Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

Safety Effects of Horizontal Curve and 

Grade Combinations on Rural Two-Lane

Highways

PUBLICATION NO. FHWA-HRT-13-077 JANUARY 2014
FOREWORD

The first edition of the American Association of State Highway and Transportation Officials 

(AASHTO) Highway Safety Manual (HSM) provides crash modification factors (CMFs) for the 

safety effects of horizontal curvature and percent grade on rural two-lane highways.

(1) However, 

the HSM does not provide a method to account for the interactions between these effects. 

Researchers have always presumed that there are interactions between the safety effects of 

horizontal and vertical alignment, but this has not been demonstrated for specific alignment 

combinations in a form useful for safety prediction.

This document is a final report highlighting research undertaken to quantify the safety effects 

of five specific combinations of horizontal and vertical alignment for rural two-lane highways. 

The research is based on data from the Federal Highway Administration’s Highway Safety 

Information System (HSIS) for Washington State, including roadway characteristics, horizontal 

and vertical alignment, traffic volume, and crashes. The outcome is a set of safety prediction 

models for fatal and injury and property damage only crashes. CMFs representing safety 

performance relative to level tangents on rural two-lane highways were developed from the 

models for each of the five combinations of horizontal and vertical alignment to present the 

results in a form suitable for incorporation in the HSM. This report should be of interest to safety 

engineers and planners as well as other AASHTO HSM users.

Monique R. Evans

Director, Office of Safety

Research and Development

Notice

This document is disseminated under the sponsorship of the U.S. Department of Transportation 

in the interest of information exchange. The U.S. Government assumes no liability for the use 

of the information contained in this document. This report does not constitute a standard, 

specification, or regulation.

The U.S. Government does not endorse products or manufacturers. Trademarks or 

manufacturers’ names appear in this report only because they are considered essential to the 

objective of the document.

Quality Assurance Statement

The Federal Highway Administration (FHWA) provides high-quality information to serve 

Government, industry, and the public in a manner that promotes public understanding. Standards 

and policies are used to ensure and maximize the quality, objectivity, utility, and integrity of its 

information. FHWA periodically reviews quality issues and adjusts its programs and processes to 

ensure continuous quality improvement.
TECHNICAL REPORT DOCUMENTATION PAGE

1. Report No.

FHWA-HRT-13-077

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

4. Title and Subtitle

Safety Effects of Horizontal Curve and Grade Combinations on Rural TwoLane Highways

5. Report Date

January 2014

6. Performing Organization Code

7. Author(s)

Karin M. Bauer and Douglas W. Harwood

8. Performing Organization Report No.

110785.01.001

9. Performing Organization Name and Address

MRIGlobal

425 Volker Blvd

Kansas City, MO 64110

10. Work Unit No. (TRAIS)

11. Contract or Grant No.

DTFH61-11-C-00050

12. Sponsoring Agency Name and Address

Federal Highway Administration

Office of Safety Research and Development

6300 Georgetown Pike

McLean, VA 22101-2296

13. Type of Report and Period Covered

Final Report

December 2010–August 2012

14. Sponsoring Agency Code

15. Supplementary Notes

This work was performed under subcontract with the University of North Carolina (UNC) Highway Safety Research 

Center for Highway Safety Information System (HSIS) under Master Agreement No. 5-555697 Contract 

No. DTFH61-06-C-000013 and under subcontract for HSIS under Master Agreement No. 5-55771 Contract 

No. DTFH61-11-C-00050. The UNC Program Manager was David L. Harkey, and the Federal Highway 

Administration HSIS Program Manager was Carol Tan, Ph.D.

16. Abstract

This report summarizes the results of research undertaken to quantify the safety effects of five types of horizontal 

and vertical alignment combinations for rural two-lane highways. The research is based on Federal Highway 

Administration Highway Safety Information System data for Washington State, including crash records from 2003 to 

2008. The outcome is a set of safety prediction models for fatal and injury and property damage only crashes.

To present the results in a form suitable for incorporation in the American Association of State Highway and 

Transportation Officials Highway Safety Manual, crash modification factors representing safety performance relative 

to level tangents were developed from these models for each of the five combinations of horizontal and vertical 

alignment—horizontal curves and tangents on straight grades, horizontal curves and tangents at type 1 crest vertical 

curves, horizontal curves and tangents at type 1 sag vertical curves, horizontal curves and tangents at type 2 crest 

vertical curves, and horizontal curves and tangents at type 2 sag vertical curves.(1)

17. Key Words

Safety, Horizontal curves, Vertical curves, Grade 

interaction, Crests, Sags, Rural two-lane highways, 

Crash modeling, Safety performance function, Crash 

modification factor

18. Distribution Statement

No restrictions. This document is available to the public 

through the National Technical Information Service, 

Springfield, VA 22161 

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

63

22. Price

N/A
SI* (MODERN METRIC) CONVERSION FACTORS 

APPROXIMATE CONVERSIONS TO SI UNITS

Symbol When You Know Multiply By To Find Symbol 

LENGTH

in inches 25.4 millimeters mm 

ft feet 0.305 meters m 

yd yards 0.914 meters m 

mi miles 1.61 kilometers km 

AREA 

in2 square inches 645.2 square millimeters mm2

ft2 square feet 0.093 square meters m2

yd2 square yard 0.836 square meters m2

ac acres 0.405 hectares ha

mi2 square miles 2.59 square kilometers km2

VOLUME 

fl oz fluid ounces 29.57 milliliters mL 

gal gallons 3.785 liters L 

ft3 cubic feet 0.028 cubic meters m3 

yd3 cubic yards 0.765 cubic meters m3 

NOTE: volumes greater than 1000 L shall be shown in m3

MASS oz ounces 28.35 grams g

lb pounds 0.454 kilograms kg

T short tons (2000 lb) 0.907 megagrams (or "metric ton") Mg (or "t") 

TEMPERATURE (exact degrees) o

F Fahrenheit 5 (F-32)/9 Celsius oC 

or (F-32)/1.8 

ILLUMINATION 

fc foot-candles 10.76 lux lx

fl foot-Lamberts 3.426 candela/m2 cd/m2

FORCE and PRESSURE or STRESS 

lbf poundforce 4.45 newtons N 

lbf/in2 poundforce per square inch 6.89 kilopascals kPa 

APPROXIMATE CONVERSIONS FROM SI UNITS 

Symbol When You Know Multiply By To Find Symbol 

LENGTH mm millimeters 0.039 inches in 

m meters 3.28 feet ft 

m meters 1.09 yards yd 

km kilometers 0.621 miles mi 

AREA mm2 square millimeters 0.0016 square inches in2 

m2 square meters 10.764 square feet ft2 

m2 square meters 1.195 square yards yd2 

ha hectares 2.47 acres ac

km2 square kilometers 0.386 square miles mi2 

VOLUME 

mL milliliters 0.034 fluid ounces fl oz 

L liters 0.264 gallons gal 

m3 cubic meters 35.314 cubic feet ft3 

m3 cubic meters 1.307 cubic yards yd3 

MASS 

g grams 0.035 ounces oz

kg kilograms 2.202 pounds lb

Mg (or "t") megagrams (or "metric ton") 1.103 short tons (2000 lb) T 

TEMPERATURE (exact degrees) o

C Celsius 1.8C+32 Fahrenheit oF 

ILLUMINATION 

lx lux 0.0929 foot-candles fc 

cd/m2 candela/m2 0.2919 foot-Lamberts fl

FORCE and PRESSURE or STRESS 

N newtons 0.225 poundforce lbf 

kPa kilopascals 0.145 poundforce per square inch lbf/in2

*SI is the symbol for the International System of Units. Appropriate rounding should be made to comply with Section 4 of ASTM E380. 

(Revised March 2003)

ii
TABLE OF CONTENTS

CHAPTER 1—INTRODUCTION...............................................................................................1

BACKGROUND ......................................................................................................................1

RESEARCH OBJECTIVES AND SCOPE...........................................................................1

REPORT ORGANIZATION..................................................................................................1

CHAPTER 2—STATE OF KNOWLEDGE LITERATURE REVIEW..................................3

GEOMETRIC DESIGN CRITERIA.....................................................................................3

SAFETY EFFECTS OF HORIZONTAL ALIGNMENT....................................................4

SAFETY EFFECTS OF VERTICAL ALIGNMENT..........................................................6

INTERACTIONS BETWEEN SAFETY EFFECTS OF HORIZONTAL AND 

VERTICAL ALIGNMENT ....................................................................................................6

CHAPTER 3—DATABASE DESCRIPTION ............................................................................7

DATABASE DEVELOPMENT .............................................................................................7

DESCRIPTIVE STATISTICS................................................................................................9

VALUE RANGE OF ROADWAY CHARACTERISTICS...............................................12

CHAPTER 4—STATISTICAL ANALYSIS.............................................................................13

ANALYSIS APPROACH......................................................................................................13

Selection of Independent Variables Considered in Models...............................................13

Assessment of Goodness-of-Fit of Final Models...............................................................15

ANALYSIS RESULTS..........................................................................................................17

Horizontal Curves and Tangents on Straight Grades.........................................................17

Horizontal Curves and Tangents at Type 1 Crest Vertical Curves....................................21

Horizontal Curves and Tangents at Type 1 Sag Vertical Curves ......................................26

Horizontal Curves and Tangents at Type 2 Crest Vertical Curves....................................30

Horizontal Curves and Tangents at Type 2 Sag Vertical Curves ......................................34

STUDY LIMITATIONS .......................................................................................................38

CHAPTER 5—CRASH MODIFICATION FACTORS ..........................................................39

HORIZONTAL CURVES AND TANGENTS ON STRAIGHT GRADES.....................39

HORIZONTAL CURVES AND TANGENTS AT TYPE 1 CREST VERTICAL 

CURVES .................................................................................................................................42

HORIZONTAL CURVES AND TANGENTS AT TYPE 1 SAG VERTICAL 

CURVES .................................................................................................................................44

HORIZONTAL CURVES AND TANGENTS AT TYPE 2 CREST VERTICAL 

CURVES .................................................................................................................................46

HORIZONTAL CURVES AND TANGENTS AT TYPE 2 SAG VERTICAL 

CURVES .................................................................................................................................48

CMFS FOR COMBINED CRASH SEVERITY LEVELS................................................50

COMPARISON OF RESULTS TO EXISTING HSM CMFS ..........................................51

CHAPTER 6—CONCLUSIONS AND RECOMMENDATIONS ..........................................53

CONCLUSIONS ....................................................................................................................53

RECOMMENDATIONS.......................................................................................................54

REFERENCES.............................................................................................................................55

iii
LIST OF FIGURES

Figure 1. Illustration. Types of vertical curves................................................................................4

Figure 2. Equation. Relationship between crash rates and curve characteristics.............................4

Figure 3. Equation. CMF for horizontal curves on rural two-lane highways..................................5

Figure 4. Equation. CMF for superelevation as a function of superelevation variance...................5

Figure 5. Equation. CMF for percent grade on rural two-lane highways........................................6

Figure 6. Equation. Probability of observing a given number of crashes......................................16

Figure 7. Equation. Probability of observing a given number of crashes expressed as a 

cumulative distribution ..................................................................................................................16

Figure 8. Equation. Percent of roadway segments with unlikely low or high crash numbers.......16

Figure 9. Illustration. Alignment combinations used in the analysis of horizontal curves and 

tangents on straight grades.............................................................................................................17

Figure 10. Equation. Predicted FI crashes on horizontal curves and tangents on straight 

grades (general form).....................................................................................................................18

Figure 11. Equation. Predicted PDO crashes on horizontal curves and tangents on straight 

grades (general form).....................................................................................................................18

Figure 12. Equation. Predicted FI crashes on horizontal curves and tangents on straight 

grades (explicit form).....................................................................................................................20

Figure 13. Equation. Predicted PDO crashes on horizontal curves and tangents on straight 

grades (explicit form).....................................................................................................................20

Figure 14. Equation. Predicted total crashes on horizontal curves and tangents on straight 

grades (general form).....................................................................................................................20

Figure 15. Illustration. Alignment combinations used in the analysis of horizontal curves 

and tangents at type 1 crest vertical curves....................................................................................22

Figure 16. Equation. Predicted FI crashes on horizontal curves and tangents at type 1

crest vertical curves (general form) ...............................................................................................23

Figure 17. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1

crest vertical curves (general form) ...............................................................................................23

Figure 18. Equation. Predicted FI crashes on horizontal curves and tangents at type 1

crest vertical curves (explicit form)...............................................................................................25

Figure 19. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1

crest vertical curves (explicit form)...............................................................................................25

Figure 20. Equation. Predicted FI crashes on horizontal curves and tangents at type 1

crest vertical curves (alternate form) .............................................................................................26

Figure 21. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1

crest vertical curves (alternate form) .............................................................................................26

Figure 22. Illustration. Alignment combinations used in the analysis of horizontal curves 

and tangents at type 1 sag vertical curves......................................................................................26

Figure 23. Equation. Predicted FI crashes on horizontal curves and tangents at type 1

sag vertical curves (general form)..................................................................................................28

Figure 24. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1

sag vertical curves (general form)..................................................................................................28

Figure 25. Equation. Predicted FI crashes on horizontal curves and tangents at type 1

sag vertical curves (explicit form) .................................................................................................29

iv
Figure 26. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1

sag vertical curves (explicit form) .................................................................................................29

Figure 27. Equation. Predicted FI crashes on horizontal curves and tangents at type 1

sag vertical curves (alternate form)................................................................................................30

Figure 28. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1

sag vertical curves (alternate form)................................................................................................30

Figure 29. Illustration. Alignment combinations used in the analysis of horizontal curves 

and tangents at type 2 crest vertical curves....................................................................................31

Figure 30. Equation. Predicted FI crashes on horizontal curves and tangents at type 2

crest vertical curves (general form) ...............................................................................................32

Figure 31. Equation. Predicted PDO crashes on horizontal curves and tangents at type 2

crest vertical curves (general form) ...............................................................................................32

Figure 32. Equation. Predicted FI crashes on horizontal curves and tangents at type 2

crest vertical curves (explicit form)...............................................................................................33

Figure 33. Equation. Predicted PDO crashes on horizontal curves and tangents at type 2

crest vertical curves (explicit form)...............................................................................................34

Figure 34. Illustration. Alignment combinations used in the analysis of horizontal curves 

and tangents at type 2 sag vertical curves......................................................................................34

Figure 35. Equation. Predicted FI crashes on horizontal curves and tangents at type 2

sag vertical curves (general form)..................................................................................................36

Figure 36. Equation. Predicted PDO crashes on horizontal curves and tangents at type 2

sag vertical curves (general form)..................................................................................................36

Figure 37. Equation. Predicted FI crashes on horizontal curves and tangents at type 2

sag vertical curves (explicit form) .................................................................................................37

Figure 38. Equation. Predicted PDO crashes on horizontal curves and tangents at type 2

sag vertical curves (explicit form) .................................................................................................37

Figure 39. Equation. FI CMF for horizontal curves and tangents on straight grades....................39

Figure 40. Equation. PDO CMF for horizontal curves and tangents on straight grades ...............39

Figure 41. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and 

tangents on straight grades.............................................................................................................40

Figure 42. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves 

and tangents on straight grades......................................................................................................40

Figure 43. Equation. FI CMF for horizontal curves and tangents at type 1 crest 

vertical curves................................................................................................................................42

Figure 44. Equation. PDO CMF for horizontal curves and tangents at type 1 crest 

vertical curves................................................................................................................................42

Figure 45. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and 

tangents at type 1 crest vertical curves ..........................................................................................43

Figure 46. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and 

tangents at type 1 crest vertical curves ..........................................................................................43

Figure 47. Equation. FI CMF for horizontal curves and tangents at type 1 sag vertical curves....44

Figure 48. Equation. PDO CMF for horizontal curves and tangents at type 1 sag vertical 

curves.............................................................................................................................................44

Figure 49. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and 

tangents at type 1 sag vertical curves.............................................................................................45

v
Figure 50. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and 

tangents at type 1 sag vertical curves.............................................................................................45

Figure 51. Equation. FI CMF for horizontal curves and tangents at type 2 crest vertical 

curves.............................................................................................................................................46

Figure 52. Equation. PDO CMF for horizontal curves and tangents at type 2 crest vertical 

curves.............................................................................................................................................46

Figure 53. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and 

tangents at type 2 crest vertical curves ..........................................................................................47

Figure 54. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and 

tangents at type 2 crest vertical curves ..........................................................................................47

Figure 55. Equation. FI CMF for horizontal curves and tangents at type 2 sag vertical curves....48

