Proposed Guidelines for Fixed Objects in the Roadside Design Guide (2022)

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53 Chapter 5. Empirical Model Development This chapter summarizes the development of empirical crash prediction models using the data assembled, as described in Chapter 4. These empirical models were developed with negative binomial regression and logistic regression. Recent experience has shown that the most appropriate approach to regression modeling with crash frequency as the dependent variable is generally negative binomial regression. Ordinary least squares regression assumes that the crash measure used as the dependent variable is distributed according to a normal distribution. In fact, the distribution of most crash data is nonnormal, so an ordinary least squares approach would be potentially biased. Poisson regression has also been tried for modeling crash data. Poisson regression assumes that the crash data follow a Poisson distribution. The key assumption in a Poisson distribution is that the mean and variance of the data are equal. In fact, most crash data have a variance substantially larger than the mean, which is referred to as overdispersion. Therefore, the use of Poisson regression for crash modeling would be potentially biased. The most suitable regression technique for modeling overdispersed (or underdispersed) data is negative binomial regression analysis. Negative binomial regression can account for the potential bias resulting from overdispersed (or underdispersed) crash data. 5.1 Model Forms Used with Negative Binomial Regression The following discussion presents the application of negative binomial regression analysis to predicting crashes in the Kentucky database for two target crash types: tree-related and utility-pole-related crashes. The dependent variables used in negative binomial regression models for crash frequency on specific roadway segments include: • total target crash frequency (including all crash severity levels combined) • fatal-and-injury target crash frequency • property-damage-only target crash frequency The negative binomial regression models were developed using a general linear modeling tool known as PROC GLIMMIX in the Version 9.4 SAS® software package. Models were developed separately for tree groups, individual trees, and individual utility poles located on the roadside within 40 ft of the traveled way. For the reasons indicated in Chapter 4, all modeling was for rural two-lane undivided highways. The independent variables considered in negative binomial regression modeling included: • AADT for the road segment on which the roadside fixed object is located • Offset distance (in feet) from the traveled way to the roadside fixed object (for multiple fixed objects, the offset for the fixed object closest to the roadway was used) • Indicator variable for horizontal curve vs. tangent (equal to one for roadside objects on horizontal curves and equal to zero for roadside objects on tangent roadways)

54 • For roadside objects located on horizontal curves only, indicator variable for roadside object located on outside vs. inside of horizontal curve (equal to one for location on outside of horizontal curve and equal to zero for roadside object on the inside of a horizontal curve) The crash prediction models also included: • consideration of the number of years of crash data available as a scale factor (in Kentucky, 5 years of crash data were used for all sites) • consideration of the frequency of roadside objects within a given roadside object as a scale factor; the frequency variable was expressed as: - for tree groups, longitudinal extent (length given in feet) of continuous trees within the roadway segment - for individual trees, number of trees within the roadway segment - for individual utility poles, number of poles within the roadway segment Negative binomial models in six forms were considered: Model Form #1 𝑁 = exp (𝑏 + (𝑏 × 𝐴𝐴𝐷𝑇) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇)) (7) Model Form #2 𝑁 = exp (𝑏 + (𝑏 × 𝐴𝐴𝐷𝑇) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇) + (𝑏 × 𝐶𝑈𝑅𝑉𝐸𝑇𝐴𝑁)) (8) Model Form #3 𝑁 = exp (𝑏 + (𝑏 × 𝐴𝐴𝐷𝑇) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇) + (𝑏 × 𝐶𝑈𝑅𝑉𝐸_𝐼𝑁𝑂𝑈𝑇)) (9) Model Form #4 𝑁 = exp (𝑏 + (𝑏 × 𝑙𝑛(𝐴𝐴𝐷𝑇)) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇)) (10) Model Form #5 𝑁 = exp (𝑏 + (𝑏 × 𝑙𝑛(𝐴𝐴𝐷𝑇)) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇) + (𝑏 × 𝐶𝑈𝑅𝑉𝐸𝑇𝐴𝑁)) (11)

