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Publication Number:  FHWA-HRT-13-077    Date:  January 2014
Publication Number: FHWA-HRT-13-077
Date: January 2014

 

Safety Effects of Horizontal Curve and Grade Combinations on Rural Two-Lane Highways

CHAPTER 4—STATISTICAL ANALYSIS

The overall statistical approach to estimating the safety effects of horizontal curve and grade combinations on rural two-lane highways is presented in this chapter along with the results for each type of combination for FI and PDO crashes.

ANALYSIS APPROACH

The safety effects of horizontal curve and grade combinations are estimated based on a cross-sectional analysis using a generalized linear model approach assuming a negative binomial (NB) distribution of crash counts and an exponential model using the combined crash data from all 6 years and selected roadway geometrics. FI and PDO crashes were modeled separately for each of the five types of horizontal curve and grade combinations.

Selection of Independent Variables Considered in Models

The parameters considered in each model may include the following:

For each type of horizontal curve and grade combination, the dataset used for modeling included the roadway segments for the relevant curve and grade combination but also all level tangents (i.e., no horizontal curvature and grade < 1 percent) to serve as the base condition.

Modeling was performed by encoding each parameter, or a transformation of it, so as to treat roadway segments on horizontal curve and grade combinations separately from tangent roadways on nonlevel grades and from level tangents (base condition). Segments for all of these types were used to develop a single model. This is equivalent to using indicator variables for each type of roadway segment. For example, since the natural log of the inverse radius was used in some of the models, the radius was recoded as ln(2 × 5,730 divided by R) based on the following reasoning:

the smallest value of 5,730 divided by R for horizontal curves in the database is 0.5. Therefore, the ratio, 5,730 divided by R , was multiplied by 2 such that the smallest value of ln(2 × 5,730 divided by R) becomes zero. This term was then set equal to zero for all tangents. This approach ultimately allows for the computation of CMFs using the developed models (discussed in further detail in chapter 5 of this report).

Within each vertical alignment type (i.e., straight grades, type 1 crest vertical curves, type 1 sag vertical curves, type 2 crest vertical curves, and type 2 sag vertical curves), the functional form of the relationship between crash frequency (FI or PDO) and the parameters listed was assessed separately. The following approach was used to explore the appropriate functional form for these relationships:

*Modified on November 16, 2014

Based on the visual assessment of these relationships, a final model form was selected using all parameters and relevant interactions. The parameters in these final model forms were continuous variables. In other words, the categorization into three groups was used only for exploring the potential functional forms for the data and was not used in the final models.

Before analyzing crashes on horizontal curves, tangents on nonlevel grades, and level tangents using a single model, the effect of vertical curve characteristics was assessed using tangents alone. This was done by comparing the effect of vertical curve characteristics on crashes/mi/year between level tangents and tangents on nonlevel grades for each type of vertical curve. This subset of data represents the largest proportion of the database. If a vertical curve parameter showed a statistically significant effect for tangents on nonlevel grades, then that parameter was included as such in the model using all three types of roadway (horizontal curves, level tangents, and tangents on nonlevel grades).

An attempt was also made to model crashes on horizontal curves and tangents by allowing separate intercepts and AADT slopes (i.e., an analysis of covariance) for each segment type to assess whether the relationship between crash frequency and AADT differs among horizontal curves, tangents on nonlevel grades, and base condition tangents. This modeling effort was inconclusive (e.g., slopes were inconsistent and counterintuitive) and therefore abandoned.

Final crash prediction models were derived for horizontal curves and tangents using the same group of level tangent sections as the base condition for all five horizontal curve and grade combinations. A stepwise approach was used where all parameters and interactions were included. The least significant interaction(s) and then the least significant parameter(s) were eliminated one at a time until all remaining interactions and parameters were significant. This is known as backwards stepwise selection. At each step, extreme data points were excluded from the data using leverage estimates, residuals, or Cook´s D criterion, all statistical criteria to evaluate the goodness-of-fit of the model to the data. In general, a 5 percent significance level associated with the type 3 chi-squared statistic was selected. All analyses were performed using a procedure for fitting generalized linear models of SAS® Version 9.3.(10)

It should be noted that additional geometric features for roadway segments, such as lane and shoulder widths, were not included in the analysis. The decision to exclude other geometric features was made because they were outside the scope of the current research. Experience with the Zegeer et al. results found that the roadway width term was dropped out of the final CMF.(3,4,6) Also, it was unlikely that the available data would support inclusion of additional terms.

