Safety Effects of Differential Speed

Limits on Rural Interstate Highways

PUBLICATION NO. FHWA-HRT-05-042 OCTOBER 2005

Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296
FOREWORD 

The Surface Transportation and Uniform Relocation Assistance Act, (STURAA) enacted on 

April 2, 1987, permitted individual States to raise rural interstate speed limits from the 

previously mandated national speed limit of 89 kilometers per hour (km/h) (55 miles per hour 

(mi/h)) to 105 km/h (65 mi/h) on rural interstate highways. Of those that changed their speed 

limits, some States raised the limits for passenger cars but not trucks while other States raised the 

limits for both passenger cars and trucks. The former category, with different speed limits for 

cars and trucks, is known as differential speed limits (DSL). The latter category, which mandates 

the same speed limits for cars and trucks, is known as uniform speed limits (USL). The 1995 

repeal of the national maximum speed limit gave States additional flexibility in setting their 

limits, such that by 2002 several States had experimented with both DSL and USL. 

This report compares the safety effects of USL for all vehicles as opposed to DSL for cars and 

heavy trucks. Detailed crash data, speed monitoring data, and traffic volumes were sought for 

rural interstate highways in 17 States for the period 1991 to 2000. The information and results of 

the study will be of particular interest to State traffic managers in making decisions about the 

application of USL or DSL in their highway systems. 

Michael Trentacoste, 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. 
1. Report No 2. Government Accession No. 3. Recipient's Catalog No. 

FHWA-HRT-05-042 N/A N/A 

4. Title and Subtitle 5. Report Date 

 October 2005 

6. Performing Organization Code 

THE SAFETY IMPACTS OF DIFFERENTIAL SPEED LIMITS ON RURAL 

INTERSTATE HIGHWAYS 

N/A 

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

Nicholas J. Garber, John S. Miller, Bo Yuan and Xin Sun N/A 

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

N/A 

11. Contract or Grant No. 

VRC-000S(007) 

13. Type of Report and Period Covered 

Virginia Transportation Research Council 

530 Edgemont Road 

Charlottesville, VA 22903

12. Sponsoring Agency Name and Address 14. Sponsoring Agency Code 

Office of Safety 

Federal Highway Administration 

6300 Georgetown Pike 

McLean, VA 22101

 

15. Supplementary Notes 

Contracting Officer’s Technical Representative: A. J. Nedzesky, HRDS-05

16. Abstract 

To compare the safety effects of a uniform speed limit (USL) for all vehicles as opposed to a differential speed limit 

(DSL) for cars and heavy trucks, detailed crash data, speed monitoring data, and traffic volumes were sought for 

rural interstate highways in 17 States for the period 1991 to 2000. Conventional statistical tests (analysis of 

variance, Tukey’s test, and Dunnett’s test) were used to study speed and crash rate changes in the four policy 

groups. A modified empirical Bayes formation was used to evaluate crash frequency changes without presuming a 

constant relationship between crashes and traffic volume. 

No consistent safety effects of DSL as opposed to USL were observed within the scope of the study. The mean 

speed, 85th percentile speed, median speed, and crash rates tended to increase over the 10-year period, regardless of 

whether a DSL or USL limit was employed. When all sites within a State were included in the analysis, temporal 

differences in these variables were often not significant. Further examination suggests that while these data do not 

show a distinction between DSL and USL safety impacts, the relationship between crashes and traffic volume 

cannot be generalized but instead varies by site within a single State. Because application of the modified empirical 

Bayes methodology suggested that crash risk increased for all four policy groups, a mathematical model that 

predicts sharp changes in crash rates based only on ADT does not appear valid at the statewide level.

17. Key Words 18. Distribution Statement 

Differential Speed Limit, Universal Speed 

Limit, Truck Speed Limit, Speed Limit 

No restrictions. This document is available to the 

Public through the National Technical Information Service; 

Springfield, VA 22161 

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

 Unclassified Unclassified 99 

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized (art. 5/94)
ii 

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 th International System of Units. Approp e riate rounding should be made to comply with Section 4 of ASTM E380. 

(Revised March 2003)
iii 

Table of Contents 

INTRODUCTION .......................................................................................................................... 1 

PURPOSE AND SCOPE................................................................................................................ 2 

LITERATURE REVIEW ............................................................................................................... 2 

Impact of DSL on Mean Speed................................................................................................... 2 

Impact of DSL on Speed Variance ............................................................................................. 3 

Impact of DSL on Crashes.......................................................................................................... 3 

Modified Empirical Bayes Methodology ................................................................................... 4 

METHODOLOGY ......................................................................................................................... 5 

Data Synthesis............................................................................................................................. 5 

Rural Interstate Speed Limits: 1991–2000 ..................................................................................... 6 

Analysis of Speed Data............................................................................................................... 7 

Analysis of Crash Data Using Conventional Statistical Approaches ......................................... 8 

Analysis of Crash Data Using the Empirical Bayes Technique ................................................. 9 

Conceptual Overview .......................................................................................................... 10 

Development of the Crash Estimation Model........................................................................... 10 

Critical Assumptions of the Crash Estimation Model .............................................................. 16 

Application of the Crash Estimation Model with the Before Data........................................... 17 

Application of the Crash Estimation Model with the After Data ............................................. 18 

Quantifying the Safety Impact of the Speed Limit Change.................................................. 19 

RESULTS, DISCUSSION, AND LIMITATIONS ...................................................................... 20 

Vehicle Speeds.......................................................................................................................... 20 

Mean Speeds: An Example of How the Data May Be Assessed.......................................... 21 

Graphical Overview of Changes in Speed Variance, 85th Percentile Speed, 

Median Speed, and Noncompliance Rates ...................................................................... 23 

Statistical Results of Changes in Speed Variance, 85th Percentile Speed, Median 

Speed, and Noncompliance Rates ................................................................................... 25 

Comparison of Six Interstate Highway Segments in Idaho................................................. 27 

Discussion of Speed Impacts ............................................................................................... 28 

Crash Rates (Analyzed by Conventional Methods).................................................................. 29 

Crashes (Analyzed by the Modified Empirical Bayes Method) ............................................... 31 

Virginia Crashes (DSL to USL)) ......................................................................................... 32 

Arkansas Crashes (USL to DSL)......................................................................................... 34 

Idaho Crashes (USL to DSL)............................................................................................... 37 

Crashes from the States of Arizona, Missouri, North Carolina, and Washington.............. 37 

Relating Speed and Crash Changes .......................................................................................... 44 

STUDY LIMITATIONS .............................................................................................................. 45 

Caveats About the Use of the Empirical Bayes Method........................................................... 45 

General...................................................................................................................................... 47 

Caveats...................................................................................................................................... 47 

CONCLUSIONS........................................................................................................................... 47 

Safety Impacts of DSL Versus USL ......................................................................................... 48 

Methodological Findings .......................................................................................................... 48 

APPENDIX A. EXAMPLES OF DATA COLLECTION LETTERS AND PROCESSING ...... 51 

APPENDIX B. EXAMPLE OF A CLARIFYING DATA REQUEST LETTER........................ 53 
iv 

APPENDIX C. CONFIRMATION OF THE NEGATIVE BINOMIAL DISTRIBUTION TO 

CRASH DATA ................................................................................................................. 55 

Verification of the Poisson Distribution ................................................................................... 55 

Verification of the Negative Binomial Distribution ................................................................. 56 

APPENDIX D: EFFECT OF CHANGING THE BASE YEAR IN Ci,y ...................................... 59 

Using the First Year as the Base Year ...................................................................................... 59 

Using the Third Year as the Base Year..................................................................................... 61 

APPENDIX E. DETERMINING THE SAMPLE SIZE............................................................... 65 

Test for Significant Differences Using the Sample Size as the Number of Sites..................... 65 

Test for Significant Differences Using the Sample Size as the Number of Vehicles............... 65 

Simulating the Speed Variances ............................................................................................... 66 

Determining the Difference in Mean Speeds Needed to Show Significant Difference 

Based on the Number of Vehicles ........................................................................................ 66 

Discussion................................................................................................................................. 67 

APPENDIX F: THEORETICAL CONSIDERATIONS IN THE COMPUTATION OF 

CONFIDENCE INTERVALS FOR THE 85th PERCENTILE SPEED .......................... 69 

APPENDIX G: EXAMINATION OF THE EFFECTS OF ADT ON TOTAL 

CRASH RATES ............................................................................................................... 71 

Histograms of ADT Versus Total Crash Rate .......................................................................... 71 

Two-Way Analysis of Variance................................................................................................ 72 

APPENDIX H. EXAMPLE APPLICATION OF THE CRASH ESTIMATION MODEL TO 

THE AFTER DATA......................................................................................................... 75 

Estimation of Expected Crash Frequency M1, M2,... My for the Before Period ......................... 75 

Prediction of My+1, My+2,...My+Z for the After Period................................................................. 78 

Evaluation of Safety Effects of Changing the Speed Limit for This Particular Site ................ 79 

Evaluation of Safety Effects of Changing the Speed Limit for All Sites ................................. 81 

ACKNOWLEDGMENTS ............................................................................................................ 85 

REFERENCES ............................................................................................................................. 87 
v 

List of Tables 

Table 1. Accident proportions by speed limit, collision type, and vehicle involvement................ 3 

Table 2. Overview of data availability for rural interstates from the various States. ..................... 6 

Table 3. Available speed data. ........................................................................................................ 6 

Table 4. Available crash data for all sites....................................................................................... 7 

Table 5. Five potential models for total number of crashes for Virginia rural 

interstate highways. ......................................................................................................... 15 

Table 6. Five models for total number of crashes on Arizona rural interstates............................ 16 

Table 7. Before/after mean speed comparisons from the ANOVA test. ...................................... 22 

Table 8. Annual mean speed comparisons.................................................................................... 22 

Table 9. Longitudinal comparison of speed variables within the States....................................... 26 

Table 10. Idaho speed limits. ........................................................................................................ 27 

Table 11. ANOVA results of mean speed and 85th percentile speed in Idaho. ........................... 28 

Table 12. Statistical Tests for Significance in Crash Rates. ......................................................... 30 

Table 13. Virginia data for the before and after periods............................................................... 32 

Table 14. Crash estimation model parameters for Virginia data. ................................................. 33 

Table 15. Total crashes for Virginia. ............................................................................................ 33 

Table 16. Virginia total fatal crashes. ........................................................................................... 34 

Table 17. Crash data for Arkansas................................................................................................ 34 

Table 18. Crash estimation model parameters for Arkansas data................................................. 35 

Table 19. Arkansas rear-end crashes. ........................................................................................... 36 

Table 20. Total crashes for Arkansas............................................................................................ 36 

Table 21. Fatal crashes for Arkansas. ........................................................................................... 36 

Table 22. Crash data for Idaho...................................................................................................... 37 

Table 23. Total crashes for Idaho. ................................................................................................ 37 

Table 24. Impact of speed limit changes according to the empirical Bayes formulation............. 39 

Table 25. Crash increases and confidence intervals according to the empirical 

Bayes formulation. ....................................................................................................... 41 

Table 26. T-Statistics for the empirical Bayes crash estimation models (before data)................. 42 

Table 27. Poisson validation description and results using the total crashes at four test sites. .... 56 

Table 28. Negative binomial validation description and results................................................... 57 

Table 29. Estimation of expected crashes using 1991 data as a base in the Ci,y ratio................... 62 

Table 30. Estimation of expected crashes using 1993 data as a base in the Ci,y ratio................... 63 

Table 31. Summary Idaho data from speed sampling sites. ......................................................... 65 

Table 32. Sample sizes required to achieve significant differences. ............................................ 67 

Table 33. 95 Percent confidence intervals for the 85th percentile speed.*................................... 70 

Table 34. ANOVA variable definitions........................................................................................ 73 

Table 35. ANOVA Arizona results............................................................................................... 73 

Table 36. ANOVA Virginia results. ............................................................................................. 74 

Table 37. Before and after crash data for a single site.................................................................. 75 

Table 38. Estimation results for the before years. ........................................................................ 78 

Table 39. Prediction results for the after years. ............................................................................ 79 

Table 40. Evaluation of the treatment for the example site. ......................................................... 80 
vi 

List Figures 

Figure 1. Chart. Speed limits throughout the 1990s on rural interstate highways.......................... 1 

Figure 2. Chart. Data analysis process flowchart. .......................................................................... 8 

Figure 3. Equation. Crash rate. ....................................................................................................... 8 

Figure 4. Chart. Fundamental steps of the empirical Bayes approach............................................ 9 

Figure 5. Equation. Crash models for Virginia............................................................................. 11 

Figure 6. Equation. Crash models for Washington....................................................................... 11 

Figure 7. Chart. Comparison of crash estimation models for Virginia and Washington State 

based on 1991–1993 data............................................................................................... 11 

Figure 8. Equation. Expected mean value of crashes. .................................................................. 12 

Figure 9. Chart. Plot of goodness of fit for the crash estimation model versus ADT................... 13 

Figure 10. Chart. Plot of goodness of fit for the crash estimation model versus length............... 13 

Figure 11. Equation. Alternative crash estimation model............................................................. 14 

Figure 12. Equation. CEM for before years.................................................................................. 17 

Figure 13. Equation. Expected crash frequency m for period 1. .................................................. 17 

Figure 14. Equation. Variance of expected crash frequency m for period 1. ............................... 18 

Figure 15. Equation. Expected crash frequency m for period y. .................................................. 18 

Figure 16. Equation. Variance of expected crash frequency m for period y. ............................... 18 

Figure 17. Equation. Would-have-been crashes, had there been no speed limit change.............. 19 

Figure 18. Equation. Actual crashes, given that the speed limit did change. ............................... 19 

Figure 19. Equation. The difference between would-have-been and actual crashes.................... 19 

Figure 20. Equation. Variance for δ.............................................................................................. 19 

Figure 21. Equation. Confidence intervals for δ........................................................................... 20 

Figure 22. Equation. Reduction in the expected number of crashes............................................. 20 

Figure 23. Equation. Ratio of actual to would-have-been crashes. .............................................. 20 

Figure 24. Equation. Variance of ratio of actual to would-have-been crashes............................. 20 

Figure 25. Equation. Confidence intervals for θ........................................................................... 20 

Figure 26. Chart. Mean speed for all vehicles. ............................................................................. 21 

Figure 27. Chart. 85th Percentile speeds and median speeds. ...................................................... 24 

Figure 28. Chart. Median speed trends. ........................................................................................ 24 

Figure 29. Chart. Speed variance rates. ........................................................................................ 25 

Figure 30. Chart. Noncompliance rates. ....................................................................................... 25 

Figure 31. Chart. Total crash rates,............................................................................................... 29 

Figure 32: Chart. Total truck-involved crash rates in Virginia interstate highways..................... 31 

Figure 33. Chart. Relationship between the Poisson and negative binomial distributions 

for crash frequencies................................................................................................... 55 

Figure 34. Chart. Comparison of Poisson distribution and actual crash distribution. .................. 56 

Figure 35. Chart. comparison of negative binomial distribution and actual crash 

distribution (probability density function).................................................................. 58 

Figure 36. Equation. Crash frequency for year 1 as base year. .................................................... 59 

Figure 37. Equation. Crash frequency for year 3 as base year. .................................................... 59 

Figure 38. Equation. Expected value of crash count for year 1.................................................... 59 

Figure 39. Equation. Variance of expected value of crash count for year 1................................. 59 

Figure 40. Equation. Estimation of estimated values of crash counts for year 1.......................... 60 
vii 

Figure 41. Equation. Variance of estimation of estimated values of crash counts for year 1....... 60 

Figure 42. Equation. Expected value of crash count for year 3.................................................... 61 

Figure 43. Equation. Expected value of crash count, year 3 as base year. ................................... 62 

Figure 44. Equation. Variance of expected value of crash count, year 3 as base year. ................ 62 

Figure 45. Equation. Statistically significant difference in mean speeds. .................................... 65 

Figure 46. Chart. Histogram based on random numbers. ............................................................. 66 

Figure 47. Equation. Formula to determine confidence intervals associated with mean speed. .. 69 

Figure 48. Equation. Example of formula in figure 47................................................................. 69 

Figure 49. Equation. Confidence interval for 85th percentile speed. ........................................... 69 

Figure 50. Equation. Binomial distribution. ................................................................................. 69 

Figure 51. Chart. Arizona total crash rate versus ADT. ............................................................... 71 

Figure 52. Chart. Virginia total crash rate versus ADT................................................................ 72 

Figure 53. Chart. Virginia total crash rate versus ADT................................................................ 72 

Figure 54. Equation. Crash estimation model............................................................................... 75 

Figure 55. Equation. Mean of the estimate for 1991. ................................................................... 76 

Figure 56. Equation. Mean of the estimate for 1992. ................................................................... 76 

Figure 57. Equation. Mean of the estimate for 1993. ................................................................... 76 

Figure 58. Equation. Calculation for ratio before year y. ............................................................. 76 

Figure 59. Equation. Ratio before year 1991................................................................................ 76 

Figure 60. Equation. Ratio before year 1992................................................................................ 76 

Figure 61. Equation. Ratio before year 1993................................................................................ 76 

Figure 62. Equation. Expected crash counts................................................................................. 76 

Figure 63. Equation. Variance of the expected crash counts for year 1. ...................................... 77 

Figure 64. Equation. Expected crash counts................................................................................. 77 

Figure 65. Equation. Variance of expected crash counts.............................................................. 77 

Figure 66. Equation. Application for 1991. .................................................................................. 77 

Figure 67. Equation. Application for variance 1991..................................................................... 77 

Figure 68. Equation. Application for 1992. .................................................................................. 77 

Figure 69. Equation. Application for variance 1992..................................................................... 77 

Figure 70. Equation. Application for 1993. .................................................................................. 77 

Figure 71. Equation. Application for variance for 1993............................................................... 77 

Figure 72. Equation. Computation of E(m1,1995)........................................................................... 78 

Figure 73. Equation. Computation of E(m1,1996)........................................................................... 78 

Figure 74. Equation. Computation of C1,1995. ............................................................................... 78 

Figure 75. Equation. Computation of C1,1996. ............................................................................... 78 

Figure 76. Expected crash counts, year y. .................................................................................... 78 

Figure 77. Variance of expected crash counts, year y. ................................................................. 79 

Figure 78. Equation. Expected crash counts, year 1995............................................................... 79 

Figure 79. Variance of expected crash counts, year 1995. ........................................................... 79 

Figure 80. Expected crash counts, year 1996. .............................................................................. 79 

Figure 81. Variance of expected crash counts, year 1996. ........................................................... 79 

Figure 82. Equation. Total would-have-been crashes for a particular site. .................................. 80 

Figure 83. Equation. Total actual crashes for a particular site...................................................... 80 

Figure 84. Equation. Safety impact for a particular site. .............................................................. 80 

Figure 85. Equation. Ratio of actual to would-have-been crashes. .............................................. 80 

Figure 86. Chart. Cumulative differences, by year, at the example site....................................... 81 
viii 

Figure 87. Total would-have-been crashes. .................................................................................. 81 

Figure 88. Total actual crashes. .................................................................................................... 81 

Figure 89. Safety impact............................................................................................................... 82 

Figure 90. Variance of the difference between would-have-been crashes and actual crashes. .... 82 

Figure 91. Standard deviation of the difference between would-have-been crashes 

and actual crashes. ...................................................................................................... 82 

Figure 92. Equation. Computation of the index of effectiveness. ................................................ 82 

Figure 93. Equation. Variance of θ............................................................................................... 82 

Figure 94. Equation. Empirical confidence bounds...................................................................... 83 
ix 

ABSTRACT 

To compare the safety effects of a uniform speed limit (USL) for all vehicles as opposed to a 

differential speed limit (DSL) for cars and heavy trucks, detailed crash data, speed monitoring 

data, and traffic volumes were sought for rural interstate highways in 17 States for the period 

1991 to 2000. Data from nine of those States were used such that they could be divided into four 

policy groups based on the type of speed limit employed during the period. These were 

maintenance of a uniform limit only, maintenance of a differential limit only, a change from a 

uniform to a differential limit, and a change from a differential to a uniform limit. Conventional 

statistical tests (analysis of variance, Tukey’s test, and Dunnett’s test) were used to study speed 

and crash rate changes in the four policy groups. A modified empirical Bayes formation was 

used to evaluate crash frequency changes without presuming a constant relationship between 

crashes and traffic volume. 

No consistent safety effects of DSL as opposed to USL were observed within the scope of the 

study. The mean speed, 85th percentile speed, median speed, and crash rates tended to increase 

over the 10-year period, regardless of whether a DSL or USL limit was employed. When all sites 

within a State were included in the analysis, temporal differences in these variables were often 

not significant. Further examination suggests that while these data do not show a distinction 

between DSL and USL safety impacts, the relationship between crashes and traffic volume 

cannot be generalized but instead varies by site within a single State. Because application of the 

modified empirical Bayes methodology suggested that crash risk increased for all four policy 

groups, a mathematical model that predicts sharp changes in crash rates based only on ADT does 

not appear valid at the statewide level. 

Any study that relies on historical data will be subject to the limitations of incomplete data sets, 

and to that extent, additional data collection may shed insights not available from an examination 

of 1990s data alone. Because the investigators believe that accurate mathematical models may 

require extensive calibration data, a future effort may be more productive if resources are 

focused on a small group of States over a period of several years, so that speed variance 

information and crash information may be obtained by individual roadway segment. 

