Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers (2022)

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B-1   Probability of Reaching the Lateral Offset of Feature j—PY(Yj) A P P E N D I X B The work presented in this appendix represents a significant enrichment to the understanding of the lateral extent of errant vehicle trajectories during an encroachment. This effort was possible because of the cooperation of four separate research projects teams. NCHRP Project 16-05, “Guidelines for Cost-Effective Safety Treatments of Roadside Ditches,” NCHRP Project 17-55, “Guidelines for Slope Traversability,” and NCHRP Project 17-43, “Long-Term Roadside Crash Data Collection Program” each contributed data to the undertaking documented herein. This research project and roadside safety have benefited from the willingness of these other research project teams at the Texas Transportation Institute and Virginia Tech to collaborate and share the data collected under those ongoing efforts. CONTENTS Chapter 1 Introduction Chapter 2 Background Chapter 3 Available Data Data Summary Chapter 4 Data Analysis Software Used Assessment of Covariates Statistical Methods Survival Competing Risks Chapter 5 Field Collected Trajectories Chapter 6 Results Chapter 7 Implementing the Results References

B-2 Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers CHAPTER 1 INTRODUCTION One objective of this research was to determine the probability of an errant vehicle’s trajectory laterally extending to a location of interest (e.g., across the median, to the barrier). The influence of encroachment speed and angle, vehicle type, median/roadside terrain, and the shape of the median on the probable lateral extent, PY(Yj), of the errant vehicle’s trajectory were studied. The analysis of PY(Yj) and these causal elements is documented below. The encroachment probability model, as implemented in RSAPv3 as well as in RSAP and BCAP before it, assesses the probability of a crash based on passenger vehicle trajectories. In the case of BCAP and RSAP, a distribution of encroachment speeds and angles were used and straight-line trajectories were assumed. RSAPv3 included a database of reconstructed vehicle trajectories assembled under NCHRP Project 17-22 and assessed each reconstructed trajectory against individual obstacles. (Mak 2010; Ray 2012) The intent for the guidelines resulting from this research was to develop a selection process that can be included in the AASHTO Roadside Design Guide (RDG) and does not require the use of software such as RSAPv3 for each design decision. A model that represents the probable distribution of vehicle trajectories was therefore the desired outcome, not the continued use of software. In this study, the statistical field of survival analysis was applied for the ability to model time to event data. Background information on this statistical field and the available data are discussed below. Descriptive methods, as well as statistical methods, were explored to represent these data. Conclusions and recommendations are formed from these analyses and presented at the close of this attachment.

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-3   CHAPTER 2 BACKGROUND The outcome under assessment in epidemiological studies is often the time to an event of interest such as relapse of cancer or relief from disease. This time is generically known as survival time regardless of whether time to death or time to cure is being studied. The field of study known as survival analysis has evolved specifically to assess survival time. Time to event, survival time, life length, and time to death are terms used interchangeably to describe the outcome variable in survival analysis. The “event” may be failure of some mechanical component, length of life after an AIDS diagnosis, time to relapse after alcohol recovery, and others. If the failure of a mechanical component were the event of interest, the study would measure time in service for that component until failure or a pre-determined end of the study data collection. The measurement of time in the example may be hours in service, revolutions, or a different appropriate measure for the said mechanical component. When patients are being studied, as in length of life after an AIDS diagnosis, time would be measured as well as likely interventions (e.g., medicines). The age and/or sex of the patient at the start of the study may also be considered causal. In this study, the statistical field of survival analysis was applied for the ability to model time to event data. The event under assessment is the maximum lateral extent of an errant vehicle’s trajectory. In this context, survival time is considered the maximum lateral distance each trajectory traveled from the encroachment location (i.e., edge of travel). The unit of measure is feet from the encroachment location. The encroachment speed and angle as well as the vehicle type have been assessed for influence on the maximum lateral extent. Intermediate terrain changes are also included in the study to determine, what if any, influence these terrain changes have on survival time. The genesis of survival analysis can be found in the study of time to death; therefore, it was common to have a data set with events that were not observed (e.g., the study ends before all participants die). In the case of vehicle trajectories, data gathering might stop after, for example, 100 feet from the travel way, but the vehicle may not have been observed stopping. The unobserved events are certain to happen if observation were to continue long enough (e.g., death is inevitable, vehicles eventually run out of momentum). These types of data are known as censored data and are specifically addressed using this analysis technique. On the other hand, death from other causes is not considered censored. Recall the maximum lateral extent of the vehicle trajectory is the “event of interest.” To put a finer point on it, we are interested in those events involving the vehicle stopping or coming to rest with all four wheels on the ground. Using this definition, if a vehicle’s maximum lateral extent is observed not from stopping, but from rolling over (i.e., other cause), this is not a censored observation. The observation is complete and known as a competing risk. (Pintilie 2006) We certainly want to capture all the vehicles which traveled each foot of the terrain, and we want to also capture how the trajectory terminated. The various descriptive and statistical methods examined using survival analysis techniques, including the consideration of the censored data and the competing risks, are summarized below after the discussion of available data.

