current regional 

regression equations might overestimate peak-streamflow 

for both the smallest and largest watersheds represented 

in the AS1997 investigation. The bias is scale-dependent 

(depends on the size of the drainage area) and can be 

reduced. 
2 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

Purpose and Scope 

The primary purpose of this report, which parallels 

the discussion of Asquith and Thompson (2005), is to use 

an alternative statistical framework to develop regression 

equations with potentially less bias and therefore enhanced 

prediction capability—in particular, enhanced prediction 

capability for small watersheds. For this report, a small 

watershed is defined as having a contributing drainage 

area less than about 10 square miles. Peak-streamflow 

frequency estimation for small undeveloped (rural) and 

ungaged watersheds in Texas is a major concern for TxDOT 

engineers. The alternative framework uses a technique 

involving the minimization of the PRESS (PRediction Error 

Sum of Squares) statistic (Helsel and Hirsch, 2002, p. 247). 

The secondary purpose of this report is to present regression 

equations based on PRESS minimization for the estimation 

of peak-streamflow frequency at ungaged sites in undeveloped watersheds in Texas. Finally, the tertiary purpose of 

this report is to present “statewide” regression equations for 

Texas lacking a context of specific geographic regions. 

The scope of the report is limited to the at-site peakstreamflow frequency values for 664 USGS streamflowgaging stations used in AS1997 and digitally tabulated in 

Asquith and Slade (1999, file tx664.dat). The alternative regression equations presented here are based on the 

entire study area (Texas and slight overlap with surrounding 

states) of AS1997. The scope of the report does not include 

consideration of the spatial dependence of peak-streamflow 

frequency beyond its association with mean annual precipitation. 

Acknowledgments 

The authors recognize the following colleagues of 

TXDOT for their contributions and support of this work: 

George R. Herrmann, 0–5521 Project Director and Amy J. 

Ronnfeldt, 0–5521 Project Advisor. 

Current (2008) Regional Regression 

Equations for Peak-Streamflow Frequency Estimation in Texas 

The current (2008) regional regression equations are 

provided by AS1997, who provide 96 equations to estimate 

the 2-, 5-, 10-, 25-, 50-, and 100-year annual peak discharge (the peak-streamflow frequency curve) for undeveloped watersheds in Texas. The equations use the watershed 

characteristics of drainage area, main-channel slope, and 

watershed shape as predictor variables. AS1997 divides 

Texas into 11 regions. The mean number of stations used 

for each equation is 36. For each region, 6 or 12 weightedleast-squares regression equations were developed using a 

forward stepwise procedure. The distinction between 6­

and 12-equation regions is elaborated upon later in this section. 

The AS1997 statistical analysis is sound, with innovative methods of equation development and presentation, 

and widely used (in a second printing); however, three 

observations regarding the AS1997 procedural framework 

are important for this report. The observations are important because they relate to application or implementation of 

the AS1997 equations by end users involved in public and 

private infrastructure design. The observations gradually 

developed over the years since publication of AS1997 and 

were refined for this investigation. The three observations 

are described in the following sections. 

Inconsistent Peak-Streamflow Frequency 

Curves by Regional Regression 

For a given region, watershed characteristics used to 

develop the AS1997 regression equations for the 2- through 

100-year equations are inconsistent; a fact that can be 

attributed to statistically inconsistent peak-streamflow frequency curve for some watersheds. By definition, a peakstreamflow frequency curve must monotonically increase 

with increasing recurrence interval. The term inconsistent 

in this context means that the computed discharge for a 

recurrence interval exceeds the discharge for a larger recurrence interval. For example, the 50-year peak streamflow 

is computed to be greater than the 100-year peak streamflow. The source of the peak-streamflow inconsistency is 

the inconsistent use of watershed characteristics within an 

equation ensemble (a set of equations for a given region). 

The inconsistency in watershed characteristics exists 

because AS1997 used a forward stepwise regression procedure and did not specifically force predictor variables into 

the equations. For example, the equations for region 11 of 

AS1997 (southeastern Texas) are listed in table 1. Mainchannel slope is not used for the 2-year recurrence interval, 

but it is for larger recurrence intervals. Watershed shape is 

used for the 2- through 10-year recurrence intervals, but it is 

not used for larger recurrence intervals. Although difficult 

to visualize, combinations of watershed characteristics can 

be substituted into the equations listed in table 1 to produce 

an inconsistent frequency curve. 

AS1997 explicitly discusses the potential for inconsistent peak-streamflow frequency curves from the equations (Asquith and Slade, 1997, p. 11). When the equations 
Current (2008) Regional Regression Equations for Peak-Streamflow Frequency Estimation in Texas 3 

and guidance on equation application originally were developed, the authors (Asquith and Slade) anticipated that end 

users would apply “hydrologic engineering judgement” to 

manually mitigate peak-streamflow inconsistencies. However, numerous end users have communicated to the senior 

author a degree of confusion or frustration in regard to 

application of the AS1997 equations, which indicates a 

need for alternative equations that will not produce, or have 

a greatly reduced potential for producing, inconsistent peakstreamflow frequency curves. 

Regional Regression Equation Applicability 

and Implementation 

AS1997 provides numerous figures (Asquith and 

Slade, 1997, figs. 4–14) in which the relations between 

drainage area, main-channel slope, and watershed shape 

are graphically depicted for each of the 11 regions. 

Superimposed on these plots are generalized “convex 

hulls”2 representing the “approximate [parameter space] 

defined by [watershed] characteristics” for each region. 

For watersheds having coordinates of drainage area, 

main-channel slope, and watershed shape outside the 

convex hull, the applicability of the equations for the 

region is uncertain, and the potential for an inconsistent 

peak-streamflow frequency curve increases. 

Since publication of AS1997, the senior author has 

learned from interaction with end users that the convex 

hulls presented in AS1997 commonly are underutilized. 

2Quoting from http://en.wikipedia.org/wiki/Convex_hull accessed 

on February 19, 2008: “For planar objects, [those] lying in the plane, the convex 

hull may be easily visualized by imagining an elastic band stretched open to encompass the given object; when released, it will assume the shape of the required convex 

hull.” 

Table 1. Asquith and Slade (1997) regression equations for 

region 11 (southeast Texas). 

[QT , peak streamflow for T-year recurrence interval in cubic feet 

per second; A, drainage area in square miles; S, main-channel 

slope in feet per mile; and H, dimensionless watershed shape. 

Sixty-six stations were used in the regression development.] 

Regression equation 

Adjusted 

Rsquared 

Residual 

standard 

error 

log10(QT )

Q2 = 159A0.669 H−0.262 0.91 0.18 

Q5 = 191A0.696 S0.130 H−0.186 .91 .18 

Q10 = 199A0.718 S0.221 H−0.151 .90 .20 

Q25 = 201A0.713 S0.313 .88 .22 

Q50 = 207A0.735 S0.380 .86 .24 

Q100 = 213A0.755 S0.442 .85 .26 

Furthermore, some end users abstracted (reproduced) for 

application only the equations from AS1997. As a result, 

important context that contributes to optimum use of the 

equations is lost. 

The apparent lack of full adherence to the entire procedural framework and caveats of the AS1997 regional 

regression equations is understandable given that AS1997 

provides 96 separate equations and considerable detail. 

Therefore, a simpler regional regression method for estimation of peak-streamflow frequency in Texas would be 

useful. 

Biased Peak-Streamflow Frequency Values 

The multiple linear regional regression equations of 

AS1997 are exclusively based on log10 transformations of 

observed peak-streamflow frequency values, drainage area, 

main-channel slope, and watershed shape. Multiple linear 

regression is based on a linear relation between the regressor variable (peak-streamflow frequency) and the predictor variables (drainage area, main-channel slope, watershed 

shape, and others). AS1997 (Asquith and Slade, 1997, 

p. 8) notes that, for some regions, peak-streamflow values (for example, the 100-year peak streamflow) have a 

discernible curvilinear relation with drainage area in log10 

space. AS1997 addresses the nonlinearity (and thus mitigates the bias) by classifying watersheds into two ranges of 

drainage area. Separate regional regression analyses were 

done for watersheds with drainage areas less than 32 square 

miles and for watersheds with areas greater than 32 square 

miles. The 32-square-mile break point was determined by 

data interpretation. The drainage-area distinction and bias 

mitigation is explicitly discussed in AS1997 (Asquith and 

Slade, 1997, p. 13). 

The drainage-area classification was not made for six 

of the 11 regions because either the number of watersheds 

was small (degrees of freedom for regression) within a 

region or an absence of a discernible curvilinear relation 

between log10-transformed peak streamflow and drainage 

area was perceived. For a region in which the drainage-area 

classification was made, 12 equations for the region were 

developed—six equations for watersheds with drainage 

areas less than 32 square miles and six equations for watersheds with drainage areas greater than 32 square miles. 

Conversely, six equations were developed for regions in 

which nonlinearity was not apparent and no drainage-area 

classification was made. 

The drainage-area classification complicates application of the equations for watersheds near the 32-square mile 

break point. AS1997 (Asquith and Slade, 1997, p. 12) provides an ad hoc procedure to prorate estimates for water­
4 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

sheds of 10 to 100 square miles between the equation 

ensemble for drainage areas less than or equal to 32 square 

miles and the ensemble for drainage areas greater than or 

equal to 32 square miles. If the proration procedure is not 

followed, “jumps” in peak streamflow at 32 square miles 

will result. 

The nonlinearity is apparent in the graphical depiction 

of the 32-square-mile classification technique to mitigate 

for nonlinearity (Asquith and Slade, 1997, figs. 3 and 15). 

Despite the measures to address the nonlinearity and thus 

mitigate bias, the AS1997 equations still have the potential 

to overestimate peak-streamflow frequency values for both 

the smallest and largest watersheds. As noted, eliminating 

or reducing the potential for inconsistent peak-streamflow 

frequency curves and making the regional regression equation method easier for end users to apply are ancillary 

reasons to develop the alternative equations shown in this 

report; but another reason for development of alternative 

equations is to remove the bias inherent in the AS1997 

equations. 

Typical regression practice to reduce underestimation 

or overestimation (peak-streamflow frequency values for 

this report) is to seek an alternative transformation on the 

regressor variable (Maindonald and Braun, 2003, p. 126– 

127). Some readers might question why an alternative transformation on drainage area (a predictor variable) is sought 

rather than an alternative transformation on the 2- through 

100-year peak-streamflow values (regressor variables). The 

authors chose to assess an alternative transformation on 

drainage area so that the residual standard errors (log10 

units of streamflow) reported are directly comparable to 

those from AS1997. 

Alternative Regression Equations 

for Estimation of Peak-Streamflow 

Frequency for Watersheds in Texas 

Regression Equations Based on Logarithmic 

Transformation of Drainage Area 

The traditional practice for development of regression equations to estimate peak-streamflow frequency is to 

transform regressor variables (the at-site peak-streamflow 

frequency values, such as the 2- through 100-year peak 

streamflows) and all the predictor variables (Stedinger and 

others, 1993, p. 18.35) by the log10 function. Drainage area, 

a measure of watershed slope, and other characteristics 

are common predictor variables. AS1997 considered six 

characteristics: 2-year 24-hour precipitation, mean annual 

precipitation (1951–80)3, drainage area, stream length, 

basin shape factor, and main-channel slope. The precipitation statistics reported in AS1997 are for the approximate 

watershed centroid. However, for the equations reported 

in AS1997, only drainage area, main-channel slope, and 

watershed shape are used. For this report, only drainage 

area, mean annual precipitation, and main-channel slope 

are used. 

Because of the ubiquitous nature of log10 transformation in hydrologic analyses, important comparative analysis 

for this report is facilitated by developing regression equations using log10 transformation on the same data used in 

AS1997. However, no designation of geographic region is 

used for this report. AS1997 considered data for 664 USGS 

streamflow-gaging stations. From preliminary data analysis 

(results not presented here), eight stations were identified 

as outliers on the basis of the relative change of exploratory 

regressions and associated diagnostics when each of the 

outlying stations were individually dropped. These stations 

were eliminated from further analysis. The summary statistics (table 2) of drainage area, mean annual precipitation, 

and main-channel slope were computed after the removal 

of the eight stations listed in table 3. 

Weighted-least squares regression on the 2-, 5-, 10-, 

25-, 50-, and 100-year peak-streamflow values for the remaining 656 streamflow-gaging stations is accomplished 

using drainage area, mean annual precipitation, and mainchannel slope as predictor variables. For comparison, the 

mean number of stations per equation in AS1997 is 36. 