Figure 56. Equation. PDO CMF for horizontal curves and tangents at type 2 sag vertical 

curves.............................................................................................................................................48

Figure 57. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and 

tangents at type 2 sag vertical curves.............................................................................................49

Figure 58. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and 

tangents at type 2 sag vertical curves.............................................................................................49

Figure 59. Equation. CMF for combined crash severity level.......................................................50

Figure 60. Graph. Comparison of CMFs developed in this study to the combined AASHTO 

HSM CMFs for horizontal curves and grades for fixed radius and varying percent grades .........51

Figure 61. Graph. Comparison of CMFs developed in this study to the combined AASHTO 

HSM CMFs for horizontal curves and grades for fixed percent grade and varying radii .............52

vi
LIST OF TABLES

Table 1. CMF5r for grade of roadway segments..............................................................................6

Table 2. Roadway length (mi) in available data from Washington HSIS database.........................9

Table 3. Exposure (MVMT) in available data from Washington HSIS database (2003–2008)......9

Table 4. Fatal and injury (FI), property damage only (PDO), and total 6-year crash 

frequencies in available data from Washington HSIS database ....................................................10

Table 5. FI, PDO, and total crash rates per MVMT in available data from Washington 

HSIS database ................................................................................................................................11

Table 6. Percent of sections with and without crashes on rural two-lane highways in 

available data from Washington HSIS database (2003–2008) ......................................................12

Table 7. Descriptive statistics for horizontal curves and tangents on straight grades ...................18

Table 8. FI and PDO crash modeling results for horizontal curves and tangents on 

straight grades................................................................................................................................19

Table 9. Descriptive statistics for horizontal curves and tangents at type 1 crest 

vertical curves................................................................................................................................22

Table 10. FI and PDO crash modeling results for horizontal curves and tangents at type 1 

crest vertical curves........................................................................................................................24

Table 11. Descriptive statistics for horizontal curves and tangents at type 1 sag 

vertical curves................................................................................................................................27

Table 12. FI and PDO crash modeling results for horizontal curves and tangents at type 1 

sag vertical curves..........................................................................................................................28

Table 13. Descriptive statistics for horizontal curves and tangents at type 2 crest 

vertical curves................................................................................................................................31

Table 14. FI and PDO crash modeling results for horizontal curves and tangents at type 2 

crest vertical curves........................................................................................................................33

Table 15. Descriptive statistics for horizontal curves and tangents at type 2 sag vertical 

curves.............................................................................................................................................35

Table 16. FI and PDO crash modeling results for horizontal curves and tangents at type 2 

sag vertical curves..........................................................................................................................36

Table 17. Example CMFs for FI and PDO crashes on horizontal curves and tangents on 

straight grades................................................................................................................................41

Table 18. Example CMFs for FI and PDO crashes on horizontal curves and tangents at 

type 1 crest vertical curves.............................................................................................................42

Table 19. Example CMFs for FI and PDO crashes on horizontal curves and tangents at 

type 1 sag vertical curves...............................................................................................................46

Table 20. Example CMFs for FI and PDO crashes on horizontal curves and tangents at 

type 2 crest vertical curves.............................................................................................................48

Table 21. Example CMFs for FI crashes on horizontal curves and tangents at type 2 sag 

vertical curves................................................................................................................................50

Table 22. Example CMFs PDO crashes on horizontal curves and tangents at type 2 sag 

vertical curves................................................................................................................................50

vii

CHAPTER 1—INTRODUCTION

This chapter presents an introduction to the report, including key background information for the 

research, research objectives and scope, and the organization of this report.

BACKGROUND

The safety effects of horizontal curves and grades on highways have been quantified separately, 

but it is not currently known whether and how the safety performance of horizontal curves and 

grades interact. Furthermore, there are established safety effects for crest and sag vertical curves,

and it is not known whether and how the safety performance of crest and sag vertical curves 

interacts with any horizontal curves that may be present. The research presented in this report 

was undertaken as part of the Federal Highway Administration (FHWA) Highway Safety 

Information System (HSIS) contract to investigate these issues and provide results in a form 

useful to highway designers and highway safety engineers.

RESEARCH OBJECTIVES AND SCOPE

The objective of the research was to quantify the safety effects of horizontal and vertical 

alignment combinations and to present the results in a form suitable for incorporation in the 

American Association of State Highway and Transportation Officials (AASHTO) Highway 

Safety Manual (HSM).(1)

The scope of the work initially included horizontal and vertical alignment for the four facility 

types whose safety performance is addressed in the first edition of the HSM:(1)

• Rural two-lane highways.

• Rural multilane undivided highways.

• Rural multilane divided highways.

• Urban and suburban arterials.

The research found that only rural two-lane highways had sufficient data for which modeling 

efforts appeared promising. 

REPORT ORGANIZATION 

The remainder of this report is organized as follows:

• Chapter 2 reviews the existing state of knowledge about the safety effects of horizontal 

curves and grades prior to the current research. 

• Chapter 3 describes the HSIS database used for analysis.

1
• Chapter 4 discusses the statistical analysis approach and presents the results of the 

safety analysis.

• Chapter 5 details how crash modification factors (CMFs) are derived. 

• Chapter 6 presents the conclusions and recommendations.

2
CHAPTER 2—STATE OF KNOWLEDGE LITERATURE REVIEW

This chapter reviews the existing state of knowledge about the safety effects of horizontal curves 

and grades prior to the current research.

GEOMETRIC DESIGN CRITERIA

Design criteria for horizontal and vertical alignment are presented in chapter 3 of the AASHTO 

Policy on Geometric Design of Highways and Streets, commonly known as the Green Book.

(2)

Many State highway agencies have their own design manuals, but they tend to closely resemble 

the AASHTO Green Book in the areas of horizontal and vertical alignment.(2)

The key design parameters for horizontal curves include the following:

• Radius of curvature.

• Length of curve.

• Superelevation.

• Transition design.

Straight road sections with no horizontal curvature are generally referred to as “tangents”

because they are usually tangent to any horizontal curves that they adjoin.

The fundamental design parameter for vertical alignment is the percent grade, which is defined 

as 100 times the change in roadway elevation per unit length of roadway measured in the 

horizontal plane. A road section with constant percent grade, regardless of its horizontal 

alignment, is generally referred to as a straight grade. Where the grade of the roadway changes, 

the straight grade sections are normally joined by a parabolic vertical curve. Figure 1 illustrates 

the four types of vertical curves (two types of crest vertical curves and two types of sag vertical 

curves) that are used in highway design. Key design parameters for vertical curves include the

following:

• Algebraic difference in grade.

• Length of curve.

• Ratio of algebraic difference in grade and length of curve (K), which represents the 

sharpness of the vertical curve.

3
Source: AASHTO. Used by permission.

Figure 1. Illustration. Types of vertical curves.

(2)

SAFETY EFFECTS OF HORIZONTAL ALIGNMENT

The most widely used safety relationship for horizontal curves on rural two-lane highways was 

developed by Zegeer et al. in the early 1990s as follows:(3,4)

N = �1.55 (LC�(V) + 0.014(D)(V) − 0.012(S)(V)](0.978)

W − 30

Figure 2. Equation. Relationship between crash rates and curve characteristics.

Where:

N = Total number of crashes on curve per year.

LC = Length of curve (mi).

V = Volume of vehicles (in millions) passing through the curve in a 1-year period.

D = Degree of curvature (5,730/radius of curvature (ft)).

S = Presence of spiral transitions, where S = 0 if no spirals exist and S = 1 if spirals do exist.

W = Width of roadway on curve (ft).

4
D, which represents the deflection angle for the curve per 100 ft of curve length, is an obsolete 

measure that is no longer used in the AASHTO Green Book.

(2) Therefore, the equivalent to 

degree of curvature expressed in terms of the radius of curvature is provided. The original 

Zegeer et al. model was expressed in terms of predicting crash frequency for a 5-year period; 

the model in figure 2 is expressed on a per-year basis for ease of comparison to other models.

No safety relationships for horizontal curvature have been developed for highway types other 

than rural two-lane highways, although such relationships for freeways have been developed in 

National Cooperative Highway Research Program Project 17-45.(5)

An assessment of literature conducted by Harwood, et al., as part of the development of the 

FHWA Interactive Highway Safety Design Model, concluded that the Zegeer, et al. model in 

figure 2 was the most useful and accurate model to account for the safety effect of horizontal 

curves on rural two-lane highways.

(6,7) Harwood et al. expressed the Zegeer, et al. model as a 

CMF in the following form:(6)

CMFHC = 1.55LC + 

80.2

R − 0.012S

1.55LC

Figure 3. Equation. CMF for horizontal curves on rural two-lane highways.

Where:

CMFHC = CMF for horizontal curvature on a rural two-lane highway.

R = Radius of curvature (ft).

The base condition for this CMF is a tangent roadway. The AASHTO HSM adopted the CMF in 

figure 3 to represent the safety effects of horizontal curvature on rural two-lane highways.(1)

The AASHTO HSM also includes a CMF for the safety effect of superelevation for horizontal 

curves of rural two-lane highways as a function of superelevation variance in the following form:

CMFSV = 1.00 for SV < 0.01

CMFSV = 1.00 + 6(SV − 0.01) for 0.01 ≤ SV < 0.02

CMFSV = 1.06 + 3(SV − 0.02) for SV ≥ 0.02

Figure 4. Equation. CMF for superelevation as a function of superelevation variance.

Where:

CMFSV = CMF for superelevation variance on a rural two-lane highway.

SV = Superelevation variance (ft/ft), which represents the design superelevation rate presented in 

the AASHTO Green Book minus the actual superelevation of the curve.

(2)

CMF was also adapted by Harwood et al. from the work of Zegeer et al.(3,4,6) The base condition

for CMF is a horizontal curve with superelevation within 0.01 ft/ft of the applicable design 

superelevation presented in the AASHTO Green Book.

(2)

No CMFs for horizontal curvature on rural multilane undivided highways, rural multilane 

divided highways, or urban and suburban arterials are included in the AASHTO HSM.(1)

5
SAFETY EFFECTS OF VERTICAL ALIGNMENT

Table 1 presents the AASHTO HSM CMF, denoted as CMF5r, which represents the safety effect 

of percent grade on rural two-lane highways.

Table 1. CMF for grade of roadway segments.

(1)

Level Grade

(≤ 3 percent)

Moderate Terrain

(3 percent < Grade

≤ 6 percent)

Steep Terrain

(> 6 percent)

1.00 1.10 1.16

The base condition for the CMFs shown in table 1 is a level roadway. Figure 5 represents the 

underlying functional form for CMF shown in table 1.

CMFG = 1.016 abs(G)

Figure 5. Equation. CMF for percent grade on rural two-lane highways. 

Where:

CMFG = CMF for percent grade on a rural two-lane highway.

abs(G) = Absolute value of percent grade.

The CMF in this form, as a continuous function, is a more useful representation of the safety 

effect of percent grade than the form shown in table 1, as the stepwise function in table 1 may 

be misleading.

The CMF for percent grade shown in table 1 and figure 5 applies only to straight grades. There 

are no CMFs in the AASHTO HSM for crest or sag vertical curves on rural two-lane highways. 

Also, there are no vertical alignment CMFs in the AASHTO HSM for facility types other than 

rural two-lane highways.

INTERACTIONS BETWEEN SAFETY EFFECTS OF HORIZONTAL AND VERTICAL 

ALIGNMENT

While the AASHTO HSM provides CMFs for the safety effects of horizontal curvature and 

percent grade on rural two-lane highways, it does not have any method for accounting for the 

interactions between these effects. In other words, in the AASHTO HSM procedures for rural 

two-lane highways, the safety effect of a horizontal curve is the same whether it is located on a 

level roadway, a straight grade, or a vertical curve. Similarly, the safety effect of a straight grade 

is the same whether it is located on a tangent roadway or on a horizontal curve. Researchers have 

always assumed that there are interactions between the safety effects of horizontal and vertical 

alignment, but this has not been demonstrated in a form useful for safety prediction. Recent 

research by Easa and You have partially addressed this issue with separate models for horizontal 

curves and horizontal tangents, but they did not tie their models back to a common base 

condition (such as a level tangent roadway) or express the modeling results in a form that 

could be considered a CMF.(8,9)

6
CHAPTER 3—DATABASE DESCRIPTION

This chapter describes the database used in the analysis. The research was performed with HSIS 

data for State highways in Washington. This is the only data source that includes system-wide 

data on curve and grade geometry that can be linked to system-wide roadway characteristics, 

traffic volume, and crash data.

DATABASE DEVELOPMENT

Research began with a review of available databases that contain roadway data (including 

horizontal and vertical alignment), traffic volume data, and crash data in a format that could be 

linked by location, with a primary focus on available HSIS data. The only dataset found with 

sufficient detail concerning horizontal and vertical alignment was HSIS data for State highways 

in Washington.

Preliminary data processing was performed to identify the following types of roadways on 

the Washington State highway system (note that the abbreviations are used in table 2 through 

table 5):

• Rural two-lane highways (R2U).

• Rural four-lane undivided highways (R4U).

• Rural four-lane divided highways (nonfreeways) (R4D).

• Urban/suburban two-lane undivided arterials (U2U).

• Urban/suburban three-lane arterials with a center two-way left-turn lane (U3T).

• Urban/suburban four-lane undivided arterials (U4U).

• Urban/suburban four-lane divided arterials (nonfreeways) (U4D).

• Urban/suburban five-lane arterials with a center two-way left-turn lane (U5T).

Roadway segments with atypical features such as passing and climbing lanes as well as roadway 

segments with transitions between grades identified as angle points were eliminated from 

consideration. Angle points most likely represent crest or sag vertical curves that were too short 

or not well enough defined to be measured properly. Finally, a limited set of roadways with 

obvious data problems, such as successive vertical curves whose lengths appeared to overlap, 

were also eliminated from consideration. The descriptive statistics for the roadway data shown 

below are based on the roadway lengths that were retained for analysis after the initial screening.

Next, each roadway segment was classified into categories by its horizontal and vertical 

alignment. Horizontal alignment was classified as follows:

• Tangent roadways.

7
• Roadways on horizontal curves.

Vertical alignment was classified as follows:

• Level roadways.

• Straight grades (constant percent grade of 1 percent or more).

• Type 1 crest vertical curves.

• Type 2 crest vertical curves.

• Type 1 sag vertical curves.

• Type 2 sag vertical curves.

Figure 1 illustrates the distinction between types 1 and 2 crest and sag vertical curves. Every 

roadway segment was defined by its horizontal alignment, vertical alignment, and combination 

of horizontal and vertical alignments. Since there were two horizontal alignment categories and 

six vertical alignment categories, there were a total of 12 possible combinations of horizontal and 

vertical alignments. Where horizontal and vertical curves overlap, their beginnings and ends may 

not coincide; therefore, a new roadway segment began at any point where the horizontal or 

vertical alignment changed. Thus, some segments might include all of a horizontal or vertical 

curve, while others might include only part of a horizontal or vertical curve. The length of 

every roadway segment was determined for use in the analysis as well as the length of any 

horizontal curve that was wholly or partially within the segment and the length of any vertical 

curve (LVC) that was wholly or partially within the segment. Additionally, each horizontal curve 

was characterized by its radius. No data on the superelevation of horizontal curves were 

available for analysis. Each straight grade was characterized by its percent grade (G). Each

vertical curve was characterized by its approach grade (G1) and departure grade (G2), its 

algebraic difference in grade (A = abs(G1 – G2)), and the ratio of its length to its algebraic 

difference in grade (K = LVC/A).

Crash data for a 6-year period (2003 to 2008) were obtained and used in the analysis. Each crash 

was assigned to a particular roadway segment, with particular horizontal and vertical alignment 

based on its assigned milepost location. Since the results of this research are intended for use in 

the roadway segment procedures of the AASHTO HSM, only nonintersection crashes were 

considered. Nonintersection crashes are those that did not occur at an intersection and were not 

classified by the investigating officer or data coder as related to the operation of an intersection. 

The traffic volume for each roadway segment was determined from available traffic volume data.

Some of the roadway segments included in the analysis may have contained minor intersections,

but crashes occurring at or related to those intersections have been excluded from the analysis. 

No major intersections were included within a roadway segment because a change in traffic

volume along the roadway would have led to the creation of a roadway segment boundary.

8
Driveway crashes are considered roadway segment crashes. As a result, they were included in 

the analysis. The amount of driveway access within segments may have varied, but all of the 

segments were located in areas classified as rural.

