Prediction of the Expected Safety Performance of Rural TwoLane Highways
3. BASE MODELS
This section of the report presents the base models used as part of the accident prediction algorithm. The base models for roadway segments and atgrade intersection are addressed separately in the following discussion. The base models were developed in separate studies by Vogt and Bared.^{(3,4,5)}The development of the base models and the choices made among alternative models that were considered are presented in appendix B.
Base Model for Roadway Segments
The base model for roadway segments is presented below:
N_{br} =

EXPO exp(0.6409 + 0.1388STATE  0.0846LW  0.0591SW + 0.0668RHR + 0.0084DD)
(SWH_{i} exp(0.0450DEG_{i})) ( WV_{j} exp (0.4652 V_{j}))(SWG_{k} exp(0.1048GR_{k}))


where:
N_{br} 
= 
predicted number of total accidents per year on a particular roadway segment; 
EXPO 
= 
exposure in million vehiclemiles of travel per year = (ADT)(365)(L)(10^{6}); 
ADT 
= 
average daily traffic volume (veh/day) on roadway segment; 
L 
= 
length of roadway segment (mi); 
STATE 
= 
location of roadway segment (0 in Minnesota, 1 in Washington); 
LW 
= 
lane width (ft); average lane width if the two directions of travel differ; 
SW 
= 
shoulder width (ft); average shoulder width if the two directions of travel differ; 
RHR 
= 
roadside hazard rating; this measure takes integer values from 1 to 7 and represents the average level of hazard in the roadside environment along the roadway segment. (For definitions of the roadside hazard rating categories, see appendix D; for the development of the roadside hazard ratings, see Zegeer.^{(6)}); 
DD 
= 
driveway density (driveways per mi) on the roadway segment; 
W_{hi} 
= 
weight factor for the i^{th} horizontal curve in the roadway segment; the proportion of the total roadway segment length represented by the portion of the i^{th} horizontal curve that lies within the segment. (The weights, WH_{i}, must sum to 1.0.); 
DEG_{i} 
= 
degree of curvature for the i^{th} horizontal curve in the roadway segment (degrees per 100 ft); 
WV_{j} 
= 
weight factor for the j^{th} crest vertical curve in the roadway segment; the proportion of the total roadway segment length represented by the portion of the j^{th} crest vertical curve that lies within the segment. (The weights, WV_{j}, must sum to 1.0.); 
V_{j} 
= 
crest vertical curve grade rate for the j^{th} crest vertical curve within the roadway segment in percent change in grade per 31 m (100 ft) = g_{j2}g_{j1}/l_{j}; 
g_{jl'}g_{j2} 
= 
roadway grades at the beginning and end of the j^{th} vertical curve (percent); 
l_{j} 
= 
length of the j^{th} vertical curve (in hundreds of feet); 
WG_{k}

= 
weight factor for the k^{th} straight grade segment; the proportion of the total roadway segment length represented by the portion of the k^{th} straight grade segment that lies within the segment. (The weights, WG_{k}, must sum to 1.0.); and 
GR_{k} 
= 
absolute value of grade for the k^{th} straight grade on the segment (percent). 
This model was developed with negative binomial regression analysis for data from 619 rural twolane highway segments in Minnesota and 712 roadway segments in Washington obtained from the FHWA HSIS. These roadway segments including approximately 1,130 km (700 mi) of twolane roadways in Minnesota and 850 km (530 mi) of roadways in Washington. The database available for model development included 5 years of accident data (19851989) for each roadway segment in Minnesota and 3 years of accident data (19931995) for each roadway segment in Washington. The model predicts the total nonintersection accident frequency for any roadway segment for which the independent variables shown in equation (5) are known. The model predictions are reliable only within the ranges of independent variables for which data were available in the database used to develop the model (see table 30 in appendix B).
When the accident prediction model is employed to predict the expected accident experience for any specified roadway section, equation (5) is used in the following manner:
 The exposure variable (EXPO) in million vehiclemiles of travel is computed using the actual ADT and segment length (L) for the roadway section and a duration of 1 year (365 days). This assures the accident frequency predicted by the base model has units of accidents per year.
 The STATE variable in base model is set equal to zero, representing Minnesota conditions. This is done for consistency with the base models for three and fourleg STOPcontrolled intersections, both of which are based solely on Minnesota data. It should be noted that the calibration procedure described later in this chapter can be used to adapt the base models to the safety conditions of any State other than Minnesota. Calibration would even be desirable to apply the algorithm in Minnesota to a time period other than the period for which the base models were developed.
 The remaining variables in the model are set to the following nominal or base conditions:
Lane width (LW) 
3.6 m (12 ft) 
Shoulder width (SW) 
1.8 m (6 ft) 
Roadside hazard rating (RHR) 
3 
Driveway density (DD) 
3 driveways per km
(5 driveways per mi) 
Horizontal curvature 
None 
Vertical curvature 
None 
Grade 
Level (0 percent) 
With the default values given above, the base model in equation (5) reduces to:
N_{br} = (ADT) (L) (365) (10^{6}) exp(0.4865)