Therefore, the degrees of freedom for the regression equations reported here are about 18 times larger than those of 

AS1997. 

Analysis of collinearity through variance inflation factors and statistical significance (results not reported here) 

strongly indicated that inclusion of watershed shape in the 

regression equations in addition to drainage area, mean 

3This period coincides with the most undeveloped peak-streamflow data and 

streamflow-gaging stations in Texas. 

Table 2. Summary statistics of basin characteristics used in 

regression analysis described in this report. 

[A, drainage area, in square miles; P, mean annual precipitation, 

in inches (1951–1981); S, main-channel slope, in feet per mile] 

Mininum

1st 

Quartile 

Median

3rd 

Quartile 

Maximum 

A 0.10 7.2 128 6,960 174,000 

P 8 22 31 41 57 

S .38 7.35 13.4 33.6 371 
Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 5 

Table 3. U.S. Geological Survey streamflow-gaging stations identified as outliers and removed from analysis. 

Station no. Station name Drainage area 

(square miles) 

08039900 Little Sandy Creek tributary near Jasper, Texas 0.46 

08080700 Running Water Draw at Plainview, Texas 382 

08089100 Elm Creek tributary near Graford, Texas 1.10 

08210400 Lagarto Creek near George West, Texas 155 

08383200 Pintada Arroyo tributary near Clines Corners, New Mexico 29.20 

08393600 North Spring River at Roswell, New Mexico 19.50 

08405050 Last Chance Canyon tributary near Carlsbad Caverns, New Mexico .20 

08434000 Toyah Creek below Toyah Lake near Pecos, Texas 3,709 

annual precipitation, and main-channel slope is not appropriate. The six regression equations are listed in table 4. 

For all six equations, the p-values for the coefficients on the 

watershed characteristics are less than .0001, which means 

that the variables and intercept are all highly statistically 

significant. 

A simple comparison between 100-year residual standard error listed in table 4 and the weighted-mean 100-year 

residual standard error from AS1997 provides perspective. 

The weighted-mean 100-year residual standard error from 

AS1997 is computed by weighting the errors in AS1997 

(Asquith and Slade, 1997, table 2) by the number of stations for each region. The weighted-mean, 100-year, residual standard error from AS1997 is about 0.27; the 100­

year residual standard error listed in table 4 is 0.34. These 

two residual standard errors are of similar magnitude. Additional comparisons of residual standard errors listed in 

table 4 to those in AS1997 indicate that all have about the 

same magnitude, although overall the errors are greater for 

the equations reported here. The conclusion from this comparison is that the six equations in table 4 have approximately the same residual standard error as the 96 equations 

reported in AS1997. 

For the equations in table 4, inclusion of mean annual 

precipitation for the watershed is useful. Mean annual 

precipitation becomes a surrogate for spatial location that 

replaces the concept of geographic region designation associated with the equations in AS1997. Mean annual precipitation was not used in AS1997 for the final equations shown 

in that report. 

Bias in multiple linear regression is well depicted in 

a residual (observed minus predicted) graph in which the 

residual for a particular data point is plotted on the vertical 

axis and the corresponding fitted value is plotted on the 

horizontal axis. If there is a discernible trend or shape in 

the graph—that is, a tendency for residuals to plot above 

or below the zero-residual line, then bias in the equation 

exists. 

Residuals for the 100-year peak-streamflow equation 

listed in table 4 are graphed in figure 1. A LOWESS 

(LOcally WEighted Scatterplot Smoothing) trend line 

(Cleveland, 1979) through the data is superimposed. 

The lowess() function of the R software package (R 

Development Core Team, 2006) with default settings was 

used. The concave-down shape of the LOWESS trend line 

indicates systematic bias in the regression. The negative 

magnitudes of the left and right segments of the LOWESS 

trend line indicate that overestimation of the 100-year 

peak-streamflow occurs for watersheds with small and 

large fitted values (the smallest and largest watersheds, 

respectively). 

The LOWESS trend line is only an indicator of bias 

and does not represent a true bias correction; however, 

interpretation of the line as a bias measure is useful. For 

example, referring to figure 1, for a fitted value of about 

2.5 (316 cubic feet per second) and a LOWESS-indicated 

bias of about −0.25, a more appropriate value might be 

2.5 − 0.25 = 2.25 (178 cubic feet per second). Therefore, the bias-corrected value, albeit ad hoc, is about 44 

percent less than the fitted value. In general, the log10 ­

exclusive regressions in table 4 have concave-down trend 

lines through the residuals. The concavity of the LOWESS 

trend line (interpreted as bias in the equations) increases 

with increasing recurrence interval (results not presented 

here). 

Hydrologic scale typically is measured by drainage 

area. Therefore, it is informative to develop second and 

third sets of log10-transformed regression equations on the 

same 656 stations using drainage area and mean annual 

precipitation (table 5) and only drainage area (table 6) as 

predictor variables. 
6 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

FITTED Q100, IN LOG10(CUBIC FEET PER SECOND)

OBSERVED MINUS PREDICTED Q100,

IN LOG10(CUBIC FEET PER SECOND

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0

0.25

0.50

0.75

1.00

1.25

LOcally-WEighted Scatterplot Smoothing (LOWESS) trend line

Regression residual

EXPLANATION

Figure 1. Residual plot of regression of 100-year peak streamflow using logarithmic transformation of drainage area using 

three predictor variables. 

For these 12 equations, the p-values for the coefficients 

on the intercept, drainage area, and mean annual precipitaiton are less than .0001. The residual standard errors associated with the equations in tables 5 and 6 are all greater 

than those listed in table 4 and because one and two fewer 

predictor variables are in the equations in tables 5 and 6, 

respectively. 

Residuals for the 100-year peak-streamflow equation 

listed in table 6 are shown in figure 2. The LOWESS 

trend line superimposed through the data has considerable 

downward concavity similar to the trend line in figure 1. 

The interpretations of the regressions in table 6 using the 

LOWESS trend line on the residual plot are the same as 

those for the regression equations in table 4. Specifically, 

peak streamflow is overestimated for watersheds with small 

fitted values (the smallest watersheds) and for watersheds 

with large fitted values (the largest watersheds). The bias 

is considerable. The concavity of the LOWESS trend line 

increases with increasing recurrence interval (results not 

presented here). 

In conclusion, systematic bias is present in the regression equations reported in tables 4–6, and by general 

method association, bias is present in the AS1997 equations. The bias exists because of the curvilinear relation 

between log10-transformed peak streamflow and drainage 

area. The bias is mitigated in the AS1997 analysis by separating regressions into two groups on the basis of watershed 

drainage area, less than or greater than 32 square miles. 

The relation between log10-transformed peak streamflow 

and drainage area becomes increasingly curvilinear with 

increasing recurrence interval. 

Regression Equations Based on PRESS Minimization and Power Transformation of 

Drainage Area 

The PRESS statistic generally is regarded as a measure 

of regression performance when the model is used to predict new data (Montgomery and others, 2001, p. 153). Prediction of new data is what analysts and engineers do when 

they estimate peak streamflow from a regression equation. 

Regression equations with small PRESS values are desirable. Thus, PRESS minimization is an appropriate goal. 

Helsel and Hirsch (2002, p. 247) state that, “Minimizing 

PRESS means that the equation produces the least error 

when making new predictions.” Conceptually, PRESS minimization identifies the appropriate transformation to “press” 

the bias out of the equations (fig. 3). 
n � �2 ei PRESS = ∑w . i 1−hii i=1 

� e(i) = yi −yi, 

Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 7 

FITTED Q100, IN LOG10(CUBIC FEET PER SECOND)

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0

0.25

0.50

0.75

1.00

1.25

LOCally-WEighted Scatterplot Smoothing (LOWESS) trend line

Regression residual

EXPLANATION

OBSERVED MINUS PREDICTED Q100,

IN LOG10(CUBIC FEET PER SECOND)

Figure 2. Residual plot of regression of 100-year peak streamflow using logarithmic transformation of drainage area using 

drainage area as the only predictor variable. 

The PRESS statistic is computed from the PRESS resid- software packages. The PRESS computation is made by 

uals, which are defined as 

(3) (1) 

where e(i) is the PRESS residual, yi is the observed ith peakstreamflow value, and y�

i is the predicted value based on the 

remaining n−1 sample points. In other words, the ith station (data point) is not used to generate the ith regression 

equation. Thus, PRESS residuals are regarded as validation 

statistics. The PRESS statistic, with inclusion of the regression weight factor (wi), is 

n

∑wie2 

(i) PRESS = . 

i=1 

Because the PRESS statistic is an overall measure of 

regression fit (like residual standard error) and is a validation statistic (unlike residual standard error), minimization of PRESS is desirable. The most “valid” regression is 

produced when the PRESS statistic is minimized. The following transformation on drainage area was selected after 

exploratory analysis: 

(2) A� = Aλ , (4)

Equation 2 is computationally intensive (n regressions 

are required). A more efficient computation of PRESS is 

made using regression residuals (ei) and leverage (hii). (The 

double subscript ii refers to the diagonal of the hat matrix.4) 

These values are readily available from modern regression 

4Quoting from Montgomery and others (2001, p. 75): The n × n matrix H =

X(X�

X)−1X is usually called the hat matrix. It maps the vector of observed values 

into a vector of fitted values. The hat matrix and its properties play a central role in 

regression analysis. 

where A� is the transformed value for the regression, A is 

drainage area, and λ is a real number. The transformation 

is referred to in this report as the power transformation. 

Three computer programs were written in R (R Development Core Team, 2006) to loop through successive noninteger values of λ and record the value that yields a 

minimum PRESS for each of the six recurrence intervals. 

Tens of thousands of regressions were done in the process of exploratory data analysis and for the final minimization reported here. The first program implemented the 
8 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

E PR SS PRESS

Appropriate transformation

Candidate transformations

Bias

Bias

Figure 3. Conceptual display of the PRESS statistic minimization. 

watershed characteristics drainage area, mean annual precipitation, and main-channel slope as predictor variables; 

the second program implemented drainage area and mean 

annual precipitation; and the third program implemented 

only drainage area as a predictor variable. 

The programs and incremental output are provided in 

appendixes 1–3. The programs and output are included in 

the report to provide an archive of the PRESS minimization

algorithm and the regression analysis results summarized 

in tables 4–6 as well as tables 7–9. 

The results of the power transformation of drainage 

area using the three predictor variables are listed in table 7. 

The value of λ is the exponent on A in the equations. The 

value of λ increases in absolute magnitude with increasing 

recurrence interval; the larger the absolute value of λ, the 

larger the amount of concavity in the trend line of residuals 

(systematic bias) that is reduced. 

In all six equations, the p-values for the coefficients 

on the watershed characteristics are less than .0001. The 

diagnostic statistics of adjusted R-squared and residual standard error in the table are greater and less than, respectively, 

those in table 4. Therefore, the equations using the power 

transformation have less uncertainty. However, the PRESS 

statistic is the more important statistic to compare. 

The PRESS statistic for a given recurrence interval is 

less when the power transformation on drainage area is 

used instead of the log10 transformation. The percentage 

changes in the PRESS statistics associated with the power 

transformation (table 7) compared to those associated with 

the log10 transformation (table 4) show that, as recurrence 

interval increases, the power transformation produces an 

increasingly more valid regression. 

Residual standard errors of the PRESS-minimized equations in table 7 are similar to those of the equations reported 

in AS1997. For example, the 100-year residual standard 

error is about 0.33 and the AS1997 weighted value is 0.27 

for the 11 regions collectively. 

Residuals for the 100-year peak-streamflow equations 

using the three predictor variables (table 7) are shown in figure 4. Downward concavity of the superimposed LOWESS 

trend line is not present, unlike the LOWESS trend line 

in figure 1. In fact, the LOWESS trend line is essentially 

flat, which indicates that systematic bias in the equation 

is reduced through use of the specified power transformation. The power transformation on drainage area effectively 

linearizes the relation between 100-year peak streamflow 

and drainage area. Minimization of the PRESS statistic 

effectively removes systematic bias. Similar results (not 

reported here) were obtained for the other five recurrence 

intervals. 

The results of the power transformation of drainage 

area using drainage area and mean annual precipitation 

and only drainage area as predictor variables are listed in 

tables 8 and 9. Again, the value of λ is the exponent on A in 

the equations. In all 12 equations, the p-values for the coefficients on the watershed characteristics are less than .0001. 

