Prediction of the Expected Safety Performance of Rural TwoLane Highways
APPENDIX B
DEVELOPMENT OF BASE MODELS
The base models for the accident prediction algorithm were developed as part of the preparation of two FHWA reports, Accident Models for TwoLane Rural Roads: Segments and Intersections, and Accident Models for Rural Intersections: 4Lane by 2Lane StopControlled and 2Lane by 2Lane Signalized.^{(3,5)} This appendix describes the data base development, the base model for roadway segments, and the base models for atgrade intersections.
Data Base Development
The base models were developed with geometric design, traffic control, traffic volume, and accident data on roadway sections and intersections on rural twolane highways in California, Michigan, Minnesota, and Washington. These data were obtained from the FHWA Highway Safety Information System (HSIS). The geometric design data in the HSIS files, from databases maintained by the States identified above, were supplemented with additional data obtained by Vogt and Bared from field measurements and photolog review. These efforts are described more fully in two FHWA reports and a published paper.^{(3,4,5)}
Base Model for Roadway Segments
In the modeling of roadway segment accidents, the dependent variable included five years of accident data (19851989) for 619 rural twolane roadway segments in Minnesota and 3 years of accident data (19931995) for 712 roadway segments in Washington. The model development excluded roadway segments within 76 m (250 ft) of an atgrade intersection and excluded the (relatively few) accidents that occurred more than 76 m (250 ft) from an intersections but were identified by the investigating officer as related to an intersection.
The independent variables representing geometric, design traffic control. and traffic volume used in modeling included:
 Exposure (million vehiclemiles of travel).
 State in which the roadway section is located (Minnesota/Washington).
 Lane width.
 Shoulder width.
 Roadside hazard rating.
 Driveway density.
 Horizontal curvature.
 Grade rate for crest vertical curves.
 Percent grade for straight grades.
All of these independent variables were found to have a statistically significant relationship to roadway section accidents.
The base model for roadway segments was developed from the HSIS roadway segment data for rural twolane highways in Minnesota and Washington. The base model is presented below:
N_{br} = EXPO exp(0.6409 + 0.1388STATE  0.0846LW  0.0591 SW + 0.0668 RHR + 0.0084DD)
(S WH_{i} exp(0.0450DEG_{i} )) (SWV_{j}exp (0.4652V_{j})) (SWG_{k} exp(0.1048GR_{k})) 
(42) 
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 the development of the roadside hazard rating, see Zegeer et al.; for definitions of individual rating levels, see Appendix D.);^{(6)} 
DD 
= 
driveway density (driveways per mi) on the roadway segment; 
WH_{i} 
= 
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.); 
DEGj 
= 
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). 
The model was developed with extended negative binomial regression analysis. This extension of the standard negative binomial regression analysis technique was devised by Miaou.^{(3)} In negative binomial models for roadway segments, the mean number of accidents in a specified time period is typically represented in the form:
EXPO exp(aX + bY + ...)
which is equivalent to:
EXPO exp(aX) exp(bY) ...
where EXPO is a measure of exposure, X and Y are measures of roadway segment characteristics, and a and b are appropriate regression coefficients.
The extended negative binomial regression analysis technique devised by Miaou replaces some of the factors exp(aX) by expressions of the form:
w_{1} exp(aX_{1}) + w_{2} exp(aX_{2}) + ... +w_{m} exp (aX_{m})
where X_{1}, X_{2}, ..., X_{m} are local variables along the roadway segment, characterizing for subsegments what X attempts to measure for the entire segment. For example, X_{1}, ..., X_{m} might represent the degree of curvature for individual horizontal curves while X is the average degree of curvature for the roadway segment as a whole.Thus, X is a composite variable, while the X_{i}s represent variation within the segment. Such variation occurs for many variables used in accident modeling of roadway segments, even on supposedly homogeneous segments. Notable examples are degree of curvature, grade, and change of grade per unit length. The variable w_{i} is the proportion of the segment length to which the value X_{i} applies. (The default value for X_{i} is assumed to be zero if there is no horizontal or vertical curve or if the grade is level, and an artificial subsegment with this value is added, if necessary, so that the weights, w_{i}, always sum to 1.0.)^{(3)}
The extended negative binomial regression model decomposes the roadway segment into subsegments within which the roadway characteristic measured by X is constant. If this is done for two or more variables (e.g., X, Y, ...), the method assumes that the variables are independent of one another, so that the value Y_{j} occurs with a particular value of X_{i}. Although such independence cannot be assured, the extended negative binomial regression model attempts to capture the effect of variation within a segment in an additive manner consistent with the basic form of the model.^{(3)}
Table 29 summarizes the model presented in equation (42) with the coefficient, standard deviation, and significance level (p) for each independent variable and the overdispersion parameter (k). The goodnessoffit measures for the roadway segment base model include R^{2}, the traditional measure of the percentage of variation in accident frequency explained by the independent variables in the model, as well as R_{k}^{2}, defined as:
Where:
k 
= 
the overdispersion parameter for the regression model; and 
k_{max} 
= 
the overdispersion parameter in a model with no covariables (the socalled “zero model”). 
This latter measure of goodness of fit, R_{k}^{2}, has been proposed by Miaou.^{(9)} For the roadway segment base model in equation (42), the values of the goodnessoffit measures are R^{2} = 0.6547 and R_{k}^{2} = 0.8291.
Table 29. Model Parameters and Goodness of Fit for Equation (42).