DESCRIPTIVE STATISTICS

The following discussion presents descriptive statistics for the roadway sections available for

analysis including roadway length (miles), exposure (million vehicle miles traveled in the 6-year 

period (MVMT)), crash frequencies, and crash rates per MVMT for specific combinations of 

horizontal and vertical alignment and roadway type. Roadway length, exposure, crash frequency, 

and crash rate are presented in table 2 through table 5, respectively.

Of the 6,944 mi of roadway in the entire Washington HSIS database, 4,785 mi (69 percent) are 

on rural two-lane highways. Of these, 3,457 mi were used for analysis. Rural two-lane highways 

with passing or climbing lanes and segments with missing or obviously incorrect alignment data 

(e.g., overlapping curves) were excluded from the study. Therefore, based on data availability, 

the analysis of the safety effects of horizontal curve and grade combinations in this report 

focused on rural two-lane highways only.

Table 2. Roadway length (mi) in available data from Washington HSIS database.

Alignment

Type

Roadway 

Element

Rural Highways Urban/Suburban Arterials

R2U R4D R4U U2U U3T U4D U4U U5T

Horizontal

Tangent 2,472.1 122.4 4.9 122.1 15.9 24.2 46.3 48.0

Curve 985.0 54.0 1.2 43.3 2.5 6.4 13.1 8.9

Total 3,457.1 176.4 6.1 165.4 18.4 30.7 59.4 56.9

Vertical

Straight grade 2,260.7 107.6 4.0 112.4 12.2 17.6 37.5 36.2

Type 1 crest 364.5 19.9 0.1 14.7 1.7 4.5 6.0 6.3

Type 2 crest 300.8 20.1 0.5 13.3 2.0 2.7 6.1 5.5

Type 1 sag 252.1 12.8 0.5 12.0 1.2 3.7 3.9 5.1

Type 2 sag 279.1 16.0 0.9 13.0 1.3 2.2 5.9 3.8

Total 3,457.1 176.4 6.1 165.4 18.4 30.7 59.4 56.9

Table 3. Exposure (MVMT) in available data from Washington HSIS database (2003–2008).

Alignment

Type

Roadway 

Element

Rural Highways Urban/Suburban Arterials

R2U R4D R4U U2U U3T U4D U4U U5T

Horizontal

Tangent 16,675.2 3,648.2 113.5 3,228.2 483.5 1,408.2 2,613.9 2,826.3

Curve 6,194.2 1,587.6 35.1 1,108.8 76.5 351.3 684.0 500.3

Total 22,869.5 5,235.8 148.6 4,336.9 560.0 1,759.5 3,297.9 3,326.5

Vertical

Straight grade 14,847.0 3,216.5 91.6 2,894.0 366.8 991.4 2,072.1 2,091.4

Type 1 crest 2,616.4 636.6 2.3 398.3 54.3 265.9 341.4 375.6

Type 2 crest 1,870.5 521.2 11.9 364.1 62.4 162.4 341.5 322.7

Type 1 sag 1,772.6 392.6 23.4 344.5 38.0 209.3 227.4 309.4

Type 2 sag 1,762.9 469.0 19.4 336.0 38.5 130.5 315.5 227.4

Total 22,869.5 5,235.8 148.6 4,336.9 560.0 1,759.5 3,297.9 3,326.5

9
Table 4. Fatal and injury (FI), property damage only (PDO), and total 6-year crash 

frequencies in available data from Washington HSIS database.

Alignment

Type

Roadway 

Element

Rural Highways Urban/Suburban Arterials

R2U R4D R4U U2U U3T U4D U4U U5T

FI Crash Frequencies in 6 Years (2003–2008)

Horizontal

Tangent 7,360 865 77 2,564 335 1,557 5,100 2,867

Curve 3,659 353 11 772 55 406 1,017 245

Total 11,019 1,218 88 3,336 390 1,963 6,117 3,112

Vertical

Straight grade 7,347 740 63 2,313 260 1,291 3,570 2,117

Type 1 crest 1,168 125 1 260 25 262 634 323

Type 2 crest 826 97 5 292 39 156 780 222

Type 1 sag 896 108 4 235 29 101 577 304

Type 2 sag 782 148 15 236 37 153 556 146

Total 11,019 1,218 88 3,336 390 1,963 6,117 3,112

PDO Crash Frequencies in 6 Years (2003–2008)

Horizontal

Tangent 10,519 1,403 213 3,753 519 2,841 9,012 4,323

Curve 4,758 621 14 1,264 77 741 1,800 426

Total 15,277 2,024 227 5,017 596 3,582 10,812 4,749

Vertical

Straight grade 10,222 1,273 185 3,451 401 2,259 6,250 3,315

Type 1 crest 1,498 229 10 397 55 527 1,208 461

Type 2 crest 1,264 162 4 437 64 310 1,390 333

Type 1 sag 1,154 177 13 383 37 227 1,037 402

Type 2 sag 1,139 183 15 349 39 259 927 238

Total 15,277 2,024 227 5,017 596 3,582 10,812 4,749

Total Crash Frequencies in 6 Years (2003–2008)

Horizontal

Tangent 17,879 2,268 290 6,317 854 4,398 14,112 7,190

Curve 8,417 974 25 2,036 132 1,147 2,817 671

Total 26,296 3,242 315 8,353 986 5,545 16,929 7,861

Vertical

Straight grade 17,569 2,013 248 5,764 661 3,550 9,820 5,432

Type 1 crest 2,666 354 11 657 80 789 1,842 784

Type 2 crest 2,090 259 9 729 103 466 2,170 555

Type 1 sag 2,050 285 17 618 66 328 1,614 706

Type 2 sag 1,921 331 30 585 76 412 1,483 384

Total 26,296 3,242 315 8,353 986 5,545 16,929 7,861

10
Table 5. FI, PDO, and total crash rates per MVMT in available data from Washington 

HSIS database.

Alignment

Type

Roadway 

Element

Rural Highways Urban/Suburban Arterials

R2U R4D R4U U2U U3T U4D U4U U5T

FI Crash Rate per MVMT

Horizontal Tangent 0.441 0.237 0.679 0.794 0.693 1.106 1.951 1.014

Curve 0.591 0.222 0.313 0.696 0.719 1.156 1.487 0.490

Vertical

Straight grade 0.495 0.230 0.688 0.799 0.709 1.302 1.723 1.012

Type 1 crest 0.446 0.196 0.432 0.653 0.460 0.985 1.857 0.860

Type 2 crest 0.442 0.186 0.421 0.802 0.625 0.961 2.284 0.688

Type 1 sag 0.505 0.275 0.171 0.682 0.763 0.483 2.537 0.982

Type 2 sag 0.444 0.316 0.775 0.702 0.961 1.173 1.762 0.642

PDO Crash Rate per MVMT

Horizontal Tangent 0.631 0.385 1.877 1.163 1.073 2.017 3.448 1.530

Curve 0.768 0.391 0.399 1.140 1.007 2.110 2.632 0.852

Vertical

Straight grade 0.688 0.396 2.020 1.192 1.093 2.279 3.016 1.585

Type 1 crest 0.573 0.360 4.323 0.997 1.012 1.982 3.539 1.227

Type 2 crest 0.676 0.311 0.337 1.200 1.025 1.909 4.070 1.032

Type 1 sag 0.651 0.451 0.555 1.112 0.974 1.085 4.560 1.299

Type 2 sag 0.646 0.390 0.775 1.039 1.013 1.985 2.938 1.047

Total Crash Rate per MVMT

Horizontal Tangent 1.072 0.622 2.556 1.957 1.766 3.123 5.399 2.544

Curve 1.359 0.614 0.712 1.836 1.726 3.265 4.118 1.341

Vertical

Straight grade 1.183 0.626 2.707 1.992 1.802 3.581 4.739 2.597

Type 1 crest 1.019 0.556 4.755 1.649 1.472 2.967 5.396 2.087

Type 2 crest 1.117 0.497 0.758 2.002 1.650 2.870 6.354 1.720

Type 1 sag 1.156 0.726 0.726 1.794 1.737 1.567 7.097 2.282

Type 2 sag 1.090 0.706 1.550 1.741 1.975 3.158 4.701 1.689

Note: Crash rates cannot be added; therefore, no totals are shown.

The majority of rural two-lane roadway segments included in the analysis experienced no 

crashes in the 6-year period. Table 6 shows the percentage of roadway segments with no crashes 

and with one or more crashes for each roadway and severity type. The data clearly show the 

highly skewed crash distribution for all severity types and all roadway types. The high 

percentages of roadway segments without crashes (78 to 89 percent for FI crashes and

72 to 84 percent for PDO crashes) provided a challenge in modeling crash frequencies.

11
Table 6. Percent of sections with and without crashes on rural two-lane highways in 

available data from Washington HSIS database (2003–2008).

Vertical and Horizontal Alignment 

Combination

FI Crashes PDO Crashes Total Crashes

Percent of Sections With:

Zero

Crashes

1+ 

Crashes

Zero

Crashes

1+

Crashes

Zero

Crashes

1+

Crashes

Straight grade Horizontal curve 84 16 81 19 72 28

Tangent on nonlevel grade 84 16 79 21 71 29

Type 1 crest Horizontal curve 84 16 82 18 73 27

Tangent on nonlevel grade 84 16 80 20 72 28

Type 2 crest Horizontal curve 88 12 84 16 78 22

Tangent on nonlevel grade 89 11 84 16 78 22

Type 1 sag Horizontal curve 86 14 84 16 76 24

Tangent on nonlevel grade 86 14 83 17 75 25

Type 2 sag Horizontal curve 86 14 84 16 76 24

Tangent on nonlevel grade 86 14 83 17 75 25

Level tangent Level tangent 78 22 72 28 63 37

VALUE RANGE OF ROADWAY CHARACTERISTICS

Prior to statistical modeling, the parameters of interest were assessed for extreme values 

(both high and low). This was done using a combination of plots of crash rates per MVMT 

versus selected parameters and distributions of the individual parameters. The following rules 

were implemented:

• Roadway segments less than 0.01 mi long were excluded from analysis (such short 

segments likely represent small overlaps in horizontal and vertical features and are 

unlikely to be useful analysis sections).

• For type 1 crest and type 1 sag vertical curves, segments where both initial (G1) and final 

(G2) grades were less than 1 percent in absolute value were excluded (such minor vertical 

curves are very close to being level).

• For type 2 crest and type 2 sag vertical curves, segments where A = abs(G1 – G2) was 

less than 1 percent were excluded (such minor vertical curves are very close to being 

straight grades).

• All records with K exceeding 1,000 were excluded (these were typically long, vertical 

curves with small grade changes and could be classified as straight grades, including 

level grades).

• All records with a curve radius exceeding 11,460 ft were excluded (these could be 

classified as tangents for all practical purposes).

• Horizontal curves with a radius less than 100 ft were included in the analysis, but the 

radius was set at 100 ft based on guidance in the AASHTO HSM.(1)

12
CHAPTER 4—STATISTICAL ANALYSIS

The overall statistical approach to estimating the safety effects of horizontal curve and grade 

combinations on rural two-lane highways is presented in this chapter along with the results for 

each type of combination for FI and PDO crashes.

ANALYSIS APPROACH

The safety effects of horizontal curve and grade combinations are estimated based on a crosssectional analysis using a generalized linear model approach assuming a negative binomial (NB) 

distribution of crash counts and an exponential model using the combined crash data from all 

6 years and selected roadway geometrics. FI and PDO crashes were modeled separately for each

of the five types of horizontal curve and grade combinations.

Selection of Independent Variables Considered in Models

The parameters considered in each model may include the following:

• Average annual daily traffic (AADT) (averaged across all 6 years).

• Segment length.

• Horizontal curve radius.

• Absolute value of percent grade.

• Horizontal curve length.

• Vertical curve length.

• Algebraic difference between the initial and final grades (A).

• Measure of the sharpness of vertical curvature (K).

• Relevant interactions of selected parameters.

For each type of horizontal curve and grade combination, the dataset used for modeling included 

the roadway segments for the relevant curve and grade combination but also all level tangents 

(i.e., no horizontal curvature and grade < 1 percent) to serve as the base condition.

Modeling was performed by encoding each parameter, or a transformation of it, so as to treat

roadway segments on horizontal curve and grade combinations separately from tangent

roadways on nonlevel grades and from level tangents (base condition). Segments for all of these 

types were used to develop a single model. This is equivalent to using indicator variables for 

each type of roadway segment. For example, since the natural log of the inverse radius was used 

in some of the models, the radius was recoded as Ɩn(2 ×

R

5,730 ) based on the following reasoning: 

13
the smallest value of 

R

5,730 for horizontal curves in the database is 0.5. Therefore, the ratio, 

R

5,730 , 

was multiplied by 2 such that the smallest value of Ɩn(2 ×

R

5,730 ) becomes zero. This term was 

then set equal to zero for all tangents. This approach ultimately allows for the computation of 

CMFs using the developed models (discussed in further detail in chapter 5 of this report).

Within each vertical alignment type (i.e., straight grades, type 1 crest vertical curves, type 1 sag

vertical curves, type 2 crest vertical curves, and type 2 sag vertical curves), the functional form 

of the relationship between crash frequency (FI or PDO) and the parameters listed was assessed 

separately. The following approach was used to explore the appropriate functional form for 

these relationships:

• Each parameter, other than AADT, was categorized into three groups, which were

typically of equal size (tertiles).

• A crash prediction model was developed including AADT and only the interaction of all 

categorized parameters (this is a standard analysis of variance (ANOVA) using an NB

distribution and a log link).

• The safety effect of one parameter was plotted against the cell means of another 

parameter, encoding the data by the levels of the third parameter. If a four-way 

interaction was included, then multiple sets of plots were generated.

• The shape of the relationship between safety effects and a given parameter across the 

levels of another parameter was assessed.

• These trends were assessed for each model to determine whether they were consistent. If 

not, an assessment was made to determine whether interactions exist.

Based on the visual assessment of these relationships, a final model form was selected using all

parameters and relevant interactions. The parameters in these final model forms were continuous 

variables. In other words, the categorization into three groups was used only for exploring the 

potential functional forms for the data and was not used in the final models.

Before analyzing crashes on horizontal curves, tangents on nonlevel grades, and level tangents 

using a single model, the effect of vertical curve characteristics was assessed using tangents 

alone. This was done by comparing the effect of vertical curve characteristics on crashes/mi/year 

between level tangents and tangents on nonlevel grades for each type of vertical curve. This 

subset of data represents the largest proportion of the database. If a vertical curve parameter 

showed a statistically significant effect for tangents on nonlevel grades, then that parameter was 

included as such in the model using all three types of roadway (horizontal curves, level tangents, 

and tangents on nonlevel grades).

An attempt was also made to model crashes on horizontal curves and tangents by allowing 

separate intercepts and AADT slopes (i.e., an analysis of covariance) for each segment type to 

assess whether the relationship between crash frequency and AADT differs among horizontal 

14
curves, tangents on nonlevel grades, and base condition tangents. This modeling effort was 

inconclusive (e.g., slopes were inconsistent and counterintuitive) and therefore abandoned.

Final crash prediction models were derived for horizontal curves and tangents using the same 

group of level tangent sections as the base condition for all five horizontal curve and grade 

combinations. A stepwise approach was used where all parameters and interactions were

included. The least significant interaction(s) and then the least significant parameter(s) were 

eliminated one at a time until all remaining interactions and parameters were significant. This is 

known as backwards stepwise selection. At each step, extreme data points were excluded from 

the data using leverage estimates, residuals, or Cook’s D criterion, all statistical criteria to 

evaluate the goodness-of-fit of the model to the data. In general, a 5 percent significance level 

associated with the type 3 chi-squared statistic was selected. All analyses were performed using

a procedure for fitting generalized linear models of SAS® Version 9.3.(10)

It should be noted that additional geometric features for roadway segments, such as lane and 

shoulder widths, were not included in the analysis. The decision to exclude other geometric 

features was made because they were outside the scope of the current research. Experience 

with the Zegeer et al. results found that the roadway width term was dropped out of the 

final CMF.

(3,4,6) Also, it was unlikely that the available data would support inclusion of 

additional terms.

