Development of Safety Performance-Based Guidelines for the Roadside Design Guide (2022)

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77   3.1 Background An objective of this research is to develop performance-based roadside safety guidance to address high-priority needs that support quantitative design decisions, and that promote con- sistency in interpretation and implementation. NCHRP Report 785 established a framework for highway designers to use in the geometric design of highways as outlined in Figure 21 (Ray et al. 2014a). This same general framework is used in this research to promote consistency and implementation across highway design disciplines. The only change to Figure 21 needed to make it applicable to roadside design is to change “establish geometric design decisions” to “establish roadside safety design decision” in Step 2 (i.e., numeral 2 inside the blue circle). The performance outcomes (i.e., step 3) to be considered should be quantifiable measures of safety or risk as will be discussed in the next section. A single evaluation methodology that measures safety or risk of each roadside safety decision will provide greatly improved consistency in the RDG. Several roadside safety research projects have explicitly used the C H A P T E R 3 Roadside Risk Design Methodology Figure 21. Framework for performance-based analysis of geometric design (Ray et al. 2014a).

78 Development of Safety Performance-Based Guidelines for the Roadside Design Guide estimation of the risk of serious and fatal crashes to develop roadside hardware selection procedures for the following: • Test level selection of bridge railings (Ray and Carrigan 2014b; Ray et al. 2014d). • Shielding bridge piers for bridge and occupant protection (Ray et al. 2018). • Test level selection of median barriers (Carrigan and Ray 2022). • Assessment of tree crash risk (Carrigan et al. 2019). The risk of serious and fatal crashes was favored as a performance measure compared to all crashes for many reasons, including: (1) ensuring the observation of catastrophic crashes was not camouflaged by large numbers of low-severity crashes, (2) because the RDG specifically states the goal of roadside design is to reduce serious and fatal crashes, and (3) the TCRS Strategic Plan called for the development of guidelines which quantify serious and fatal (KA) crash reduction and are explicit in the communication of the design objectives. 3.2 Safety Performance Goal The RDG and MASH do not explicitly define the design objective of designing a roadside or roadside appurtenance (AASHTO 2011b; AASHTO 2016). According to the FHWA, “guard- rails cannot completely protect drivers in every situation” (FHWA 2019). No barrier can be designed to accommodate every possible impact condition or every possible type of vehicle. What the FHWA says about guardrails is true of designing the roadside in general: no roadside design can completely protect drivers in every situation. The predecessor to MASH, Report 350, states that “the approach used in formulating the recommended test conditions is to evaluate the devices for cases that are very severe, yet practical” (Ross et al. 1993). Report 350 refers to this design strategy as the practical worst-case impact scenario. The worst-case scenario would be the absolute and most demanding situation possible. Designing to protect against such extreme con- ditions would be technologically challenging if not impossible. Practical worst-case scenarios are demanding impact conditions that are likely to be encountered routinely in the field and, therefore, address the majority of crashes. NHTSA has traditionally established vehicle design objectives using occupant injury limits based on the probability of observing an Abbreviated Injury Scale (AIS) value greater than three, where values greater than three are considered life-threatening (Kuppa 2006). These probabili- ties are represented in so-called dosage curves for the head injury criteria (HIC), rib deflection, pelvis acceleration, and abdominal force. A dosage curve plots the injury measure (e.g., head injury criteria, rib deflection, pelvis acceleration, abdominal force, etc.) versus the probability of observing a particular level of that measure. For example, the probability of observing an AIS greater than three injury, when the HIC is 1,000, is about 0.2. Observing a HIC of 1,000 is not a guarantee that a life-threatening injury (i.e., AIS > 3) will not occur but an estimate that the probability of such an injury is 20% or more. Injury tolerance measures are designed for use with anthropometric test devices (ATD) that are used in some types of crash tests to assess the risk of life-threatening injury to vehicle occupants. ATDs are not typically used in roadside hardware crash tests so the linkage to the ADT measurements and the probability of the dosage curves are not generally available in roadside design. Michie, the author or coauthor of the four crash test procedures prior to the MASH proce- dures, stated that “since complete safety is an unattainable ideal, safety performance is mea- sured in terms of the degree of risk experienced by occupants when the vehicle collides with a roadside appurtenance” (Michie 1981b). Michie developed the flail-space method to measure the “degree of risk” in roadside hardware crash tests using two measures: the occupant impact velocity (OIV) and occupant ride-down acceleration (ORA). Michie also suggested the toler- ance limits for use with the flail-space occupant risk method. These tolerances have been used as acceptance criteria in crash testing evaluation procedures since 1981 in NCHRP Report 230,

Roadside Risk Design Methodology 79   NCHRP Report 350, and even in the most current crash test procedures, MASH. Michie explains that “in line with current Federal Motor Vehicle Safety Standard (FMVSS) 208, an upper design limit for occupant protection falls between [AIS] codes 3 and 4. That is, severe injury is accepted as long as it is not life-threatening” (Michie 1981b). In other words, roadside safety hardware is not designed to prevent all possible injuries, but to minimize the chance of experiencing life- threatening injuries. Life-threatening in this context means an AIS greater than three. Vehicle occupants that strike roadside hardware may be injured and may even experience serious injuries since the design objective is to limit the loss of life. As discussed above, the implicit objective of roadside hardware design is to maximize the safety of highway users who may leave the roadway. The inverse of maximizing safety would, therefore, be minimizing risk by roadside design. Before risk or safety can be optimized, how- ever, they must be quantified in some measurable way. In evaluating the performance of road- side features or a roadway segment, the proportion of impacts with poor outcomes (e.g., fatal and serious injury crashes) must be found with respect to all possible outcomes with the same feature. The World Health Organization (WHO) defines risk as “the proportion by which the incidence rate of the outcome in the entire population would be reduced if exposure were eliminated” (Bonita et al. 2006). This proportion, or risk, is therefore defined as the poor out- comes divided by all outcomes. Nearly all state and local law enforcement agencies use the so-called KABCO injury scale on their police crash reports. The KABCO scale is a measure of the outcome of the crash, so it fits the WHO definition given above. The KABCO scale is defined in the “Model Minimum Uni- form Crash Criteria” published by NHTSA and is summarized in Table 6 (NHTSA and GHSA Code Description K Fatal A fatal injury is any injury that results in death within 30 days after the motor vehicle crash in which the injury occurred. If the person did not die at the scene but died within 30 days of the motor vehicle crash in which the injury occurred, the injury classification should be changed from the attribute previously assigned to the attribute “Fatal Injury.” A Suspected Serious Injury A suspected serious injury is any injury other than fatal that results in one or more of the following: • Severe laceration resulting in exposure of underlying tissues/muscle/organs or resulting in significant loss of blood. • Broken or distorted extremity (arm or leg). • Crush injuries. • Suspected skull, chest, or abdominal injury other than bruises or minor lacerations. • Significant burns (second- and third-degree burns over 10% or more of the body). • Unconsciousness when taken from the crash scene • Paralysis. B Suspected Minor Injury A minor injury is any injury that is evident at the scene of the crash, other than fatal or serious injuries. Examples include a lump on the head, abrasions, bruises, and minor lacerations (i.e., cuts on the skin surface with minimal bleeding and no exposure of deeper tissue/muscle). C Possible Injury A possible injury is any injury reported or claimed which is not a fatal, suspected serious, or suspected minor injury. Examples include momentary loss of consciousness, a claim of injury, limping, or complaint of pain or nausea. Possible injuries are those that are reported by the person or are indicated by his/her behavior, but no wounds or injuries are readily evident. O No Apparent Injury No apparent injury is a situation where there is no reason to believe that the person received any bodily harm from the motor vehicle crash. There is no physical evidence of injury, and the person does not report any change in normal function. Table 6. MMUCC definitions of crash injury status (after NHTSA and GHSA 2017).

80 Development of Safety Performance-Based Guidelines for the Roadside Design Guide 2017). Some states use slight variations on these definitions. For example, Washington State sub- divides the fatal category into several subcategories (e.g., dead at scene, dead on arrival, and died at hospital). Various states use the terms serious injury, severe injury, and incapacitating injury synonymously to refer to what the Model Minimum Uniform Crash Criteria (MMUCC) defines as an A-level injury. Many states refer to crashes where there is no apparent injury (i.e., “O” in the MMUCC) as property-damage-only crashes. This report will use the MMUCC definitions. Like most states and other highway agencies, Maine DOT states that “a core part of Maine DOT’s mission is to provide a safe transportation system for all users” (MDOT n.d.). This typi- cal statement by Maine DOT suggests that safety should be maximized which, in turn, implies that risk should be minimized. Minimizing risk is a direct approach to meeting highway agency goals like that described above by Maine DOT. As described above, the implicit design objec- tive of both the roadside design and roadside hardware crash testing has long been to minimize the probability of life-threatening injuries. A quantifiable explicit design objective for both the RDG and MASH is provided in Table 7 where the objective of a roadside design is to minimize the expected number of serious injury and fatal crashes (KA) per mile of roadway edge per year. 3.3 Safety Performance Design 3.3.1 Method The basic design philosophy presented in the 1974 Yellow Book has guided roadside designers for more than 40 years and is still the foundational design philosophy for all editions of the RDG through the present 2011 edition (AASHTO 2011b). In the 2011 RDG, the forgiving roadside four-step process of the 1974 Yellow Book has been rearranged and expanded to the following six-step process: 1. Remove the fixed object. 2. Redesign the fixed object so that it can be safely traversed. 3. Relocate the fixed object to a point where it is less likely to be struck. 4. Reduce the impact severity by using an appropriate breakaway device or impact attenuator. 5. Redirect a vehicle by shielding the obstacle with a longitudinal traffic barrier. 6. Delineate the fixed object if the previous options are not appropriate (AASHTO 2011b). More recent practices of roadside design and the outline for the pending update to the RDG (from NCHRP Project 20-07/Task 383, “Review and Update of the AASHTO Roadside Design Guide”) suggest a slightly modified variation of the more recent RDG six-point list: • Retain vehicles on the roadway (e.g., use rumble strips, delineation, safety edge treatment, and geometric improvements). • Reduce crashes by making the roadside traversable and free of fixed objects (e.g., removing fixed objects, relocating fixed objects, regrading slopes and ditches). • Reduce the severity of functional features of the roadside (e.g., signs, luminaire supports, utility poles, etc.) by using breakaway supports. • Reduce the severity of crashes by redirecting vehicles away from fixed objects and untravers- able terrain within the clear zone with longitudinal barriers or shielding fixed objects within the clear zone with crash cushions or terminals. The objective of a roadside design is to minimize the expected number of serious injur ies and fatal crashes (KA) per mile of roadway edge per year. Table 7. Explicit roadside design objective.