Independent Variable 
Parameter 
Intercept 
State 
LW 
SW 
RHR 
DD 
DEG 
V 
GR 
Overdipersion
parameter (k) 
Coefficient 
0.6409 
0.1388 
0.0846 
0.0591 
0.0668 
0.0084 
0.0450 
0.4652 
0.1048 
0.3056 
Standard deviation 
0.5008 
0.0659 
0.0425 
0.0114 
0.0211 
0.0026 
0.0078 
0.1260 
0.0287 
0.0331 
Significance level (p) 
0.2006 
0.0351 
0.0465 
0.0001 
0.0015 
0.0011 
0.0001 
0.0002 
0.0003 
0.0001 
Note: The values of the goodnessoffit measures are R^{2} = 0.6547 and R_{k}^{2} = 0.8291
Table 30 presents descriptive statistics for the variables in the roadway segment model.
Table 30. Descriptive Statistics for Roadway Segments Used in Modeling.
Variable 
Mean 
Standard
deviation 
Minimum 
25th
percentile 
Median 
75th
percentile 
Maximum 
MINNESOTA (619 roadway segments) 
ADT (veh/day) 
2,402 
1,937 
208 
1,176 
1,866 
2,900 
15,162 
L (mi) 
1.14 
1.30 
0.10 
0.26 
0.66 
1.50 
8.24 
LW (ft) 
11.54 
0.67 
10.00 
11.00 
12.00 
12.00 
12.00 
SW (ft) 
7.08 
2.44 
0.00 
6.00 
8.00 
8.00 
12.00 
RHR 
2.14 
0.98 
1.00 
1.00 
2.00 
3.00 
6.00 
DD (mi^{1}) 
6.58 
10.25 
0.00 
0.89 
3.73 
7.68 
100.00 
DEG (degrees/100 ft) 
0.51 
0.95 
0.00 
0.00 
0.08 
0.62 
7.50 
V (percent/100 ft) 
0.066 
0.092 
0.00 
0.007 
0.037 
0.086 
0.888 
GR (percent) 
0.38 
0.52 
0.00 
0.10 
0.24 
0.45 
4.46 
WASHINGTON (712 roadway segments) 
ADT (veh/day) 
3,352 
3,199 
159 
1,261 
2,239 
4,455 
17,766 
L (mi) 
0.75 
0.83 
0.10 
0.27 
0.554 
0.948 
13.23 
LW (ft) 
11.37 
0.56 
9.00 
11.00 
11.00 
12.00 
12.00 
SW (ft) 
5.01 
2.35 
0.00 
3.00 
5.00 
7.00 
10.00 
RHR 
3.67 
1.57 
1.00 
2.00 
3.00 
6.00 
7.00 
DD (mi^{1}) 
10.12 
12.41 
0.00 
2.07 
6.12 
13.61 
85.07 
DEG (degrees/100 ft) 
1.03 
2.13 
0.00 
0.00 
0.32 
1.31 
30.55 
V (percent/100 ft 
0.068 
0.127 
0.000 
0.000 
0.026 
0.083 
1.997 
GR (percent) 
0.92 
1.17 
0.00 
0.20 
0.49 
1.13 
6.92 
Conversion: 1 ft = 0.305 m; 1 mi = 1.61 km
The exposure variable, EXPO, was treated as a scale factor in the development of the model. Therefore, even though multiple years of accident data were used in developing the model, the expected annual accident frequency can be determined from the model if EXPO is determined as (ADT)(365)(L)(10^{6}).
Other variables investigated in the development of the roadway segment base models included posted speed limit, truck percentage, and intersection density (i.e., number of intersections per mile). Posted speed limit was found to be negatively correlated with accident frequency, while truck percentage and intersection density were positively correlated with accident frequency. However, none of these three variables was statistically significant in regression models that included the variables listed above. Grade change per unit length of roadway was also considered for sag vertical curves and for all vertical curves (i.e. both sags and crests), but the version of the variable for crest vertical curves (V_{j}) had the greatest statistical significance and was, therefore, retained in the final model. Two weatherrelated variables were also investigated in the modeling for the Minnesota roadway segments—number of rain days and number of snow days per year. These variables were based on the climate district in which each roadway segment was located. These variables were found to be negatively correlated with accident frequency. Because these variables were not sufficiently local (i.e., they represented climate districts rather than the climate of individual roadway segments) and because they were only marginally statistically significant, they were not collected for the Washington data and were omitted from the final model.
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.
Models for each of these intersection types were developed using three different modeling approaches, described below. The available database used in modeling included 5 years of accident data (198589) for selected STOPcontrolled intersections in Minnesota and 3 years of accident data (19931995) for selected signalized intersections in California and Michigan. Following the description of the modeling approaches, the models developed for each intersection type are presented.
The available accident data for atgrade intersections generally included accidents within 76 m (250 ft) of the intersection on the major road, typically a State highway. For minor roads that are also State highways, data were generally available for all accidents that occur within 76 m (250 ft) of the intersection. For minor roads that are not State highways, accidents on the minor road that are classified as intersectionrelated are typically assigned the milepost of the intersection on the major road and are therefore included in the available data. In Michigan, all minorroad accidents within 31 m (100 ft) of the intersection and, in California, all accidents within 76 m (250 ft) of the intersection are included in the available data. In Minnesota, all accidents that occurred on minorroad approaches and were identified as related to the intersection are included in the available data.
Modeling Approaches
Three different modeling approaches were employed to develop candidate base models for each intersection type. The approaches are:
 The dependent variable (number of accidents in a 5year period) included only accidents that occurred within the curbline limits of a particular intersection or within 76 m (250 ft) of that intersection and were identified by the investigating officer as being related to the intersection.
 The dependent variable included only accidents that occurred within the curbline limits of a particular intersection or within 76 m (250 ft) of that intersection and were of accident types that are generally related to intersection operations.
 The dependent variable included all accidents that occurred within the curbline limits of a particular intersection or within 76 m (250 ft) of that intersection, whether those accidents were related to the intersection or not. However, the modeling technique used an offset to the observed accident frequency applied iteratively to account for the frequency of roadway section accidents that would be expected to occur on 152 m (500 ft) of the major roadway (76 m or 250 ft on either side of the intersection), as predicted by equation (37).
Each of these modeling techniques is described in greater detail in the following discussion. A fourth modeling technique that was considered but not used is explained at the end of this appendix.
The research reports by Vogt and Bared from the FHWA project that developed the base models and the related paper include only the models based on accidents that were identified by the investigating officer as being intersection related.^{(3,4,5)} The models based on accident types that are generally related to intersection operations and on the iterative offset approach were developed as part of the same project but are documented only in this report.
Limitation of the Analysis to IntersectionRelated Accidents
The first modeling approach used to develop base models for atgrade intersections was to limit the dependent variable to those accidents that occurred within the curbline limits of the intersection or within 76 m (250 ft) of the intersection on any approach and were identified by the investigating officer as related to the intersection. This definition is generally reasonable because the vast majority of intersectionrelated accidents on a rural twolane highway would be expected to occur within 76 m (250 ft) of the intersection to which they are related. For example, traffic queues that extend more than 76 m (250 ft) from an intersection and which might lead to rearend collisions remote from the intersection are much less common in rural than in urban areas.
Not all accidents within 76 m (250 ft) of an intersection would be expected to be related to the operation of that intersection. For example, collisions between motor vehicles and animals and collisions involving vehicles entering or leaving a driveway may occur within 76 m (250 ft) of an intersection but may have no particular relationship to the presence of that intersection. In other words, some accidents would be expected to occur on the roadway near the intersection whether the intersection were present or not. the first approach to modeling distinguishes between intersectionrelated and nonintersectionrelated accidents based on judgments made by the investigating officer or by an accident coder and recorded in the State’s computerized accident record system. Nominally, such judgments make exactly the distinction that is desired, but there is concern that investigating officers and accident coders may not make such judgments consistently.
Once the dependent variable was defined, as described above, predictive models were developed using extended negative binomial regression in a manner similar to the development of the roadway segment model in equation (42).
Limitation of the Analysis to Selected Accident Types
The second modeling approach used was to limit the dependent variable to accidents that occurred within the curbline limits of the intersection and are of accident types that are generally related to intersection operations. The accident types classified as related to the intersection were:
 Rearend collisions.
 Rightangle collisions.
 Leftturn collisions.
 Rightturn collisions.
 Sideswipe, samedirection collisions.
All other accidents types were classified as unrelated to the intersection.
The list of accident types classified as intersectionrelated and nonintersectionrelated were established through two special studies that involved a review of hard copy police accident reports and classification of accidents as intersectionrelated or nonintersectionrelated based on the judgment of safety experts.
This approach to modeling eliminates the concern raised in the first modeling approach about the potential misjudgment in classifying accidents made by investigating officers or accident coders. However, the approach based on classification of particular accident types as intersectionrelated has a similar concern since not every accident of the types identified above is actually related to an intersection and some accidents of other types are, in fact, intersectionrelated. For example, the proposed classification scheme based on accident types would classify all reported rightturn accidents within 76 m (250 ft) of the intersection as intersectionrelated, including accidents that are, in fact, related to turning movements at nearby driveways. The effectiveness of this classification could be improved by an agency by utilizing other fields of the accident record such as a driveway involvement indicator, if available, in classifying accidents.
Once the dependent variable was defined, as described above, predictive models were developed using extended negative binomial regression in a manner similar to the development of the roadway segment model in equation (42).
Use of All Accidents with an Offset for Expected RoadwaySegment Accidents
The third modeling technique uses as the dependent variable all accidents that occur within the curbline limits of a particular intersection and all accidents that occur within 76 m (250 ft) of that intersection, regardless of the accident type and regardless of the judgment made by the investigating officer or accident coder concerning the relationship of the accident to the intersection. Since this dependent variable includes some accidents that are clearly unrelated to the intersection, the expected frequency of roadway segment accidents, as predicted by equation (42), is used as an offset factor in the model development. This modeling used an iterative technique that makes successive approximations to the model coefficients.
The iterative offset modeling approach was performed as follows. Let:
Z_{i} = exp(k_{i} + a_{i}X + b_{i}Y + ...) 
(44) 
denote an equation with integer subscript i estimating the mean number of intersectionrelated accidents per unit time, Z^{i}, in terms of intersection characteristics (X, Y, ...). Let N denote the mean number of nonintersectionrelated accidents on a 152 m (500 ft) section of roadway containing the intersection (i.e., 76 m or 250 ft on either side of the intersection), as predicted by a roadway section model such as equation (42), as applied to Minnesota (STATE=0).
Let A be the mean number of accidents of all kinds per unit time within 76 m (250 ft) of the intersection. Let:
OFFSET1 = ln 
ln 