Assessment of Goodness-of-Fit of Final Models

Once a model for a specific horizontal and vertical alignment combination was finalized, its 

goodness-of-fit to the observed crash data was evaluated. For each alignment combination and 

severity level, predicted crash frequencies versus observed frequencies were plotted and assessed 

to determine how well the data followed the line of equality. However, it should be recognized 

that perfect or near perfect agreement between predicted and observed crash frequencies should 

not be expected no matter how good the methodology is. The predicted crash frequencies, at 

their best, represent an estimate of the long-term average crash frequency for similar roadway 

segments. The observed crash frequency is simply one observation from a random process whose 

mean is estimated by the predicted crash frequency. There is no reason to expect that one 

observation from a random process should exactly equal the long-term mean. In addition, the 

methodology cannot predict a crash frequency of zero because each model has a positive 

intercept. This is reasonable because no roadway segment can ever be expected to be crash-free 

in the long term. However, in any given time period, it is reasonable to expect that many 

roadway segments, particularly lower volume segments, will experience zero observed crashes.

To assess the goodness-of-fit of each model, the level of agreement between predicted and 

observed crash frequencies was estimated based on the percentage of extreme observations in 

the 6-year period at the 5 percent significance level. Consideration of the likelihood of extreme 

observed crash frequencies, either high or low, is a method that directly takes into account the 

parameters of the NB distribution (i.e., mean and dispersion) of crash frequencies at a particular 

type of roadway segment.

Let Oi denote the number of observed crashes of a given type (FI or PDO) on a roadway 

segment, i, during the 6-year period. The likelihood of observing Oi crashes in 6 years is then 

15
computed under the assumption that Oi is an observation from a NB distribution with mean μi

and dispersion parameter k. The mean is the predicted number of crashes in the 6-year period 

calculated using the prediction model applicable for roadway segment i. The dispersion 

parameter is obtained when developing the final model.

The likelihood, pi, of observing Oi or fewer crashes at roadway segment i can then be written 

as follows:

pi = prob(Number of crashes ≤ Oi)

Figure 6. Equation. Probability of observing a given number of crashes.

Or:

pi = � f

i

(x)

Oi

x = 0

Figure 7. Equation. Probability of observing a given number of crashes expressed as a 

cumulative distribution.

Where fi(x) is the probability distribution function for an NB with μi and k and values of x = 0, 1, 

2, ..., Oi. If pi ≥ 0.5, then pi was calculated as 1 − pi. This approach is equivalent to calculating 

the area under the cumulative distribution curve at either low or high tail of the distribution.

The following two final single-value criteria used to assess how well the observed crash 

frequencies can be estimated by the proposed methodology are proposed:

• The percent of roadway segments of a given alignment combination with observed crash 

frequencies outside the upper and lower 2.5 percentile of the theoretical distribution. This 

value, Pctunlikely, is calculated as follows:

Pctunlikely = 100 (Number of segments where pi ≤ 0.025/Total number of segments)

Figure 8. Equation. Percent of roadway segments with unlikely low or high 

crash numbers.

On average, the value of Pctunlikely would be close to 5 percent if the methodology is to 

fit the observed data reasonably well because one would expect that on average, 5

percent of the observed values be outside the middle 95 percent of the distribution. Such 

observed crash frequencies would be considered extreme (low or high) crash frequencies 

under the estimated distribution.

• The average probability of more extreme than observed crash frequency. This is the 

average pi across all segments i of a given roadway type for a particular crash type (FI or 

PDO). On average, the mean would be close to 0.25 if the methodology is to fit the 

observed data reasonably well.

16
ANALYSIS RESULTS

This section presents the final modeling results for the five alignment categories for rural 

two-way highways. Each subsection is organized as follows:

• Basic description of database used.

• Final predictive regression equations.

• ANOVA tables.

• Goodness-of-fit results.

For each alignment category considered, a level tangent roadway serves as the base condition.

Horizontal Curves and Tangents on Straight Grades

The following three alignment combinations shaded in figure 9 were included in the analysis:

• Horizontal curves on straight grades (including both level and nonlevel alignments)

(number of roadway segments (N) = 8,095 and total roadway length = 595 mi).

• Tangents on nonlevel grades (grade ≥ 1 percent) (N = 7,569 and total roadway length 

= 727 mi).

• Level tangents—base condition (grade < 1 percent) (N = 5,701 and total roadway length 

= 779 mi).

Straight Grades Vertical Curves

Level

Abs(Grade)

< 1%

Non level

Abs(Grade)

≥ 1%

Type 1

Crest

Type 1

Sag

Type 2

Crest

Type 2

Sag

Tangents

Base

condition

Horizontal

Curves

Data in the three shaded cells were used in this analysis, while data in the blank 

cells were not used in this analysis.

Figure 9. Illustration. Alignment combinations used in the analysis of horizontal curves 

and tangents on straight grades.

Basic descriptive statistics such as sample size (i.e., number of roadway sections); total roadway

length; and minimum, maximum, mean, and median values for specific parameters) are shown in 

table 7 for each of the three alignment types included in the analysis.

17
Table 7. Descriptive statistics for horizontal curves and tangents on straight grades.

Parameter Minimum Maximum Mean Median

Horizontal Curves on Straight Grades

(N = 8,095; total roadway length = 595 mi)

AADT (vehicles/day) 169 26,088 2,695 1,664

Section length (mi) 0.01 0.75 0.07 0.05

Horizontal curve length (mi) 0.01 1.19 0.15 0.11

Curve radius (ft) 100 11,459 2,067 1,433

Grade (percent) 0 9.67 2.11 1.53

FI crashes per MVMT 0 39.50 0.75 0

PDO crashes per MVMT 0 46.26 0.91 0

Total crashes per MVMT 0 54.62 1.66 0

Tangents on Nonlevel Grades 

(N = 7,569; total roadway length = 727 mi)

AADT (vehicles/day) 169 26,088 2,700 1,644

Section length (mi) 0.01 0.99 0.10 0.06

Horizontal curve length (mi)

Curve radius (ft)

Grade (percent) 1.00 10.85 3.10 2.64

FI crashes per MVMT 0 39.33 0.61 0

PDO crashes per MVMT 0 44.14 0.80 0

Total crashes per MVMT 0 53.48 1.42 0

Level Tangents—Base Condition 

(N = 5,701; total roadway length = 779 mi)

AADT (vehicles/day) 169 26,088 3,285 2,153

Section length (mi) 0.01 0.98 0.14 0.09

Horizontal curve length (mi)

Curve radius (ft)

Grade (percent)

FI crashes per MVMT 0 34.21 0.46 0

PDO crashes per MVMT 0 39.50 0.67 0

Total crashes per MVMT 0 55.38 1.13 0

Note: No roadway segments exist in the shaded cells.

The final crash prediction models for FI and PDO crashes are as follows:

NFI = exp �b0 + b1 ln(AADT) + b2G + b3 ln �2 ×

5,730

R � × IHC + b4 �

1

R

� � 1

LC

� × IHC�

Figure 10. Equation. Predicted FI crashes on horizontal curves and tangents on straight 

grades (general form).

NPDO = exp �b0 + b1 ln(AADT) + b2G + b3 ln �2 ×

5,730

R � × IHC + b4 �

1

R� � 1LC

� × IHC�

Figure 11. Equation. Predicted PDO crashes on horizontal curves and tangents on straight 

grades (general form).

18
Where:

NFI = FI crashes per mile per year.

NPDO = PDO crashes per mile per year.

AADT = Vehicles per day.

G = Absolute value of percent grade (0 percent for level tangents; ≥ 1 percent otherwise).

R = Curve radius (ft) (missing for tangents).

IHC = Horizontal curve indicator (1 for horizontal curves; 0 otherwise).

LC = Horizontal curve length (mi) (not applicable for tangents).

ln = Natural logarithm function.

b0,..., b4 = Regression coefficients.

The regression results, including the coefficient estimate, dispersion parameter, standard error, 

confidence limit, chi-squared statistic, and significance level for all statistically significant 

parameters and interaction are shown in table 8.

Table 8. FI and PDO crash modeling results for horizontal curves and tangents 

on straight grades.

Parameter 

Description

Regression 

Coefficient

Coefficient 

Estimate

Standard 

Error

Lower 

95 Percent

Confidence 

Limit

Upper 

95 Percent

Confidence 

Limit

ChiSquared

Statistic

Significance

Level

FI Crashes/Mi/Year

Intercept b0 -8.76 0.15 -9.05 -8.46 N/A N/A

ln(AADT) b1 1.00 0.02 0.96 1.03 3,052.7 < 0.0001

Grade b2 0.044 0.01 0.03 0.06 27.5 < 0.0001

1/radius 

terma

b3 0.19 0.02 0.16 0.22 116.3 < 0.0001

1/R × 1/LC

interaction

b4 4.52 0.79 2.97 6.07 26.8 < 0.0001

Dispersion N/A 0.85 0.04 0.77 0.94 N/A N/A

PDO Crashes/Mi/Year

Intercept b0 -8.63 0.14 -8.89 -8.36 N/A N/A

ln(AADT) b1 1.03 0.02 1.00 1.06 4,003.5 < 0.0001

Grade b2 0.040 0.01 0.03 0.05 29.1 < 0.0001

1/radius 

terma

b3 0.13 0.02 0.10 0.16 67.4 < 0.0001

1/R × I/LC 

interaction

b4 3.80 0.84 2.15 5.45 17.3 < 0.0001

Dispersion N/A 0.80 0.03 0.73 0.87 N/A N/A a

1/radius term = ln(2 × 5,730/R).

N/A = Not applicable.

Applying figure 8, the percentage of roadway segments with extremely high observed FI crash 

frequencies was 6.09 percent across all roadway segments, which was only slightly above 

the expected 5 percent. The percentages were 6.12 percent for segments on level tangents, 

6.25 percent for segments on tangents on nonlevel grades, and 5.91 percent for segments on 

19
horizontal curves on straight grades, indicating that a few roadway segments in these roadway 

categories experienced unusually high FI crash frequencies given the prediction model used. 

None of the segments experienced extremely low FI crash frequencies under the assumed model 

at the 5 percent significance level.

Similarly, the percentage of roadway segments with extremely high observed PDO crash 

frequencies was 6.56 percent across all roadway segments, which was slightly higher than that 

for FI crashes. The percentages were 6.71 percent for segments on level tangents, 6.90 percent 

for segments on tangents on nonlevel grades, and 6.14 percent for segments on horizontal 

curves on straight grades, indicating that a few roadway segments in these roadway categories 

experienced unusually high PDO crash frequencies given the prediction model used. None of the 

segments experienced extremely low PDO crash frequencies under the assumed model at the 

5 percent significance level.

The average probability of predicting a more extreme than observed FI crash frequency was 

calculated using figure 7. Across all roadway segments, the average probability was 0.13. The 

average probability was 0.14 for segments on level tangents, 0.11 for segments on tangents on 

nonlevel grades, and 0.13 for segments on horizontal curves on straight grades, all considerably 

lower than the theoretically expected value of 0.25.

Similarly, the average probability of predicting a more extreme than observed PDO crash 

frequency was 0.14 across all roadway segments. The average probability was 0.16 for segments 

on level tangents, 0.13 for segments on tangents on nonlevel grades, and 0.15 for segments on 

horizontal curves on straight grades, all considerably lower than the theoretically expected value 

of 0.25. For both FI and PDO crashes, these low probabilities are an indication that the model 

might not provide an adequate fit to the data. This is not too surprising given the large number of 

roadway segments with a wide range of AADTs and geometrics and a high percentage of 

segments with zero crashes.

Substituting the regression coefficients in figure 10 and figure 11 with their corresponding 

estimates in table 8, the prediction models for FI and PDO crashes/mi/year are as follows:

NFI = exp �-8.76 + 1.00 ln(AADT) + 0.044G + 0.19 ln �2 ×

5,730

R � × IHC + 4.52 �

1

R

� � 1

LC

� × IHC�

Figure 12. Equation. Predicted FI crashes on horizontal curves and tangents on straight 

grades (explicit form).

NPDO = exp �-8.63 + 1.03 ln(AADT) + 0.040G + 0.13 ln �2 ×

5,730

R � × IHC + 3.80 �

1

R� � 1LC

� × IHC�

Figure 13. Equation. Predicted PDO crashes on horizontal curves and tangents on straight 

grades (explicit form).

NTotal = NFI + NPDO

Figure 14. Equation. Predicted total crashes on horizontal curves and tangents on straight 

grades (general form).

20
Table 8 clearly shows that AADT is the most predominant predictor of crashes as indicated by its 

high chi-squared statistic. Other parameters and interaction are one or two orders of magnitude 

smaller and indicate that once the variability due to traffic volume is accounted for, the 

remaining parameters explain only a small portion of the remaining variability in the data. 

This is a consistent trait across all models developed in this project.

These final models include only two significant parameters and an interaction from the list of 

parameters and interactions originally considered. This is in large part because all safety effects 

are estimated relative to level tangents as the base condition. Crash frequencies in this group of 

roadway segments alone (5,701 segments for a total of 779 mi) exhibit considerable variability 

(e.g., FI crash rates range from zero to 34.21 with a mean of 0.46 and a median of zero crashes

per MVMT as shown in table 7). To detect a significant effect of any horizontal curve 

characteristic on straight grades, the effect of such a characteristic would need to be large relative 

to the variability in the base condition set. This, in effect, is the challenge in finding statistically 

significant safety effects of practical engineering relevance.

Figure 12 and figure 13 show that crash frequency increases with increasing percent grade and 

decreases with increasing curve radius, as expected. The interaction term between radius and 

curve length represents an additional effect on safety for short and sharp horizontal curves—as 

the radius decreases and the curve shortens, the last term in figure 12 and figure 13 increases, 

adding to the crash frequency. For long horizontal curves and curves with larger radii, this term 

approaches zero and thus will have little impact on the predicted crash frequency.

Of interest is the fact that the effects of percent grade, curve radius, and the interaction between 

radius and curve length is more pronounced for FI crashes than for PDO crashes.

Initial modeling effects indicated that there might be an interaction between horizontal curve 

radius and percent grade on straight grade segments, suggesting that the effect of curve radius on 

crash frequency might change with increasing percent grade. However, this effect was found to 

be an artifact of a few data points that were clearly outliers. These outliers were eliminated from 

the dataset used to produce the final models presented in figure 12 and figure 13. Thus, the 

analysis did not find a statistically significant interaction between horizontal curve radius and 

percent grade. The results of the study presented in this report are as follows:

• The results show an interaction between horizontal curve radius and length, such that 

short and sharp horizontal curves have particularly high crash frequencies.

• The effects of horizontal curve radius, horizontal curve length, and percent grade were fit 

in the same model rather than in separate research studies, as is the case with the curve 

and grade effects currently in the HSM.

Horizontal Curves and Tangents at Type 1 Crest Vertical Curves

The following three alignment combinations shaded in figure 15 were included in the analysis:

• Horizontal curves at type 1 crest vertical curves (N = 1,219 and total roadway length

= 87 mi).

21
• Tangents at type 1 crest vertical curves (N = 2,089 and total roadway length = 200 mi).

• Level tangents—base condition (grade < 1 percent) (N = 5,743 and total roadway 

length = 833 mi).

Straight Grades Vertical Curves

Level

Abs(Grade) 

<1 percent

Nonlevel

Abs(Grade)

≥ 1 percent

Type 1

Crest

Type 1

Sag

Type 2

Crest

Type 2

Sag

Tangents

Base

Condition

Horizontal

Curves

Note: Data in the three shaded cells were used in this analysis, while data in the 

blank cells were not used in this analysis.

Figure 15. Illustration. Alignment combinations used in the analysis of horizontal curves 

and tangents at type 1 crest vertical curves.

Basic descriptive statistics such as sample size (i.e., number of roadway sections); total roadway 

length; and minimum, maximum, mean, and median values for specific parameters are shown in 

table 9 for each of the three alignment types included in the analysis.

Table 9. Descriptive statistics for horizontal curves and tangents at type 1 

crest vertical curves.