B-4 Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers CHAPTER 3 AVAILABLE DATA The NCHRP Project 17-22 data contains 787 reconstructed trajectories from run-off-road crashes. (Mak 2010) Also of interest, the ongoing NCHRP Project 17-43 data is augmenting the NCHRP Project 17-22 data. (Gabler 2020) A beta version of the NCHRP Project 17-43 data set was made available for use in this effort. Both data sets include reconstructed trajectories which resulted in a crash, prematurely ending the vehicle trajectory. While neither of these data sets includes the full distribution of the probable lateral extent of a vehicle trajectory when the roadside is free of other obstacles, both data sets do provide a distribution of encroachment angle and speed. Capturing the full lateral extent of vehicle trajectories (i.e., trajectories not involved in a crash) necessitates applying the findings to a more general roadside/median environment. The influence that fixed object or other roadside features such as barriers have on a crash is captured in a separate portion of the encroachment probability model; therefore, it is paramount that this model captures exclusively the influence of terrain, not other roadside features such as fixed objects or barriers. The ongoing NCHRP Project 16-05, “Guidelines for Cost-Effective Safety Treatments of Roadside Ditches,” included the simulation of vehicle trajectories through a variety of median ditches. (Sheikh 2021) The recently completed NCHRP Project 17-55 resulted in NCHRP Research Report 911: Guidelines for Traversability of Roadside Slopes that reports on simulated vehicle trajectories on a range of infinite slopes. (Sheikh 2019) Using these simulated data and assuming the simulation results are identical for roadsides and medians, one could determine the probable lateral extent of a vehicle when navigating a host of terrains while not being subjected to other roadside/median features such as fixed objects or barriers. These simulated data certainly are favored for the richness of information, including both slopes with terrain features such as ditches and simple continuous slopes without complex terrain features. These simulated data, however, do not capture the actual distribution of vehicle encroachment angles and speeds or vehicle types. The NCHRP Projects 17-22 and 17-43 data do include these distributions but are much smaller data sets where the trajectories stop due to a crash (i.e., right-truncated). The favored course of action was to use the strengths of both data sets. The field-collected data were then used to weight the causal elements of the simulated data only after the elements were determined to be influential. The goal of this approach is to maximize the utility of each data set and minimize the complexity of the resulting model. The simulated data were used as two data sets: (1) a combined data set of simple and complete roadside terrains; and (2) a limited data set of simple terrain without ditches. This dual approach allows for the unambiguous examination of complex terrain features. The simulated data were evaluated for the influence of vehicle type, encroachment angle, and speed. Data Summary The TTI research team provided the 57,600 trajectories from NCHRP Project 16-05, which studied trajectories through ditches, and the 43,200 trajectories from NCHRP Project 17- 55, which studied trajectories on slopes. Both research projects simulated trajectories for the Ford Taurus, MASH Small Car, Ford Explorer, and MASH Pick up. The TTI data were gathered in SI units and converted to English units for guideline development before analysis. The variable names used throughout this document and the summary statistics for the Projects 16-05 and 17-55 data sets are shown in Table B-1 for the variables treated as factors and Table B-2 for the continuous variables.

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-5   Table B-1 Factor Level Summary Statistics and Variable Names Covariate Variable Name Levels Number of Trajectories 16-05 17-55 Foreslope (1V:xH) FS −10, −6, −4, −3, and −2 11,520 at each level 8,640 at each level Foreslope width FSW 8 ft 16 ft 32 ft 105 ft 28,800 28,800 0 0 10,800 10,800 10,800 10,800 Ditch bottom width BtW 0 ft 4 ft 10 ft 19,200 19,200 19,200 43,200 0 0 Backslope (1V:xH) BS 0 2 3 4 6 14,400 14,400 14,400 14,400 14,400 43,200 0 0 0 0 Backslope width BSW 8 ft 16 ft 28,800 28,800 43,200 0 Encroachment speed (mi/h) Spd 25 35 45 55 65 75 0 0 14,400 14,400 14,400 14,400 7,200 7,200 7,200 7,200 7,200 7,200 Encroachment angle (degrees) EncAng 5 10 15 20 25 30 0 19,200 0 19,200 0 19,200 7,200 7,200 7,200 7,200 7,200 7,200 Driver input number (See Appendix A) DriverInput 1, 2, 3, 4, and 5 11,520 at each level 8,640 at each level Outcome Outcome Gone.Far Overturn Returns Stops Time.exceeded 6,803 22,425 16,224 9,061 3,087 10,766 6,213 13,008 10,725 2,488 Vehicle type Veh Pickup_Truck FordTaurus Small_car Explorer2002-v1 14,400 at each level 10,800 at each level

B-6 Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers Table B-2 shows that in both data sets, the maximum recorded lateral extent in feet (MaxLatF) is above 100 ft by a few decimal places, but the study design included the censoring of data at 100 ft. This minor discrepancy was likely introduced by either converting between units of measure or from translating the vehicle center of gravity to the maximum point of the vehicle on the terrain. The data have been limited to reflect a maximum 100 ft MaxLatF value to ensure the calculations performed for censored data do not overcompensate for this one-half inch value. Thus, the results of this study should be considered to have an accuracy of ± half an inch. Table B-2 Continuous Variable Summary Statistics and Variable Names Covariate Variable Summary Statistics 16-05 17-55 Maximum lateral extent (feet) as measured from the edge of travel MaxLatF Minimum 1st quartile Median Mean 3rd quartile Maximum 3.814 26.552 37.642 46.792 66.277 100.042 0.296 16.436 46.169 51.396 99.797 100.029 The frequency distribution of the combined Project 16-05 and 17-55 data sets is shown in Figure B-1a. The frequency distribution in Figure B-1b is limited to the Project 17-55 data set (i.e., simple slope data). Notice the large number of trajectories that travel at least 100 feet. The pre-determined stoppage of data collection when the center of gravity of the vehicle traveled 100 ft laterally accounts for this fact. This histogram confirms these data are right-censored, as is often the case in survival analysis. The tools for addressing censoring are discussed more below. These frequency distributions also confirm the complex terrains represented in Figure B-1a have a higher instance of trajectories stopping between 30 and 40 feet, where many of the ditches are introduced. The data elements were reviewed to determine how closely increases in an element correlate with increases in another data element using the Pearson and Spearman’s correlation coefficients as shown in Table B-3 for the combined Projects16-05 and 17-55 data sets and Table B-4 for only the 17-55 data set. A value of one indicates that an element is a linear function of the other (e.g., when an element is compared to itself). A value of zero indicates the data elements are not correlated. Data elements with higher values are considered more correlated. Negative values indicate inverse correlation. Pearson’s correlation coefficient assumes: (1) the data elements are normally distributed and (2) if a relationship exists between the elements, it is linear. The Spearman test does not make either of these assumptions but is interpreted in the same manner (i.e., values approaching unity are more closely correlated, zero are not correlated and negative values are inversely correlated). By inspection, EncAng, Spd, DriverInput, and Outcome have the highest correlations to MaxLatF. These elements have higher correlations than the slope of the terrain. Most interestingly, the sign for the correlation of MaxLatF and FS changes between Table B-3 and Table B-4. In other words, the correlation was negative when the complex terrains were considered, but positive when the simple sloped terrain was considered.

a) b) Figure B-1 Frequency distribution for the maximum lateral extent of the trajectory data sets with a) combined Projects 16-05 and 17-55 data sets and b) only the Project 17-55 data set.