N + Z_{i} 

(45) 

Z_{i} 
The modeling approach used is iterative. Equation (44) implies that the following relationship must be valid for iteration i+1:
Z_{i+1} = exp(k_{i+1} + a_{i+1}X + b_{i+1}Y + ...) 
(46) 
A negative binomial model for A is sought of the form:
A = exp(OFFSET_{i} + k_{i+1} + a_{i+1}X + b_{i+1}Y + ...) 
(47) 
which can be expressed as:
A = exp(OFFSET_{i}) exp(k_{i+1} + a_{i+1}X + b_{i+1}Y) 
(48) 
A = exp(OFFSET_{i}) Z_{i+1} 
(49) 
A = 
N + Z_{i} 
Z_{i + 1} 
(50) 

Z_{i} 
The initial negative binomial model for intersectionrelated accidents is represented by an equation for Z_{0} in the form of equation (44). The offset technique represented by equations (47) through (50) is applied repeatedly to obtain a sequence of new models for Z_{1}, Z_{2}, Z_{3}, ... . This process is continued until the coefficients k_{n}, a_{n}, b_{n}, ... cease to change appreciably; i.e., until Zn+1 is approximately the same as Z_{n}. The appropriate model for the mean number of intersectionrelated accidents is then:
Z = exp(k_{n} + a_{n}X + b_{n}Y + ...) 
(51) 
In principle, A = N + Z, or Z = A  N. Thus from, the model for roadway segment accidents used to estimate N and the initial model for intersectionrelated accidents Z_{0}, the offset technique yields a final model for intersectionrelated accidents, Z. This final model depends on the choice of the model for N, but should not generally depend on the initial model for Z_{0}. Thus, initial estimates or starting values of the regression coefficients k_{0}, a_{0}, b_{0}, ... can be selected through engineering judgment or an alternative preliminary model.
Models Developed
The following discussion presents the candidate base models developed for threeleg STOPcontrolled intersections, fourleg STOPcontrolled intersections, and fourleg signalized intersections.
ThreeLeg STOPControlled Intersections
Candidate base models were developed for threeleg intersections with STOP control on the minorroad approach. The dependent variables used in these models have been described above in the discussion of modeling techniques. All of the models for threeleg STOPcontrolled intersections used a data sets of approximately 382 intersections in Minnesota including 5 years of accident data (19851989). There were minor variations in sample size from one model to the next because of small amounts of missing data. The candidate independent variables considered in predicting accidents at threeleg STOPcontrolled intersections were:
 Average daily traffic volume on the major road.
 Average daily traffic volume on the minor road.
 Intersection skew angle.
 Posted speed limit on major road.
 Presence of a rightturn lane on the major road.
 Number of driveways on the major road within 76 m (250 ft) of the intersection.
 Grade rate for crest vertical curves on the major road within 76 m (250 ft) of the intersection.
 Radius of horizontal curves on the major road within 76 m (250 ft) of the intersection.
 Roadside hazard rating along the major road within 76 m (250 ft) of the intersection.
Table 31 presents descriptive statistics for these variables.
A candidate model for threeleg STOPcontrolled intersections developed using the intersectionrelated accident definition based on the investigating officer’s assessment of each accident is:
N_{bi =} 
exp(12.99 + 0.8.05ln ADT_{1} + 0.504ln ADT_{2} + 0.290VCI + 0.034HI
+ 0.029SPDI + 0.173RHRI + 0.27RT + 0.0045SKEW_{3}) 
(52) 
where:
N_{bi} 
= 
predicted number of total accidents per year at a particular intersection and within 76 m (250 ft) in either direction along the major road; 
ADT_{1} 
= 
average daily traffic volume (veh/day) on the major road; if the ADTs differ between the majorroad legs, they should be averaged; 
ADT_{2} 
= 
average daily traffic volume (veh/day) on the minor road; 
VI 
= 
crest vertical curve grade rate on the major road within 76 m (250 ft) of the ^{intersection = (1/m) SV}i ^{for all crest vertical curves wholly or partly within 76} m (250 ft) of the intersection; 
m 
= 
number of crest vertical curves wholly or partly within 76 m (250 ft) of the intersection; 
HI 
= 
horizontal curvature change rate on the major road within 76 m (250 ft) of the ^{intersection = (1/n) SDEG}i ^{for all horizontal curves wholly or partly within 76} m (250 ft) of the intersection; 
n 
= 
number of horizontal curves within 76 m (250 ft) of the intersection; 
SPDI 
= 
posted speed limit on the major road (mi/h); 
RHRI 
= 
roadside hazard rating within 76 m (250 ft) of the intersection on the major road [see description of the variable RHR in Equation (42)]; 
RT 
= 
presence of rightturn lane on the major road (0 = no rightturn lane present; 1 = rightturn lane present); and 
SKEW_{3} 
= 
intersection angle (degrees) minus 90 for the angle between the majorroad leg in the direction of increasing stations and a leg to the right; 90 minus intersection angle (degrees) for the angle between the majorroad leg in the direction of increasing stations and a leg to the left. 
Table 31. Descriptive Statistics for 382 ThreeLeg STOP Controlled Intersections in Minnesota Used Modeling.
Variable 
Mean 
Standard
deviation 
Minimum 
25th
percentile 
Median 
75th
percentile 
Maximum 
ADT_{1} (veh/day) 
3,718 
3,725 
201 
1,239 
2,333 
4,627 
19,413 
ADT_{2} (veh/day) 
408 
531 
5 
103 
237 
478 
4206 
ln ADT_{1} 
7.81 
0.91 
5.30 
7.12 
7.75 
8.44 
9.87 
ln ADT_{2} 
5.40 
1.14 
1.51 
4.64 
5.47 
6.17 
8.34 
VCI (percent/100 ft) 
0.142 
0.300 
0 
0 
0 
0 
4.39 
HI (degrees/100 ft) 
1.22 
2.52 
0 
0 
0 
2 
29 
SPDI (mph) 
52.8 
4.6 
22.5 
52.5 
55 
55 
55 
RHRI 
2.10 
0.88 
1 
1 
2 
3 
5 
RT 
0.42 
0.49 
0 
0 
0 
1 
1 
SKEW_{3} (degrees) 
0.67 
24.98 
90 
0 
0 
0 
85.1 
ND_{1} 
1.24 
1.44 
0 
0 
1 
2 
9 
Conversion: 1 ft = 0.305 m
Equation (52) includes all of the candidate independent variables except the number of driveways which was dropped because its coefficient was not statistically significant (p=0.5405) and because coefficient had a negative sign, which is opposite to the direction expected. Table 32 summarizes the model parameters and goodness of fit for the model in equation (52). Goodness of fit for models like equation (52) is measured by R^{2} and R_{k}^{2}, as noted earlier for equation (42), as well as by R_{PD}^{2}. This last goodnessoffit measure, R_{PD}^{2}, has been proposed by Fridstrøm for use with negative binomial models.^{(43)} A description of this goodnessoffit measure is also provided by Vogt and Bared.^{(3)}
Table 32. Model Parameters and Goodness of Fit for Equation (52).