1 

INTRODUCTION 

The Surface Transportation and Uniform Relocation Assistance Act (STURAA), enacted on 

April 2, 1987, permitted individual States to raise rural interstate speed limits from the 

previously mandated national speed limit of 89 kilometers per hour (km/h) (55 miles per hour 

(mi/h)) to 105 km/h (65 mi/h) on rural interstate highways. Of those that changed their speed 

limits, some States raised the limits for passenger cars but not trucks while other States raised the 

limits for both passenger cars and trucks. The former category, with different speed limits for 

cars and trucks, is known as differential speed limits (DSL). The latter category, which mandates 

the same speed limits for cars and trucks, is known as uniform speed limits (USL). The 1995 

repeal of the national maximum speed limit gave States additional flexibility in setting their 

limits, such that by 2002, several States had experimented with both DSL and USL. Figure 1 

shows the types of speed limits throughout the 1990s for rural interstate highways used in this 

study. 

Figure 1. Map. Speed limits throughout the 1990s on rural interstate highways. 

Proponents of DSL note that heavy trucks require a larger stopping distance and stopping time 

than passenger cars. Thus, a lower speed for trucks would reduce the risk of crashes that would 

otherwise be associated with these larger vehicles traveling at high speeds. Proponents of USL 

contend that different limits could increase speed variance, resulting in more conflicts between 

trucks and other types of vehicles, especially rear-end and sideswipe collisions. Advocates of a 
2 

uniform limit also suggest that the higher driver position in a truck provides a greater sight 

distance than would be the case for a passenger car, giving truck drivers more time to stop. 

Studies conducted during the early 1990s to compare the safety impacts of USL and DSL were 

constrained because of the limited data available at that time. Most such studies compared effects 

at different physical sites, such as Interstate 64 in the western portion of Virginia and the 

adjacent section of Interstate 64 in the eastern portion of West Virginia, where, in 1990 the 

former had a uniform limit and the latter had a differential limit. With more than a decade having 

elapsed since the passage of STURAA, however, the Federal Highway Administration (FHWA) 

requested that a longitudinal study be conducted, focusing especially on States that had changed 

their limits from USL to DSL or vice versa. 

PURPOSE AND SCOPE 

The purpose of this study is to compare the safety impacts of DSL and USL on rural interstate 

highways. Thus, the study’s objectives are twofold: 

• To compare the effect of DSL and USL on crashes. 

• To compare the effect of DSL and USL on vehicle speeds. 

The scope of this study is limited to crash and speed data available in the U.S. on rural interstate 

highways between the period 1991and 2000. 

LITERATURE REVIEW 

One of FHWA’s justifications for a longitudinal study was the lack of consistent findings from 

previous research. Previous authors had focused on three main indicators of safety: speed, speed 

variance, and the crashes themselves. Some studies found no difference between DSL and USL, 

and some studies found one or the other to be a better policy choice. Because it was thought that 

a new methodological approach might help resolve these inconsistencies, literature suggesting 

the empirical Bayes approach was also reviewed. 

Impact of DSL on Mean Speed 

In 1990, Freedman and Williams analyzed speed data collected at 54 sites in 11 Northeastern 

States to determine the effect of DSL on mean and 85th percentile speeds.(1) Six States had 

retained a uniform limit of 89 km/h (55 mi/h), three had raised speed limits for all vehicles to a 

uniform value of 105 km/h (65 mi/h), and two States employed a differential limit for cars and 

trucks of 105/89 km/h (65/55 mi/h), respectively. The results showed that for passenger cars, the 

mean speed and 85th percentile speed for the two DSL States were not significantly different 

from the States with a uniform limit of 105 km/h (65 mi/h). Further, the mean and 85th 

percentile truck speeds in DSL states were close to those of the USL States. Similar results were 

obtained when comparing the percentage of vehicles complying with the speed limit. Harkey and 

Mera also found there to be no significant difference between passenger car and truck mean 

speeds when comparing DSL and USL.(2) Garber and Gadiraju, however, did find a significant 

difference between truck mean speeds under DSL and USL, as well as an increase in passenger 

cars’ mean speed when the speed limits for those vehicles were raised.(3)
3 

Impact of DSL on Speed Variance 

Garber and Gadiraju also found that speed variance for all types of vehicles were significantly 

greater at DSL sites than at non-DSL sites.(3) The implication of increased variance is that of 

increased interactions between vehicles and, thus, a potential in some types of crashes. While 

researchers found differences in truck speed variance in 10 of 13 comparisons between a USL 

site and DSL site, Harkey and Mera found no significant differences were between car speed 

variances at the DSL and USL sites.(2) Furthermore, they found no difference between the speed 

distributions for both cars and trucks for the 105/97 km/h (65/60 mi/h) and 105/105 km/h (65/65 

mi/h) speed limits. 

Impact of DSL on Crashes 

Harkey and Mera also looked at crash results from 26 sites in 11 States, where the sites were 

grouped into pairs.(2) Each pair was comprised of a USL site with a speed of 105 km/h (65 mi/h) 

or 89 km/h (55 mi/h) and a DSL site with a speed of 105/89 km/h (65/55 mi/h) or 105/97 km/h 

(65/60 mi/h). The study investigated the percentage of different collision types for the total 

number of crashes for each of four types of speed limits. Three types of collisions were taken 

into consideration: rear end, sideswipe, and all other crashes. Table 1 shows that a higher 

proportion of car-into-truck and truck-into-car crashes occurred in USL States than in DSL 

States, with the exception of rear-end crashes where more car-into-truck collisions happened in 

the DSL group. 

Table 1. Accident proportions by speed limit, collision type, and vehicle involvement. 

Rear-End Sideswipe Other 

Speed Limit CarintoTruck

TruckintoCar CarintoTruckTruckintoCar CarintoTruck TruckintoCar 

USL: 105 km/h (65 mi/h) and 89 km/h 

(55 mi/h) 

10.91 10.78 22.12 21.07 2.57 2.01

DSL: 105/89 km/h (65/55 mi/h) and 

105/97 km/h (65/60 mi/h) 

13.70 6.86 21.52 14.96 2.07 0.99

In contrast, Garber and Gadiraju, who had compared sites in three DSL States (California, 

Michigan, and Virginia) against two USL States (Maryland and West Virginia) found no 

statistically significant differences between crash rates when stratifying by collision type and 

crash severity.(3) Council, Duncan, and Khattack in 1998 found that for rear-end collisions 

between cars and trucks, a high speed differential increases the severity of the crash.(4) A 

simulation study by Garber reported a potential for an increase in crash rates for facilities using 

DSL, especially in the case of high vehicle volumes and truck percentages.(5) Further, a 1991 

study found no evidence indicating that the increase of the speed limit to 105 km/h (65 mi/h) for 

trucks at the test sites resulted in a significant increase in fatal, injury and overall accident 

rates.(6) In that study, comparisons of crash rates in the adjacent States of Virginia (DSL) and 

West Virginia (USL) showed relatively more rear-end crashes in Virginia, suggesting that DSL 

might have a negative impact on safety. 
4 

Lending credence to the use of speed variance as a surrogate for crashes, Garber and Gadiraju 

found that crash rates increased with increasing speed variance for all classes of roads.(7) A 1974 

study by Hall and Dickinson showed that speed differences contributed to crashes, primarily 

rear-end and lane-change collisions.(8) The existence of a posted DSL, however, was not found to 

be related to the occurrence of truck crashes. The study also noted that lower rates of truck 

crashes could be expected with higher speed limits and hence the study recommended an 

increase of truck speed limits to 105 km/h (65 mi/h) for highways carrying a higher truck 

percentage. Finally, an evaluation conducted by the Idaho Department of Transportation found 

that a change from USL to DSL did not increase crashes.(9)

Modified Empirical Bayes Methodology 

Because of the discrepancies in findings in the literature with respect to the safety impacts of 

DSL versus USL, the investigators considered a new conceptual approach that had been refined 

during the past decade. The empirical Bayes method, developed by Ezra Hauer and modified by 

the investigators because of data issues specific to this analysis, was applied because recent 

literature suggested it could delineate between random variation and variation that resulted from 

some policy change (such as DSL to USL) more clearly than is the case with conventional 

methods.(10) Specifically, four advantages of the empirical Bayes approach have been cited in the 

literature: 

• It employs the correct mathematical distribution for crashes. Many conventional statistical 

tests are predicated on the assumption of normality; however, previous studies have indicated 

that the error or residuals (difference between predicted values and actual values) structure of 

the negative binomial distribution is a better description of the variation of crash frequency 

between sites. (See references 10, 11, 12, and 13.) 

• It does not assume that other conditions remain constant. The estimated number of crashes at 

a particular site usually does not remain constant from one year to the next because of 

variation in traffic volumes, traffic flow characteristics, weather conditions, driver attitudes, 

and a host of other factors beyond the control of the researcher. While conventional methods 

strive to control for these factors by either judiciously selecting sites that are common in 

most characteristics except that being studied (such as USL versus DSL) or using a control 

group, the empirical Bayes technique provides for the use of trend data. It has been shown 

that this method can be used to evaluate safety impacts even when yearly data are not 

available. Although yearly trend information is lost, researchers still benefit from the use of 

reference groups to pinpoint which effects are significant.(14) 

• It does not assume a proportional relationship between crashes and average daily traffic 

(ADT). Instead, the number of crashes, or crash frequency, is used to reflect highway safety, 

given that empirical studies have suggested that a percentage change in ADT will not 

necessarily have the same percentage change in crashes, even if all other factors were 

constant. 

• Crash estimation models are used to account for the fact that the observed crash frequency is 

just a point observation from some underlying distribution. In sum, the actual number of 
5 

crashes observed may not be an unbiased estimator of the expected number of crashes. Thus, 

in lieu of this point estimate, a crash estimation model (CEM) is employed to statistically 

predict the best estimator of crash frequency. The generalized linear modeling (GLM) 

technique has been suggested as a way of determining parameters for the crash estimation 

model, although recently the generalized estimating equation (GEE) has been proposed as an 

alternative. (See references 10, 11, 14, 15, 16, and 17.) 

METHODOLOGY 

Four general steps comprised the methods used to compare the safety impacts of DSL and USL. 

First, crash and speed data were synthesized from 17 States that had been recommended by 

FHWA, were recommended by other researchers, or were thought to have changed their speed 

limits at least once during the 1990s from USL to DSL or vice versa. Second, conventional 

statistical approaches, such as the analysis of variance (ANOVA) were used to analyze speed 

monitoring data from these States. Third, comparable approaches were used to evaluate crash 

data from these States. Last, the empirical Bayes procedure was applied to these crash data. 

Data Synthesis 

As shown in appendix A, crash, speed, and volume data were solicited from multiple States by 

phone and e-mail. Raw data formats varied widely. Some data were only available in hardcopy 

format and were manually entered. Moreover, the detail of electronic data sets also varied 

widely. For example, some States provided detailed speed data in 8.0 km/h (5 mi/h) bins, 

whereas other States only provided a mean speed. During this data synthesis, data records that 

appeared likely to contain errors were removed. As an illustration, consider the 24-hour ADTs 

available from one State. The available data were the number of axles per vehicle, the vehicle 

speed, and the distance between the axles. There were a few records that showed both a speed of 

less than 8.0 km/h (5 mi/h), and a total axle-distance less than 1.2 meters (m) (4 feet); thus, these 

were removed from the data. In some cases, State data were not used because of concerns about 

the data quality. For example, after data had been partially synthesized for one State, 

practitioners from that State told investigators that the location system for the speed limits was so 

imprecise that the data were not reliable; thus, that State was omitted from the analysis. In other 

cases, local knowledge suggested that a facility should be excluded. For example, an experienced 

Virginia Transportation Research Council (VTRC) technician noted that although Interstate 66 

had been designated as a “rural” section, the high volumes on that segment caused it to function 

as an urban section. In a few cases, the data were of good quality but the time required for 

reformatting the data for the purposes of this analysis was prohibitive. However, repeated 

telephone calls were helpful for clarifying the meaning of the data elements within the individual 

data sets. 

Table 2 shows that the States that provided data used in this study may be divided into four 

policy groups based on their speed limits during the 1990s: 

1. States that maintained a uniform limit for cars and trucks. 

2. States that maintained a differential limit for cars and trucks. 

3. States that changed from a uniform to a differential limit. 

4. States that changed from a differential to a uniform limit. 
6 

Table 2. Overview of data availability for rural interstates from the various States. 

Rural Interstate Speed Limits: 1991–2000 Crash Data Speed Data 

Policy Group 1: Maintained USL 

Arizonaa 121 km/h (75 mi/h) Y N 

Iowa 105 km/h (65 mi/h) N Y 

North Carolina 105 km/h (65 mi/h) before 1996 

113 km/h (70 mi/h) after 1996 

Y N 

Policy Group 2: Maintained DSL (passenger cars/trucks) 

Illinois 113/105 km/h (70/65 mi/h) N Y 

Indiana 105/97 km/h (65/60 mi/h) N Y 

Washington b 105/97 km/h (65/60 mi/h) Y N 

Policy Group 3: Changed from USL to DSL (passenger cars/trucks) 

Arkansas From: 105 km/h (65 mi/h) 

To: 113/105 km/h (70/65 mi/h) 1996 

Y N 

Idaho From: 105 km/h (65 mi/h) 

To: 121 km/h (75 mi/h) 1996 

To: 121/105 km/h (75/65 mi/h) 1998 

Y Y 

Policy Group 4: Changed from DSL to USL (passenger cars/trucks 

Virginia From: 105/89 km/h (65/55 mi/h) 

To: 105 km/h (65 mi/h), 1994 

Y Y 

a

 Prior to December 1995, Arizona’s uniform limit was 105 km/h (65 mi/h). The State raised the limit by route to 

121 km/h (75 mi/h) between December 1995 and the summer of 1996. 

b

 The Washington State limits shown refer to nine sections on I-90, which comprise the entire Washington data set 

used for this study. During inclement weather, speed limits on those sections drop to 40.2 km/h (25 mi/h). 

As shown in table 3, the number of sites with speed data could vary from year to year. 

Table 3. Available speed data. 

Policy

Group State Speed Data Availability Number of Sites Years of Data 

 Mean 

Speed 

Speed 

Variance 

85% 

Speed

Median

Speed

Noncompliance 

1 Iowa x N/A N/A N/A N/A 1 to 27 1991–2000

2 Illinois x x x x x 4 1993, 1994, 1997–

1999

 Indiana x N/A x x N/A 4, 3 1991, 2000

3 Idaho x N/A x N/A N/A 24 to 38 1991–1999

4 Virginia x x x x x 3 to 7 1991,1993,1995,2000,

2001

Note: “x” indicates that the correspondent data are available. 
7 

Table 4 shows the categories of crash data obtained from the different States used in this study. 

Table 4. Available crash data for all sites. 

Crash Data Availability 

Crash Rate for all 

Vehicles 

Crash Rates for Trucks Policy

Group State 

Total Fatal RearEnd Total Fatal Rear-End

Number 

of Sites 

Years of 

Data 

North x x x x x x 26 1991–1995 

Carolina 1 1997–2000 

Arizona x x x x x x 278 1991–2000 

Idaho x x x x x x 29 1991–2000 

3 Arkansas x x x x N/A N/A 10 1991–1995 

 1997–1999 

x x x x x N/A 267 1991–1993 4 Virginia 

 1995–1999 

Note: “x” indicates that the correspondent data are available. 

Analysis of Speed Data 

To the extent possible with available data, annual changes in five speed variables (mean speed, 

speed variance, 85th percentile speed, median speed, and noncompliance rates) were compared 

within individual States. The Analysis of Variance (ANOVA) was used to determine if the 

change between the USL and DSL periods was significant. For those States that never changed 

their policy, the data were categorized into two virtual groups, 1990–1995 and 1996–2000, to 

determine whether significant changes occurred over time even without a policy shift between 

USL and DSL. 

ANOVA was also used to look at speed changes on a yearly basis within every State. When a 

significant difference at the 5 percent confidence level was detected for these yearly changes, 

then, as shown in figure 2, three additional statistical tests were employed. Levene’s test served 

as a screening procedure to determine if the groups had equal variances. If so, then Tukey’s test 

was used to determine whether the differences were significant, while Dunnett’s test was used 

for samples with unequal variances, since the former assumes equal variances and the latter does 

not.(18,19,20) The Levene, Tukey, and Dunnett tests were performed with the SPSS software 

following ANOVA test; hence, they are referred to as the “post hoc analysis” in figure 2. 
8 

Figure 2. Chart. Data analysis process flowchart. 

Analysis of Crash Data Using Conventional Statistical Approaches 

The same procedure employed for evaluating the speed data was employed for evaluating the 

crash data, using the ANOVA, Levene, Tukey, and Dunnett tests. Six types of crash rates were 

studied: total, fatal only, rear-end only, total truck-involved, fatal truck involved, and truckinvolved rear-end. The crash rate was computed as shown in figure 3, where the annual crash 

frequency was simply the annual number of crashes; ADT was the average daily traffic.(21)

( )( )

( )( )( ) ADT Section Length

Annual Crash Frequency Crash rate

365

100,000,000 =

Figure 3. Equation. Crash rate. 
9 

Analysis of Crash Data Using the Empirical Bayes Technique 

A fundamental reason for applying the empirical Bayes technique is that a traditional before/after 

test, which simply compares the number of crashes on a facility before and after some type of 

treatment, may not necessarily have been the result of the treatment. Figure 3 illustrates how the 

use of a reference population can assist in making this determination, provided the reference 

group is judiciously selected.(10) To understand the impact of assumptions, it is helpful to trace 

how the available crash data can be mapped to figure 4. 

Figure 4. Chart. Fundamental steps of the empirical Bayes approach. 

 REFERENCE POPULATION

TREATED ENTITIES

Before After

Multivariate model giving E (m) and VAR (m) for years 1, 

2, 3..., Y, Y+1, Y+2,..., Y+Z 

Data for a representative sample of entities for 

years 1, 2, 3..., Y, Y+1, Y+2,..., Y+Z

Crash counts (K) and covariate values for years 1, 2,

 3..., Y, Y+1, Y+2,..., Y+Z

Estimates of m for years 1, 

2, 3..., Y 

Predictions of m for years Y+1, 

Y+2, ..., Y+Z 
10 

Conceptual Overview 

Consider a State such as Arkansas that changed from a uniform limit to a differential limit in the 

mid 1990s. If the shift to DSL is viewed as a treatment, then figure 4 shows that years 1 to Y

refer to the before-treatment portion and years Y+1 to Y+Z refer to the after-treatment portion, 

such that the before period has a duration of Y years and the after period has a duration of Z

years. For each roadway segment within the State, there is sequence of crash counts K for year 1 

to year Y; thus, the number of crashes at a particular site i during a year y may be designated as 

Ki,y. To evaluate how this treatment affected safety, there must be a prediction as to what the 

expected crash frequency m would have been in the after period had there had been no such 

treatment and then compare this would-have-been value to the actual number of crashes that 

occurred during the after period. 

To determine this would-have-been value, a single multivariate crash estimation model (CEM)

was developed to estimate the mean E(mi,y) and the variance VAR(mi,y) of the expected crash 

frequency m for each year y and at each site i during the before and after periods. For the treated 

entity with covariate values available, the multivariate model was applied to calculate the E(mi,y) 

and VAR(mi,y) for the before and after years. The expected crash frequency m was then 

calculated from the E(mi,y) from the before year values. Finally, these frequencies (m) of the 

before years then serve as a basis to obtain the predictions of the frequencies (m) of the after 

years of treated entities, with the use of the multivariate model. These steps are illustrated in the 

subsections that follow. 

Development of the Crash Estimation Model 

The CEM predicts the mean of the expected frequency of crashes E(mi,y) and is especially 

relevant for predicting would-have-been crashes. The ideal CEM will account for all sources of 

variation other than the treatment being studied, which in this case are the DSL and USL 

policies. Thus, the ideal CEM would account for effects such as operational changes (e.g., 

volume growth, enforcement modifications, and the installation of safety service patrols), 

geometric changes (e.g., work zones, median barriers, or new interchange construction), and 

driver changes pertaining to behavior, licensing, and vehicle maintenance. Unfortunately, the 

data for these other sources of variation either were not available, or at best were very limited. 

The research team, therefore, had to develop the crash estimation models based on the available 

data. For example, in Virginia, the before data were from 1991–1994 and the after data were 

from 1995–1999. Unfortunately, there is no perfect technique for selecting the data from which 

to build the crash estimation model for the after period. Given this, there are two options for 

acquiring these data: 

• Use data from the same time period but from a different State. In Virginia’s case, the 

researchers could select crash data for the period from 1995 to 1999 from a different State 

with comparable geometric and volume characteristics. The advantage is that if there is some 

temporal trend that applies nationwide, the reference group will capture that trend. The 

disadvantage is that the chosen State would need characteristics comparable to those of 

Virginia. 
11 

• Use data from the before period but for the same State. The advantages and disadvantages 

are the reverse of the previous option. In this case, the researchers are guaranteed of having 

the right geometric characteristics, but may miss temporal trends that are present in the after 

years, but not present in the before years. 

The investigators chose the latter course of action, deeming the disparity between States as being 

greater than the disparity between time periods and using as an example a comparison of two 

crash models from the before periods for Virginia and Washington State from the same time 

period of 1991–1993. Both States maintained a differential speed limit at the time. 

E(m) = 0.022(Length)0.622 (ADT)0.548 

Figure 5. Equation. Crash models for Virginia. 

E(m) = 0.531(Length)0.440 (ADT)0.340 

Figure 6. Equation. Crash models for Washington. 

Visual inspection of the plots of the crash estimation models, shown in figure 7 below for a fixed 

length of 8 km (5 mi) and a range of ADTs, confirms that a different number of crashes at a 

given site with a given volume could be predicted. For example, if an 8-km (5-mi) site registered 

an ADT of 10,000, then the CEM for Virginia would have predicted approximately 9 crashes, 

whereas the CEM for Washington would have predicted almost 25 crashes. 