Parameter Minimum Maximum Mean Median

Horizontal Curves at Type 1 Crest Vertical Curves 

(N = 1,219; total roadway length = 87 mi)

AADT (vehicles/day) 175 26,088 3,059 1,877

Section length (mi) 0.01 0.72 0.07 0.06

Horizontal curve length (mi) 0.02 1.00 0.16 0.12

Curve radius (ft) 100 11,459 2,102 1,433

Vertical curve length (ft) 100 4,000 824 600

A 1.0 14.7 5.2 4.9

K 11.1 985.2 186.2 147.9

FI crashes per MVMT 0 23.10 0.55 0

PDO crashes per MVMT 0 28.12 0.66 0

Total crashes per MVMT 0 28.12 1.21 0

Tangents at Type 1 Crest Vertical Curves 

(N = 2,089; nonlevel total roadway length = 200 mi)

AADT (vehicles/day) 169 26,088 3,105 1,858

Section length (mi) 0.01 0.59 0.10 0.08

Horizontal curve length (mi)

Curve radius (ft)

Vertical curve length (ft) 60 4,000 776 600

22
Parameter Minimum Maximum Mean Median

A 1.0 14.7 4.7 4.3

K 5.4 985.2 192.4 151.5

FI crashes per MVMT 0 20.85 0.40 0

PDO crashes per MVMT 0 25.43 0.57 0

Total crashes per MVMT 0 33.85 0.98 0

Level Tangents—Base Condition 

(N = 5,743; total roadway length = 833 mi)

AADT (vehicles/day) 169 26,088 3,287 2,160

Section length (mi) 0.01 2.10 0.15 0.09

Horizontal curve length (mi)

Curve radius (ft)

Vertical curve length (ft)

A

K

FI crashes per MVMT 0 34.21 0.46 0

PDO crashes per MVMT 0 39.50 0.67 0

Total crashes per MVMT 0 39.50 1.13 0

Note: No roadway segments exist in the shaded cells.

The final crash prediction models for FI and PDO crashes are as follows:

NFI = exp �b0 + b1 ln(AADT) + b2 �

5,730

R � A × IVC × HC�

Figure 16. Equation. Predicted FI crashes on horizontal curves and tangents at type 1 

crest vertical curves (general form).

NPDO = exp �b0 + b1 ln(AADT) + b2 �

5,730

R � A × IVC × HC�

Figure 17. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1 

crest vertical curves (general form).

Where:

A = abs(G1 – G2) (percent); not applicable for level tangents.

G1 = Initial grade (percent) (positive for upgrade; negative for downgrade).

G2 = Final grade (percent) (positive for upgrade; negative for downgrade).

IVC × HC = Combined vertical and horizontal curve indicator (1 for combined vertical and 

horizontal curves; 0 otherwise).

The regression results, including the significant interaction, are shown in table 10.

23
Table 10. FI and PDO crash modeling results for horizontal curves and tangents at type 1 

crest vertical curves.

Parameter

Description

Regression

Coefficient

Coefficient

Estimate

Standard

Error

Lower 

95 Percent

Confidence 

Limit

Upper 

95 Percent

Confidence 

Limit

ChiSquared

Statistic

Significance

Level

FI Crashes/Mi/Year

Intercept b0 -9.56 0.23 -10.01 -9.11 N/A N/A

ln(AADT) b1 1.09 0.03 1.04 1.15 1,661.0 < 0.0001

1/R × A

interactiona

b2 0.0088 0.003 0.004 0.014 11.1 0.001

Dispersion N/A 0.70 0.05 0.60 0.81 N/A N/A

PDO Crashes/Mi/Year

Intercept b0 -8.46 0.20 -8.85 -8.08 N/A N/A

ln(AADT) b1 1.01 0.02 0.96 1.05 1,858.8 < 0.0001

1/R × A

interactiona

b2 0.0046 0.002 0.001 0.008 6.4 0.011

Dispersion N/A 0.72 0.04 0.64 0.82 N/A N/A a

1/R × A interaction = (5,730/R) × A.

N/A = Not applicable.

Using figure 8, the percentage of roadway segments with extremely high observed FI crash 

frequencies was 6.50 percent across all roadway segments, which was slightly above the 

expected 5 percent. The percentages of roadway segments with extremely high observed FI crash 

frequencies were 6.37 percent for level tangents, 6.52 percent for tangents at type 1 crests, and 

7.08 percent for horizontal curves on type 1 crests, indicating that a few roadway segments in 

these roadway categories experienced unusually high FI crash frequencies given the prediction 

model used. None of the segments experienced extremely low FI crash frequencies under the 

assumed model at the 5 percent significance level.

Similarly, the percentage of roadway segments with extremely high observed PDO crash 

frequencies was 6.70 percent across all roadway segments, similar to that for FI crashes. The 

percentages were 7.00 percent for level tangents, 5.94 percent for tangents at type 1 crests, and 

6.59 percent for horizontal curves on type 1 crests, indicating that a few roadway segments in 

these roadway categories experienced unusually high FI crash frequencies given the prediction 

model used. None of the segments experienced extremely low FI crash frequencies under the 

assumed model at the 5 percent significance level.

The average probability of predicting a more extreme than observed FI crash frequency was 

calculated using figure 7. Across all roadway segments, the average probability was 0.13. The 

probability was 0.13 for level tangents, 0.11 for tangents at type 1 crests, and 0.11 for horizontal 

curves on type 1 crests, which are all considerably lower than the theoretically expected value 

of 0.25.

Similarly, the average probability of predicting a more extreme than observed PDO crash 

frequency was 0.15 across all roadway segments. The probability was 0.16 for level tangents, 

0.14 for tangents on type 1 crests, and 0.13 for horizontal curves on type 1 crests, which are all 

24
considerably lower than the theoretically expected value of 0.25. For both FI and PDO crashes, 

these low probabilities indicate that the model might not provide an adequate fit to the data. 

Substituting the regression coefficients in figure 16 and figure 17 with their corresponding 

estimates shown in table 10, the prediction models for FI and PDO crashes/mi/year are obtained 

as follows:

NFI = exp �-9.56 + 1.09 ln(AADT) + 0.0088 �

5,730

R � A × IVC × HC�

Figure 18. Equation. Predicted FI crashes on horizontal curves and tangents at type 1 

crest vertical curves (explicit form).

NPDO = exp �-8.46 + 1.01 ln(AADT) + 0.0046 �

5,730

R � A × IVC × HC�

Figure 19. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1 

crest vertical curves (explicit form).

For horizontal curves and tangents at type 1 crest vertical curves, only the interaction term 

between curve radius (a characteristic of horizontal curves) and A, the algebraic difference in 

initial and final grades (a vertical curve characteristic), was found to be statistically significant. 

Crash frequency increased with the combination of decreasing curve radius and an increasing 

difference in grades.

Of interest is the fact that the marginal effect of the interaction, once the effect of AADT is 

accounted for, is very small as indicated by the extremely small chi-squared value in comparison 

to that for AADT (shown in table 10). After traffic volume was accounted for, there was little 

difference among the three roadway groups (i.e., level tangents, tangents on nonlevel grades, and 

horizontal curves at type 1 crest vertical curves). The argument can be made that the crash 

variability in all three groups was large (as seen in table 9). Additionally, the horizontal curves 

group consisted of only 87 mi of roadway as compared to the 833 mi of level tangents and 

200 mi of tangents at type 1 crests.

In summary, the predictive model developed for type 1 crest vertical curves includes a term 

representing the interaction between the horizontal curve radius and the algebraic difference in 

grade for the crest vertical curve. The main effects of horizontal curve radius and algebraic 

difference in grade were not statistically significant. The interaction term indicates that the 

crash frequency increased as the horizontal curve radius decreased, the length of the vertical 

curve increased, and the sharpness of the vertical curve increased (i.e., as K decreases). 

This interaction term, while not nearly as strongly related to crash frequency as AADT, is 

potentially useful in the AASHTO HSM because it currently does not contain CMFs for crest 

vertical curves.(1)

A model with a main effect or interaction for K, representing the sharpness of the vertical curve, 

would potentially be more useful to designers than a model containing the algebraic difference in 

grade. However, neither the main effect of K nor any interactions involving K were statistically 

significant for type 1 crest vertical curves. The models in figure 18 and figure 19 can also be 

written with LVC/K substituted for A. The effect expressed in this form makes engineering sense 

25
as it indicates that crash frequency would increase as the length of the vertical curve increases 

and as the sharpness of the vertical curve increases (i.e., as K gets smaller). Therefore, it is 

suggested that figure 18 and figure 19 be recast as follows:

NFI = exp �b0 + b1 ln(AADT) + b2 �

5,730

R � �

LVC

K � × IVC × HC�

Figure 20. Equation. Predicted FI crashes on horizontal curves and tangents at type 1 

crest vertical curves (alternate form).

NPDO = exp �b0 + b1 ln(AADT) + b2 �

5,730

R � �

LVC

K � × IVC × HC�

Figure 21. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1 

crest vertical curves (alternate form).

Horizontal Curves and Tangents at Type 1 Sag Vertical Curves

The following three alignment combinations shaded in figure 22 were included in the analysis:

• Horizontal curves at type 1 sag vertical curves (N = 982 and total roadway length

= 57 mi).

• Tangents at type 1 sag vertical curves (N = 1,973 and total roadway length = 145 mi).

• Level tangents—base condition (grade < 1 percent) (N = 5,744 and total roadway length

= 833 mi).

Straight Grades Vertical Curves

Level

Abs(Grade) 

<1 percent

Nonlevel

Abs(Grade) 

≥ 1 percent

Type 1

Crest

Type 1

Sag

Type 2

Crest

Type 2

Sag

Tangents

Base

condition

Horizontal

Curves

Note: Data in the three shaded cells were used in this analysis, while data 

in the blank cells were not used in this analysis.

Figure 22. Illustration. Alignment combinations used in the analysis of horizontal curves 

and tangents at type 1 sag vertical curves.

Basic descriptive statistics such as sample size (i.e., number of roadway sections); total roadway 

length; and minimum, maximum, mean, and median values for specific parameters are shown in 

table 11 for each of the three alignment types included in the analysis.

26
Table 11. Descriptive statistics for horizontal curves and tangents at type 1 

sag vertical curves.

Parameter Minimum Maximum Mean Median

Horizontal Curves at Type 1 Sag Vertical Curves 

(N=982; total roadway length = 57 mi)

AADT (vehicles/day) 169 19,373 3,074 1,821

Section length (mi) 0.01 0.31 0.06 0.05

Horizontal curve length (mi) 0.01 1.00 0.15 0.12

Curve radius (ft) 100 11,459 2,085 1,433

Vertical curve length (ft) 92 2,200 545 500

A 1.0 13.0 4.4 3.8

K 10.4 966.2 153.2 116.4

FI crashes per MVMT 0 36.61 0.71 0

PDO crashes per MVMT 0 21.35 0.81 0

Total crashes per MVMT 0 52.06 1.53 0

Tangents at Type 1 Sag Vertical Curves 

(N=1,973; total roadway length = 145 mi)

AADT (vehicles/day) 175 26,088 3,098 1,828

Section length (mi) 0.01 0.51 0.07 0.06

Horizontal curve length (mi)

Curve radius (ft)

Vertical curve length (ft) 60 2,800 523 400

A 1.0 15.1 4.2 3.6

K 6.8 969.7 153.0 120.2

FI crashes per MVMT 0 46.26 0.48 0

PDO crashes per MVMT 0 40.11 0.65 0

Total crashes per MVMT 0 70.19 1.12 0

Level Tangents—Base Condition 

(N=5,744; total roadway length = 833 mi)

AADT (vehicles/day) 169 26,088 3,287 2,160

Section length (mi) 0.01 2.10 0.15 0.09

Horizontal curve length (mi)

Curve radius (ft)

Vertical curve length (ft)

A

K

FI crashes per MVMT 0 34.21 0.46 0

PDO crashes per MVMT 0 39.50 0.67 0

Total crashes per MVMT 0 55.38 1.14 0

Note: No roadway segments exist in the shaded cells.

27
The final crash prediction models for FI and PDO crashes are as follows:

NFI = exp �b0 + b1 ln(AADT) + b2

1

K

× IVC + b3 �

5,730

R � A × IVC × HC�

Figure 23. Equation. Predicted FI crashes on horizontal curves and tangents at type 1 

sag vertical curves (general form).

NPDO = exp �b0 + b1 ln(AADT) + b2

1

K

× IVC + b3 �

5,730

R � A × IVC × HC�

Figure 24. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1 

sag vertical curves (general form).

Where:

K = LVC/A; not applicable for level tangents.

LVC = Vertical curve length (ft).

IVC = Vertical curve indicator (1 for vertical curves; 0 otherwise).

The regression results, including all statistically significant parameters and interaction, are 

shown in table 12.

Table 12. FI and PDO crash modeling results for horizontal curves and tangents at type 1 

sag vertical curves.

Parameter

Description

Regression

Coefficient

Coefficient

Estimate

Standard

Error

Lower 

95 percent

Confidence 

Limit

Upper 95

percent

Confidence 

Limit

ChiSquared 

Statistic

Significance

Level

FI Crashes/Mi/Year

Intercept b0 -9.55 0.24 -10.02 -9.08 N/A N/A

ln(AADT) b1 1.10 0.03 1.04 1.15 1,516.6 < 0.0001

1/K b2 10.51 5.18 0.36 20.66 3.9 0.048

1/R × A

interactiona

b3 0.011 0.003 0.005 0.017 12.3 0.0005

Dispersion N/A 0.86 0.06 0.75 0.99 N/A N/A

PDO Crashes/Mi/Year

Intercept b0 -8.63 0.20 -9.03 -8.24

ln(AADT) b1 1.03 0.03 0.98 1.08 1,776.9 < 0.0001

1/K b2 8.62 4.41 -0.02 17.26 3.7 0.055

1/R × A

interactiona

b3 0.010 0.002 0.005 0.014 16.7 < 0.0001

Dispersion N/A 0.79 0.05 0.70 0.89 N/A N/A a

1/R × A interaction = (5,730/R) × A.

N/A = Not applicable.

Using figure 8, the percentage of roadway segments with extremely high observed FI crash 

frequencies was 5.90 percent across all roadway segments, slightly above the expected 5 percent. 

The percentages of roadway segments with extremely high observed FI crash frequencies were 

28
6.03 percent for level tangents, 5.23 percent for tangents at type 1 sags, and 6.51 percent for 

horizontal curves on type 1 sags, indicating that a few roadway segments in these roadway 

categories experienced unusually high FI crash frequencies given the prediction model used. 

None of the segments experienced extremely low FI crash frequencies under the assumed model 

at the 5 percent significance level.

Similarly, the percentage of roadway segments with extremely high observed PDO crash 

frequencies was 6.60 percent across all roadway segments, slightly higher than that for 

FI crashes. The percentages were 6.79 percent for level tangents, 6.39 percent for tangents at 

type 1 sags, and 5.90 percent for horizontal curves on type 1 sags, indicating that a few roadway 

segments in these roadway categories experienced unusually high FI crash frequencies given the 

prediction model used. None of the segments experienced extremely low FI crash frequencies 

under the assumed model at the 5 percent significance level.

The average probability of predicting a more extreme than observed FI crash frequency was 

calculated using figure 7. Across all roadway segments, the average probability was 0.13. The 

probability was 0.14 for level tangents, 0.11 for tangents at type 1 sags, and 0.12 for horizontal 

curves on type 1 sags, which were all considerably lower than the theoretically expected value 

of 0.25.

Similarly, the average probability of predicting a more extreme than observed PDO crash 

frequency was 0.15 across all roadway segments. The probability was 0.16 for level tangents, 

0.14 for tangents on type 1 sags, and 0.14 for horizontal curves on type 1 sags, which were all 

considerably lower than the theoretically expected value of 0.25. For both FI and PDO crashes, 

these low probabilities indicate that the model might not provide an adequate fit to the data.

Substituting the regression coefficients in figure 23 and figure 24 with their corresponding 

estimates in table 12, the prediction models for FI and PDO crashes/mi/year are as follows:

NFI = exp �-9.55 + 1.10 ln(AADT) + 10.51

1

K

× IVC + 0.011 �

5,730

R � A × IVC × HC�

Figure 25. Equation. Predicted FI crashes on horizontal curves and tangents at type 1 

sag vertical curves (explicit form).

NPDO = exp �-8.63 + 1.03 ln(AADT) + 8.62

1

K

× IVC + 0.010 �

5,730

R � A × IVC × HC�

Figure 26. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1 

sag vertical curves (explicit form).

For horizontal curves and tangents at type 1 sag vertical curves, the inverse of K, a measure of 

vertical curvature, and the interaction term between curve radius (a characteristic of horizontal 

curves) and A (a vertical curve characteristic) were found to be statistically significant in both 

models. These models show that crash frequency increases with decreasing K (i.e., with sharper 

sags). The interaction term between R and A shows that crash frequency increases with the

decreasing curve radius and increasing difference in grades. In both models, the effect of the 

interaction is greater than that of K as reflected in the comparison of their chi-squared values in 

table 12.

29
As noted earlier for the crash model at type 1 crest vertical curves shown in figure 18, figure 19, 

and table 10, the marginal effect of the parameters in each model after the effect of AADT is 

accounted for is small, as indicated by the small chi-squared values in comparison to that for 

AADT (see table 12). This is an indication that after traffic volume is accounted for, there is little 

difference among the three groups of roadway (i.e., level tangents, tangents on nonlevel grades, 

and horizontal curves at type 1 sag vertical curves). The argument can be made that the crash 

variability in all three groups is large (see in table 11). Additionally, the horizontal curves group 

consists of only 57 mi of roadway as compared to the 833 mi of level tangents and 145 mi of 

tangents at type 1 sags.