Independent variable 
Over
dispersion
parameter(k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ADT _{2} 
VCI 
HI 
SPDI 
RHRI 
RT 
SKEW_{3} 
Coefficient 
12.99 
0.805 
0.504 
0.290 
0.034 
0.029 
0.173 
0.27 
0.0045 
0.481 
Standard deviation 
1.15 
0.064 
0.071 
0.294 
0.033 
0.018 
0.068 
0.140 
0.0032 
0.100 
Significance level 
0.000 
0.0001 
0.0001 
0.323 
0.3004 
0.107 
0.0108 
0.0561 
0.1578 
0.0001 
Note: The values of the goodnessoffit measures are R^{2} = 0.4409, R_{k}^{2} = 0.7805, and R_{PD}^{2} = 0.6279
The model in equation (52) was reevaluated including only those variables that were statistically significant with a significance level (p) of 0.150 or less. This model is presented below:
N_{bi} = exp(11.28 + 0.79ln ADT_{1} + 0.49ln ADT_{2} + 0.19RHRI + 0.28RT) 
(53) 
Table 33 summarizes the model parameters and goodness of fit for the model in equation (53).
A candidate model developed for threeleg STOPcontrolled intersections using only those accident types generally related to intersection operations is:
N_{bi} = exp(12.82 + 1.001ln ADT_{1} + 0.406ln ADT_{2} + 0.22RHRI + 0.33RT + 0.0040SKEW_{3}) 
(54) 
Equation (54) includes all of the candidate independent variables except speed limit (p=0.41), number of driveways (p=0.56), horizontal curvature (p=0.62), and vertical curvature (p=0.40) which were dropped because their coefficients were not statistically significant. Table 34 summarizes the model parameters and goodness of fit for the model in equation (54).
Table 33. Model Parameters and Goodness of Fit for Equation (53).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ADT _{2} 
RHRI 
RT 
Coefficient 
11.28 
0.79 
0.49 
0.19 
0.28 
0.54 
Standard deviation 
0.063 
0.062 
0.068 
0.067 
0.14 
0.102 
Significance level (p) 
0.0001 
0.0001 
0.0001 
0.0035 
0.0402 
0.0001 
Note: The values of the goodnessoffit measures are R^{2} = 0.3955, R_{k}^{2} = 0.7546, and R _{PD}^{2} = 0.6109.
Table 34. Model Parameters and Goodness of Fit for Equation (54).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ADT _{2} 
RHRI 
RT 
SKEW_{3} 
Coefficient 
12.82 
1.001 
0.406 
0.22 
0.33 
0.0040 
0.46 
Standard deviation 
0.73 
0.072 
0.073 
0.073 
0.15 
0.0029 
0.106 
Significance level 
0.0001 
0.0001 
0.0001 
0.0024 
0.032 
0.17 
0.0001 
Note: The values of the goodnessoffit measures are R^{2} = 0.4181, R_{k}^{2} = 0.8070, and R _{PD}^{2} = 0.7228.
The model in equation (54) was reevaluated including only those variables that were statistically significant with a significance level of 0.150 or less. This model is presented below:
N_{bi} = exp(13.01 + 1.015ln ADT_{1} + 0.42ln ADT_{2} + 0.23RHRI + 0.29RT) 
(55) 
Table 35 summarizes the model parameters and goodness of fit for the model in equation (55).
Table 35. Model Parameters and Goodness of Fit for Equation (55).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ADT _{2} 
RHRI 
RT 
Coefficient 
13.01 
1.015 
0.42 
0.23 
0.29 
0.49 
Standard deviation 
0.72 
0.072 
0.071 
0.073 
0.14 
0.108 
Significance level (p) 
0.0001 
0.0001 
0.0001 
0.0020 
0.041 
0.0001 
Note: The values of the goodnessoffit measures are R^{2} = 0.4008, R _{k}^{2} = 0.7953, and R _{PD}^{2} = 0.7162
A candidate model for threeleg STOPcontrolled intersections developed with the iterative offset technique is:
N_{bi} = exp(12.40 + 0.74ln ADT_{1} + 0.53ln ADT_{2} + 0.36 VCI + 0.028SPDI + 0.14RHRI + 0.0063 SKEW_{3}) 
(56) 
Equation (56) includes all of the candidate independent variables except number of driveways (p=0.35), presence of rightturn lane (p=0.47), and horizontal curvature (p=0.50) which were dropped because their coefficients were not statistically significant. Table 36 summarizes the model parameters and goodness of fit for the model in equation (56). It should be noted that the goodnessoffit measure, R_{PD}^{2}, is not directly applicable to models developed with the iterative offset technique and, therefore, is not presented in Table 36. Moreover, R^{2} must be interpreted with caution since it measures the goodnessoffit for the combined roadway segment and intersection model for all accidents with 76 m (250 ft) of the intersection. Since most of these accidents are intersectionrelated, R^{2} at least roughly measures the goodness of fit for intersectionrelated accidents
Table 36. Model Parameters and Goodness of Fit for Equation (56)