Length = 5 Miles

0

5

10

15

20

25

30

35

40

0 5000 10000 15000 20000 25000 30000 35000 40000

ADT

Crashes Estimated by CEM

Virginia

Washington

5 mi = 8 km 

Figure 7. Chart. Comparison of crash estimation models for Virginia 

and Washington State based on 1991–1993 data. 
12 

Generally, the variation between States in terms of the crash estimation models was found to be 

relatively large, even among States with similar speed limit policies. Thus, for this specific 

application, it appeared that capture of nontreatment sources of variation was best accomplished 

by using CEMs developed within the same State, which is a significant departure from the 

empirical Bayes formulation as given in the literature. For that reason, the nomenclature 

modified empirical Bayes will be used for the remainder of this report to acknowledge that the 

selection of comparison sites herein deviates from that original methodology. 

The crash estimation model took the form shown in figure 8, where the expected mean value of 

crashes for a roadway segment during a given year was the function of the length and ADT. As 

recommended in the literature, the maximum likelihood technique was used to estimate the 

parameters since the crash distribution was thought to follow a negative binomial distribution.(17)

Note parameters β1 and β2 do not imply a proportionate effect of ADT and length unless equal to 

unity, since others have suggested that such an effect cannot be assumed.(22,23) 

( ) ( ) ( ) α β1 β 2 E m = Length ADT

Figure 8. Equation. Expected mean value of crashes. 

Estimation of the parameters was done with the GENSTAT software package (release 4.2.1). 

The procedure also gives the variance, VAR(mi,y) of expected crash frequency for each section, 

which enables researchers to determine an aggregation parameter, shown as k, which describes 

the underlying crash distribution as k approaches infinity, when the distribution moves from 

negative binomial to Poisson. Theoretical explanations of k and techniques for estimating k are 

given in the literature. (See references 12, 13, 17, and 24.) Software for estimating k used for this 

effort was developed by Persaud and Lord.(17) 

It should be noted that the equation for this effort shown in figure 8 was chosen after 

investigators considered a variety of formulations and determined that this formulation could be 

applied most consistently for each State’s data set. To ascertain whether the model was 

appropriate, the method of cumulative residuals may be used.(17) (The cumulative residual is the 

difference between actual crash counts and model estimated crash counts. If the cumulative 

residuals oscillate around zero within the range of the two plots of two standard deviations, a 

good quality of fit is reflected.)(17,25) Figures 9 and 10, for example, show the cumulative residual 

plots with respect to section length and annual ADT (AADT) for Virginia. Although there 

existed several sites where the cumulative residuals exceed the range of two standard deviations, 

the overall figures show a good quality of fit. 
13 

-80

-60

-40

-20

0

20

40

60

80

100

120

2925.32154183.33337052.53167666.66679666.666710333.33310987.42111166.66712166.66712368.38312583.68112833.33313012.579

13335.55613574.539

14333.33314478.58915299.62115973.62116693.654

18787.634

31386.19

57833.333

AADT

cumulative residuals

residuals

+ 2 standard deviations

- 2 standard deviations

Note: top line is +2 standard deviations, middle line is residuals, bottom line is −2 deviations. 

Figure 9. Chart. Plot of goodness of fit for the crash estimation model versus ADT. 

-80

-60

-40

-20

0

20

40

60

80

100

120

1.052.493.073.513.914.314.674.975.175.415.705.946.166.446.767.187.708.53

length

cumulative residuals

residuals

+ 2 standard deviations

- 2 standard deviations

Note: top line is +2 standard deviations, middle line is residuals, bottom line is −2 deviations. 

Figure 10. Chart. Plot of goodness of fit for the crash estimation model versus length.

An alternative crash estimation model that was explored in the course of this work entailed the 

use of yearly trend analysis and is given in figure 10, where ay indicates the yearly trend. This 

expression was ultimately not used because reliable estimates could not be obtained for the after 

years for the States that changed from USL to DSL or from DSL to USL. That is, for a State such 

as Virginia, which changed from DSL to USL in 1994, calibration of this model would indeed 
14 

yield a coefficients for the before period (a1991, a1992, a1993, and a1994); however, data were not 

available to estimate the appropriate coefficients for the after years, e.g., a1997, a1998, and a1999. In 

short, the formulation shown in figure 8 proved advantageous over the formulation below 

because it required less data. 

( ) ( ) ( ) 1 2 E β β m = αy Length ADT

Figure 11. Equation. Alternative crash estimation model. 

In terms of specific software, general estimating equation (GEE) and generalized linear modeling 

(GLM) can accommodate both expressions, but the investigators used GEE for all States because 

it considers the possibility of temporal correlation between the years of the analysis. 

In sum, several considerations governed the crash estimation model formulation used in this 

study. First, using total crashes for Virginia as a case study, four different formulations were 

considered (the equation in figure 8 using GEE and figure 8 using GLM, figure 11 using GEE 

and using GLM), and it was found with the Virginia case that the equation in figure 7 using the 

GEE approach was most suitable. As shown in tables 5 and 6, the parameters for these 

formulations were quite similar, but GEE was chosen over GLM because it accounts for 

temporal correlation; and the equation in figure 8 was chosen over the one in figure 11 because it 

meshed better with the data available for the study. Second, the method of cumulative residuals 

showed that, for total crashes in Virginia, both variables were indeed appropriate. The Genstat 

software used to implement the GEE method also showed that the t-statistics for length and ADT 

exceeded the critical values; thus both variables are significant. Third, the equation in figure 8 

using the GEE software was applied for the remaining States and crash types. 

Tables 5 and 6 illustrate the parameters for the equations in figures 8 and 11 with GLM and GEE 

respectively. A fifth formulation based on 3-year averages was also considered in the course of 

this study as shown in the rightmost columns of tables 5 and 6; however, this model ultimately 

was not used because ADT variation by year is not included therein. 
15 

Table 5. Five potential models for total number of crashes 

for Virginia rural interstate highways. 

Number of Sites: 267 

Years of Data: 1991, 1992, 1993 

Figure 10 Figure 10 Figure 7 Figure 7 Not Shown in 

the Report 

GEE with trend GLM with trend GEE without 

trend 

GLM without 

trend 

GLM without 

trend 

Yearly data Yearly data Yearly data Yearly data 3-year averaged 

data 

Parameters 

E(m)= αy (Length)β

1

(ADT)β

2

E(m)= αy (Length)β

1

(ADT)β

2

E(m)= α

(Length)β

1

(ADT)β

2

E(m)= α

(Length)β

1

(ADT)β

2

ave_E(m)= α

(Length)β

1

(ave_ADT)β

2

 Estimate Standard 

error 

Estimate Standard

error 

EstimateStandard

error 

EstimateStandard 

error 

Estimate Standard

error 

LN(α1) −3.774 0.5271 −3.774 0.41 – – – – – – 

LN(α2) −3.848 0.5277 −3.848 0.415 – – – – – – 

LN(α3) −3.732 0.528 −3.732 0.416 – – – – – – 

LN(α) −3.78467 0.5276 −3.78467 0.413667 −3.828 0.5274 −3.829 0.41 −3.791 0.592 

β1 0.631 0.1167 0.6309 0.0526 0.632 0.1163 0.6323 0.0528 0.6117 0.0737

β2 0.545 0.0588 0.5447 0.042 0.549 0.0585 0.5492 0.0418 0.5481 0.0607

K 5.62 5.62 5.61 5.61 5.56 
16 

Table 6. Five models for total number of crashes on Arizona rural interstates. 

Number of Sites: 556 

Years of Data: 1991–2000 

Parameters Figure 11 Figure 11 Figure 8 Figure 8 Not Shown in 

the Report 

 GEE with trend GLM with trend

GEE without 

trend 

GLM without 

trend 

GLM without 

trend 

yearly data 

(5560pts) 

yearly data 

(5560pts) 

yearly data 

(5560pts) 

yearly data 

(5560pts) 

10-year data 

(556pts) 

E(m)= αy

(Length)β

1

(ADT)β

2

E(m)= αy

(Length)β

1

(ADT)β

2

E(m)= α

(Length)β

1

(ADT)β

2

E(m)= α

(Length)β

1

(ADT)β

2

E(m)= α

(Length)β

1

(ADT)β

2

Estimate Standard 

error 

Estimate Standard 

error 

Estimate Standard 

error 

Estimate Standard 

error 

Estimate Standard 

error 

LN(α1) −1.963 0.6397 −1.963 0.238 – – – – – – 

LN(α2) −1.933 0.6373 −1.934 0.238 – – – – – – 

LN(α3) −1.908 0.6381 −1.908 0.238 – – – – – – 

LN(α4) −1.819 0.6381 −1.819 0.24 – – – – – – 

LN(α5) −1.853 0.6434 −1.853 0.241 – – – – – – 

LN(α6) −1.808 0.6511 −1.809 0.243 – – – – – – 

LN(α7) −1.687 0.6506 −1.687 0.243 – – – – – – 

LN(α8) −1.621 0.6493 −1.621 0.244 – – – – – – 

LN(α9) −1.649 0.6523 −1.649 0.246 – – – – – – 

LN(α10) −1.657 0.6572 −1.657 0.249 – – – – – – 

LN(α) −1.7898 0.64571 −1.79 0.242 −2.328 0.6196 −2.33 0.233 −1.68 0.831

β1 1.114 0.0609 1.1144 0.0242 1.133 0.0606 1.1325 0.0242 1.151 0.0826

β2 0.24 0.0668 0.2403 0.024 0.297 0.0639 0.2974 0.0234 0.2242 0.0842

k 0.73 0.73 0.73 0.73 0.73 

The equation shaded in tables 5 and 6 was ultimately used in this study. The standard error are 

lower than those shown for GLM because GEE accounts for the possibility of correlation 

between years. 

Critical Assumptions of the Crash Estimation Model 

It is important to recognize two critical characteristics of the crash frequency distribution that 

affect the form of the crash estimation model. 

• For any particular site i, such as an 8-km (5-mi) segment of rural interstate highway, the 

annual number of crashes for year y will follow the Poisson rather than the normal 

distribution. Thus, in the equations that follow, the crash frequency variable Ki,y obeys the 

Poisson distribution. 
17 

• For any particular year y, such as 1995, the crash frequency variable Ki,y for sites i follows the 

negative binomial distribution rather than the Poisson distribution. 

Although recent studies have indicated that the negative binomial distribution is a better 

descriptor for crash frequencies between sites, the investigators confirmed this by comparing the 

actual crash frequencies for Arizona, Idaho, North Carolina, and Virginia with theoretical 

frequencies using the Poisson and negative binomial distributions. Appendix C shows that the 

negative binomial distribution is a valid descriptor of these data sets. 

Application of the Crash Estimation Model with the Before Data 

After CEM was developed, the expected crash frequency m1, m2,... my for the treated segments for 

the after years were determined. To accomplish this, the following steps were undertaken in 

sequence: 

• CEM was applied to the before years data to obtain E(mi,y), as shown in figure 8. (Recall that 

E(mi,y) is the mean of the estimated crash frequency of site i during year y.) 

• The ratio Ci,y was then computed for each of these before years from y = 1, 2, ....Y. Thus Ci,y

is the ratio of the current E(mi,y) to the first year E(mi,1). 

( )

( )

, ,1

,

i

i y

E m

E m C i y =

Figure 12. Equation. CEM for before years. 

Note that in the application of figure 12, the literature points out it is not essential that the first 

year of the before period be the denominator.(10) Appendix D confirms this view by illustrating 

that the use of another year rather than the first year for Virginia would not affect the results. 

• Recall that k was determined from the calibration process for the crash estimation model and 

essentially reflects the type of crash distribution reflected in the model. (On the other hand, 

recall that Ki,y represents the actual crashes observed at site i in year y.) The expected crash 

frequency mi,y and its variance VAR(mi,y) for years of y = 1, 2, ....Y are then calculated from 

the next four equations.(10) 

∑

∑

=

=

+

+

= Y

y

i y E mi

k

Y

y

i y

C

k K

mi

1

, ) ,1 (

1

,

,1

 Figure 13. Equation. Expected crash frequency m for period 1. 
18 

∑ ∑

∑

= =

=

+ +

+

= = Y

y

i y E mi

k

i

Y

y

i y E mi

k

Y

y

i y

C

m

C

k K

VAR mi

1

, 1 ) , (

,1

2

1

, 1 ) , (

1

,

( )

,1 ( )

Figure 14. Equation. Variance of expected crash frequency m for period 1. 

mi,y= Ci,y mi,1

Figure 15. Equation. Expected crash frequency m for period y. 

VAR(mi,y) = (Ci,y)

2

VAR(mi,1) 

Figure 16. Equation. Variance of expected crash frequency m for period y. 

Application of the Crash Estimation Model with the After Data 

With the expected crash frequency m1, m2,... my estimated for before period for each treated entity 

from figure 15, investigators can compute the “would have been crash frequency” mY+1,

mY+2,...mY+Z for the after period as if there had been no such treatment using the following 

sequential steps. 

• The crash estimation model from figure 8 is applied to the after years data in order to obtain 

E(mi,y) for from y = Y + 1 to y = Y + Z. 

• The ratios Ci,y are be computed using the equation in figure 12 for the years from y = Y + 1 to 

y = Y + Z. 

• The values for mi,y and VAR(mi,y) are computed using equations 6 and 7 for the years from 

y = Y + 1 to y = Y + Z. These mi,y are the crashes that would have occurred had no treatment 

been made. 

Thus, this process is very similar to application of the crash estimation model for the before 

years, with the exception of figures 13 and 14, used to compute mi,1 and its variance VAR(mi,1) 

do not need to be repeated since the quantities mi,1 and VAR(mi,1) have already been determined. 

(That is, mi,1 and VAR(mi,1) refer to year 1 as designated in the subscript mi,1.) 
19 

Quantifying the Safety Impact of the Speed Limit Change 

The effect of the treatment, which in this case is a change in the speed limit for Arkansas from 

uniform to differential, is quantified by comparing the would-have-been crashes (shown above as 

mi,y for y ranging from Y + 1 to Y + Z) to the actual crashes in the after period (shown above as 

Ki,y with y ranging from Y + 1 to Y + Z).* The would-have-been crashes at each site i during the 

after period are denoted as πi while the actual crashes at each site during the after period are 

denoted as λi. Figures 17 and 18 are used to sum the crashes from the individual sites as π and λ. 

π = ∑π

i

i

 Figure 17. Equation. Would-have-been crashes, had there been no speed limit change. 

= ∑

i

λ λi

 

Figure 18. Equation. Actual crashes, given that the speed limit did change. 

Then, the literature suggests two alternative formulations for assessing the safety impact, based 

on a comparison of π and λ.

10

• Reduction in the expected number of crashes. Figure 19 computes δ as the difference 

between would-have-been and actual crashes, such that a positive value of δ indicates that 

the treatment had a desirable effect of reducing the number of crashes. Figures 20 and 21 

show the computation of the variance and the resultant confidence intervals for δ. 

δ = π - λ

Figure 19. Equation. The difference between would-have-been and actual crashes. 

 

Var(δ) = Var(π) + Var(λ) = ΣVar(πi) + ΣVar(λi) 

Figure 20. Equation. Variance for δ. 

 

*

 The investigators acknowledge that, in theory, the Ki,y are themselves random observations that 

could be replaced by a crash estimation model that smoothes the annual data. The investigators 

felt that using CEM would add additional confusion when interpreting the results; thus the actual 

Ki,y were used as indicators of the actual crashes in the after period. 
20 

empirical confidence bounds for δ = δ ± 2[Var(δ)]0.5

Figure 21. Equation. Confidence intervals for δ. 

• Ratio of actual to would-have-been crashes. Figure 22 computes this ratio, also known as the 

index of effectiveness (θ), such that a value of θ less than 1.0 indicates that the speed limit 

change improved safety. The unbiased estimator, however, is not the equation in figure 22 

but rather the one in figure 23 as shown below.(10) Figures 24 and 25 show the variance and 

confidence bounds, respectively. 

θ = λ/π(“actual crashes” divided by “would have been” crashes) 

Figure 22. Equation. Reduction in the expected number of crashes. 

θ = (λ/π)/{1+Var(π)/π

2

} 

Figure 23. Equation. Ratio of actual to would-have-been crashes. 

Var(θ)=θ2{[var(λ)/λ2]+[ var(π)/π

2

]}/[1+ var(π)/π

2

]

2

Figure 24. Equation. Variance of ratio of actual to would-have-been crashes. 

θ is θ ± [2Var(θ)]0.5

Figure 25. Equation. Confidence intervals for θ. 

The confidence bounds, shown in figures 21 and 25, are used to determine whether the values for 

δ and θ show a statistically significant safety impact. If the confidence bounds for δ and θ

contain 0 and 1 respectively, then the safety impact computed by figures 20 and 24 are not 

significant; thus, it cannot be said that the treatment had a measurable effect. 

RESULTS, DISCUSSION, AND LIMITATIONS 

The results of the analysis are presented across three key areas: vehicle speeds as collected from 

speed monitoring data and evaluated using conventional statistical approaches, crash rates that 

were evaluated using conventional statistical approaches, and crashes as evaluated with the 

empirical Bayes technique. 

Vehicle Speeds 

The five types of speed data (mean speeds, speed variance, 85th percentile speeds, median 

speeds, and noncompliance rates) were analyzed for all States where such speed monitoring data 
21 

were readily available. Furthermore, because Idaho provided a large number of speed monitoring 

sites and was a State that had changed its type of speed limit (from USL to DSL), speed data 

from that State were studied in detail. 

Mean Speeds: An Example of How the Data May Be Assessed 

Figure 26 illustrates the trends in mean speeds among five States from which speed data were 

analyzed. Two observations that arise from examination of figure 26 are, that with the exception 

of Virginia, speeds appear to be increasing over time (whether or not the differences are 

practically or statistically significant) and, unfortunately, data cannot always be obtained for all 

time periods. 

Mean Speed Trend

58

60

62

64

66

68

70

72

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Year

Mean Speed (mi/h)

Iowa

Illinois

Indiana

Idaho

Virginia

 1 mi/h = 1.6 km/h

Figure 26. Chart. Mean speed for all vehicles. 

Tables 7 and 8 illustrate the two types of analyses that were conducted for all speed and crash 

data: a before-after analysis to determine whether the speeds from the before period were 

significantly different from the after period at the 95 percent confidence level, and a year-by-year 

analysis, to determine whether individual years showed a significant difference. For this report, p

values of 0.05 or lower were considered significant and are designated by an asterisk (*). 
22 

Table 7. Before/after mean speed comparisons from the ANOVA test. 

Policy Group State Before-After p

Group 1 (uniform limit) Iowa 0.000*(+) 

Group 2 (differential limit) Illinois 0.626(+) 

Indiana 0.537(+) 

Group 3 (uniform to 

differential) 

Idaho a 0.000* (+) 

(uniform to uniform) 

0.790(−) 

(uniform to differential) 

Group 4 (differential to 

uniform) 

Virginia 0.318(−) 

*Designates a significant difference at the 0.05 level.

a

 Throughout this report, when before/after comparisons were done for the State of Idaho, the first statistic reflects a 

before group with a uniform speed limit of 105 km/h (65 mi/h) for all vehicles and an after group with a uniform 

limit of 121 km/h (75 mi/h) for all vehicles. The second statistic reflects a before group with the same uniform limit 

of 121 km/h (75 mi/h) and an after group with a differential speed limit of 121/105 km/h (75/65 mi/h) for cars and 

trucks, respectively. 

Table 7 shows two sets of results for Idaho since its speed limits were changed twice: first, a 

raising of its uniform limits and second, a lowering of truck speeds only. In table 8, the Levene 

test showed the variances to be significantly different; Dunnett’s test was used to compare annual 

mean speeds because it does not require comparison groups to have similar variances. 

Table 8. Annual mean speed comparisons. 

State p values for tests of significant 

differences in the variances 

(based on Levene’s test) 

Years where the means are 

significantly different at the 0.05 level 

(based on Dunnett’s Test) 

Iowa .000* 1995<2000, 1996<2000, 1997<2000 

Idaho .000* 1991<1996, 1992<1996, 1993<1996, 

1994<1996, 

1995<1996, 1996<1997, 1996<1998, 

1996<1999 

Virginia .004* 1995>2001 

*Denotes significance at the .05 confidence level. 

Examination of the statistics in table 7 shows that the mean speed of Iowa, a State that 

maintained a uniform limit, increased by 3.6 km/h (2.2 mi/h), which was significant. The mean 

speed change for Idaho, which maintained a uniform speed limit but increased that limit by 16 

km/h (10mi/h) for all vehicles, was also significant with an increase of 9.2 km/h (5.7 mi/h). In 

Illinois and Indiana, which maintained DSL, their mean speeds increased by 1.2 km/h (0.8 mi/h) 

and 3 km/h (1.9 mi/h), respectively, which were not significant. Also, Idaho’s second change, 

which was a shift from USL to DSL, resulted in a slight decrease in the mean speed of 0.18 km/h 

(0.11 mi/h), but this decrease was not significant. The mean speed in Virginia, which changed 

from DSL to USL by increasing the speed limit for trucks by 16 km/h (10 mi/h), decreased by 

1.15 km/h (0.71 mi/h), which was insignificant. In examining the results of the year-to-year 
23 

analysis shown in table 8, the mean speed in Iowa for 1999, after the change from USL to DSL, 

was significantly higher than that for 1996 before the change, and that for 2001 was significantly 

lower than for 1995. 

