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U.S. Department of

Transportation

Federal Railroad

Administration

A New Model for Highway-Rail Grade Crossing

Accident Prediction and Severity

Office of Research,

Development,

and Technology

Washington, DC 20590

DOT/FRA/ORD-20/40 Final Report

October 2020

NOTICE

This document is disseminated under the sponsorship of the

Department of Transportation in the interest of information exchange.

The United States Government assumes no liability for its contents or

use thereof. Any opinions, findings and conclusions, or

recommendations expressed in this material do not necessarily reflect

the views or policies of the United States Government, nor does

mention of trade names, commercial products, or organizations imply

endorsement by the United States Government. The United States

Government assumes no liability for the content or use of the material

contained in this document.

NOTICE

The United States Government does not endorse products or

manufacturers. Trade or manufacturers’ names appear herein solely

because they are considered essential to the objective of this report.

i

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Final Report 10/1/2019 – 7/31/2020

4. TITLE AND SUBTITLE

New Model for Highway-Rail Grade Crossing Accident Prediction and Severity

5. FUNDING NUMBERS

6. AUTHOR(S)

Daniel Brod 0000-0002-1213-0908 -, David Gillen 0000-0001-7047-2669

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

Decisiontek, LLC

11821 Parklawn Drive, Ste. 260

Rockville, MD 20852

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REPORT NUMBER

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

U.S. Department of Transportation

Federal Railroad Administration

Office of Railroad Policy and Development

Office of Research, Development, and Technology

Washington, DC 20590

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AGENCY REPORT NUMBER

DOT/FRA/ORD-20/40

11. SUPPLEMENTARY NOTES

Project Manager: Karen McClure

12a. DISTRIBUTION/AVAILABILITY STATEMENT

This document is available to the public through the FRA website.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

The U.S. Department of Transportation Accident Prediction and Severity (APS) model has been used by Federal, State, and local

authorities to assess accident risk at highway-rail grade crossings since the late 1980s. The Federal Railroad Administration

funded research for the development of a new model that employs current consensus analysis methods and recent data trends. The

new model also seeks to address several limitations of the current APS model and to provide a more robust tool for analysts.

This report presents the stages of the new model development, the statistical estimation of the new model, and validations

comparing the performance of the new model with the APS. The research shows that the new model described here out-performs

the APS by multiple measures.

The new model will support grade crossing management by enabling: more accurate risk ranking of grade crossings, more rational

allocation of resources for public safety improvements at grade crossings, and the ability to assess the statistical significance of

variances in the measured risk at grade crossings.

14. SUBJECT TERMS

Highway-rail grade crossing; risk assessment; accident rates; accident severity; accident types;

high-speed rail; highway safety; highway travel; highways; improvements; highway vehicles;

railroad crashes; railroad safety; railroad trains

15. NUMBER OF PAGES

62

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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. 239-18

298-102

ii

METRIC/ENGLISH CONVERSION FACTORS

ENGLISH TO METRIC

METRIC TO ENGLISH

LENGTH (APPROXIMATE) LENGTH (APPROXIMATE)

1 inch (in) = 2.5 centimeters (cm) 1 millimeter (mm) =

0.04 inch (in)

1 foot (ft) = 30 centimeters (cm) 1 centimeter (cm) =

0.4 inch (in)

1 yard (yd) = 0.9 meter (m) 1 meter (m) =

3.3 feet (ft)

1 mile (mi)

=

1.6 kilometers (km)

1 meter (m)

=

1.1 yards (yd)

1 kilometer (km) =

0.6 mile (mi)

AREA

(APPROXIMATE)

AREA

(APPROXIMATE)

1 square inch (sq in, in2) = 6.5 square centimeters (cm2) 1 square centimeter (cm2) =

0.16 square inch (sq in, in2)

1 square foot (sq ft, ft2) = 0.09 square meter (m2) 1 square meter (m2) =

1.2 square yards (sq yd, yd2)

1 square yard (sq yd, yd2) = 0.8 square meter (m2) 1 square kilometer (km2) =

0.4 square mile (sq mi, mi2)

1 square mile (sq mi, mi2) = 2.6 square kilometers (km2) 10,000 square meters (m2) =

1 hectare (ha) = 2.5 acres

1 acre = 0.4 hectare (he)

=

4,000 square meters (m

2

)

MASS - WEIGHT

(APPROXIMATE)

MASS - WEIGHT

(APPROXIMATE)

1 ounce (oz) = 28 grams (gm) 1 gram (gm) =

0.036 ounce (oz)

1 pound (lb)

=

0.45 kilogram (kg)

1 kilogram (kg)

=

2.2 pounds (lb)

1 short ton = 2,000 pounds

(lb)

= 0.9 tonne (t) 1 tonne (t) =

=

1,000 kilograms (kg)

1.1 short tons

VOLUME (APPROXIMATE) VOLUME (APPROXIMATE)

1 teaspoon (tsp) = 5 milliliters (ml) 1 milliliter (ml) =

0.03 fluid ounce (fl oz)

1 tablespoon (tbsp) = 15 milliliters (ml) 1 liter (l) =

2.1 pints (pt)

1 fluid ounce (fl oz) = 30 milliliters (ml) 1 liter (l) =

1.06 quarts (qt)

1 cup (c)

=

0.24 liter (l)

1 liter (l)

=

0.26 gallon (gal)

1 pint (pt) = 0.47 liter (l)

1 quart (qt) = 0.96 liter (l)

1 gallon (gal) = 3.8 liters (l)

1 cubic foot (cu ft, ft3) = 0.03 cubic meter (m3) 1 cubic meter (m3) =

36 cubic feet (cu ft, ft3)

1 cubic yard (cu yd, yd

3

)

=

0.76 cubic meter (m

3

)

1 cubic meter (m

3

)

=

1.3 cubic yards (cu yd, yd

3

)

TEMPERATURE

(EXACT)

TEMPERATURE

(EXACT)

[(x-32)(5/9)]

°

F =

y

°

C [(9/5) y + 32]

°

C

=

x

°

F

QUICK INCH - CENTIMETER LENGTH CONVERSION

1

02 3 4 5

Inches

Centimeters

0 1 3 452 6 1110987 1312

QUICK FAHRENHEIT - CELSIUS TEMPERATURE CONVERSION

-40° -22° -4° 14° 32° 50° 68° 86° 104° 122° 140° 158° 176° 194° 212°

°F

°C -40° -30° -20° -10° 0° 10° 20° 30° 40° 50° 60°

70° 80° 90° 100°

For more exact and or other conversion factors, see NIST Miscellaneous Publication 286, Units of Weights and

Measures. Price $2.50 SD Catalog No. C13 10286 Updated 6/17/98

iii

Acknowledgements

The authors appreciate the insightful review comments by Federal Railroad Administration

(FRA) staff Karen McClure and Francesco Bedini. We also acknowledge the ongoing support of

Sam Al-Ibrahim of FRA’s Office of Train Control and Communication.

iv

Contents

Executive Summary ............................................................................................................................... 1

1. Introduction ........................................................................................................................ 3

1.1 Background ........................................................................................................................ 3

1.2 Objectives .......................................................................................................................... 7

1.3 Overall Approach .............................................................................................................. 7

1.4 Scope ................................................................................................................................. 9

1.5 Organization of the Report ................................................................................................ 9

2. Preliminary Data Review ................................................................................................. 11

2.1 Risk by Warning Device Types ....................................................................................... 11

3. Data Selection and Analysis ............................................................................................ 18

3.1 Data Sources .................................................................................................................... 18

3.2 Data Selection .................................................................................................................. 18

3.3 Candidate Variables ......................................................................................................... 20

4. Accident Prediction Model .............................................................................................. 25

4.1 The Accident Prediction Model ....................................................................................... 26

4.2 ZINB Regression ............................................................................................................. 27

4.3 Predicting Accidents from the Regression Outputs ......................................................... 30

4.4 Empirical Bayes Prediction Adjustment .......................................................................... 30

5. Accident Severity Model ................................................................................................. 33

5.1 Description of the Data .................................................................................................... 33

5.2 The Accident Severity Model .......................................................................................... 33

5.3 Multinomial Logistic Regression .................................................................................... 35

5.4 Accident Severity Forecast Equations ............................................................................. 36

6. Validation ........................................................................................................................ 39

6.1 Accident Prediction – Cumulative Risk .......................................................................... 39

6.2 Accident Prediction – Risk at Crossings by Accident Count Groups ............................. 42

6.3 Accident Prediction – Accident Risk for Groups of High-Risk Crossings ...................... 44

6.4 Accident Severity – Model Comparisons ........................................................................ 44

7. Conclusion ....................................................................................................................... 49

8. References ........................................................................................................................ 51

Appendix A. Interpreting Regression Outputs ..................................................................................... 52

Appendix B. Application of the New Model ....................................................................................... 54

v

Figures

Figure 2-1. Accidents by Warning Device Type ........................................................................... 13

Figure 2-2. Risk per Exposure (Accident Rate) by Warning Device Type ................................... 15

Figure 2-3. Relative Risk Levels by Warning Device ................................................................... 16

Figure 3-1. Boxplot of Normalized Crossing Accidents by Grade Crossing Surface Type .......... 22

Figure 3-2. Boxplot of Normalized Crossing Accidents by Grade Crossing Angle ...................... 23

Figure 3-3. Boxplot of Normalized Crossing Accidents by Maximum Timetable Speed Deciles 24

Figure 3-4. Boxplot of Normalized Crossing Accidents by Percent Truck Deciles ...................... 24

Figure 4-1. ZINB Predicted Accidents by Warning Device Type ................................................. 30

Figure 4-2. ZINB+EB Predicted Accidents by Warning Device Type .......................................... 32

Figure 5-1. Severity Predictions for 50 Crossings with the New Model ....................................... 38

Figure 6-1. Model Comparison (2014–2018, all crossings in sample) .......................................... 40

Figure 6-2. Model Comparison (2015–2019, all crossings in sample) .......................................... 40

Figure 6-3. Model Comparison (2014–2018, crossings in sample with non-zero accidents) ........ 41