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ADT _{2} 
VCI 
SPDI 
RHRI 
SKEW_{3} 
Coefficient 
12.40 
0.74 
0.53 
0.36 
0.028 
0.14 
0.0063 
0.52 
Standard deviation 
0.93 
0.063 
0.064 
0.28 
0.013 
0.070 
0.0029 
0.090 
Significance level (p) 
0.0001 
0.0001 
0.0001 
0.20 
0.033 
0.053 
0.032 
0.0001 
Note: The values of the goodnessoffit measures are R^{2} = 0.4163, and R k^{2} = 0.5809.
The model in equation (56) was reevaluated including only those variables that were statistically significant with a significance level of 0.150 or less. This model is presented below:
N_{bi} = exp(12.25 + 0.75ln ADT_{1} + 0.52ln ADT_{2} + 0.026SPDI + 0.15RHRI + 0.0059SKEW_{3)} 
(57) 
Table 37 summarizes the model parameters and goodness of fit for the model in equation (57).
A decision was made to use in the accident prediction algorithm the models containing only those independent variables that are statistically significant at a significance level of 0.15 or less, like equations (53), (55), and (57).
Table 37. Model Parameters and Goodness of Fit for Equation (57).

Independent Variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ADT _{2} 
SPDI 
RHRI 
SKEW_{3} 
Coefficient 
12.25 
0.75 
0.52 
0.026 
0.15 
0.0059 
0.52 
Standard deviation 
0.92 
0.062 
0.064 
0.013 
0.065 
0.0029 
0.089 
Significance level (p) 
0.0001 
0.0001 
0.0001 
0.046 
0.019 
0.043 
0.0001 
Note: The value of goodnessoffit measures are R^{2} = 0.4069 and R_{k}^{2} = 0.5765.
FourLeg STOPControlled Intersections
Candidate base models were developed for fourleg intersections with STOP control on the minorroad approaches. The dependent variables used in these models have been described above in the discussion of modeling techniques. All of the models for fourleg STOPcontrolled intersections used a data sets of approximately 324 intersections in Minnesota including 5 years of accident data (19851989). There were minor variations in sample size from one model to the next because of small amounts of missing data. The candidate independent variables considered in predicting accidents at fourleg STOPcontrolled intersections were:
 Average daily traffic volume on the major road.
 Average daily traffic volume on the minor road.
 Intersection skew angle.
 Posted speed limit on major road.
 Presence of rightturn lanes on the major road.
 Number of driveways within 76 m (250 ft) of the intersection.
 Grade rate for crest vertical curves within 76 m (250 ft) of the intersection.
 Radius of horizontal curves within 76 m (250 ft) of the intersection.
 Roadside hazard rating.
Table 38 presents descriptive statistics for these variables.
A candidate model for fourleg STOPcontrolled intersections developed using the intersectionrelated accident definition is:
N_{bi} = exp(10.43 + 0.603ln ADT_{1} + 0.609ln ADT_{2} + 0.29 VCI + 0.045 HI + 0.019 SPDI + 0.12ND_{1} .0049  SKEW_{4}) 
(58) 
where:
ND_{1} 
= 
number of driveways on the major road with 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. 
Table 38. Descriptive Statistics for 324 FourLeg STOPControlled Intersections in Minnesota.
Variable 
Mean 
Standard
deviation 
Minimum 
25th
percentile 
Median 
75th
percentile 
Maximum 
ADT _{1} (veh/day) 
2,216 
1,966 
174 
972 
1,739 
2,611 
14,611 
ADT _{2} (veh/day) 
304 
383 
7 
105 
191 
365 
3,414 
ln ADT _{1} 
7.42 
0.75 
5.16 
6.88 
7.46 
7.87 
9.59 
ln ADT _{2} 
5.25 
0.97 
1.93 
4.65 
5.25 
5.90 
8.14 
VCI (percent/100 ft) 
0.146 
0.280 
0 
0 
0.023 
0.207 
2.942 
HI (degrees/100 ft) 
0.46 
1.08 
0 
0 
0 
0.25 
8.00 
SPDI (mi/h) 
54.0 
3.3 
30 
55 
55 
55 
55 
ND_{1} 
0.61 
1.14 
0 
0 
0 
1 
6 
SKEW_{4} (degrees) 
0.14 
18.34 
60.00 
0.44 
0.00 
0.58 
75.00 
Conversion: 1 ft = 0.305 m; 1 mi = 1.61 km
Equation (58) includes all of the candidate independent variables except roadside hazard rating (p=0.28) and presence of rightturn lanes (p=0.66) which were not statistically significant. Table 39 summarizes the model parameters and goodness of fit for the model in equation (58).
Table 39. Model Parameters and Goodness of Fit for Equation (58).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
^{lnADT} 1 
^{lnADT} 2 
VCI 
HI 
SPDI 
ND_{1} 
SKEW_{4} 
Coefficient 
10.43 
0.603 
0.609 
0.29 
0.045 
0.019 
0.12 
0.0049 
0.205 
Standard deviation 
1.32 
0.084 
0.069 
0.26 
0.047 
0.018 
0.05 
0.0033 
0.065 
Significance level 
0.000 
0.000 
0.000 
0.26 
0.34 
0.29 
0.01 
0.13 
0.0016 
Note: The values of goodnessoffit are R^{2} = 0.5944, R _{k}^{2} = 0.8336, and R _{PD}^{2} = 0.7364
The model in equation (58) was reevaluated including only those variables that were statistically significant with a significance level of 0.150 or less. This model is presented below:
N_{bi} = exp(9.34 + 0.60ln ADT_{1} + 0.61ln ADT_{2} + 0.13ND_{1 } .0054 SKEW_{4}) 
(59) 
Table 40 summarizes the model parameters and goodness of fit for the model in equation (59).
Table 40. Model Parametere and Goodness of Fit for Equation (59).