Appendix E discusses, in detail, whether the sample size used in testing for statistical 

significance should be the number of speed monitoring sites or the number of vehicles. As 

explained in that appendix, the investigators chose to use the number of sites for speed 

monitoring, although this decision had a pragmatic rather than a theoretical justification. 

Graphical Overview of Changes in Speed Variance, 85th Percentile Speed, Median Speed, and 

Noncompliance Rates 

Figures 26–29 give a pictorial understanding of how speeds changed from 1991 to 2000 for the 

States where data were available. These figures facilitated the observations of three general 

phenomena. First, there appear to be some correlations among the speed variables where data are 

available. For example, the mean and median speeds for Virginia are similar to one another, and 

the same can be said for Indiana. Likewise, the mean speeds and 85th percentile speeds for Idaho 

show similar trends. Although this is not surprising, it should not be taken for granted, since 

differences can sometimes be observed in the 85th percentile speed even though no differences 

are observed in mean speeds.(26) Second, there is not a clear difference in behavior between 

uniform and differential States. For example, Iowa (always a uniform limit) and Illinois (always 

a differential limit) show similar peaks in 1999, followed by a decreasing trend with respect to 

speed variance. Similarly, most States tended to show an overall increase in speeds. Virginia (the 

one State that went from differential to uniform) did show a decrease in mean speeds. However, 

table 6 shows that this difference was not significant. (Furthermore, table 8 shows that decreases 

in Virginia for the 85th percentile and median speeds were also not significant). Last, figure 29 

suggests, but does not prove, that there may be a correlation between speed variance and 

noncompliance rates. The small amount of data available do not justify firm conclusions, but do 

suggest a relationship worthy of further study. 
24 

85th Percentile Speed Trend

64

66

68

70

72

74

76

78

80

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Year

85th Percentile Speed 

(mi/h)

Iowa

Illinois

Indiana

Idaho

Virginia

 1 mi/h = 1.6 km/h 

Figure 27. Chart. 85th Percentile speeds and median speeds. 

Median Speed Trend

58

60

62

64

66

68

70

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

Year

Median Speed (mi/h)

Iowa

Illinois

Indiana

Idaho ('96,'98)

Virginia ('94)

 1 mi/h = 1.6 km/h 

Figure 28. Chart. Median speed trends. 
25 

Speed Variance Trend

0

20

40

60

80

100

1991

1992199319941995199619971998199920002001

Year

Speed Variance (mi/h)

 1 mi/h = 1.6 km/h 

Figure 29. Chart. Speed variance rates. 

Noncompliance Trend

0

20

40

60

80

100

199119921993

1994

1995

1996

19971998

1999

20002001

Year

Noncompliance Speed 

(mi/h)

Iowa

Illinois

Indiana

Idaho ('96,'98)

Virginia ('94)

 1 mi/h = 1.6 km/h 

Figure 30. Chart. Noncompliance rates. 

Statistical Results of Changes in Speed Variance, 85th Percentile Speed, Median Speed, and 

Noncompliance Rates 

Table 9 highlights the results of the statistical analysis and suggests that while increasing speeds 

were observed in most cases, these were not usually significant. (Idaho and Iowa are exceptions, 

which definitely do show a significant increase in speeds; it should be noted, however, that Iowa 

had a small number of sites.) The fact that no State showed a significant decrease in speeds and 

that States from all four policy groups showed either a significant or insignificant increase lends 

weight to the idea that changes in speed as a result of a differential or uniform speed limit were 

not supported by this study. 
26 

Table 9. Longitudinal comparison of speed variables within the States. 

Policy Group State Variable p value

Effect from 

Before to After 

Period 

Significant at the 

5% confidence level

Mean Speed 0.000 Increase Y 

Speed Variance 0.878 Increase N 

85th-Percentile 

Speed 

 

Median Speed 

1: USL Iowa 

Noncompliance 

Mean Speed 0.626 Increase N 

Speed Variance 0.250 Increase N 

85th-Percentile 

Speed 

0.171 Increase N 

Median Speed 0.535 Increase N 

Illinois 

Noncompliance 0.350 Increase N 

Mean Speed 0.537 Increase N 

Speed Variance 

85th-Percentile 

Speed 

0.338 Increase N 

Median Speed 0.608 Increase N 

2: DSL 

Indiana 

Noncompliance 

Mean Speed 0.000, 

0.790 

Increase, 

Decrease 

Y, N 

Speed Variance 

85th-Percentile 

Speed 

0.000, 

0.563 

Increase, 

Decrease 

Y, N 

Median Speed 

3: USL to DSL Idahoa

Noncompliance 

Mean Speed 0.318 Decrease N 

Speed Variance 0.136 Increaseb N 

85th-Percentile 

Speed 

0.356 Decrease N 

Median Speed 0.209 Decrease N 

4: DSL to USL Virginia 

Noncompliance 0.000 Decrease Y a

As stated previously, when before/after comparisons were made for the State of Idaho, the first statistic reflects a 

before group with the uniform speed limit of 105 km/h (65 mi/h) for all vehicles and an after group with a uniform 

limit of 121 km/h (75 mi/h) for all vehicles. The second statistic reflects a before group with the same uniform limit 

of 121 km/h (75 mi/h) and an after group with a differential speed limit of 121/105 km/h (75/65 mi/h) for cars and 

trucks, respectively. 

b

As shown in figure 28, Virginia speed variance calculations are based on only 3 years of data. 

Noncompliance rate should be considered as an example of the influence of other factors. In 

Virginia, these data were only available for 3 years: 1991, 2000, and 2001, with the percentage 

of drivers exceeding the speed limit being 77 percent, 55.2 percent, and 50 percent, respectively. 
27 

Differences between the before DSL period (1991) and the after DSL period (2000 and 2001) 

were significant, whereas differences between 2000 and 2001 were not significant. Statistically, 

a strong case can be made that this State’s results suggest that elimination of DSL was correlated 

with a lowering of the noncompliance rate in this one particular State. Given the sparse data 

available from Virginia, which are more than that available from other States, however, this 

inference is tenuous, given the small data points shown in figure 29. 

Comparison of Six Interstate Highway Segments in Idaho 

It could be argued that the five different States shown in table 9 may have masked impacts of 

speed limit changes because of differences among them. Fortunately, one State—Idaho—had six 

sites where speed data were available; some of these sites had DSL and some had USL. Thus, the 

changes in the 85th percentile speed and mean speeds were compared at these sites. As shown in 

table 10, a shift from a uniform limit to a differential limit occurred at the three rural sites meant 

that researchers could look closely at how speeds at those sites changed after 1998. 

Table 10. Idaho speed limits. 

Site (Interstate 

and Milepost) 

Speed Limit Change 

(mi/h) 

Speed Limit Change 

(km/h) 

Year of 

Change 

Type of 

Area 

I-84 MP 14.9 65/65–75/75–75/65 104/104–120/120–121/105 1996, 1998 Rural 

I-84 MP 19.1 65/65–75/75–75/65 104/104–120/120–121/105 1996, 1998 Rural 

I-90 MP 35.59 65/65–75/75–75/65 104/104–120/120–121/105 1996, 1998 Rural 

I-84 MP 51 55/55–65/65 105/105–120/120 1996 Urban 

I-90 MP 61.6 65/65–70/70 105/105–112/112 1996 Urban 

I-90 MP 86.2 65/65–70/70 105/105–112/112 1996 Urban 

Statistical tests of these data show that, while significant differences were observed in the Idaho 

speeds when comparing yearly data, none of the differences were significant when comparing 

DSL to USL, as shaded in tables 10 and 11. The finding is that, although some factors clearly 

caused Idaho interstate mean speeds and 85th percentile speeds to change, a decrease in the truck 

speed limit (changing that limit from USL to DSL) for the three rural segments clearly did not 

affect these speeds. For example, for the first site shown in table 11 (Interstate 84 at milepost 

14.9), it is clear that the first speed limit change, which raised the USL by 16.1 km/h (10 mi/h), 

significantly increased the mean and 85th percentile speeds (p = 0.000). Yet the second speed 

limit change for that site, which is shown in table 10, a change to DSL, did not significantly 

change the mean or 85th percentile speeds as shaded in table 11. Shaded results indicate that 

there was no significant change in the mean or 85th percentile speeds. 
28 

Table 11. ANOVA results of mean speed and 85th percentile speed in Idaho. 

Interstate Number of 

Data Points Variable Before-After ANOVA 

Mean Speed 0.000(+)* 

I-84 MP 0.099(−) 

14.9 3–14 85th Percentile Speed 0.000(+)* 

0.084(−) 

Mean Speed 0.000(+)* 

I-84 MP 0.710(−) 

19.1 8–12 85th Percentile Speed 0.000(+)* 

0.241(−) 

Mean Speed 0.000(+)* 

I-90 MP 0.935(−) 

35.59 8–12 85th Percentile Speed 0.000(+)* 

0.937(−) 

Mean Speed 0.000(+)* I-84 MP 51 6–12 85th Percentile Speed 0.000(+)* 

I-90 MP Mean Speed 0.000(+)* 

61.6 3–12 85th Percentile Speed 0.000(+)* 

I-90 MP Mean Speed 0.000(+)* 

86.2 6–12 85th Percentile Speed 0.000(+)* 

*The asterisk indicates that there was a significant increase in the mean and 85th percentile speeds. 

Given that this is a regulatory change and assuming the statistical test is accurate, three 

implications are possible: the lowering of the speed limit was ignored by trucks; trucks were a 

relatively small percentage of the ADT such that the behavior of trucks did not influence speeds 

in an observable way; or increases in car speeds offset decreases in truck speeds. (Table 11 could 

be viewed as evidence that a shift to DSL can decrease speeds, since all changes in speed limits 

lead to statistically significant increases except the change to DSL. However, the only way to 

make this determination would be to evaluate changes at non-DSL sites during 1998.)

Discussion of Speed Impacts 

Several inferences become apparent in examining these speed data: 

• Significant increases in mean speed for Idaho and Iowa were observed over the study period. 

The other States tended to show insignificant increases. This trend suggests that the mean 

speed tended to increase over the 1990s regardless of whether the speed limit was uniform or 

differential. The 85th percentile speeds and median speeds showed a comparable upward 

trend, although fewer data were available for those two variables. 

• Idaho’s first speed limit change, from a uniform limit of 105 km/h (65 mi/h) to a uniform 

limit of 121 km/h (75 mi/h), resulted in a significant increase in both the mean speed and the 

85th percentile speed. 
29 

• Idaho’s second speed limit change, from a uniform limit of 121 km/h (75 mi/h) to a 

differential limit of 121/105 km/h (75/65 mi/h) did not significantly change either the mean 

speed or the 85th percentile speed, although these did decrease slightly. 

• In Illinois, where no speed limit changes were implemented, the noncompliance rate 

increased steadily. Given the increasing mean speed of Illinois, this could be due to drivers 

desiring to travel at a higher speed. 

Crash Rates (Analyzed by Conventional Methods) 

Table 4 showed that crash rate data could be obtained from five States (Arizona, Arkansas, 

Idaho, North Carolina, and Virginia). Six types of crash rates (total crash rate, fatal crash rate, 

rear-end crash rate, total truck-involved crash rate, truck-involved fatal crash rate, and truckinvolved rear-end crash rate) were evaluated. As was done with the speed data, both a beforeafter comparison and a year-pair comparison were performed. (Again, for those States that never 

changed their policy, the data were categorized into two virtual groups, 1990–1995 and 1996–

2000.) 

Figure 31 graphically compares the total crash rate in the five States for all sites, while table 12 

indicates which changes were significant. Of the two States that maintained USL, one (North 

Carolina) showed a significant increase in the total crash rate, whereas only Arizona showed an 

insignificant increase. None of the other States, all of which had either changed from DSL to 

USL or from USL to DSL, showed a significant change in the total crash rate. 

Total Crash Rates

0

20

40

60

80

100

120

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000

Year

Total Crash Rates (Crashes 

Per 100 Million VMT)

AZ

NC

ID

AR

VA

 1 mi/h = 1.6 km/h 

Figure 31. Chart. Total crash rates, 

Note that speed limits changed in Arkansas (1996), Idaho (1996, 1998), and Virginia (1994) 

Table 12 shows the before versus after period change in the six types of crash rates of the five 

States, with the p values from the ANOVA test given in parentheses. In examining table 12, 
30 

there were 27 cases where both an ADT and a crash rate were available. There was a significant 

difference between the before and after period in 3 of those 27 cases. 

Table 12. Statistical Tests for Significance in Crash Rates. 

Policy State Type of Crash Rate Difference Significance (p) 

Total + N (0.583) 

Fatal + N (0.140) 

Rear-end + N (0.052) 

Total truck-involved + N (0.949) 

Truck-involved fatal + N (0.134) 

Arizona 

Truck-involved rear-end + N (0.406) 

Total + Y (0.007) 

Fatal + N (0.100) 

Rear-end + Y (0.035) 

Total truck-involved + N (0.504) 

Truck-involved fatal − N (0.525) 

Group 1: 

Maintained 

Uniform 

North 

Carolina**

Truck-involved rear-end + N (0.366) 

Total − N (0.935) 

Fatal + N (0.495) 

Rear-end + N (0.258) 

Total truck-involved + N (0.250) 

Truck-involved fatal 

Arkansas 

Truck-involved rear-end 

Total −, + N, N (0.539,0.153) 

Fatal −, + N, N (0.336,0.192) 

Rear-end −, + N, N (0.539, 0.327) 

Total truck-involved −, + N, N (0.473,0.139) 

Truck-involved fatal* −, 0 N, N (0.656,1.000) 

Group 3: 

Changed from 

Uniform to 

Differential 

Idaho 

Truck-involved rear-end −, + N, N (0.820,0.370) 

Total + N (0.425) 

Fatal − N (0.270) 

Rear-end + N (0.119) 

Total truck-involved + Y (0.000) 

Truck-involved fatal + N (0.665) 

Group 4: 

Changed from 

Differential to 

Uniform 

Virginia 

Truck-involved rear-end *

Note: The number of truck-involved fatal crashes was 0 in Idaho, which is why “1.000” is shown in that cell. **North Carolina maintained a uniform limit but also raised this limit for both passenger cars and trucks. 

Table 12 also shows that, as was the case with the speed analysis, there is no consistent trend in 

crash rates matching the change in speed limits. For example, Virginia and Arkansas, two States 

that were diametrically opposed in terms of their policies (Arkansas changed to DSL and 

Virginia changed from DSL), both showed statistically insignificant increases in rear-end crashes 

with almost identical p-values. 
31 

On the basis of the limited data available, there is potential evidence that the change to USL (as 

in Virginia) resulted in an increase in the number of rear-end crashes since the number of these 

crashes increased in Virginia, albeit insignificantly. This viewpoint, however, is tempered by two 

observations: first, such an increase was insignificant; and second, there was no corresponding 

decrease in the States that shifted from USL to DSL. In fact, the States that made such a shift 

(Idaho and Arkansas) saw an increase (albeit insignificant) in the number of rear-end crashes. 

Virginia observed a reduction in the fatal crash rates after the change from DSL to USL, and 

Idaho observed an increase in the fatal crash rate after changing from USL to DSL, although 

both of these changes were insignificant. 

The most striking feature of table 12 is that, despite all four combinations of speed limit policies 

(maintaining USL, maintaining DSL, changing from USL to DSL, or changing from DSL to 

USL), not a single State saw a significant decrease in any of the various crash rate categories. 

Crash rates either did not change significantly or significantly increase. 

Because crash data were available for five interstate segments in Virginia, these crash data were 

scrutinized to see if patterns could be gleaned from an examination of the sites grouped by 

interstate. A snapshot of these data is shown in figure 32, which illustrates that no single segment 

dominated the statewide average for total truck-involved crash rate. Figure 31 does, however, 

show a slight increase over time for the crash rates on the interstate segments. 

Total Truck Involved Crash Rates in Virginia

0

5

10

15

20

1991 1992 1993 1994 1995 1996 1997 1998 1999

Year

Crash Rate

State

I-64

I-77

I-81

I-85

I-95

Figure 32: Chart. Total truck-involved crash rates in Virginia interstate highways.

Crashes (Analyzed by the Modified Empirical Bayes Method) 

Because the empirical Bayes approach is best suited for a before/after scenario where one State 

undergoes some type of treatment—such as a change in the speed limits—the results are 

presented first for the policy group 3 and 4 States, where States changed from either a 

differential limit to a uniform limit or from a uniform to a differential limit. 
32 

Virginia Crashes (DSL to USL)) 

For the 266 sites that comprised the Virginia study data, the length of the sites ranged from 

1.7 km to 22.9 km (1.05 mi to 14.25 mi) in length, and substantial variation in ADT was 

observed, as shown in table 13. The minimum, maximum, and total crashes per segment are 

shown for each crash type and reveal substantial variation, as well. 

Table 13. Virginia data for the before and after periods. 

Year ADT Total 

Crash 

Fatal 

Crash 

RearEnd 

Crash 

Total 

Crash 

with 

Truck 

Fatal 

Crash 

with 

Truck 

 Min-Max-Average Min-maxtotal 

Minmaxtotal Minmaxtotal Minmaxtotal 

Minmax-total

1991 3,067-55,050-

12,719 

0-55-

2,888 0-2-59 0-23-554 

0-11-

528 0-1-10 

1992 2,634-145,000-

14,968 

0-43-

2,839 0-2-51 0-22-524 

0-21-

592 0-2-17 

1993 3,075-62,558-

14,638 

1-53-

3,322 0-3-58 0-35-580 

0-17-

648 0-1-20 

 

1995 3,422-63,204-

15,500 

0-82-

3,608 0-3-77 0-50-727 

0-17-

772 0-1-17 

1996 3,669-65,860-

16,135 

1-73-

3,964 0-3-57 0-47-732 

0-17-

867 0-2-17 

1997 4,216-74,644-

16,940 

0-58-

3,735 0-4-69 0-34-713 

0-18-

846 0-3-19 

1999 3,600-61,132-

17,693 

1-76-

4,070 0-2-53 0-56-798 

0-17-

948 0-2-26 

Calibration of the parameters for the crash estimation model of the form shown in figure 8 as 

() ( ) ( ) α β1 β 2 E m = Length ADT are shown in table 14 for the Virginia data. The fact that β2 is less than 

1.0 for all crash types means that the model presumes that ADT does not have an equal 

proportional effect on crashes. Instead, a certain percentage increase in ADT will result in a 

small percentage increase in the number of crashes. In short, an increase in ADT should yield a 

lower crash rate according to the model. Similarly, the β1 values below 1.0 mean that an increase 

in section length, according to the model, will increase crashes by a smaller percentage. (For all 

States, the β1 exponent usually was also almost always less than 1.0, meaning that as the section 

length increases, the number of crashes forecasted by the crash estimation model will correspond 

to a lower crash rate.) 
33 

Table 14. Crash estimation model parameters for Virginia data. 

Parameter Total Fatal Rear-End Truck Total Truck 

Fatal 

k 5.9 5.58 1.71 3.99 15.0 

α 0.02243 0.00119 0.000325 0.000316 0.000017

β1 0.622 0.842 0.536 0.819 1.226 

β2 0.548 0.398 0.829 0.788 0.639 

The application of equations in figures 17–20 is reflected in table 15. The application of figure 

18 results in the actual crash data for each year being shown in the “λ” column with the 

cumulative values of λ shown in the “Cumu λ” column. Similarly, the application of the 

equation in figure 17 shows the predicted would-have-been crash data for each year during the 

after period listed in the fourth column (“π”), with the cumulative values of these would-havebeen crashes shown in the fifth column (“Cumu π”). The variance of π was calculated for each 

year during the after period and listed in the sixth column (“VAR(π)”), with the cumulative 

values of VAR(π) listed in the next column (“Cumu VAR(π)”). The equation in figure 19 means 

that the evaluation was conducted using the difference between the would-have-been after 

crashes and the actual after crashes, shown as “Excess δ” and tabulated by year in the “Cumu δ” 

column. The evaluation was also investigated using figure 22—by obtaining the ratio of the 

actual after crashes to the would-have-been after crashes for each year (“θ”) and the total values 

of (“Cumu θ”). Variances for both the difference δ and the ratio θ are also calculated and listed 

to the right of table 15 as an application of figures 20 and 25. 

Table 15. Total crashes for Virginia. 

YEAR λ Cumu 

λ π Cumu π VAR(π) Cumu VAR(π) 

Excess 

δ

Cumu 

δ

Ratio 

θ

Cumu 

θ Var(δ) Var(θ) Cumu Var(δ) 

Cumu 

Var(θ) 

1995 3,608 3,608 3,223 3,223 986 986 -385 -385 1.119 1.119 4,594 0.000466 4,594 0.000466

1996 3,964 7,572 3,296 6,519 1,031 2,017 -668 -1,053 1.202 1.161 4,995 0.000502 9,589 0.000242

1997 3,735 11,307 3,383 9,903 1,088 3,105 -352 -1,404 1.104 1.142 4,823 0.000442 14,412 0.000156

1999 4,070 15,377 3,463 13,366 1,142 4,247 -607 -2,011 1.175 1.150 5,212 0.000471 19,624 0.000117

Ave 3,844 3,341 4,247 -503 1.150 

Table 15 shows that the ratio θ for total crashes in Virginia was larger than 1.0. Furthermore, 

when the empirical confidence bounds are selected in accordance with the equation in figure 25, 

it is clear that the upper and lower bounds for θ are very close to 1.15 and certainly do not 

include 1.0 within that range. Thus, according to the empirical Bayes technique, because the ratio 

of the actual after crashes (λ) to the would-have-been after crashes (π) is greater than 1.0, then 

the treatment (a change from a differential limit to a uniform limit) resulted in an increase in the 

number of crashes. In fact, for all Virginia crash types, θ is greater than 1.0. If the analysis is 

restricted to Virginia alone and used only the empirical Bayes method, then the interpretation 

would be that the change to a uniform speed limit increased the number of crashes. However, as 

is explained in the following sections, data from other States do not support this interpretation. 
34 

Table 16 shows a comparable analysis except that it is restricted to fatal crashes in Virginia. 