Table B-3 Correlation Matrix for Data Elements in the Combined Projects 16-05 and 17-55 Data Sets Pearson’s Correlation MaxLatF Veh Outcome DriverInput EncAng Spd BtW BS BSW FS FSW MaxLatF 1.0000 -0.0377 -0.5024 -0.5041 0.5051 0.2727 0.0180 0.0063 -0.0497 -0.0513 0.1541 Veh 1.0000 -0.0557 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Outcome 1.0000 0.1693 -0.3482 -0.3103 0.0167 -0.0412 -0.0052 -0.0499 -0.0962 DriverInput 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 EncAng 1.0000 0.0489 0.0877 0.1259 0.0805 0.0000 -0.0711 Spd 1.0000 0.1986 0.2852 0.1824 0.0000 -0.1610 BtW 1.0000 0.5110 0.3268 0.0000 -0.2885 BS 1.0000 0.4692 0.0000 -0.4143 BSW 1.0000 0.0000 -0.2649 FS 1.0000 0.0000 FSW 1.0000 Spearman’s Correlation MaxLatF Veh Outcome DriverInput EncAng Spd BtW BS BSW FS FSW MaxLatF 1.0000 -0.0320 -0.5026 -0.5029 0.5557 0.2779 0.0293 0.0202 -0.0320 -0.0343 0.1304 Veh 1.0000 -0.0600 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Outcome 1.0000 0.2067 -0.3530 -0.3202 0.0116 -0.0479 -0.0107 -0.0779 -0.0438 DriverInput 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 EncAng 1.0000 0.0477 0.0970 0.1321 0.0805 0.0000 -0.0673 Spd 1.0000 0.2144 0.2919 0.1779 0.0000 -0.1487 BtW 1.0000 0.5934 0.3617 0.0000 -0.3023 BS 1.0000 0.4922 0.0000 -0.4115 BSW 1.0000 0.0000 -0.2508 FS 1.0000 0.0000 FSW 1.0000

Table B-4 Correlation Matrix for Data Elements Limited to the Project 17-55 Data set Pearson’s Correlation MaxLatF Veh Outcome DriverInput EncAng Spd FS FSW MaxLatF 1.0000 -0.0333 -0.5713 -0.5018 0.5245 0.4036 0.0371 0.1794 Veh 1.0000 -0.0366 0.0000 0.0000 0.0000 0.0000 0.0000 Outcome 1.0000 0.1994 -0.2898 -0.3636 -0.0447 -0.1830 DriverInput 1.0000 0.0000 0.0000 0.0000 0.0000 EncAng 1.0000 0.0000 0.0000 0.0000 Spd 1.0000 0.0000 0.0000 FS 1.0000 0.0000 FSW 1.0000 Spearman’s Correlation MaxLatF Veh Outcome DriverInput EncAng Spd FS FSW MaxLatF 1.0000 -0.0298 -0.5315 -0.5149 0.5513 0.4111 0.0410 0.1663 Veh 1.0000 -0.0431 0.0000 0.0000 0.0000 0.0000 0.0000 Outcome 1.0000 0.1995 -0.2574 -0.3643 -0.0543 -0.1509 DriverInput 1.0000 0.0000 0.0000 0.0000 0.0000 EncAng 1.0000 0.0000 0.0000 0.0000 Spd 1.0000 0.0000 0.0000 FS 1.0000 0.0000 FSW 1.0000

B-10 Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers CHAPTER 4 DATA ANALYSIS Time, as measured in feet to the trajectory terminus, is the response variable (T). The survival function is the probability that an observation survives longer than t, . The cumulative distribution can be expressed as . Therefore, at time equal to zero, survival is 100% (i.e., when t=0, S(t)=1), and as time approaches infinity, survival approaches zero (i.e., when t= , S(t)=0). In terms of trajectories, at the point where a trajectory exits the travel way (i.e., time equal to zero), the survival of that trajectory is 100% (i.e., when t=0, S(t)=1), and as the maximum lateral extent of the trajectory approaches infinity, the trajectory survival approaches zero (i.e., when t= , S(t)=0). The trajectory may stop for a variety of reasons (e.g., lack of energy, rollover, and exceeded study measurement period). It is generally the goal of data collection to collect complete observations for each trajectory. “Two mechanisms can lead to incomplete observations of time: censoring and truncation. A censored observation is one whose value is incomplete due to factors that are random for each subject. A truncated observation is incomplete due to a selection process inherent in the study design.” (Hosmer 2011) The trajectories that exceed the measurement period are said to be right-censored data. Conversely, left censoring occurs when a trajectory does not originate at the same beginning as the other trajectories. These are simulated data collected as part of a designed study where each of the trajectories originated at the edge of travel (i.e., time zero). Left censoring of these raw data is not considered an issue. The Projects 16-05 and 17-43 data sets of reconstructed crash trajectories discussed above are examples of right-truncated trajectories (i.e., the data collection stopped when a trajectory was involved in a crash). Software Used The statistical computing software and language R was used for the model selection, visual inspection, and model development. (R 2017) The survival package (Therneau 2015) available in R was used for the Kaplan-Meier, Cox Proportional Hazard, and Weibull estimates. The SurvRegCensCov package (Hubeaux 2015) available in R was used to interpret the estimated Weibull model. The survminer package (Kassambara 2017) available in R was used to develop faceted plots of data. The cmprsk package (Fine 1999) was used to estimate and evaluate the cumulative incident function (CIF). The riskRegression package (Fine 1999) was used to develop a formula for the CIF. (Gerds 2018) Assessment of Covariates The cumulative probability of survival to any point in time can be found through a univariate description of the data. The K-M estimator of the survival function is a univariate, nonparametric estimate of time to event. Assuming each trajectory is independent, S(t) is simply calculated directly from the trajectories: Where: S(tj) Probability of continued movement at measurement tj nj Number of trajectories with continued movement just before tj dj Number of trajectories where movement stopped by measurement tj