Assessment of Goodness-of-Fit of Final Models

Once a model for a specific horizontal and vertical alignment combination was finalized, its goodness-of-fit to the observed crash data was evaluated. For each alignment combination and severity level, predicted crash frequencies versus observed frequencies were plotted and assessed to determine how well the data followed the line of equality. However, it should be recognized that perfect or near perfect agreement between predicted and observed crash frequencies should not be expected no matter how good the methodology is. The predicted crash frequencies, at their best, represent an estimate of the long-term average crash frequency for similar roadway segments. The observed crash frequency is simply one observation from a random process whose mean is estimated by the predicted crash frequency. There is no reason to expect that one observation from a random process should exactly equal the long-term mean. In addition, the methodology cannot predict a crash frequency of zero because each model has a positive intercept. This is reasonable because no roadway segment can ever be expected to be crash-free in the long term. However, in any given time period, it is reasonable to expect that many roadway segments, particularly lower volume segments, will experience zero observed crashes.

To assess the goodness-of-fit of each model, the level of agreement between predicted and observed crash frequencies was estimated based on the percentage of extreme observations in the 6-year period at the 5 percent significance level. Consideration of the likelihood of extreme observed crash frequencies, either high or low, is a method that directly takes into account the parameters of the NB distribution (i.e., mean and dispersion) of crash frequencies at a particular type of roadway segment.

Let Oi denote the number of observed crashes of a given type (FI or PDO) on a roadway segment, i, during the 6-year period. The likelihood of observing Oi crashes in 6 years is then computed under the assumption that Oi is an observation from a NB distribution with mean and dispersion parameter k. The mean is the predicted number of crashes in the 6-year period calculated using the prediction model applicable for roadway segment i. The dispersion parameter is obtained when developing the final model.

The likelihood, pi, of observing Oi or fewer crashes at roadway segment i can then be written as follows shown in figure 6 and figure 7*:

*Modified on November 16, 2014

p subscript i equals prob times open parenthesis number of crashes is less than or equal to O subscript i closed parenthesis.

 

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

Or:

p subscript i equals the sum of f subscript i times open parenthesis x closed parenthesis from x equals zero to O subscript i.

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

Where fi(x) is the probability distribution function for an NB with μi and k and values of x = 0, 1, 2, …, Oi. If pi = 0.5, then pi was calculated as 1 - pi. This approach is equivalent to calculating the area under the cumulative distribution curve at either low or high tail of the distribution.

The following two final single-value criteria used to assess how well the observed crash frequencies can be estimated by the proposed methodology are proposed:

ANALYSIS RESULTS

This section presents the final modeling results for the five alignment categories for rural two-way highways. Each subsection is organized as follows:

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

Horizontal Curves and Tangents on Straight Grades

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

*Modified on November 16, 2014

. This illustration shows alignment combinations used in the analysis of horizontal curves and tangents on straight grades. Data used in the analysis are level (i.e., grade less than 1 percent in absolute value) tangents (base condition), tangents on nonlevel (i.e., grades greater than or equal to 1 percent in absolute value) straight grades, and horizontal curves on straight grades.

Figure 9. Illustration. Alignment combinations used in the analysis of horizontal curves and tangents on straight grades.

Basic descriptive statistics such as sample size (i.e., number of roadway sections); total roadway length; and minimum, maximum, mean, and median values for specific parameters) (i.e., number of roadway sections; total roadway length; and minimum, maximum, mean, and median values for specific parameters)* are shown in table 7 for each of the three alignment types included in the analysis.

*Modified on November 16, 2014

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

Parameter

Minimum

Maximum

Mean

Median

Horizontal Curves on Straight Grades 
(N = 8,095; total roadway length = 595 mi)

AADT (vehicles/day)

169

26,088

2,695

1,664

Section length (mi)

0.01

0.75

0.07

0.05

Horizontal curve length (mi)

0.01

1.19

0.15

0.11

Curve radius (ft)

100

11,459

2,067

1,433

Grade (percent)

0

9.67

2.11

1.53

FI crashes per MVMT

0

39.50

0.75

0

PDO crashes per MVMT

0

46.26

0.91

0

Total crashes per MVMT

0

54.62

1.66

0

Tangents on Nonlevel Grades
(N = 7,569; total roadway length = 727 mi)

AADT (vehicles/day)

169

26,088

2,700

1,644

Section length (mi)

0.01

0.99

0.10

0.06

Horizontal curve length (mi)

 

 

 

 

Curve radius (ft)

 

 

 

 

Grade (percent)

1.00

10.85

3.10

2.64

FI crashes per MVMT

0

39.33

0.61

0

PDO crashes per MVMT

0

44.14

0.80

0

Total crashes per MVMT

0

53.48

1.42

0

Level Tangents—Base Condition
(N = 5,701; total roadway length = 779 mi)

AADT (vehicles/day)

169

26,088

3,285

2,153

Section length (mi)

0.01

0.98

0.14

0.09

Horizontal curve length (mi)

 

 

 

 

Curve radius (ft)

 

 

 

 

Grade (percent)

 

 

 

 

FI crashes per MVMT

0

34.21

0.46

0

PDO crashes per MVMT

0

39.50

0.67

0

Total crashes per MVMT

0

55.38

1.13

0

       Note: No roadway segments exist in the shaded cells.