Adjusted R-squared and residual standard error for regression based on power transformation are greater and less 

than, respectively, for those regressions based exclusively 

on log10 transformation (tables 5 and 6). Therefore, the 

equations using the power transformation have less uncertainty. The PRESS statistic for a given recurrence interval 

is less when the power transformation on drainage area is 

used instead of the log10 transformation. The percentage 

changes in the PRESS statistic associated with the power 

transformation (tables 8 and 9) compared to those associated with the log10 transformation (tables 5 and 6) show 

that, as recurrence interval increases, the power transformation produces an increasingly more valid regression. 

Residuals for the 100-year peak-streamflow equations 

in table 9 are graphed in figure 5. The concave-down shape 

of the superimposed LOWESS trend line in the residuals 

graph from the log10 transformation (fig. 2) is not present 

in the graph derived from the power transformation. In fact, 

the LOWESS trend line is essentially flat (fig. 5), which indicates that systematic bias in the equation has been reduced. 

The authors conclude that the power transformation on 

drainage area effectively linearizes the relation between 

100-year peak streamflow and drainage area. Minimization of PRESS effectively removes systematic bias. Similar 

results (not reported here) were obtained for the other five 

recurrence intervals. 

The PRESS statistics for the equations in tables 7 and 9 

are shown graphically by recurrence interval in figure 6 (the 

PRESS statistics for the equations in table 8 are not shown). 

From the figure it is clear that the power transformation 

with PRESS minimization produces PRESS statistics less 

than those from the log10-exclusive equations. PRESS minimization becomes increasingly important as recurrence 

interval increases because the log10 transformation does 

not produce a linear relation between peak streamflow and 

drainage area for the larger recurrence-interval events. 
Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 9 

FITTED Q100, IN LOG10(CUBIC FEET PER SEOCOND)

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0

0.25

0.50

0.75

1.00

1.25

LOcally-WEighed Scatterplot Smoothing (LOWESS) trend line

Regression residual

EXPLANATION

OBSERVED MINUS PREDICTED Q100,

IN LOG10(CUBIC FEET PER SECOND)

Figure 4. Residual plot of regression of 100-year peak streamflow using power transformation of drainage area using three 

predictor variables. 

FITTED Q100, IN LOG10(CUBIC FEET PER SECOND)

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

-1.50

-1.25

-1.00

-0.75

-0.50

-0.25

0

0.25

0.50

0.75

1.00

1.25

LOcally-WEighted Scatterplot Smoothing (LOWESS) trend line

Regression residual

EXPLANATION

OBSERVED MINUS PREDICTED Q100,

IN LOG10(CUBIC FEET PER SECOND)

Figure 5. Residual plot of regression of 100-year peak streamflow using power transformation of drainage area using drainage 

area as the only predictor variable. 
Qlog10 103.318A0.5094 y-axis in 100 = ≡ QPRESS 106.800−3.659A−0.0934 figure 7. 100 

10 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

Although the smallest PRESS statistics occur for the 5­

year recurrence interval, the PRESS statistics for the 2-year 

recurrence interval are not exceeded until the 25-year and 

larger recurrence intervals are reached. An interpretation 

of the PRESS statistic is that estimation of the 2-year peak 

streamflow using watershed characteristics is more difficult 

than estimation for the 5-year and 10-year peak streamflow. 

This observation is consistent with residual standard errors 

reported in AS1997. 

Finally, the magnitude and extent of the bias between 

the log10-exclusive regression and the PRESS-minimized 

regression is informative. The magnitude of the bias can be 

expressed as the ratio (the bias ratio) of the log10 equations 

(tables 4 or 6) to the PRESS-minimized equations (tables 7 

or 9). For example, the bias ratio for the 100-year peak 

streamflow for the drainage-area-only equations is 

(5)

When the ratio is greater than 1, the log10-exclusive 

regression overestimates peak streamflow relative to the 

PRESS-minimized regression. Similar equations of the bias 

ratio for other recurrence intervals are easily defined. Together, the six equations defining the bias ratio document 

the inherent differences between the log10-exclusive peakstreamflow equations and the PRESS-minimized equations. 

The extent of the bias ratio is shown by the ratio as a 

function of drainage area. An example, by recurrence interval, for the regressions using drainage area as the only 

predictor variable is shown in figure 7 (see eq. 5). An interpretation of the figure is that the log10-exclusive regressions 

overestimate peak-streamflow frequency for drainage areas 

less than about 8 square miles and drainage areas greater 

than about 2,000 square miles. The overestimation for 

drainage areas less than about 2 square miles is substantial. The overestimation for drainage areas less than about 

0.5 square mile exceeds 100 percent for all but the 2-year 

peak streamflow. Alternatively, the log10-exclusive regressions slightly underestimate peak-streamflow frequency for 

drainage areas between about 8 and 2,000 square miles. 

Summary 

Peak-streamflow frequency estimates are needed for 

flood-plain management; for objective assessment of flood 

risk; for cost-effective design of dams, levees, other floodcontrol structures; and for design of roads, bridges, and 

culverts. Peak-streamflow frequency represents the collective peak streamflow for recurrence intervals of 2, 5, 10, 25, 

50, and 100 years. 

The U.S. Geological Survey (USGS), in cooperation 

with the Texas Department of Transportation and in partnership with Texas Tech University, investigated a refinement 

of the regional regression method and developed alternative 

equations for estimation of peak-streamflow frequency for 

undeveloped watersheds in Texas. A common model for 

estimation of peak-streamflow frequency is based on the 

regional regression method, which relates peak-streamflow 

frequency to watershed characteristics. 

The current (2008) regional regression equations (96 

separate equations) for 11 geographic regions of Texas 

are based on log10 transformations on all regression variables (the peak-streamflow values and the watershed characteristics of drainage area, main-channel slope, and watershed shape). The log10 transformation does not fully linearize the relations between the variables, which is a major 

assumption in linear regression analysis. As a result, some 

systematic bias remains in the current equations. The primary source of the bias is the discernible curvilinear relation between peak streamflow and drainage area in log10 

space. The bias results in overestimation of peak streamflow for both the smallest and largest watersheds, and the 

bias increases with increasing recurrence interval. 

To demonstrate the extent of the bias, equations using 

log10(drainage area) for the study area (Texas and slight 

overlap with surrounding states) are reported. Separate 

regional distinction is not made for this report. Mean 

annual precipitation provides a surrogate for spatial location that replaces the concept of geographic region designation associated with the current equations. The use of 

mean annual precipitation reduces the number of equations 

for a given number of predictor variables (three, two, or 

one) from 96 to 6—one equation for each of the six recurrence intervals. To address the bias, a statistical framework 

based on minimization of the PRESS statistic through power 

transformation on drainage area is described. 

The PRESS statistic is an important measure of regression performance. It is a validation-type statistic, and small 

values are desirable. Minimization of PRESS is appropriate 

for peak-streamflow frequency analysis because the equations are used in hydrologic engineering practice to predict 

new data. 

Compared to log10(drainage area) equations, the equations derived from PRESS minimization have PRESS statistics and residual standard errors less than the log10(drainage 

area) equations. Selected residual plots for the PRESSminimized equations demonstrate that the systematic bias 
Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 11 

in regional regression equations for peak-streamflow frequency estimation in Texas can be reduced. Because the 

overall error is similar to the overall error associated with 

the equations currently in use and bias is reduced, the 

PRESS-minimized equations reported here provide alternative equations for peak-streamflow frequency estimation. 

References 

Asquith, W.H., and Slade, R.M., 1997, Regional equations for estimation of peak-streamflow frequency 

for natural basins in Texas: U.S. Geological Survey 

Water-Resources Investigations Report 96–4307, 68 p. 

[http://pubs.usgs.gov/wri/wri964307/] 

Asquith, W.H., and Slade, R.M., 1999, Site-specific 

estimation of peak-streamflow frequency using 

generalized least-squares regression for natural 

basins in Texas: U.S. Geological Survey WaterResources Investigations Report 99–4172, 19 p. 

[http://pubs.usgs.gov/wri/wri994172/] 

Asquith, W.H., and Thompson, D.B., 2005, Alternative regression equations for estimation of annual 

peak-streamflow frequency for undeveloped watersheds in Texas using PRESS minimization: Lubbock, 

Center for Multidisciplinary Research in Transportation, Texas Tech University, Texas Department of 

Transportation Research Report 0–4405–2, 27 p. 

[http://library.ctr.utexas.edu/pdf/4405-2.

pdf] 

Cleveland, W.W., 1979, Robust locally weighted regression and smoothing scatterplots: Journal of the American Statistical Association, v. 80, p. 829–836. 

Helsel, D.R., and Hirsch, R.M., 2002, Statistical methods in water resources: U.S. Geological Survey Techniques of Water-Resources 

Investigations, book 4, chap. A3, 510 p. 

[http://pubs.usgs.gov/twri/twri4a3/] 

Maindonald, J., and Braun, J., 2003, Data analysis and 

graphics using R—An example-based approach: Cambridge, United Kingdom, Cambridge University Press, 

362 p. 

Montgomery, D.C., Peck, E.A., and Vining, G.G., 2001, 

Introduction to linear regression analysis (3rd ed.): New 

York, Wiley, 641 p. 

R Development Core Team, 2006, R—A language and environment for statistical computing: Vienna, Austria, R Foundation for Statistical Computing, ISBN 3–900051–07–0, http://

www.R-project.org and direct link to report is 

[http://www.cran.r-project.org/doc/manuals/

refman.pdf] 

Stedinger, J.R., Vogel, R.M., and Foufoula-Georgiou, E., 

1993, Frequency analysis of extreme events, chapter 18, 

in Maidment, D.A., ed., Handbook of hydrology: New 

York, McGraw-Hill, p. 18.1–18.66. 

Thompson, D.B., 2006, The rational method, regional 

regression equations, and site-specific flood frequency relations: Lubbock, Center for Multidisciplinary Research in Transportation, Texas Tech 

University, Texas Department of Transportation Research Report 0–4405–1, 92 p., http://www.techmrt.

ttu.edu/reports.php and direct link to report is 

[http://www.techmrt.ttu.edu/Reports/

Complete%20Reports/4405-1.pdf] 
12 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas

Table 4. Regression equations based on logarithmic transformation of drainage area using three predictor variables.

[QT , peak streamflow for T-year recurrence interval in cubic feet per second; A, drainage area in square miles; P, mean annual 

precipitation in inches; and S, main-channel slope in feet per mile.] 

Regression equation 

Adjusted 

Rsquared 

Residual 

standard 

error 

log10(QT )

PRESS 

statistic 

Q2 = 10−0.5240 A0.6565 P1.474 S0.3525 0.8282 0.2866 54.36 

Q5 = 10−0.2204 A0.6790 P1.376 S0.4828 .8414 .2686 47.76 

Q10 = 10−0.04207 A0.6896 P1.317 S0.5421 .8310 .2778 51.12 

Q25 = 100.1501 A0.7005 P1.256 S0.6005 .8086 .2993 59.34 

Q50 = 100.2748 A0.7073 P1.218 S0.6359 .7887 .3186 67.25 

Q100 = 100.3879 A0.7133 P1.183 S0.6660 .7675 .3393 76.25 

Table 5. Regression equations based on logarithmic transformation of drainage area using two predictor variables. 

[QT , peak streamflow for T-year recurrence interval in cubic feet per second; A, drainage area in square miles; and P, mean annual 

precipitation in inches.] 

Regression equation 

Adjusted 

Rsquared 

Residual 

standard 

error 

log10(QT )

PRESS 

statistic 

Q2 = 100.8330 A0.5534 P0.9732 0.8153 0.2971 58.29 

Q5 = 101.639 A0.5378 P0.6896 .8157 .2895 55.32 

Q10 = 102.045 A0.5311 P0.5469 .7987 .3032 60.69 

Q25 = 102.462 A0.5249 P0.4029 .7698 .3282 71.10 

Q50 = 102.723 A0.5214 P0.3140 .7463 .3491 80.44 

Q100 = 102.952 A0.5186 P0.2366 .7224 .3707 90.72 

Table 6. Regression equations based on logarithmic transformation of drainage area using one predictor variable. 

[QT , peak streamflow for T-year recurrence interval in cubic feet per second; and A, drainage area in square miles.] 

Regression equation 

Adjusted 

Rsquared 

Residual 

standard 

error 

log10(QT )

PRESS 

statistic 

Q2 = 102.339 A0.5158 0.7642 0.3357 74.22 

Q5 = 102.706 A0.5111 .7889 .3099 63.28 

Q10 = 102.892 A0.5100 .7820 .3156 65.63 

Q25 = 103.086 A0.5093 .7612 .3343 73.67 

Q50 = 103.209 A0.5092 .7414 .3525 81.88 

Q100 = 103.318 A0.5094 .7199 .3724 91.39 
Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 13 

Table 7. Regression equations based on power transformation of drainage area using three predictor variables. 