Roadside Risk Design Methodology 81   These four steps also correspond to the three countermeasures for a strategic approach to roadway departure safety promoted by the FHWA and illustrated in Figure 22 where the third and fourth steps are combined under “reduce crash severity.” Treatments like rumble strips, lane striping, and reducing speed limits in curves can reduce the probability of leaving the traveled way (i.e., “keep vehicles on the roadway” in Figure 22). Moving objects as far as practical from the edge of the lanes or removing them entirely will reduce the probability of striking an object (i.e., “provide for safe recovery” in Figure 22). If an object is struck, it can be designed to mini- mize the risk of injury to vehicle occupants (i.e., “reduce crash severity” in Figure 22) and if roadside hardware contains and redirects errant vehicles the vehicles will be less likely to interact with other off-road objects (i.e., also “reduce crash severity” in Figure 22). Each of these four components is a conditional probability, and the method outlined in the following sections presents step-by-step instructions for calculating each of these components. This sequence of subevents can be represented by the following equation where OUTCOMES is the result that is to be minimized by roadside design: OUTCOME OUTCOME 1S j j 1 N ∑[ ]= = ∏ ( )=       − δ         = = − • • • • • • •OUTCOME BEF EAF L 5,280 P THR P 1 THR PSL 65 2 OUTCOME ENCR CRASH SEV j S S S cj i i 1 j 1 SEV j j s 3 3 j j j j j where: OUTCOMES = The total number of crashes with the specified outcome on segment S involv- ing all roadside features on the segment. OUTCOMEj = The number of crashes with the specified outcome involving feature j (e.g., the number of serious injury or fatal crashes involving impacts with a tree) per edge mile per year. j = Feature number from 1 to N where N is the total number of features evaluated on the segment. BEFS = The expected annual number of encroachments expected on a segment in edge encroachments/mile/year assuming base conditions as a function of AADT volume. encroachment = Highway and traffic characteristic encroachment adjustment factors for adjustment the highway segment of interest, S (EAFS). factors (EAFS) Figure 22. FHWA’s roadway departure safety countermeasures (FHWA 2019).

82 Development of Safety Performance-Based Guidelines for the Roadside Design Guide LS = Segment length in feet. Pcj = The conditional probability of a vehicle interacting with a roadside feature given an encroachment occurs. The length ratios are the probability of leav- ing the roadway in the given proportion of the roadway under the assump- tion that encroachments are equally likely anywhere on the segment. The form of Pcj depends on the type of object as shown below: Continuous Features (e.g., guardrails, median barriers, terrain, etc.) Discreet Features (e.g., trees, poles, bridge piers, water bodies, etc.) PSEVj = The conditional probability of observing the severity of interest given that there is an interaction with roadside feature j. THRi and THRj = The conditional probability of passing through, over, or under feature i or j given the vehicle interacts with feature i or j. δj = 1 if only interactions with the feature that do not pass through the feature lead to an increase in harm (e.g., terrain). = 0 if all interactions with the feature lead to an increase in harm regardless of whether the feature is passed through (e.g., longitudinal barriers). PSLs = Posted speed limit on the segment in mph. Lj = The effective length of an individual feature j along the segment in feet. Continuous Features (e.g., longitudinal barriers, terrain, medians, etc.) The length of a continuous feature measured parallel to the roadway in feet where Lj ≤ LS. Single Discreet Features For single discreet features such as trees or utility poles, this is equal to the dimension of the feature parallel to the road, or the diameter measured in feet. Add Wv sin θ85 to the length or diameter for a fixed object (FO). Multiple Discreet Features For features like a line of poles or series of bridge piers, the effective length is the length in feet from the upstream traffic face of the first feature to the downstream face of the last feature plus WV sin θ85 as long as the spacing between features is less than WB/tan θ15 and Lj ≤ LS. If the spacing between features is less than + θ θ W WB FO V cos tan 15 15 then treat multiple features as single isolated features. Py(Yj) = Cumulative probability density function of the lateral extent of encroach- ment when lateral offset y = Y. Px(Xj) = Sum of the cumulative probability density function of the maximum longi- tudinal extent of encroachment. WBj = The distance in feet from the edge of the traveled way measured laterally to the farthest point of feature j plus WV cos(θ15) for discreet features. ( )=     •P L L P Wcj j S y Fj ( ) ( ) ( )( )( )=     +     − •P L L P W L L P L P W P Wcj j S y Fj TMax S x TMax y Fj y Bj

Roadside Risk Design Methodology 83   WFj = The distance in feet from the edge of the traveled way to the closest face (i.e., traffic side) of feature j. For foreslopes, the distance is measured to the bottom of the foreslope. WV = Typical passenger vehicle width in feet (e.g., 6.5 ft). LTMax = The length in feet of the longest trajectory in the data base of trajectories used to calculate Px(Xj) and Py(Yj) [i.e., 1,000 ft (from NCHRP Project 17-43, “Long-Term Roadside Crash Data Collection Program”)]. θ15 = The 15th percentile encroachment angle in degrees [e.g., 5 degrees (from NCHRP Project 17-43, “Long-Term Roadside Crash Data Collection Program”)]. θ85 = The 85th percentile encroachment angle in degrees [e.g., 22 degrees (from NCHRP Project 17-43, “Long-Term Roadside Crash Data Collection Program”)]. Equation 2 has three bracketed terms, one each for the three FHWA countermeasures: • Keep vehicles on the roadway ENCRj = =     • • ENCR BEF EAF L 5,280j S S S • Provide for safe recovery CRASHj = ∏  = − •P THRj cj i i 1 j 1 • Reduce crash severity SEVj = ( )= − δ     • •SEV P 1 THR PSL 65j SEV j j s 3 3j The procedure for the safety performance assessment of a roadside or median design is more succinctly described in Table 8. 3.3.2 Calculate Feature Risk (OUTCOMEj) The frequency of fatal and serious injury ROR crashes involving each roadside feature (or OUTCOMEj) is calculated as the product of the three bracketed terms in Equation 2. Each bracket corresponds to one of the FHWA countermeasures for improving roadside safety as follows: • Keep vehicles on the roadway → ENCR = =    • • ENCR BEF EAF L 5,280 S S S • Provide for safe recovery → CRASH = ∏   = − •P THRcj i i 1 j 1 • Reduce crash severity → SEV = ( )= − δ     • •SEV P 1 THR PSL 65SEV j j s 3 3j Each of these terms and countermeasures are discussed in the following sections. 3.3.2.1 Keep Vehicles on the Road The first of the three FHWA countermeasures is to keep vehicles on the roadway. Stated another way, the objective of this countermeasure is to minimize the number of encroachments where the number of encroachments expected is given by: =    • • ENCR BEF EAF L 5,280j S S S

84 Development of Safety Performance-Based Guidelines for the Roadside Design Guide 3.3.2.1.1 Base Encroachment Frequency. The estimated annual base encroachment fre- quency on a segment (BEFS) represents the number of vehicles that will depart an edge of a highway and enter the roadside or median as a function of the two-way total AADT volume. The roadside risk governing equation (i.e., Equation 2) is based on a theory of how crashes occur. A crash is separated into a series of conditional events, and each event is modeled as a condi- tional probability. These conditional events include: the encroachment probability, the prob- ability of a roadside feature interaction given an encroachment, and the severity of the roadside inter action if an interaction occurs. The probability of an encroachment (i.e., vehicle leaving the traveled way) has been the focus of many studies in the last 60 years, however, very little successful data collection on the frequency of encroachments has been accomplished. Limitations on the data collected by Hutchinson and Kennedy and later by Cooper have received much attention in the past, but there are few alternate sources of encroachment data (Cooper 1981; Cooper 1980; Hutchinson 1962; Hutchinson and Kennedy 1966). There is currently an NCHRP project underway to collect Find: The expected average annual frequency of serious injury and fatal crashes on a roadway segment edge for the existing conditions and proposed alternatives (i.e., OUTCOMES) and compare them to the safety performance goal (i.e., OUTCOMEGOAL). Given: The traffic and site characteristics for each edge of the roadway where a vehicle might encroach: Segment the roadway of interest into homogeneous sections and determine each segment length (LS) where S is the segment number. A homogeneous section is one where all the roadway characteristics (e.g., lane width, curvature, grade, etc.) are the same. 1) Determine the total number of roadside or median features (N) as well as their location, offset, size, and type. 2) Calculate the expected average annual frequency of serious injury and fatal collisions for feature j (or OUTCOMEj) on each segment edge. a. Find the total base encroachment frequency on a segment (BEFS) given the highway type (i.e., divided or undivided) and AADT from Table 29. b. Find the encroachment adjustment factors of a segment (EAFS) from Table 30. Note that for horizontal curves and grades the adjustment will be different for each direction of travel. c. Find the conditional probability of a vehicle striking feature j based on its type (i.e., continuous or discreet) and lateral offset from the traveled way [i.e., Py(WFj) and Py(WBj)] from Table 44. Continuous Features (e.g., guardrails, median barriers, terrain, etc.) Discreet Features (e.g., trees, poles, bridge piers, water bodies, etc.) d. Find the conditional probability of the outcome of interest (PSEVj, e.g., a KA crash) given an interaction with feature j from Table 48. e. Find the proportion of interactions that pass through feature j from Table 45 through Table 47 (THRj). f. Let: δj = 1 For all-terrain features and other geometric features where the harm is only associated with those vehicles that do not make it through the feature. δj = 0 For all longitudinal barriers, breakaway devices, crash cushions, and guardrail terminals where the harm will be the same whether the vehicle passes through it or not. g. Calculate the feature risk from: IF j < N, THEN Go to the next feature by returning to Step 3a with j = j + 1 ELSE Continue to Step 4. 3) Calculate the risk for the entire segment from: 4) IF OUTCOMES ≤ OUTCOMEGOAL THEN The safety performance of the evaluated design for segment S meets the safety performance goal. Table 8. Risk-based safety performance design procedure.

Roadside Risk Design Methodology 85   new encroachment data over a wider range of traffic conditions which, it is hoped, will improve knowledge about the relationship between encroachments, traffic conditions, and highway geometry (from NCHRP Project 17-88, “Roadside Encroachment Database Development and Analysis”). The encroachment probability model with the program ROADSIDE used an encroachment rate based on the Hutchinson-Kennedy data, and RSAP through version 2.0.3 used the Cooper data to model encroachments (Cooper 1981; Cooper 1980; Hutchinson 1962; Hutchinson and Kennedy 1966). RSAPv3 used the Cooper data, but the data was reanalyzed in an attempt to resolve some long-standing concerns about the previous modeling (Ray et al. 2012b). Carrigan and Ray used HSIS crash data to develop a roadside crash prediction model (Carrigan and Ray 2018a). The safety performance function (SPF) of that model can be loosely interpreted as a ROR encroachment model. Both the results of the reanalysis of the Cooper data for RSAPv3 and Carrigan’s HSM ROR prediction model used the following base conditions for modeling the encroachment and crash frequencies: • Straight (i.e., degree of curve ± 10 or greater) road segment. • Flat (i.e., ± 3% grade or flatter) road segment. • Twelve-foot-wide lanes. • One lane in each direction for undivided roadways and two lanes in each direction for divided highways. • Zero major access points/mile. • Posted speed limit of 65 mph. • Generally flat terrain (i.e., as per the Highway Capacity Manual definition) (Highway Capacity Manual 2016). Deviating from these base conditions requires the use of segment EAFS to calibrate the encroachment frequency to the specific site conditions. The derivation and use of these adjust- ment factors are described later in Section 3.3.2.1.2: Encroachment Adjustment Factors (EAFS). This section describes encroachment data collection and modeling efforts that have taken place over the past 60 years. As such, a wide variety of nomenclatures, analysis techniques, and terminology have been used which often can cause confusion when comparing one study to another. In order to try and avoid such confusion, the results of all the studies discussed have been transformed such that they use the same definitions and nomenclatures. First, as shown in Figure 23, there are four possible encroachment directions for any highway. Primary right (PR) and opposing right (OR) encroachments leave the right travel edge of either divided or undivided highways in the PR and OR directions. On divided highways, vehicles leaving the highway on the left edge enter the median on either the primary left (PL) side or the opposing left (OL) side. On undivided highways, both left encroachments (PL and OL) first cross the centerline and may or may not reach the other side. The direction indicators PR, PL, OR, and OL are used throughout this report to denote the encroachment direction. However, most of the discussion will focus on the PR encroachments since the others can be derived from this single value. Previous researchers have sometimes limited data collection to either left or right encroach- ment directions depending on the data collection technique used. Hutchinson and Kennedy, for example, studied only left encroachments on divided highways so their data includes only PL and OL encroachments (Hutchinson 1962). Cooper, on the other hand, studied only right-side departures (PR and OR) on both divided and undivided highways (Cather 1978). In both cases, either set of data would have to be either multiplied by two to get the total encroachments on the section or divided by two to get a single encroachment direction.