The final crash prediction models for FI and PDO crashes are as follows shown in figure 10 and figure 11*:

*Modified on November 16, 2014

N subscript FI equals exponent open bracket b subscript 0 plus b subscript 1 times natural logarithm of open parenthesis AADT closed parenthesis plus b subscript 2 times G plus b subscript 3 times natural logarithm of open parenthesis 2 times 5,730 divided by R closed parenthesis times I subscript HC plus b subscript 4 times open parenthesis 1 divided by R closed parenthesis times open parenthesis 1 divided by L subscript C closed parenthesis times I subscript HC closed bracket.

Figure 10. Equation. Predicted FI crashes on horizontal curves and tangents on straight grades (general form).

N subscript PDO equals exponent open bracket b subscript 0 plus b subscript 1 times natural logarithm of open parenthesis AADT closed parenthesis plus b subscript 2 times G plus b subscript 3 times natural logarithm of open parenthesis 2 times 5,730 divided by R closed parenthesis times I subscript HC plus b subscript 4 times open parenthesis 1 divided by R closed parenthesis times open parenthesis 1 divided by L subscript C closed parenthesis times I subscript HC closed bracket.

Figure 11. Equation. Predicted PDO crashes on horizontal curves and tangents on straight grades (general form).

Where:
NFI = FI crashes per mile per year.
NPDO = PDO crashes per mile per year.
AADT = Vehicles per day.
G = Absolute value of percent grade (0 percent for level tangents; ≥ 1 percent otherwise).
R = Curve radius (ft) (missing for tangents).
IHC = Horizontal curve indicator (1 for horizontal curves; 0 otherwise).IHC = Horizontal curve indicator (1 for horizontal curves; 0 otherwise).
LC = Horizontal curve length (mi) (not applicable for tangents).
ln = Natural logarithm function.
b0,…, b4 = Regression coefficients.

The regression results, including the coefficient estimate, dispersion parameter, standard error, confidence limit, chi-squared statistic, and significance level for all statistically significant parameters and interaction are shown in table 8.

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

Parameter
Description

Regression
Coefficient

Coefficient
Estimate

Standard
Error

Lower
95 Percent
Confidence
Limit

Upper
95 Percent
Confidence
Limit

Chi-
Squared
Statistic

Significance
Level

FI Crashes/Mi/Year

Intercept

b0

-8.76

0.15

-9.05

-8.46

N/A

N/A

ln(AADT)

b1

1.00

0.02

0.96

1.03

3,052.7

< 0.0001

Grade

b2

0.044

0.01

0.03

0.06

27.5

< 0.0001

1/radius terma

b3

0.19

0.02

0.16

0.22

116.3

< 0.0001

1/R × 1/LC interaction

b4

4.52

0.79

2.97

6.07

26.8

< 0.0001

Dispersion

N/A

0.85

0.04

0.77

0.94

N/A

N/A

PDO Crashes/Mi/Year

Intercept

b0

-8.63

0.14

-8.89

-8.36

N/A

N/A

ln(AADT)

b1

1.03

0.02

1.00

1.06

4,003.5

< 0.0001

Grade

b2

0.040

0.01

0.03

0.05

29.1

< 0.0001

1/radius terma

b3

0.13

0.02

0.10

0.16

67.4

< 0.0001

1/R × I/LC interaction

b4

3.80

0.84

2.15

5.45

17.3

< 0.0001

Dispersion

N/A

0.80

0.03

0.73

0.87

N/A

N/A

a1/radius term = ln(2 × 5,730/R).
N/A = Not applicable

Applying figure 8, the percentage of roadway segments with extremely high observed FI crash frequencies was 6.09 percent across all roadway segments, which was only slightly above the expected 5 percent. The percentages were 6.12 percent for segments on level tangents, 6.25 percent for segments on tangents on nonlevel grades, and 5.91 percent for segments on horizontal curves on straight grades, indicating that a few roadway segments in these roadway categories experienced unusually high FI crash frequencies given the prediction model used. None of the segments experienced extremely low FI crash frequencies under the assumed model at the 5 percent significance level.

Similarly, the percentage of roadway segments with extremely high observed PDO crash frequencies was 6.56 percent across all roadway segments, which was slightly higher than that for FI crashes. The percentages were 6.71 percent for segments on level tangents, 6.90 percent for segments on tangents on nonlevel grades, and 6.14 percent for segments on horizontal curves on straight grades, indicating that a few roadway segments in these roadway categories experienced unusually high PDO crash frequencies given the prediction model used. None of the segments experienced extremely low PDO crash frequencies under the assumed model at the 5 percent significance level.