[QT , peak streamflow for T-year recurrence interval in cubic feet per second; A, drainage area in square miles; P, mean annual 

precipitation in inches; and S, main-channel slope in feet per mile. The exponent of A is the power λ.] 

Regression equation 

Adjusted 

Rsquared 

Residual 

standard 

error 

log10(QT )

PRESS 

statistic 

Percent 

change from 

PRESS in 

table 4 

Q2 = 1035.60−36.09A−0.0082 P1.448 S0.3472 0.8286 0.2863 54.27 −0.17 

Q5 = 1011.16−11.28A−0.0299 P1.279 S0.4640 .8461 .2646 46.37 −2.9 

Q10 = 109.047−8.950 A−0.0400 P1.188 S0.5172 .8396 .2707 48.57 −5.0 

Q25 = 107.949−7.628 A−0.0497 P1.096 S0.5699 .8217 .2889 55.32 −6.8 

Q50 = 107.554−7.090 A−0.0553 P1.039 S0.6021 .8048 .3062 62.18 −7.5 

Q100 = 107.307−6.714A−0.0601 P0.9883 S0.6295 .7862 .3253 70.19 −7.9 

Table 8. Regression equations based on power transformation of drainage area using two predictor variables. 

[QT , peak streamflow for T-year recurrence interval in cubic feet per second; A, drainage area in square miles; and P, mean annual 

precipitation in inches. The exponent of A is the power λ.] 

Regression equation 

Adjusted 

Rsquared 

Residual 

standard 

error 

log10(QT )

PRESS 

statistic 

Percent 

change from 

PRESS in 

table 5 

Q2 = 1017.36−16.51A−0.0157 P0.9429 0.8162 0.2965 58.02 −0.46 

Q5 = 108.080−6.403A−0.0451 P0.6065 .8226 .2841 53.27 −3.7 

Q10 = 107.200−5.107A−0.0596 P0.4397 .8103 .2943 57.18 −5.8 

Q25 = 106.849−4.329A−0.0740 P0.2729 .7870 .3157 65.80 −7.5 

Q50 = 106.777−3.991A−0.0828 P0.1706 .7670 .3345 73.88 −8.2 

Q100 = 106.776−3.758 A−0.0903 P0.08212 .7458 .3548 82.98 −8.5 

Table 9. Regression equations based on power transformation of drainage area using one predictor variable. 

[QT , peak streamflow for T-year recurrence interval in cubic feet per second; and A, drainage area in square miles. The exponent of A is 

the power λ.] 

Regression equation 

Adjusted 

Rsquared 

Residual 

standard 

error 

log10(QT )

PRESS 

statistic 

Percent 

change from 

PRESS in 

table 6 

Q2 = 108.280−6.031 A−0.0465 0.7710 0.3309 72.06 −2.9 

Q5 = 107.194−4.614 A−0.0658 

.8030 .2994 59.00 −6.8 

Q10 = 106.961−4.212 A−0.0749 

.8002 .3021 60.10 −8.4 

Q25 = 106.840−3.914 A−0.0837 

.7834 .3184 66.77 −9.4 

Q50 = 106.806−3.766 A−0.0890 

.7659 .3354 74.08 −9.5 

Q100 = 106.800−3.659 A−0.0934 

.7462 .3545 82.78 −9.4 
14 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas PRESS STATISTIC

45

50

55

60

65

70

75

80

85

90

95

ANNUAL NONEXCEEDANCE PROBABILITY, IN PERCENT

50 60 70 80 85 90 95 98 99

PRESS statistic from logarithmic transformation of drainage area regression using

three predictor variables

PRESS statistic from power transformation of drainage area regression using

three predictor variables

PRESS statistic from logarithmic transformation of drainage area regression using

drainage area as predictor variable

PRESS statistic from power transformation of drainage area regression using

drainage area as predictor variable

EXPLANATION

RECURRENCE INTERVAL, IN YEARS

2 5 10 25 50 100

more error

less error

Figure 6. Comparison of PRESS statistics from regression analysis. 

DRAINAGE AREA, IN SQUARE MILES

RATIO OF LO

GARITHMIC REGRES

SIO

N

T

O PRES

S-MINIMIZED REGRES

SIO

N

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

2

5

10

25

50 100

Recurrence interval

0.1 0.5 1 5 10 50 100 500 1,000 5,000

Figure 7. Relation between bias ratio and drainage area by recurrence interval for the regressions using drainage area as the 

only predictor variable. 
Appendix 1—Computational Script using PRESS 

Minimization and Drainage Area, 

Mean Annual Precipitation, and 

Main-Channel Slope 
Blank Page
Appendix 1—Computational Script using PRESS Minimization and Drainage Area, Mean Annual Precipitation, and . . . 17 

R : Copyright 2006 , The R Foundation for Statistical Computing

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Type ’ demo () ’ for some demos , ’ help () ’ for on - line help , or
 

’ help . start () ’ for an HTML browser interface to help .
 

Type ’q () ’ to quit R.
 

> # In R , it appears necessary to ensure that the sum of the

> # weight factors equals the length of the weight factor

> # vector . Otherwise , one artificially inflates the

> # residual standard error and hence prediction limits . Other

> # diagnostics are changed , but remain in relative proportion

> # with one another - - so influence conclusions remain the same ?

> # Also , the coefficients of the regression are correct regardless

> # of the summation constraint needed on the weights .

> MLRweights <- function ( vector ) {

+ tmp = length ( vector ) / sum ( vector )
 

+ return ( tmp * vector )
 

+ }
 

>
 

> # PRESS statistics
 

> PRESS <- function ( model ) {
 

+ if ( is . null ( model $ terms )) stop ( " invalid �’lm ’� object :� no � terms ")
 

+ sum ( ( weighted . residuals ( model )/ (1- hatvalues ( model )) ) ^2 )
 

+ }
 

>
 

>
 

> DATA <- read . csv ( " tx664 . csv " , header =T )
 

> attach ( DATA )
 

> names ( DATA )
 

[1] " Station " " LatD " " LatM " " LatS " " LonD " " LonM " " LonS "

[8] " EqYrs " " CDA " " MAP " " P224 " " Slope " " Shape " " Q2"

[15] " Q5" " Q10 " " Q25 " " Q50 " " Q100 " " C2" " C25 "
 

[22] " C100 "
 

> outliers <- c (212 ,323 ,358 ,602 ,614 ,620 ,628 ,637)
 

> CDA <- CDA [- outliers ]
 

> Q2 <- Q2 [ - outliers ]
 

> Q5 <- Q5 [ - outliers ]
 

> Q10 <- Q10 [- outliers ]
 

> Q25 <- Q25 [- outliers ]
 

> Q50 <- Q50 [- outliers ]
 

> Q100 <- Q100 [ - outliers ]
 

> MAP <- MAP [- outliers ]
 

> Slope <- Slope [ - outliers ]
 

> WEIGHTS <- MLRweights ( EqYrs [- outliers ])
 

>
 

>
 

> WLS2 _ 2. OUT <- lm ( Q2~ CDA + MAP + Slope , weights = WEIGHTS )
 

> WLS2 _ 5. OUT <- lm ( Q5~ CDA + MAP + Slope , weights = WEIGHTS )
 

> WLS2 _ 10. OUT <- lm ( Q10 ~ CDA + MAP + Slope , weights = WEIGHTS )
 

> WLS2 _ 25. OUT <- lm ( Q25 ~ CDA + MAP + Slope , weights = WEIGHTS )
 

> WLS2 _ 50. OUT <- lm ( Q50 ~ CDA + MAP + Slope , weights = WEIGHTS )
 

> WLS2 _ 100. OUT <- lm ( Q100 ~ CDA + MAP + Slope , weights = WEIGHTS )
 

>
 

> PRESS ( WLS2 _ 2. OUT )
 

[1] 54.36495
 

> summary ( WLS2 _ 2. OUT )
 

Call :
 

lm ( formula = Q2 ~ CDA + MAP + Slope , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3 Q Max

-1.119685 -0.171447 -0.009884 0.188213 1.104813

Coefficients :

Estimate Std . Error t value Pr ( >| t |)
18 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

( Intercept ) -0.52401 0.22148 -2.366 0.0183 *

CDA 0.65645 0.01762 37.250 < 2e -16 ** *

MAP 1.47417 0.09926 14.851 < 2e -16 ** *

Slope 0.35245 0.04981 7.076 3.84e -12 * **

---

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

Residual standard error : 0.2866 on 652 degrees of freedom

Multiple R- Squared : 0.829 , Adjusted R- squared : 0.8282

F- statistic : 1054 on 3 and 652 DF , p- value : < 2.2e -16

>

> PRESS ( WLS2 _5. OUT )

[1] 47.75871

> summary ( WLS2 _5. OUT )

Call :

lm( formula = Q5 ~ CDA + MAP + Slope , weights = WEIGHTS )

Residuals :

Min 1 Q Median 3Q Max

-0.97622 -0.17392 -0.01896 0.14456 0.86452

Coefficients :

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

( Intercept ) -0.22042 0.20756 -1.062 0.289

CDA 0.67897 0.01652 41.111 <2e -16 ** *

MAP 1.37583 0.09303 14.790 <2e -16 ** *

Slope 0.48282 0.04668 10.344 <2e -16 * **

---

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

Residual standard error : 0.2686 on 652 degrees of freedom

Multiple R- Squared : 0.8422 , Adjusted R- squared : 0.8414

F- statistic : 1160 on 3 and 652 DF , p- value : < 2.2e -16

>

> PRESS ( WLS2 _ 10. OUT )

[1] 51.11918

> summary ( WLS2 _ 10. OUT )

Call :

lm( formula = Q10 ~ CDA + MAP + Slope , weights = WEIGHTS )

Residuals :

Min 1Q Median 3Q Max

-1.01418 -0.19171 -0.01982 0.13608 0.84651

Coefficients :

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

( Intercept ) -0.04207 0.21472 -0.196 0.845

CDA 0.68963 0.01708 40.365 <2e -16 ** *

MAP 1.31742 0.09623 13.690 <2e -16 ** *

Slope 0.54210 0.04829 11.227 <2e -16 * **

---

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

Residual standard error : 0.2778 on 652 degrees of freedom

Multiple R- Squared : 0.8318 , Adjusted R- squared : 0.831

F- statistic : 1075 on 3 and 652 DF , p- value : < 2.2e -16

>

> PRESS ( WLS2 _ 25. OUT )

[1] 59.34262

> summary ( WLS2 _ 25. OUT )

Call :

lm( formula = Q25 ~ CDA + MAP + Slope , weights = WEIGHTS )

Residuals :

Min 1Q Median 3Q Max

-1.03979 -0.20721 -0.02706 0.13507 1.02047

Coefficients :

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

( Intercept ) 0.15008 0.23132 0.649 0.517

CDA 0.70048 0.01841 38.057 <2e -16 ** *

MAP 1.25642 0.10368 12.119 <2e -16 ** *

Slope 0.60053 0.05202 11.544 <2e -16 * **

---
Appendix 1—Computational Script using PRESS Minimization and Drainage Area, Mean Annual Precipitation, and . . . 19 

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

Residual standard error : 0.2993 on 652 degrees of freedom
 

Multiple R - Squared : 0.8095 , Adjusted R - squared : 0.8086
 

F - statistic : 923.4 on 3 and 652 DF , p - value : < 2.2 e -16
 

>
 

> PRESS ( WLS2 _ 50. OUT )
 

[1] 67.2532

> summary ( WLS2 _ 50. OUT )

Call :
 

lm ( formula = Q50 ~ CDA + MAP + Slope , weights = WEIGHTS )
 

Residuals :

Min 1Q Median 3Q Max

-1.09952 -0.22645 -0.03173 0.13983 1.09056

Coefficients :

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

( Intercept ) 0.27484 0.24625 1.116 0.265

CDA 0.70729 0.01959 36.097 <2e -16 ** *

MAP 1.21778 0.11037 11.034 <2e -16 ** *

Slope 0.63587 0.05538 11.482 <2e -16 * **

---

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

Residual standard error : 0.3186 on 652 degrees of freedom
 

Multiple R - Squared : 0.7896 , Adjusted R - squared : 0.7887
 

F - statistic : 815.8 on 3 and 652 DF , p - value : < 2.2 e -16
 

>
 

> PRESS ( WLS2 _ 100. OUT )
 

[1] 76.25372

> summary ( WLS2 _ 100. OUT )