86 Development of Safety Performance-Based Guidelines for the Roadside Design Guide is report denotes base encroachment frequency using the variable symbol BEF and a sub- script to indicate the highway type. For example, BEFUNDIV PR describes encroachments from the PR encroachment direction on an undivided highway, and BEFDIV OL describes encroach- ments from the opposing le encroachment direction on a divided highway. e encroachment frequency has units of either encroachments/mile/year or encroachments/km/year. e base encroachment rate (BER) is the derivative of the base encroachment frequency with respect to trac volume in units of either million vehicle miles traveled (MVMT) or million vehicle kilo- meters traveled (MVKT). is recognizes that the rate is essentially the slope of the frequency relationship although graphs are plotted with the trac volume as the AADT since that is more conventional. 3.3.2.1.1.1 e Hutchinson-Kennedy Encroachment Model. In the late 1950s and early 1960s Hutchinson and Kennedy conducted a direct observation study of encroachments (i.e., not crashes) on 40-foot medians in Illinois to “determine the signicance and nature of vehicle encroachments on certain types of medians under selected eld conditions . . .” to better under- stand the safety performance of medians (Hutchinson 1962; Hutchinson and Kennedy 1966). Encroachment locations and the extent of encroachment were identied through observation of vehicle tracks in snow-covered medians. Supplemental data was also gathered from avail- able police crash reports and construction plans. Data was collected in the years 1957 through 1963. e 1957 through 1960 portion of the study included only nine miles of highway. Another 198 miles were added in the 1960 through 1963 phase of the study. Hutchison and Kennedy observed a total of 332 encroachments on two highways (i.e., US 66 and FAI 74) with approxi- mately 207 miles of road. Hutchinson concluded that the frequency of encroachments can be related to trac volume below practical capacity. Furthermore, “as trac reaches capacity, the rate of encroachment becomes constant” as shown in Figure 24 (Hutchinson 1962). As shown in Figure 25, Hutchinson and Kennedy found that there were essentially two constant encroachment rates. e rst is for trac volumes less than about 4,000 veh/day where the rate was 4.33 le-side encroachments per MVMT. e second is for trac volumes Undivided Highway Divided Highway PR = Primary Right PL=Primary Left OR=Opposing Right OL=Opposing Left Figure 23. The four possible encroachment directions on undivided (left) and divided (right) highways (Carrigan and Ray 2017).

Roadside Risk Design Methodology 87   greater than about 6,500 veh/day where the rate was about 1.2 left-side encroachments/MVMT. Hutchinson and Kennedy provided Figure 25 to demonstrate the relationship between AADT and encroachments. Hutchinson and Kennedy proposed the following equation to model the BER into the median on divided highways after the first phase of the study (i.e., 1957 through 1960) (Hutchinson 1962): =+ −•BER 705 10 . DIVPL OL 0.0000466 AADT The Hutchinson-Kennedy data was used as the basis for the constant encroachment rate of 0.0005 encroachments/mile/year in the 1989 RDG (RDG) program ROADSIDE as well as in Figure 25. Hutchinson and Kennedy encroachment frequency (Hutchinson and Kennedy 1966). Figure 24. Hutchinson and Kennedy encroachment rate (Hutchinson and Kennedy 1966).

88 Development of Safety Performance-Based Guidelines for the Roadside Design Guide the bridge railing cost-benefit program BCAP (AASHTO 1989a; AASHTO 1989b). This value was apparently derived by taking the highest encroachment frequency at the highest AADT (i.e., 14.1 encroachments on the Kingery Expressway divided by the AADT of 31,253 veh/day = 0.0005 both left-side encroachments/mile/year/AADT). On a single left-edge encroachment basis this would be equivalent to: • BERDIV PL = 2.165 left encroachments/MVMT and BEFDIV = 7.9(10)−4 • AADT left encroachments/ mile/year for 0 < AADT < 2,500 veh/day and • BERDIV PL = −0.6 left encroachments/MVMT and BEFDIV = −2.19(10)−4 • AADT + 2.5 left encroachments/mile/year for 2,500 < AADT < 6,500 veh/day. These expressions are the basis of the often-reproduced encroachment frequency chart shown below in Figure 26 (note: the equations use length units of miles whereas Figure 26 uses kilo- meters). A plot of the PL encroachment direction for divided highways based on the Hutchinson- Kennedy data is shown in Figure 29. 3.3.2.1.1.2 The Cooper Encroachment Model. The next attempt at collecting encroachment data was undertaken in 1978 in five Canadian provinces by Cooper (Cooper 1980). Data col- lection took place over four months on 59 road sections. Approximately 20% of the segments were divided highways, and the remainder were two-lane undivided highways with speed limits of about 50 mph. The traffic volumes ranged from 6,000 to 45,000 veh/day for the divided high- ways and from about 1,000 to 13,000 veh/day for the undivided highways. Encroachments that occurred in the median area were not collected. Cooper analyzed the data and developed the relationship presented in Figure 27. The data collection efforts and supplemental office data linkage with roadway and environ- mental characteristics for the Copper data were documented in multiple reports and are only briefly summarized here; full details are available in the original reports (Cather 1978; Cooper 1981; Cooper 1980). Twelve data collection teams distributed throughout Canada were recruited and trained in June of 1978. Field data collection took place from July to October in 1978. Over the data collection period, tire marks and objects struck by vehicles on the roadside Figure 26. Typical approximation of the Hutchinson-Kennedy data (Mak and Sicking 2003).

Roadside Risk Design Methodology 89   were monitored, marked, measured, and graphed, in order to count and characterize vehicle encroachments. According to the field report, the field crews were mindful that some tire tracks might be generated by vehicles performing highway and railroad maintenance work, and efforts were made to exclude those encroachments. The data were collected on 59 road sections, each between 60 and 100 kilometers in length. The road sections were selected from five geographi- cally dispersed Canadian provinces. The study sections were not homogeneous in key attributes, including the posted speed limit, AADT, and paved shoulder width. The posted speed limits of subsections ranged from 45 mph to 60 mph. The field report provides an account of the planning, operation, and execution of the data collection efforts, and it documents experience gained throughout the collection process. It touches on several field data collection issues and discusses actions taken when problems were encountered. Overall, the report gives a glimpse of the nuances and potential issues that such a field data collection effort may experience (Cather 1978). The efforts made and procedures used to supplement each field-collected encroachment case with the related traffic, roadway and roadside geometrics, and weather data were reported (Cooper 1981; Cooper 1980; Weaver et al. 1975a). These reports provided the rationale and procedures used to delineate long road sections into shorter road segments for developing encroachment rate models. The Cooper Study pointed out some challenges in collecting encroachment data that would complicate the development of encroachment rate and frequency models. Each road section was Figure 27. Cooper encroachment frequency (Cooper 1980).

90 Development of Safety Performance-Based Guidelines for the Roadside Design Guide surveyed within a one-day period at one-week intervals for the duration of the study period. For two- and three-lane undivided highways, encroachments that encroached onto both edges of the undivided highways, including all four travel directions and encroachment-side com- binations, were collected. For four-lane divided highways, only those encroachments to the right of the edge line of the through lanes were collected (i.e., encroachments that occurred in the median area were not collected). A small number of encroachments that crossed both the median and the opposing travel lanes were detected and included in the data set. The encroachment data was based on monitoring tire tracks, and the field crews drove through each road section once a week; therefore, any encroachment that did not go beyond, or went only slightly beyond, the paved or graveled shoulders would be very difficult to detect. Recall that a vehicle roadside encroachment is defined as a vehicle inadvertently traveling beyond the edge of a traveled way, which includes vehicles that do not leave the shoulder. An operational definition of encroachments was adopted as being beyond the paved and graveled shoulder to allow the field crew to focus their attention on the area beyond the shoulder when they drove through study road sections (Cooper 1980). The encroachment rate could be expected to be higher under inclement weather conditions for any road segment. More specifically, under wet, slippery road surface conditions or lim- ited sight distance visibility, a higher proportion of drivers might be expected to lose traction or control of their vehicles and run off the road. Since the field collection of the Cooper data took place only during the summer months, it is expected that the number of weather-related encroachments in the Cooper data set would be under-represented when the data are expanded to represent the encroachments for the whole year. In short, annual encroachment rates gener- ated from the Cooper data are expected to be lower than the true encroachment rates because of the temporal constraint of the field collection. Despite the due diligence carried out by the field crews, some encroachments were inevitably missed. The field report shows evidence that field crews were trying to spot as many encroach- ments as possible while driving the study sections. Capturing 90% of the encroachments appeared to be the goal. For example, one of the teams reported that “to ensure that 90% of the off-road accidents are being found the van should travel along the shoulder at a speed no greater than 50 kilometers per hour and that the team members should change positions every 20 kilometers” (Cather 1978). Another difficulty in collecting encroachment data is detecting the encroachment and deter- mining the intentionality of the encroachment. Some encroachments are intentional in that the driver intentionally leaves the road whereas others, those that are the concern of the road- side safety engineer, involve unintentional and uncontrolled encroachments. Also, detecting encroachments on a compacted gravel or paved shoulder proved difficult. The study targeted the following three parameters for each detected encroachment: Maximum extent of lateral encroachment: The perpendicular distance measured from the edge line of the rightmost through lane to the farthest point of the lateral extent of the trajectory or the point where the first fixed object is struck. Longitudinal distance: The parallel distance of the trajectory along the roadway from the first point where the vehicle leaves the edge of the through lane to the point where the maximum extent of lateral encroachment occurs. Encroachment angle: The angle of departure measured from the tangent to the edge line of the traveled way at the estimated point of departure to the line connecting the tire-track path, at the point where the vehicle first leaves “the shoulder” and the estimated point of departure.