Figure 6-4. Model Comparison (2015–2019, crossings in sample with non-zero accidents) ........ 41

Figure 6-5. Model Comparison, Accident Counts, and Predictions (2014–2018) ......................... 42

Figure 6-5. Model Comparison, Accident Counts, and Predictions (2014–2018) ......................... 43

Figure 6-6. Model Comparison, Accident Counts, and Predictions (2015–2019) ......................... 43

Figure 6-7. Comparison of Predictions for Riskiest Crossings ...................................................... 44

Figure 6-8 Distribution of Predicted Accident Severities with the New Model ............................ 46

Figure 6-9. Distribution of Predicted Accident Severities with APS ............................................ 46

vi

Tables

Table 1-1. Estimates Required for Different Types of Analysis Focus ........................................... 8

Table 2-1. Warning Device Type Codes and Descriptions ............................................................ 12

Table 3-1. Summary of Data Selection .......................................................................................... 20

Table 3-2 Candidate Variables for Inclusion in the New Model ................................................... 20

Table 3-3 Final Data Selection....................................................................................................... 25

Table 4-1. ZINB Regression Output .............................................................................................. 28

Table 5-1. Accident Severity Multinomial Logistic Regression Output ....................................... 36

Table 6-1. Predicted Severity (Percent of Total) by the New Model and APS ............................. 45

Table 6-2. Summary of Severity Category Skewness by Model ................................................... 47

1

Executive Summary

This report presents research on a new model as an alternative to the U.S. Department of

Transportation grade crossing Accident Prediction and Severity (APS) model, which dates back

to 1986. This report follows the steps in developing the new model, presents the modeling

results, and validates the new model in comparison to the APS.

In the nomenclature of AASHTO’s Highway Safety Manual, the new model is a safety

performance function (SPF). SPFs generate metrics (e.g., predicted accidents by severity type)

indicating safety (or risk, insofar as more safety means less risk) and have been applied to a

range of highway facilities. The SPF approach is applicable to grade crossings, individually and

to aggregated collections (i.e., populations)1, as well.

The new model derives from a policy perspective on grade crossing safety, a review of the data,

statistical analysis, and validation. The authors conclude that the new model outperformed the

APS, and its adoption would result in more accurate risk ranking of grade crossings, more

rational allocation of resources for public safety improvements at grade crossings, and the ability

to assess the statistical significance of variances in the measured risk at grade crossings.

Key Conclusions

The preliminary data review indicates a new model could replace the APS based on the key

drivers of exposure and grade crossing warning device type (i.e., the data show that risk

increases with exposure, and decreases with a more protective warning device type).

There is justification for a single model with warning device type category as a variable rather

than separate models for each of the three warning device type categories.

In the U.S. there are 105,377 grade crossings that are public, not closed, not grade separated, and

that have non-missing, non-erroneous values for exposure and warning device type. From 2014–

2018, there were 8,467 accidents at these grade crossings.

An aggregate analysis of these grade crossings shows that relative to a passive crossing, an

average lights crossing had 73 percent less risk per exposure than a passive crossing. An average

gated crossing had 63 percent less risk per exposure than a lights crossing.

The findings of the above analysis indicate a functional form with exposure, warning device

type, and other grade crossing characteristics.

Model estimation using the zero-inflated negative binomial (ZINB) regression yielded

parameters of the expected sign and magnitude, and had strong statistical significance.

The empirical Bayes (EB) method accounted for accident history while correcting for

“regression to the mean” bias. Adjusted results with EB produced predictions that more closely

track the actual counts than did the APS with its (non-EB) adjustment process for accident

history.

1 The “population” of grade crossings refers to all public grade crossings in the U.S. that are not closed or grade

separated. The analysis sample is a large subset (over 100,000) of all grade crossings.

2

The new model severity component determines the probabilities that an accident will be of one

of three severity types: fatal, injury, or property damage only. The severity component of the

new model was derived using multinomial logistic regression (MNL) on the accidents in a 6-year

period, 2014–2019. In this period there were 11,131 accidents at public crossings. Of these, there

were 9,870 at grade crossings with non-missing, non-erroneous data.

These 9,870 accidents were included in the severity model estimation. The MNL regression

shows that the best results were obtained with explanatory variables: rural or urban, maximum

time table speed, number of daily trains, and whether a crossing had a lights warning device.

Validations indicate the new model outperformed the APS. One of the validations looks at

cumulative risk at crossings, with crossings ordered from greatest to least risk (i.e., accident

count). The riskiest crossings in the data sample include 7,822 accidents at 6,409 crossings in

2014-2018. Applying each model (new and APS) to the data, the new model predicted 4,853.3

accidents (62.0 percent of the actual count) whereas the APS predicted 2360.2 accident (30.2

percent of the actual count).

3

1. Introduction

1.1 Background

1.1.1 About the APS Model

The U.S. Department of Transportation (DOT) Accident Prediction and Severity (APS)2 model

has been used to assess accident risk at highway-rail grade crossings by all levels of government

since the late 1980s. The assessments of accident risk at grade crossings are foundational

information that guide the management of grade crossings, the identification of high-risk

crossings (“hotspots”), and the allocation of resources for improving grade crossing safety.

The APS model was developed in 1986 based on grade crossing and accident data from the

preceding 20 years.

Additional modeling efforts intended to support and supplement the APS were conducted more

recently by the Volpe National Transportation System Center (Volpe) and FRA. Volpe

developed a High-Speed Rail (HSR) Accident Severity Model in 20003 to predict accidents and

their severity by types of traffic on the highway and railroad. In 2005 the FRA published the

final Train Horn Rule (49 CFR 222), which specified “supplementary safety measures” and their

impacts on risk reduction. Such measures include: four-quad gates, median barriers, mountable

curbs, and new technologies like photo enforcement.

Among its enhancements for assessing grade crossing risk, FRA’s GradeDec.Net online tool

gives users access to the HSR Accident Severity Model, and complements the APS model with

the supplementary safety measure impacts from the Train Horn Rule.

While these improvements are notable, a new replacement model for the APS is still required to

ensure that U.S. DOT, State Departments of Transportation, and local governments efficiently utilize

resources for reducing risk at grade crossings.

1.1.2 Grade Crossing Accident Trends and the APS

Grade crossing accidents declined sharply in the 25 years following APS development (from

about 3,000 per year to about 2,000 per year). This reduction was due to a number of factors,

indicating the relationship between grade crossing characteristics and accidents has likely

shifted.

FRA periodically updates the APS normalizing constants4 so that the national aggregate number

of predicted accidents equals the actual number of accidents in the most recently ended calendar

year. While the normalizing constants are applied uniformly within each warning device type

2 Farr (1987) describes the APS.

3 See Mironer, et al. (2000) at https://rosap.ntl.bts.gov/view/dot/8433.

4 See Farr (1987), 3-7.

4

group, they do not account for the many factors influencing accident risk that have changed in

recent years, namely: rail and highway environments, technology, traffic trends, etc.

On the rail side, freight trains are longer, which causes longer block times at crossings. The

expansion of intermodal traffic and the growth of intermodal facilities have led to choke points

on highways in the vicinity of some major intermodal facilities. Longer waits at crossings

contribute to “incentivizing” risky behavior (e.g., driving around lowered gates) by some

highway users. In recent years there has been an uptick in grade crossing accidents.

One would also expect changes in highway user behavior to impact safety at crossings. Trends

toward larger vehicles (e.g., SUVs and light trucks replacing smaller cars) result in slower queue

dispersal at crossings. Changes in traffic mix, increases in number of delivery vehicles, and the

rise of ride-sharing – would all contribute to changes in crossing safety and its prediction based

on characteristics of grade crossings and traffic volumes by mode.

Moreover, since 1986, new technologies and traffic management measures have been deployed at

many crossings, including: constant warning time (CWT) devices, signal pre-emption, and queue

cutters.

1.1.3 APS Limitations

State and local government agencies have alerted the FRA Office of Research and Development

that the APS produces very similar results for a majority of crossings within their jurisdictions,

making it difficult to identify the highest-risk highway-rail grade crossings. Limited variance

among APS-generated assessments is attributed to the predominance of crossings with no

accidents in the preceding 5 years, and similar-site specific characteristics (like traffic counts and

warning devices). New consensus methods of analysis (see the Accident Prediction Model

section) directly address these issues.

The APS includes three separate models for accident prediction – one for each of the three major

grade crossing warning device type categories: passive (signage), flashing lights, and gates.

There is no clear rationale for splitting accident prediction into three separate models, as opposed

to treating the warning device type as a grade crossing characteristic in a single model for all

crossings.

Moreover, the separate models can generate inconsistent outcomes. For example, for some

combinations of grade crossing characteristics, the APS calculates higher risk for crossings with

the same characteristics except for a more protective warning device. It is easy to see how an

analysis of grade crossing risk in a corridor or region could yield results with measures of

relative risk between similar crossings with different warning device types that are highly

suspect.

Similarly, if seeking to estimate the effect of a warning device upgrade (say, from lights to

gates), one could not use the models, segregated by device type category, to estimate risk

reduction. The APS resource allocation procedure is to work around this issue by applying a

crash modification factor (CMF).5 A CMF reduces the risk of the unimproved grade crossing by

a fixed percentage. The workaround uses the CMF-reduced risk result in place of the APS result

for the assessed risk of the improved crossing. The CMF method, while accepted practice, has

5 Farr (1987), p. 11, calls these “effectiveness factors.” The term crash modification factor was adopted later.

5

been critiqued in the safety research literature.6 Regardless, the model should enable

recalculation of the risk at the crossing corresponding to a warning device upgrade without

relying on external methods.

Another limitation of the APS model is that it provides no method to determine if risk measures

at different crossings differ with statistical significance7 (e.g., two crossings with predicted

annual accidents of, say, 0.21 and 0.23, respectively). If the difference in measured risk at two

crossings is not statistically significant, there is no evidentiary basis for treating these crossings

differently (e.g., applying an improvement to one of the crossings and not the other). The APS is

essentially a scoring model where a statistical model is needed (see the example in Appendix B,

Application of the New Model).