Call :
 

lm ( formula = Q100 ~ CDA + MAP + Slope , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3 Q Max

-1.16119 -0.23060 -0.03985 0.14572 1.20525

Coefficients :

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

( Intercept ) 0.38788 0.26221 1.479 0.140

CDA 0.71333 0.02086 34.190 <2e -16 ** *

MAP 1.18327 0.11752 10.069 <2e -16 ** *

Slope 0.66604 0.05897 11.295 <2e -16 * **

---

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

Residual standard error : 0.3393 on 652 degrees of freedom

Multiple R - Squared : 0.7686 , Adjusted R - squared : 0.7675

F - statistic : 721.7 on 3 and 652 DF , p - value : < 2.2 e -16

>

>

> doQt <- function (Q , type ) {

+ smallpress <- 10000

+ smallpower <- 10000

+ # Q <- 10^ Q

+ for ( power in seq ( -.1 ,.1 , by =0.0001) ) { # 0.007 , 0.08 , by =0.0001

+ if ( power == 0) next

+ power <- -1 * power

+ CDA1 <- 10^ CDA

+ CDA1 <- CDA1 ^ power

+ WLS . OUT <- lm ( Q~ CDA1 + MAP + Slope , weights = WEIGHTS )

+ press <- PRESS ( WLS . OUT )

+ if ( press < smallpress ) {

+ smallpress <- press

+ smallpower <- power

+ }

+ # plot ( fitted ( WLS . OUT ) , residuals ( WLS . OUT ) , ylim = c ( -2 ,2) , col =2)

+ # sm <- lowess ( fitted ( WLS . OUT ) ,y = residuals ( WLS . OUT ) )

+ # lines ( sm , lwd =2)

+ # lines ( c ( -10 ,10) ,c (0 ,0) )

+ }

+ print ( c( type , smallpower , smallpress ))

+ return (c ( type , smallpower , smallpress ) )
---

20 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

+ }
 

>
 

>
 

> vals2 <- doQt (Q2 ,2)
 

[1] 2.00000 -0.00820 54.26502
 

> vals5 <- doQt (Q5 ,5)
 

[1] 5.00000 -0.02990 46.36806
 

> vals10 <- doQt ( Q10 ,10)
 

[1] 10.00000 -0.04000 48.56881
 

> vals25 <- doQt ( Q25 ,25)
 

[1] 25.00000 -0.04970 55.31596
 

> vals50 <- doQt ( Q50 ,50)
 

[1] 50.0000 -0.0553 62.1783
 

> vals100 <- doQt ( Q100 ,100)
 

[1] 100.00000 -0.06010 70.18746
 

>
 

> vals2
 

[1] 2.00000 -0.00820 54.26502
 

> vals5
 

[1] 5.00000 -0.02990 46.36806
 

> vals10
 

[1] 10.00000 -0.04000 48.56881
 

> vals25
 

[1] 25.00000 -0.04970 55.31596
 

> vals50
 

[1] 50.0000 -0.0553 62.1783
 

> vals100
 

[1] 100.00000 -0.06010 70.18746
 

>
 

>
 

> finalQt <- function (Q , power , type ) {
 

+ CDA1 <- 10^ CDA
 

+ CDA1 <- CDA1 ^ power
 

+ WLS . OUT <- lm (Q ~ CDA1 + MAP + Slope , weights = WEIGHTS )
 

+ # plot (CDA , residuals ( WLS . OUT ) , ylim = c ( -3 ,3) )

+ # plot ( fitted ( WLS . OUT ) , residuals ( WLS . OUT ) , pch =16 , col =2 , ylim = c ( -1.5 ,1) )

+ # sm <- lowess ( fitted ( WLS . OUT ) ,y = residuals ( WLS . OUT ) )

+ # lines ( sm , lwd =2)

+ # lines ( c ( -10 ,10) ,c (0 ,0) )

+ cat (c (" POWER :�" , power ," \n") )

+ print ( summary ( WLS . OUT ))

+

+ W <- diag ( WEIGHTS )

+ X = model . matrix ( WLS . OUT )

+ Xt = t( X)

+

+ # Perform manual WLS regression and hat matrix

+ tmp <- chol2inv ( chol ( Xt % *% W % *% X ) )

+ wlshat1 <- X %* % tmp %* % Xt

+ print ( tmp ) # inverted covariance matrix

+ print ( max ( diag ( wlshat1 ) )) # the maximum leverage

+ m . wls . out <- tmp %* % Xt % *% W % *% Q

+ print (m . wls . out ) # the regression coefficients

+

+ PRESS ( WLS . OUT )

+ return ( WLS . OUT )

+ }

>

>

> F2 . OUT <- finalQt (Q2 , vals2 [2] ,2)

POWER : -0.0082

Call :
 

lm ( formula = Q ~ CDA1 + MAP + Slope , weights = WEIGHTS )
 

Residuals :

Min 1Q Median 3 Q Max

-1.134364 -0.169019 -0.008506 0.190237 1.097330

Coefficients :

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

( Intercept ) 35.59867 0.78972 45.078 < 2e -16 * **

CDA1 -36.09370 0.96760 -37.302 < 2e -16 * **

MAP 1.44788 0.09866 14.676 < 2e -16 * **

Slope 0.34723 0.04963 6.996 6.53 e -12 ** *

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

Residual standard error : 0.2863 on 652 degrees of freedom

Multiple R - Squared : 0.8293 , Adjusted R - squared : 0.8286
Appendix 1—Computational Script using PRESS Minimization and Drainage Area, Mean Annual Precipitation, and . . . 21 

F - statistic : 1056 on 3 and 652 DF , p - value : < 2.2 e -16

[ ,1] [ ,2] [ ,3] [ ,4]

[1 ,] 7.6093394 -9.2195944 0.55243205 0.36807256

[2 ,] -9.2195944 11.4232854 -0.80660576 -0.48359566

[3 ,] 0.5524321 -0.8066058 0.11875864 0.04243762

[4 ,] 0.3680726 -0.4835957 0.04243762 0.03005193

[1] 0.1053207

[ ,1]

[1 ,] 35.5986711

[2 ,] -36.0936961

[3 ,] 1.4478828

[4 ,] 0.3472258

> F5 . OUT <- finalQt (Q5 , vals5 [2] ,5)

POWER : -0.0299

Call :
 

lm ( formula = Q ~ CDA1 + MAP + Slope , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3 Q Max

-0.933747 -0.170028 -0.005652 0.142126 0.836051

Coefficients :

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

( Intercept ) 11.16386 0.14688 76.01 <2e -16 ** *

CDA1 -11.27975 0.26878 -41.97 <2e -16 * **

MAP 1.27941 0.08982 14.24 <2e -16 * **

Slope 0.46404 0.04544 10.21 <2e -16 ** *

---

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

Residual standard error : 0.2646 on 652 degrees of freedom

Multiple R - Squared : 0.8468 , Adjusted R - squared : 0.8461

F - statistic : 1201 on 3 and 652 DF , p - value : < 2.2 e -16

[ ,1] [ ,2] [ ,3] [ ,4]

[1 ,] 0.30819904 -0.3799637 -0.01210666 0.03055440

[2 ,] -0.37996370 1.0321015 -0.23486075 -0.14338346

[3 ,] -0.01210666 -0.2348608 0.11524761 0.04091835

[4 ,] 0.03055440 -0.1433835 0.04091835 0.02949867

[1] 0.1047414

[ ,1]

[1 ,] 11.1638574

[2 ,] -11.2797452

[3 ,] 1.2794083

[4 ,] 0.4640428

> F10 . OUT <- finalQt ( Q10 , vals10 [2] ,10)

POWER : -0.04

Call :
 

lm ( formula = Q ~ CDA1 + MAP + Slope , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3 Q Max

-0.949383 -0.169673 -0.007528 0.145502 0.782993

Coefficients :

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

( Intercept ) 9.04660 0.11863 76.26 <2e -16 * **

CDA1 -8.94992 0.21392 -41.84 <2e -16 ** *

MAP 1.18822 0.09121 13.03 <2e -16 ** *

Slope 0.51724 0.04625 11.18 <2e -16 * **

---

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

Residual standard error : 0.2707 on 652 degrees of freedom

Multiple R - Squared : 0.8403 , Adjusted R - squared : 0.8396

F - statistic : 1143 on 3 and 652 DF , p - value : < 2.2 e -16

[ ,1] [ ,2] [ ,3] [ ,4]

[1 ,] 0.192009986 -0.1216301 -0.06356396 -0.000679897

[2 ,] -0.121630080 0.6244004 -0.17970515 -0.110637868

[3 ,] -0.063563963 -0.1797051 0.11352360 0.040132667

[4 ,] -0.000679897 -0.1106379 0.04013267 0.029183281

[1] 0.1041505

[ ,1]

[1 ,] 9.0465969

[2 ,] -8.9499203

[3 ,] 1.1882238
22 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

[4 ,] 0.5172355

> F25 . OUT <- finalQt (Q25 , vals25 [2] ,25)

POWER : -0.0497

Call :

lm( formula = Q ~ CDA1 + MAP + Slope , weights = WEIGHTS )

Residuals :

Min 1 Q Median 3Q Max

-1.01532 -0.18941 -0.01757 0.14541 0.94158

Coefficients :

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

( Intercept ) 7.94932 0.11881 66.91 <2e -16 ** *

CDA1 -7.62804 0.19053 -40.03 <2e -16 ** *

MAP 1.09555 0.09660 11.34 <2e -16 ** *

Slope 0.56992 0.04906 11.62 <2e -16 * **

---

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

Residual standard error : 0.2889 on 652 degrees of freedom

Multiple R- Squared : 0.8225 , Adjusted R- squared : 0.8217

F- statistic : 1007 on 3 and 652 DF , p- value : < 2.2e -16

[ ,1] [ ,2] [ ,3] [ ,4]

[1 ,] 0.16917355 -0.01930396 -0.09202358 -0.01816880

[2 ,] -0.01930396 0.43507942 -0.14753723 -0.09156878

[3 ,] -0.09202358 -0.14753723 0.11183416 0.03934196

[4 ,] -0.01816880 -0.09156878 0.03934196 0.02885127

[1] 0.1034083

[ ,1]

[1 ,] 7.9493152

[2 ,] -7.6280350

[3 ,] 1.0955540

[4 ,] 0.5699208

> F50 . OUT <- finalQt (Q50 , vals50 [2] ,50)

POWER : -0.0553

Call :

lm( formula = Q ~ CDA1 + MAP + Slope , weights = WEIGHTS )

Residuals :

Min 1Q Median 3Q Max

-1.10808 -0.20353 -0.02085 0.14901 1.00273

Coefficients :

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

( Intercept ) 7.55371 0.12581 60.04 <2e -16 ** *

CDA1 -7.08958 0.18527 -38.27 <2e -16 ** *

MAP 1.03865 0.10195 10.19 <2e -16 ** *

Slope 0.60210 0.05183 11.62 <2e -16 * **

---

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

Residual standard error : 0.3062 on 652 degrees of freedom

Multiple R- Squared : 0.8057 , Adjusted R- squared : 0.8048

F- statistic : 901.1 on 3 and 652 DF , p- value : < 2.2e -16

[ ,1] [ ,2] [ ,3] [ ,4]

[1 ,] 0.16878887 0.01314134 -0.10338009 -0.02523111

[2 ,] 0.01314134 0.36601926 -0.13398330 -0.08354382

[3 ,] -0.10338009 -0.13398330 0.11084897 0.03887227

[4 ,] -0.02523111 -0.08354382 0.03887227 0.02864815

[1] 0.1029075

[ ,1]

[1 ,] 7.5537081

[2 ,] -7.0895811

[3 ,] 1.0386544

[4 ,] 0.6021032

> F100 . OUT <- finalQt (Q100 , vals100 [2] ,100)

POWER : -0.0601

Call :

lm( formula = Q ~ CDA1 + MAP + Slope , weights = WEIGHTS )

Residuals :

Min 1Q Median 3Q Max

-1.21129 -0.21871 -0.03113 0.15723 1.08687

Coefficients :
---

Appendix 1—Computational Script using PRESS Minimization and Drainage Area, Mean Annual Precipitation, and . . . 23 

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

( Intercept ) 7.30661 0.13476 54.218 <2e -16 * **

CDA1 -6.71364 0.18421 -36.445 <2e -16 ** *

MAP 0.98827 0.10790 9.159 <2e -16 ** *

Slope 0.62951 0.05489 11.468 <2e -16 * **

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

Residual standard error : 0.3253 on 652 degrees of freedom

Multiple R - Squared : 0.7872 , Adjusted R - squared : 0.7862

F - statistic : 803.9 on 3 and 652 DF , p - value : < 2.2 e -16

[ ,1] [ ,2] [ ,3] [ ,4]

[1 ,] 0.17159632 0.03242555 -0.11114151 -0.03010189

[2 ,] 0.03242555 0.32062551 -0.12431136 -0.07782179

[3 ,] -0.11114151 -0.12431136 0.11000107 0.03846329

[4 ,] -0.03010189 -0.07782179 0.03846329 0.02846809

[1] 0.1024386

[ ,1]

[1 ,] 7.3066130

[2 ,] -6.7136406

[3 ,] 0.9882687

[4 ,] 0.6295080
Blank Page
Appendix 2—Computational Script Using PRESS 

Minimization and Drainage Area and 

Mean Annual Precipitation 
Blank Page
Appendix 2—Computational Script Using PRESS Minimization and Drainage Area and Mean Annual Precipitation 27 

R : Copyright 2006 , The R Foundation for Statistical Computing

Version 2.3.0 (2006 -04 -24)

ISBN 3 -900051 -07 -0

R is free software and comes with ABSOLUTELY NO WARRANTY .