Roadside Risk Design Methodology 91   In statistical terms, the distance data were left truncated in that, with very few exceptions, only those encroachments that traveled beyond shoulders were recorded. The level of truncation varies from site to site as shoulder width varies. The distance data are also “interval censored” in that for each encroachment case the location of the maximum lateral distance is given in the form of a two-meter square grid. That is, the extent of lateral encroachment can only be known to occur within a 2x2 meter square grid but the exact location within the grid is unknown. Further complicating the interpretation of the data, the data are also “right-censored” in two ways: • About 37% of the encroachments struck fixed objects, and the lateral distances were measured from the edge line to the first fixed object struck. For these encroachment cases, the distance is right-censored because if the roadside were free of objects the extent of lateral encroachment would have been longer than the distance recorded. • When the lateral extent of encroachment exceeded a certain distance, the same code rep- resenting a catch-all-beyond distance category was recorded. The same coding scheme was used for longitudinal distances. For example, lateral distances of 16 meters, 21 meters, and 40 meters were all recorded as occurring in the 16–18 meter interval, which was the last interval allowed for recording the lateral distance. Longitudinal distances beyond 196 meters were all recorded as occurring in the 196–198 meter interval, the last interval allowed for recording the longitudinal distance. About 9.3% of the encroachments had a maximum lateral distance greater than 16 meters, and less than 1% of the cases had associated longitudinal distances beyond 196 meters. To summarize, the encroachment distance data in Cooper are left-truncated for almost all encroachments and, depending on whether an encroached vehicle struck fixed objects and the distances traveled, they are either right-censored or interval-censored. Statistical techniques to handle this type of data were not available in Cooper’s time but have been researched vigor- ously in the biomedical science and reliability engineering fields in the last three decades. If the truncation and censoring natures of the data are not properly formulated into the statistical estimation procedure, the modeling results are likely to be biased (Klein and Moeschberger 2003). This is especially true for the Cooper distance data, where almost all data are left-truncated, and a significant portion of the data are right-censored. The data were stored in three electronic files: an encroachment case file, a segment data file, and a section data file. The encroachment case file contains 1,949 records, each of which rep- resented an individual encroachment case detected and investigated in the field. Each record contains variables that quantify the characteristics of the encroachment such as encroachment angle, encroachment distances, and objects struck, as well as those variables that describe the traffic, roadway, and roadside conditions of the site where the encroachment occurred. The road section file contains 37 records with identifiers and attributes for 37 of the 59 sections surveyed. The road segment file contains 1,512 records including identifiers and variables for 756 road segments with one record per direction, which were delineated from 54 of the 59 sur- veyed sections. Five of the sections, all surveyed by one team (Team #9), were eliminated from the segment data due to questionable quality. Out of the 756 road segments, 575 segments were two- or three-lane undivided highways, and 181 segments were four-lane divided highways. A total of 1,881 encroachments occurred on these segments. Most of the highway characteristics are not homogeneous within a segment. For example, the general descriptors for highway alignments provided in the data indicate average alignment characteristics over the whole segment. For the two- or three-lane undivided highway segments, the length-weighted posted speed limit varied from 72.5 km/h to 97.5 km/h excluding those segments with missing values, with close to 60% of the segments under 80 km/h. The four-lane divided highways varied from 77.5 km/h to 97.5 km/h, with over 76% of the segments over 90 km/h.

92 Development of Safety Performance-Based Guidelines for the Roadside Design Guide The study road segments ranged in traffic volumes from about 1,000 to 13,000 vehicles per day (veh/day) for the two- and three-lane undivided highways and 6,000 to 45,000 veh/day for the four-lane divided highways. Cooper used an exploratory statistical “clustering” technique to delineate 54 of the study’s roadway sections to create a road segment file that contains 756 road segments for estimating encroachment rates. Cooper provided two main reasons for the need to delineate “sections” into shorter “segments”: (a) to increase the number of “data points” (or sample size), which would be beneficial statistically, and (b) to reduce the diluted effects of mixing various geometric features in these relatively long sections. For these reasons, Cooper stated, “it was thus considered neces- sary to subdivide the sections, and the means in which this is done is perhaps the most critical stage in the data reduction process” (Cooper 1980). The clustering technique basically grouped similar sites within a section to form segments based on a subjective similarity measure. Cooper adopted a similarity measure that was based primarily on the spacing between encroachment cases and secondarily on traffic volume. In other words, within a section, sites with encroach- ment cases that were closer in space, and perhaps similar in traffic volume, were grouped to create segments. Cooper’s method of using the clustering technique created a statistically flawed segment data file because it made the determination of analysis units (i.e., road segments) dependent on the outcome variable (i.e., encroachments). This dependency artificially inflated the highs and deflated the lows of encroachment frequencies among the delineated road segments. The con- sequence is that any relationship developed from the segment data regarding the encroachment frequency or rate is likely overstated. 3.3.2.1.1.3 Special Report 214. TRB Special Report 214 (SR214) was published in 1987 and deals with a wide variety of topics regarding designing roadsides for so-call 3R projects (i.e., resurfacing, restoration and rehabilitation) (TRB 1987). Appendix F deals with the relationships between crashes and specific roadside features and includes an encroachment model as part of a conditional probability model for predicting the frequency of crashes involving specific road- side features. The first term (i.e., Ex(E) in SR214 Appendix F) is the “expected annual number of encroachments on the highway segment encompassing the hazard” (TRB 1987). The expected frequency of encroachments is given as: = •BEF A1 AADTUNDIVPR A2 where A1 and A2 are regression coefficients. SR214 assumes that the encroachment frequency is only a function of the traffic volume. Utility pole crashes were used to back-calculate the cali- bration coefficients A1 and A2 (called a and b in SR214), but the values obtained greatly over- predicted all prior estimates of encroachment rates. Miaou suggests that one of the reasons is that an ordinary least-squares method was used to estimate the parameters after the logarithmic transformation. The least-squares method is based on an assumption that the distribution is normal so after the transformation, the distributions would be log-normal. This assumption, however, lacks “the distributional property to describe adequately random, discrete, non- negative, and typically sporadic, vehicle accident events on the road” (Miaou and Lum 1993). The SR214 model is also sensitive to segment length. S. P. Miaou and H. Lum concluded that Poisson regression models would be a better choice for modeling rare and “sporadic” events like traffic crashes. Paul Jovanis and Hsin-Li Chang came to a similar conclusion, and the later development of the HSM predictive models confirmed this choice since the negative binomial distribution is in the Poisson class of probability distributions (AASHTO 2010a; Jovanis and Chang 1986).

Roadside Risk Design Methodology 93   3.3.2.1.1.4 The Miaou Encroachment Model. Miaou investigated the possibility of combining both the encroachment and crash prediction approaches in a 2001 report for the FHWA (Miaou 2001). Miaou used a database of crashes and roadway characteristics data from Minnesota and Washington State reported by A. Vogt and J.G. Bared (Vogt and Bared 1998). The data represented crashes and road characteristics in 1985–1989 in Minnesota and 1993–1995 in Washington State for generally rural undivided highways. Miaou limited his study to roadways with AADT at less than 12,000 veh/day and the horizontal degrees of curvature less than 30. The model takes the following form for PR encroachments on flat, straight, undivided highway segments with 12-foot-wide lanes: BER e 2 . RURALUNDIVPR 0.42 0.04 AADT 1,000 0.45 =         − −   +     where: BERRURAL UNDIV PR = The encroachment for one right edge on undivided highways in encroachments/MVMT. AADT = Two-way average annual daily traffic between 1,000 to 12,000 veh/day. Miaou does not explicitly state if this model is for both edges of the road or only a single edge; therefore, it is assumed all encroachment directions are included, and the model included both undivided edges and should be divided by two for only one edge. It is also not clear if Miaou only included single vehicle ROR crashes or all crashes that occurred on the roadside regardless of the number of vehicles involved. A plot of the PR encroachment direction for rural undivided highways based on Miaou’s data compared to the other models in this section is shown in Figure 29. 3.3.2.1.1.5 RSAP 2.0.3 Encroachment Model. RSAP 2.0.3, released in 2002 and included with the 2002 and 2006 RDG, used the Cooper data to model encroachments (AASHTO 2002a; AASHTO 2006; Mak and Sicking 2003). NCHRP Report 492, the RSAP Engineer’s Manual, states that several adjustments were made to the Cooper data for use in RSAP 2.0.3. First, the authors believed encroachments were underreported, especially for roadways with 12 feet or less paved shoulders, so they eliminated all encroachments with a lateral extent of 12 feet or less. They then performed the encroachment modeling and extrapolated backward based on the probability of lateral extent distribution. This resulted in increasing the encroachments by a factor of 2.466 for two-lane undivided highways and 1.878 for divided highways. Second, as noted by Cooper, it was difficult to discern the controlled encroachments from the uncontrolled encroachments. The developers of RSAP 2.0.3 decreased the encroachment frequency by 60% based on a 1961 California study of barrier collisions by K. Moskowitz and W.E. Schaefer (Moskowitz and Schaefer 1961). NCHRP Report 492 does not describe the modeling effort for the encroachment frequency or the form of the model, but the values were inferred from Figure 28 and plotted for the primary right-side encroachment consistent with the other models in this section and plotted in Figure 29. 3.3.2.1.1.6 RSAPv3 Encroachment Model. Miaou returned to his analysis of the Cooper data as part of the update to the RSAPv3. Using more up-to-date statistical techniques he re-examined the statistical modeling of the Cooper data to resolve the left-truncated and right- censored nature of the original data collection. The negative binomial (NB) or Poisson-gamma regression model has become by far the most popular class of statistical models used to model highway crashes as evidenced by its choice for the development of safety predictive models in the HSM (AASHTO 2010a; Miaou 2001; Miaou and Lum 1993; Miaou 1997).

94 Development of Safety Performance-Based Guidelines for the Roadside Design Guide Default base encroachment frequencies from the Cooper data for two-lane undivided high- ways were developed using a negative binomial model. Summary statistics of highway and traffic variables for two- and three-lane undivided highway segments are presented in Table 9. Many of these variables were selected for the final stage of model development for RSAPv3. There were 1,353 encroachments observed during the field survey period for the 575 seg- ments. The total vehicle kilometers during the period were 1,985 MVKT. The overall encroach- ment rate was, therefore, 0.68 encroachments/MVKT (1.09 encroachments/MVMT). About 21.6% of the encroachments on two- and three-lane undivided highways were described as left-side departures which means that the vehicle encroached on the centerline and then crossed into the opposing travel lanes. The estimated base encroachment rate should, therefore, be reduced by 21.6% to approximate the encroachment rate for right-side only encroachments in the direction of travel. The mean encroachment rate for two-lane undivided highways can be expressed as: =+ − + + − −   +    BER e . . . . . . UNDIVPR OR 0.8528 0.3531 PSL 1.015 Rolling 0.8194 Mountain 0.2805 3LN 0.2092 AADT 1,000 0.6393 AD where: BERUNDIV PR+OR = Encroachments per MVKT for the primary and opposing right (OR) encroach- ment directions. PSL = 1 if posted speed limit > 90 km/h and zero otherwise. Rolling = 1 if rolling terrain and zero otherwise. Mountain = 1 if mountainous terrain and zero otherwise. 3LN = 1 if three-lane highways and zero for two-lane highways. AADT = AADT in veh/day. AD = Major access-point density in number of major road and highway access points per kilometer. The results of the encroachment rate model are presented in Table 10. It contains the mean and standard error of the estimated model parameters and goodness-of-fit statistics of the model. Among the variables in the model, access density, AADT, and terrain type are the most statistically Figure 28. RSAP 2.0.3 encroachment frequency (Mak and Sicking 2003).