1.1.4 Purpose of a New Model

The overarching purpose of a new grade crossing safety model, an alternative to the APS, is to

effect evidence-based safety management of grade crossings. The new grade crossing safety

model should enable users to:

1. Estimate safety and risk at grade crossings.

2. Estimate safety gains due to prospective improvements to crossings and support the

estimation of benefits from these gains.

3. Screen for high risk crossings and develop strategies and programs for safety improvements.

4. Account for statistical significance of differences in measured risk at crossings.

5. Estimate changes in safety at crossings due to changes in some variable value (e.g., growth

of AADT over time).

1.1.5 Policy Perspective of Grade Crossing Safety

Grade crossings are “safety hotspots.” Fatalities in grade crossing accidents numbered 2608 in

2018. While this may seem small in comparison to total U.S. highway fatalities (36,5609 in

2018), fatalities and accidents at grade crossings are highly significant when considering the

amount of highway travel that actually traverses grade crossings.

Transportation agencies at all levels recognize that grade crossings are a significant source of

risk and have been singled-out for special programs and safety countermeasures over the years.

Accident risk at grade crossings is eliminated by closure or grade separation (closure, however,

could possibly re-direct the risk from the closed crossing to other grade crossings). Additional

measures like warning device upgrades, supplementary safety measures, and other engineering

solutions have been shown to significantly reduce risk at grade crossings.

6 See Hauer (2015), 186-188.

7 This is similar to asking whether the risk measures of the two crossings are within the “margin of error.”

8 https://safetydata.fra.dot.gov/OfficeofSafety/publicsite/Query/AccidentByRegionStateCounty.aspx

9 https://cdan.nhtsa.gov/tsftables/National%20Statistics.pdf

6

There is a definitional relationship between risk and exposure. Exposure is a measure of

opportunities for accidents to occur. The exposure10 metric for grade crossing usage is based on

coincident arrivals of trains and highway vehicles at a crossing. It is not surprising to find that

more heavily trafficked grade crossings, in general, have more protection from warning devices.

The analysis in this report examines the relationship between accidents, exposure, and the

principal warning device type categories.11

The current U.S. DOT APS model has three accident prediction models, one for each warning

device type category. For some ranges of input variables, APS calculates higher risk than with a

more protective warning device type. (For example, with exposure of 1,000 and maximum

timetable speed of 79 mph, the APS predicts more accidents at a gated crossing than at a lights-

only crossing.) This should give pause when considering APS predictions in a region or corridor.

If two crossings have similar data with the exception of the warning device type, do we have

confidence in the relative measure of their predicted accidents? Moreover, would proposed

improvements for the corridor or region be allocated to their most effective use? The new model,

based on modern techniques, replaces the three APS models with a single prediction model that

incorporates warning device type category as a variable. Its predictions consistently preserve

relative magnitudes of risk with different warning devices.

Moreover, the APS resource allocation procedure relies on “effectiveness values”12 to account

for risk reduction with a warning device upgrade (in recent years, these have been renamed

“crash modification factors”). The road safety literature indicates that such mixed methods can

result in methodological inconsistencies.13

The assessment of grade crossing risk and the planning and budgeting for improvements are the

sole responsibility of State and local authorities.14 The public authority assessing grade crossing

risk relies on a model like the APS15 and bases management decisions for improvements,

accordingly. The quality of those decisions will rely to a great extent on the quality of the risk

assessment.

The new model developed here as an alternative to APS seeks to address the issue of risk

assessment quality by:

•Relying upon current data, appropriate data analysis, and statistical methods

•Examining the relationship between exposure, warning device type, and other key grade

crossing characteristics

10 Exposure, or exposure to risk, is defined for grade crossings as average annual daily trains times average annual

daily highway vehicles at a crossing. This definition is imperfect because accident risk should consider the

correlation of vehicle arrivals by mode, accounting for both seasonality and diurnal distributions of traffic.

11 The APS is defined in Farr (1987). Warning device type categories are: passive, lights, and gates.

12 Farr (1987) p. 11, Table 3 “Effectiveness Values for Crossing Warning Devices.”

13 See, for example Hauer (2015), Appendix L.

14 Upon request, the owning railroad grants the public authority easement to build and maintain the road that

traverses its track. The railroad bears full responsibility for maintaining warning devices and any equipment within

the grade crossing right-of-way.

15 FRA maintains the APS and provides a web-based version at https://safetydata.fra.dot.gov/webaps/.

7

•Properly accounting for accident history

•Presenting a fully transparent model that allows for: single crossing estimates, estimates

of risk for groups of crossings, and determining whether differences in grade crossing

risk warrant similar or different treatment based on statistical measures.

1.2 Objectives

The objectives of the research are as follows:

•Develop a new model to serve as an alternative to the current U.S. DOT APS.

•Document the full development process of the model.

•Demonstrate that the model satisfies statistical criteria and is practical for practitioner

use.

•Validate the new model by comparing its performance against the APS and actual

accident data.

1.3 Overall Approach

1.3.1 About Safety Performance Functions

Since the late 1990s, there has been substantial progress in consensus methods for developing

safety prediction models. These new approaches are presented in AASHTO’s Highway Safety

Manual.16 In the current mode of thinking, the APS is a type of “safety performance function”

(SPF), which yields a metric indicating the safety of a grade crossing. That metric can be either

the annual expected number of accidents at a crossing or expected accidents by severity type

(e.g., fatal, injury, property damage only – the APS accident severity types).

The SPF is derived in a multi-stage process. The key sources of data for this process are: 1) a set

of traits that characterize the facilities under consideration and 2) the 5-year accident history at

the grade crossings. The database of traits is the U.S. DOT Grade Crossing Inventory System

(GCIS). The database of U.S. DOT Form 57 (a form must be submitted for each highway-rail

crossing accident) captures the grade crossing accident history.

The SPF development involves: First, screen the data in the inventory to eliminate irrelevant or

erroneous data. Second, discover via analysis the functional forms that best describe the data,

and offer hints regarding possible relationships between accidents and traits. Third, derive the

safety model from a suitable statistical estimation procedure. Fourth, adjust the number of

predicted accidents at each crossing to account for the accident history using empirical Bayes

(EB) estimators, which derive from another statistical procedure.

This research covers the development of a new model, namely: the derivation of the SPF, its

validation, and the process for estimating safety risk at grade crossings as an alternative to the

APS.

16 AASHTO (2010).

8

1.3.2 Information the SPF Provides

The model, or SPF, provides estimates of four elements for a given set, or population, of grade

crossings:

1. E[μi], the expected or predicted number of accidents at crossing i

2. σ[μi], the standard deviation of the predicted number of accidents at crossing i

3. E{μ}, the mean of all the μs in a population (all crossings or a subset of crossings)

4. σ{μ}, the standard deviation of all the μs in a population

The following table shows situations for which the above estimates are needed:

Table 1-1. Estimates Required for Different Types of Analysis Focus

Analysis Focus

Average safety E{μ} for subsets of grade

crossings

Safety (μ,

σ

) of specific grade crossings

What is normal for grade crossings with given

traits?

Is the crossing “unsafe” or has unusually

high risk?

How do the E{μ} vary across subsets of crossings

(e.g., by states or region, by device type)?

Can we rank a collection of crossings and

divide into high- and low-risk groupings?

What would be the aggregate effect of making an

improvement over a population of crossings (e.g.,

eliminate humped crossings)?

What might be the safety effect and

benefit of applying some improvement to

a crossing?

Need E{μ} and σ{μ} to answer the questions

Need E[μ] and σ[μ] to answer the

questions

Source: based on Hauer (2015).

The estimate of the standard deviation of the safety metric is needed in the case of specific

crossings in order to determine whether:

• Predicted accidents are different from zero with statistical significance.17

• Safety measures of two crossings are statistically different from one another (i.e., if

crossings A and B, say, have predicted accidents of 0.21 and 0.23, respectively, should

they be treated differently or with different priority on the basis of the evidentiary data).

To achieve an SPF, data about grade crossing characteristics, or traits, need to be cast as

statistical models that explain the accident counts at crossings. In developing a safety model for

crossings, there are two clues that the model needs to exploit:

• The first clue is the characteristics (or traits) of the grade crossing. These traits contain

information regarding the common features of grade crossings that contribute to

accidents.

• The second clue available for developing a safety model is the accident history. Accident

history captures the unique qualities of each crossing contributing to safety and risk.

17 “Statistical significance” means that a relationship between two or more variables is caused by something besides

chance. If the ratio of a crossing’s mean predicted accidents to its standard deviation exceeds a threshold value (e.g.,

1.65) then the predicted accidents is said to be “statistically significant at the (e.g.) 90% level.” This is equivalent to

saying that there is a 10 percent probability of a Type I error (falsely rejecting the null hypothesis).

9

As a general approach, the safety model will account for both clues by first predicting accidents

based on characteristics, and then adjust the outcome to account for accident history.

The principles outlined in this section guided the development of the new model for grade

crossing accident prediction and severity.

1.4 Scope

The analysis of the accident and GCIS data and the development of the new model focused on

methods described in AASHTO’s Highway Safety Manual.18 The approach the project researchers

followed sought to:

•Make best use of their understanding of historical trends, the policy environment, and

practice in using the APS.

•Maximize the number of grade crossings included in the regression analysis.

Researchers did not conduct an exhaustive search of alternative approaches, such as: artificial

intelligence (AI) methods, like “k nearest neighbors” (KNN); methods for “slicing and dicing”

the data into smaller subsets; non-multiplicative (i.e., non-linear in logs) functional forms, etc.

The research team believes that alternative approaches may have merits, but also drawbacks in

comparison with the chosen approach.

The focus of the research was on developing the model. The team recognizes that additional

work is needed to further operationalize the model and provide guidance for use of the new

model by practitioners.19

1.5 Organization of the Report

Section 2 is a preliminary data review. The section discusses well-established relationships (e.g.,

exposure drives risk, upgrading the warning device type at a crossing reduces risk). It concludes

with a generic functional form based on the principal drivers of risk (exposure and warning

device type) and accommodates additional variables as warranted by data analysis and the

estimation process.