Independent variable 
Overdispersion parameter (k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ADT _{2} 
ND_{1} 
SKEW_{4} 
Coefficient 
9.34 
0.601 
0.61 
0.13 
0.0054 
0.24 
Standard deviation 
0.72 
0.078 
0.069 
0.039 
0.0034 
0.071 
Significance level (p) 
0.0001 
0.0001 
0.0001 
0.0009 
0.108 
0.0008 
Note: The values of goodnessoffit measures are R^{2} = 0.5662, R _{k}^{2} = 0.8081, and R _{PD}^{2} = 0.7326
A candidate model developed using only those accident types generally related to intersection operations is:
N_{bi} = exp(9.40 + 0.55ln ADT_{1} + 0.65ln ADT_{2} + 0.31 VCI + 0.14ND_{1}  0.0049 SKEW_{4}) 
(60) 
Equation (60) includes all of the candidate independent variables except speed limit (p=0.25 to 0.50), presence of rightturn lane (p=0.59), and horizontal curvature (p=0.48) which were not statistically significant, and roadside hazard rating (p=0.12) which was marginally statistically significant but whose coefficient was negative, which is opposite to the direction expected. Table 41 summarizes the model parameters and goodness of fit for the model in equation (60).
The model in equation (60) was reevaluated including only those variables that were statistically significant with a significance level of 0.150 or less. This model is presented below:
N_{bi} = exp(9.30 + 0.53ln ADT_{1} + 0.67ln ADT_{2} + 0.15ND_{1}  0.0049 SKEW_{4}) 
(61) 
Table 41. Model Parameters and Goodness of Fit for Equation (60).

Independent variable 

Overdispersion
parameter (k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ADT _{2} 
VCI 
ND_{1} 
SKEW_{4} 
Coefficient 
9.40 
0.55 
0.65 
0.31 
0.14 
0.0049 
0.253 
Standard deviation 
0.77 
0.085 
0.073 
0.27 
0.05 
0.0038 
0.079 
Significance level (p) 
0.0001 
0.0001 
0.0001 
0.25 
0.004 
0.1997 
0.0014 
Note: The values of goodnessoffit measures are R^{2} = 0.5495, R _{k}^{2} = 0.8131, and R _{PD}^{2} = 0.7183
Table 42. Model Parameters and Goodness of Fit for Equation (61).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ADT _{2} 
ND_{1} 
SKEW_{4} 
Coefficient 
9.30 
0.53 
0.67 
0.15 
0.0057 
0.293 
Standard deviation 
0.78 
0.085 
0.074 
0.043 
0.0039 
0.086 
Significance level (p) 
0.0001 
0.0001 
0.0001 
0.0004 
0.14 
0.0007 
Note: The values of goodnessoffit measures are R^{2}= 0.5020, R k^{2}= 0.7835, and R _{PD}^{2}= 0.7047
Table 42 summarizes the model parameters and goodness of fit for the model in equation (61).
A candidate model for fourleg STOPcontrolled intersections developed with the iterative offset technique is:
N_{bi} = exp(11.25 + 0.69ln ADT_{1} + 0.56ln ADT_{2} + 0.033 SPDI + 0.11ND_{1}  0.21 RT  0.0059 SKEW_{4}) 
(62) 
Equation (62) includes all of the candidate independent variables except horizontal curvature (p=0.51) and vertical curvature (p=0.30 to 0.41) which were not statistically significant, and roadside hazard rating (p=0.11) which was marginally statistically significant but whose coefficient was negative, which is opposite to the direction expected. Table 43 summarizes the model parameters and goodness of fit for the model in Equation (62). No value of R _{PD}^{2} is computed for equation (62) because this goodnessoffit measure is not directly applicable to iterative offset technique.
Table 43. Model Parameters and Goodness of Fit for Equation (62).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ADT _{2} 
SPDI 
ND_{1} 
RT 
SKEW_{4} 
Coefficient 
11.25 
0.69 
0.56 
0.033 
0.107 
0.21 
0.0059 
0.203 
Standard deviation 
1.59 
0.080 
0.073 
0.021 
0.054 
0.12 
0.0031 
0.065 
Significance level 
0.0001 
0.0001 
0.0001 
0.11 
0.048 
0.082 
0.057 
0.0018 
Note: The values of goodnessoffit measures are R^{2} = 0.5563, and Rk^{2} = 0.7984
All of the variables in equation (62) are statistically significant at the 0.15 significance level. Therefore, no alternative version of equation (62) was developed.
FourLeg Signalized Intersections
Candidate base models were developed for fourleg signalized intersections. The dependent variables used in these models were those described above for the intersectionrelated and selectedaccidenttype modeling techniques. All of the models for fourleg signalized intersections used a data sets of 49 intersections, 18 in California and 31 in Michigan, with 3 years of accident data (19931995) for each intersection. The candidate independent variables considered in predicting accidents at fourleg signalized intersections on rural twolane highways were:
 Average daily traffic volume on the major road.
 Average daily traffic volume on the minor road.
 Presence of protected leftturn signal phase on the major road.
 Grade rate for all vertical curves (including both crests and sags) within 244 m (800 ft) of the intersection along the major and minor roads.
 Percentage of minorroad traffic turning left at the signal for the morning and evening peak hours combined.
 Percentage of trucks (vehicles with more than four wheels) in the traffic stream entering the intersection from all approaches for morning and evening peak hours combined.
 Number of driveways within 76 m (250 ft) of the intersection on the majorroad leg.
Table 44 presents descriptive statistics for these variables.
Table 44. Descriptive Statistics for 49 FourLeg Signalized Intersections in California and Michigan Used in Modeling.
Variable 
Mean 
Standard
deviation 
Minimum 
25th
percentile 
Median 
75th
percentile 
Maximum 
ADT _{1} (veh/day) 
10,491 
4,331 
4,917 
7,568 
8,900 
13,133 
25,133 
ADT _{2} (veh/day) 
4,367 
2,369 
940 
2,800 
3,670 
5,080 
12,478 
ln ADT _{1} 
9.18 
0.39 
8.50 
8.93 
9.09 
9.48 
10.13 
ln ADT _{2} 
8.26 
0.49 
6.85 
7.93 
8.21 
8.53 
9.43 
SUMLADT 
17.44 
0.65 
16.34 
16.94 
17.49 
17.78 
19.14 
PCTLEFT_{2} (percent) 
28.4 
15.1 
2.5 
19.0 
25.7 
35.7 
75.70 
VEICOM (percent/100 ft)) 
1.88 
1.87 
0.00 
0.50 
1.43 
2.54 
8.13 
PTRUCK (percent) 
9.0 
6.7 
2.7 
5.0 
7.7 
11.2 
45.4 
ND_{1} 
3.00 
3.00 
0.00 
0.00 
3.00 
4.00 
15.00 
PROTLT ^{a} 
 
 
 
 
 
 
 