Roadside Risk Design Methodology 95   Variable Mean Std Dev. Min Max Total Distribution Number of Encroachments (all four encroachment directions and encroachment- side combinations included) 2.4 2.0 0 20 1,353 AADT Segment Length (km) Exposure (MVKT) 4,794 6.4 3.5 2,355 10.8 6.4 1,000 0.5 0.1 12,903 80.0 65.4 3,698 1,985 Number of Lanes 2 Lanes = 441 segments (76.7%) 3 Lanes = 134 segments (23.3%) Posted Speed Limit (PSL) 82.3 6.4 72.5 97.5 - Length-Weighted PSL (km/h) 72.5 to 80.0 km/h = 336 segments (58.4%) 80.0 to 90.0 km/h = 184 segments (32%) 90.0 to 97.5 km/h = 55 segments (9.6%) Terrain Type (Based on Vertical Segment Geometry) Flat = 163 segments (28.4%) Rolling = 305 segments (53%) Mountainous = 107 segments (18.6%) Major Access-Point Density (Number of major road and highway access points/km) 1.2 1.5 0.002 9.4 - Curve Severity Moderate-Long Radius = 495 segments (86%) Short-Moderate Radius = 40 segments (4%) Unknown = 40 segments (4%) Horizontal Segment Geometry Long Tangent + Curve = 398 segments (69%) Reverse Curve = 177 segments (31%) Shoulder Width: (Paved + gravel (m) weighted by segment length) 2.9 0.8 0.2 6.2 - - - - - - - - Note: Overall encroachment rate = 1,353/1,985 = 0.68 ENCR/MVKT = 1.09 ENCR/MVMT. Table 9. Two- and three-lane undivided highway summary statistics. Table 10. Estimated parameters and statistics of two-lane undivided highways. Variable Estimated Coefficient Offset = (Exposure in MVKM) = vi (νi = 365⋅AADT⋅Segment Length/1,000,000) - Intercept = βo 0.8528 (±0.15) Posted Speed Limit (PSL in km/h) = β1 (β1 = 1 if PSL > 90; β1 = 0 otherwise) -0.3531 (±0.19) Rolling Terrain = β2 (β2 = 1 if Yes; β2 = 0, otherwise) 1.015 (±0.12) Mountainous Terrain = β3 (β3 = 1 if Yes; β3 = 0, otherwise) 0.8194 (±0.18) Three-Lanes= β4 (β4 = 1 if Yes; β4 = 0, otherwise) -0.2805 (±0.14) AADT = β5 (in 1,000s of veh/day) -0.2092 (±0.02) Major Access-Point Density = β6 (integer number of major road and highway access points/km) 0.6393 (±0.04) Inverse Dispersion Parameter Inverse Dispersion Parameter (ψ) 1.349 (±0.11) Inverse Dispersion Parameter for Model with βo only 0.6741 (±0.05) Goodness-of-Fit Measures (Hilbe 2011) RM2 = 1− 1 ψ 1 ψo 0.50 Notes: 1. Parameters (i.e., βs and ψ) were estimated using Markov chain Monte Carlo (MCMC) techniques, and the values shown in the table are the posterior means. 2. Values in parentheses are the estimated standard error of parameters based on the posterior density of the parameter.

96 Development of Safety Performance-Based Guidelines for the Roadside Design Guide significant variables. Given the variables included in the model, the following explicit base con- ditions were chosen: • Number of lanes = 2. • Posted speed limit = 104 km/h (i.e., 65 mph). • Terrain = Flat (i.e., Rolling = 0 and Mountain = 0). • Access Density = Zero major intersecting roads or highways (i.e., AD = 0). Considering the conditions under which the data were collected and those variables that were not found to be statistically significant, implicit base conditions should include good weather, good pavement conditions, lane width of about 12 feet, and heavy trucks under 15% of total traffic volume. With the base conditions listed above, the equation reduces to: = =+ − −       −      BER e e . . . UNDIVPR OR 0.8528 0.3531 1 0.2092 AADT 1,000 0.4997 0.2092 AADT 1,000 The above equation represents the encroachments on the two right-side edges of an undivided highway. A two-lane undivided highway has four encroachment possibilities: • Primary direction encroaching right, • Primary direction encroaching left (i.e., crossing the center line), • Opposing direction encroaching right, and • Opposing direction encroaching left (i.e., crossing the center line). As discussed earlier, 21.6% of the right-edge encroachments started as left-side departures that crossed the opposing lane before encroaching. It is assumed that left-side encroachments are as probable as right-side encroachments so the equation above should be reduced by 78.4% (i.e., 100 − 21.6 = 78.4) to represent right-side encroachments from the direction of travel, and then that value should be divided by two to obtain the frequency of one right-side edge encroachment. The base encroachment frequency is tabulated as one encroachment direction in units of right encroachments per edge-mile per year. Making the appropriate conversions, multiplying by 0.784 to take out the left-side encroachments in the right-side departures and then dividing by two to get one encroachment direction and multiplying by 1.6093 to convert kilometers to edge- miles results in the following equation for two-lane undivided highways with traffic volumes 1,000 ≤ AADT ≤ 13,000: =         −       • • BEF 0.784 1.6093 2 365 AADT 1,000,000 e . UNDIVPR 0.4997 0.2092 AADT 1,000 where: BEFUNDIV PR = The base encroachment frequency in right-side encroachments/edge-mile/year. AADT = Average annual daily traffic in veh/day. A plot of the PR encroachment direction for two-lane undivided highways based on Miaou’s reanalysis of the Cooper data is shown in Figure 29 where the solid blue line represents the model over the range of observed data and the dashed blue line is an extrapolation beyond the limits of the observed data. The Cooper encroachment data was also used to develop an encroachment rate model for divided highways for the development of RSAPv3. Summary statistics of the highway and traffic variables for the four-lane divided highway segments are presented in Table 11. There were 528 encroachments observed during the field survey period for the 181 divided highway

Roadside Risk Design Methodology 97   segments. The total vehicle kilometers during the period were 1,243 MVKT. Thus, the overall encroachment rate was 528/1,243 = 0.42 ENCR/MVKT or 0.68 enc/MVMT. About 6.7% of the encroachments on four-lane divided highways were described as left-side departure meaning they encroached on the left side of the divided highway and then crossed both the median and the opposing travel lanes. The estimated base encroachment rate should, therefore, be reduced by 6.7% to obtain the rate for right-side only encroachments. The mean encroachment rate for four-lane divided highways can be expressed as follows: =+ − −   +    BER e . . DIVPR OR 0.2104 0.04128 AADT 1,000 1.145 AD where: BERDIV PR+OR = Encroachments per MVKT. AADT = Annual average daily traffic in veh/day. AD = Major access-point density in number of major road and highway access points per kilometer. Among the variables considered, only AADT and access density were found to be statistically significant in the final stage of modeling. The estimation results of the final divided highway model are presented in Table 12 which contains the mean and standard error of the estimated model parameters and goodness-of-fit statistics of the model. Despite the fact that only two covariates were included in the model, the overall model goodness-of-fit, as indicated by a R2M value of 0.81, was good. Given that AADT and access density are the only variables included, the only explicit base condition that needs to be selected is the access density. The assumed base condition is that Variable Mean Std Dev Min Max Total Distribution Number of Encroachments (Observed on right-side of each travel direction but not in the median) 2.9 2.6 0 15 528 - AADT Segment Length (km) Exposure (MVKT) 21,564 3.4 6.9 9,098 5.6 8.8 5,954 0.5 0.5 44,930 34.0 67.2 - 620 1,243 - - - Posted Speed Limit (PSL) 95.9 3.8 77.5 97.5 - Length-Weighted PSL (km/h) 77.5 to 90.0 = 24 segments (13.3%) 90 to 97.0 = 10 segments (5.5%) Greater than 97.5 = 147 segments (81.2%) Terrain Type Flat = 73 segments (40.3%) Rolling = 71 segments (39.2%) Mountainous = 37 segments (20.5%) Major Access-Point Density (Number of major road and highway access points/km) 0.9 0.7 0.018 2.8 - - Curve Severity Moderate-Long Radius = 154 segments (85%) Short-Moderate Radius = 19 segments (11) Unknown = 8 segments (4%) Horizontal Segment Geometry Long Tangent + Curve = 163 segments (90%) Reverse Curve = 18 segments (10%) Shoulder Width (Paved + gravel (m) weighted by segment length) 3.7 0.8 0.1 6.2 - - Note: Overall encroachment rate = 528/1,243 = 0.42 ENCR/MVKT = 0.68 ENCR/MVMT Table 11. Four-lane divided highway summary statistics.

98 Development of Safety Performance-Based Guidelines for the Roadside Design Guide there are no major intersecting roads or highways (i.e., AD = 0). Considering the conditions under which the data were collected and the range of those variables that were not found to be statistically significant, implicit base conditions should also include a posted speed limit of 65 mph, level terrain, good weather, good pavement conditions, lane width of about 12 feet, and heavy trucks under 15% of total traffic volume. With these base conditions, the equation reduces to the following: =+ − −      BER e . DIVPR OR 0.2104 0.0413 AADT 1,000 Similar to the discussion of two-lane undivided highways, the above equation represents the encroachments on the two right-side edges of a four-lane divided highway. A four-lane divided highway has four encroachment possibilities: • Primary direction encroaching right, • Primary direction encroaching left (i.e., into the median), • Opposing direction encroaching right, and • Opposing direction encroaching left (i.e., into the median). As discussed earlier, 6.7% of the right-edge encroachments started as left-side departures that crossed the median and opposing lanes before encroaching on a right edge. It is assumed that left-side encroachments are as probable as right-side encroachments so the equation above should be multiplied by 1 − 0.067 = 0.933 to represent only the right-side encroachments from the direction of travel, and then that value should be divided by two to yield only one right-side encroachment direction. Making the appropriate unit conversions to obtain the base encroach- ment frequency, multiplying by 0.933 to take out the left-side encroachments in the right-side departures, dividing by two to get only one right edge, and converting from kilometers to miles results in the following equation for the base encroachment frequency for one right edge of a four-lane divided highway for 6,000 ≤ AADT ≤ 45,000: =         − −    • • BEF 0.933 1.6093 2 365 AADT 1,000,000 e . DIV PR 0.2104 0.0413 AADT 1,000 Variable Estimated Coefficient Offset = (Exposure in MVKM) = νi (νi = 365⋅AADT⋅Segment Length/1,000,000) - Intercept = βo -0.2104 (±0.15) AADT = β1 (in 1,000s of veh/day) -0.0413 (±0.007) Major Access-Point Density = β2 (integer number of major road and highway access points/km) 1.1450 (±0.08) Inverse Dispersion Parameter Inverse Dispersion Parameter (ψ) 8.549 (±5.28) Inverse Dispersion Parameter for Model with βo only 1.628 (±0.28) Goodness-of-Fit Measures (Hilbe 2011) 0.81RM2 = 1− 1 ψ 1 ψo Notes: 1. Parameters (i.e., βs and ψ) were estimated using MCMC techniques, and the values shown in the table are the posterior means. 2. Values in parentheses are the estimated standard error of parameters based on the posterior density of the parameter. Table 12. Estimated parameters and statistics of four-lane divided highways.

Roadside Risk Design Methodology 99   where: BEFDIV PR = The base encroachment frequency in encroachments/edge-mile/year. AADT = Average annual daily traffic in veh/day. A plot of the PR encroachment direction for four-lane divided highways based on Miaou’s reanalysis of the Cooper data is shown in Figure 29 where the solid blue line represents the model over the range of observed data and the dashed blue line is an extrapolation beyond the limits of the observed data. 3.3.2.1.1.7 The Carrigan Encroachment Model. Like Miaou and Special Report 214, Carrigan used an indirect approach to estimating encroachments by developing an SPF for an HSM-style ROR crash prediction model (Carrigan and Ray 2018a). The method is indirect because it models crashes rather than encroachments. While not all encroachments become crashes, all crashes start out as encroachments. If the probability of interacting with a roadside feature is captured in a CMF, the SPF should closely correspond to the frequency of encroaching on the roadside. Carrigan used the Washington State and Ohio HSIS data from the 2002 to 2011 time period. This data set included almost 140,000 highway segments that were used for devel- oping the models. This data set is the largest ever used to estimate encroachments and, as will be shown below, included a very wide range of traffic conditions (i.e., AADT, MVMT, PSL, and PT). One of the unique and important differences of Carrigan’s modeling using crash data is that it was based on all vehicles that left the road rather than assuming that all ROR crashes were single vehicle crashes. Many ROR crashes begin as on-road interactions or even collisions between multiple vehicles, one or more of which subsequently leave the road. For example, a vehicle passing another vehicle may sideswipe the passed vehicle while returning to its lane, causing the vehicle to lose control, leave the roadway, roll over on a roadside slope, and strike a tree. A cross-median crash where the vehicle encroached left and entered the opposing lanes interact- ing with a second vehicle is also a multi-vehicle crash. These crashes would not be included as a single vehicle ROR crash even though the rollover and interaction with the tree or cross-median crash may have been the most harmful event in the collision. Including all vehicles that run off the road is a more appropriate strategy for evaluating roadside designs. Carrigan considered and explored many model forms and covariates. Posted speed, for example, was expected to be an important predictive variable, but the correlation analysis revealed that speed is not a good predictor of ROR crashes. On the other hand, the land-use vari- able (i.e., urban or rural) was found to have important effects on predicting ROR crashes. PT was also found to influence crash frequency significantly and differently in rural and urban areas although it is generally ignored in other HSM models. Regression models which included traditional HSM exposure terms such as AADT and seg- ment length but did not include MVMT or PT were compared to forms which did. Models were explored where AADT was considered as the sole independent variable and segment length was the offset (i.e., length had no coefficient). Models were explored with both AADT and VMT and a segment length offset. Models with only VMT and a segment length offset were reviewed. Finally, models with AADT as the independent variable and VMT offset were also reviewed. Rural and urban divided roadways are best represented by a model having a log-transformed AADT as an explanatory variable, PT as an explanatory variable, and segment length as the offset. Rural and urban undivided roadways are best represented by a model which included AADT and PT as the explanatory variables while including VMT as the offset. The undivided models, therefore, include AADT twice, whereas the divided models include AADT only once. This additional AADT term is believed to be needed to represent the cross-centerline vehicles, which may initiate or prevent a ROR crash (i.e., avoiding a sideswipe may lead to running off the