As in the case for type 1 crest vertical curves, it makes engineering sense to replace A with LVC/K

in figure 25 and figure 26. In this form, both the main effect and the interaction term include K

in the denominator rather than having an inconsistency with the main effect including K and 

the interaction including A. Thus, it is recommended that figure 25 and figure 26 be recast 

as follows:

NFI = exp �b0 + b1 ln(AADT) + b2

1

K

× IVC + b3 �

5,730

R � �

LVC

K � × IVC × HC�

Figure 27. Equation. Predicted FI crashes on horizontal curves and tangents at type 1 

sag vertical curves (alternate form).

NPDO = exp �b0 + b1 ln(AADT) + b2

1

K

×IVC + b3 �

5,730

R � �

LVC

K � × IVC × HC�

Figure 28. Equation. Predicted PDO crashes on horizontal curves and tangents at type 1 

sag vertical curves (alternate form).

In summary, the predictive model developed for type 1 sag vertical curves includes a term 

representing the sharpness of the vertical curve as a main effect and an interaction between the 

horizontal curve radius, the vertical curve length, and the sharpness of the vertical curve. The 

model indicates that crash frequency increases as the horizontal curve radius decreases, the 

length of the vertical curve increases, and the sharpness of the vertical curve increases (i.e., as K

decreases). The main effect and the interaction, while not as strongly related to crash frequency 

as AADT, are potentially useful in the AASHTO HSM because it does not currently contain 

CMFs for sag vertical curves.(1)

Horizontal Curves and Tangents at Type 2 Crest Vertical Curves

The following three alignment combinations shaded in figure 29 were included in the analysis:

• Horizontal curves at type 2 crest vertical curves (N = 1,071 and total roadway length

= 62 mi).

• Tangents at type 2 crest vertical curves (N = 1,922 and total roadway length = 132 mi).

• Level tangents—base condition (grade < 1 percent) (N = 5,742 and total roadway length

= 833 mi).

30
Straight Grades Vertical Curves

Level

Abs(Grade) 

<1 percent

Nonlevel

Abs(Grade) 

≥ 1 percent

Type 1

Crest

Type 1

Sag

Type 2

Crest

Type 2

Sag

Tangents

Base

condition

Horizontal

Curves

Note: Data in the three shaded cells were used in this analysis, while 

data in the blank cells were not used in this analysis.

Figure 29. Illustration. Alignment combinations used in the analysis of horizontal curves 

and tangents at type 2 crest vertical curves.

Basic descriptive statistics such as sample size (i.e., number of roadway sections); total roadway

length; and minimum, maximum, mean, and median values for specific parameters are shown in

table 13 for each of the three alignment types included in the analysis.

Table 13. Descriptive statistics for horizontal curves and tangents at type 2 

crest vertical curves.

Parameter Minimum Maximum Mean Median

Horizontal Curves at Type 2 Crest Vertical Curves 

(N=1,071; total roadway length = 62 mi)

AADT (vehicles/day) 202 20,931 2,603 1,607

Section length (mi) 0.01 0.34 0.06 0.05

Horizontal curve length (mi) 0.01 1.09 0.16 0.12

Curve radius (ft) 100 11,459 1,960 1,433

Vertical curve length (ft) 75 2,400 543 400

A 1.0 8.3 2.8 2.4

K 15.9 952.4 227.0 178.8

FI crashes per MVMT 0 28.16 0.63 0

PDO crashes per MVMT 0 30.02 0.87 0

Total crashes per MVMT 0 30.02 1.50 0

Tangents at Type 2 Crest Vertical Curves 

(N=1,922; total roadway length = 132 mi)

AADT (vehicles/day) 175 21,825 2,741 1,652

Section length (mi) 0.01 0.38 0.07 0.06

Horizontal curve length (mi)

Curve radius (ft)

Vertical curve length (ft) 60 2,400 498 400

A 1.0 8.0 2.6 2.2

K 16.2 985.9 222.4 176.3

FI crashes per MVMT 0 36.12 0.42 0

PDO crashes per MVMT 0 27.05 0.61 0

31
Parameter Minimum Maximum Mean Median

Total crashes per MVMT 0 36.12 1.03 0

Level Tangents—Base Condition 

(N=5,742; total roadway length = 833 mi)

AADT (vehicles/day) 169 26,088 3,287 2,160

Section length (mi) 0.01 2.10 0.15 0.09

Horizontal curve length (mi)

Curve radius (ft)

Vertical curve length (ft)

A

K

FI crashes per MVMT 0 34.21 0.46 0

PDO crashes per MVMT 0 31.60 0.66 0

Total crashes per MVMT 0 36.34 1.12 0

Note: No roadway segments exist in the shaded cells.

The final crash prediction models for FI and PDO crashes are as follows:

NFI = exp �b0 + b1 ln(AADT) + b2ln �2 ×

5,730

R � × IHC�

Figure 30. Equation. Predicted FI crashes on horizontal curves and tangents at type 2

crest vertical curves (general form).

NPDO = exp �b0 + b1 ln(AADT) + b2ln �2 ×

5,730

R � × IHC�

Figure 31. Equation. Predicted PDO crashes on horizontal curves and tangents at type 2

crest vertical curves (general form).

The regression results, including all statistically significant parameters, are shown in table 14.

There were no statistically significant interactions for type 2 crest vertical curves.

Using figure 8, the percentage of roadway segments with extremely high observed FI crash 

frequencies was 6.46 percent across all roadway segments, slightly above the expected 

percentage, which was 5 percent. The percentages of roadway segments with extremely high 

observed FI crash frequencies were 6.46 percent for level tangents, 5.84 percent for tangents at 

type 2 crests, and 7.55 percent for horizontal curves on type 2 crests, indicating that a few 

roadway segments in these roadway categories experienced unusually high FI crash frequencies 

given the prediction model used. None of the segments experienced extremely low FI crash 

frequencies under the assumed model at the 5 percent significance level.

32
Table 14. FI and PDO crash modeling results for horizontal curves and tangents at type 2 

crest vertical curves.

Parameter

Description

Regression

Coefficient

Coefficient

Estimate

Standard

Error

Lower 

95 percent

Confidence 

Limit

Upper 

95 percent

Confidence 

Limit

ChiSquared

Statistic

Significance

Level

FI Crashes/Mi/Year

Intercept b0 -9.52 0.24 -9.99 -9.05 N/A N/A

ln(AADT) b1 1.09 0.03 1.03 1.14 1,470.5 < 0.0001

1/R terma b2 0.20 0.04 0.12 0.28 20.4 < 0.0001

Dispersion N/A 0.67 0.06 0.57 0.79 N/A N/A

PDO Crashes/Mi/Year

Intercept b0 -8.38 0.20 -8.78 -7.99 N/A N/A

ln(AADT) b1 1.00 0.02 0.95 1.05 1,699.0 < 0.0001

1/R terma b2 0.10 0.04 0.03 0.18 6.6 0.010

Dispersion N/A 0.65 0.05 0.57 0.74 N/A N/A a

1/R term = ln(2 × 5,730/R).

N/A = Not applicable.

The percentage of roadway segments with extremely high observed PDO crash frequencies 

was 6.93 percent across all roadway segments, slightly higher than that for FI crashes. The 

percentages were 6.99 percent for level tangents, 6.61 percent for tangents at type 2 crests, and 

7.16 percent for horizontal curves on type 2 crests, indicating that a few roadway segments in 

these roadway categories experienced unusually high FI crash frequencies given the prediction 

model used. None of the segments experienced extremely low FI crash frequencies under the 

assumed model at the 5 percent significance level.

The average probability of predicting a more extreme than observed FI crash frequency was 

calculated using figure 7. Across all roadway segments, the average probability was 0.12. The 

probability was 0.13 for level tangents, 0.09 for tangents at type 2 crests, and 0.11 for horizontal 

curves on type 2 crests, which were all considerably lower than the theoretically expected value 

of 0.25.

Similarly, the average probability of predicting a more extreme than observed PDO crash 

frequency was 0.14 across all roadway segments. The probability was 0.16 for level tangents, 

0.12 for tangents on type 2 crests, and 0.12 for horizontal curves on type 2 crests, which were all 

considerably lower than the theoretically expected value of 0.25. For both FI and PDO crashes, 

these low probabilities indicate that the model might not provide an adequate fit to the data.

Substituting the regression coefficients in figure 30 and figure 31 with their corresponding 

estimates in table 14, the prediction models for FI and PDO crashes/mi/year are as follows:

NFI = exp �-9.52 + 1.09 ln(AADT) + 0.20ln �2×

5,730

R � × IHC �

Figure 32. Equation. Predicted FI crashes on horizontal curves and tangents at type 2

crest vertical curves (explicit form).

33
NPDO = exp �-8.38 + 1.00 ln(AADT) + 0.10ln �2 ×

5,730

R � × IHC�

Figure 33. Equation. Predicted PDO crashes on horizontal curves and tangents at type 2

crest vertical curves (explicit form).

The horizontal curves group consists of only 62 mi of roadway as compared to the 833 mi of 

level tangents and 132 mi of tangents at type 2 crests. For horizontal curves and tangents at 

type 2 crest vertical curves, curve radius (a characteristic of horizontal curves) was the only 

parameter found to be statistically significant in both models. Although its effect is more 

pronounced for FI than for PDO crashes, it is small compared to that of AADT. It is interesting 

to note that the effect of curve radius for horizontal curves and tangents at type 2 crest vertical 

curves is comparable to the effect of curve radius for horizontal curves on straight grades 

(compare coefficient estimates in table 14 to those in table 8).

The models developed for type 2 crest vertical curves include a main effect for horizontal curve 

radius but no effect for any vertical curve elements. This makes the models for type 2 crest 

vertical curves less useful than the models for type 1 crest vertical curves. Sensitivity analyses 

are needed to establish whether the straight grade models in figure 10 and figure 11 might be 

applied to type 2 crest vertical curves, with G set equal to the average of G1 and G2 instead of 

figure 32 and figure 33.

Horizontal Curves and Tangents at Type 2 Sag Vertical Curves

The following three alignment combinations shaded in figure 34 were included in the analysis:

• Horizontal curves at type 2 sag vertical curves (N = 1,217 and total roadway length

= 63 mi).

• Tangents at type 2 sag vertical curves (N = 2,174 and total roadway length = 129 mi).

• Level tangents—base condition (grade < 1 percent) (N = 5,741 and total roadway length

= 833 mi).

Straight Grades Vertical Curves

Level

Abs(Grade) 

<1 percent

Nonlevel

Abs(Grade) 

≥ 1 percent

Type 1

Crest

Type 1

Sag

Type 2

Crest

Type 2

Sag

Tangents

Base

condition

Horizontal

Curves

Note: Data in the three shaded cells were used in this analysis, while data 

in the blank cells were not used in this analysis.

Figure 34. Illustration. Alignment combinations used in the analysis of horizontal curves 

and tangents at type 2 sag vertical curves.

34
Basic descriptive statistics such as sample size (i.e., number of roadway sections); total roadway

length; and minimum, maximum, mean, and median values for specific parameters are shown in

table 15 for each of the three alignment types included in the analysis.

Table 15. Descriptive statistics for horizontal curves and tangents at type 2 sag vertical 

curves.

Parameter Minimum Maximum Mean Median

Horizontal Curves at Type 2 Sag Vertical Curves 

(N=1,217; total roadway length = 63 mi)

AADT (vehicles/day) 175 21,825 2,691 1,742

Section length (mi) 0.01 0.30 0.05 0.04

Horizontal curve length (mi) 0.01 1.09 0.17 0.13

Curve radius (ft) 100 11,459 1,964 1,433

Vertical curve length (ft) 60 1,600 424 400

A 1.0 7.7 2.7 2.5

K 9.7 917.4 185.6 149.3

FI crashes per MVMT 0 26.06 0.60 0

PDO crashes per MVMT 0 27.82 0.95 0

Total crashes per MVMT 0 27.82 1.54 0

Tangents at Type 2 Sag Vertical Curves 

(N=2,174; total roadway length = 129 mi)

AADT (vehicles/day) 169 23,334 2,909 1,776

Section length (mi) 0.01 0.38 0.06 0.05

Horizontal curve length (mi)

Curve radius (ft)

Vertical curve length (ft) 60 2,000 400 400

A 1.0 7.6 2.6 2.2

K 16.2 970.9 184.3 148.1

FI crashes per MVMT 0 27.82 0.44 0

PDO crashes per MVMT 0 28.27 0.61 0

Total crashes per MVMT 0 38.20 1.05 0

Level Tangents—Base Condition 

(N=5,741; total roadway length = 833 mi)

AADT (vehicles/day) 169 26,088 3,288 2,160

Section length (mi) 0.01 2.10 0.15 0.09

Horizontal curve length (mi)

Curve radius (ft)

Vertical curve length (ft)

A

K

FI crashes per MVMT 0 34.21 0.46 0

PDO crashes per MVMT 0 25.79 0.66 0

Total crashes per MVMT 0 36.34 1.12 0

Note: No roadway segments exist in the shaded cells.

35
The final crash prediction models for FI and PDO crashes are:

NFI = exp �b0 + b1 ln(AADT) + b2ln �2 ×

5,730

R � × IHC�

Figure 35. Equation. Predicted FI crashes on horizontal curves and tangents at type 2

sag vertical curves (general form).

NPDO = exp �b0 + b1 ln(AADT) + b2 �

5,730

R � A × IVC × HC�

Figure 36. Equation. Predicted PDO crashes on horizontal curves and tangents at type 2

sag vertical curves (general form).

The regression results, including all statistically significant parameters and interaction, are 

shown in table 16.

Table 16. FI and PDO crash modeling results for horizontal curves and tangents at type 2 

sag vertical curves.

Parameter 

Description

Regression 

Coefficient

Coefficient 

Estimate

Standard 

Error

Lower 

95 percent

Confidence 

Limit

Upper 

95 percent

Confidence 

Limit

ChiSquared 

Statistic

Significance 

Level

FI Crashes/Mi/Year

Intercept b0 -9.42 0.24 -9.90 -8.95 N/A N/A

ln(AADT) b1 1.08 0.03 1.02 1.13 1,427.2 < 0.0001

1/R terma b2 0.188 0.04 0.11 0.27 18.2 < 0.0001

Dispersion N/A 0.76 0.06 0.65 0.88 NA NA

PDO Crashes/Mi/Year

Intercept b0 -8.30 0.20 -8.69 -7.90 N/A N/A

ln(AADT) b1 0.99 0.02 0.94 1.03 1,648.2 < 0.0001

1/R × A

interactionb

b2 0.022 0.005 0.013 0.031 20.8 < 0.0001

Dispersion N/A 0.64 0.05 0.56 0.73 N/A N/A a

1/R term = ln(2 × 5,730/R). b

1/R × A interaction = (5,730/R) × A.

N/A = Not applicable.

Using figure 8, the percentage of roadway segments with extremely high observed FI crash 

frequencies was 6.08 percent across all roadway segments, which was slightly above the 

expected percentage of 5 percent. The percentage of roadway segments with extremely high 

observed FI crash frequencies was 6.24 percent for level tangents, 5.53 percent for tangents at 

type 2 sags, and 6.32 percent for horizontal curves on type 2 sags, indicating that a few roadway 

segments in these categories experienced unusually high FI crash frequencies given the 

prediction model used. None of the segments experienced extremely low FI crash frequencies 

under the assumed model at the 5 percent significance level.

Similarly, the percentage of roadway segments with extremely high observed PDO crash 

frequencies was 6.99 percent across all roadway segments, which was slightly higher than that 

36
for FI crashes. The percentages were 7.10 percent for level tangents, 6.46 percent for tangents at 

type 2 sags, and 7.42 percent for horizontal curves on type 2 sags, indicating that a few roadway 

segments in these roadway categories experienced unusually high FI crash frequencies given the 

prediction model used. None of the segments experienced extremely low FI crash frequencies 

under the assumed model at the 5 percent significance level.

The average probability of predicting a more extreme than observed FI crash frequency was 

calculated using figure 7. Across all roadway segments, the average probability was 0.12. The 

probability was 0.13 for level tangents, 0.09 for tangents at type 2 sags, and 0.10 for horizontal 

curves on type 2 sags, which were all considerably lower than the theoretically expected value 

of 0.25.