Conversion: 1 ft = 0.305 m
^{a} 43 percent of the fourleg signalized intersections have protected leftturn signal phases and 57 percent do not.
Negative binomial models were developed for predicting accident experience at fourleg signalized intersections on rural twolane highways using the modeling technique based on the investigating officer’s identification of intersectionrelated accidents. A candidate model developed using this approach is:
N_{bi} = exp(6.12 + 0.46 SUMLADT 0.61 PROTLT  0.013 PCTLEFT_{2} + 0.12VEICOM + 0.030PTRUCK) 
(63) 
where:
SUMLADT 
= 
ln ADT _{1} + ln ADT _{2}; 
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 leftturning traffic at the signal for the morning and evening peak hours combined; 
VEICOM 
= 
grade rate for all vertical curves (crests and sags) any portion of which is within 244 m (800 ft) of the intersection averaged for the major and minor road legs of the intersection; 
PTRUCK 
= 
percentage of trucks (vehicles with more than four wheels) entering the intersection for morning and evening peak hours combined; 
The variable PROTLT indicates the presence of either a fully protected leftturn signal phase or a protectedpermitted phase.
An alternative to equation (63) using the major and minorroad traffic volumes separately:
N_{bi} = exp(6.95 + 0.62ln ADT_{1} + 0.39ln ADT_{2}  0.68 PROTLT  0.014 PCTLEFT_{2} + 0.13 VEICOM + 0.032PTRUCK) 
(64) 
Tables 45 and 46 summarize the model parameters and goodness of fit for equations (63) and (64), respectively.
Negative binomial models were developed for predicting accident experience at fourleg signalized intersections on rural twolane highways using the modeling technique based on the accident types generally considered to be intersectionrelated. A model using the sum of the major and minorroad traffic volume variables is:
N_{bi} = exp(4.96 + 0.39 SUMLADT  0.38 PROTLT  0.015 PCTLEFT_{2} + 0.103 VEICOM + 0.027PTRUCK) 
(65) 
An alternative form of equation (65) using separate variables for the major and minorroad traffic volumes is:
N_{bi} = exp(6.084 + 0.60ln ADT_{1} + 0.29ln ADT_{2}  0.47 PROTLT  0.017 PCTLEFT_{2} + 0.11 VEICOM + 0.029PTRUCK) 
(66) 
Negative binomial models comparable to equations (65) and (66) including an independent variable representing the number of driveways within 76 m (250 ft) of the intersection were also developed. A model using the sum of the major and minorroad traffic volume variables is:
N_{bi} = exp(4.11 + 0.33 SUMLADT 0.30 PROTLT  0.016 PCTLEFT_{2} + 0.100 VEICOM + 0.023PTRUCK + 0.035 ND_{1}) 
(67) 
Table 45. Model Parameters and Goodness of Fit for Equation (63).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
SUMLADT 
PROTLT 
PCTLEFT_{2} 
VEICOM 
PTRUCK 
Coefficient 
6.12 
0.46 
0.611 
0.013 
0.12 
0.030 
0.12 
Standard deviation 
2.60 
0.15 
0.151 
0.0048 
0.051 
0.014 
0.032 
Significance level (p) 
0.018 
0.0017 
0.0001 
.0052 
0.014 
0.033 
0.0002 
Note: The values of goodnessoffit measures are R^{2} = 0.5208, R _{k}^{2} = 0.6414, and R _{PD}^{2} = 0.2550
Table 46. Model Parameters and Goodness of Fit for Equation (64).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ADT _{2} 
PROTLT 
PCTLEFT_{2} 
VEICOM 
PTRUCK 
Coefficient 
6.95 
0.62 
0.39 
0.68 
0.014 
0.13 
0.032 
0.12 
Standard deviation 
2.79 
0.25 
0.17 
0.18 
0.0047 
0.045 
0.014 
0.032 
Significance level 
0.013 
0.013 
0.023 
0.0002 
.0023 
0.0039 
0.028 
0.0003 
Note: The values of goodnessoffit measures are R^{2} = 0.5053, R _{k}^{2} = 0.6490, and R _{PD}^{2} = 0.2362
where:
ND_{1} 
= 
number of driveways with 76 m (250 ft) of the intersection on the major road. 
An alternate form of equation (67) using separate variables for the major and minorroad traffic volumes is:
N_{bi} = exp(5.46 + 0.60ln ADT_{1} + 0.20ln ADT_{2}  0.40PROTLT  0.018 PCTLEFT_{2} + 0.11 VEICOM + 0.026PTRUCK + 0.041 ND_{1}) 
(68) 
Tables 47 through 50 summarize the model parameters and goodness of fit for equations (65) through (68), respectively.
A negative binomial model for predicting accident experience at fourleg signalized intersections on rural twolane highways was also developed using the iterative offset technique. This model uses a different functional form for the major and minor road ADT variables than was used in the preceding models. This model is:
N_{bi} = ADT_{1}^{0.307} exp(0.0000376 ADT_{1}) ADT_{2}^{0.461 LTLN1 RTLN1} 
(69) 
where:
LTLN_{1} 
= 
factor for number of majorroad leftturn lanes present at the intersection
(RTLN_{1} = 1.000 for zero leftturn lanes, 0.934 for one leftturn lane, and 0.737 for two leftturn lanes.); and 
RTLN_{1} 
= 
The value of the overdispersion parameter for this model is 0.26. 
Table 47. Model Parameters and Goodness of Fit for Equation (65).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
SUMLADT 
PROTLT 
PCTLEFT_{2} 
VEICOM 
PTRUCK 
Coefficient 
4.96 
0.39 
0.38 
0.015 
0.103 
0.027 
0.14 
Standard deviation 
3.078 
0.18 
0.17 
0.006 
0.042 
0.013 
0.039 
Significance level (p) 
0.107 
0.0309 
0.022 
0.0101 
0.013 
0.040 
0.0005 
Note: The values of goodnessoffit measures are R^{2}= 0.3913, R _{k}^{2}= 0.5382, and RPD^{2}= 0.1770
Table 48. Model Parameteres and Goodness of Fit for Equation (66).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
ln ADT _{1} 
ln ^{ADT} 
2 PROTLT 
PCTLEFT_{2} 
VEICOM 
PTRUCK 
Coefficient 
6.084 
0.60 
0.29 
0.47 
0.017 
0.113 
0.029 
0.13 
Standard deviation 
3.39 
0.28 
0.20 
0.20 
0.0057 
0.037 
0.013 
0.039 
Significance level 
0.072 
0.037 
0.14 
0.019 
.0036 
0.002 
0.028 
0.0008 
Note: The values of goodnessoffit measures are R^{2} = 0.3650, R _{k}^{2} = 0.5521, and R _{PD}^{2} = 0.1816
Table 49. Model Parameters and Goodness of Fit for Equation (67).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
SUMLADT 
PROTLT 
PCTLEFT_{2} 
VEICOM 
PTRUCK 
ND_{1} 
Coefficient 
4.11 
0.332 
0.302 
0.016 
0.100 
0.023 
0.035 
0.12 
Standard deviation 
2.95 
0.172 
0.175 
0.0055 
0.038 
0.012 
0.027 
0.037 
Significance level (p) 
0.163 
0.053 
0.083 
0.0038 
0.009 
0.055 
0.199 
0.0011 
Note: The values of goodnessoffit measures are R^{2}= 0.4563, R _{k}^{2}= 0.2834, and RPD^{2} = 0.1822
Table 50. Model Parameters and Goodness of Fit for Equation (68).