55 Model Form #6 𝑁 = exp (𝑏 + (𝑏 × 𝑙𝑛(𝐴𝐴𝐷𝑇)) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇) + (𝑏 × 𝐶𝑈𝑅𝑉𝐸_𝐼𝑁𝑂𝑈𝑇)) (12) where: Nx = placeholder for NTOT, NFI, or NPDO NTOT = predicted number of total crashes of all severity levels per year per foot of roadside edge with trees for groups, per year per tree for individual trees, or per year per utility pole for individual utility poles NFI = predicted number of fatal-and-injury crashes per year per foot of roadside edge with trees for groups, per year per tree for individual trees, or per year per utility pole for individual utility poles NPDO = predicted number of property damage only per year per foot of roadside edge with trees for groups, per year per tree for individual trees, or per year per utility pole for individual utility poles b0, b1, b2, b3, and b4 = regression coefficients AADT = annual average daily traffic volume (veh/day) OFFSET = offset distance from traveled way to roadside object (ft) CURVETAN = indicator variable for curve vs. tangent (=0 for roadside fixed objects on tangents; =1 for roadside fixed objects on a horizontal curve) CURVE_INOUT = indicator variable for inside of curve vs. outside of curve (=0 for roadside fixed objects on the inside of a horizontal curve; =1 for roadside fixed objects on the outside of a horizontal curve These six model forms were used for negative binomial regression with Nx representing the crash frequency for total crashes (NTOT), fatal-and-injury crashes (NFI), and property-damage-only crashes (NPDO). 5.2 Model Forms Used with Logistic Regression Logistic regression was used because a large percentage of the roadway segments experienced either zero or one target crash in the five-year study period. Logistic regression models the outcome probability of a binary process; i.e., one with two and only two outcomes. In this case, the two outcomes modeled were that there were zero crashes during the five-year study period or there were a nonzero number of crashes during the five-year study period. The six model forms considered for the logistic regression analysis, analogous to those used for negative binomial regression were: Model Form #7 𝑃 = 1 − 𝑆𝐹 × exp (𝑏 + (𝑏 × 𝐴𝐴𝐷𝑇) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇)) (13)

56 Model Form #8 𝑃 = 1 − 𝑆𝐹 × exp (𝑏 + (𝑏 × 𝐴𝐴𝐷𝑇) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇) + (𝑏 × 𝐶𝑈𝑅𝑉𝐸𝑇𝐴𝑁)) (14) Model Form #9 𝑃 = 1 − 𝑆𝐹 × exp (𝑏 + (𝑏 × 𝐴𝐴𝐷𝑇) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇) + (𝑏 × 𝐶𝑈𝑅𝑉𝐸_𝐼𝑁𝑂𝑈𝑇)) (15) Model Form #10 𝑃 = 1 − 𝑆𝐹 × exp (𝑏 + (𝑏 × 𝑙𝑛(𝐴𝐴𝐷𝑇)) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇)) (16) Model Form #11 𝑃 = 1 − 𝑆𝐹 × exp (𝑏 + (𝑏 × 𝑙𝑛(𝐴𝐴𝐷𝑇)) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇) + (𝑏 × 𝐶𝑈𝑅𝑉𝐸𝑇𝐴𝑁)) (17) Model Form #12 𝑃 = 1 − 𝑆𝐹 × exp (𝑏 + (𝑏 × 𝑙𝑛(𝐴𝐴𝐷𝑇)) + (𝑏 × 𝑂𝐹𝐹𝑆𝐸𝑇) + (𝑏 × 𝐶𝑈𝑅𝑉𝐸_𝐼𝑁𝑂𝑈𝑇)) (18) where: Px = placeholder for PTOT, PFI, or PPDO PTOT = probability that a road segment 327-ft in length with specified characteristics will experience nonzero total crash frequency for all severity levels per year per foot of roadside edge with trees for tree groups, per year per tree for individual trees, or per year per utility pole for individual utility poles PFI = probability that a road segment 327-ft in length with specified characteristics will experience nonzero fatal-and-injury crash frequency per year per foot of roadside edge with trees for tree groups, per year per tree for individual trees, or per year per utility pole for individual utility poles PPDO = probability that a road segment 327-ft in length with specified characteristics will experience nonzero property-damage only crash frequency per year per foot of roadside edge with trees for tree groups, per year per tree for individual trees, or per year per utility pole for individual utility poles SF = Scale factor appropriate to the type of roadside object to which the model is applied (LTG for tree groups; NTREE for individual trees; NPOLES for utility poles) LTG = Length of roadside edge (ft) with tree groups present within a 327-ft roadway segment (NOTE: maximum value of LTG within a 327-ft roadway segment is 2 x 327 = 654 ft) NTREE = Number of individual trees on the roadside for both sides of the road combined within a 327-ft roadway segment NPOLE = Number of individual utility poles on the roadside for both sides of the road combined within a 327-ft roadway segment