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-11   The Kaplan-Meier survival curves essentially provide a visual representation of the collected trajectory data, as shown in Figure B-2a for the combined Projects 16-05 and 17-55 data sets and Figure B-2b for only the Project 17-55 data set. The above function can also be used to generate a lookup table of numeric values. “The Kaplan-Meier method is the most common as well as the most controversial technique in the competing risks framework.” (Pintilie 2006) Parametric inference can be more informative than methods that assume no form for the distribution. Multivariate analysis allows for the consideration of how factors jointly impact survival. A statistical model with multiple covariates, therefore, provides a tool to assess clinical differences, joint influence, and competing risks such as rollover. Several covariates were assessed before undertaking the modeling. The covariates were assessed using the Kaplan-Meier method, a log-rank test, and Gray's test. Covariate correlation with the maximum lateral extent was previously discussed. Figure B-3 shows one of these visual comparisons between trajectory outcome and vehicle type for the combined Projects 16-05 and 17-55 data sets and Figure B-4 when the data are limited to only the Project 17-55 data set. Notice that the probability of returning to the road decreased for each vehicle type the further the vehicle travels from the road. Also notice that when the censored categories (e.g., gone too far, time exceeded) are considered individually, a probability of survival cannot be calculated. Differences in survival between groups (e.g., vehicle type, outcome) can be assessed using the log-rank test and/or visually using Kaplan-Meier survival curves. A log-rank test for differences in survival between groups was conducted for the study. This method calculates at each event time, for each group, the number of events one would expect since the previous event if there were no difference between the groups. “While the log-rank test provides a P-value for the differences between groups, it offers no estimate of the actual effect size; in other words, it offers a statistical, but not a clinical, assessment of the factor’s impact.” (Bradburn 2003) The results are summarized in Table B-5. The p-value for each covariate is less than 0.05; therefore, there is a statistically significant difference between the complete survival curves for each covariate. The statistical models will be used to determine the size effect of these differences. (Clark 2003)

a) b) Figure B-2 Kaplan-Meier estimate with 95% confidence bounds for a) combined Projects 16-05 and 17-55 data sets and b) only the Project 17-55 data set.

Figure B-3 Survival probability by vehicle type and trajectory outcome for the combined Projects 16-05 and 17-55 data sets.

Figure B-4 Survival probability by vehicle type and trajectory outcome limited to only the Project 17-55 data set.

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-15   Table B-5 Log-Rank Test of Each Covariate. Covariate 16-05 & 17-55 Data sets 17-55 Data set x2 DF p-value x2 DF p-value Veh 155 3 0.0000 59.5 3 7.46e-13 Outcome 78514 4 0.0000 42099 4 0.0000 FS 86.3 4 0.0000 99.2 4 0.0000 BS 1584 4 0.0000 --- --- --- BtW 279 2 0.0000 --- --- --- BSW 82.5 1 0.0000 --- --- --- FSW 1677 3 0.0000 --- --- --- EncAng 34658 5 0.0000 14214 5 0.000 Spd 11364 8 0.0000 7022 5 0.000 It is important to appreciate any possible proportional relationship, which can be accomplished through visual inspection of the data. If the survival curves do not cross, but rather are generally parallel, then the covariates are proportional and could be represented by multipliers. If proportionality exists, there is potential to use a model which is simpler within the guidance documents. If the curves cross, this proportional assumption is violated. Figure B-5 shows the survival curves by encroachment angle. Recall encroachment angle is the variable most highly correlated with maximum lateral extent (see Tables B-3 and B-4). Table B-5 shows that encroachment angle is a significant predictor of the maximum lateral extent. When the data sets are combined such that complex roadside terrains are included with simple roadside slopes as shown in Figure B-5a, increases to the encroachment angle are not proportionally related to the maximum lateral extent. Figure B-5b, however, shows that when simple slopes are considered alone, the encroachment angle is proportionally related. The encroachment angles appear to influence survivability differently between these two data sets, or it is more likely that the interpretation is marred by the complex terrain. The survival probability by encroachment speed is shown in Figure B-6. Encroachment speed is also one of the more correlated covariates with maximum lateral extent. Note that some of the curves cross in Figure B-6a on the left, while the curves become parallel in Figure B-6b on the right. As with encroachment angle, encroachment speed appears to influence survivability differently between these two data sets, or else complex terrains are impacting the interpretation. When limited to simple roadside slopes, encroachment speed has a multiplicative effect on the maximum lateral extent. When complicated by complex terrains, this multiplicative relationship dissolves. Vehicle type is the least correlated covariate with maximum lateral extent; therefore, it has the least influence on increases or decreases in value. The p-value for vehicle type is less than 0.05, which indicates a statistically significant difference between curves, but the practical difference is negligible in both data sets, as shown in Figure B-7. Recall Figures B-3 and B-4 where the possible outcomes were assessed by vehicle type and no apparent difference in outcome by vehicle type was observed. These data indicate that the distinctions among passenger vehicles, when modeling maximum lateral extent, are not necessary.

a) b) Figure B-5 Kaplan-Meier survival curves by encroachment angle with a) combined Projects 16-05 and 17-55 data sets and b) only the Project 17-55 data set.

a) b) Figure B-6 Kaplan-Meier survival curves by encroachment speed with a) combined Projects 16-05 and 17-55 data sets and b) only the Project 17-55 data set.

a) b) Figure B-7 Kaplan-Meier survival curves by vehicle type with a) combined Projects 16-05 and 17-55 data sets and b) only the Project 17-55 data set.