You are welcome to redistribute it under certain conditions .

Type ’ license () ’ or ’ licence () ’ for distribution details .

Natural language support but running in an English locale

R is a collaborative project with many contributors .
 

Type ’ contributors () ’ for more information and
 

’ citation () ’ on how to cite R or R packages in publications .
 

Type ’ demo () ’ for some demos , ’ help () ’ for on - line help , or
 

’ help . start () ’ for an HTML browser interface to help .
 

Type ’q () ’ to quit R.
 

> # In R , it appears necessary to ensure that the sum of the

> # weight factors equals the length of the weight factor

> # vector . Otherwise , one artificially inflates the

> # residual standard error and hence prediction limits . Other

> # diagnostics are changed , but remain in relative proportion

> # with one another - - so influence conclusions remain the same ?

> # Also , the coefficients of the regression are correct regardless

> # of the summation constraint needed on the weights .

> MLRweights <- function ( vector ) {

+ tmp = length ( vector ) / sum ( vector )
 

+ return ( tmp * vector )
 

+ }
 

>
 

> # PRESS statistics
 

> PRESS <- function ( model ) {
 

+ if ( is . null ( model $ terms )) stop ( " invalid �’lm ’� object :� no � terms ")
 

+ sum ( ( weighted . residuals ( model )/ (1- hatvalues ( model )) ) ^2 )
 

+ }
 

>
 

>
 

> DATA <- read . csv ( " tx664 . csv " , header =T )
 

> attach ( DATA )
 

> names ( DATA )
 

[1] " Station " " LatD " " LatM " " LatS " " LonD " " LonM " " LonS "

[8] " EqYrs " " CDA " " MAP " " P224 " " Slope " " Shape " " Q2"

[15] " Q5" " Q10 " " Q25 " " Q50 " " Q100 " " C2" " C25 "
 

[22] " C100 "
 

> outliers <- c (212 ,323 ,358 ,602 ,614 ,620 ,628 ,637)
 

> CDA <- CDA [- outliers ]
 

> Q2 <- Q2 [ - outliers ]
 

> Q5 <- Q5 [ - outliers ]
 

> Q10 <- Q10 [- outliers ]
 

> Q25 <- Q25 [- outliers ]
 

> Q50 <- Q50 [- outliers ]
 

> Q100 <- Q100 [ - outliers ]
 

> MAP <- MAP [- outliers ]
 

> Slope <- Slope [ - outliers ]
 

> WEIGHTS <- MLRweights ( EqYrs [- outliers ])
 

>
 

>
 

> WLS2 _ 2. OUT <- lm ( Q2~ CDA + MAP , weights = WEIGHTS )
 

> WLS2 _ 5. OUT <- lm ( Q5~ CDA + MAP , weights = WEIGHTS )
 

> WLS2 _ 10. OUT <- lm ( Q10 ~ CDA + MAP , weights = WEIGHTS )
 

> WLS2 _ 25. OUT <- lm ( Q25 ~ CDA + MAP , weights = WEIGHTS )
 

> WLS2 _ 50. OUT <- lm ( Q50 ~ CDA + MAP , weights = WEIGHTS )
 

> WLS2 _ 100. OUT <- lm ( Q100 ~ CDA + MAP , weights = WEIGHTS )
 

>
 

> PRESS ( WLS2 _ 2. OUT )
 

[1] 58.28556
 

> summary ( WLS2 _ 2. OUT )
 

Call :
 

lm ( formula = Q2 ~ CDA + MAP , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3 Q Max

-1.09718 -0.18756 -0.01553 0.18068 1.10729

Coefficients :

Estimate Std . Error t value Pr ( >| t |)
28 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

( Intercept ) 0.83297 0.11490 7.25 1.19e -12 ** *

CDA 0.55340 0.01029 53.78 < 2e -16 ** *

MAP 0.97323 0.07215 13.49 < 2e -16 ** *

---

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

Residual standard error : 0.2971 on 653 degrees of freedom

Multiple R- Squared : 0.8159 , Adjusted R- squared : 0.8153

F- statistic : 1447 on 2 and 653 DF , p- value : < 2.2e -16

>

> PRESS ( WLS2 _5. OUT )

[1] 55.31802

> summary ( WLS2 _5. OUT )

Call :

lm( formula = Q5 ~ CDA + MAP , weights = WEIGHTS )

Residuals :

Min 1 Q Median 3Q Max

-0.86588 -0.20147 -0.02265 0.14310 0.86740

Coefficients :

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

( Intercept ) 1.63852 0.11196 14.635 <2e -16 ** *

CDA 0.53779 0.01003 53.638 <2e -16 ** *

MAP 0.68959 0.07030 9.809 <2e -16 ** *

---

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

Residual standard error : 0.2895 on 653 degrees of freedom

Multiple R- Squared : 0.8163 , Adjusted R- squared : 0.8157

F- statistic : 1451 on 2 and 653 DF , p- value : < 2.2e -16

>

> PRESS ( WLS2 _ 10. OUT )

[1] 60.68633

> summary ( WLS2 _ 10. OUT )

Call :

lm( formula = Q10 ~ CDA + MAP , weights = WEIGHTS )

Residuals :

Min 1Q Median 3Q Max

-0.91736 -0.22632 -0.02745 0.14438 1.00583

Coefficients :

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

( Intercept ) 2.04511 0.11726 17.441 < 2e -16 * **

CDA 0.53112 0.01050 50.577 < 2e -16 ** *

MAP 0.54692 0.07363 7.428 3.47e -13 ** *

---

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

Residual standard error : 0.3032 on 653 degrees of freedom

Multiple R- Squared : 0.7993 , Adjusted R- squared : 0.7987

F- statistic : 1300 on 2 and 653 DF , p- value : < 2.2e -16

>

> PRESS ( WLS2 _ 25. OUT )

[1] 71.10338

> summary ( WLS2 _ 25. OUT )

Call :

lm( formula = Q25 ~ CDA + MAP , weights = WEIGHTS )

Residuals :

Min 1Q Median 3Q Max

-0.97147 -0.24335 -0.03816 0.13714 1.17710

Coefficients :

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

( Intercept ) 2.46222 0.12692 19.400 < 2e -16 * **

CDA 0.52489 0.01137 46.181 < 2e -16 ** *

MAP 0.40288 0.07970 5.055 5.59e -07 ** *

---

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

Residual standard error : 0.3282 on 653 degrees of freedom

Multiple R- Squared : 0.7705 , Adjusted R- squared : 0.7698
Appendix 2—Computational Script Using PRESS Minimization and Drainage Area and Mean Annual Precipitation 29 

F - statistic : 1096 on 2 and 653 DF , p - value : < 2.2 e -16
 

>
 

> PRESS ( WLS2 _ 50. OUT )
 

[1] 80.44262

> summary ( WLS2 _ 50. OUT )

Call :
 

lm ( formula = Q50 ~ CDA + MAP , weights = WEIGHTS )
 

Residuals :

Min 1Q Median 3Q Max

-1.00636 -0.26998 -0.04396 0.13322 1.28007

Coefficients :

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

( Intercept ) 2.72306 0.13499 20.173 < 2e -16 * **

CDA 0.52137 0.01209 43.130 < 2e -16 ** *

MAP 0.31401 0.08476 3.705 0.000230 ** *

---

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

Residual standard error : 0.3491 on 653 degrees of freedom
 

Multiple R - Squared : 0.7471 , Adjusted R - squared : 0.7463
 

F - statistic : 964.5 on 2 and 653 DF , p - value : < 2.2 e -16
 

>
 

> PRESS ( WLS2 _ 100. OUT )
 

[1] 90.72161

> summary ( WLS2 _ 100. OUT )

Call :
 

lm ( formula = Q100 ~ CDA + MAP , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3 Q Max

-1.03776 -0.28890 -0.05912 0.15130 1.41052

Coefficients :

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

( Intercept ) 2.95226 0.14334 20.596 < 2e -16 * **

CDA 0.51858 0.01284 40.398 < 2e -16 ** *

MAP 0.23661 0.09001 2.629 0.00877 **

---

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

Residual standard error : 0.3707 on 653 degrees of freedom

Multiple R - Squared : 0.7233 , Adjusted R - squared : 0.7224

F - statistic : 853.4 on 2 and 653 DF , p - value : < 2.2 e -16

>

>

> doQt <- function (Q , type ) {

+ smallpress <- 10000

+ smallpower <- 10000

+ # Q <- 10^ Q

+ for ( power in seq ( -.1 ,.1 , by =0.0001) ) { # 0.007 , 0.08 , by =0.0001

+ if ( power == 0) next

+ power <- -1 * power

+ CDA1 <- 10^ CDA

+ CDA1 <- CDA1 ^ power

+ WLS . OUT <- lm ( Q~ CDA1 + MAP , weights = WEIGHTS )

+ press <- PRESS ( WLS . OUT )

+ if ( press < smallpress ) {

+ smallpress <- press

+ smallpower <- power

+ }

+ # plot ( fitted ( WLS . OUT ) , residuals ( WLS . OUT ) , ylim = c ( -2 ,2) , col =2)

+ # sm <- lowess ( fitted ( WLS . OUT ) ,y = residuals ( WLS . OUT ) )

+ # lines ( sm , lwd =2)

+ # lines ( c ( -10 ,10) ,c (0 ,0) )

+ }
 

+ print ( c( type , smallpower , smallpress ))
 

+ return (c ( type , smallpower , smallpress ) )
 

+ }
 

>
 

>
 

> vals2 <- doQt (Q2 ,2)
 

[1] 2.00000 -0.01570 58.02393
 

> vals5 <- doQt (Q5 ,5)
 
---

30 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

[1] 5.00000 -0.04510 53.26502

> vals10 <- doQt ( Q10 ,10)

[1] 10.00000 -0.05960 57.17684

> vals25 <- doQt ( Q25 ,25)

[1] 25.00000 -0.07400 65.79596

> vals50 <- doQt ( Q50 ,50)

[1] 50.00000 -0.08280 73.87932

> vals100 <- doQt ( Q100 ,100)

[1] 100.00000 -0.09030 82.98081

>

> vals2

[1] 2.00000 -0.01570 58.02393

> vals5

[1] 5.00000 -0.04510 53.26502

> vals10

[1] 10.00000 -0.05960 57.17684

> vals25

[1] 25.00000 -0.07400 65.79596

> vals50

[1] 50.00000 -0.08280 73.87932

> vals100

[1] 100.00000 -0.09030 82.98081

>

>

> finalQt <- function (Q , power , type ) {

+ CDA1 <- 10^ CDA

+ CDA1 <- CDA1 ^ power

+ WLS . OUT <- lm (Q ~ CDA1 + MAP , weights = WEIGHTS )

+ # plot (CDA , residuals ( WLS . OUT ) , ylim = c ( -3 ,3) )

+ # plot ( fitted ( WLS . OUT ) , residuals ( WLS . OUT ) , pch =16 , col =2 , ylim = c ( -1.5 ,1) )

+ # sm <- lowess ( fitted ( WLS . OUT ) ,y = residuals ( WLS . OUT ) )

+ # lines ( sm , lwd =2)

+ # lines ( c ( -10 ,10) ,c (0 ,0) )

+ cat (c (" POWER :�" , power ," \n") )

+ print ( summary ( WLS . OUT ))

+

+ W <- diag ( WEIGHTS )

+ X = model . matrix ( WLS . OUT )

+ Xt = t( X)