57 The results of the crash prediction modeling are presented below separately by roadside object type and modeling approach. All statements concerning statistical significance in this report apply to the 5- percent significance level (i.e., the 95-percent confidence level), unless otherwise stated. A total of 216 regression analyses were performed with the Kentucky data. These regression analyses consisted of models for: • 3 object types (tree groups, individual trees, and individual utility poles) • 2 regression modeling types (negative binomial regression and logistic regression) • 3 crash severity levels (total crashes, fatal-and-injury crashes, and property-damage only crashes) • 2 crash categories (primary crashes; and combined primary and secondary crashes) • 6 model types [see Equations (7) through (12) and Equations (13) through (18) presented above] All combinations of these factors were considered in modeling, thus, constituting 3 × 2 × 3 × 2 × 6 = 216 regression analyses conducted. 5.3 Modeling Results for Tree Groups Using Negative Binomial Regression A total of 36 models were tried for tree groups in Kentucky using negative binomial regression analysis. For the 12 models tried for total tree-related crashes at tree groups (NTOT) as the dependent variable, there were no models for any of the Model Forms #1 through #6 for which the AADT term and the offset distance terms were both statistically significant, even at the 20-percent significance level (80-percent confidence level). The same result was found for the 12 models tried for fatal-and-injury tree-related crashes (NFI) as the dependent variable. For the 12 models tried for property-damage-only tree-related crashes at tree groups (NPDO), there was one model for which both the AADT and OFFSET terms were statistically significant and the effects were in the expected direction. The expected direction is a positive coefficient b1, indicating that tree-related crashes increase as traffic volume increases, and a negative coefficient b2, indicating that tree-related crashes decrease when the trees are further from the traveled way. This model uses Model Form #4. Table 28 presents the coefficients developed for this model form and their standard errors and statistical significance. Table 28. Negative Binomial Regression Analysis Results for Property-Damage-Only Primary Tree-Related Crashes on Roadway Segments with Roadside Tree Groups Symbol Parameter estimate Standard error DF t-value p-value Statistically Significant? b0 -16.0825 1.9855 1095 -8.10 <0.0001 YES b1 0.8367 0.2897 1095 2.89 0.0039 YES b2 -0.08096 0.0428 1095 -1.89 0.0588 YESa NOTE: This model is based on Model Form #4 [see Equation (10)] a statistically significant at 10% significance level (90% confidence level)

58 The values for b0 through b2 in the parameter estimate column in Table 28 are the values that are intended for use in the model shown in Equation (10) to compute an estimate of the target crash frequency, NT. The model intercept (b0), and the coefficients for AADT and offset from the traveled way to roadside trees (b1 and b2, respectively) are all statistically significant. The positive value of the parameter estimate b1 indicates that, as would be expected, the target crash frequency increases as the traffic volume increases. The negative value of the parameter estimate for b2 indicates that the target crash frequency decreases as the offset from the traveled way to roadside trees increases. The Model #1 results, based on Equation (10) and Table 28, can be shown as: 𝑁 = exp (−16.0825 + (0.8367 × 𝑙𝑛(𝐴𝐴𝐷𝑇)) + (−0.08096 × 𝑂𝐹𝐹𝑆𝐸𝑇)) (19) In summary, only 1 of 36 negative binomial regression models for tree groups was statistically significant and that model was for PDO crashes. Table 29 shows that relative crash frequencies that can be used as crash modification factors (CMFs) for tree groups can be derived from Equation (19), but such CMFs are of limited applicability since they address only PDO crashes, the lowest priority of the three crash severity levels considered. Table 30 shows the variation of predicted tree-related crashes with offset from the traveled way to the tree group for this example based on Equation (19), again keeping in mind the limited applicability of this case for PDO crashes. Table 29. Predicted Property-Damage-Only Primary Tree-Related Crash Frequency as a Function of AADT for a Rural Two-Lane Undivided Highway Based on the Model Shown in Equation (19) AADT (veh/day) Predicted crash frequency for assumed conditions PDO tree-related crash frequency per mile per year Relative crash frequencya 1,000 0.301 1.00 2,000 0.538 1.79 3,000 0.755 2.51 4,000 0.961 3.19 5,000 1.158 3.84 6,000 1.349 4.49 7,000 1.535 5.09 8,000 1.716 5.70 9,000 1.894 6.29 10,000 2.068 6.87 Assumed conditions: rural two-lane undivided highway with continuous trees 2 ft outside the traveled way on both sides of the roadway. NOTE: a Ratio of crash frequency for specified AADT to Crash frequency for base condition (AADT = 1,000 veh/day).