Base Models for AtGrade Intersections
Base models have been developed for three types of atgrade intersections on rural twolane highways. These are:
 Threeleg intersections with STOP control on the minorroad approach.
 Fourleg intersections with STOP control on the minorroad approach.
 Fourleg signalized intersections.
The base models for each of these intersection types predict total accident frequency per year for intersectionrelated accidents within 76 m (250 ft) of a particular intersection.
These models address intersections that have only two lanes on both the major and minorroad legs. Intersections on multilane highways or intersections between a twolane highway and a multilane highway may be addressed in a future improvement to the accident prediction algorithm. The base models for each of the three intersection types are presented below.
ThreeLeg STOPControlled Intersections
The base model for threeleg intersections with STOP control on the minorroad leg is presented below:
N_{bi} = exp(11.28 + 0.79ln ADT_{1} + 0.49ln ADT_{2} + 0.19RHRI + 0.28RT)


where:
ADT_{1} 
= 
average daily traffic volume (veh/day) on the major road; 
ADT_{2} 
= 
average daily traffic volume (veh/day) on the minor road; 
RHRI 
= 
roadside hazard rating within 76 m (250 ft) of the intersection on the major road [see description of the variable RHR in equation (5)]; and 
RT 
=

presence of rightturn lane on the major road (0 = no rightturn lane present; 1 = rightturn lane present). 
This model was developed with negative binominal regression analysis from data for 382 threeleg STOPcontrolled intersections in Minnesota. The data base available for model development included 5 years of accident data (19851989) for each intersection. The model predicts the total intersectionrelated accident frequency for any threeleg STOPcontrolled intersection for which the independent variables shown in equation (7) are known. The model predictions are reliable only within the ranges of independent variables for which data were available in the data base used to develop the model (see table 31 in appendix B).
When the accident prediction model is employed to predict the expected accident frequency for any specified threeleg STOPcontrolled intersection on a twolane highway, equation (7) is used in the following manner:
 The traffic volume variables (ADT_{1} and ADT_{2}) are set equal to the actual ADTs of the major and minorroad legs. If the ADTs differ between the two majorroad legs, they should be averaged.
 The remaining variables in the model should be set equal to the following nominal or base conditions:
Roadside hazard rating (RHRI) 
2 
Presence of rightturn lane
on the major road (RT) 
None present (0) 
With the default values of given above, the base model in equation (7) reduces to:
N_{bi} = exp(10.9 + 0.79ln ADT_{1} + 0.49ln ADT_{2})


FourLeg STOPControlled Intersections
The base model for fourleg intersections with STOP control is presented below:
N_{bi} = exp(9.34 + 0.60ln ADT_{1} + 0.61ln ADT_{2} + 0.13 ND_{1}  0.0054SKEW_{4})


where:
ND_{1} 
= 
number of driveways on the majorroad legs within 76 m (250 ft) of the intersection; and 
SKEW_{4} 
= 
intersection angle (degrees) expressed as onehalf of the angle to the right minus onehalf of the angle to the left for the angles between the majorroad leg in the direction of increasing stations and the right and left legs, respectively. 
This model was developed with negative binominal regression from data for 324 fourleg STOPcontrolled intersections in Minnesota. The database available for model development included 5 years of accident data (19851989) for each intersection. The model predicts the total intersectionrelated accident frequency for any fourleg STOPcontrolled intersection for which the independent variables shown in equation (9) are known. The model predictions are reliable only within the ranges of independent variables for which data were available in the database used to develop the model (see table 38 in appendix B).
When the accident prediction model is employed to predict the expected accident frequency for any specified fourleg STOPcontrolled intersection on a twolane highway, equation (9) is used in the following manner:
 The traffic volume variables (ADT_{1} and ADT_{2}) are set equal to the actual ADTs of the major and minorroad legs, respectively. If the ADTs differ between either the two major or minorroad legs, they should be averaged.
 The remaining variables in the model should be set equal to the following nominal or base conditions:
Number of driveways within 76 m (250 ft) of the intersection on the major rad (ND_{1}) 
No driveways 
Intersection skew angle (SKEW_{4}) 
0 degrees 
With the default values of ND_{1} and SKEW_{4} given above, the base model in equation (9) reduces to:
N_{bi} = exp(9.34 + 0.60ln ADT_{1} + 0.61ln ADT_{2})

(10)