Section 3 describes the data selection and data analysis.

Section 4, the Accident Prediction Model, presents the functional form of the new model

accident prediction, its estimation using the zero-inflated negative binomial (ZINB) regression

method, and the application of the EB method. The section concludes with the new model

equations for accident prediction.

Section 5, the Accident Severity Model, presents the accident severity component of the new

model. It describes the multinomial logistic (MNL) regression method used to develop the

model.

18 18 AASHTO (2010).

19 For example, guidance should provide rules for treating missing data or replacing data from the GCIS with more

current or more relevant estimates.

11

2. Preliminary Data Review

In this section, the research team identifies known relationships or well-supported theories

relating accident risk at grade crossings to grade crossing traits.

The team explored whether a single model could internalize warning device types and thus avoid

having separate models for each class of device. A unified model would ensure that a device

upgrade will be accompanied by accurate risk reduction measurements of accidents at grade

crossings. This would eliminate the need for employing a “crash modification factor” (CMF)20

approach to estimate the effect of a device upgrade.

It is intuitively clear, and supported by research21, that upgrading a warning device type to one

that provides a higher level of protection reduces the accident risk at a crossing (given that all

other factors remain the same). That said, it does not follow that a device upgrade is cost-

beneficial or even a cost-effective way to improve safety at a crossing.

There are three warning device type categories: passive, lights, and gates. Within each category,

there are several warning device types with somewhat differing risk characteristics than the main

category. These will be discussed below.

It is also well understood that risk increases with exposure (although not at a uniform rate for

every level of exposure). As one would expect, for a given crossing the greater the exposure and

risk, the more likely it is that a local authority will (in coordination with the owning railroad)

upgrade the warning device. Consequently, nearly all very low-exposure crossings have passive

devices and nearly all very high-exposure crossings have gates. The researchers expected to

observe a high correlation between device type and exposure at crossings.

This section examines the relationship between accidents, exposure, and device types and

concludes with a general functional form for the accident prediction model.

2.1 Risk by Warning Device Types

Table 2-1 shows the warning device codes by super-category (passive, lights, gates) and their

meaning in GCIS.

20 The CMF approach, often based on before-and-after crash studies, provides a factor associated with risk reduction

for a particular safety countermeasure. For example, a CMF of 0.12 means that predicted accidents after applying

the safety countermeasure will equal predicted accidents before such application times one minus the CMF, i.e.,

Aafter=Abefore*(1-CMF).

21 Elvik, R. and Vaa, T. (2004).

12

Table 2-1. Warning Device Type Codes and Descriptions

Code

Description of Warning Device Type

PASSIVE

1

No sign or signal

2

Other signs or signals

3

Stop signs

4

Crossbucks

LIGHTS

5

Non-train-activated special protection

6

Highway traffic signals, wigwags or bells

7

Flashing lights

GATES

8

Gates

9

4-quadrant gates

Figure 2-1 shows the filtered crossings in the inventory grouped by device type category. The

bars indicate the number of crossings with the specified device type having the number of

accidents in the period shown on the x-axis. Note that the y-axis uses a log scale.

13

Figure 2-1. Accidents by Warning Device Type

2.1.1 Aggregate Risk Adjusted for Exposure by Warning Device Type

To support an accident prediction model with exposure and warning device type as core

variables, the research team examined aggregate risk at crossings by warning device type and

accident rates (i.e., accident count divided by exposure).

Accident per exposure is the most common way to express accident rates on a facility.22 Note

that the accidents are for 5 years. The exposure data in GCIS23 are for a typical day. Exposure for

the 5-year period is given by:

22 For example, “Highway Statistics 2018, Federal Highway Administration” gives fatality rates in terms of

“fatalities per 100 million VMT (vehicle-miles traveled).” VMT is the measure of exposure for general highway use.

23 As a caveat, note that the GCIS data are reported by State and local agencies with varying data quality. Moreover,

some data fields are not maintained as vigorously as others. For example, data for warning device type are, for the

most part, current and accurate. Data for the railroad and highway environments at crossings (e.g., AADT, train

traffic) tend to be less current and may be out-of-date.

14

Equation 1. Exposure in the Analysis Period (2014–2018)

where:

xp

Exposure in 5-year period

aadt

Average annual daily traffic

dt

Daily trains at the crossing

300

Number of annual traffic days

5

Number of years

Figure 2-2 shows the crossing risk divided by exposure for each device type category. The data

points (colored purple and orange) show the risk per exposure at each crossing grouped by

warning device type. The risk values are shown at the data points in bold and by the left y-axis.

The bars are the number of crossings in each group and their values are represented by the right

y-axis. Note also the number below the risk value, which is the count of accidents in the period

for each grouping of crossings.

Focusing for now on the orange data points, these represent the largest groupings in each of the

three super-categories: passive, lights, and gates. A lights crossing has 73 percent less risk per

exposure in comparison to the passive crossing. Compared to a lights crossing, the gated

crossings have 63 percent less risk per exposure.

The orange points were singled-out because they represent: 1) the main grouping in the super-

category, and 2) in each, there is a substantial number of crossings and accidents. The “Stop

Signs” category is also sizable and its risk per exposure is not that different than the risk per

exposure of the crossbucks grouping (1.122 vs 1.479; in other words, crossbucks are about 75

percent as risky per exposure as stop signs). Moreover, there are over 10,000 crossings in the

“Stop Signs” category and initial inspection indicates that it will likely be advantageous to merge

the two categories into the “passive” category.

The other warning device type categories within each super-category are somewhat small

samples of crossings and accidents with widely different risk characteristics than the main

grouping. The crossings with codes for these groupings (1, 2, 5, 6, 9) will be omitted from the

analysis. (For accident prediction of these device types, we would use the super-category and

then apply a CFM to scale the risk given the best available information).

15

Figure 2-2. Risk per Exposure (Accident Rate) by Warning Device Type

2.1.2 A Generic Functional Form for Accident Prediction

The following generic functional form follows from the above discussion.

Equation 2. Generic Functional Form for Accident Prediction

NOTES TO FIGURE 2-2

The bars in the chart are the number of grade crossings (shown on the right y-axis) for each

warning device type (shown on the x-axis). In the x-axis labels, the letters in parentheses indicate

the principal warning device type category (P – passive, L – lights, G – gates).

The square markers represent the average number of accidents per exposure at crossings with the

warning device type (shown on the left y-axis). Markers are colored orange for the warning

device type with the largest number of grade crossings in the warning device type category.

Upper number: Accident rate

Lower number: Accidents in sample

16

where:

xp

Exposure (= daily trains * aadt)

x

Other variables (vector)

D2

1 if crossing warning device is lights, 0 otherwise

D3

1 if crossing warning device is gates, 0 otherwise

Note: If D2 = D3 = 0 then the warning device at the crossing is passive

From an understanding of the impacts of exposure and warning device types on accident risk, the

parameter estimates of coefficients from a statistical estimation process would yield the

following:

that is, a crossing with lights warning device has less risk than a crossing with passive device,

and a gates crossing has less risk than a lights crossing. (The “hat” diacritical indicates an

estimated coefficient of the model.)

The following chart shows the relative risk of an example grade crossing for different warning

device types and at different levels of exposure. Note that for very low levels of exposure all

crossings have passive warning devices, and at very high levels of exposure grade crossings are

gated. Grade crossings with lights fall in the middle range of exposure.

Figure 2-3. Relative Risk Levels by Warning Device24

24 The elasticity of risk with respect to exposure (set to a value of 0.35) is drawn from the current APS and

preliminary data analysis. Elasticity is the percent change in one variable (e.g., accident risk) when another variable

(e.g., exposure) varies by 1 percent.

0 >

̂

2

>

̂

3

17

The following sections show how this general form, together with additional model variables,

will combine in the new accident prediction model.

18

3. Data Selection and Analysis

The section describes the process of data selection for the development of the new model that

will serve as an alternative to APS. The goal was to produce a model that defines an SPF for

grade crossings. The first focus was on a model predicting accident occurrence, and later in this

document address accident severity prediction given an accident.

Following data analysis and selection of traits for inclusion in the new model, additional filters

may be applied to the data to account for missing/erroneous values for the new model traits. An

additional consideration that accompanied the data analysis was to retain as many grade

crossings in the dataset for model estimation as practical.

The research team sought variables that were likely to support a model. Since the researchers

proceeded from the assumption that key drivers are represented by exposure and warning device

type, they further assumed that f(x) from Equation 2 in the previous section was linear in its

variables (which were the explanatory variables the team sought to identify for inclusion in the

model).

3.1 Data Sources

The two sources of data for the development of the new model are:

•Grade Crossing Inventory System (GCIS) data. The reference document for the data is

“FRA Instructions for Electronic Submission of U.S. DOT Crossing Inventory Data,

Grade Crossing Inventory System (GCIS), v2.9.0.0, Released: 7/2/2019.” Grade crossing

data updates are electronic submissions of Form FRA F 6180.71 by railroads, transit

agencies, and States. GCIS uses Open Data (OData), a RESTful (REpresentational state

transfer), for data downloads. OData downloads provide a single table that includes all

five parts of the inventory – including header information. The data contain one row for

each grade crossing in the inventory representing the most current data per the submitting

agency’s most recent submission.

•The FRA safety data website provides downloading accident data by year. The accident

data source is Form 6180.57, which railroads submit to FRA following each grade

crossing accident. The Form 6180.57 data download as a single table (in Excel or Access

formats) with each accident represented as a single row in the table. For the analysis,

researchers looked at accidents in the 5-year period 2014–2018.

We downloaded and inserted the data into SQL server database tables. The tables were merged

into a single table with an additional column for total accidents in the period (2014–2018).