Statistically, application of the equation in figure 25 shows that the cumulative value of θ is 1.06 

for total fatal crashes. Close examination of table 16 shows inconsistent performance from year 

to year. Although θ is greater than 1.0 for 1995 and 1997, it is less than 1.0 for 1996 and 1999. 

Table 16. Virginia total fatal crashes. 

YEAR λ Cumu 

λ π Cumu π VAR(π) Cumu VAR(π)

Excess

δ

Cumu 

δ

Ratio 

θ

Cumu 

θ Var(δ) Var(θ) Cumu Var(δ) 

Cumu 

Var(θ) 

1995 77 77 58.65 58.65 2.28 2.28 -18.36 -18.36 1.31 1.31 79.28 0.02 79.28 0.02347 

1996 57 134 59.57 118.22 2.36 4.64 2.57 -15.79 0.96 1.13 59.36 0.02 138.64 0.01000 

1997 69 203 60.82 179.04 2.46 7.10 -8.18 -23.97 1.13 1.13 71.46 0.02 210.10 0.00661 

1999 53 256 61.82 240.85 2.53 9.62 8.82 -15.15 0.86 1.06 55.53 0.01 265.62 0.00460 

Ave 64 60.21 9.62 -3.79 1.06 

Arkansas Crashes (USL to DSL) 

Arkansas is of special interest because it is the reverse of Virginia in that the State changed from 

USL to DSL. The Arkansas data shown in table 17 are from 10 interstate sections in Arkansas, 

each 16 km (10 mi) in length. 

Table 17. Crash data for Arkansas. 

ADT Total Crash Fatal 

Crash 

RearEnd 

Crash 

Total 

Crash 

with 

Year Truck 

Min-max-average Min-maxtotal 

Minmaxtotal Minmaxtotal 

Min-maxtotal 

1991 18,000-26,220-

21,406 14-36-275 0-2-5 3-12-74 5-14-107 

1992 18,968-22,860-

21,956 15-65-326 0-2-6 1-25-85 5-23-118 

1993 16,250-28,500-

23,313 25-54-380 0-5-11 4-23-88 9-21-153 

1994 15,780-31,000-

24,600 25-53-374 0-1-4 5-24-113 8-24-143 

1995 15,000-28,504-

24,133 15-57-387 0-2-5 4-23-99 5-32-150 

 
35 

Table 17. Crash data for Arkansas—continued. 

ADT Total Crash Fatal 

Crash 

RearEnd 

Crash 

Total 

Crash 

with 

Year Truck 

Min-max-average Min-maxtotal 

Minmaxtotal 

Minmaxtotal 

Min-maxtotal 

1997 26,288-33,746-

29,711 26-61-414 0-2-6 7-22-113 8-30-194 

1998 26,000-39,339-

30,182 31-59-459 0-5-14 6-21-119 14-28-197 

1999 27,000-35,312-

29,675 26-74-476 0-3-10 3-44-173 11-32-195 

As shown in table 18, the β1 parameter was assumed to be 1.0 since all sections were of the 

same length and the β2 parameter was found to be less than 1.0 for all crash types except that of 

rear-end crashes. The 1.774 coefficient from table 18 means that, according to the model, a 

certain increase in ADT would increase crashes by a larger proportion. In short, at a given site, 

increasing the ADT will increase the crash rate. 

Table 18. Crash estimation model parameters for Arkansas data. 

Parameter Total Fatal Rear-End Truck Total 

k 19.03 25.0 14.86 18.2 

α 0.00267 0.0598 0.000000016 0.0239 

β1 1.0 1.0 1.0 1.0 

β2 0.714 0.00355 1.774 0.401 

Except for rear-end crashes, all crash types for Arkansas showed that θ was significantly 

greater than 1.0, meaning that the shift (from USL to DSL) increased the number of crashes. 

Rear-end crashes were the only crash type that did not follow this trend because such crash 

types showed θ less than 1.0 in table 19; moreover, as shown in table 18, only rear-end crashes 

had a β2 ADT coefficient that was greater than 1.0. Table 20 shows the application of the 

empirical Bayes approach for the total number of crashes on Arkansas rural interstate highways. 

Another important finding, which supports the statistical analysis, is that although both Virginia 

and Arizona showed an increase in the fatal crashes after the changes in the speed policies, the 

crashes in Virginia increased by only 6 percent. (See table 21 that shows Arkansas had a 60-

percent increase when the State changed from DSL to USL. There is some variation in the 

Arkansas results, however, with θ being greater than 1.0 for 2 of the 3 years.) 
36 

Table 19. Arkansas rear-end crashes. 

YEAR λ Cumu 

λ π Cumu π VAR(π) CumuVAR(π)

Excess 

δ

Cumu

δ

Ratio 

θ

Cumu 

θ Var(δ) Var(θ) Cumu Var(δ) 

Cumu 

Var(θ) 

1997 113 113 143.31 143 34.34 34.34 30.31 30.31 0.787 0.787 147.34 0.00650 147 0.00650 

1998 119 232 148.01 291 37.48 71.82 29.01 59.33 0.803 0.796 156.48 0.00649 304 0.00326 

1999 173 405 141.66 433 33.52 105.34 -31.34 27.98 1.219 0.935 206.52 0.01104 510 0.00265 

Ave 135 144.33 105.34 9.33 0.936 

Table 20. Total crashes for Arkansas. 

YEAR λ Cumu 

λ π Cumu 

π VAR(π) Cumu 

VAR(π)

Excess

δ

Cumu 

δ

Ratio 

θ

Cumu

θ Var(δ) Var(θ) Cumu 

Var(δ) 

Cumu 

Var(θ) 

1997 414 414 419.24 419 91.34 91.34 5.24 5.24 0.987 0.987 505.34 0.00286 505 0.00286

1998 459 873 425.14 844 94.16 185.49 -33.86 -28.62 1.079 1.034 553.16 0.00314 1058 0.00150

1999 476 1,349 417.70 1262 90.64 276.13 -58.30 -86.92 1.139 1.069 566.64 0.00340 1625 0.00104

Ave 450 420.70 276.13 -28.97 1.068 

Finally, the total number of fatal crashes for Arkansas was only 30 during the after period. 

Certainly, this small sample size contributes to the width of the corresponding confidence 

interval shown in table 25. Yet, the exceptionally low value of calibration parameter β2 shown in 

table 24 may be a result of the fact that, for the specific case of Arkansas fatal crashes, the low 

calibration parameter of β2 for Arkansas was not significant, as shown in table 26. Overall, the 

effect of that low value is to make the effect of increasing ADT almost negligible in terms of 

increasing crash risk. Arguably, this is a reasonable impact for fatal truck crashes, and the 

extraordinarily low value of that parameter relative to that of the other crash types makes one 

hesitant to draw conclusions from that set of truck-involved fatal crashes for Arkansas. (For 

Arkansas only, all β1 values were set to 1.0 because lengths for all Arkansas sites were identical.) 

Table 21. Fatal crashes for Arkansas. 

YEAR λ Cumu 

λ π Cumu 

π VAR(π) Cumu 

VAR(π)

Excess

δ

Cumu 

δ

Ratio 

θ

Cumu

θ Var(δ) Var(θ) Cumu 

Var(δ) 

Cumu 

Var(θ) 

1997 6 6 6.204 6.204 0.137 0.137 0.204 0.204 0.964 0.964 6.137 0.157 6.137 0.157 

1998 14 20 6.205 12.409 0.137 0.274 -7.795 -7.591 2.248 1.609 14.137 0.376 20.274 0.134 

1999 10 30 6.204 18.613 0.137 0.411 -3.796 -11.387 1.606 1.700 10.137 0.265 30.411 0.089 

Ave 10 6.204 0.411 -3.7957 1.606 
37 

Idaho Crashes (USL to DSL) 

Similar to Arkansas, Idaho changed its speed limit from uniform to differential. Table 22 shows 

that 1996 was chosen as the before period and 1999–2000 were chosen as the after period, such 

that 32 sections from Interstates 15, 84, 86, and 90 comprised the study group. Section lengths 

ranged from 0.16 km to 139 km (0.1 mi to 86.1 mi). 

Table 22. Crash data for Idaho. 

Year ADT Total 

Crash 

Rear-End 

Crash 

Total Crash 

with Truck 

Rear-End 

with Truck 

 Min-max-average Min-maxtotal 

Min-maxtotal Min-maxtotal Min-max-total 

1997 2,800-36,782-11,006 0-15-152 0-10-33 0-2-8 0-1-2 

 

1999 3,050-40,522-11,881 0-44-237 0-42-70 0-2-21 0-2-6 

2000 3,100- 41,686-12,131 0-23-184 0-18-55 0-4-21 0-2-5 

With β1 and β2 parameters of 0.81 and 0.75 respectively for total crashes, Idaho data suggest that 

the total number of crashes was approximately 29 percent higher than would have occurred, had 

the uniform limit been retained. The variation of θ, shown in the far right column of table 23, 

suggests that these results are statistically significant. In fact, θ values were greater than 1.0 and 

significant for all types of Idaho crashes. There were cases where the variance of θ was not 

always so low. For example, for Idaho truck rear-end crashes, the variance of θ was computed as 

approximately 0.5. In that instance, however, θ had been found to be about 2.35, meaning that 

the impact of a speed limit change on the number of Idaho truck rear-end crashes was still 

significant according to the empirical Bayes method. 

Table 23. Total crashes for Idaho. 

YEAR λ Cumu 

λ π Cumu π VAR(π) Cumu 

VAR(π)

Excess 

δ

Cumu 

δ

Ratio 

θ

Cumu 

θ Var(δ) Var(θ) Cumu 

Var(δ)

Cumu 

Var(θ) 

1999 237 237 161.6 161.6 93.76 93.76 -75.41 -75.41 1.461 1.461 330.8 0.0166 330.8 0.0166 

2000 184 421 162.9 324.5 95.55 189.31 -21.06 -96.47 1.125 1.295 279.6 0.0114 610.3 0.00697 

Ave 210.5 162.3 189.3 -48.23 1.293 

Crashes from the States of Arizona, Missouri, North Carolina, and Washington 

Table 24 shows the values of θ that were obtained when evaluating the effect of maintaining the 

same speed limit in four other States. For comparison purposes, the values from Arkansas, Idaho, 

and Virginia are shown as well. In most cases, table 24 shows that θ was greater than 1.0. On 

examining these θ values for individual States, it seems that, although the ratio for total crashes 
38 

in Virginia (changed from DSL to USL) is higher than that for Arkansas (changed from USL to 

DSL), it is lower than that for Idaho, which, like Arkansas, also changed from USL to DSL. An 

important factor in the case of Virginia is that the total fatal crash ratio is 1.06, whereas that for 

Arkansas was 1.61. All ratios for Idaho were much higher than those for Virginia. This suggests 

that, although there was an overall trend for an increase in crashes, the percentage of increases, 

particularly for fatal crashes, tended to be higher in States that changed from USL to DSL. 

On the other hand, the data from table 24 are not consistent. For total crashes, θ remained 

approximately 1.0 for the one State that maintained DSL (Washington), whereas it was greater 

than 1.0 for States that maintained USL, Arizona and North Carolina. At first glance, the results 

might be interpreted to mean that maintaining DSL caused no change in crashes while 

maintaining USL caused an increase in the number of crashes. Examination of the fatal crashes 

shows inconsistency in those States that maintained USLs where fatal crash data were available; 

in these cases, θ was greater than 1.0 for Arizona, but less than 1.0 for North Carolina. 

Table 25 presents confidence intervals for the expected percentage crash increase for each type 

of crash and for each State according to the empirical Bayes method. For example, consider the 

category of “total crash with truck involved.” Table 24 shows θ values of 1.31 and 1.25 for 

Arkansas and Virginia, respectively, suggesting that there was a 31 percent increase in Arkansas 

and a 25 percent increase in Virginia as a result of the changes in the speed limit policies. 

However, these percentage increases are not perfect, given the variability that can occur in 

crashes. Thus, table 25 is used to suggest that the confidence interval for Arkansas is between 

18.9 percent and 42.8 percent, and between 20.0 percent and 29.8 percent for Virginia. Clearly, 

larger sample sizes tended to lead to smaller confidence intervals. 

Generally, the standard interpretation of confidence intervals is to declare the change 

insignificant if the confidence interval includes zero. Using the Virginia example, it can be 

inferred that according to this application of the empirical Bayes methodology, the change from 

DSL to USL in Virginia did not affect truck-involved fatal crashes (since that confidence interval 

runs from –12.5 percent to +38.4 percent and thus includes zero), but that it did cause total 

crashes to significantly increase (since the corresponding confidence interval is 12.9 percent to 

17.2 percent and thus does not include zero.) 

Interestingly, confidence intervals associated with the use of the empirical Bayes method in table 

25 showed either increases in crashes or no change in crashes for all States and for all categories 

of crashes regardless of the speed policy change, with only one exception—fatal crashes in North 

Carolina, a State which maintained USL. 
39 

Table 24. Impact of speed limit changes according to 

the empirical Bayes formulation. 

Policy 

Group State Crash type Ratio θ β2

Inference of Modified 

Empirical Bayes Method 

Only* 

Total crash 1.15 0.548 

Total crash with truck 

involved 

1.25 0.788 

Rear-end crash 1.16 0.829 

Fatal crash 1.06 0.398 

Fatal crash with truck 

involved 

1.13 0.639 

Group 4: 

Changed 

from DSL to 

USL 

Virginia

Average 1.15 

According to Virginia data, 

the actual number of 

crashes during the after 

period was higher than 

would have been expected 

without the change to USL.

Total crash 1.07 0.714 

Total crash with truck 

involved 

1.31 0.401 

Rear-end crash 0.93 1.774 

Fatal crash 1.61 0.00355

Arkansas 

Average 1.23 

According to Arkansas 

data, the actual number of 

crashes during the after 

period was higher than 

would have been expected 

without the change to DSL 

except for rear-end crashes.

Total crash 1.29 0.745 

Total crash with truck 

involved 

2.46 0.717 

Rear-end crash 1.62 1.717 

Rear-end crash with 

truck involved 

2.36 1.698 

Group 3: 

Changed 

from USL to 

DSL 

Idaho 

Average 1.93 

According to Idaho data, 

the actual number of 

crashes during the after 

period was higher than 

would have been expected 

without the change to DSL.
40 

Table 24. Impact of speed limit changes according to 

the empirical Bayes formulation—continued. 

Policy 

Group State Crash type Ratio θ β2

Inference of Modified 

Empirical Bayes Method 

Only* 

Total crash 1.26 0.127 

Total crash with truck 

involved 

1.16 0.208 

Rear-end crash 1.2 0.757 

Rear-end crash with 

truck involved 

1.07 0.537 

Fatal crash 1.33 0.120 

Fatal crash with truck 

involved 

1.63 0.180 

Arizona 

Average 1.28 

According to Arizona data, 

the actual number of 

crashes during the after 

period was higher than 

would have been expected 

even though USL was 

maintained. 

Total crash 1.26 0.827 

Total crash with truck 

involved 

0.91 1.704 

Rear-end crash 1.002 1.619 

Rear-end crash with 

truck involved 

0.97 1.834 

Fatal crash 0.74 2.878 

Group 1: 

Maintained 

USL 

North 

Carolina

Average 0.98 

According to North 

Carolina data, the actual 

number of total crashes 

during the after period was 

higher than would have 

been expected even though 

USL was maintained, 

except for some specific 

crash types where the 

opposite was observed to 

increase total crashes and 

decrease fatal crashes. 

Group 2: 

Maintained 

DSL 

Washington Total crash 0.99 0.340 According to Washington 

data, the decision to 

maintain DSL had no effect 

on crashes. 

*The inferences shown here are findings that would be drawn if each State were examined in isolation and the modified 

empirical Bayes method as applied were the only analysis technique available. As discussed later in this paper, a 

consideration of all States together using all available methods leads to different conclusions than those shown in table 24. **North Carolina maintained its uniform limit but also raised this limit for both passenger cars and trucks. 
41 

Table 25. Crash increases and confidence intervals according to 

the empirical Bayes formulation. 

Policy Group State Crash Type Lower Bound 

Increase 

Upper Bound 

Increase 

Statistical 

Interpretation

Total crash 12.9% 17.2% Increase 

Total crash with 

truck involved 20.0% 29.8% Increase 

Rear-end crash 11.2% 21.1% Increase 

Fatal crash −7.3% 19.9% No change 

Group 4: 

Changed from 

DSL to USL 

Virginia 

Fatal crash with 

truck involved 

−12.5% 38.4% No change

Total crash 0.4% 13.4% Increase 

Total crash with 

truck involved 18.9% 42.8% Increase 

Rear-end crash −16.7% 3.9% No change

Arkansas 

Fatal crash 1.5% 121.3% Increase 

Total crash 13.2% 46.7% Increase 

Total crash with 

truck involved 68.6% 224.9% Increase 

Rear-end crash 30.6% 94.8% Increase 

Group 3: 

Changed from 

USL to DSL 

Idaho 

Rear-end crash 

with truck 

involved 

−7.4% 281.1% No change
42 

Table 25. Crash increases and confidence intervals according to 

the empirical Bayes formulation—continued.

Policy Group State Crash Type Lower Bound 

Increase 

Upper Bound 

Increase 

Statistical 

Interpretation

Total crash 24.2% 28.6% Increase 

Total crash with 

truck involved 12.1% 20.7% Increase 

Rear-end crash 14.8% 25.3% Increase 

Rear-end crash 

with truck 

involved 

−1.5% 14.7% No change

Fatal crash 20.6% 45.2% Increase 

Arizona 

Fatal crash with 

truck involved 30.0% 97.4% Increase 

Total crash 19.9% 31.9% Increase 

Total crash with 

truck involved −19.7% 1.5% No change

Rear-end crash −12.6% 13.1% No change

Rear-end crash 

with truck 

involved 

−23.4% 18.6% No change

Group 1: 

Maintained 

USL 

North 

Carolina 

Fatal crash −50.1% −2.1% Decrease 

Group 2: 

Maintained 

DSL 

Washington Total crash −6.6% 5.0% No change

Table 26. T-Statistics for the empirical Bayes crash estimation models (before data). 

Policy 

Group State Crash Type 

T -

Statistic 

for β1

(length)

T -

Statistic 

for β2

(ADT) 

Degrees 

of 

Freedom

Critical 

TStatistic 

Were the

Values 

Significant? 

Total crash 10.69 11.93 

Total crash with 

truck involved 9.45 12.45 

Rear-end crash 4.07 7.91 

Fatal crash 3.63 2.54 

Group 4: 

Changed 

from DSL 

to USL 

Virginia 

Fatal crash with 

truck involved 2.86 2.54 

795 1.96 

Yes, for all 

crash types. 
43 

Table 26. T-Statistics for the empirical Bayes crash estimation models 

(before data)—continued.

Policy 

Group State Crash Type 

T -

Statistic 

for β1

(length)

T -

Statistic 

for β2

(ADT) 

Degrees 

of 

Freedom

Critical 

Tstatistic 

Were the

Values 

Significant? 

Total crash 2.29 

Total crash with 

truck involved 1.02 

Rear-end crash 3.25 

Arkansas 

Fatal crash 

Lengths 

were 

same 

length 

0.00 

48 2.01 

Yes for total 

crash and rearend crash. 

No for total 

crash with 

truck involved 

and fatal crash.

Total crash 4.77 3.96 

Total crash with 

truck involved 2.97 1.66 

Rear-end crash 2.95 6.05 

Group 3: 

Changed 

from USL 

to DSL 

Idaho 

Rear-end crash 

with truck 

involved 

2.59 2.81 

29 2.05 

Yes, except β2

was not 

significant for 

total crash with 

truck involved.

Total crash 25.32 2.97 

Total crash with 

truck involved 19.46 3.65 

Rear-end crash 13.72 9.93 

Rear-end crash 

with truck 

involved 

11.74 5.20 

Fatal crash 8.93 0.93 

Arizona 

Fatal crash with 

truck involved 4.4 0.57 

1,665 1.96 

Yes, except β2

was not 

significant for 

fatal crash and 

fatal crash with 

truck involved.

Total crash 5.27 9.86 

Total crash with 

truck involved 0.43 8.27 

Rear-end crash 2.25 7.25 

Rear-end crash 

with truck 

involved 

0.21 4.91 

Group 1: 

Maintained 

USL 

North

Carolina

Fatal crash 0.75 4.22 

122 1.98 

β2 was always 

significant, but 

β1 was only 

significant for 

total crash and 

rear-end crash.

Group 2: 

Maintained 

DSL 

Washington Total crash 1.49 1.17 87 1.99 Neither β1 nor 

β2 was 

significant. 

*

Shaded values were not significant at the 95 percent confidence level 
44 

Finally, table 26 shows the T-statistics associated with the β1 and β2 values used in the crash 

estimation model. An interpretation of table 26 is that, for all States and crash types, either 

section length, section ADT, or both were significant determinants of the number of crashes with 

three exceptions: total crashes in Washington, total crashes with truck involved in Arkansas, and 

fatal crashes in Arkansas. For the State of Washington, the corresponding confidence interval 

from table 25 had shown no change in crashes; in that sense, the results from table 26 are not 

surprising. For the case of Arkansas fatal crashes, the results from table 26 show that the crash 

estimation model for that particular case was probably spurious. In that sense, the decision from 

table 24 (that, because of the abnormally low value for β2 observed for Arkansans fatal crashes 

in table 24, one should be cautious about inferences for this particular case) is confirmed by the 

corresponding T-statistic in table 26.