+

+ # Perform manual WLS regression and hat matrix

+ tmp <- chol2inv ( chol ( Xt % *% W % *% X ) )

+ wlshat1 <- X %* % tmp %* % Xt

+ print ( tmp ) # inverted covariance matrix

+ print ( max ( diag ( wlshat1 ) )) # maximum leverage

+ m . wls . out <- tmp %* % Xt % *% W % *% Q

+ print (m . wls . out ) # the regression coefficients

+

+ PRESS ( WLS . OUT )

+ return ( WLS . OUT )

+ }

>

>

> F2 . OUT <- finalQt (Q2 , vals2 [2] ,2)

POWER : -0.0157

Call :
 

lm ( formula = Q ~ CDA1 + MAP , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3Q Max

-1.11654 -0.18172 -0.01350 0.18093 1.09514

Coefficients :

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

( Intercept ) 17.35866 0.27228 63.75 <2e -16 * **

CDA1 -16.51148 0.30614 -53.93 <2e -16 * **

MAP 0.94286 0.07183 13.13 <2e -16 * **

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

Residual standard error : 0.2965 on 653 degrees of freedom

Multiple R - Squared : 0.8167 , Adjusted R - squared : 0.8162

F - statistic : 1455 on 2 and 653 DF , p - value : < 2.2 e -16

[ ,1] [ ,2] [ ,3]

[1 ,] 0.84355557 -0.88041997 -0.02485273

[2 ,] -0.88041997 1.06646053 -0.06598735

[3 ,] -0.02485273 -0.06598735 0.05871136

[1] 0.02790216
---

---

Appendix 2—Computational Script Using PRESS Minimization and Drainage Area and Mean Annual Precipitation 31 

[ ,1]

[1 ,] 17.3586632

[2 ,] -16.5114820

[3 ,] 0.9428574

> F5 . OUT <- finalQt (Q5 , vals5 [2] ,5)

POWER : -0.0451

Call :
 

lm ( formula = Q ~ CDA1 + MAP , weights = WEIGHTS )
 

Residuals :

Min 1Q Median 3Q Max

-0.91606 -0.20427 -0.02218 0.14384 0.83370

Coefficients :

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

( Intercept ) 8.07960 0.11743 68.806 <2e -16 * **

CDA1 -6.40329 0.11665 -54.894 <2e -16 ** *

MAP 0.60654 0.06857 8.846 <2e -16 ** *

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

Residual standard error : 0.2841 on 653 degrees of freedom

Multiple R - Squared : 0.8231 , Adjusted R - squared : 0.8226

F - statistic : 1519 on 2 and 653 DF , p - value : < 2.2 e -16

[ ,1] [ ,2] [ ,3]

[1 ,] 0.17084538 -0.09552241 -0.06531660

[2 ,] -0.09552241 0.16858758 -0.02473015

[3 ,] -0.06531660 -0.02473015 0.05825606

[1] 0.0277843

[ ,1]

[1 ,] 8.0796013

[2 ,] -6.4032928

[3 ,] 0.6065365

> F10 . OUT <- finalQt ( Q10 , vals10 [2] ,10)

POWER : -0.0596

Call :
 

lm ( formula = Q ~ CDA1 + MAP , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3 Q Max

-0.98237 -0.22307 -0.01623 0.14879 0.93836

Coefficients :

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

( Intercept ) 7.19977 0.11028 65.284 < 2e -16 * **

CDA1 -5.10664 0.09729 -52.490 < 2e -16 ** *

MAP 0.43974 0.07091 6.201 9.94 e -10 ** *

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

Residual standard error : 0.2943 on 653 degrees of freedom

Multiple R - Squared : 0.8109 , Adjusted R - squared : 0.8103

F - statistic : 1400 on 2 and 653 DF , p - value : < 2.2 e -16

[ ,1] [ ,2] [ ,3]

[1 ,] 0.14038398 -0.05084317 -0.07034334

[2 ,] -0.05084317 0.10924813 -0.01930732

[3 ,] -0.07034334 -0.01930732 0.05804055

[1] 0.02768903

[ ,1]

[1 ,] 7.199766

[2 ,] -5.106641

[3 ,] 0.439738

> F25 . OUT <- finalQt ( Q25 , vals25 [2] ,25)

POWER : -0.074

Call :
 

lm ( formula = Q ~ CDA1 + MAP , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3 Q Max

-1.05059 -0.23898 -0.02263 0.15452 1.09630

Coefficients :

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

( Intercept ) 6.84851 0.11304 60.586 < 2e -16 * **

CDA1 -4.32907 0.08916 -48.551 < 2e -16 ** *
32 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

MAP 0.27291 0.07593 3.594 0.00035 ** *

---

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

Residual standard error : 0.3157 on 653 degrees of freedom

Multiple R- Squared : 0.7877 , Adjusted R- squared : 0.787

F- statistic : 1211 on 2 and 653 DF , p- value : < 2.2e -16

[ ,1] [ ,2] [ ,3]

[1 ,] 0.12817981 -0.03022921 -0.07326873

[2 ,] -0.03022921 0.07975385 -0.01598832

[3 ,] -0.07326873 -0.01598832 0.05783358

[1] 0.02757124

[ ,1]

[1 ,] 6.8485138

[2 ,] -4.3290741

[3 ,] 0.2729137

> F50 . OUT <- finalQt (Q50 , vals50 [2] ,50)

POWER : -0.0828

Call :

lm( formula = Q ~ CDA1 + MAP , weights = WEIGHTS )

Residuals :

Min 1 Q Median 3Q Max

-1.11361 -0.24415 -0.02832 0.14935 1.19162

Coefficients :

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

( Intercept ) 6.77654 0.11793 57.461 <2e -16 ** *

CDA1 -3.99103 0.08744 -45.643 <2e -16 ** *

MAP 0.17064 0.08037 2.123 0.0341 *

---

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

Residual standard error : 0.3345 on 653 degrees of freedom

Multiple R- Squared : 0.7677 , Adjusted R- squared : 0.767

F- statistic : 1079 on 2 and 653 DF , p- value : < 2.2e -16

[ ,1] [ ,2] [ ,3]

[1 ,] 0.12426816 -0.02270511 -0.07450586

[2 ,] -0.02270511 0.06831574 -0.01451140

[3 ,] -0.07450586 -0.01451140 0.05771085

[1] 0.02748834

[ ,1]

[1 ,] 6.7765427

[2 ,] -3.9910263

[3 ,] 0.1706351

> F100 . OUT <- finalQt (Q100 , vals100 [2] ,100)

POWER : -0.0903

Call :

lm( formula = Q ~ CDA1 + MAP , weights = WEIGHTS )

Residuals :

Min 1Q Median 3Q Max

-1.18053 -0.25296 -0.04201 0.15569 1.27209

Coefficients :

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

( Intercept ) 6.77645 0.12386 54.711 <2e -16 * **

CDA1 -3.75755 0.08748 -42.953 <2e -16 ** *

MAP 0.08212 0.08509 0.965 0.335

---

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

Residual standard error : 0.3545 on 653 degrees of freedom

Multiple R- Squared : 0.7469 , Adjusted R- squared : 0.7461

F- statistic : 963.4 on 2 and 653 DF , p- value : < 2.2e -16

[ ,1] [ ,2] [ ,3]

[1 ,] 0.12205140 -0.01801320 -0.07534347

[2 ,] -0.01801320 0.06088392 -0.01347027

[3 ,] -0.07534347 -0.01347027 0.05760862

[1] 0.02741139

[ ,1]

[1 ,] 6.77645185

[2 ,] -3.75754823

[3 ,] 0.08211505
Appendix 3—Computational Script Using PRESS 

Minimization and Drainage Area 
Blank Page
Appendix 3—Computational Script Using PRESS Minimization and Drainage Area 35 

R : Copyright 2006 , The R Foundation for Statistical Computing

Version 2.3.0 (2006 -04 -24)

ISBN 3 -900051 -07 -0

R is free software and comes with ABSOLUTELY NO WARRANTY .

You are welcome to redistribute it under certain conditions .

Type ’ license () ’ or ’ licence () ’ for distribution details .

Natural language support but running in an English locale

R is a collaborative project with many contributors .
 

Type ’ contributors () ’ for more information and
 

’ citation () ’ on how to cite R or R packages in publications .
 

Type ’ demo () ’ for some demos , ’ help () ’ for on - line help , or
 

’ help . start () ’ for an HTML browser interface to help .
 

Type ’q () ’ to quit R.
 

> # In R , it appears necessary to ensure that the sum of the

> # weight factors equals the length of the weight factor

> # vector . Otherwise , one artificially inflates the

> # residual standard error and hence prediction limits . Other

> # diagnostics are changed , but remain in relative proportion

> # with one another - - so influence conclusions remain the same ?

> # Also , the coefficients of the regression are correct regardless

> # of the summation constraint needed on the weights .

> MLRweights <- function ( vector ) {

+ tmp = length ( vector ) / sum ( vector )
 

+ return ( tmp * vector )
 

+ }
 

>
 

> # PRESS statistics
 

> PRESS <- function ( model ) {
 

+ if ( is . null ( model $ terms )) stop ( " invalid �’lm ’� object :� no � terms ")
 

+ sum ( ( weighted . residuals ( model )/ (1- hatvalues ( model )) ) ^2 )
 

+ }
 

>
 

>
 

> DATA <- read . csv ( " tx664 . csv " , header =T )
 

> attach ( DATA )
 

> names ( DATA )
 

[1] " Station " " LatD " " LatM " " LatS " " LonD " " LonM " " LonS "

[8] " EqYrs " " CDA " " MAP " " P224 " " Slope " " Shape " " Q2"

[15] " Q5" " Q10 " " Q25 " " Q50 " " Q100 " " C2" " C25 "
 

[22] " C100 "
 

> outliers <- c (212 ,323 ,358 ,602 ,614 ,620 ,628 ,637)
 

> CDA <- CDA [- outliers ]
 

> Q2 <- Q2 [ - outliers ]
 

> Q5 <- Q5 [ - outliers ]
 

> Q10 <- Q10 [- outliers ]
 

> Q25 <- Q25 [- outliers ]
 

> Q50 <- Q50 [- outliers ]
 

> Q100 <- Q100 [ - outliers ]
 

> MAP <- MAP [- outliers ]
 

> Slope <- Slope [ - outliers ]
 

> WEIGHTS <- MLRweights ( EqYrs [- outliers ])
 

>
 

>
 

> WLS2 _ 2. OUT <- lm ( Q2~ CDA , weights = WEIGHTS )
 

> WLS2 _ 5. OUT <- lm ( Q5~ CDA , weights = WEIGHTS )
 

> WLS2 _ 10. OUT <- lm ( Q10 ~ CDA , weights = WEIGHTS )
 

> WLS2 _ 25. OUT <- lm ( Q25 ~ CDA , weights = WEIGHTS )
 

> WLS2 _ 50. OUT <- lm ( Q50 ~ CDA , weights = WEIGHTS )
 

> WLS2 _ 100. OUT <- lm ( Q100 ~ CDA , weights = WEIGHTS )
 

>
 

> PRESS ( WLS2 _ 2. OUT )
 

[1] 74.22438
 

> summary ( WLS2 _ 2. OUT )
 

Call :
 

lm ( formula = Q2 ~ CDA , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3 Q Max

-1.391650 -0.189314 0.009216 0.194396 1.339501

Coefficients :

Estimate Std . Error t value Pr ( >| t |)
36 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

( Intercept ) 2.33920 0.03059 76.47 <2e -16 ** *

CDA 0.51577 0.01119 46.09 <2e -16 ** *

---

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

Residual standard error : 0.3357 on 654 degrees of freedom

Multiple R- Squared : 0.7646 , Adjusted R- squared : 0.7642

F- statistic : 2124 on 1 and 654 DF , p- value : < 2.2e -16

>

> PRESS ( WLS2 _5. OUT )

[1] 63.27671

> summary ( WLS2 _5. OUT )

Call :

lm( formula = Q5 ~ CDA , weights = WEIGHTS )

Residuals :

Min 1 Q Median 3Q Max

-1.05357 -0.20297 -0.03195 0.16029 1.03193

Coefficients :

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

( Intercept ) 2.70576 0.02824 95.82 <2e -16 ** *

CDA 0.51113 0.01033 49.48 <2e -16 ** *

---

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

Residual standard error : 0.3099 on 654 degrees of freedom

Multiple R- Squared : 0.7892 , Adjusted R- squared : 0.7889

F- statistic : 2449 on 1 and 654 DF , p- value : < 2.2e -16

>

> PRESS ( WLS2 _ 10. OUT )