3.2 Data Selection

This section describes the process for filtering the data so as to include those crossings that are

the focus of the analysis, while eliminating from analysis those crossings that are not of interest

(e.g., closed or grade separated). Researchers also filtered out data that had missing or erroneous

values for several key analysis variables. Table 3-1 summarizes the data filters along with the

number of crossings, accidents, and number of crossings with accidents remaining after applying

each filter. The team sought to keep the number of grade crossings in the selection as large as

19

possible so that its practical application in prediction would not require an extensive set of rules

to account for missing or erroneous data. For example, if a variable seemed promising for

inclusion, yet only, say, 30 percent of grade crossings had data for the variable – researchers

opted to exclude it.

3.2.1 Public Crossings Only

GCIS identifies public crossings as those having a value of 3 in the TypeXing field. For private

crossings, the roadway is maintained by a private individual or entity. There is no legal

obligation for the road maintainers at private crossings to submit data to GCIS. Each year, on

average, 14 to 15 percent of accidents occur at private crossings. However, the data of crossing

characteristics at private crossings are extremely sparse. Consequently, these have been excluded

from the analysis.

3.2.2 At-Grade Crossings Only

Crossings that are grade separated pose no risk of collision between trains and highway vehicles,

hence these crossings are excluded. The field PosXing with value set to 1 identifies a crossing at-

grade.

3.2.3 Closed Crossings

GCIS identifies closed crossings when the ReasonID (reason for submitting a data update) field

is set to value 16. Crossings with ReasonID = 16 have been eliminated from the analysis. Note

that it may be the case that a closed crossing was subsequently updated for a different reason, in

which case there would be no indicator in GCIS that the crossing was closed.

3.2.4 Missing or Erroneous Values for AADT

Without a value for average annual daily traffic (AADT), risk exposure at the crossing could not

be evaluated (defined as AADT times the number of daily trains). Note that AADT, like other

variables in GCIS, may be out-of-date.

3.2.5 Missing or Erroneous Values for Number of Daily Trains

As with AADT, crossings that have missing or erroneous data for total number of daily trains

have been excluded.

3.2.6 Missing or Erroneous Values for Highway Lanes and Tracks

These two variables are the key descriptors of infrastructure at crossings and may be important

predictors of accidents.

20

Table 3-1. Summary of Data Selection

Filter Criterion

(with previous

filters)

Number of

Crossings

Remaining after

Filter

Total Number of

Accidents 2014-2018

at Remaining

Crossings

Of Remaining

Crossings, Number

with Accidents

None

429,463

10,675

8,814

Public only

266,304

9,147

7,538

At-grade only

220,289

9,110

7,503

Exclude closed

130,107

8,986

7,390

Exclude 0, missing,

erroneous AADT

128,378

8,922

7,334

Exclude 0, missing,

erroneous highway

lanes

127,755

8,895

7,308

Exclude 0, missing,

erroneous daily trains

105,383

8,467

6,944

Exclude 0, missing,

erroneous total tracks

105,362

8,465

6,942

3.3 Candidate Variables

Variables in the GCIS that were considered candidates for explaining accidents are shown in the

table below. Researchers eliminated from the list variables that are already accounted for in the

exposure variable (i.e., trains and AADT) and those that are likely highly correlated with these

variables. Warning devices were also excluded, as the team included these by default in the new

model. The variables are divided into two groups: discrete and continuous.

The analysis assesses whether a variable is a likely candidate for inclusion in the model.

Table 3-2 Candidate Variables for Inclusion in the New Model

Discrete

Continuous

Approach angle

Percent truck

Development type

Passenger train count

Main track?

Hwy speed

Traffic lane type

Max timetable speed

Paved/unpaved

Crossing surface type

Urban/rural

21

Discrete

Continuous

Highway functional class

Advanced warning

3.3.1 Discrete Explanatory Variables

The discrete variables are essentially category variables that indicate a crossing belongs to a

particular category among two or more possibilities. The variables are represented in the data as

integer values. However, there is no ordered relationship among the categories represented by

the integers.

The method for evaluating the discrete variables for inclusion in the model was to consider

crossings with 5-year accident history greater than 0. Researchers then examined a boxplot chart

of accidents normalized for exposure and warning device types25, grouped by the variable by its

different levels. If the boxplot indicated significant variance across groupings (i.e., the groupings

displayed different medians and other measures indicating variance), then the variable would be

considered for inclusion in estimation. If the boxplot displayed no such variance, the team

concluded that the variable did not have a strong impact on accident prediction and would be

excluded.

As an example, the following chart shows the boxplot for the variable of grade crossing surface

type. Researchers aggregated the two categories of “Concrete” and “Concrete and Rubber.” This

variable displays variance across its categories, so it was flagged for inclusion in the new model.

25 “Accidents normalized for exposure and warning device types” means accidents in 5-year history divided by the

product of exposure and a risk factor for the warning device type. The risk factors used were: passive = 1.0, lights =

0.3 and gates = 0.1. These values are based on the analysis of the previous Section.

22

Figure 3-1. Boxplot of Normalized Crossing Accidents by Grade Crossing Surface Type

23

The following chart shows the boxplot for the variable of grade crossing angle. There is very

little variance across the groupings. Consequently, this variable was excluded from the model.

Figure 3-2. Boxplot of Normalized Crossing Accidents by Grade Crossing Angle

Following the review of the discrete variables, it was found that the following variables

warranted inclusion in the model: 1) Crossing surface type, and 2) RuralUrban.

3.3.2 Continuous Explanatory Variables

The grade crossings characteristics that are continuous variables were ordered (i.e., all variable

values are comparable, and if values are different, then one is greater than the other). Each can

assume a range of values, not necessarily integers. However, data specifications typically restrict

the values to integers (e.g., maximum timetable speeds can assume values from 1 to 99).

The method for evaluating the continuous variables for inclusion in the model was to consider

crossings with 5-year accident history greater than 0. Researchers then examined a boxplot chart of

accidents normalized for exposure and warning device types, grouped by the variable for each of its

10 deciles. If the boxplot indicated a good distribution of the variable, and an observed functional

relationship across deciles, then the variable would be considered for inclusion in estimation,

otherwise it was not.

The following chart shows the boxplot for the variable of maximum timetable speed.

There was a clear increasing trend for increasing decile. Consequently, this variable was included in

the model.

24

Figure 3-3. Boxplot of Normalized Crossing Accidents by Maximum Timetable Speed Deciles

The following chart shows the boxplot for the variable of percent truck of highway traffic.

There was no clear relationship that changes over deciles of the variable. Consequently, this

variable was excluded from the model.

Figure 3-4. Boxplot of Normalized Crossing Accidents by Percent Truck Deciles

25

Following the identification of variables for inclusion in the model estimation, researchers

further filtered the remaining crossings to exclude from the regression analysis crossings that

have a) non-standard warning device codes or b) missing or erroneous values for included

explanatory variables.

Table 3-3. Final Data Selection

Filter Criterion (with

previous filters)

Number of

Crossings

Remaining after

Filter

Total Number of

Accidents 2014-2018 at

Remaining Crossings

Of Remaining

Crossings,

Number with

Accidents

Exclude non-standard

warning device codes

(1, 2, 5, 6, 9). See

Section 2.2.1

102,054

8,204

6,743

RuralUrban missing or

erroneous values

101,838

8,187

6,730

XSurfaceIds2 missing

or erroneous values

94,033

7,822

6,409

MaxTtSpd missing or

erroneous values

94,029

7,822

6,409

26

4. Accident Prediction Model

This section presents the selected accident prediction model, its regression with the ZINB

estimation procedure, and the EB adjustment of the ZINB-predicted values.

ZINB is one type of zero-inflated models. It is used for count variables (e.g., accidents) that

exhibit excess zeroes. “Excess zeroes” means that of the many crossings with no accidents in the

preceding 5 years, some of those were crossings effectively had no risk of an accident.

The ZINB model assumes that:

•Each crossing has some non-zero probability of being a no-risk crossing.

•Each crossing has an expected number of annual accidents.

•Accident counts for the population of crossings conform to a negative binomial

distribution (the standard deviation of accidents for the population is greater than the

mean, indicating overdispersion).

ZINB has been adopted in numerous accident studies and is well-suited for the analysis of grade

crossing accidents.

The EB method adjusts the estimate of the expected number of accidents so as to account for

history, and correct for “regression to the mean”26 bias. The equation relies on the ZINB

regression outputs to estimate a weighting factor. The EB-adjusted estimate is a linear

combination of the predicted accidents (from ZINB) and the actual count of accidents. If the

accident history indicates no accidents, then the EB adjustment will adjust the expected value of

accidents downwards toward zero. For crossings with non-zero accident history, EB will adjust

the expected value (usually upward) so that it is closer to the actual count.

R software was used in the model estimation.

4.1 The Accident Prediction Model

Based on the analysis described in the previous sections, the selected accident prediction mode is

shown below. The model has two components: 1) a count model and 2) a zero-inflated model.

Equation 3. The ZINB Count Model

26 “Regression to the mean” basically means that if a variable is extreme the first time you measure it, it will be

closer to the average the next time you measure it. For example, if we randomly selected a crossing that had several

accidents in its 5-year history (that is, a very high risk grade crossing), the next random selection would be a

crossing whose risk was much closer to the mean for all grade crossings.

27

Equation 4. The ZINB Zero-Inflated Model

Equation 5. The ZINB Combined Model

where:

N

CountPredicted

Predicted accidents of count model (data for left-hand side of regression are

counts of accidents at crossings in 5-year period 2014–2018)

P

InflatedZero

The probability that the grade crossing is an “excess zero”

N

Predicted

Predicted accidents after accounting for excess zeroes

lExpo1

Exposure, equal to average annual daily traffic times daily trains

D

2

If warning device type is lights =1, 0 otherwise

D

3

If warning device type is gates =1, 0 otherwise

(note: if both D2 and D3 are zero, then warning device type is passive)

RurUrb

If Rural = 0, if Urban = 1

XSurfID2s

Timber = 1, Asphalt = 2, Asphalt and Timber OR Concrete OR Rubber = 3,

Concrete and Rubber = 4

lMaxTtSpd1

Maximum timetable speed (integer value between 0 and 99)

lAadt1

Average annual daily traffic

lTotalTrains1

Total number of daily trains

1These variables have been transformed as follows: lx = log(1+αx), where x is the original

variable and α is a factor. The factor α was selected so that for the median value of x, ln(1+αx) =

ln(x)

4.2 ZINB Regression

The ZINB regression model has two components: the count model and the zero-inflated model.