Relating Speed and Crash Changes 

The logical question is whether any relationship exists between crashes or crash rate changes and 

changes in speed. Accordingly, researchers may compare the speed results shown in table 9 to 

the crash results shown in table 24, and/or the crash rate results shown in table 12. The States 

that intersect both speed-related data in table 8 and either of the crash tables (tables 12 or 24) are 

Idaho and Virginia. From those two States, investigators can glean three sets of changes in speed 

policy: Idaho increasing its USL, Idaho changing from USL to DSL, and Virginia changing from 

DSL to USL. Yet even these results are inconclusive: 

• When Idaho raised its USL, there was an increase in both the mean speed and the 85th 

percentile speed (other speed data were not available), as expected. Yet there was no 

significant change in the crash rate. 

• When Idaho changed USL to DSL, there was no significant change in its speed or its crash 

rate. There was, on the other hand, an increase in the number of crashes as reflected in tables 

24 and 25. (Except for the category of truck-involved rear-end crashes, the upper and lower 

confidence bounds for all Idaho crash rate categories were positive. To the extent that having 

the 95 percent confidence interval not include zero indicates statistical significance, therefore, 

one would say that the Idaho crash rate increases for all categories except rear-end with truck 

involved were significant. For the category of rear-end with truck involved, research could 

deduce no significant change because the confidence interval includes zero. The lower 

confidence interval is less than zero, and the upper confidence interval is greater than zero.) 

• When Virginia changed DSL to USL, there were no significant changes in any speed 

measures except for one—the rate of noncompliance significantly dropped (generally viewed 

as a positive development). There was a significant increase in two of five crash rate 

categories (rear-end and total truck involved) and an increase in three of the five categories 

for the number of crashes via the empirical Bayes method (the other two categories showed 

no change in crash rates). 

Looking at only these three bullets, a logical inference based on these changes in speed policy 

would find no relationship between changes in speed and safety changes. Idaho’s first change 

saw an increase in speeds but no adverse safety impacts, whereas Idaho’s second change and 

Virginia’s change each saw no change in most measures of speed and adverse safety impacts. 
45 

The only exception was Virginia’s decreased rate of noncompliance—accompanied by an 

increase in the number of crashes. Certainly, it is plausible that the indicators of speed used in 

this study are simply not useful measures of crash risk. 

Acceptance of this outcome—that there is no relationship between changes in speed indicators 

and crash changes—however, requires two logical steps. First, there must be the tacit acceptance 

that crashes (as measured by the empirical Bayes method) are a better indicator of safety than 

crash rates (thereby simplifying the second and third bullets), but that crash rates are a better 

measure of safety than nothing (thereby simplifying the first bullet). Second, tables 24 and 25 

must be accepted as true even though they generally tended to show increasing crash risk, 

regardless of the policy speed limit change. In sum, this study offers some tendency for speed 

indicators and crash indicators to not be correlated. However, because these inferences are based 

on observations from only two States, such results prevent investigators from using this study to 

make that conclusion. 

STUDY LIMITATIONS 

Caveats About the Use of the Empirical Bayes Method 

Several data limitations may have influenced the values shown in tables 24 and 25. These data 

limitations apply to the investigators’ use of the modified empirical Bayes method and not 

necessarily to the application of the empirical Bayes method in other situations. 

First, table 24 shows the corresponding exponent β2 for each crash type and State, where β2

reflects the impact of a change in ADT according to the crash estimation model developed for 

that State and crash type. A plausible explanation for this is that, in most States, ADT increases 

over time, so with a β2 less than 1.0 the would-have-been crashes predicted by the crash 

estimation model will correspond to a lower crash rate. As β2 approaches zero, large increases in 

ADT tend to only increase crashes by a disproportionately small amount, with the effect being 

that the would-have-been crash rate is so low that it is virtually impossible for any policy change 

to show a θ less than 1.0, which would be an improvement in safety. In sum, a low value of β2

far from unity renders the model very insensitive to changes in ADT from the before scenario (in 

the sense that a low value of β2 means that large additional ADT should only increase crashes 

slightly). Thus, the interpretation that Arkansas fatal crashes increased as a result of the change 

from USL to DSL should be tempered by the very low β2 value of 0.00355 shown in table 24, 

which would mean that almost any change would probably result in a θ value greater than 1.0. 

Alternatively, one can cite that the insignificance of the crash estimation model, based on table 

26, for the specific case of Arkansas fatal crashes, is also a reason not to rely solely on inferences 

from Arkansas fatal crashes. 

Second, comparison groups are imperfect. Ideally, the comparison (control) group would have 

been selected from the same State at the same time as the studied group. For example, in 

Virginia, if after the statewide differential speed limit of the early 1990s was repealed, one 

section of Interstate 81 could have been kept at DSL and another section of the highway could 

have been changed to DSL. Unfortunately, not only were comparison groups composed of 

different roadways, but they also drew from different States altogether. Thus, since the actual 

Virginia reference group was an extrapolation of the temporal trend that occurred during the 
46 

before period from 1991 to 1993, later aberrations in this trend may not have been identified. In 

other words, if some significant change occurred in one year that increased the crashes but had 

nothing to do with speed limits policy change, then unfortunately this change would not be 

captured in the comparison group and therefore would not be reflected in the models. In short, a 

problem would arise if the relationship between ADT and crashes had drastically changed from 

the before to the after period. When this relationship is based on multiple years of data and 

multiple sites, one can have some confidence in the model, but temporal changes in the after year 

are possible that might alter this relationship. 

Third, although speed-monitoring data were available to clarify general speed trends throughout 

a State, specific speeds often were not available on every section of interstate that was a segment 

in the crash analysis. (In other words, for a given 322-km (200-mi) stretch of interstate, annual 

data indicating overall speed trends might be available from two speed-monitoring sites. While 

analyzing crashes on those interstates, however, if the interstates were divided into 20 sections 

for analysis, it would be best to have 20 speed-monitoring sites to include actual speeds in the 

crash estimation models.) 

Next, the crash estimation models used identical treatment and comparison sites, which, as 

discussed earlier in the paper, was done to reduce error in the prediction of crashes by CEM. In 

traditional before/after studies, such a decision could possibly subject the study to what Hauer 

describes as “regression-to-mean (RTM) bias” or “selection bias.”(10) As explained by Hauer, 

RTM arises when “there is a link between the decision to treat an entity and its accident 

history.”(10) If the basis of site selection was those sites with the largest number of crashes in a 

single year, then it is possible that subsequent crash reductions would be erroneously attributed 

to the treatment when in fact such reductions were truly the result of random variation that would 

have transpired even without the treatment.† Two characteristics of this study likely eliminate the 

possibility of regression to the mean bias. First, the sites studied were generally not chosen by 

persons with an interest in testing the effects of USL and DSL; rather, the sites were those that 

had available data. Second, the sites did not show behavior expected in a study where regression 

to the mean occurs. Instead of seeing dramatic crash reductions, States generally saw crash 

increases. For the rate-based method, the increases were often not significant and for the 

modified empirical Bayes approach, the increases were sometimes significant. Had the bias been 

present, researchers would have expected to see crash reductions in lieu of increases. 

Finally, the crash estimation model used only two variables, AADT and section length. This 

could have been the result of too few variables, in case other factors, such as the number of 

interchanges per mile, could have influenced crash rates. Other variables besides AADT and 

section length also may have been relevant. To mitigate the impact of this last problem, goodness 

of fit tests were conducted for the crash estimation model for total crashes for Virginia, as shown 

in figures 9 and 10. Based on the Virginia results, this model formulation was applied for other 

 

† In practice, RTM occurs when an agency identifies, based on a short period of data, the most unsafe locations and 

then makes subsequent engineering modifications to those locations. Since crash data have an element of probability, 

choosing the worst locations based on a short period of data may mean that investigators simply have identified sites 

that, at random, happened to exhibit a large number of crashes in a given year and would probably exhibit a lower 

number of crashes the following year, even if no change were made. Investigators seek to avoid this bias by 

selecting sites that have good and bad crash records and studying those accordingly with a new treatment.
47 

States. However, additional data elements, such as the number of interchanges per mile, might 

have helped reduce the amount of noise used throughout this analysis. 

General Caveats 

There are six additional limitations that, unlike the concept of the comparison reference groups 

cited above, apply to all three sets of results: speed, crashes, and crash rates. As was the case 

with the limitations described for the application of the modified empirical Bayes method, these 

caveats arise because of the limited data available. Firstly, the sample size varied by State. For 

example, there may be less certainty for the North Carolina crash results (based on 26 sites) 

being representative of North Carolina than that of the Arizona crash results (based on 278 sites) 

being representative of Arizona. For the empirical Bayes method, Washington, which maintained 

a differential speed limit, comprised the only State in policy group 2 with nine sites. Secondly, 

the selected sites may be an unbiased sample; however, the investigators cannot control site 

selection; their randomness is a function of how individual States set up their individual speed 

monitoring programs. Thirdly, the durations used in this study are relatively short. Certainly 3 

years is normal for a before/after study, but some of these States, notably Idaho, only had a speed 

limit in effect for 1 or 2 years. Fourthly, the rural interstates were analyzed at an annual level of 

detail, without stratification by time period or season. Fortunately, since congestion is usually 

lower for rural interstates, this annual approach should not have been as significant a problem as 

it would have been for urban interstates. Next, the speed data shown are based on all vehicles, 

not just trucks. Except for speed variance, researchers ideally want to know how truck speeds, 

not just all vehicle speeds, were affected by differential versus uniform speed limit policy. 

Similarly, when looking at truck-involved crashes, investigators would want to be able to 

delineate between crash types that are directly affected by DSL versus USL policy (e.g., car-intotruck collisions) and those that are indirectly affected by such a policy (e.g., truck-into-truck 

collisions).27

Finally, the sample size used in the statistical tests associated with the speed analysis was defined 

as the number of speed monitoring sites. Although the investigators decided that this was 

appropriate to determine whether speed changes were meaningful, an argument can be made that 

the sample size should have been estimated as the number of vehicles at all of the sites combined. 

Acceptance of this latter view would give significance to many of the statistical tests shown in 

table 11 as insignificant, although there still would be no clear pattern as to the effect of DSL 

versus USL for speed. (A minor item is the need to make some assumptions regarding the 

underlying vehicle distribution since individual vehicle speeds were not available; appendix E 

discusses this issue further.) An extension of this argument arises in the computation of 

confidence intervals for the 85th percentile speed; hence, theoretical issues associated with 

determining significant differences in 85th percentile speed are discussed in appendix F but are 

not believed to influence the outcome of this study.(28) 

CONCLUSIONS 

The original purpose of this study was to compare the safety impacts of DSL and USL, with 

safety impacts being assessed through crashes and speeds, and to that extent, findings may be 

presented across both those areas. An unforeseen outcome of this study, however, was to identify 

considerations in the application of the empirical Bayes methodology, both with respect to the 
48 

formulation of the crash estimation model and the types of data necessary to conduct the 

analysis. 

Safety Impacts of DSL Versus USL 

1. Speed characteristics were generally unaffected by a DSL versus USL policy. Except for 

Virginia, the mean, 85th-percentile, and median speeds tended to increase over the 1990s 

regardless of whether the State maintained USL, maintained DSL, or changed from one to 

the other. In some cases, the difference was significant; in other cases, it was not. 

2. Crash rates, when compared using conventional statistical methods, did not show an obvious 

relationship to the type of speed limit chosen. When States were stratified into four policy 

groups (USL, DSL, shift from USL to DSL, and vice versa), the changes in crash rates and 

crash rate types did not all correspond to one group. 

3. Actual number of crashes, when compared using the modified empirical Bayes approach 

indicated that the percentage of increase of the fatal actual crashes over that for the wouldhave-been fatal crashes was higher for the States that changed from USL to DSL than for 

Virginia, which changed from DSL to USL. On the other hand, the same empirical Bayes 

approach showed that the percentage increase of the total actual crashes over the wouldhave-been total crashes was higher for the States that maintained USL than for the State that 

maintained DSL. 

4. Measurable variation within speeds and crash rates by year and by State may confound any 

statistical tests employed. The performance of Illinois annual speed variances, as shown in 

figure 17, is indicative of the noise associated with random variation, where the annual speed 

variance has an insignificant but observable upwards and downwards trend despite the fact 

that Illinois made no policy changes to its speed limits. 

Methodological Findings 

1. In this study, most of the crash estimation models developed by the investigators for the 

modified empirical Bayes approach were very sensitive to changes in ADT. In most cases for 

this study, as time passed, the actual number of crashes for the after period was larger than 

the predicted would-have-been crashes. The investigators believe this to be the result of the 

less than unity exponent associated with the equation in figure 8. 

2. Even within a single State, different trends are observed on different rural interstates. 

Examination of figure 19, for example, suggests that the between-interstate variation adds 

quite a bit of noise to any attempt to study statewide temporal trends. The reality may be that 

DSL and USL have different safety impacts, as might be implied by an imagined 

extrapolation of the shaded Idaho data in table 11 (beyond 1998) or the Virginia mean speed 

downward trend in figure 26 (beyond 2000 to 2001). This particular study, however, could 

not identify such a difference. It may be the case, therefore, that a detailed crash-by-crash 

evaluation at a few interstate segments may be able to reveal a more precise analysis, 

especially after a few more years of data have been collected at States such as Idaho that 

have recently made changes form USL to DSL. 
49 

3. The proportionality of crashes to ADT has been questioned but not resolved. The crash 

estimation models developed for this study showed a nonlinear effect of crashes to ADT, 

such that it was often the case that a large increase in ADT would, according to the crash 

estimation models, yield a small increase in the number of crashes. For that reason, 

investigators can state that a proportional relationship between crashes and ADT, which is 

presumed in the use of crash rates, is questionable. Yet, the same approach yielded 

contradictory results on a State-by-State basis. Some CEMs predicted, for example, that one 

policy adversely affected safety, whereas others predicted that the opposite policy adversely 

affected safety. For this reason, the highly nonproportionate effect of crashes to ADT may be 

overstated by the CEMs developed for this study—but these results cannot prove or disprove 

that assessment. Thus, because of the contradictory nature of these CEMs, the proportionality 

of crashes to ADT is not resolved. 

4. To assist decisionmakers under conditions of sparse data availability, it may be productive to 

analyze safety impacts through both the traditional method of crash rates and the newer 

method of the empirical Bayes (or modified empirical Bayes) approach. If the aggregate 

results from the two methods are the same, then results can be presented using the method 

that is more familiar. If the aggregate results from the two methods are contradictory, then 

researchers have an indication that there may be a flaw in the analysis. For example, consider 

the States analyzed in this study: 

• Application of the modified empirical Bayes method showed no consistent trend 

among the States. Regardless of speed limit policy change, table 24 suggests a 

statistically significant increase in crashes for each State. The aggregate result of the 

modified empirical Bayes method, therefore, is that, because the State-by-State 

results are inconsistent, the modified empirical Bayes method does not show that 

changes in speed limit policy have a statistically significant impact on crashes. 

• Application of the traditional crash rate method, shown in table 12, also indicates that 

most State-by-State results were statistically insignificant. In the few cases where a 

statistically significant rate was found, results were not consistent among States in the 

same group. For example, one State that maintained USL saw a significant increase, 

whereas the other two States that maintained USL did not see a significant increase. 

The aggregate results of the traditional crash rate method may be stated thus: 

“Because most States saw insignificant changes in the crash rate and because there 

was no discernible pattern in the States that did have a significant change in the crash 

rate, the traditional crash rate method does not show that the changes in speed limit 

policy have a statistically significant impact on crash rates.” 

Because the two bulleted methods show the same conclusions, the investigator may have greater 

confidence that the analysis is correct than would be the case had the two bulleted methods 

yielded contradictory conclusions. 

51 

APPENDIX A. EXAMPLES OF DATA COLLECTION LETTERS AND PROCESSING 

Appendix A shows the initial data request letter. 

Dear Mr. Baldwin, 

As I mentioned on the phone, the Virginia Transportation Research Council is working with the 

Federal Highway Administration to evaluate the potential safety impacts of differential speed 

limits for cars and trucks on interstate facilities. I would like to request your assistance with 

obtaining crash and speed data (on interstate highways) that can help us with this study. While 

we already have some limited data for the sites shown at the bottom of this letter, I’d like to 

obtain some additional data pertaining to these and other sites. 

I realize that obtaining data can be time consuming so I am certainly willing to do whatever is 

possible to make it easier for you to fulfill our request. If at all possible, we would like to obtain 

the following crash and speed data in an electronic format. The crash data and the speed data 

may come from the same sites or they may come from different sites, whichever is easier (but all 

should be on interstate highways, with the speed limit shown. At sites with differential speed 

limits please list the speed limits for passenger cars and trucks separately). 

For each year from 1991 to 2000, we would like to obtain the following speed data elements at 

each site: 

1. Average speed. 

2. Individual vehicle speeds or speed bins (e.g., x vehicles between 51-55 mph, y vehicles 56-

60 mph, etc.). 

3. Individual truck speeds or average truck speeds (if available). 

4. Individual car speeds and average car speeds (if available). 

5. Critical geometric data such as: 

 a. The number of lanes 

b. The number of interchanges (or the number of interchanges per mile) 

The speed data sites that interest us are these plus any additional sites you recommend. 

Route 29 Northbound Milepost 29 

Route 35 Northbound Milepost 14 

For each year from 1991 to 2000, I would also like to obtain the following crash data elements: 

1. Total number of crashes 

 a. All crashes that do not involve trucks. 

 b. All crashes that do involve trucks. 

c. All fatal crashes (regardless of truck involvement). 

d. All fatal crashes (of truck involvement). 

e. Total number of rear-end crashes that do involve trucks. 

f. Total number of rear-end crashes that do not involve trucks. 
52 

The crash sites should be the same from year to year, and can either be the sites shown above or 

be different from the speed sites. Each crash site should be a homogeneous section that can show 

some crashes (e.g., whether a crash site is 1 mi long or 10 mi long, it should be (a) big enough to 

obtain some crashes annually, yet (b) small enough such that speeds and geometric 

characteristics for the site are homogeneous). We would like, if possible, up to 10 crash sites 

altogether. 

If possible, we would like the data to be in the following format. But we are also happy to have 

any format of data you sent to us. 

Finally, I would like to confirm that you have had since 1991 a uniform limit, that is, the same 

speed limit for cars and trucks. 

Again, I sincerely appreciate your assistance. I would also be delighted to provide you with 

additional information about the purpose of this study, as well as any findings that result. 

Route Year Beginning 

Mile Post 

Ending 

Mile Post AADT Total Crash

Fatal 

Crash

Rear

-End

Truck 

Total

Truck 

Fatal 

Truck 

RE 

Mean 

Speed

85th

Speed

 
53 

APPENDIX B. EXAMPLE OF A CLARIFYING DATA REQUEST LETTER 

Appendix B shows an example of a follow up letter providing clarification. 

Dear Mr. Wyatt, 

Thank you very much for the data you sent to us! Once again we have another question. The 

accident data we got from you are from 1991 to 2000 as shown below 

I77, From Future I-74 to Va. State Line 

I77,From Yadkin Co. to Future I-74 

I85,Randolph 

I85,Vance 

I40,Johnston 

I40,Duplin 

I40, Pender 

I-95, Halifax 

I-95,Nash 

I-95,Wilson 

I-85,Granville 

[The names above refer to counties.] 

I have four quick questions: 

1. Can we get the ADT by year, from 1991 to 2000, for the accident sites shown above? 

2. Do you have any crash data for: 

(a) Rear-end crashes involving trucks, and 

(b) fatal crashes involving trucks? 

3. When tabulating crashes, some States include only crashes on the mainline, and others 

include ramp crashes as well (regardless of causality). For North Carolina, which should we 

assume? If it is the case that ramp crashes are included, then do you know roughly how the 

number of ramp crashes compares to the number of mainline crashes? 

4. Is the speed limit for cars and trucks 65 on North Carolina interstates, and has the limit 

changed since 1990? 

55 

APPENDIX C. CONFIRMATION OF THE NEGATIVE BINOMIAL DISTRIBUTION 

TO CRASH DATA 

As stated in the literature, the assumption in the application of the empirical Bayes formulation 

as done herein is that for a particular site i, the distribution of the number of crashes Ki,y over the 

years y obeys the Poisson distribution. Further, for a particular year y, the distribution of the 

number of crashes Ki,y between different i sites follows the negative binomial distribution. Based 

on these two assumptions, the expected number of crashes of a group mi1 are Gamma 

distributed.10,11,12, 13 Figure 33 illustrates these concepts, where Ki,y is the actual crash count for 

site i and year y and mi,y is the expected crash counts for site i and year y.

Site 1991 1992 1993 1994 1995 1996 

 K11 m11 K12 m12 K13 m13 K14 m14 K15 m15 K16 m16

2 K21 m21 K22 m22 K23 m23 K24 m24 K25 m25 K26 m26

3 K31 m31 

4 K41 m41

5 K51 m51

6 K61 m61

Figure 33. Chart. Relationship between the Poisson and negative binomial 

distributions for crash frequencies. 

The Poisson and negative binomial distributions were tested with the data sets for selected states 

as described herein. 