[1] 65.62669

> summary ( WLS2 _ 10. OUT )

Call :

lm( formula = Q10 ~ CDA , weights = WEIGHTS )

Residuals :

Min 1Q Median 3Q Max

-0.99571 -0.22175 -0.03330 0.15717 0.95890

Coefficients :

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

( Intercept ) 2.89156 0.02875 100.57 <2e -16 * **

CDA 0.50998 0.01052 48.48 <2e -16 ** *

---

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

Residual standard error : 0.3156 on 654 degrees of freedom

Multiple R- Squared : 0.7823 , Adjusted R- squared : 0.782

F- statistic : 2351 on 1 and 654 DF , p- value : < 2.2e -16

>

> PRESS ( WLS2 _ 25. OUT )

[1] 73.67088

> summary ( WLS2 _ 25. OUT )

Call :

lm( formula = Q25 ~ CDA , weights = WEIGHTS )

Residuals :

Min 1Q Median 3Q Max

-1.0580 -0.2404 -0.0421 0.1519 1.1425

Coefficients :

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

( Intercept ) 3.08574 0.03046 101.3 <2e -16 * **

CDA 0.50931 0.01114 45.7 <2e -16 ** *

---

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

Residual standard error : 0.3343 on 654 degrees of freedom

Multiple R- Squared : 0.7616 , Adjusted R- squared : 0.7612

F- statistic : 2089 on 1 and 654 DF , p- value : < 2.2e -16

>

> PRESS ( WLS2 _ 50. OUT )
Appendix 3—Computational Script Using PRESS Minimization and Drainage Area 37 

[1] 81.88232

> summary ( WLS2 _ 50. OUT )

Call :
 

lm ( formula = Q50 ~ CDA , weights = WEIGHTS )
 

Residuals :

Min 1Q Median 3Q Max

-1.09259 -0.27013 -0.04421 0.14452 1.25313

Coefficients :

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

( Intercept ) 3.20903 0.03212 99.92 <2e -16 * **

CDA 0.50923 0.01175 43.34 <2e -16 ** *

---

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

Residual standard error : 0.3525 on 654 degrees of freedom
 

Multiple R - Squared : 0.7418 , Adjusted R - squared : 0.7414
 

F - statistic : 1879 on 1 and 654 DF , p - value : < 2.2 e -16
 

>
 

> PRESS ( WLS2 _ 100. OUT )
 

[1] 91.3865

> summary ( WLS2 _ 100. OUT )

Call :
 

lm ( formula = Q100 ~ CDA , weights = WEIGHTS )
 

Residuals :

Min 1Q Median 3Q Max

-1.1200 -0.2832 -0.0552 0.1460 1.3960

Coefficients :

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

( Intercept ) 3.31846 0.03393 97.81 <2e -16 * **

CDA 0.50944 0.01241 41.05 <2e -16 ** *

---

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

Residual standard error : 0.3724 on 654 degrees of freedom

Multiple R - Squared : 0.7204 , Adjusted R - squared : 0.7199

F - statistic : 1685 on 1 and 654 DF , p - value : < 2.2 e -16

>

>

> doQt <- function (Q , type ) {

+ smallpress <- 10000

+ smallpower <- 10000

+ # Q <- 10^ Q

+ for ( power in seq ( -.1 ,.1 , by =0.0001) ) { # 0.007 , 0.08 , by =0.0001

+ if ( power == 0) next

+ power <- -1 * power

+ CDA1 <- 10^ CDA

+ CDA1 <- CDA1 ^ power

+ WLS . OUT <- lm ( Q~ CDA1 , weights = WEIGHTS )

+ press <- PRESS ( WLS . OUT )

+ if ( press < smallpress ) {

+ smallpress <- press

+ smallpower <- power

+ }

+ # plot ( fitted ( WLS . OUT ) , residuals ( WLS . OUT ) , ylim = c ( -2 ,2) , col =2)

+ # sm <- lowess ( fitted ( WLS . OUT ) ,y = residuals ( WLS . OUT ) )

+ # lines ( sm , lwd =2)

+ # lines ( c ( -10 ,10) ,c (0 ,0) )

+ }
 

+ print ( c( type , smallpower , smallpress ))
 

+ return (c ( type , smallpower , smallpress ) )
 

+ }
 

>
 

>
 

> vals2 <- doQt (Q2 ,2)
 

[1] 2.00000 -0.04650 72.06471
 

> vals5 <- doQt (Q5 ,5)
 

[1] 5.00000 -0.06580 59.00068
 

> vals10 <- doQt ( Q10 ,10)
 

[1] 10.00000 -0.07490 60.09731
 

> vals25 <- doQt ( Q25 ,25)
 

[1] 25.0000 -0.0837 66.7667
 

> vals50 <- doQt ( Q50 ,50)
 
---

38 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

[1] 50.0000 -0.0890 74.0839

> vals100 <- doQt ( Q100 ,100)

[1] 100.00000 -0.09340 82.78375

>

> vals2

[1] 2.00000 -0.04650 72.06471

> vals5

[1] 5.00000 -0.06580 59.00068

> vals10

[1] 10.00000 -0.07490 60.09731

> vals25

[1] 25.0000 -0.0837 66.7667

> vals50

[1] 50.0000 -0.0890 74.0839

> vals100

[1] 100.00000 -0.09340 82.78375

>

>

> finalQt <- function (Q , power , type ) {

+ CDA1 <- 10^ CDA

+ CDA1 <- CDA1 ^ power

+ WLS . OUT <- lm (Q ~ CDA1 , weights = WEIGHTS )

+ # plot (CDA , residuals ( WLS . OUT ) , ylim = c ( -3 ,3) )

+ # plot ( fitted ( WLS . OUT ) , residuals ( WLS . OUT ) , pch =16 , col =2 , ylim = c ( -1.5 ,1) )

+ # sm <- lowess ( fitted ( WLS . OUT ) ,y = residuals ( WLS . OUT ) )

+ # lines ( sm , lwd =2)

+ # lines ( c ( -10 ,10) ,c (0 ,0) )

+ cat (c (" POWER :�" , power ," \n") )

+ print ( summary ( WLS . OUT ))

+

+ W <- diag ( WEIGHTS )

+ X = model . matrix ( WLS . OUT )

+ Xt = t( X)

+

+ # Perform manual WLS regression and hat matrix

+ tmp <- chol2inv ( chol ( Xt % *% W % *% X ) )

+ wlshat1 <- X %* % tmp %* % Xt

+ print ( tmp ) # inverted covariance matrix

+ print ( max ( diag ( wlshat1 ) )) # maximum leverage

+ m . wls . out <- tmp %* % Xt % *% W % *% Q

+ print (m . wls . out ) # the regression coefficients

+

+ PRESS ( WLS . OUT )

+ return ( WLS . OUT )

+ }

>

>

> F2 . OUT <- finalQt (Q2 , vals2 [2] ,2)

POWER : -0.0465

Call :
 

lm ( formula = Q ~ CDA1 , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3Q Max

-1.43150 -0.18074 0.01207 0.20405 1.28629

Coefficients :

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

( Intercept ) 8.2799 0.1002 82.63 <2e -16 ** *

CDA1 -6.0308 0.1284 -46.97 <2e -16 ** *

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

Residual standard error : 0.3309 on 654 degrees of freedom

Multiple R - Squared : 0.7713 , Adjusted R - squared : 0.771

F - statistic : 2206 on 1 and 654 DF , p - value : < 2.2 e -16

[ ,1] [ ,2]

[1 ,] 0.09169293 -0.1165254

[2 ,] -0.11652540 0.1505865

[1] 0.01885060

[ ,1]

[1 ,] 8.279869

[2 ,] -6.030849

> F5 . OUT <- finalQt (Q5 , vals5 [2] ,5)

POWER : -0.0658

Call :
 

lm ( formula = Q ~ CDA1 , weights = WEIGHTS )
 
Appendix 3—Computational Script Using PRESS Minimization and Drainage Area 39 

Residuals :

Min 1Q Median 3 Q Max

-1.064167 -0.195349 0.003766 0.160790 0.957834

Coefficients :

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

( Intercept ) 7.19379 0.06350 113.29 <2e -16 * **

CDA1 -4.61402 0.08928 -51.68 <2e -16 ** *

---

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

Residual standard error : 0.2994 on 654 degrees of freedom

Multiple R - Squared : 0.8033 , Adjusted R - squared : 0.803

F - statistic : 2671 on 1 and 654 DF , p - value : < 2.2 e -16

[ ,1] [ ,2]

[1 ,] 0.04499556 -0.06218392

[2 ,] -0.06218392 0.08895185

[1] 0.02071799

[ ,1]

[1 ,] 7.193790

[2 ,] -4.614025

> F10 . OUT <- finalQt ( Q10 , vals10 [2] ,10)

POWER : -0.0749

Call :
 

lm ( formula = Q ~ CDA1 , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3 Q Max

-1.069176 -0.212532 -0.002709 0.158296 0.889868

Coefficients :

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

( Intercept ) 6.96115 0.05610 124.08 <2e -16 * **

CDA1 -4.21241 0.08222 -51.23 <2e -16 ** *

---

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

Residual standard error : 0.3021 on 654 degrees of freedom

Multiple R - Squared : 0.8005 , Adjusted R - squared : 0.8002

F - statistic : 2625 on 1 and 654 DF , p - value : < 2.2 e -16

[ ,1] [ ,2]

[1 ,] 0.03449016 -0.04941901

[2 ,] -0.04941901 0.07408408

[1] 0.02164636

[ ,1]

[1 ,] 6.961146

[2 ,] -4.212410

> F25 . OUT <- finalQt ( Q25 , vals25 [2] ,25)

POWER : -0.0837

Call :
 

lm ( formula = Q ~ CDA1 , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3 Q Max

-1.14066 -0.23002 -0.01563 0.16378 1.06665

Coefficients :

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

( Intercept ) 6.84023 0.05276 129.64 <2e -16 * **

CDA1 -3.91357 0.08038 -48.69 <2e -16 ** *

---

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

Residual standard error : 0.3184 on 654 degrees of freedom

Multiple R - Squared : 0.7838 , Adjusted R - squared : 0.7834

F - statistic : 2370 on 1 and 654 DF , p - value : < 2.2 e -16

[ ,1] [ ,2]

[1 ,] 0.02746451 -0.04066401

[2 ,] -0.04066401 0.06374535

[1] 0.02257363

[ ,1]

[1 ,] 6.840234

[2 ,] -3.913570

> F50 . OUT <- finalQt ( Q50 , vals50 [2] ,50)
 

 

40 Alternative Regression Equations for Estimation of Peak-Streamflow Frequency for Watersheds in Texas 

POWER : -0.089
 

Call :
 

lm ( formula = Q ~ CDA1 , weights = WEIGHTS )
 

Residuals :

Min 1 Q Median 3Q Max

-1.18085 -0.24244 -0.02337 0.16170 1.17321

Coefficients :

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

( Intercept ) 6.80600 0.05219 130.4 <2e -16 ** *

CDA1 -3.76582 0.08133 -46.3 <2e -16 ** *

---

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

Residual standard error : 0.3354 on 654 degrees of freedom

Multiple R - Squared : 0.7662 , Adjusted R - squared : 0.7659

F - statistic : 2144 on 1 and 654 DF , p - value : < 2.2 e -16

[ ,1] [ ,2]

[1 ,] 0.02422098 -0.03653889

[2 ,] -0.03653889 0.05882340

[1] 0.02314619

[ ,1]

[1 ,] 6.806003

[2 ,] -3.765824

> F100 . OUT <- finalQt ( Q100 , vals100 [2] ,100)

POWER : -0.0934

Call :
 

lm ( formula = Q ~ CDA1 , weights = WEIGHTS )
 

Residuals :

Min 1Q Median 3Q Max

-1.21299 -0.25175 -0.03704 0.15891 1.26325

Coefficients :

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

( Intercept ) 6.80019 0.05252 129.49 <2e -16 ** *

CDA1 -3.65879 0.08336 -43.89 <2e -16 ** *

---

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

Residual standard error : 0.3545 on 654 degrees of freedom

Multiple R - Squared : 0.7465 , Adjusted R - squared : 0.7462

F - statistic : 1926 on 1 and 654 DF , p - value : < 2.2 e -16

[ ,1] [ ,2]

[1 ,] 0.02194667 -0.03360600

[2 ,] -0.03360600 0.05530056

[1] 0.0236296

[ ,1]

[1 ,] 6.800189

[2 ,] -3.658793

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