Similarly, the average probability of predicting a more extreme than observed PDO crash 

frequency was 0.14 across all roadway segments. The probability was 0.16 for level tangents, 

0.12 for tangents on type 2 sags, and 0.12 for horizontal curves on type 2 sags, which were all 

considerably lower than the theoretically expected value of 0.25. For both FI and PDO crashes, 

these low probabilities indicate that the model might not provide an adequate fit to the data.

Substituting the regression coefficients in figure 35 and figure 36 with their corresponding 

estimates in table 16, the prediction models for FI and PDO crashes/mi/year are as follows:

NFI = exp �-9.42 + 1.08 ln(AADT) + 0.188ln �2 ×

5,730

R � × IHC�

Figure 37. Equation. Predicted FI crashes on horizontal curves and tangents at type 2

sag vertical curves (explicit form).

NPDO = exp �-8.30 + 0.99 ln(AADT) + 0.022 �

5,730

R � A × IVC × HC�

Figure 38. Equation. Predicted PDO crashes on horizontal curves and tangents at type 2

sag vertical curves (explicit form).

The horizontal curves group consists of only 63 mi of roadway as compared to 833 mi of level 

tangents and 129 mi of tangents at type 2 sags. For horizontal curves and tangents at type 2 sag

vertical curves, curve radius (a characteristic of horizontal curves) was the only parameter found 

to be statistically significant in the FI crash prediction model. The model in figure 37 shows that 

FI crash frequency increases with decreasing curve radius, as expected. For the PDO crash 

prediction model, only the interaction term between curve radius (a characteristic of horizontal 

curves) and A (a vertical curve characteristic), was found to be statistically significant. The

model in figure 38 shows that crash frequency increases with the combination of decreasing 

curve radius and increasing difference in grades.

In either model, the effect of the significant parameter or interaction of parameters is more 

pronounced than in previous models as reflected in the slightly higher chi-squared values. The 

magnitude of the coefficients is consistent between this set of models and previous models.

There is a substantial inconsistency between the FI and PDO crash models for type 2 sag vertical 

curves in that the PDO crash model includes a statistically significant interaction between 

horizontal curve radius and algebraic difference in grade, but the FI crash model does not. This 

37
is the only situation in the study where different parameters or functional forms had to be used 

for the corresponding FI and PDO crash models. This inconsistency makes the models for type 2 

sag vertical curves less useful than the models for type 1 sag vertical curves. LVC/K could be 

substituted for A in figure 38, as was done for type 1 crest vertical curves and type 1 sag vertical 

curves, but it seems pointless to do this in the PDO crash model when the more important FI

crash model does not include a similar interaction term. Sensitivity analyses might establish 

whether, as an alternative to the inconsistent models in figure 37 and figure 38, the straight grade 

models in figure 10 and figure 11 might be applied to type 2 sag vertical curves with the value of 

G set equal to the average of G1 and G2.

STUDY LIMITATIONS

The major limitation of the study results is that data from only one State were used in the 

analysis. It would have been desirable to use data from additional States, but Washington is the 

only known State for which curve and grade data are available for the entire State highway 

system in a form that can be linked to crash data.

The study divided the roadway into segments for analysis. A new segment started at each point 

where a horizontal curve or vertical alignment feature (straight grade or vertical curve) began or 

ended. Segmenting the roadway in this way resulted in a number of very short segments. The 

shortest of these segments were dropped from the analysis, but some of the segments that were 

analyzed may be shorter than the expected accuracy of reported crash locations. Nevertheless, 

this approach to segmenting roadways appears to be the best method to the authors of this report 

for analyzing these data because the geometrics of each analysis section are known to be 

homogeneous with respect to horizontal and vertical alignment features. Alternative analysis 

approaches that use fixed-length study sections (typically 0.5 or 1.0 mi long) and model the 

percentage of section length over which particular geometric features (or combinations of 

features) are present do not appear likely to provide satisfactory results for this application.

As noted earlier, the analysis considered the effects of horizontal and vertical geometrics on 

crashes but did not consider other features known to affect crashes such as lane and shoulder 

widths. In fact, the AASHTO HSM includes 12 factors that are known to affect safety on rural 

two-lane highways.(1) Consideration of all these factors was not beyond the scope of this current 

study. A future study is necessary to develop safety prediction models for rural two-lane 

highways considering all 12 factors and their interactions.

38
CHAPTER 5—CRASH MODIFICATION FACTORS

CMFs used in the AASHTO HSM can be derived from the predictive models developed in 

chapter 4 of this current report. A CMF is a factor that represents the effect on crash frequency

for a given crash severity level of varying geometric design or traffic control feature of interest 

(or a particular combination of geometric design or traffic control feature). Each CMF has a 

nominal value of 1.0 for a specified base condition. A CMF with a value greater than 1.0 

represents a condition for which more crashes would be expected for the base condition. A CMF 

with a value less than 1.0 represents a condition for which fewer crashes would be expected than 

for the base condition. The base condition for all CMFs developed in this research is a level 

tangent roadway.

For each combination of alignment type (and for FI crashes and PDO crashes), CMFs were 

calculated as the ratio of the predicted crash frequency for a given horizontal curve and grade 

combination to the predicted crash frequency for the level tangent base condition. The following 

subsections provide equations for each CMF and figures illustrating the relationships of each 

roadway parameter to crash frequency and to the relevant CMF. Sample CMF calculations are 

also presented.

HORIZONTAL CURVES AND TANGENTS ON STRAIGHT GRADES

CMFs for horizontal curves and tangents on straight grades can be derived from figure 12 and

figure 13 as follows:

CMFSG,FI = 

⎩

⎨

⎧exp �0.044G + 0.19 ln �2×

5,730

R � +4.52 �

1

R� � 1LC

�� for horizontal curves

exp[0.044G] for tangents on nonlevel grades

1.0 for level tangents (base condition)⎭

⎬

⎫

Figure 39. Equation. FI CMF for horizontal curves and tangents on straight grades.

CMFSG,PDO =

⎩

⎨

⎧exp �0.040G + 0.13 ln �2×

5,730

R � +3.80 �

1

R� � 1

LC

��for horizontal curves

exp[0.040G] for tangents on nonlevel grades

1.0 for level tangents (base condition)

 

 

 

 ⎭

⎬

⎫

Figure 40. Equation. PDO CMF for horizontal curves and tangents on straight grades.

The functional relationships shown in figure 12 (crashes/mi/year) and figure 39 (CMF) for FI

crashes are illustrated in figure 41 for combinations of horizontal curve lengths and percent 

grades. Curve radius ranged from 100 to 11,460 ft; AADT was fixed at 2,000 vehicles/day; the 

median traffic volume was for rural two-lane roadways in the database; the horizontal curve 

length was set at 0.05, 0.10, 0.15, and 0.20 mi; and grade was set at level (i.e., 0 percent) and 1 to 

6 percent in increments of 1 percent. Similarly, figure 42 illustrates the relationships shown in 

figure 13 and figure 40 for PDO crashes.

39
Figure 41. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and 

tangents on straight grades.

Figure 42. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and 

tangents on straight grades.

40
In figure 41 and figure 42, crashes/mi/year are shown on the left y-axis, and the corresponding 

CMFs are shown on the right y-axis. The dotted blue line corresponds to a base condition tangent 

with an AADT of 2,000 vehicles/day and therefore has a CMF of 1.0.

To calculate CMF for FI or PDO crashes for a given horizontal curve on a level or nonlevel 

grade or a tangent on a nonlevel grade, G (percent), R (ft), and LC (mi) are substituted in 

figure 39 or figure 40. Example CMFs were calculated for rural two-lane roadways with a 

R of 1,433 or 5,730 ft; LC of 0.05, 0.10, and 0.50 mi; and G ranging from level to 6 percent. 

The results are shown in table 17 for rural two-lane roadways with AADTs from 200 to

26,000 vehicles/day.

Table 17. Example CMFs for FI and PDO crashes on horizontal curves and tangents on 

straight grades.

Grade

(Percent)

Tangent

on 

Nonlevel 

Grade

R = 1,433 ft R = 5,730 ft

Horizontal Curve Length (mi) Horizontal Curve Length (mi)

0.05 0.10 0.50 0.05 0.10 0.50

CMFs for FI Crashes

Level 

(< 1 percent) 1.00 1.57 1.53 1.49 1.15 1.15 1.14

1 1.04 1.64 1.59 1.56 1.20 1.20 1.19

2 1.09 1.71 1.67 1.63 1.25 1.25 1.25

3 1.14 1.79 1.74 1.70 1.31 1.31 1.30

4 1.19 1.87 1.82 1.78 1.37 1.36 1.36

5 1.25 1.95 1.9 1.86 1.43 1.42 1.42

6 1.30 2.04 1.98 1.94 1.49 1.49 1.48

CMFs for PDO Crashes

Level 

(< 1 percent) 1.00 1.38 1.35 1.32 1.10 1.10 1.10

1 1.04 1.44 1.40 1.38 1.15 1.14 1.14

2 1.08 1.49 1.46 1.43 1.20 1.19 1.19

3 1.13 1.56 1.52 1.49 1.25 1.24 1.24

4 1.17 1.62 1.58 1.55 1.30 1.29 1.29

5 1.22 1.69 1.65 1.62 1.35 1.35 1.34

6 1.27 1.76 1.72 1.69 1.41 1.40 1.40

In any given column in table 17, the CMFs show the increasing effect on crashes of an increasing 

grade for a horizontal curve or tangent on a straight grade. In any given row for a given curve 

radius, the CMFs show the increasing effect on crashes of horizontal curve length. The 

decreasing effect on crashes of longer curve radii is shown by comparing any two columns for 

the same curve length between the two radii. The effect of a short and sharp curve (due to the last 

term in figure 39 and figure 40) is reflected in the high CMFs in the third column of the table. As 

the curve radius increases and becomes a tangent for all practical purposes, the CMFs for the 

curve approach in value those of a tangent with the same percent grade. When percent grade 

nears zero (level roadway) and the radius becomes infinite (tangent roadway), the roadway 

becomes a level tangent, and CMF becomes 1.

41
HORIZONTAL CURVES AND TANGENTS AT TYPE 1 CREST VERTICAL CURVES

CMFs for horizontal curves and tangents at type 1 crest vertical curves can be derived from 

figure 18 and figure 19 as follows:

CMFC1,FI = �

exp �0.0088 �

5,730

R �

LVC

K � for horizontal curves

1.0 for tangents at type 1 crests

1.0 for level tangents (base condition)

 

 

 

 ⎭

⎬

⎫

Figure 43. Equation. FI CMF for horizontal curves and tangents at type 1 crest vertical 

curves.

 

 

 

 ⎭

⎬

⎫

Figure 44. Equation. PDO CMF for horizontal curves and tangents at type 1 

crest vertical curves.

The functional relationships shown in figure 18 (crashes/mi/year) and figure 43 (CMF) for FI

crashes are illustrated in figure 45 for selected values of K. Curve radius ranged from 100 to 

11,460 ft, AADT was fixed at 2,000 vehicles/day, and LVC was fixed at 500 ft, the median LVC in 

the database. Values of K were set at 250, 125, 83, 63, and 50, which correspond to a grade 

difference of 2, 4, 6, 8, and 10 percent, respectively, for a curve length of 500 ft. Similarly, 

figure 46 illustrates the relationships shown in figure 19 and figure 44 for PDO crashes.

To calculate CMF for FI or PDO crashes for a given horizontal curve at a type 1 crest vertical 

curve, the actual values of R (ft), LVC (ft), and parameter K (ft/percent) are substituted in 

figure 43 or figure 44. Example CMFs were calculated for rural two-lane roadways with an 

LVC of 500 ft, R of 1,433 or 5,730 ft, and K values of 250, 125, 83, 63, and 50 ft/percent using 

figure 43 and figure 44. The results are shown in table 18 for rural two-lane roadways with 

AADTs from 200 to 26,000 vehicles/day and LVC of 500 ft.

Table 18. Example CMFs for FI and PDO crashes on horizontal curves and tangents at 

type 1 crest vertical curves.

K

Tangent

at Crest

FI CMFs PDO CMFs

R = 1,433 ft R = 5,730 ft R = 1,433 ft R = 5,730 ft

250 1.0 1.07 1.02 1.04 1.01

125 1.0 1.15 1.04 1.08 1.02

83 1.0 1.24 1.05 1.12 1.03

63 1.0 1.33 1.07 1.16 1.04

50 1.0 1.42 1.09 1.20 1.05

42
Figure 45. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and 

tangents at type 1 crest vertical curves.

Figure 46. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and 

tangents at type 1 crest vertical curves.

43
In any given column in table 18, the CMFs show the increasing effect on crashes of a decreasing 

K (steeper crest) of a horizontal curve at a type 1 crest vertical curve. This effect is less 

pronounced for PDO crashes than for FI crashes. The combined effect of a sharp horizontal 

curve on a steep vertical crest is reflected in the last rows of the third and fifth columns. When 

K becomes infinite (i.e., level roadway) and R becomes infinite (i.e., tangent roadway), the 

roadway becomes a level tangent, and CMF becomes 1.0.

HORIZONTAL CURVES AND TANGENTS AT TYPE 1 SAG VERTICAL CURVES

CMFs for horizontal curves and tangents at type 1 sag vertical curves can be derived from 

figure 25 and figure 26 as follows:

CMFS1,FI =

⎩

⎪

⎨

⎪

⎧exp �10.51 

1

K

+ 0.011 �

5,730

R �

LVC

K � for horizontal curves

exp �10.51 

1

K� for tangents at type 1 sags

1.0 for level tangents (base condition)

 

 

 

 ⎭

⎬

⎫

Figure 47. Equation. FI CMF for horizontal curves and tangents at type 1 

sag vertical curves. 

CMFS1,PDO =

⎩

⎪

⎨

⎪

⎧exp �8.62

1

K

+0.010 �

5,730

R �

LVC

K � for horizontal curves

exp �8.62 

1

K

� for tangents at type 1 sags

1.0 for level tangents (base condition)

 

 

 

 ⎭

⎬

⎫

Figure 48. Equation. PDO CMF for horizontal curves and tangents at type 1 

sag vertical curves.

The functional relationships shown in figure 25 (crashes/mi/year) and figure 47 (CMF) for FI

crashes are illustrated in figure 49 for K values of 250, 125, 83, 63, and 50. R ranged from 

100 to 11,460 ft, AADT was fixed at 2,000 vehicles/day, and LVC was fixed at 500 ft. Similarly, 

figure 50 illustrates the relationships shown in figure 26 and figure 48 for PDO crashes.

To calculate CMF for FI or PDO crashes for a given horizontal curve at a type 1 sag vertical 

curve, the actual values of R (ft), LVC (ft), and parameter K (ft/percent) are substituted in 

figure 47 or figure 48. Example CMFs were calculated for rural two-lane roadways with a 

vertical curve length of 500 ft, an R of 1,433 or 5,730 ft, and K values of 250, 125, 83, 63, and 

50 using figure 47 and figure 48. The results are shown in table 19 for rural two-lane roadways 

with AADTs from 200 to 26,000 vehicles/day.

.

44
Figure 49. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and 

tangents at type 1 sag vertical curves.

Figure 50. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and 

tangents at type 1 sag vertical curves.

45
Table 19. Example CMFs for FI and PDO crashes on horizontal curves and tangents at 

type 1 sag vertical curves.

K

FI CMFs PDO CMFs

Tangent 

at Sag R = 1,433 ft R = 5,730 ft

Tangent 

at Sag R = 1,433 ft R = 5,730 ft

250 1.04 1.14 1.07 1.04 1.12 1.05

125 1.09 1.30 1.14 1.07 1.25 1.11

83 1.13 1.49 1.21 1.11 1.39 1.17

63 1.18 1.68 1.29 1.15 1.55 1.24

50 1.23 1.93 1.38 1.19 1.74 1.31

In any given column in table 19, CMFs show the increasing effect on crashes of decreasing K

(sharper sag) for a horizontal curve or tangent at a type 1 sag vertical curve. This effect is 

slightly less pronounced for PDO than for FI crashes. When K becomes very large (i.e., level 

roadway), and R becomes infinite (i.e., tangent roadway), the roadway becomes a level tangent,

and CMF becomes 1.0.

HORIZONTAL CURVES AND TANGENTS AT TYPE 2 CREST VERTICAL CURVES

CMFs for horizontal curves and tangents at type 2 crest vertical curves can be derived from 

figure 32 and figure 33 as follows:

CMFC2,FI =�

exp �0.20 ln �2×

5,730

R �� for horizontal curves

1.0 for tangents at type 2 crests

1.0 for level tangents (base condition)

 

 

 

 ⎭

⎬

⎫

Figure 51. Equation. FI CMF for horizontal curves and tangents at type 2 crest vertical 

curves. 