100 Development of Safety Performance-Based Guidelines for the Roadside Design Guide road or a head-on crash on the road may preclude the lane departure from becoming a ROR crash). The divided and undivided highway SPFs developed by Carrigan take the form here: = = • • • • • • • • SPF e e e AADT 365 L SPF AADT e e L . . . UNDIV A1 AADT A2 PT A3 DIV A4 A5 PT A6 where: SPFi = Frequency of ROR crashes by segment edge per year. AADT = Annual Average Daily Traffic in veh/day. PT = Percentage of Trucks (%). L = Segment Length (miles). Ai = Regression coefficients. Models were developed for each of the four encroachment directions (i.e., PR, PL, OR, and OL) as well as for each of the encroachment edges (i.e., PRE and ORE for undivided roadways and PRE, PLE, ORE, and OLE for divided roadways). Only the modeling for the PR encroachment direction is discussed herein to be consistent with the discussion of other encroachment models. For each crash, the encroachment direction and edge were identified using the crash data. The data set for modeling encroachments on rural divided highways included data from the HSIS for 2002–2010 in Ohio and 2002–2007 in Washington State. The resulting data set included 124,458 rural divided highway segment edges. The data were filtered as follows where the number following the colon is the number of segment edges remaining after that filter: • Consider only segments ≤ 2 miles and ≥ 0.1miles: 58,210 segments. • Consider only segments AADT > 0: 58,196 segments. • Consider only 12-foot lanes: 54,200 segments. • Consider only flat segments (i.e., |Grade| ≤ 3%): 47,810 segments. • Consider only straight segments: 39,386 segments. • Consider only 8-foot or greater right shoulders: 36,916 segments. • Consider only segments with two lanes per barrel (i.e., 4 lanes): 32,306 segments. • Consider only segments with four-foot left shoulders: 25,414 segments. A summary of the filtered data set is provided in Table 13. Each data element was reviewed to determine how closely increases in one element correlate with increases in another data element. For example, if the AADT increases do vehicle crashes increase? This correlation analysis was Statistic Min. 1st Qu. Median Mean 3rd Qu. Max. L 0.1 0.16 0.27 0.4324 2.00 AADT 710 11,260 15,732 19,053 67,390 Median Width 2 40 60 78.34 9,999 PT 0 0 13.64 16.45 63.66 PSL 35 60 65 64.51 70 PRE 0 0 0 0.2421 10 PLE 0 0 0 0.2211 13 UNK 0 0 0 0.1984 22 All† 0 0 0 0.5196 20 VMT‡ 44,676 890,469 1,614,461 3,096,834 0.54 24,262 76 27.65 70 0 0 0 1 3,485,969 42,202,760 † All=PRE+PLE+UNK/2 ‡ VMT= 365∙AADT∙L Note: L = segment length; AADT = average annual daily traffic; PT = percentage of trucks; PSL = posted speed limit; PRE = primary right edge; PLE = primary left edge; UNK = unknown; VMT = vehicle miles traveled. Table 13. Descriptive statistics of filtered rural divided highway data set.

Roadside Risk Design Methodology 101   conducted using Pearson and Spearman’s correlation coefficients. The results for the divided highway data set are shown in Table 14. A value of zero indicates the data elements are not correlated, a value of one indicates perfect correlation, and negative values indicate inverse correlation. Data elements with higher values should be considered for use in the development of the SPF. 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). Both correlation measures are shown in Table 14. By inspection, the VMT appears to have the highest correlation, and posted speed limit (PSL) appears to be inversely correlated. As shown in Table 13, the observed AADT values ranged from 710 to almost 70,000 veh/day, the percentage of trucks ranged from 0% to over 60%, and the posted speed limits from 35 mph to 70 mph. These data, therefore, had a much broader range of traffic conditions than all previous studies of roadside encroachment and crash frequency. While Carrigan developed models for all four encroachment directions and all four encroach- ment edges, only the results of the PR encroachment analysis are shown here since predicting the PR encroachment is the focus of this section. The coefficients and goodness-of-fit statistics for the model for PR encroachments on rural divided highways are shown in Table 15a and 15b. The pseudo-R2 statistic was determined for each model. The pseudo- R2 is not interpreted the same way as the coefficient of determination is for an ordinary least-squares regression. A low value of the pseudo-R2 can indicate a lack of fit while higher values carry no such indication. There is no definition of a low value. The Akaike Information Criterion (AIC) goodness-of-fit statistic provides comparative information, with lower values indicating a better fitting model than the model it is compared to. Bayesian Information Criterion (BIC) is interpreted the same way. Both are calculated from the likelihood function (Hilbe 2011). Pearson Correlation Index All PRE PLE L AADT PT PSL VMT All 1.0000 PRE 0.7909 1.0000 PLE 0.7917 0.3557 1.0000 L 0.4404 0.3616 0.3489 1.0000 AADT 0.2782 0.1974 0.2336 0.0539 1.0000 PT 0.1823 0.1490 0.1559 0.2543 0.1704 1.0000 PSL -0.0434 -0.0472 -0.0227 -0.1569 0.1732 -0.2301 1.0000 VMT 0.5646 0.4348 0.4534 0.7619 0.5109 0.2944 -0.0107 1.0000 Spearman Correlation Index All PRE PLE L AADT PT PSL VMT All 1.0000 PRE 0.7561 1.0000 PLE 0.7173 0.2597 1.0000 L 0.3820 0.3104 0.2955 1.0000 AADT 0.2542 0.1845 0.2134 -0.0098 1.0000 PT 0.1932 0.1600 0.1618 0.2619 0.0792 1.0000 PSL -0.0982 -0.0919 -0.0624 -0.1885 0.1298 -0.3731 1.0000 VMT 0.4442 0.3445 0.3479 0.7869 0.5694 0.2611 -0.0748 1.0000 Note: PRE = primary right edge; PLE = primary left edge; L= segment length; AADT = average annual daily traffic; PT = percentage of trucks; PSL = posted speed limit; VMT = vehicle miles traveled. Table 14. Correlation matrix for rural divided highway ROR events and data elements.

102 Development of Safety Performance-Based Guidelines for the Roadside Design Guide Negative binomial model parameters are estimated using the maximum likelihood method, where the parameters of the probability distribution that characterize the data are estimated. The log of the likelihood function is used to determine which parameters make the model most likely for the data considered. Through an iterative process, the derivative of the log likelihood (LL) function is taken and set to zero to estimate the parameters. When the difference between iterative values is less than a specified tolerance (e.g., 10−6), the iterations stop, and the values are at the maximum likelihood estimated values. The log likelihood is also reported with the models; however, it is only useful when calculating other goodness-of-fit statistics (e.g., AIC and BIC). The pseudo-R2 is not interpreted the same way as the coefficient of determination is for an ordinary least-squares regression. A low value of the pseudo-R2 can indicate a lack of fit while higher values carry no such indication. There is no definition of a low value. Any measurement has uncertainty which should be documented. This uncertainty in statis- tical analysis can be conveyed by noting the standard error or the confidence interval along with the measurements. The standard error is a measure of how much the estimate could change within the model. The 95% confidence interval is essentially the same type of statistic as standard error; the 95% confidence interval limits indicate that the analyst is 95% confident the true value of the coefficient is within the stated range. It is important to note the 95% con- fidence interval is equal to twice the standard error for normally distributed error. Negative binomial models are assumed to have normally distributed errors. Carrigan also developed CMFs for a variety of roadway and roadside features. One such roadway feature was the right-shoulder width. The base condition for the right-shoulder width was eight feet so extrapolating the shoulder width back to zero to obtain encroachments yields a CMF of 1.3629 for rural divided highways. Using Carrigan’s model form shown earlier for encroachment frequency, inserting the coefficients from Table 15a, using a unit length of one mile, multiplying by the shoulder-width CMF, and converting the frequency to an encroach- ment rate results in the following: BEF 1.3629 AADT e 365 AADT 10 BER BEF . RURALDIVPR A1 A2 A3 PT 6 RURALDIVPR RURALDIVPR =     = [ ]+• • • Statistic Value AIC: 26,827 Theta: 1.594 Std. Err.: BIC: LL: 0.11 26,851 -13,410.36 Pseudo-R2: 0.4444 Note: AIC = Akaike Information Criterion; BIC = Bayesian Information Criterion; LL = Log Likelihood. Table 15b. Rural divided highway encroachment model goodness-of-fit statistics. Regression Coefficients Estimate Std. Error z value Pr(>|z|) AADT Coeff. (A1) 0.7546 0.02 29.14 <2e-16 Intercept (A2) -8.0913 0.25 -32.01 <2e-16 PT Coeff. (A3) 0.0042 0.00 4.06 4.99e-05 Table 15a. Rural divided highway encroachment model coefficients.

Roadside Risk Design Methodology 103   365 AADT 10 BER 1.3629 AADT e BER 10 365 AADT 1.3629 AADT e BER 10 365 1.3629 AADT e BER 1.1433 AADT . . . 6 RURALDIVPR A1 A2 A3 PT RURALDIVPR 6 A1 A2 A3 PT RURALDIVPR 6 A1 1 A2 A3 PT RURALDIVPR 0.2454     = =     =     = [ ] [ ] [ ]( ) + + − + − • • • • • • • A plot of the PR encroachment direction for rural divided highways is shown in Figure 29. The data set for modeling encroachment on urban divided highways included HSIS data from 2002–2010 in Ohio and 2002–2011 in Washington State. The resulting multi-state (i.e., Washington and Ohio) data set included 244,050 urban divided highway segment edges. The data were filtered as follows where the number following the colon is the number of segment edges remaining after that filter: • Consider only segments ≤ 2 miles and ≥ 0.1miles: 105,802 segments. • Consider only segments AADT > 0: 105,318 segments. • Consider only 12-foot lanes: 89,380 segments. • Consider only flat segments (i.e., |Grade| ≤ ±3%): 82,070 segments. • Consider only straight segments: 70,222 segments. • Consider only 8-foot or greater right shoulders: 64,292 segments. • Consider only greater than 0 mph: 63,963 segments. • Consider only segments with 2 lanes per barrel 39,920 segments. (i.e., 4 lanes): • Consider only segments with 4-foot left shoulders: 24,025 segments. A summary of the filtered data set is provided in Table 16. The correlation analysis was conducted using Pearson and Spearman’s correlation coefficients as shown in Table 17 and described earlier. As shown in Table 16, the observed AADT values ranged from 780 to more than 155,000 veh/day, the percentage of trucks ranged from 0% to 55%, and the posted speed limits from 20 mph to 70 mph. Like the rural divided data, these data had a much broader range of traffic conditions than all previous studies of roadside encroachment and crash frequency. Statistic Min. 1st Qu. Median Mean 3rd Qu. Max. L 0.1 0.16 0.27 0.4152 0.52 2.00 AADT 780 18,181 29,883 34,464 45,611 155,340 Median Width 2 40 44 83.92 64 9,999 PT 0 6.59 11.04 13.86 19.22 55 PSL 20 55 60 60.31 65 70 PRE 0 0 0 0.3669 0 14 PLE 0 0 0 0.3691 0 11 UNK 0 0 0 1.555 1 141 All† 0 0 0 1.456 2 74 VMT‡ 34,164 1,554,352 2,974,969 5,072,729 6,342,094 50,969,272 † All=PRE+PLE+UNK/2 ‡ VMT= 365∙AADT∙L Note: L = segment length; AADT = average annual daily traffic; PT = percentage of trucks; PSL = posted speed limit; PRE = primary right edge; PLE = primary left edge; UNK = unknown; VMT = vehicle miles traveled. Table 16. Descriptive statistics of filtered multi-state urban divided highway data set.