FourLeg Signalized Intersections
The base model for fourleg signalized intersections is presented below:
Nbi =

exp(5.46 + 0.60ln ADT_{1} + 0.20ln ADT_{2}  0.40PROTLT  0.018PCTLEFT_{2}
+ 0.11VEICOM + 0.026PTRUCK + 0.041ND_{1})


where:
PROTLT 
= 
presence of protected leftturn signal phase on one or more majorroad approaches; = 1 if present; = 0 if not present 
PCTLEFT _{2} 
= 
percentage of minorroad traffic that turns left at the signal during the morning and evening hours combined 
VEICOM 
= 
grade rate for all vertical curves (crests and sags) within 76 m (250 ft) of 
PTRUCK 
= 
the intersection along the major and minor roads percentage of trucks (vehicles with more than four wheels) entering the intersection for the morning and evening peak hours combined 
ND_{1} 
= 
number of driveways within 76 m (250 ft) of the intersection on the major road. 
This model was developed with negative binominal regression from data for 49 fourleg signalized intersections, 18 in California and 31 in Michigan. The data base available for model development included three years of accident data (19931995) for each intersection. The model predicts total intersectionrelated accident frequency for any fourleg signalized intersection for which the independent variables shown in equation (11) are known. The model predictions are reliable only within the ranges of independent variables for which data were available in the data base used to develop the model (see table 44 in appendix B).
When the accident prediction model is employed to predict the expected accident frequency for any specified fourleg intersection on a twolane highway, equation (11) is used in the following manner:
 The traffic volume variables (ADT_{1}and ADT_{2}) are set equal to the actual ADTs of the
major and minorroad legs, respectively. If the ADTs differ between either the majoror minorroad legs, they should be averaged.
 The remaining variables in the model should be set equal to the following nominal or base conditions:
Presence of protected leftturn
signal phase (PROTLT) 
No leftturn phase 
Percentage of minorroad traffic
turning left (PCTLEFT_{2}) 
28.4 percent 
Grade rate for vertical curves
within 76 m (250 ft) of the
intersection (VEICOM) 
No vertical curves 
Percentage of trucks entering
the intersection (PTRUCK) 
9.0 percent 
Number of driveways within
76 m (250 ft) of the intersection
on the major road (ND _{1)} 
0 driveways 
With the nominal or base values of PROTLT, PCTLEFT_{2}, VEICOM, and PTRUCK given above, the base model in equation (11) reduces to:
N_{bi} = exp(5.73 + 0.60ln ADT_{1} + 0.20ln ADT_{2})

(12)

Calibration Procedure
The accident prediction algorithm is intended for use by highway agencies throughout the United States. Accident frequencies, even for nominally similar roadway sections or intersections, are known to vary widely from agency to agency. These variations are of two types, those that can be directly accounted for by the accident prediction algorithm and those that cannot.
States differ markedly both in terrain and in the history of the development of their highway system, resulting in statetostate differences in roadway alignment, cross section, and intersection design. However, differences of this type can be accounted for by the AMFs in the accident prediction algorithm.
States also differ markedly in climate, animal population, driver populations, accident reporting threshold, and accident reporting practices. These variations may result in some States experiencing substantially more reported traffic accidents on rural twolane highways than others. Such variations cannot be directly accounted for by the accident prediction algorithm. Therefore, a calibration procedure has been developed to allow highway agencies to adjust the accident prediction algorithm to suit the safety conditions present in their State.
The calibration procedure is implemented by a highway agency by determining the value of calibration factors for roadway segments and atgrade intersections from comparison of their own data to estimates from the accident prediction algorithm. The calibration factors are incorporated in equations (13) and (14) in the following fashion for roadway segments and atgrade intersections, respectively:
N_{rs} = N_{br} C_{r} (AMF_{1r} AMF_{2r} · · · AMF_{nr})

(13)

N_{int} = N_{bi} C_{i} (AMF_{1i} AMF_{2i} · · · AMF_{ni})


where:
C_{r} 
= 
calibration factor for roadway segments developed for use by a particular highway agency; and 
C_{i} 
= 
calibration factor for atgrade intersections developed for use by a particular highway agency. 
The calibration factors (C_{r} and C_{i}) will have values greater than 1.0 for highway agencies whose roadways, on the average, experience more accidents than the roadways used in the development of the accident prediction algorithm. The calibration factors for highway agencies whose roadways, on the average, experience fewer accidents than the roadways used in the development of the accident prediction algorithm will have values less than 1.0. The calibration factor for atgrade intersections (C_{i}) may have different values for each of the three intersectiontypes for which base models have been developed. The calibration procedures for application by highway agencies is presented in appendix C.
It is generally expected that the calibration factors (C_{r} and C_{i}) would be determined by highway agencies based on statewide data. In larger and more diverse States, a highway agency might choose to develop separate calibration factors for individual highway districts or climate regions. It is also possible for users to provide a local calibration factor for smaller areas with distinct driver populations or climate conditions. However, use of the local calibration factor would require a special study to determine the safety performance of roads in that specific local area relative to the statewide or districtwide expected values.
In addition to estimates of accident frequency, the accident prediction algorithm includes default distributions of accident severity and accident type for rural twolane highway roadway sections and intersections. These default distributions have been presented in tables 1 and 2 of this report. The calibration procedure presented in appendix C includes a capability for highway agencies who use the accident prediction algorithm to modify the default distributions of accident severity and accident type to match their own experience on rural twolane highways.
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