Verification of the Poisson Distribution 

Using data from Virginia and Arizona, two techniques were used to verify that the Poisson 

distribution is appropriate. Firstly, theoretical versus actual frequencies were compared 

graphically. Secondly, the chi-square test was used to determine whether a statistically 

significant difference existed between the actual and theoretical distributions for the Kiy over 

time. 

Ki1: Negative Binomial K1y : Poisson 

 mi1: Gamma 
56 

Figure 34 compares the actual crash frequency distribution and the Poisson distribution using 

one site on Interstate 85 in Virginia between milepost 19.52 and milepost 24.73, looking at the 

annual number of crashes between 1991 and 1999. 

0

1

2

3

4

5

6

7

8

9

10

123456789

actual

statistic

Figure 34. Chart. Comparison of Poisson distribution and actual crash distribution. 

Table 27 shows that the calculated χ2 value is less than the critical (tabulated) χ2 value, which 

means that the assumed distribution is accepted. Theoretically, the computed chi-square value 

(which represents error in, or divergence from, the Poisson distribution) is less than the tabulated 

chi-square value; therefore, the hypothesis that the distributions are different cannot be proven at 

the 5 percent confidence level. 

Table 27. Poisson validation description and results using 

the total crashes at four test sites. 

State Test Sites Sample 

Size 

Data 

χ2

calculated 

χ2

sta, 

0.05 

χ2

sta, 

0.01

Arizona I-8 mp 42.06 to 54.96 10 1991–2000 9.530 14.07 18.47

Arizona I-10 mp 19.79 to 26.65 10 1991–2001 6.738 15.51 20.08

Virginia I-85 mp 19.52 to 24.73 5 1995–1999 8.764 11.07 15.09

Virginia I-81 mp 206.04 to 213.48 6 1995–2000 6.468 7.82 11.33

Verification of the Negative Binomial Distribution 

A similar procedure was used to test the validity of the negative binomial distribution, except 

that crash rates as defined in figure 3 rather than the total number of crashes, was used to as the 

variable of interest. Table 28 highlights the result of the chi-square test and visual inspection of 

figure 35 suggests that the negative binomial distribution is appropriate for these data. (Crash 

rates rather than the number of crashes was used because of variation in the section lengths.) 

Poisson Cumulative Frequency distribution 
57 

Table 28. Negative binomial validation description and results. 

State Year Sample 

Size Test Site Crash Type χ2

cal 

χ2

sta, 

0.05 

χ2

sta, 0.01

Result at 

5% Level 

Result 

at 1% 

Level 

VA 1991 91 85n,95n,81n total 42.326 44.8 60.1 Yes yes

VA 1992 90 85n,95n,81n total 14.516 19.68 24.75 Yes yes

VA 1993 91 85n,95n,81n total 10.730 20.08 14.07 Yes yes

VA 1995 117 85, 95, 81n total 21.386 22.37 27.72 Yes yes

VA 1996 116 85, 95,81n total 18.153 18.31 23.91 Yes yes

VA 1997 116 85, 95,81n total 14.135 19.68 24.76 Yes yes

VA 1999 84 85n, 85s, 81n total 29.852 27.59 33.44 no(close)* yes

NC 1993 26 40,95,77,85 total 4.005 6 9.22 Yes yes

NC 1999 25 40,95,77 total 7.385 11.07 15.09 Yes yes

ID 1992 32 84,86,90,15 total 28.327 38.89 45.67 Yes yes

ID 1994 32 84,86,90,16 total 28.305 37.66 44.34 Yes yes

ID 1999 32 84,86,90,16 total 36.322 42.57 49.61 Yes yes

ID 2000 32 84,86,90,16 total 26.209 32.68 38.96 Yes yes

AZ 1991 277 8,10,15,17,19,40 total 24.151 21.03 26.25 no(close)* yes

AZ 1993 277 8,10,15,17,19,40 total 20.392 23.37 27.71 Yes yes

AZ 1994 278 8,10,15,17,19,40 total 17.697 26.3 32.03 Yes yes

AZ 1995 277 8,10,15,17,19,40 total 21.232 22.37 27.71 Yes yes

AZ 1996 278 8,10,15,17,19,40 total 19.977 22.37 27.71 Yes yes

AZ 1998 279 8,10,15,17,19,41 total 21.164 22.37 27.71 Yes yes

AZ 1999 280 8,10,15,17,19,42 total 20.149 23.69 29.17 yes yes

AZ 2000 281 8,10,15,17,19,43 total 24.187 27.59 33.44 yes yes *

The significance level of a chi-square test is actually a proof that a theoretical distribution does not fit the data. 

Thus, if a calculated chi-square value is sufficiently large such that it exceeds the 5 percent chi-square value, then it 

can be said that “researchers are 95 percent certain that the two distributions are different.” In the two rows with 

asterisks, there is a 95 percent certainty that the two distributions are different but not 99 percent certain. In all other 

cases, it cannot be proved at the 95 percent level that the theoretical and actual distributions are different; therefore, 

it is presumed they are the same. 
58 

0

5

10

15

20

25

30

35

40

45

50

10

30

50

70

90

110

130

150

170

190

Negative binomial distribution

Crash rates

actual

stastic

Figure 35. Chart. comparison of negative binomial distribution and actual crash 

distribution (probability density function). 
59 

APPENDIX D: EFFECT OF CHANGING THE BASE YEAR IN CI,Y

Figure 12 showed that the ratio of the expected crash frequency for a given year y to the expected 

crash frequency during the “base” year is given as the expression Ci,y. The question may arise as 

to whether the use of a different base year would significantly influence the results. Both a 

mathematical derivation and a data-driven experiment suggest that the selection of the base year 

will not influence the analysis. 

In the derivation that follows, the ratio using the first year as base in the denominator as 1 Ci, y , 

while the ratio with the third year as base in the denominator is 3 Ci, y . Thus, the equation in figure 

12 may be rewritten for each case as: 

1

( ) ,

( )

,1

,

,1

,

E m i y

E m

m

m

C

i

i y

i

i y

= =

Figure 36. Equation. Crash frequency for year 1 as base year. 

3

( ) ,

( )

, 3

,

, 3

,

E m i y

E m

m

m

C

i

i y

i

i y

= =

Figure 37. Equation. Crash frequency for year 3 as base year. 

Using the First Year as the Base Year 

If the first year is used as a base, the expected value of the first year crash count is estimated first 

as following: 

∑

∑

=

=

+

+

= Y

y

i y E mi

k

Y

y

i y

C

k K

mi

1

1

, ) ,1 (

1

,

,1

Figure 38. Equation. Expected value of crash count for year 1. 

∑ ∑

∑

= =

=

+ +

+

= = Y

y

i y E mi

k

i

Y

y

i y E mi

k

Y

y

i y

C

m

C

k K

VAR mi

1

1

, ) ,1 (

,1

2

1

1

, ) ,1 (

1

,

( )

,1 ( )

Figure 39. Equation. Variance of expected value of crash count for year 1. 
60 

The estimation of the expected values of crash counts of the other years was then calculated by 

multiplying the first year expected estimation of its changing ratio using the following 

expressions. 

( ) ,1

( )

,1

1

, , ,1

,

E m i

E m

m i y C i y m i m i

i y

= =

Figure 40. Equation. Estimation of estimated values of crash counts for year 1. 

,1

2

( )

( )

,1

1 2

, , ( ) ( ) [ ] ,1

,

E m i

E m

VAR m i y C i y m i m i

i y

= =

Figure 41. Equation. Variance of estimation of estimated values 

of crash counts for year 1. 

For example, applying these equations for the third year expected value yields the following 

equation in figure 42. 

( ) i,1

( )

mi,3 Ci,3 mi,1 m ,1

,3

i

i

E m

E m = =

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

∑

=

=

=

=

=

=

=

=

=

=

=

=

+

+

+

+

+

+

+

+

+

+

+

+

= =

=

=

= =

Y

y

i y E mi

k

Y

y

i y

Y

y

E mi

E mi y

E mi

k

Y

y

i y

Y

y

i y

Y

y

i y

Y

y

E mi

E mi y

E mi

k i

Y

y

i y

Y

y

E mi

E mi y

E mi

k

Y

y

i y

i

i

Y

y

i y E mi

k

Y

y

i y

i

i

C

k K k K

k E m

k K

i

E m

k K

i

k K

E m

E m

C

k K

E m

E m

E m

E m

1

3

, ) ,3 (

1

,

1

) ,3 (

) , (

) ,3 (

1

,

1

,

1

,

1

) ,1 (

) , (

) ,1 1 ( ,

1

,

1

1 ) , (

) , (

1 ) , (

1

,

,1

,3

1

1

, 1 ) , (

1

,

,1

,3

( )

,3

( )[ ]

,3

( )

( )

( )

( )

( )

( )

Continues on next page. 
61 

∑

∑

∑

∑

=

=

=

=

+

+

+

+

= =

= =

Y

y

E mi

E mi y

E mi

k

Y

y

i y

i

i

Y

y

i y E mi

k

Y

y

i y

i

i

i

i

k K

E m

E m

C

k K

E m

E m

E m i

E m VAR mi Ci VAR mi VAR m

1

2

1) , (

) , (

1) , (

1

,

,1

,3

2

1

1

, 1) , (

1

,

,1

,3

,1

,3

[ ]

2

( )

( )

[ ]

2

( )

( )

,1

2

( )

( )

,1

1 2

,3 ,3

[ ] [ ]

( ) ( ) ( ) [ ] ( )

 

2

1

) ,1 (

) , (

) ,1 (

2 ,1

1

,

[ ( )] [ ]

2

,3 [ ( )]

∑

∑

=

=

+

+

= Y

y

E mi

E mi y

E mi

k

i

Y

y

i y

E m

k K

E mi

2

1

3

, ) , 3 (

1

,

1

2

3 ) , (

) , (

3 ) , (

1

,

2

1

,

1

,

[ ]

[ ]

[ ( ) ]

2

, 3 [ ( )]

∑

∑

∑

∑

∑

∑

=

=

=

=

=

=

+

+

+

+

+

+

=

=

=

Y

y

i y E m i

k

Y

y

i y

Y

y

E m i

E m i y

E m i

k

Y

y

i y

Y

y

i y

Y

y

i y

C

k K

k K

k E m

k K

E m i

Figure 42. Equation. Expected value of crash count for year 3. 

Using the Third Year as the Base Year 

If the third year is used as a base in the denominator, then the expected value of the third year 

crash count was estimated first as following: 
62 

∑

∑

=

=

+

+

= Y

y

i y E mi

k

Y

y

i y

C

k K

mi

1

3

, ) ,3 (

1

,

,3

 Figure 43. Equation. Expected value of crash count, year 3 as base year. 

∑ ∑

∑

= =

=

+ +

+

= = Y

y

i y E mi

k

i

Y

y

i y E mi

k

Y

y

i y

C

m

C

k K

VAR mi

1

3

, ) ,1 (

,1

2

1

3

, ) ,1 (

1

,

( )

,3 ( )

Figure 44. Equation. Variance of expected value of crash count, year 3 as base year. 

Comparing the two results, it is evident that the expected crash result using the third year as a 

base year is the same as that which would be calculated using the first year as a base year and 

then multiplying this by the Ci,y ratio for third year. 

As an empirical example, a 9.8-km (6.13-mi) section of Interstate 64 East in Virginia was 

selected, and the crash estimation model was established. The results obtained from using the 

1991 as the first year in the denominator of Ci,y is shown in table 29, and the results from using 

the 1993 as the denominator of Ci,y are shown in table 30. The results are identical. 

Table 29. Estimation of expected crashes using 1991 data as a base in the Ci,y ratio. 

Year Ki,y Ei,y C1

i,y mi,y VAR(mi,y)

1991 7 11.454 1 7.4 2.137 

1992 4 10.372 0.906 6.701 1.753 

1993 9 12.21 1.066 7.888 2.429 

1994 — — — — — 

1995 13 11.35 0.991 7.332 2.099 

1996 15 11.35 0.991 7.332 2.099 

1997 6 12.428 1.085 8.029 2.516 

1998 10 12.94 1.13 8.36 2.728 

1999 10 13.435 1.173 8.68 2.941 
63 

Table 30. Estimation of expected crashes using 1993 data as a base in the Ci,y ratio. 

Year Ki,y Ei,y C3

i,y mi,y VAR(mi,y)

1991 7 11.454 0.938 7.4 2.137 

1992 4 10.372 0.849 6.701 1.753 

1993 9 12.21 1 7.888 2.429 

1994 — — — — — 

1995 13 11.35 0.93 7.332 2.099 

1996 15 11.35 0.93 7.332 2.099 

1997 6 12.428 1.018 8.029 2.516 

1998 10 12.94 1.06 8.36 2.728 

1999 10 13.435 1.1 8.68 2.941 

However, as is the case with any data set, it is always possible that a single year could be an 

outlier. Thus, it should be clarified that appendix D only tests the effect of changing the E(mi,1) 

shown in the denominator from year 1 to another year. It does not test for the effect of removing 

1991 from the data set entirely. 

65 

APPENDIX E. DETERMINING THE SAMPLE SIZE 

Although the speeds of individual vehicles are not available, mean speeds from speed monitoring 

stations in Idaho are shown in table 31 below. For example, in 1991, vehicle speeds were 

monitored at 24 sites in Idaho, where each site measured an unknown number of vehicles. The 

mean speeds from these 24 sites were 104 km (64.66 mi/h). 

Table 31. Summary Idaho data from speed sampling sites. 

Year Number of 

Sites 

Mean Speed from 

the Sites 

1991 24 64.66 

1992 36 64.89 

1993 36 65.61 

1994 27 65.37 

1995 36 65.59 

1996 37 68.45 

1997 36 70.92 

1998 32 71.01 

1999 38 70.81 

Test for Significant Differences Using the Sample Size as the Number of Sites 

Visually, figure 1 generally shows an upward trend in speeds. Statistically, however, there is not 

always a significant different in mean speeds of the individual site means. For example, compare 

the two shaded rows that contrast 1991 and 1995. With Nx = 24, Ny = 36, and Ux – Uy = 65.59 – 

64.66, there is no statistically significant difference as measured by the equation in figure 45. 

The logical conclusion is that there is no statistically significant difference in the mean speed of 

speed sampling site means. 

2 2

1.96 ? x y

x y

x y

S S

Is U U

n n

−> +

Figure 45. Equation. Statistically significant difference in mean speeds. 

Test for Significant Differences Using the Sample Size as the Number of Vehicles 

While the last sentence in the above paragraph is correct, it is not very meaningful in practice. 

The problem is that the sites shown in figure 45 (e.g., the 24 sites that comprise the initial 1991 

year) are not individual speeds. Rather, they are means of individual vehicles. That is, the 24 

sites from 1991 do not represent 24 vehicles. Rather, they represent far more vehicles. 

Unfortunately, there are two pieces of information that prevent researchers from performing an 

exact statistical test on the means of individual speeds: 
66 

• The number of vehicles represented by each site is unknown. 

• The variation (or even the distribution) of those vehicles at each site is unknown. 

Simulating the Speed Variances 

To estimate the types of individual speed distributions that might give rise to those shown in 

figure 45, the speeds of individual vehicles were simulated for two hypothetical sites. Each site 

initially had 1,365 speeds generated within a predefined range. Because normality could not be 

guaranteed at the site, the individual vehicle speeds were generated in the form of a nonnormal 

distribution. Figure 46 shows a histogram for the individual speeds at site X and site Y, with 

each bin being 1.6 km/h (1 mi/h) in width. 

-5

0

5

10

15

20

25

30

35

40

30 40 50 60 70 80 90

Bins

Frequency

X Y

Figure 46. Chart. Histogram based on random numbers. 

This simulation showed standard deviations of 14.61 when 25 speeds were used and standard 

deviation of 14.31 when 1,365 speeds were used. Thus, for a normal distribution, the standard 

deviation would likely be lower. 

Determining the Difference in Mean Speeds Needed to Show Significant Difference Based 

on the Number of Vehicles 

Returning to figure 3 and using the larger standard deviation of 14.61, researchers may ask at 

what point a difference in mean speeds leads to a significant difference. Table 32 summarizes 

these results. With just 100 vehicles, for example, a difference of 3.94 mi/h is required to show a 
67 

significant difference, whereas with 10,000 vehicles, the difference need be only 0.62 km/h 

(0.39) mi/hour. 

Table 32. Sample sizes required to achieve significant differences. 

n Difference (km/h) Difference (mi/h) 

100 6.34 3.94 

200 4.49 2.79 

300 3.66 2.27 

1000 2.01 1.25 

2000 1.42 0.88 

3000 1.16 0.72 

5000 0.90 0.56 

10000 0.63 0.39 

20000 0.20 0.28 

Researchers can compare these data to the number of vehicles available for each State, 

summarized as follows: 

Idaho: Between 5,874 and 19,255 vehicles per site. 

Illinois: No ADT data. 

Indiana: Between 3,978 and 23,000 vehicles per site. 

Iowa: No ADT data. 

Virginia: Between 10,312 and 20,071 vehicles per site. 

Thus, if researchers were to base the sample size on the number of vehicles, the results for 

Indiana and Virginia would change from an insignificant difference to a significant difference, as 

noted in the body of the report. 

Discussion 

This analysis considers only statistical significance, which is a different issue from practical 

significance. In fact, table 32 may be carried to its illogical extreme in the sense that a sample 

size of 100,000 would mean that a mean speed difference of about one tenth of one mi/h 

indicated a statistically significant difference. Yet, intuitively, researchers would argue that such 

a difference is not meaningful because the difference is so tiny that it cannot be readily observed. 

As researchers extend table 32 to a larger number of vehicles, there comes a point at which the 

test of statistical significance is not a useful indicator for what is being measured: whether a 

change in speeds has significant meaning. The question becomes, then, at what point does that 

occur? One way to address this issue is to realize that by using the number of sites as the sample 

size n, the researchers prevent themselves from reaching this point. In short, using the number of 

sites gives a practical way of ensuring that statistically significant differences have a practical 

meaning. Another way of summarizing this issue is to state that by using the number of sites as 

n, researchers raise the bar for the test of significant differences. 

69 

APPENDIX F: THEORETICAL CONSIDERATIONS IN THE COMPUTATION OF 

CONFIDENCE INTERVALS FOR THE 85TH PERCENTILE SPEED 

It was informally suggested to the investigators that a question arises as to how to determine the 

confidence intervals for the 85th percentile speed.(28) The team considered this question and 

developed the following derivation. To determine confidence intervals associated with a mean 

speed, for example, investigators generally use the formula shown in figure 47: 

n

zs

x ±

Figure 47. Equation. Formula to determine confidence intervals 

associated with mean speed. 

Thus, for a sample of n = 200 vehicles, a standard deviation of 4.82 km/h (3 mi/h), a mean speed 

of 80.5 km/h (50 mi/h), the 95 percent confidence interval becomes: 

( )( ) 49.58 to 50.42 mi/h

200

1.96 3 50 ± =

Figure 48. Equation. Example of formula in figure 47. 

Suppose for an instant that speeds are represented perfectly by the normal distribution, such that 

the 85th percentile speed is 85.4 km/h (53.11 mi/h). If the investigator applies this same equation 

to compute a 95 percent confidence interval for that 85th percentile speed, then the investigator 

computes the confidence interval as being: 

( )( ) 52.69 to 53.53 mi/h

200

1.96 3 53.11± =

Figure 49. Equation. Confidence interval for 85th percentile speed. 

If the investigator can assume that the central limit theorem still applies to the question of 

determining an 85th percentile speed, then figure 49 should still be applicable. However, it may 

be the case that the investigator cannot presume that the central limit theorem will hold at 

relatively small n; for example, it may be the case that at the upper tail of the normal distribution, 

the odds of observing a vehicle are relatively small. One way to explore the implications of this 

is to use the binomial distribution, where the investigator says that there are two groups of 

speeds: group 1 (the vehicles traveling below the 85th percentile speed) and group 2 (the 

vehicles traveling above the 85th percentile speed). Thus, assuming an observed 85th percentile 

speed of 85.4 km/h (53.11 mi/h), a vehicle has an 85 percent probability of being in group 1 and 

a 15 percent probability of being in group 2. The investigator can establish 95 percent confidence 

bounds for the binomial distribution using figure 50. 

0.8005 to 0.8995

200

0.85(1 0.85) 0.85 1.96

n

p(1 p) p z = − = ± − ±

Figure 50. Equation. Binomial distribution. 
70 

The interpretation of figure 50 is that while investigators may guess that about 0.8500 of the 

vehicles fall into group 1, they are 95 percent confident that at least 0.8005 of the vehicles fall 

into this group and that no more than 0.8995 of the vehicles fall into this group. The investigators 

can then map each of these proportions shown in figure 50 back to the normal distribution. For 

example, the researcher finds that the value on the normal distribution curve corresponding to a 

cumulative frequency of 0.8005 of the vehicles (assuming a mean of 80.5 km/h (50 mi/h) and a 

standard deviation of 4.8 km/h (3 mi/h)) is 84.5 km/h (52.53 mi/h). Similarly, the investigator 

finds that the value on the normal distribution curve corresponding to a cumulative frequency of 

0.8995 of the vehicles is 86.6 km/h (53.84 mi/h). Thus, the 95 percent confidence bounds are 

84.5 km/h (52.53 mi/h) to 86.6 km/h (53.84 mi/h) and shaded in table 33 for a sample size of 

200. 

It is apparent that the bounds from the binomial assumption (84.5 km/h (52.53 mi/h) to 86.6 

km/h (53.84 mi/h)) are wider than the range based on the normal assumption in figure 49, 

presuming a sample size of n = 200. Table 33 shows that as n increases, the range given by the 

normal assumption and the binomial assumption become very similar. 