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-19   Competing Risks It was found with the Kaplan-Meier (K-M) estimator, tests for correlation, and the log- rank test that the encroachment angle and encroachment speed influence survivability differently between these two data sets. This is believed to be a result of the complex terrains simulated under NCHRP Project 16-05. Until this point, the measurement for maximum lateral extent has not differentiated between the assorted reasons a vehicle trajectory may terminate. These data include five events of interest coded in the data under “Outcome” as shown here: • Gone far • Overturn • Returns • Stops • Time exceeded If a vehicle trajectory has exceeded the measurement period (i.e., “Gone Far” or “Time exceeded”), then the data are censored. There is a desire, however, to capture the differences between the vehicle stopping, overturning, or returning to the traveled way. This additional information can better support the application of the encroachment probability model and policy decisions. For example, if there is a 0.70 probability of an errant vehicle traveling 20 ft from the travel way, how does the vehicle come to rest? This vehicle may simply stop or return to the roadway without an increase in crash probability. The vehicle may also overturn which by itself is an increase in crash probability (i.e., overturn crash). While it might be desirable to locate barriers far from the road to minimize the probability of an errant vehicle getting to the barrier and having a crash, this should be balanced with any increased probability of rollover. The K-M estimator discussed above is a univariate, non-parametric representation of the data appropriate when competing risks (e.g., rollover, stops, returns to road, censored) are not modeled. The CIF is appropriate for modeling competing risks. The CIF is also a non-parametric approach to survival analysis. The CIF is the probability of remaining event-free measured laterally from the encroachment location. The CIF allows for the consideration of how overturning contributed to the probable maximum lateral extent. The CIF is a joint probability of each of the possible outcomes being observed (or not observed in the case of the censored data). The CIF estimator for each event depends on both the number of trajectories experiencing an event at a time point and the number of trajectories not experiencing any other event at the same timepoint. The sum of the CIFs for each event at each time represents the joint probability at that time of the maximum lateral extent of the trajectory. The distance to the first observed event was considered. A censoring variable (cens) was created from the outcomes and coded as 0 when no events were observed (i.e., gone too far or time exceeded); 1 if the vehicle stopped on the terrain; 2 if the vehicle returned to the road; and 3 if an overturn was observed. Figure B-8 provides the CIF for the combined data (i.e., the Projects 16-05 and 17-55 data sets) and the data limited to continuous slopes (i.e., only the Project 17-55 data set).

a) b) Figure B-8 CIF for maximum lateral extent with a) combined Projects 16-05 and 17-55 data sets and b) only the Project 17-55 data set.

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-21   Comparing the CIFs considers each type of event and does not assume independence between the time to the different types of events. (Pintilie 2006) These figures are cumulative risks. Notice that the “risk” of returning to the road quickly ascends to approximately 20%, but no further increase is seen the further the trajectory journeys from the traveled way. Conversely, notice that rollover risk is essentially zero for the first twenty feet, then the risk increases the longer the vehicle stays on the roadside. In Figure B-8a, where the data that include complex terrains are shown, the risk of overturning is not linearly related to time on the terrain but is likely reflective of the introduction of the complex ditch elements. In Figure B-8b, where only simple slopes are shown, the risk of overturning appears to be a linear relationship (after 20 ft) that increases with time on the slope. This would indicate that simple exposure to the slope increases overturning risk while the exposure to the complex terrain elements should also be captured. The “risk” of the errant vehicle stopping is virtually linearly related to distance. Interestingly, the risk of overturning becomes greater than either the vehicle stopping or returning, but never greater than the combined risks of the vehicle stopping or returning to the traveled way. This certainty merits further consideration. The CIF is a non-parametric estimate. Neither encroachment angles, nor encroachment speeds are explicitly modeled using this approach. Each technique provides insight. The log-rank test shown in Table B-5 is a test based on the cause-specific risk where the different outcomes are ignored. Each variable captured in this study is considered statically significant using the log- rank test. Variations in outcome by, for example, encroachment speed and angle are not captured in the log-rank test. While the log-rank test is informative, it should be not considered alone. Gray’s test for encroachment angle or speed by outcome is not shown here. Neither encroachment angle nor speed was found to be significant when predicting the differences between possible outcomes (i.e., competing risks) within the combined data, however, when the data is limited to simple slopes, both encroachment angle and speed are highly significant predictors of outcome. Recall encroachment angle and speed are significant when predicting the maximum lateral extent in both data sets. These differences suggest the competing risks are also quite different between groups and databases. Summary of Covariate Assessment This covariate assessment considered the covariates in two overlapping data sets. When the data sets are combined such that complex roadside terrains are included with simple roadside slopes, increases to the encroachment angle and speed were not proportionally related to the maximum lateral extent. When simple slopes are considered alone, however, both encroachment angle and speed are proportionally related to maximum lateral extent. Neither encroachment angle nor speed was found to be significant when predicting the differences between outcomes (i.e., competing risks) within the combined data. Both encroachment angle and speed, however, are significant for predicting the various outcomes for the simple slopes. Due to these obvious differences between these data sets for these two highly correlated covariates, the simple slopes data (i.e., the Project 17-55 data set) were used to develop the relationship to represent maximum lateral extent. It is recommended that complex slopes data (i.e., the Project 16-05 data set) be used to explore adjustment factors for the introduction of these complex terrain elements. Vehicle type was found to be the least correlated covariate with maximum lateral extent and found to have a negligible clinical difference in both data sets. The distinction between vehicles within the passenger vehicle fleet is not justified by these data.