59 Table 30. Predicted Property-Damage-Only Primary Tree-Related Crash Frequency as a Function of Offset Distance to Roadside Objects for a Rural Two-Lane Undivided Highway Based on the Model Shown in Equation (19) Offset Distance from Traveled Way to Roadside Trees (ft) Predicted crash frequency for assumed conditions PDO tree-related crash frequency per mile per year Relative crash frequencya 2 0.00166 1.00 3 0.00153 0.92 5 0.00130 0.78 10 0.00087 0.52 15 0.00058 0.35 20 0.00039 0.23 25 0.00026 0.16 30 0.00017 0.10 35 0.00011 0.07 40 0.00008 0.05 Assumed conditions: rural two-lane undivided highway with AADT of 1,000 veh/day and continuous trees at specified distances outside the traveled way on both sides of the roadway. NOTE: a Ratio of crash frequency for specified offset to crash frequency for base condition (offset distance to roadside trees = 2 ft). 5.4 Modeling Results for Individual Trees Using Negative Binomial Regression There were no negative binomial models with statistically significant AADT and OFFSET effects among the 36 models tried for individual trees using the Kentucky data, even at the 20-percent significance level (80-percent confidence level). 5.5 Modeling Results for Individual Utility Poles Using Negative Binomial Regression Even at the 20-percent significance level (80-percent confidence level), there was only one negative binomial model with even marginally statistically significant AADT and OFFSET effects for individual utility poles out of 36 models tried using the Kentucky data for individual utility poles (see Table 31). And, this model has an AADT term that is statistically significant only at the 15-percent significance level and an offset effect that is statistically significant only at the 20-percent significance level. Furthermore, this model does not appear useful because the coefficient of the OFFSET term is positive. It is counterintuitive to believe that the frequency of utility-pole-related crashes increases as the offset distance from the traveled way to the tree increases. Table 31. Negative Binomial Regression Analysis Results for Property-Damage-Only Primary and Secondary Utility-Pole-Related Crashes on Roadway Segments with Individual Utility Poles Symbol Parameter Estimate Standard error DF t-value p-value Statistically Significant? b0 -12.9861 3.4097 1205 -3.81 0.0001 YES b1 0.6468 0.4484 1205 1.44 0.1494 YESa b2 0.05476 0.0421 1205 1.30 0.1931 YESb NOTE: This model is based on Model Form #4 [see Equation (10)]. a statistically significant at 15% significance level (85% confidence level). b statistically significant at 20% significance level (80% confidence level).

60 5.6 Modeling Results for Tree Groups Using Logistic Regression Because of the limited results obtained with negative binomial regression, logistic regression models were also tried. Logistic regression estimates the probability that a nonzero crash frequency will occur for given roadway and roadside conditions. Out of 36 logistic regression models tried, only two statistically significant models were found, both for PDO crashes involving tree groups. The first of these models is a logistic regression model for PDO primary tree crashes using Model Form #10. Table 32 presents the coefficients developed for this model and their standard errors and statistical significance. Table 32. Logistic Regression Analysis Results for Property-Damage-Only Primary Tree-Related Crashes on Roadway Segments with Roadside Tree Groups Symbol Parameter Estimate Standard error DF t-value p-value Statistically Significant? b0 0.5926 2.2559 1095 0.26 0.7928 YES b1 -0.5472 0.3209 1095 -1.71 0.0885 YESa b2 0.0763 0.0428 1095 1.78 0.0755 YESa NOTE: This model is based on Model Form #10 [see Equation (A-16)]. a statistically significant at 10% significance level (90% confidence level). The results shown in Table 32, based on the model form shown in Equation (16), can be shown as: 𝑃 = 1 − 𝐿𝑇𝐺 × exp (0.5936 + (−0.5472 × 𝑙𝑛(𝐴𝐴𝐷𝑇)) + (0.0763 × 𝑂𝐹𝐹𝑆𝐸𝑇)) (20) The intercept term (b0), as shown in Table 32 is not statistically significant, so the model would need to be refit without an intercept term if it were to be used. The second model is also a logistic regression model for PDO crashes, in this case primary and secondary tree-related crashes using Model Form #10. Table 33 presents the coefficients developed for this model and their standard errors and statistical significance. Table 33. Logistic Regression Analysis Results for Property-Damage-Only Primary and Secondary Tree- Related Crashes on Roadway Segments with Roadside Tree Groups Symbol Parameter estimate Standard error DF t-value p-value Statistically Significant? b0 0.7891 1.8802 1095 0.42 0.6748 YES b1 -0.5551 0.2670 1095 -2.08 0.0379 YES b2 0.0443 0.0328 1095 1.35 0.1765 YESa NOTE: This model is based on Model Form #10 [see Equation (A-16)]. a statistically significant at 20% significance level (80% confidence level). The results shown in Table 33, based on the model form shown in Equation (16), can be shown as: 𝑃 = 1 − 𝐿𝑇𝐺 × exp (0.7891 + (−0.5551 × 𝑙𝑛(𝐴𝐴𝐷𝑇)) + (0.0443 × 𝑂𝐹𝐹𝑆𝐸𝑇)) (21) As was the case in Equation (20), the intercept term (b0) in Equation (21), as shown in Table 33, is not statistically significant, so the model would need to be refit without an intercept term if it were to be used.