The count model is for predicted accidents before considering the probability of excess zeroes.

The zero-inflation model is for estimating the probability of an inflated zero. (An “inflated zero”

is a zero accident count that does not derive from a grade crossing’s traits; rather, it is zero

because the crossing accident risk is effectively 0.) Note that the explanatory variable for the

zero-inflated model is the total number of trains; that is, the fewer trains at a grade crossing the

higher the probability of an excess zero.

The predicted (fitted) values of the model are given by f(x)*(1-g(s)), where f is the count model

(operating on the vector of inputs x for each observation) and g is the zero-inflation model

(operating on the vector of inputs s for each observation).

28

The following table shows the output for the zero-inflated negative binomial regression for the

model in the previous section.

The final set of crossing data used in the regression included 94,029 grade crossings with 7,822

accidents at 6,409 crossings in 2014-2018 (see Table 3-3).

Table 4-1. ZINB Regression Output

Count model (negative binomial with log link)

Variable

Estimate

Std. Error

z-Value

Pr(>|z|)

(p-value)

Confidence

Level

(Intercept)

−8.3592

0.3208

−26.059

< 2e-16

> 99.99

lExpo

0.1902

0.0287

6.638

3.18e-11

> 99.99

D2

–0.2848

0.0481

−5.926

3.10e-09

> 99.99

D3

−0.8577

0.0409

−20.976

< 2e-16

> 99.99

RurUrb

0.3935

0.0316

12.444

< 2e-16

> 99.99

XSurfaceID2s

0.1318

0.0172

7.686

1.52e-14

> 99.99

lMaxTtSpd

0.6876

0.6876

22.702

< 2e-16

> 99.99

lAadt

0.1063

0.1063

3.511

0.000446

> 99.99

Log(θ)

−0.2593

0.0887

−2.925

.003447

> 99.00

Zero-inflated model (negative binomial with log link)

Variable

Estimate

Std. Error

z-Value

Pr(>|z|)

(p-value)

Confidence

Level

(Intercept)

1.1708

0.1900

6.1620

7.19e-10

> 99.99

lTotalTr

−1.0109

0.0845

−11.9610

< 2e-16

> 99.99

Summary Statistics

Log-Likelihood

AIC

–2.462e+04

49260.26

Pearson Residuals

Minimum

1st Quartile

Median

3rd Quartile

Maximum

−0.6559

−0.2742

−0.2072

−0.1504

28.5137

Notes to the regression output:

•The values in the “Estimate” column are estimates of the model coefficients and

correspond to the βs from the count model equation (Equation 3) and γs from the zero-

inflation model equation (Equation 4).

•The column “Std. Error” shows the standard error of the coefficient to the left.

•The “z-value” column is the coefficient divided by the standard error (larger absolute

values of z indicate that the coefficient has greater statistical significance).

•“Pr(>|z|)” is the probability of exceeding the absolute value of the z-value (smaller values

indicate greater statistical significance).

29

•The rightmost column shows the confidence level of the coefficient.

•θ27 is the inverse of the overdispersion parameter (α) of the count model. The estimate of

θ is 0.7716 (and the imputed value of α=1.296). α was expected to be greater than 1.

•AIC is the Akaike Information Criteria for model quality given the dataset.

Key points to note from the regression output:

•The coefficients for lExpo and lAadt have positive signs with expected magnitudes.

•The coefficients for D2 and D3 are negative (i.e., compared to passive devices, lights,

and gates reduce risk). The coefficient of D3 is about three times that of D2, which

conforms to expectations.

•The signs and magnitudes of other coefficients in the count model seem to correspond to

expectations.

•The coefficient of lTotalTr (i.e., total trains) in the zero-inflation model is negative, i.e.,

the probability of an excess zero decreases with the number of trains, as expected.

•All the coefficients have strong statistical significance.28

•The Akaike Information Criterion (AIC)29 is the least value for all tested models.

•The estimated mean and standard deviations for the population are:

oMean: 0.08316

oStandard deviation: 0.21377

Figure 4-1 is a chart of the ZINB predicted values grouped by device type. The vertical lines on

the chart indicate the average log of exposure for each grouping. The horizontal lines on the

chart indicate the average predicted 5-year accidents for each grouping. The vertical line

indicates the average log of exposure for each grouping.

27 θ is the Greek letter “theta.”

28 “Strong statistical significance” for an estimated coefficient means there is a very small probability of falsely

rejecting the null hypothesis (i.e., the hypothesis that the coefficient is actually 0).

29 From Wikipedia: The Akaike information criterion (AIC) is an estimator of out-of-sample prediction error and

thereby relative quality of statistical models for a given set of data. For a statistical model, let k be the number of

estimated parameters in the model. Let L be the maximum value of the likelihood function for the model. Then the

AIC value of the model is the following: AIC = 2k - 2*ln(L)

30

Figure 4-1. ZINB Predicted Accidents by Warning Device Type

4.3 Predicting Accidents from the Regression Outputs

One can apply Equation 3, Equation 4, and Equation 5 above to calculate the predicted accident

at a grade crossing (prior to applying the EB adjust described in the following section). The

predicted accidents are the fitted values (i.e., Ŷ) of the model.

The βs in the equations are the ZINB count model coefficient estimates and the γs are the ZINB

zero-inflated model coefficients estimates.

4.4 Empirical Bayes Prediction Adjustment

The EB adjustment intends to correct the prediction for “regression to the mean” bias while

adjusting the expected value to account for accident history. The process is described in Hauer.30

For each grade crossing, the expected number of accidents is given by:

Equation 6. Empirical Bayes Adjustment

30 E. Hauer, The Art of Regression Modeling in Road Safety, Springer 2015.

31

where:

N

Expected

The adjusted number of predicted accidents

N

Predicted

The number of predicted accidents from the ZINB regression procedure

N

Observed

The number of observed accidents (i.e., count of accidents at the grade crossing)

and the weighting factor w is given by:

Equation 7. EB Weighting Factor

The variance of NPredicted is given by:

Equation 8. Variance of Crossing's Predicted Number of Accidents

where theta, as noted above, is the inverse of the overdispersion parameter α from the ZINB

regression (θ is estimated to be 0.7716).

Note that the underlying assumptions of the model indicate that the accident count data for a

population of crossings is best described by the NB distribution. The overdispersion parameter

describes the overdispersion of data relative to a Poisson distribution (where mean and variance

are assumed equal). R software defines the variance of the count variable as μ+μ2/θ.31 Given this

definition of variance, θ should be less than 1 and greater than 0.

Figure 4-2 shows the predicted values grouped by device type, with this chart showing the

predicted values including the EB adjustment.

Compared to Figure 4-1, this chart shows the predicted values clustered around the values that

represent the accident counts in each grade crossing’s 5-year accident history.

31 Most other software packages (e.g., SAS, Stata, Limdep, SPSS, etc.) define the variance of the count variable as

μ+ α · μ2. R’s θ is equivalent to 1/ α in the other packages. α is the overdispersion parameter of the negative

binomial distribution, as defined in these other packages and most of the academic literature.

32

Figure 4-2. ZINB+EB Predicted Accidents by Warning Device Type

33

5. Accident Severity Model

Grade crossing management in the U.S. considers three severity categories: fatal, injury and

property damage only (PDO). A fatal accident is one with at least one fatality; an injury accident

has at least one injury; and a PDO accident has no injuries or fatalities.

The accident severity model seeks to determine the probabilities of prospective accidents at

grade crossings belonging to each severity category. The process for predicting accident severity

is one of allocating predicted accidents to each severity category. In the APS, there is no process

to calibrate accident severity. Over time, accident severity has been fairly stable: fatal accidents

are about 10 to 12 percent of the total, injury accidents about 27 percent, and PDO accidents

about 61 percent.

The remainder of the section describes the data, the logistic regression process used in the model

estimation, and the model results. Some comparisons of the new model with the APS are

discussed in the next section.

R software was used in the model estimation.

5.1 Description of the Data

Federal law requires filing a Form 57 accident report for each grade crossing accident. The

analysis used the Form 57 report database and GCIS. Researchers examined accidents in the

period 2014–2019 (6 years) during which there were 12,983 accidents. They excluded from the

model estimation process accidents from the following crossings:

•Private crossings

•Crossings where traits were missing data for key explanatory variables.

There were 11,131 accidents at public crossings. Of these, 9,870 contained all the data for key

explanatory variable, and these were included in the model estimation. Of the 9,870 accidents,

1,355 (13.7 percent) were fatal, 2,768 (28.0 percent) were injury accidents, and 5,747 (58.2

percent) were PDO.

These accidents were matched with the grade crossing data from GCIS for each crossing where

an accident occurred.

5.2 The Accident Severity Model

For the accident severity model, the researchers sought to estimate the probabilities that given an

accident, the accident will be one of three types: fatal, injury or PDO. The explanatory variables

for these estimates are grade crossing characteristics. The research sought, therefore, to model

three variables:

Equation 9. Probabilities to Estimate – Fatal

34

Equation 10. Probabilities to Estimate – Injury

Equation 11. Probabilities to Estimate – PDO

keeping in mind the following constraint:

Equation 12. Constraint that Severity Probabilities Sum to 1

One of the statistical methods that is well-suited for this type of problem is multinomial logistic

regression, which is described in the following section. “Multinomial” refers to the fact that the

left-hand side variable of the regression, accident type, can assume more than two values – in

this case, three values (fatal, injury, PDO).

The regression analysis requires the selection of a reference level for the accident type variable,

and the team selected “fatal.” The regression estimates the probability of each of the other two

values (injury, PDO) relative to the reference level (fatal). The fact that the three probabilities

sum to 1 enables one to readily derive forecast equations (see the section below on forecast

equations).