B-22 Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers The consideration of various statistical models for (1) competing risks and (2) parametric representation of encroachment speed and angle to allow for weighting these data are discussed below. Statistical Methods Models may be justified either on the physics of the failure mode or due to empirical success. In addition to the non-parametric CIF and K-M estimator, models considered include: • Weibull, • Extreme Value Distribution, • Accelerated Failure Time (AFT), • Cox Proportional Hazard (PH), and • Competing Risk Regression (CRR). The Weibull method was considered because it is well suited for engineering manufacturing reliability analysis. It is best suited for extremely small samples (e.g., two or three failures). The database used in this analysis is quite large, therefore it was not examined further. The Extreme Value Distribution model is used when the variable of interest (i.e., maximum lateral extent) can be positive or negative. Maximum lateral extent is only measured in one direction. After the vehicle returns to the road, the maximum lateral extent had already been achieved at some previous point along the trajectory. Negative values, therefore, are not encroachments and this model is not appropriate. The AFT model is parametric. The AFT model assumes that the effect of the covariates is to accelerate the life of the trajectories. The AFT model is often favored in engineering studies where mechanical processes are studied for this underlying assumption. The covariates and failure times follow the survival function: The AFT model has a baseline survival rate, S0, which is expressed as a function. The term is the acceleration factor. Expressed in a log-linear form, the log of failure time is related to the mean µ, the acceleration factor, and the error term, , as shown here: The Cox PH model is the most widely used multivariate approach for modeling survival data in medical research. (Bradburn 2003) The Cox PH model has a baseline hazard function, h0(t), which can be specified as in any other model. The hazard model, h(t), and survival model, S(t), are related as follows: Where: h(t) = Hazard function h0(t) = Baseline hazard function

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-23   xj = Covariates collected during data collection (e.g., slope, vehicle type) bj = Coefficients determined through modeling S(t) = Probability of continued movement at measurement tj The baseline hazard function, h0(t), is the value of the hazard if all the covariates are equal to zero (i.e., not present) and the baseline hazard function varies with time. The baseline hazard function is estimated nonparametrically and can be thought of as the intercept term that varies with time. Hazard ratios, , do not vary with time but depend on each of the covariates. The hazard ratio of events is equivalent to the relative risk of events. The Cox PH model has an inherent assumption that the effects of the covariates upon survival are constant over time and that each trajectory only experiences a single event. The standard Cox PH model, however, treats competing risks of the event of interest as censored. (Scrucca 2010) It is desirable to model the time to event of each of the possible outcomes (i.e., overturn, stops, returns to road) while also considering the censored observations (i.e., gone too far and time exceeded). Fine and Gray, among others, proposed directly regressing the effect of covariates for competing risks, known as CRR. (Fine 1999) CRR models cause (r=1, …, k) for each trajectory considering multiple covariates represented by vector X. The baseline subdistribution hazard of cause r is . A partial likelihood approach is applied to estimate the semiparametric PH model for the subdistribution where βr is the vector of estimated coefficients for the covariates and the model takes the following form: When the event of interest is maximum lateral extent, one should also consider how to represent the means of achieving maximum lateral extent. For example, did the trajectory come to a stop, return to the road, or overturn? If the vehicle came to a stop or returned to the road, there is no harm caused by the encroachment alone. If the vehicle overturned, however, there is harm caused by overturning. As was shown above, some trajectories exceeded the study period (e.g., exceeded by measurement or by time). These censoring events prevent the observation of the trajectory stopping, returning to the road, or overturning. One option is to assume that those who are censored have the same chance of overturning as those who were observed. This is not believed to be the case. Therefore, overturning is a competing risk event that eliminates the chance of stopping or returning to the road. Restated, there are multiple modes of failure. In summary, a single model is not appropriate. Both the survival function of the trajectory data and the cumulative incidence of the competing risks should be captured. The Cox PH was used to characterize the survival function. A CIF was used to represent the competing risks on the simple slopes. A third model (e.g., CRR) could be used in the ongoing research project NCHRP Project 16-05, “Guidelines for Cost-Effective Safety Treatments of Roadside Ditches” to capture the risk introduced by ditches. A model of competing risks for ditches developed under NCHRP Project 16-05 could be easily integrated into the guidelines developed under this research and the ongoing NCHRP Project 15-65 research to improve the representation of roadside terrain throughout the AASHTO RDG.

B-24 Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers Survival Event of interest is the maximum lateral extent when the vehicle stops or returns to the road. Overturning is treated as a competing risk event that eliminates the chance of either stopping or returning to the road. Recall the hazard model, h(t), and survival model, S(t), are related, as shown here: The baseline hazard function, h0(t), varies with time. The hazard ratio of events is equivalent to the relative risk of events and is found by exponentiating the estimated coefficients. The hazard ratios do not vary with time. The coefficients have been estimated using the Cox PH model limiting the data set to the trajectories where an event was observed. The resulting estimates for the coefficients of EncAng and Spd as well as the hazard ratios and the corresponding confidence intervals are shown in Table B-6. The model is shown graphically in Figure B-9. Table B-6 Estimated Survival Function Coefficient (bi) S.E. z Pr(>|z|) exp(coef) lower .95 upper .95 EncAng -0.0677 0.00 -92.28 < 2e-16 0.9346 0.9332 0.9359 Spd -0.0270 0.00 -75.91 < 2e-16 0.9734 0.9727 0.9740 FS-6 -0.0566 0.02 -3.12 0.0018 0.9450 0.9120 0.9792 FS-4 -0.1218 0.02 -6.68 2.42e-11 0.8853 0.8542 0.9175 FS-3 -0.1298 0.02 -7.12 1.12e-12 0.8783 0.8475 0.9103 FS-2 -0.1469 0.02 -8.08 6.66e-16 0.8634 0.8332 0.8947 The hazard ratios indicate that for every one-unit increase in encroachment angle, the maximum lateral extent where an event is not observed will decrease by approximately 6.5% (i.e., 1 − 0.9347). Similarly, for every one unit increase in encroachment speed, the maximum lateral extent when an event is not observed will decrease by 2.5%. “One of the primary reasons for using a regression model is to include multiple covariates to adjust statistically for possible imbalances in the observed data for making statistical inferences.” (Hosmer 2011) This model will be used below to scale these data using the NCHRP Project 17-43 real-world distributions.