61 The models shown in Equations (20) and (21) are very similar, as they address the same roadside object type and crash severity level with the same model form. They differ only in that the model in Equation (20) predicts probabilities for primary crashes with tree groups and the model in Equation (21) predicts probabilities for both primary and secondary crashes with tree groups. The models in Equations (20) and (21) have limited applicability, as does the model in Equation (19) because these equations address only PDO crashes related to tree groups. 5.7 Modeling Results for Individual Trees Using Logistic Regression Among the 36 logistic regression models tried using the Kentucky data for individual trees, there were no models with statistically significant AADT and OFFSET effects, even at the 20-percent significance level (80-percent confidence level). 5.8 Models for Individual Utility Poles Using Logistic Binomial Regression Among the 36 logistic regression models tried using the Kentucky data for individual utility poles, there were no models with statistically significant AADT and OFFSET effects, even at the 20-percent significance level (80-percent confidence level). 5.9 Discussion of Results As shown above, statistically significant models were found in the Kentucky data for only four of the 216 cases for which models were developed. Specifically, statistically significant models were found for only one roadside object type (tree groups) and only one crash severity level (PDO crashes). Tree groups experienced many more crashes on rural two-lane undivided highways than individual trees or utility poles, since tree groups include so many more roadside objects than individual trees or utility poles. In addition, there were more roadway segments with tree groups than with individual trees or utility poles. Therefore, it is not surprising that a crash prediction model for tree groups could be developed where crash prediction models for individual trees and utility poles could not. The limited models developed for the Kentucky data are of limited practical use since they represent only PDO crashes, while predicting fatal-and- injury crashes is of much higher priority for roadside design. There were no models where the horizontal curve vs. tangent effect or the inside of horizontal curve vs. outside of horizontal curve effects were statistically significant, although both previous research and the summary tables of statistics for crashes related to tree groups in Section 4.3 suggest that curves experience more roadside fixed-object collisions that tangents and that roadside fixed objects on the outside of curves experience more collisions than roadside fixed objects on the inside of curves. The results of the Washington modeling for trees were even more limited in usefulness than the Kentucky results in that no statistically significant models were found. For the Washington utility pole data, there were a number of models with statistically significant effects for the offset distance from the outside edge of the traveled way to utility poles, including models for total crashes (all severity levels combined), fatal-and-injury crashes, and property-damage-only crashes. These effects were in the expected direction; i.e., the number of crashes decreased as the

62 distance from the traveled way to the utility pole increased. However, in all of these models, the AADT term was either not statistically significant or was statistically significant in the opposite direction to the expected direction; i.e., some models indicated that crashes decreased as AADT increased. This may be the result of a correlation between variables that would be difficult to separate; i.e., it may be that utility poles tend to be closer to the road (and, thus, more likely to be struck) on roads with lower AADTs. The results from empirical modeling with the Kentucky and Washington data are disappointing. It appears likely that a much larger database would be needed to develop empirical models for tree- and utility-pole-related crashes. A second approach to design guidance development, based on adapting and using the existing RAP models, is presented in Chapters 6 and 7. This approach based on the RAP models appears more promising than empirical modeling.