Independent variable 
Overdispersion
parameter (k) 
Parameter 
Intercept 
ADT _{1} 
ln ADT _{2} 
PROTLT 
PCTLEFT_{2} 
VEICOM 
PTRUCK 
ND_{1} 
Coefficient 
5.46 
0.599 
0.202 
0.404 
0.018 
0.111 
0.026 
0.098 
0.11 
Standard 
3.19 
0.279 
0.192 
0.188 
0.005 
0.033 
0.012 
0.041 
0.040 
Significance level 
0.087 
0.032 
0.293 
0.032 
0.0005 
0.0006 
0.028 
0.025 
0.0043 
Note: The values of goodnessoffit measures are R^{2} = 0.4327, R _{k}^{2} = 0.6094, and R _{PD}^{2} = 0.1816
Assessment of Alternative Base Models
A careful examination and comparison was made of the alternative base models presented above to select the most suitable models for use in the accident prediction algorithm. The final selections were made based on a sensitivity analysis and an assessment of the combined roadway segment and intersection models to predict total accidents within 76 m (250 ft) of the set of study intersections as a whole, including both intersectionrelated and nonintersectionrelated accidents.
Equation (42) was the only candidate roadway segment model and that model was found to be satisfactory for use as the base model for roadway segments. This model also appears as equation (5) in the main text of the report.
Six candidate models were developed for threeleg STOPcontrolled intersections. Of these six models, the models developed using the officer’s assessment of intersectionrelated accidents were found, together with the roadway segment model, to provide the best estimate of total accidents with in 76 m (250 ft) of the 382 study intersections. In choosing between equations (52) and (53), preference was given to equation (53) which includes only those variables found to be statistically significant with a significance level (p) of 0.015 or less. Therefore, equation (53) was selected as the base model for threeleg STOPcontrolled intersections and it also appears as equation (7) in the main text of this report.
Similar logic was used to select equation (59) as the base model for fourleg STOPcontrolled intersections. Equation (59) also appears as equation (9) in the main text of this report.
The assessment of the candidate models for fourleg signalized intersections was difficult because all seven candidate models were based on a relatively small sample size (49 intersections) and none of the models was found to be completely satisfactory. Equation (69) was appealing because it contains the same independent variables for which AMFs have been provided in the accident prediction algorithm. However, the functional form selected for equation (69) is such that the predicted accident frequency begins to decrease when the majorroad ADT exceeds approximately 10,000 veh/day. This does not appear reasonable, so equation (69) was eliminated from consideration. The best of the remaining candidate models was equation (68), which also appears in the main text of the report as equation (11).
Another Alternative Approach for Future Consideration
The accident prediction algorithm for twolane highways is based on the concept that accidents in the vicinity of an intersection can be predicted by separate models for intersectionrelated accidents and roadwaysegmentrelated (or nonintersectionrelated) accidents. The models used to predict roadway segment and intersection accidents are independent; this is based on the assumption that there is a certain “background” level of roadway segment accidents that would occur on the majorroad legs of the intersection whether the intersection was present or not. The accident prediction algorithm does not consider the possibility that the presence of the intersection could affect the frequency, severity, or collision types for accidents that are classified as nonintersectionrelated.
An alternative concept for considering the interaction described above was proposed during the development of the accident prediction algorithm. While this approach was, in the end, judged to have sufficient conceptual problems to be unworkable, this alternative approach is documented here for future reference in case a workable variation can be developed in the future.
The proposed concept was that a 152 m (500 ft) roadway segment along the major road (76 m or 250 ft on either side of the intersection) operates as an intersection approach during the time when traffic is present on the minorroad approach and operates as a roadway segment the rest of the time. Thus, if one could estimate the proportions of time when the segment operated as a roadway segment and as an intersection, its safety performance could be predicted as:
N_{t} = (1p) N_{rs} + (p) N_{int} 
(70) 
where:
N_{t} 
= 
Total predicted accident frequency for a 152 m (500 ft) roadway segmentcontaining an intersection; and 
p 
= 
proportion of time during which the roadway segment operates as an intersection (0£p£1). 
Values for N_{rs} and N_{int} would could be determined from the accident prediction algorithm as shown in Equations (13) and (14). Furthermore, it was proposed that the proportion p could be estimated as follows:
p = 
t_{minor}ADT_{2} 
, p £ 1.0 
(71) 

86,400 
where:
t_{minor} 
= 
average time each minorroad vehicle is present at or near the intersection (sec/veh); and 
ADT _{2} 
= 
average daily traffic volume on the minor road (veh/day). 
The average time each minorroad vehicle is present at or near the intersection, t_{minor}, could be estimated as some representatives value, such as 10 to 20 sec. The constant value of 86,400 in the denominator of equation (66) represents the number of seconds in a 24hour day.
The approach represented by equations (70) and (71) has substantial conceptual appeal because it incorporates a direct proportionality to the minorroad ADT. The approach could be strengthened conceptually by accounting for traffic turning from the major road onto the minor road and by adjusting for time periods when more than one minorroad vehicle is present at the intersection at the same time. Furthermore, the concept might also be adapted to the prediction of driveway accidents by assuming very low values of ADT_{2} for driveways. This could allow the accident prediction algorithm to address driveway accidents much more explicitly than is permitted by the driveway density AMF shown in equation (22).
The approach represented by equations (70) and (71) has conceptual disadvantages as well. First, it is based on a presumption that when a minorroad vehicle is present at the intersection there is no risk of an accident occurring elsewhere on the adjacent 152 m (500 ft) roadway segment. This presumption appears unrealistic. There must be some risk of an accident elsewhere on the major road when a minorroad vehicle is present at the intersection. Indeed, there is simply no evidence as to whether the presence of a minorroad vehicle at an intersection affects the risk of accidents elsewhere on the major road at all. Second, the form of equation (71) is such that, depending on the value of t_{minor} that is selected, there is some level at which p becomes equal to 1.0. At and above that ADT level, equation (70) implies that there would be no roadwaysectionrelated accidents within 152 m (500 ft) of the intersection. Such a proposition is also unrealistic.
^{On balance, this approach to intersection accident prediction discussed above was found to be unworkable and was abandoned. The concept has been presented here to document that it was considered and to suggest an alternative that might be adapted for use in the future.}
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