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-25   Figure B-9 Estimated survival function. Competing Risks The estimated cumulative incidence (risk over time) of overturning in the presence of the other event types (e.g., stopping or returning to the road) was estimated. This estimate of attrition due to the occurrence of the competing risk, overturn by slope, is shown in Figure B-10. Notice the steeper the slope, the higher the rate of attrition.

B-26 Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers Figure B-10 Estimated competing risk of overturn.

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-27   CHAPTER 5 FIELD COLLECTED TRAJECTORIES NCHRP Project 17-11: Determination of Safe/Cost Effective Roadside Slopes and Associated Clear Distances was completed in 2004 by TTI at Texas A&M. This data set contained 485 National Automotive Sampling System (NASS) Crashworthiness Data System (CDS) cases from 1997 through 1999. (Bligh 2004) NCHRP Project 17-22, “Identification of Vehicular Impact Conditions Associated with Serious Ran-Off-Road Crashes,” was completed in 2009 at the University of Nebraska-Lincoln. This data set contained 392 NASS CDS cases from 2000 and 2001. (Sicking 2009) NCHRP Report 665: Identification of Vehicular Impact Conditions Associated with Serious Ran-off-Road Crashes combined the two data sets from NCHRP Projects 17-11 and 17-22. (Mak 2010) As mentioned earlier in this document, NCHRP Project 17-43, “Long-Term Roadside Crash Data Collection Program,” is in progress at Virginia Polytechnic Institute and State University. (Gabler 2020) The beta version of the data set has been reviewed as part of this effort. A compilation of the average departure speed and angle from each of these studies is provided in Table B-7.

B-28 Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers Table B-7 Comparison of Departure Speed and Angle for Specific Field-Collected Crash Databases Reference Number of Cases 50th Percentile of Data Set Departure Speed Departure Angle NCHRP Project 17-11 485 19.9° NCHRP Project 17-11* 485 48.9 mph 16.9° NCHRP Project 17-22 392 49.7 mph 17.2° NCHRP Report 665 877† 49.3 mph 16.9° NCHRP Project 17-43 1124 48.6 mph 13.8° * After reconstruction and manual reviews by the NCHRP Project 17-22 group. † Combination of the NCHRP Project 17-11 reconstructed data set and the NCHRP Project 17-22 data set.

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-29   CHAPTER 6 RESULTS The mean encroachment speed of the data simulated in the Project 17-55 data set was 50 mph and the mean encroachment angle was 17.5 degrees. Considering the field-collected data, it would appear the mean encroachment speed is 48.6 mph while the encroachment angle is closer to 13.8 degrees. The survival function tabulated in Table B-6 was used to scale the simulated data to better represent field-collected encroachment speeds and angles, as shown in Figure B- 11. The solid line represents the unadjusted model developed from the simulated data where the dashed line has been adjusted to represent the field-collected data. It is recommended that the green adjusted line be used in guideline development. No assumptions have been made about median symmetry; this model can be used from either direction of travel and is applicable to the roadside. This representation can be extended in the future to include the competing risk of ditches. This effort is currently underway in NCHRP Project 16-05. Recall the estimated cumulative incidence (risk over time) of overturning in the presence of the other event types (e.g., stopping or returning to the road) was also estimated. This estimate of attrition due to overturn was shown above in Figure B-10.

B-30 Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers Figure B-11 Estimated survival function (solid) and scale maximum lateral extent model (dashed).

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-31   CHAPTER 7 IMPLEMENTING THE RESULTS NCHRP Project 15-65 defines “PY j [a]s the conditional probability of a vehicle reaching a lateral offset of Y given an encroachment.” (Ray 2018) This research effort is coordinating terminology with NCHRP Project 15-65 to facilitate implementing the resulting guidance in an upcoming update to the AASHTO RDG. The results of this modeling effort to represent the maximum lateral extent have been tabulated, therefore, as PY(Yj), as shown in Figure B-12. Numeric values are provided to the left of the figure for ease of use.

B-32 Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers Yj (ft) PY(Yj) 0 1.0000 1 0.9761 2 0.9431 3 0.9090 4 0.8844 5 0.8650 10 0.7737 15 0.7191 20 0.6741 25 0.6238 26 0.6120 27 0.6014 28 0.5908 29 0.5815 30 0.5699 35 0.5082 40 0.4603 45 0.4063 50 0.3622 55 0.3254 60 0.2887 65 0.2531 70 0.2307 75 0.2115 80 0.1918 85 0.1752 90 0.1624 95 0.1515 100 0.1416 Figure B-12 Recommended scaled maximum lateral extent PY(Yj). NCHRP Project 15-65 also states that “THRj is a variable that represents the conditional probability of passing through feature j given the vehicle interacts with feature j.” (Ray 2018) For example, a vehicle may travel on a median slope, interact with and penetrate a median barrier, enter the opposing lanes where it may be struck by another vehicle. The proportion that passes through for each category of roadside feature (i.e., the first slope and the median barrier) is dependent on variables that are unique to the specific type of feature. The vehicles that roll over on the slope do not pass through, THR. The probability of rollover (i.e., do not pass THR) was shown above in Figure B-10. Table B-8 shows a table of values that are one minus the values shown for rolling over. One minus rollover provides the probability of passing THR.