The average probability of predicting a more extreme than observed FI crash frequency was calculated using figure 7. Across all roadway segments, the average probability was 0.13. The average probability was 0.14 for segments on level tangents, 0.11 for segments on tangents on nonlevel grades, and 0.13 for segments on horizontal curves on straight grades, all considerably lower than the theoretically expected value of 0.25.

Similarly, the average probability of predicting a more extreme than observed PDO crash frequency was 0.14 across all roadway segments. The average probability was 0.16 for segments on level tangents, 0.13 for segments on tangents on nonlevel grades, and 0.15 for segments on horizontal curves on straight grades, all considerably lower than the theoretically expected value of 0.25. For both FI and PDO crashes, these low probabilities are an indication that the model might not provide an adequate fit to the data. This is not too surprising given the large number of roadway segments with a wide range of AADTs and geometrics and a high percentage of segments with zero crashes.

Substituting the regression coefficients in figure 10 and figure 11 with their corresponding estimates in table 8, the prediction models for FI and PDO crashes/mi/year are as follows shown in figure 12, figure 13, and figure 14*:

*Modified on November 16, 2014

N subscript FI equals exponent open bracket -8.76 plus 1.00 times natural logarithm of open parenthesis AADT closed parenthesis plus 0.044 times G plus 0.19 times natural logarithm of open parenthesis 2 times 5,730 divided by R closed parenthesis times I subscript HC plus 4.52 times open parenthesis 1 divided by R closed parenthesis times open parenthesis 1 divided by L subscript C closed parenthesis times I subscript HC closed bracket.

Figure 12. Equation. Predicted FI crashes on horizontal curves and tangents on straight grades (explicit form).

N subscript PDO equals exponent open bracket -8.63 plus 1.03 times natural logarithm of open parenthesis AADT closed parenthesis plus 0.040 times G plus 0.13 times natural logarithm of open parenthesis 2 times 5,730 divided by R closed parenthesis times I subscript HC plus 3.80 times open parenthesis 1 divided by R closed parenthesis times open parenthesis 1 divided by L subscript C closed parenthesis times I subscript HC closed bracket.

Figure 13. Equation. Predicted PDO crashes on horizontal curves and tangents on straight grades (explicit form).

N subscript Total equals N subscript FI plus N subscript PDO.

Figure 14. Equation. Predicted total crashes on horizontal curves and tangents on straight grades (general form).

Table 8 clearly shows that AADT is the most predominant predictor of crashes as indicated by its high chi-squared statistic. Other parameters and interaction are one or two orders of magnitude smaller and indicate that once the variability due to traffic volume is accounted for, the remaining parameters explain only a small portion of the remaining variability in the data. This is a consistent trait across all models developed in this project.

These final models include only two significant parameters and an interaction from the list of parameters and interactions originally considered. This is in large part because all safety effects are estimated relative to level tangents as the base condition. Crash frequencies in this group of roadway segments alone (5,701 segments for a total of 779 mi) exhibit considerable variability (e.g., FI crash rates range from zero to 34.21 with a mean of 0.46 and a median of zero crashes per MVMT as shown in table 7). To detect a significant effect of any horizontal curve characteristic on straight grades, the effect of such a characteristic would need to be large relative to the variability in the base condition set. This, in effect, is the challenge in finding statistically significant safety effects of practical engineering relevance.

Figure 12 and figure 13 show that crash frequency increases with increasing percent grade and decreases with increasing curve radius, as expected. The interaction term between radius and curve length represents an additional effect on safety for short and sharp horizontal curves—as the radius decreases and the curve shortens, the last term in figure 12 and figure 13 increases, adding to the crash frequency. For long horizontal curves and curves with larger radii, this term approaches zero and thus will have little impact on the predicted crash frequency.

Of interest is the fact that the effects of percent grade, curve radius, and the interaction between radius and curve length is more pronounced for FI crashes than for PDO crashes.

Initial modeling effects indicated that there might be an interaction between horizontal curve radius and percent grade on straight grade segments, suggesting that the effect of curve radius on crash frequency might change with increasing percent grade. However, this effect was found to be an artifact of a few data points that were clearly outliers. These outliers were eliminated from the dataset used to produce the final models presented in figure 12 and figure 13. Thus, the analysis did not find a statistically significant interaction between horizontal curve radius and percent grade. The results of the study presented in this report are as follows:

Horizontal Curves and Tangents at Type 1 Crest Vertical Curves
The following three alignment combinations shaded in figure 15 were included in the analysis:

 

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