104 Development of Safety Performance-Based Guidelines for the Roadside Design Guide Regression Coefficients Estimate Std. Error z value Pr(>|z|) AADT Coeff. (A1) 0.6623 0.02 30.76 <2e-16 Intercept (A2) -7.1613 0.23 -31.57 <2e-16 PT Coeff. (A3) 0.0114 0.00 9.40 <2e-16 Table 18a. Urban divided highway encroachment model coefficients. Like the previous rural divided highway model, Carrigan developed models for all four encroachment directions and all four encroachment edges, but only the results of the PR encroachment analysis are shown here. The coefficients and goodness-of-fit statistics for the model for PR encroachments on rural divided highways are shown in Table 18a and 18b. The base condition for the right-shoulder width was eight feet, so extrapolating the shoulder width back to zero with Carrigan’s shoulder-width CMF results in a CMF of 1.2221 for urban divided highways. Using Carrigan’s model form shown earlier for encroachment frequency, Pearson Correlation Index All PRE PLE L AADT PT PSL VMT All 1.0000 PRE 0.6187 1.0000 PLE 0.6064 0.4063 1.0000 L 0.4560 0.3850 0.3816 1.0000 AADT 0.1423 0.1389 0.1504 -0.0504 1.0000 PT 0.0963 0.0796 0.0974 0.1072 -0.0941 1.0000 PSL 0.0359 0.0885 0.1040 0.0646 0.1946 0.2318 1.0000 VMT 0.5073 0.4509 0.4544 0.7490 0.4516 0.0658 0.1623 1.0000 Spearman Correlation Index All PRE PLE L AADT PT PSL VMT All 1.0000 PRE 0.6565 1.0000 PLE 0.6554 0.3163 1.0000 L 0.4523 0.3348 0.3228 1.0000 AADT 0.1902 0.1505 0.1769 -0.0289 1.0000 PT 0.1524 0.0984 0.1061 0.1574 -0.0630 1.0000 PSL 0.0970 0.0983 0.1195 0.0840 0.2513 0.3004 1.0000 VMT 0.4791 0.3588 0.3671 0.7329 0.6283 0.0553 0.2250 1.0000 Note: PRE = primary right edge; PLE = primary left edge; L = segment length; AADT = average annual daily traffic; PT = percentage of trucks; PSL = posted speed limit; VMT = vehicle miles traveled. Table 17. Correlation matrix for multi-state urban divided highway data set. Table 18b. Urban divided highway encroachment model goodness-of-fit statistics. Statistic Value AIC: Theta: Std. Err.: BIC: LL: Pseudo-R2: 33,344 1.2643 0.06 33,368 -16,668.99 0.11 Note: AIC = Akaike Information Criterion; BIC = Bayesian Information Criterion; LL = Log Likelihood.

Roadside Risk Design Methodology 105   inserting the coefficients from Table 18a, setting the percentage of trucks to zero and using the right-shoulder CMF, using a unit length of one mile, and converting the frequency to an encroachment rate results in the following: BEF 1.0810 AADT e 365 AADT 10 BER BEF 1.0810 AADT e BER 10 365 1.0810 AADT e BER 2.2984 AADT . . . URBANDIVPR A1 A2 A3 PT 6 URBANDIVPR URBANDIVPR A1 A2 A3 PT URBANDIVPR 6 A1 1 A2 A3 PT URBANDIVPR 0.3370 =     = = =     = [ ] [ ] [ ]( ) + + − + − • • • • • • A plot of the PR encroachment direction for urban divided highways is shown in Figure 29. As shown in Figure 29, the encroachment rate, for both rural and urban divided highways, asymp totically decays quickly from an initial value of unity and after an AADT of about 10,000 veh/day is roughly linear and nearly constant when plotted on a log-linear scale. There is very little difference between the urban and rural divided highway encroachment model. The encroachment frequency increases monotonically and nearly linearly with a slope of around 2.5(10)−5 right-edge encroachments/veh/day or 0.07 right-edge encroachments/MVMT. The data set for modeling encroachment on rural undivided highways included HSIS data from 2002–2010 in Ohio and 2002–2007 in Washington State. The resulting data set included 2,058,268 rural segment edges. The data were filtered as follows: • Consider only segments ≤ 2 miles and ≥ 0.1miles: 637,060 segments. • Consider only segments AADT > 0: 635,464 segments. • Consider only 12-foot lanes: 204,162 segments. • Consider only flat segments (i.e., |Grade| ≤ 3%): 163,948 segments. • Consider only straight segments: 137,310 segments. • Consider only 8-foot or greater right shoulders: 39,556 segments. • Consider only segments with a posted speed value > 0: 39,520 segments. • Consider only segments with 2 lanes: 38,974 segments. A summary of the filtered data set is provided in Table 19. The Pearson and Spearman’s correlation analysis is shown in Table 20. As was found for divided highways, segment length and VMT are most closely correlated with crash frequency. As shown in Table 19, the observed Statistic Min. 1st Qu. Median Mean 3rd Qu. Max. L 0.1 0.14 0.23 0.3618 2.00 AADT 40 2,795 4,514 5,142 27,540 PT 0.00 6.92 12.02 13.94 67.15 PSL 25 55 55 54.22 65 PR 0 0 0 0.0835 5 OL 0 0 0 0.0549 4 UNK 0 0 0 0.0759 8 All† 0 0 0 0.1523 9 VMT‡ 6,643 207,959 386,316 672,453 0.42 6,430 18.59 60 0 0 0 0 787,296 8,886,108 † All=PRE+PLE+UNK/2 ‡ VMT= 365∙AADT∙L Note: L = segment length; AADT = average annual daily traffic; PT = percentage of trucks; PSL = posted speed limit; PR = primary right; OL = opposing left; UNK = unknown; VMT = vehicle miles traveled. Table 19. Descriptive statistics of filtered rural undivided highway data set.

106 Development of Safety Performance-Based Guidelines for the Roadside Design Guide AADT values ranged from a low of 40 to almost 28,000 veh/day, the percentage of trucks ranged from 0% to over 67%, and the posted speed limits from 25 mph to 65 mph. Like the rural and urban divided highway data, these data had a much broader range of traffic conditions than all previous studies of roadside encroachment and crash frequency. The coefficients and goodness-of-fit statistics for the model for PR encroachments on rural divided highways are shown in Tables 21a and 21b. The base condition for the right-shoulder width was eight feet so extrapolating the shoulder width back to zero with Carrigan’s shoulder- width CMF results in a CMF of 1.8126 for rural undivided highways. Using Carrigan’s model Pearson Correlation Index All PR OL L AADT PT PSL VMT All 1.0000 PR 0.7846 1.0000 OL 0.6568 0.1515 1.0000 L 0.3307 0.2625 0.2341 1.0000 AADT 0.0925 0.0807 0.0377 -0.0162 1.0000 PT -0.0195 -0.0156 -0.0078 0.0440 -0.1043 1.0000 PSL -0.0045 0.0019 0.0058 0.0467 -0.0966 0.2281 1.0000 VMT 0.3589 0.2920 0.2301 0.8001 0.4156 0.0087 0.0034 1.0000 Spearman Correlation Index All PR OL L AADT PT PSL VMT All 1.0000 PR 0.7659 1.0000 OL 0.6285 0.1311 1.0000 L 0.2559 0.1992 0.1804 1.0000 AADT 0.1116 0.0928 0.0598 -0.0097 1.0000 PT -0.0424 -0.0353 -0.0218 -0.0017 -0.0740 1.0000 PSL -0.0550 -0.0473 -0.0285 -0.0185 -0.1040 0.4053 1.0000 VMT 0.2704 0.2129 0.1793 0.7335 0.6299 -0.0570 -0.1008 1.0000 Note: PR = primary right; OL = opposing left; L = segment length; AADT = average annual daily traffic; PT = percentage of trucks; PSL = posted speed limit; VMT = vehicle miles traveled. Table 20. Correlation matrix for rural undivided highway ROR events and data elements. Table 21b. Resulting rural undivided highway SPFEDGE goodness-of-fit statistics. Statistic Value AIC: Theta: Std. Err.: BIC: LL: Pseudo-R2: 20,431 1.296 0.15 20,457 -10,212.54 0.3717 Note: AIC = Akaike Information Criterion; BIC = Bayesian Information Criterion; LL = Log Likelihood. Regression Coefficients Estimate Std. Error z value Pr(>|z|) Intercept (A1) -15.4 0.04 -315.65 < 2e-16 AADT Coeff. (A2) -5.706(10)-5 0.00 -10.11 < 2e-16 PT Coeff. (A3) -8.974(10)-3 0.00 -4.63 3.75e-06 Table 21a. Resulting rural undivided highway SPFEDGE coefficients.