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-33   Table B-8 Recommended representation of passing THR terrain (THRTERRAIN) Survived the Terrain Lateral Extent THRFORESLOPE ft -10:1 -6:1 -4:1 -3:1 -2:1 0 1.0000 1.0000 1.0000 1.0000 1.0000 1 1.0000 1.0000 1.0000 1.0000 1.0000 2 1.0000 1.0000 1.0000 1.0000 1.0000 3 1.0000 1.0000 1.0000 1.0000 1.0000 4 1.0000 1.0000 1.0000 1.0000 1.0000 5 1.0000 1.0000 1.0000 1.0000 1.0000 10 1.0000 1.0000 1.0000 1.0000 0.9995 15 0.9992 0.9993 0.9998 0.9997 0.9985 20 0.9963 0.9962 0.9957 0.9966 0.9948 25 0.9921 0.9911 0.9885 0.9887 0.9835 26 0.9900 0.9896 0.9867 0.9869 0.9802 27 0.9892 0.9887 0.9851 0.9840 0.9762 28 0.9890 0.9876 0.9847 0.9815 0.9736 29 0.9884 0.9867 0.9831 0.9803 0.9696 30 0.9876 0.9851 0.9811 0.9782 0.9659 35 0.9804 0.9784 0.9712 0.9643 0.9356 40 0.9755 0.9731 0.9640 0.9516 0.9092 45 0.9687 0.9639 0.9557 0.9381 0.8813 50 0.9638 0.9567 0.9446 0.9252 0.8577 55 0.9579 0.9507 0.9382 0.9139 0.8320 60 0.9543 0.9451 0.9298 0.9018 0.8073 65 0.9487 0.9384 0.9181 0.8852 0.7832 70 0.9428 0.9330 0.9113 0.8757 0.7670 75 0.9416 0.9296 0.9058 0.8638 0.7514 80 0.9393 0.9264 0.8976 0.8550 0.7392 85 0.9340 0.9227 0.8903 0.8453 0.7267 90 0.9307 0.9168 0.8846 0.8377 0.7186 95 0.9295 0.9139 0.8805 0.8323 0.7068 100 0.9266 0.9104 0.8756 0.8275 0.7001

B-34 Selection and Placement Guidelines for Test Level 2 Through Test Level 5 Median Barriers REFERENCES Bligh, Roger P. and Shaw-Pin Miaou, "Determination of Safe/Cost Effective Roadside Slopes and Associated Clear Distances [Completed Project]," 2004. Bradburn, M. J., T. G. Clark, S. B. Love, and D. G. Altman, "Survival Analysis Part II: Clark, T. G., M. J. Bradburn, S. B. Love, and D. G. Altman, "Survival Analysis Part I: Basic concepts and first analyses," British Journal Of Cancer, Vol. 89, 2003. Fine, Jason P. and Robert J. Gray, "A Proportional Hazards Model for the Subdistribution of a Competing Risk," Journal of the American Statistical Association, Vol. 94, No. 446, 1999. Gabler, H. Clay, "Long-Term Roadside Crash Data Collection Program [Project]," Virginia Polytechnic Institute and State University, 2020. Gerds, Thomas Alexander, Paul Blanche, Ulla Brasch Mogensen, and Brice Ozenne, Risk Regression Models and Prediction Scores for Survival Analysis with Competing Risks. https://cran.r-project.org/web/packages/riskRegression/riskRegression.pdf. Accessed June 29, 2018, 2018. Hosmer, D. W., S. Lemeshow, and S. May, Applied Survival Analysis: Regression Modeling of Time-to-Event Data, Wiley, 2011. Hubeaux, Stanislas and Kaspar Rufibach, SurvRegCensCov: Weibull Regression for a Right- Censored Endpoint with a Censored Covariate. https://cran.r- project.org/web/packages/SurvRegCensCov/index.html https://cran.r-project.org/web/packages/SurvRegCensCov/SurvRegCensCov.pdf Kassambara, Alboukadel, Marcin Kosinski, and Przemyslaw Biecek, survminer: Drawing Survival Curves using 'ggplot2'. http://www.et.bs.ehu.es/cran/web/packages/survminer/index.html http://www.et.bs.ehu.es/cran/web/packages/survminer/survminer.pdf Mak, King K., Dean L. Sicking, Francisco Daniel Benicio de Albuquerque, and Brian A. Coon, "Identification of Vehicular Impact Conditions Associated with Serious Ran-off- Road Crashes," Transportation Research Board, NCHRP Report 665, 2010. Pintilie, Melania, Competing risks: a practical perspective, John Wiley & Sons, 2006. R Foundation for Statistical Computing, R: A Language and Environment for Statistical Computing, Vienna, Austria, 2017. https://www.R-project.org Ray, Malcolm H. and Christine E. Carrigan, NCHRP Project 15-65, "Development of Safety Performance Based Guidelines for the Roadside Design Guide," 2018. Ray, Malcolm H., Christine E. Carrigan, and C. A. Plaxico, NCHRP Project 22-27, "Roadside Safety Analysis Program (RSAP) Update, Appendix B: Engineer's Manual RSAP," Transportation Research Board, Washington, DC, 2012Scrucca, L., A. Santucci, and F. Aversa, "Regression modeling of competing risk using R: an in depth guide for clinicians," Bone Marrow Transplantation, Vol. 45, 2010. Sheikh, Nauman M., Roger P. Bligh, Sofokli Cakalli, and Shaw-Pin Miaou, NCHRP Project 17- 55, "Guidelines for Traversability of Roadside Slopes," Transportation Research Board, Washington, DC, 2019. Multivariate data analysis – an introduction to concepts and methods," British Journal Of Cancer, Vol. 89, 2003.

Probability of Reaching the Lateral Offset of Feature j—PY(Yj) B-35   Sheikh, Nauman, Roger P. Bligh, D. Lance Bullard Jr., and Shaw-Pin Miaou, NCHRP Web-Only Document 296: Guidelines for Cost-Effective Safety Treatments of Roadside Ditches, Washington, DC, 2021 Sicking, Dean L., NCHRP Project 17-22, "Identification of Vehicular Impact Conditions Associated with Serious Ran-Off-Road Crashes," 2009. Therneau, Terry M., A Package for Survival Analysis in S. https://CRAN.R- project.org/package=survival