Table 33. 95 Percent confidence intervals for the 85th percentile speed.* 

n = 30 n = 200 n = 2000 

Lower 

Bound 

Upper 

Bound 

Lower 

Bound 

Upper 

Bound 

Lower 

Bound 

Upper 

Bound 

Normal 

Assumption 

52.04 54.18 52.69 53.53 52.98 53.24 

Binomial 

Assumption 

51.77 56.03 52.53 53.84 52.91 53.32 

*Assumes a mean speed of 80.5 km/h (50 mi/h) and a standard deviation of 4.8 km/h (3 mi/h). 

Examination of table 33 shows that if the “binomial assumption” based on figure 52 is correct, 

yet the investigator uses the normal assumption as was done in this study, then the error the 

investigator may make is to assume significant differences when, in fact, such differences are not 

significant. Fortunately, this study, which used the normal assumption, tended to not find 

significant differences in the 85th percentile speed, as shown in tables 9 and 11, except in the 

cases of Idaho where the p value was an extremely low 0.000. Thus, it appears that despite some 

theoretical imperfections in consideration of the 85th percentile speed, the use of the analysis of 

variance to detect significant differences was practically appropriate. 
71 

APPENDIX G: EXAMINATION OF THE EFFECTS OF ADT ON TOTAL CRASH 

RATES 

Table 14 showed that while the removal of sites with extremely high and low ADT affects the 

results of the statistical tests, the relationship between ADT and crashes is still unclear. 

Researchers may speculate that various types of relationships exist. For example, an investigator 

could argue that a high ADT/low speed situation results in more vehicle interaction, which 

would therefore increase the likelihood of a property damage only crash, but lessen the 

likelihood of a serious crash. 

Histograms of ADT Versus Total Crash Rate 

The data employed in these histograms were the total crash rates and related ADTs from two 

States: Arizona and Virginia. The highest 5 percent and lowest 5 percent of ADTs were removed 

from the data set, because they were regarded as extreme conditions. Figures 51–53 relate ADT 

to crash rates for Arizona (where no speed limit change was instituted), the Virginia data when 

DSL was in place, and the Virginia data where USL was in place. 

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

0

4,000

8,000

12,00016,00020,00024,00028,00032,00036,00040,00044,00048,00052,00056,00060,00064,000

ADT

Total Crash Rate

Figure 51. Chart. Arizona total crash rate versus ADT. 
72 

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

0

4,000

8,000

12,000

16,000

20,000

24,000

28,000

32,000

36,000

ADT

Total Crash Rate

Figure 52. Chart. Virginia total crash rate versus ADT 

(DSL in place). 

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

160.0

0

4,000

8,000

12,000

16,000

20,000

24,000

28,000

32,00036,000

ADT

Total Crash Rate

Figure 53. Chart. Virginia total crash rate versus ADT 

(USL in place).

Figure 53 suggests that, in Arizona, as the ADT increases, the total crash rate decreases, which 

would in fact support a β2 ADT exponent less than 1.0. However, figures 52 and 53 do not show 

a comparable trend. 

Two-Way Analysis of Variance 

The effect of two independent variables (ADT and speed limit type) was assessed on the total 

crash rate for both Arizona and Virginia, as illustrated in table 34 and the ANOVA results in 

tables 35 and 36 for Arizona and Virginia, respectively. In table 35, no significant difference was 
73 

found between the before and after period, noting the p values of 0.137 and 0.129 for Arizona 

and Virginia. However, ADT did have a significant influence on total crash rates in both States. 

Meanwhile, the interaction of these two variables was significant in Arizona but not in Virginia. 

Table 34. ANOVA variable definitions. 

Variable State Level Definition Note 

Virginia 1 1991–1993 Year 

2 1995–1999 Year 

1 1991–1995 Year 

Speed Limits 

Arizona 

2 1996–1999 Year 

1 0–14,999 ADT Value 

2 15,000–27,499 ADT Value 

Virginia 

3 27,500–39,999 ADT Value 

1 0–14,999 ADT Value 

2 15,000–27,499 ADT Value 

3 27,500–39,999 ADT Value 

4 40,000–52,499 ADT Value 

ADT 

Arizona 

5 52,500–65,000 ADT Value 

Table 35. ANOVA Arizona results. 

Source Type III Sum 

of Squares df Mean Square F Significance 

Corrected 

Model 

692,674.348a 9 76,963.816 22.487 0.000

Intercept 365,693.055 1 365,693.055 106.848 0.000

Speed Limit 

Type (DSL or 

USL) 

7,570.345 1 7,570.345 2.212 0.137

ADT 480,476.984 4 120,199.246 35.096 0.000

Speed Limit 

Type (ADT) 

36,564.203 4 9,141.051 2.671 0.031

Error 8,529,027.298 2,492 3,422.563 — — 

Total 18,507,548.500 2,502 — — — 

Corrected 

Total 

9,221,701.646 2,501 — — — 
74 

Table 36. ANOVA Virginia results. 

Source Type III Sum 

of Squares df Mean Square F Significance 

Corrected 

Model 

59,074.774a 5 11,814.955 9.438 0.000

Intercept 1,034,578.071 1 1,034,578.071 826.446 0.000

Speed Limit 

Type (DSL or 

USL) 

2,891.319 1 2,891.319 2.310 0.129

ADT 45,023.284 2 22,511.642 17.983 0.000

Speed Limit 

Type (ADT) 

746.468 2 373.234 0.298 0.742

Error 2,121,868.836 1,695 1,251.840 — — 

Total 6,480,839.204 1,701 — — — 

Corrected 

Total 

2,180,943.611 1,700 — — — 
75 

APPENDIX H. EXAMPLE APPLICATION OF THE CRASH ESTIMATION MODEL 

TO THE AFTER DATA

Once the crash estimation model of the form shown in figure 8 has been developed using the 

before data, the next step is to apply the equations in figures 17–25 to determine whether the 

policy change has had a net reduction in the expected number of crashes. Appendix H illustrates 

this procedure, using a single site as an example. 

In 1994, Virginia changed speed limits on rural interstate highways from a differential to a 

uniform limit. For this illustration, suppose that the investigator has only one site in the group, 

such that the before data are represented by the years 1991, 1992, and 1993. Because of data 

limitations, the only after data that are available are 1995, 1996, and 1997. 

Thus, the investigator begins with the crash estimation model, shown as figure 54, which from 

just one site has been calibrated as shown below, using the Virginia data from 1991–1993. Table 

37 shows the before and after crash and volume data for this site, which measures 7.16 mi in 

length. 

E(m)= 0.02242775* (Length)0.62225 (ADT)0.5480

Figure 54. Equation. Crash estimation model. 

Table 37. Before and after crash data for a single site. 

Before Data After Data 

Year ADT Total 

Crashes Year ADT Total Crashes 

1991 4000 8 1995 4500 8 

1992 4250 11 1996 4650 7 

1993 4300 7 1997 4800 10 

Sum 26 1999 4825.56 5 

 Sum 30 

Estimation of Expected Crash Frequency M1, M2,... My for the Before Period 

 

The crash estimation model shown above is used with the data in table 37 to calculate the 

E(m1,y), that is, the mean of the estimated crash frequency of site i for each year, as shown in 

figures 55–57. (Normally, i will range from 1 to the number of sites (e.g., for Virginia, with 266 

sites, there would be equations with i = 1, i = 2, ... i = 266. In this example with only one site, 

however, i will always be 1.)
76 

E(m1,1991)= 0.02242775* (7.16)0.62225 (4000)0.5480 =7.191

Figure 55. Equation. Mean of the estimate for 1991. 

E(m1,1992)= 0.02242775* (7.16)0.62225 (4250)0.5480 =7.434

Figure 56. Equation. Mean of the estimate for 1992.

E(m1,1993)= 0.02242775* (7.16)0.62225 (4300)0.5480 =7.481 

Figure 57. Equation. Mean of the estimate for 1993.

Next, the ratios C1,y which are the ratios of E(m1,y) to E(m1,1) for each before year y, are 

calculated using the form of figure 58 and applied in figures 59–61. 

( )

( )

, ,1

,

i

i y

E m

E m C i y =

Figure 58. Equation. Calculation for ratio before year y. 

C1,1991= E(m1,1991)/ E(m1,1991) = 1 

Figure 59. Equation. Ratio before year 1991. 

C1,1992= E(m1,1992)/ E(m1,1991)= 1.033782 

Figure 60. Equation. Ratio before year 1992.

C1,1993= E(m1,1993)/ E(m1,1991)= 1.040429 

Figure 61. Equation. Ratio before year 1993. 

The next step is to calculate the expected crash counts mi,y on this site for each before year with 

their variance VAR(mi,y) using the equations in the figures below. 

∑

∑

=

=

+

+

= Y

y

i y E mi

k

Y

y

i y

C

k K

mi

1

, ) ,1 (

1

,

,1

Figure 62. Equation. Expected crash counts. 
77 

∑ ∑

∑

= =

=

+ +

+

= = Y

y

i y E mi

k

i

Y

y

i y E mi

k

Y

y

i y

C

m

C

k K

VAR mi

1

, ) ,1 (

,1

2

1

, ) ,1 (

1

,

( )

,1 ( )

Figure 63. Equation. Variance of the expected crash counts for year 1. 

mi,y= Ci,y mi,1

Figure 64. Equation. Expected crash counts. 

VAR(mi,y) = Ci,y

2

 VAR(mi,1) 

Figure 65. Equation. Variance of expected crash counts. 

The application of these four expressions is shown in the equations in figures 66–71. 

m1,1991=(5.9+26)/(5.9/7.191+3.074211)= 8.190599 

Figure 66. Equation. Application for 1991. 

VAR(m1,1991)=8.190599/(5.9/7.191+3.074211)= 2.103007

Figure 67. Equation. Application for variance 1991. 

m1,1992= C1,1992 m1,1991=1.033782*8.190599=8.467292 

Figure 68. Equation. Application for 1992. 

VAR(m1,1992)= C1,1992

2

* VAR(m1,1991)= 1.033782^2*2.103007=2.247493 

Figure 69. Equation. Application for variance 1992. 

m1,1993= C1,1993 m1,1991=1.040429*8.190599=8.521739 

Figure 70. Equation. Application for 1993. 

VAR(m1,1993)= C1,1993

2

* VAR(m1,1991)= 1.040429^2*2.103007=2.27649 

Figure 71. Equation. Application for variance for 1993. 

These results are summarized in table 38. 
78 

Table 38. Estimation results for the before years. 

Year ADT Total 

Crashes E(m1,y) C1, y m1,y VAR(m1,y) 

1991 4000.000 8 7.191 1.0 8.190599 2.103007 

1992 4250.000 11 7.434 1.033782 8.467292 2.247493 

1993 4300.000 7 7.481 1.040429 8.521739 2.27649 

Sum 26 3.074211 

Prediction of My+1, My+2,...My+Z for the After Period 

The next step is to use the crash estimation model from figure 56 to compute the mean of the 

expected would-have-been crash frequency for the after period years. That is, even though 

Virginia changed from a differential to a uniform limit in 1994, there question remains of “what 

would the mean of the expected crash frequency have been had Virginia not changed its speed 

limit policy.” Computation of these E(m1,y) values for the after period, as shown in figure 72 and 

73, answers this question. 

E(m1,1995)= 0.02242775* (7.16)0.62225 (4500)0.5480 = 7.670 

Figure 72. Equation. Computation of E(m1,1995). 

E(m1,1996)= 0.02242775* (7.16)0.62225 (4650)0.5480 = 7.809 

Figure 73. Equation. Computation of E(m1,1996). 

This process is repeated for the 1997 and 1999 years. 

One then computes the ratio Ci,y using figure 60, as illustrated in figures 74 and 75. 

C1,1995= E(m1,1995)/ E(m1,1991)= 1.066677 

Figure 74. Equation. Computation of C1,1995. 

C1,1996= E(m1,1996)/ E(m1,1991)= 1.086018 

Figure 75. Equation. Computation of C1,1996. 

Finally, figures 76 and 77 allow one to predict the would-have-been expected crash frequencies 

for the after years. Application of these methods is shown in figures 78–81. 

mi,y= Ci,y mi,1

Figure 76. Expected crash counts, year y. 
79 

VAR(mi,y) = Ci,y

2

 VAR(mi,1) 

Figure 77. Variance of expected crash counts, year y. 

m1,1995= C1,1995 m1,1991=1.066677*8.190599=8.73672 

Figure 78. Equation. Expected crash counts, year 1995. 

VAR(m1,1995)= C1,1995

2

* VAR(m1,1991)= 1.0666772*2.103007=2.392799 

Figure 79. Variance of expected crash counts, year 1995. 

m1,1996= C1,1996 m1,1991=1.086018*8.190599=8.895134 

Figure 80. Expected crash counts, year 1996. 

VAR(m1,1996)= C1,1996

2

* VAR(m1,1991)= 1.0860182*2.103007=2.480358 

Figure 81. Variance of expected crash counts, year 1996.

These results are summarized in table 39. 

Table 39. Prediction results for the after years. 

Year ADT aKi,y

b

E(m1,y) cC1, y

d

m1,y eVAR(m1,y) m1,y − Ki,y Ki,y /m1,y

1995 4500 8 7.670 1.066677 8.73672 2.392799 0.73672 0.915675 

1996 4650 7 7.809 1.086018 8.895134 2.480358 1.895134 0.786947 

1997 4800 10 7.946 1.105079 9.051255 2.568189 −0.94874 1.104819 

1999 4825.56 5 7.970 1.1083 9.077637 2.583182 4.077637 0.550804 

Sum 30 31.39535 4.366072 35.76075 10.02453 5.760747 3.358246 

Average 7.5 7.849 1.091518 8.940187 2.506132 1.440187 0.839561 a

Ki,y: The actual after crashes for year y. b

E(m1,y): The mean of the expected would-have-been after crashes for year y. c

C1, y: The changing ratio for the would-have-been after crashes. 

d

m1,y: The expected would-have-been after crashes for year y. e

VAR(m1,y): The variance of the expected would-have-been after crashes for year y.

Evaluation of Safety Effects of Changing the Speed Limit for This Particular Site 

The effect of the treatment (that is, changing from a differential speed limit to a uniform speed 

limit) is determined by comparing the actual after crashes Ki,y with the predicted after crashes 

mi,y for each year of the after period. The cumulative differences, shaded in table 40, are 

computed by applying the equations in figures 82–84. 
80 

πi = ∑ m 1 , y =8.73672+8.895134+9.051255+9.077637=35.76075 

Figure 82. Equation. Total would-have-been crashes for a particular site. 

λi = ∑ K 1 , y =8+7+10+5=30 

Figure 83. Equation. Total actual crashes for a particular site. 

πi - λi =35.76075 – 30 = 5.76 

Figure 84. Equation. Safety impact for a particular site. 

Table 40. Evaluation of the treatment for the example site. 

Year Ki,y

Cumulative 

K1,y

m1,y

Cumulative

m1,y

VAR(m1,y) Excess m1,y −

Ki,y

Cumulative

Excess Ki,y /m1,y

1995 8 8 8.73672 8.73672 2.392799 0.73672 0.73672 0.915675

1996 7 15 8.895134 17.63185 2.480358 1.895134 2.631854 0.786947

1997 10 25 9.051255 26.68311 2.568189 −0.94874 1.683114 1.104819

1999 5 30 9.077637 35.76075 2.583182 4.077637 5.760751 0.550804

Average 7.5 8.940187 2.506132 1.440187 0.839561

An interpretation of table 40 is that, over the 4-year after period, the actual number of crashes at 

this site was 5.76 less than the predicted number of crashes that would have resulted had there 

been no change in the speed limit. Alternatively, the investigator could use the equation in figure 

85 to indicate that the speed limit change decreased crashes by approximately 16 percent at the 

site, since the ratio of the actual crashes to “would-have-been crashes had no change occurred” is 

0.84. 

λi /πi = 30/35.76075 = 0.84 = 84% 

Figure 85. Equation. Ratio of actual to would-have-been crashes. 

The investigator can graphically portray these cumulative differences as shown in figure 86. 
81 

0

5

10

15

20

25

30

35

40

1995 1996 1997 1999

cumulative actual crash

cumulative predicted crash

cumulative excess

Crashes

Figure 86. Chart. Cumulative differences, by year, at the example site. 

Evaluation of Safety Effects of Changing the Speed Limit for All Sites 

The example shown above has been applied to only one particular site, but realistically a 

researcher would apply these concepts to all sites (e.g., in Virginia, 266 sites). Thus, using 

similar computations from all 266 sites, the investigator compares (a) the actual after crashes for 

the after period 1995 through 1999 that resulted under the uniform speed limit imposed in 1994, 

and (b) the would-have-been crashes for the after period 1995 through 1999 that would have 

resulted had the speed limit not been changed in 1994. Two basic performance measures, the 

reduction in the expected number of crashes (δ) and the index of effectiveness (θ), are computed 

and tested for statistical significance. 

The equations in figures 87–88 compute, respectively, the would-have-been crashes (those that 

would have occurred had no changes taken place) and the actual crashes that did occur. Since the 

equation in figure 89 shows that the actual number of crashes (λ) is larger than these wouldhave-been crashes (π), the negative value of δ suggests that the speed limit change had an 

adverse impact on safety, and it would have been better not to make the change. This statement, 

however, needs to be tested for statistical significance.

=∑ = 13,365.91

i

π π i

Figure 87. Total would-have-been crashes. 

= ∑ = 15,377

i

λ λi

Figure 88. Total actual crashes. 
82 

δ = π - λ = 13,365.91–15,377= -2011.09 

Figure 89. Safety impact. 

To determine statistical significance, the starting point is the variance as calculated in figure 92. 

The term ΣVar(πi) is obtained by summing the values of VAR(mi,y) for each site and each year. 

Thus, the italicized values shown in the sixth column of table 40 (e.g., 2.39, 2.48, 2.57, and 2.58) 

would be computed for each site and for each year, and with 266 sites and 4 years of data, the 

1,064 values of VAR(mi,y) are summed to equal 4,246.913. Because of statistical properties 

appropriate to the Poisson distribution, the summation of the λi values are equivalent to the 

variance of these values, which is 15,377 as tabulated in figure 89 above. These two items are 

used in figure 90 below to obtain Var(δ), which intuitively may be described as the variation 

associated with the difference between would-have-been crashes and actual crashes. 

Var(δ) = Var(π) + Var(λ) = ΣVar(πi) + ΣVar(λi) = 4,246.91 + 15,377 =19,623.91 

Figure 90. Variance of the difference between would-have-been crashes and actual crashes. 

The standard deviation is thus the square root of this variance in figure 91, such that: 

σ(δ) = {Var(δ)}0.5 =140.0854 

Figure 91. Standard deviation of the difference between would-have-been crashes 

and actual crashes. 

Empirical confidence bounds are thus δ ± 2σ(δ) or –2011 ± 2(140). 

Computation of the index of effectiveness is accomplished via the equation below as: 

θ = (λ/π)/{1+Var(π)/π

2

} 

θ = (15,377/13,365.91)/(1 + 4,246.913/13,365.912

) 

θ = 1.150437, in other words, about a 15 percent increase 

Figure 92. Equation. Computation of the index of effectiveness. 

The variance of θ is given below as: 

Var(θ) = θ2{[var(λ)/λ2]+[ var(π)/π

2

]}/[1+ var(π)/π

2

]

2

Var(θ) = 1.1504372

*{15,377/15,3772+4,246.913/13,365.912}/[1+4,246.913/13,365.912]

 2

Var(θ) = 0.000118 

Figure 93. Equation. Variance of θ. 
83 

The empirical confidence bounds are shown below: 

θ ± 2Var(θ)

0.5

1.15 ± 2(0.000118) 0.5 in absolute units, or 

15% ± 2(0.0118 %)0.5 if expressed as a percentage, or 

15% ± 2(1.08%), or 

12.9% to 17.2% 

Figure 94. Equation. Empirical confidence bounds. 

Thus, the change from DSL to USL in Virginia increased the number of crashes by about 15 

percent, with the full response being that this 15 percent is not a perfect estimator but instead 

should be given as a range, such that the increase was between approximately 12.9 percent and 

17.2 percent according to this application of the empirical Bayes method. Assuming all the other 

factors remained constant except for the change of speed limit from differential to uniform, 

therefore, the value of θ being greater than 1.0 means that the change had an adverse impact on 

safety, as reflected in the number of crashes on rural interstates in Virginia. 

85 

ACKNOWLEDGMENTS 

The investigators gratefully acknowledge the many persons who provided data, insights, 

assistance, and other contributions to this effort. Any person who has worked with State crash 

and speed data will recognize that obtaining these data, especially data from as far back as a 

dozen years ago, is no small undertaking. Thus, assistance from K. Bonds (AR), J. Williams 

(AZ), C. Haack and J. Benton (CA), N. Brizzi (ID), L. Rust (IN), W. Baldwin, G. Groat, M. 

Pawlovich, and T. Welch (IA), J. Schaefer (MO), P. Jomini and V. Gonzales (MT), S. Cosper 

and A. Wyatt (NC), L. Evans, D. Gardner, J. Hausman, and S. Shokouhi (OH), S. Reed (OR), R. 

Collins and M. Moore (TX), B. Siddiqui (UT), T. Rickman and K. Tanner (WA), J. Legg, R. 

Lewis, and R. Russell (WV) proved essential to this study. Insights from D. Lord and E. Hauer 

also proved essential for application of the empirical Bayes method. Finally, the team also could 

not have accomplished this work without the assistance of L. Woodson (VA), whose expertise in 

data reduction was needed. Inclusion of the above names and States does not imply, however, 

agreement with the results of this study.

87 

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