Based on analysis of the data, the following is the new accident severity model, with “fatal”

selected as the reference level of the left-hand side variable for the regression:

Equation 13. Accident Severity Model – Injury Relative to Fatal

Equation 14. Accident Severity Model – PDO Relative to Fatal

35

where:

P(acctype = fatal | A)

The probability of a fatal accident given an accident A

P(acctype = injury | A)

The probability of an injury accident given an accident A

P(acctype = PDO | A)

The probability of a PDO accident given an accident A

lMaxTtSpd

Natural log of the maximum (rail) timetable speed at the crossing

lTrains

Natural log of the total number of daily trains at the crossing

RuralUrban

1 if crossing is in a rural (non-urban) environment, 0 if in urban

D2

Has value 1 if warning device type is lights, 0 otherwise

5.3 Multinomial Logistic Regression

Multinomial logistic regression is used when the dependent variable in question is nominal

(equivalently categorical, meaning that it falls into any one of a set of categories that cannot be

ordered in any meaningful way) and for which there are more than two categories. In this study,

researchers studied the probability that an accident will be one of three accident types.

The problem is one of statistical classification. There is a dependent variable to be predicted that

comes from one of a limited set of accident types. There is, as well, a set of independent

variables (also known as features, explanators, traits, etc.), which are used to predict the

dependent variable. Multinomial logistic regression is a particular solution to classification

problems that uses a linear combination of the observed features and some problem-specific

parameters to estimate the probability of each particular value of the dependent variable.

The multinomial logistic regression output for the model described in the previous section is

shown in Table 5-1.

36

Table 5-1. Accident Severity Multinomial Logistic Regression Output

(with “fatal” as the selected reference level)

Part A – Given accident, probability of an injury accident (relative to a fatal accident)

Variable

Estimate

Std. Error

z-value

Pr(|z|)>0

(p-value)

Confidence

Level

(Intercept)

5.2486

0.35511

14.7803

< 2e-16

> 99.99

lMaxTtSpd

–0.9254

0.09794

–9.4488

< 2e-16

>99.99

lTrains

–0.2833

0.04246

–6.6716

2.53e-11

> 99.99

RuralUrban

–0.2741

0.07289

–3.7604

0.00017

> 99.99

D2

0.4894

0.14104

3.4696

0.000521

> 99.99

Part B – Given accident, probability of a PDO accident (relative to a fatal accident)

Variable

Estimate

Std. Error

z-value

Pr(|z|)>0

(p-value)

Confidence

Level

(Intercept)

6.9571

0.33902

20.5216

< 2e-16

> 99.99

lMaxTtSpd

–1.2313

0.09291

–13.2528

< 2e-16

> 99.99

lTrains

–0.2211

0.03941

–5.6113

2.01e-08

> 99.99

RuralUrban

–0.2409

0.06719

–3.5846

0.000338

> 99.95

D2

0.3305

0.13577

2.4342

0.014925

> 98.50

Part C – Summary Statistics

Residual Deviance

AIC

17986.35

18006.35

Most coefficient estimates exhibit a high level of confidence (greater than 99.99 percent). The

confidence levels for lTrains and RuralUrban are slightly less, but still high (high level of

confidence coincides with a low probability of a Type I error32). The value for the AIC is the

least among all of the variable combinations tested.

Interpreting the coefficient estimates can be tricky. The estimates (except for the intercept and

D2) are negative, indicating that the relative risk of injury or PDO to fatal decreases – which

means that the risk of a fatal accident (the most severe) increases.

5.4 Accident Severity Forecast Equations

Equations 15–17 show the forecast equations for the accident severity model.

Equation 15. Accident Severity Forecast Equations - Fatal

32 A Type I error occurs when rejecting a true null hypothesis.

Pr(

= |) = 1

1 + ∑

∙

3

=1

37

Equation 16. Accident Severity Forecast Equations - Injury

Equation 17. Accident Severity Forecast Equations - PDO

Notes to equations:

•The subscript k indicates accident type: fatal=1, injury=2, PDO=3.

•The subscript i indicates a grade crossing.

•Yi is the variable indicating accident type (fatal, injury or PDO).

•βs are the vectors of coefficient estimates.

•β2 is the coefficient estimate vector for probability of injury accident relative to fatal, and

Part A of Table 5-1 contains the coefficient estimate values.

•β3 is the coefficient estimate vector for probability of PDO accident relative to fatal, and

Part B of Table 5-1 contains the coefficient estimate values.

•Xi is the vector of crossing traits that explain accident severity.

Pr(

= |) =

∙

1 + ∑

∙

3

=1

Pr(

= |) =

∙

1 + ∑

∙

3

=1

38

The following chart shows forecast severity for 50 accidents with the new model:

Figure 5-1. Severity Predictions for 50 Crossings with the New Model

39

6. Validation

The section presents validations for the new model (estimated with the ZINB and EB methods).

Note here that the term “prediction” means the expected value of accidents at the crossing. In

general, accidents are rare and the (annualized) expected value of accidents at a crossing will be

a real value between 0 and 1. A non-zero accident count will be larger in most cases than the

expected value of accidents at a crossing, which reflects the fact that the observed count in a

previous year is not expected to repeat frequently in subsequent years.

The first validation compares cumulative predicted accidents by the new model and the APS

with the actual risk as measured by accident counts.

The second validation shows the predicted accidents for the new model and the APS for

crossings grouped by accident count.

The third comparison examines the model results (the new model and APS) for different

groupings of high-risk crossings and shows the results in a chart. In this case, researchers

counted accidents at the 50 highest-risk crossings (and then at the subsequent groupings of

highest-risk crossings). The better of the two models will predict accidents at the groupings of

crossings that is closer to the actual accident counts.

For the severity model, this report shows comparisons of the model performance with that of the

APS.

6.1 Accident Prediction – Cumulative Risk

For this validation we order the grade crossings from high risk to low risk (according to total

accidents in 5-year history). The y-axis on the charts below shows the actual cumulative risk and

the predicted risk with each model. The better model is the one that tracks closer to the actual

cumulative risk.

The four charts below represent two cases and two periods. The first case displays cumulative

accident count and predictions for all crossings in the estimation sample (which includes 94,029

crossings). The second case focuses on the crossings with non-zero accidents. The first period is

the estimation period 2014–2018. The second period is the following year, which covers 5-year

accidents from 2015–2019.

The vertical line indicates the boundary between those crossings with non-zero accidents in the

period (to the left of the line) and those with zero accidents in the period (to the right of the line).

Figures 6-1 and 6-2 show the counts and predictions, ordered from high to low risk, for the

complete set of crossings in the estimation sample. Figure 6-1 is for the period 2014–2018.

Figure 6-2 is for the period 2015–2019.

Figures 6-3 and 6-4 show the same chart data as Figures 6-1 and 6-2, but limit the data displayed

to those crossings with non-zero accident history.

The charts demonstrate that the new model was the better predictor of accident risk than the

APS.

40

Figure 6-1. Model Comparison (2014–2018, all crossings in sample)

Figure 6-2. Model Comparison (2015–2019, all crossings in sample)

41

Figure 6-3. Model Comparison (2014–2018, crossings in sample with non-zero accidents)

Figure 6-4. Model Comparison (2015–2019, crossings in sample with non-zero accidents)

On the riskiest crossings, the new model (ZINB+EB) predicted cumulative accident risk much

better than APS.

42

6.2 Accident Prediction – Risk at Crossings by Accident Count Groups

In the second validation, researchers grouped the crossings by the number of accidents in the 5-

year history. The chart shows the number of accidents in the grouping on the x-axis

The orange square markers show mean predicted accidents with the APS given traits at the

crossings with the specified accident history (shown on the x-axis). The square blue markers

show mean predicted accidents with the new model. The lines below and above the markers

indicate the 10th and 90th percentiles, respectively. The lines also indicate the bounds of the 80

percent confidence interval of the prediction for crossings in the period.

For example, in Figure 6-5 below (displaying the period 2014–2018) at crossings having three

accidents the new model predicted between 1.6 and 2.0 accidents. The APS predicted 0.5 to 1.4.

The new model better predicted the crashes at crossings for each level of accident risk than the

APS.

Figure 6-6 shows the results for the period 2015–2019.

Figure 6-5. Model Comparison, Accident Counts, and Predictions (2014–2018)

43

Figure 6-6. Model Comparison, Accident Counts, and Predictions (2014–2018)

Figure 6-7. Model Comparison, Accident Counts, and Predictions (2015–2019)

44

6.3 Accident Prediction – Accident Risk for Groups of High-Risk Crossings

The third validation examines the model results (APS and new model) for groupings of high-risk

crossings and shows the results in a chart. The better of the two models will predict accidents at

each grouping of crossings that is closer to the actual accident counts.

Crossings in the estimation sample were ordered by decreasing risk, and then divided into groups

of 50. In the figure below, the x-axis shows groupings 1 to 20 (20 groups of 50 equals total of

1,000). The y-axis shows the actual and predicted crossings by model (new model and APS) for

each grouping.

For each grouping, the new model performed better than the APS. For the top 1,000 high-risk

crossings in 2014–2018 the accident count was 2,578 accidents. The APS predicted 791.3

accidents while the new model predicted 1,518.0 accidents at these 1,000 high-risk crossings.

Figure 6-8. Comparison of Predictions for Riskiest Crossings

6.4 Accident Severity – Model Comparisons

The table below shows the predicted accident severity for all accidents and by each accident type

in the severity estimation sample.

45

Table 6-1. Predicted Severity (Percent of Total) by the New Model and APS

With the new model, the aggregate percentage of accidents of each accident type exactly equaled

the percentages in the sample (as expected). The APS predictions in the aggregate diverged

somewhat from the sample data; for example, APS predicted the percent of fatal accidents to be

half of the actual percentage.

An indicator of the predictive performance of the severity model is to estimate the predicted

percentage of a severity category while only considering those accidents in that category. That

value should well exceed the percentage of a severity category when considering all accidents.

Table 6-1 shows that predicted fatal accidents with the new model increased from a mean of 13.7

percent for all accidents to 18.7 (a 36 percent increase). When considering only accidents that

were actually fatal. The comparable change with APS was 6.9 to 7.2 (a 4 percent increase).