CMFC2,PDO = �

exp �0.10 ln �2 ×

5,730

R �� for horizontal curves

1.0 for tangents at type 2 crests

1.0 for level tangents (base condition)

 

 

 

 ⎭

⎬

⎫

Figure 52. Equation. PDO CMF for horizontal curves and tangents at type 2 crest vertical 

curves.

The functional relationships shown in figure 32 (crashes/mi/year) and figure 51 (CMF) for FI

crashes are illustrated in figure 53 for an R range from 100 to 11,460 ft and AADTs of 1,000, 

2,000, 4,000, and 7,000 vehicles/day (corresponding approximately to the 25th, 50th, 75th, and 

90th percentile AADTs for rural two-lane highways in the database). Similarly, figure 54

illustrates the relationships shown in figure 33 and figure 52 for PDO crashes.

To calculate CMF for FI or PDO crashes for a given horizontal curve at a type 2 crest vertical 

curve, the actual value of R (ft) is substituted in figure 51 or figure 52. Example CMFs were 

calculated for rural two-lane roadways with an R of 1,433, 5,730, or 11,460 ft using figure 51

46
and figure 52. The results are shown in table 20 for rural two-lane roadways with AADTs from 

200 to 26,000 vehicles/day.

Figure 53. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and 

tangents at type 2 crest vertical curves.

Figure 54. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and 

tangents at type 2 crest vertical curves.

47
Table 20. Example CMFs for FI and PDO crashes on horizontal curves and tangents at 

type 2 crest vertical curves.

FI CMFs PDO CMFs

R = 1,433 ft R = 5,730 ft R = 11,460 ft R = 1,433 ft R = 5,730 ft R = 11,460 ft

1.52 1.15 1.00 1.23 1.07 1.00

Only R of the horizontal curve at a type 2 vertical crest has an effect on crash rates, with 

increasing radii producing smaller CMFs. The effect of R is more pronounced for FI crashes than 

for PDO crashes. When R becomes infinite, the roadway becomes a tangent, and CMF becomes

1.0 regardless of the vertical curve characteristics.

Because this CMF lacks any measure of vertical alignment, consideration may be given to 

replacing this CMF with a CMF based on figure 10 and figure 11.

HORIZONTAL CURVES AND TANGENTS AT TYPE 2 SAG VERTICAL CURVES

CMFs for horizontal curves and tangents at type 2 sag vertical curves can be derived from 

figure 37 and figure 38 as follows:

CMFS2,FI = �

exp �0.188 ln �2 ×

5,730

R �� for horizontal curves

1.0 for tangents at type 2 sags

1.0 for level tangents (base condition)

 

 

 

 ⎭

⎬

⎫

Figure 55. Equation. FI CMF for horizontal curves and tangents at type 2 sag vertical 

curves.

CMFS2,PDO= �

exp �0.022 �

5,730

R � A� for horizontal curves

1.0 for tangents at type 2 sags

1.0 for level tangents (base condition)

 

 

 

 ⎭

⎬

⎫

Figure 56. Equation. PDO CMF for horizontal curves and tangents at type 2 sag vertical 

curves.

The functional relationships shown in figure 37 (crashes/mi/year) and figure 55 (CMF) for FI

crashes are illustrated in figure 57 for an R range from 100 to 11,460 ft and AADTs of 1,000, 

2,000, 4,000, and 7,000 vehicles/day. Similarly, figure 58 illustrates the relationships shown in 

figure 38 and figure 56 for PDO crashes for K values of 250, 125, 83, 63, and 50; curve radii

ranging from 100 to 11,460 ft; and AADT fixed at 2,000 vehicles/day.

To calculate CMF for FI crashes for a given horizontal curve at a type 2 sag vertical curve, the 

actual value of R (ft) is substituted in figure 55. To calculate CMF for PDO crashes, the actual 

value of R (ft), LVC, and K are substituted in figure 56. Example FI CMFs were calculated for 

rural two-lane roadways with an R of 1,433, 5,730, or 11,460 ft using figure 55. Example PDO 

CMFs were calculated using figure 56 for rural two-lane roadways with an R of 1,433 or 

5,730 ft, an LVC of 500 ft, and A values ranging from 2 to 10 percent. The results are shown in 

48
table 21 for FI crashes and table 22 for PDO crashes for rural two-lane roadways with AADTs 

from 200 to 26,000 vehicles/day.

Figure 57. Graph. Predicted FI crashes/mi/year and CMFs for horizontal curves and 

tangents at type 2 sag vertical curves.

Figure 58. Graph. Predicted PDO crashes/mi/year and CMFs for horizontal curves and 

tangents at type 2 sag vertical curves.

49
Table 21. Example CMFs for FI crashes on horizontal curves and tangents at type 2 

sag vertical curves.

FI CMFs

R = 1,433 ft R = 5,730 ft R = 11,460 ft

1.48 1.14 1.00

Table 22. Example CMFs PDO crashes on horizontal curves and tangents

at type 2 sag vertical curves.

A

PDO CMFs

R = 1,433 ft R = 5,730 ft R = 11,460 ft

All tangents 1.00 1.00 1.00

2 1.19 1.04 1.02

4 1.42 1.09 1.04

6 1.69 1.14 1.07

8 2.02 1.19 1.09

10 2.40 1.24 1.12

Only R of the horizontal curve at a type 2 vertical sag has an effect on FI crash rates, with 

increasing radii producing smaller CMFs (see table 21). When R becomes infinite, the roadway 

becomes a tangent and CMF becomes 1.0 regardless of the characteristics of the vertical curve. 

Only the interaction term of A and inverse R has an effect on PDO crash rates (see table 22), 

highlighting the joint effect of sharp horizontal curves at steep vertical curves on PDO crashes. 

When the initial and final grade difference, A, nears zero (i.e., level roadway) and the R becomes 

infinite (i.e., tangent roadway), the roadway becomes a level tangent, and CMF becomes 1.0.

Because this CMF (at least for FI crashes) lacks any measure of vertical alignment, consideration 

may be given to replacing this CMF with a CMF based on figure 10 and figure 11.

CMFs FOR COMBINED CRASH SEVERITY LEVELS

This study provides separate CMFs for FI and PDO crashes. At some future time, the AASHTO 

HSM may be structured to provide separate CMFs by severity level for all CMFs. In the 

meantime, if users want a CMF for total crashes (i.e., all crash severity levels combined), it can 

be computed from the results in this report as follows:

CMFTOT = [(CMFFI – 1.0) × PFI + (CMFPDO – 1.0) × PPDO] + 1.0

Figure 59. Equation. CMF for combined crash severity level.

Where:

CMFTOT = CMF for total crashes (i.e., all severity levels combined).

CMFFI = CMF for FI crashes.

CMFPDO = CMF for PDO crashes.

PFI = FI crashes expressed as a proportion of total crashes.

PPDO = PDO crashes expressed as a proportion of total crashes.

50
Values used for PFI and PPDO must always sum to 1.0. Values of PFI and PPDO indicated for rural 

two-lane highways in AASHTO HSM table 10-3 (PFI = 0.321 and PPDO = 0.679) may be used, or 

users may develop values for PFI and PPDO from their agencies’ data.(1)

COMPARISON OF RESULTS TO EXISTING HSM CMFS

The current AASHTO HSM presents separate CMFs for horizontal curves and straight grades, 

as seen as CMFHC in figure 3 and CMFG in figure 5, respectively.(1) The combined effect of 

horizontal curves and grades is represented in the AASHTO HSM as the product of CMFHC and 

CMFG. Figure 60 illustrates a comparison of CMFs for horizontal curves on straight grades 

developed in this study, and shown in figure 39 and figure 40 for FI and PDO crashes, 

respectively, to the combined AASHTO HSM CMF, holding length of horizontal curve and 

radius constant while varying percent grade. Figure 61 is an analogous plot, keeping the length 

of horizontal curve and percent grade constant while varying the radius of horizontal curve.

The plots show that the new CMF for FI crashes is consistently larger than the new CMF for 

PDO crashes. This represents an advance in knowledge over the AASHTO HSM, which treats 

CMFs for all severity levels as equal.(1) The plots also show that the new CMFs are generally 

larger than the combined AASHTO HSM CMFs, except that the new CMF for PDO crashes is 

smaller than the existing CMFs for horizontal curves with short radii. No other comparisons 

between the HSM CMFs and the CMFs developed in this study are relevant because the 

AASHTO HSM does not address the safety effects of crest or sag vertical curves. CMF

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Grade (%)

0 1 2 3 4 5 6 7 8 9 10

Horizontal Curves on

 Straight Grades

Lc = 0.10 mi

Radius = 2,000 ft

 CMF (FI) CMF (PDO) Combined HSM CMFs

Figure 60. Graph. Comparison of CMFs developed in this study to the combined

AASHTO HSM CMFs for horizontal curves and grades for fixed radius and varying 

percent grades.

(1)

51
CMF

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Radius (ft)

0 2,000 4,000 6,000 8,000 10,000 12,000

Horizontal Curves on

 Straight Grades

Lc = 0.10 mi

Grade = 2%

 CMF (FI) CMF (PDO) Combined HSM CMFs

Figure 61. Graph. Comparison of CMFs developed in this study to the combined AASHTO 

HSM CMFs for horizontal curves and grades for fixed percent grade and varying radii.

(1)

52
CHAPTER 6—CONCLUSIONS AND RECOMMENDATIONS

This chapter presents the conclusions and recommendations based on the results of the study.

CONCLUSIONS

Conclusions for this study are as follows:

• For tangents and horizontal curves on straight grades, prediction models for crash 

frequency by severity level are presented in figure 10 and figure 11, with parameter 

estimates presented in table 8. These models include a main effect for AADT, a main 

effect for horizontal curve radius, a main effect for percent grade, and an interaction 

between horizontal curve radius and length of curve. The models indicate that crash 

frequency increases with decreasing horizontal curve radius, decreasing horizontal 

curve length, and increasing percent grade. The interaction term shows that short sharp 

horizontal curves are associated with higher crash frequencies. CMFs corresponding to 

the crash prediction models are presented in figure 39, figure 40, and table 17.

• For tangents and horizontal curves at type 1 crest vertical curves, prediction models for 

crash frequency by severity level are presented in figure 16 and figure 17, with parameter 

estimates presented in table 10. Alternate prediction models are shown in figure 20

and figure 21. These models include a main effect for AADT and an interaction between 

horizontal curve radius and the difference between initial and final grade (alternatively, 

an interaction between horizontal curve radius, vertical curve length, and K). The models 

indicate that crash frequency increases with decreasing horizontal curve radius and 

increases with increasing grade difference. The interaction term shows that short 

horizontal curves at sharp crest vertical curves are associated with higher crash 

frequencies. CMFs corresponding to the crash prediction models are presented in 

figure 43, figure 44, and table 18.

• For tangents and horizontal curves at type 1 sag vertical curves, prediction models for 

crash frequency by severity level are presented in figure 23 and figure 24, with parameter 

estimates presented in table 12. Alternate prediction models are shown in figure 27 and 

figure 28. These models include a main effect for AADT, a main effect for K, and an 

interaction between horizontal curve radius and the difference between initial and final 

grade (alternatively, an interaction between horizontal curve radius, vertical curve length, 

and K). The models indicate that crash frequency increases with decreasing K, decreasing 

horizontal curve radius, and increasing grade difference. The interaction term shows that 

short horizontal curves at sharp sag vertical curves are associated with higher crash 

frequencies. CMFs corresponding to the crash prediction models are presented in 

figure 47, figure 48, and table 19.

• For tangents and horizontal curves at type 2 crest vertical curves, prediction models for 

crash frequency by severity level are presented in figure 30 and figure 31, with parameter 

estimates presented in table 14. These models include only two main effects: a main 

effect for AADT and a main effect for horizontal curve radius. The models indicate that 

53
crash frequency increases with decreasing horizontal curve radius. CMFs corresponding 

to the crash prediction models are presented in figure 51, figure 52, and table 20.

Consideration will be given to using figure 10 and figure 11, with grade set equal to 

the average of G1 and G2 in place of figure 30 and figure 31.

• For tangents and horizontal curves at type 2 sag vertical curves, prediction models 

for crash frequency by severity level are presented in figure 35 and figure 36, with 

parameter estimates presented in table 16. The FI crash prediction model includes only 

two main effects: a main effect for AADT and a main effect for horizontal curve radius. 

This model indicates that FI crash frequency increases with decreasing horizontal curve 

radius. The PDO crash prediction model includes a main effect for AADT, an interaction 

between horizontal curve radius, and the difference between G1 and G2. The PDO model 

indicates that crash frequency increases with decreasing horizontal curve radius and 

increases with increasing grade difference. The interaction term shows that short 

horizontal curves at sharp sag vertical curves are associated with higher crash 

frequencies. CMFs corresponding to the crash prediction models are presented in 

figure 55, figure 56, table 21, and table 22. Consideration will be given to using 

figure 10 and figure 11 with grade set equal to the average of G1 and G2 in place of 

figure 30 and figure 17.

• For all five horizontal and vertical alignment combinations, the coefficient of the 

AADT term in the crash prediction models is nearly equal to 1.0, indicating that crash 

frequencies are proportional to AADT.

• The AADT coefficients in the models range from 0.99 to 1.10. Consideration will be 

given as to whether the AADT coefficient should be set equal to 1.0 for consistency with 

the current AASHTO HSM.

(1)

RECOMMENDATIONS

Recommendations based on the study results are as follows:

• CMFs developed in this report should be considered for incorporation in the next edition 

of the AASHTO HSM.

• It would be desirable to validate the CMFs developed in this report using data from one 

or more additional States. The current emphasis being placed on roadway inventories for 

asset management may lead to additional suitable databases.

• Ultimately, it would be desirable to have safety prediction models for rural two-lane 

highways for use in the AASHTO HSM that consider the safety effects of a full range of 

variables of interest and all of their combinations and interactions. This will require more 

comprehensive databases than those currently available.

54
REFERENCES

1. American Association of State Highway and Transportation Officials. (2010). Highway 

Safety Manual, 1st Edition, AASHTO, Washington, DC.

2. American Association of State Highway and Transportation Officials. (2011). A Policy on 

Geometric Design of Highways and Streets, AASHTO, Washington, DC.

3. Zegeer, C.V., Stewart, J.R., Council, F.M., Reinfurt, D.W., and Hamilton, E. (1992). “Safety 

Effects of Geometric Improvements on Horizontal Curves,” Transportation Research Record

1356, Transportation Research Board, Washington, DC.

4. Zegeer, C.V., Stewart, J.R., Reinfurt, D.W., Council, F.M., Neuman, T., Hamilton, E., 

Miller, T., and Hunter, W. (1991). Cost-Effective Geometric Improvements for Safety 

Upgrading of Horizontal Curves, Report No. FHWA-RD-90-021, Federal Highway 

Administration, Washington, DC.

5. Bonneson, J.A., Geedipally, S., Pratt, M.P., and Lord, D. (2012). Safety Prediction 

Methodology and Analysis Tool for Freeways and Interchanges, Final Report, National 

Cooperative Highway Research Program Project 17-45, Texas Transportation Institute, 

College Station, TX.

6. Harwood, D.W., Council, F.M., Hauer, E., Hughes, W.E., and Vogt, A. (2000). Prediction of 

the Expected Safety Performance of Rural Two-Lane Highways, Report No. FHWA-RD-99-

207, Federal Highway Administration, Washington, DC.

7. U.S. Department of Transportation. Interactive Highway Safety Design (IHSDM) Software 

Download Website, Washington, DC. Obtained from: http://www.ihsdm.org/. Site last 

accessed July 14, 2012.

8. Easa, S.M. and You, Q.C. (2009). “Collision Prediction Models for Three-Dimensional TwoLane Highways: Horizontal Curves,” Transportation Research Record 2092, Transportation 

Research Board, Washington, DC.

9. You, Q.C. and Easa, S.M. (2009). Collision Prediction Models for Three-Dimensional TwoLane Highways: II. Horizontal Tangents, Presented at the 88th Annual Meeting of the 

Transportation Research Board, Washington, DC.

10. SAS Institute Inc. (2011). SAS/STAT® 9.3 User’s Guide, SAS Institute Inc., Cary, NC.

55


HRDS-20/01-14(750)E Recycled

Recyclable