Overall, the new model performed better, with more significant movements in the correct

direction when restricting to accidents of a particular type.

Figures 6-8 and 6-9 below show boxplot charts of predicted accident severities for the new

model and APS.

For Accidents of Severity Type

All Accidents

Fatal

Injury

PDO

New Model

Predictions

Fatal

13.7

18.7

13.1

12.9

Injury

28.4

27.3

28.4

28.0

PDO

58.2

54.0

58.4

59.1

APS

Predictions

Fatal

6.9

7.2

6.9

6.9

Injury

27.3

26.9

27.4

27.4

PDO

65.7

65.9

65.6

65.7

46

Figure 6-9 Distribution of Predicted Accident Severities with the New Model

Figure 6-10. Distribution of Predicted Accident Severities with APS

47

The two charts indicate:

•The mean value from APS for fatal was about half that of the new model, while the

means for injury and PDO accidents were similar.

•The new model had a higher variance for the fatal and PDO categories, with smaller

variance for the injury category. (Standard deviations were 0.08182, 0.02986, and

0.07511 for fatal, injury, and PDO, respectively.)

•APS had a small variance for fatal, somewhat larger for injury, and a bit larger still for

PDO. (Standard deviations were 0.0208, 0.03704, and 0.0455 for fatal, injury, and PDO,

respectively.)

•APS had more and more unbalanced outliers. The injury category skewed downward, and

the PDO category skewed up. The table below shows a summary of the skewness values:

Table 6-2. Summary of Severity Category Skewness by Model

New Model

APS

Fatal

0.3132

0.5734

Injury

0.2782

-0.9523

PDO

0.4544

0.6588

48

7. Conclusion

The preliminary data review indicates that a new model could replace the APS based on the key

drivers of exposure and grade crossing warning device type. In other words, the data show that

risk increases with exposure and more protective warning device type reduces risk.

Other findings include:

•There is justification for a single model with category of warning device type as a

variable rather than separate models for each of the three warning device type categories.

•Grade crossings that are public, not closed, not grade separated, and that have non-

missing, non-erroneous values for exposure and warning device type, number 105,377

nationally. In the period 2014–2018 there were 8,467 accidents at these grade crossings.

•An aggregate analysis of these grade crossings showed that relative to a passive crossing,

a lights crossing had 73 percent less risk per exposure. A gated crossing had 63 percent

less risk per exposure than a lights crossing.

•The findings of the above analysis indicate a functional form with exposure, warning

device type, and other grade crossing characteristics.

•The analysis indicates additional variables that are likely to explain accident occurrence:

grade crossing is in rural or urban area, maximum timetable speed, and grade crossing

surface types.

•Model estimation using ZINB regression yielded parameters of the expected sign and

magnitude, and had strong statistical significance.

•Including the number of daily trains and the AADT at the crossing, which are

components of the exposure metric, improved the regression results as indicated by the

AIC.

•The EB method accounts for accident history while correcting for “regression to the

mean” bias. Adjusted results with EB produced predictions that more closely track the

actual counts than did the APS adjustment process for accident history.

•The new model severity component determined the probabilities that an accident would

be of one of three severity types: fatal, injury or PDO.

•The severity component of the new model was derived using multinomial logistic

regression on the accidents in the 6-year period 2014–2019.

•In the period there were 11,131 accidents at public crossings. Of these, the crossings

where these accidents occurred had non-missing, non-erroneous data for 9,870 grade

crossings. The accidents at these crossings were included in the severity model

estimation.

•The multinomial logistic regression showed that the best results were obtained with

explanatory variables: rural or urban, maximum time table speed, number of daily trains,

and whether a crossing has a lights warning device.

49

•Validations showed that the new model performed better than the APS by multiple

measures.

50

8. References

1. Farr, E.H. (1987). Summary of the DOT Rail-Highway Crossing Resource Allocation

Procedure – Revisited [DOT/FAR/OS-87/05].

2. Hauer, E. (2015). The Art of Regression Modeling in Road Safety. Springer

3. Hauer, E. (2001). Overdispersion in modeling accidents on road sections and in

Empirical Bayes estimation. Accident Analysis and Prevention 33, 799–808.

4. Hauer, E. (2004). Statistical Safety Modeling. Transportation Research Record 1897.

Washington, DC: National Academies Press, 81–87.

5. American Association of State Highway and Transportation Officials. (2010). Highway

Safety Manual, 1st edition.

6. Brod, D., Weisbrod, G., Moses, S.B., Gillen, D., & Martland, C.D. (2013)

Comprehensive Costs of Highway-Rail Grade Crossing Crashes. NCHRP Report #755.

Transportation Research Board.

7. Federal Railroad Administration. (2019). GradeDec.Net Reference Manual.

8. Mironer, M., Coltman, M. & McCown, R. (2000). Assessment of Risks for High-Speed

Rail Grade Crossings on the Empire Corridor [DOT-Volpe-FRA-00-03]. Washington,

DC: U.S. Department of Transportation.

9. Elvik, R. & Vaa, T. (2014). Handbook of Road Safety Measures. Oxford, UK: Elsevier.

10. R Core Team. (2020). R: A language and environment for statistical computing. R

Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/

51

Appendix A. Interpreting Regression Outputs

A regression analysis is a set of statistical processes for estimating the relationships between a

dependent variable and one or more independent variables. A dataset contains a number of

observations for each variable.

The independent variable is often called the left-hand side (LHS) variable because it is written to

the left of the equals sign. The dependent variables (also called explanatories) are the right-hand

side (RHS) variables.

In regression analysis, the analyst develops a model linking the LHS with RHS variables and

“runs” a regression. A statistical program examines the dataset and finds the values of model

coefficients that meet optimization criteria.33

The regression output table contains general statistics along with coefficient estimates and

statistics.

The following describes the columns in the regression output table that relate to the coefficient

estimates:

Column Name

Column Description

Variable

Each row contains the name of a model variable. If the model has a constant,

the row will usually say “constant” or “intercept,” depending upon the

software used.

Estimate

The estimate of the variable model coefficient (in this report, coefficients are

subscripted and shown in model equations as lowercase Greek letters β

(beta) and γ (gamma)

Std. Error

The standard deviation of the coefficient estimate

z-value

This is the estimate divided by the standard error.

Pr(>|z|)

(p-value)

In statistical significance testing, the p-value is the largest probability of

obtaining test results at least as extreme as the results actually observed,

under the assumption that the null hypothesis is correct (i.e., assuming the

coefficient is actually 0). This is equivalent to the probability of falsely

rejecting the null hypothesis (also called a Type I error).

33 The two broad classes of regression techniques are least squares (LS) and maximum likelihood estimation (MLE).

With LS, the regression minimizes the sum of squared residuals (“residuals” are the differences between the LHS

values and the “fitted” calculated values of the model). With MLE, the regression seeks the point of maximum of a

likelihood function that is constructed from all the data observations. The datasets under consideration will usually

determine which technique is most appropriate.

52

Column Name

Column Description

Confidence

Level

This is the confidence level of the parameter estimate. It is one minus the p-

value (i.e., if the p-value is .01, then the confidence level is 0.99 – or, 99.0

percent).

The general statistics include descriptive statistics of the regression and its residuals. This study

examines the AIC, which enables model quality comparison and whose value is least for the

better model specification with the given set of data.

53

Appendix B. Application of the New Model

The APS enables risk ranking of grade crossings (in a corridor or region). However, it cannot

inform when two grade crossings with similar risk scores (e.g., predicted annual accidents)

should be treated the same or differently. The new model provides descriptive statistics of the

population of grade crossings, and these can be used to determine if scores are close enough to

warrant same or different treatment.34

For example, suppose we have two grade crossings A and B, and the new model estimates they

have predicted annual accidents of 0.21 and 0.26, respectively. From the analysis of data in

developing the model, we know that:

1. Mean value of 5-year accidents for the population of grade crossings is E{μ} = 0.08319

2. The variance of 5-year accidents for the population of grade crossings is V{k} =

0.1220627.

3. The standard deviation of 5-year accidents for the population of grade crossings is:

Since the standard deviation is for 5-year accidents, divide by 5 for the standard deviation of

predicted annual accidents:

Crossing A has predicted annual accidents of 0.21, then adding the standard deviation to the

value 0.21 + 0.03945124 = 0.24945124. Crossing B has predicted annual accidents of 0.26,

which is greater than the previous value and outside a band of one standard deviation from the

mean value of predicted annual accidents of A. We would conclude that the predicted annual

accidents of the two crossings differ significantly and, therefore, the two warrant different

treatment based on the new model.

34 Following Hauer (2015) Chapter 2, “A Safety Performance Function for Real Populations.”

54

Abbreviations and Acronyms

AADT

Average Annual Daily Traffic

AASHTO

American Association of State Highway and Transportation Officials

AIC

Akaike Information Criterion (a measure of the relative quality of a model for a given

set of data)

APS

Accident Prediction and Severity

CMF

Crash Modification Factor (a safety countermeasure’s ability to reduce crashes and

crash severity)

CFR

Code of Federal Regulations

CWT

Constant Warning Time (device at grade crossings with active warning devices that

ensures the time between initial warning and a train’s arrival at the roadway is

constant, regardless of the speed of the train)

DOT

Department of Transportation

EB

Empirical Bayes (procedure for statistical inference in which prior distributions are

derived from data)

FRA

Federal Railroad Administration

GCIS

Grade Crossing Inventory System

GX

Grade crossing (used in this document’s figures)

HSR

High-Speed Rail

MLE

Maximum Likelihood Estimation (a class of model estimation procedures)

MNL

Multinomial Logistic (a regression analysis method)

NB

Negative Binomial (a probability distribution)

PDO

Property Damage Only (a severity type of train-highway vehicle accident at a grade

crossing)

SPF

Safety Performance Function (a function for evaluating the safety of a transportation

facility, or population of facilities, from a set of facility traits and accident history)

TRB

Transportation Research Board

Volpe

Volpe National Transportation Systems Center

ZINB

Zero-Inflated Negative Binomial (a regression analysis method)