Roadside Risk Design Methodology 107   form shown earlier for encroachment frequency and inserting the coefficients from Table 21a, using a unit length of one mile and converting the frequency to an encroachment rate results in the following: = = = [ ] ( ) ( ) + + − − −  −  − − • • • BER 1.8126 e BER 1.8126 e BER 0.3680 e . . . . . RURALUNDIV PR A1 A2 AADT A3 PT RURALUNDIV PR 15.41 5.706 10 AADT 0.00897 PT RURALUNDIV PR 5.706 10 AADT 5 5 A plot of the PR encroachment direction for rural undivided highways is shown in Figure 29. The data set for modeling encroachments on urban undivided highways included HSIS data from 2002–2010 in Ohio and 2002–2011 in Washington State. The resulting data set included 485,898 urban segment edges. The data were filtered as follows: • Consider only segments ≤ 2 miles and ≥ 0.1miles: 181,340 segments. • Consider only segments AADT > 0: 179,788 segments. • Consider only 12-foot lanes: 57,468 segments. • Consider only flat segments (i.e., |Grade| ≤ 3%): 49,970 segments. • Consider only straight segments: 46,148 segments. • Consider only greater than 0 mph: 44,652 segments. • Consider only segments with two lanes: 37,140 segments. • Consider only 8-foot-wide or greater right shoulders: 10,727 segments. A summary of the filtered data set is provided in Table 22. The Pearson and Spearman’s cor- relation analysis is shown in Table 23. As shown in Table 22, the observed AADT values ranged from a low of 357 to almost 43,000 veh/day, the percentage of trucks ranged from 0% to just over 50%, and the posted speed limits from 25 mph to 60 mph. Like the rural and urban divided highway data, these data had a much broader range of traffic conditions than all previous studies of roadside encroachment and crash frequency. The coefficients and goodness-of-fit statistics for the model for PR encroachments on rural divided highways are shown in Tables 24a and 24b. The base condition for the right-shoulder width was eight feet so extrapolating the shoulder width back to zero with Carrigan’s shoulder- width CMF results in a CMF of 1.2221 for urban undivided highways. Using Carrigan’s model form shown earlier for encroachment frequency, setting the percentage of trucks to zero, using Statistic Min. 1st Qu. Median Mean 3rd Qu. Max. L 0.1 0.14 0.22 0.3304 0.38 1.94 AADT 357 5,696 8,640 9,480 11,790 42,836 PT 0.00 3.68 6.13 8.18 11.04 50.11 PSL 25 40 50 48.22 55 60 PR 0 0 0 0.1077 0 5 OL 0 0 0 0.0761 0 4 UNK 0 0 0 0.5520 0 47 All† 0 0 0 0.4145 0 26 VMT‡ 14,334 406318 691,420 1,122,688 1,312,350 11,504,070 † All=PRE+PLE+UNK/2 ‡ VMT= 365∙AADT∙L Note: L = segment length; AADT = average annual daily traffic; PT = percentage of trucks; PSL = posted speed limit; PR = primary right; OL = opposing left; UNK = unknown; VMT = vehicle miles traveled. Table 22. Descriptive statistics of filtered multi-state urban undivided highway data set.

108 Development of Safety Performance-Based Guidelines for the Roadside Design Guide Statistic Value AIC Theta Std. Err. BIC LL Pseudo-R2 6,849 1.205 0.21 6,871 -3,421.34 0.09 Note: AIC = Akaike Information Criterion; BIC = Bayesian Information Criterion; LL = Log Likelihood. Table 24b. Urban undivided highway encroachment model goodness-of-fit statistics. Pearson Correlation Index All PR OL L AADT PT PSL VMT All 1.0000 PR 0.4901 1.0000 OL 0.4420 0.2085 L 0.3747 0.2884 1.0000 AADT 0.0373 0.0330 -0.0347 1.0000 PT -0.0272 0.0142 0.0017 -0.1363 1.0000 PSL 0.0274 0.0321 0.0763 -0.0246 0.1841 1.0000 VMT 0.3475 0.2674 0.7899 0.4279 -0.0589 0.0278 1.0000 Spearman Correlation Index All PR OL L AADT PT PSL VMT All 1.0000 PR 0.6160 1.0000 OL 0.5280 0.1878 L 0.3373 0.2100 1.0000 AADT 0.0538 0.0408 -0.0511 1.0000 PT -0.0417 -0.0091 -0.0112 -0.0316 -0.0729 1.0000 PSL 0.0123 0.0197 0.0462 -0.0658 0.2365 1.0000 VMT 0.3138 0.2064 1.0000 0.2896 0.0187 0.0058 0.0468 0.2502 1.0000 0.2167 0.0286 0.0415 0.2026 0.7366 0.5937 -0.0698 0.0123 1.0000 Note: PR = primary right; OL = opposing left; L = segment length; AADT = average annual daily traffic; PT = percentage of trucks; PSL = posted speed limit; VMT = vehicle miles traveled. Table 23. Correlation matrix for multi-state urban undivided highway data set. Regression Coefficients Estimate Std. Error z value Pr(>|z|) Intercept (A1) -15.50 0.09 -181.38 < 2e-16 AADT Coeff. (A2) -6.464(10)-5 0.00 -9.85 < 2e-16 PT Coeff. (A3) 6.034(10)-3 0.04 1.42 0.156 Table 24a. Urban undivided highway encroachment model coefficients.

Roadside Risk Design Methodology 109   the urban undivided right-shoulder width CMF, inserting the coefficients from Table 24a, using a unit length of one mile, and converting the frequency to an encroachment rate results in the following: = = = [ ] ( ) ( ) + + − − −  −  − − • • • BER 1.2221 e BER 1.2221 e BER 0.2267 e . . . . . URBANUNDIV PR A1 A2 AADT A3 PT URBANUNDIV PR 6.464 10 AADT 0.006034 PT 15.50 URBANUNDIV PR 6.464 10 AADT 5 5 A plot of the PR encroachment direction for urban undivided highways is shown in Figure 29. As shown in Figure 29, the encroachment rate for both rural and urban undivided highways is nearly linear when plotted on log-linear axes. The encroachment rate curves for undivided highways are nearly parallel with a slope of about 0.065 right-edge encroachments/MVMT. The encroachment rate is essentially the slope of the encroachment frequency curve, so the frequen- cies are gently mounded shapes with a peak at 25,000 veh/day for urban undivided highways and 18,000 veh/day for rural undivided highways. 3.3.2.1.1.8 Comparison of Encroachment Models and Recommendations. The results of the re-analyses discussed in the previous sections and the resulting encroachment rates and frequencies are plotted in Figure 29 where the solid portions of the lines represent the model over the range of the observed data and the dashed lines represent extrapolations beyond the observed data. Where the distribution of the AADT was reported, the mean AADT is indicated with a diamond. Some general observations can be made about these data collected over the past 60 years and the efforts at modeling them. First, two general methods have been used to collect encroachment data. Hutchinson- Kennedy and Cooper used direct observations of physical evidence of encroachments like tire tracks in snowy medians or tire tracks in the grass or on gravel shoulders. SR214, Miaou, and Carrigan used indirect methods where crash data were used to hypothesize about encroachments using a conditional probability approach. Direct observation has the advantage of measuring the feature of interest directly but is limited by the amount of data that can reasonably be collected. Such studies are laborious and time-consuming and likely underreport encroachments when physical evidence is not apparent. For example, if the data collection team does not visit the site soon enough after the encroachment, the tire tracks in grass or gravel may have disappeared. Similarly, as Cooper himself noted, it is not always possible to separate intentional encroach- ments from unintentional encroachments. The tire tracks left on the shoulder by a motorist pulling to the side of the road to look at a map or make a phone call can be indistinguishable from those made by a vehicle swerving out of control onto the shoulder. Indirect methods have the advantage of having vastly more cases available over a much wider range of traffic conditions but risk underpredicting encroachments because the out-of-control vehicle came to a stop on the roadside without experiencing any adverse collision event. Second, the Hutchinson-Kennedy data were collected between 1957 and 1963, the Cooper data was collected in 1978, the data used by Miaou was collected between 1985 and 1995, and, finally, Carrigan’s data was collected in the 2002 to 2011 timeframe. The encroachment rate decreased for both divided and undivided highways in each study, indicating that encroachment rates seem to be generally decreasing with time. Although this is a general trend that has not been evaluated statistically, the trend makes some sense. Vehicles today have greatly improved suspensions and handling characteristics and are automated with traction and stability control such that drivers likely maintain control more often in emergency situations. In addition, the drivers in the earliest studies (e.g., Hutchinson-Kennedy) were driving on new controlled-access highways at a time when these were unfamiliar to drivers whereas today, nearly all drivers are

0.01 0.10 1.00 10.00 0 10,000 20,000 30,000 40,000 50,000 Average Daily Traffic (in veh/day) Undivided Highways Encroachment Rates RSAP 2.0.3 UNDIV PR Miaou Rural UNDIV PR Miaou-Cooper UNDIV PR Carrigan Rural UNDIV PR Carrigan Urban UNDIV PR Recommended 0.01 0.10 1.00 10.00 0 10,000 20,000 30,000 40,000 50,000 En cr oa ch m en ts /M V M T fo r O ne En cr oa ch m en t D ire ct io n Average Daily Traffic (in veh/day) Divided Highways Encroachment Rates Hutchinson-Kennedy DIV PL RSAP 2.0.3 DIV PR Miaou-Cooper DIV PR Carrigan Rural DIV PR Carrigan Urban DIV PR Recommended 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 0 10,000 20,000 30,000 40,000 50,000 Average Daily Traffic (in veh/day) Undivided Highways Encroachment Frequencies RSAP 2.0.3 UNDIV PR Miaou Rural UNDIV PR Miaou-Cooper UNDIV PR Carrigan Rural UNDIV PR Carrigan Urban UNDIV PR Recommended 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 0 10,000 20,000 30,000 40,000 50,000 En cr oa ch m en ts /m i/y r f or O ne En cr oa ch m en t D ire ct io n Average Daily Traffic (in veh/day) Divided Highways Encroachment Frequencies Hutchinson-Kennedy DIV PL RSAP 2.0.3 DIV PR Miaou-Cooper DIV PR Carrigan Rural DIV PR Carrigan Urban DIV PR Recommended PR E nc ro ac hm en ts /m i/y r PR E nc ro ac hm en ts /M V M T Figure 29. Encroachment rate (top) and frequency (bottom) for one encroachment direction by bi-directional AADT.

Roadside Risk Design Methodology 111   more familiar with driving at high speeds on high-volume controlled-access facilities. This trend seems likely to continue as vehicle suspension, handling, and control systems become ever more sophisticated at keeping vehicles from leaving the roadway. Third, encroachment modeling is based on the assumption that traffic is in a free-flow condi- tion according to the Highway Capacity Manual which implies service levels of A or B (Highway Capacity Manual 2016). For most of the studies discussed, the frequency of encroachment was determined on a segment of the highway where the segment is assumed to have the base char- acteristics listed earlier. Modifying the base encroachments is done by multiplying by encroachment adjustment factors and will be discussed later in Section 3.3.2.1.2: Encroachment Adjustment Factors (EAFS). According to the simplified FHWA method for highway capacity calculation, highest traffic volume for a two-lane rural undivided highway with the base conditions listed in Section 3.3.2.1.1 for service level B is 24,200 vehicles/day (Margiotta and Washburn 2017). There is a slight dif- ference between urban and rural four-lane divided highways, but, generally speaking, the highest traffic volume that still satisfies service level B is about 45,000 veh/day for highways with the listed base conditions (Margiotta and Washburn 2017). As shown in the top right portion in Fig- ure 29, the encroachment rates reduce to a small essentially constant value at low service levels for divided highways, and, as shown in the top left portion in Figure 29, the encroachment rates reduce more or less linearly as traffic volume increases and service level decreases for undivided highways. This may be reasonable for several reasons. First, the adjacent lanes are, by definition, occupied by other vehicles so vehicles are essentially shielded from crossing over lanes to depart the roadway. Ironically, it seems that traffic itself is an excellent traffic barrier. Second, as service level decreases, travel speed decreases as well. In essence, there will be fewer roadside encroach- ments if traffic is at a standstill or moving very slowly in poor service level conditions. Similarly, in low service level conditions vehicles may be more likely to strike other vehicles than to leave the road such that on-road vehicle-to-vehicle crashes may increase while ROR crashes decrease. On the other hand, however, even highways that experience poor service level conditions do not remain in those conditions all day every day. A highway may experience service level D condi- tions during commuting hours but service level A conditions late at night or in the early morning hours. In other words, while encroachments are less likely in low service level conditions all highways have periods when they are operating at high service levels so, from a design perspec- tive, these higher service level periods cannot be ignored. Analysis of the Cooper data, whether by Cooper himself or later by Mak and Sicking or later still by Miaou, all show a curve that increases at a decreasing rate until a peak is reached. After the peak, the encroachments decrease as shown in the bottom portions of Figure 29. As shown in Figure 27, the peak in Cooper’s original unfiltered and unmodeled data occurs at 6,000 veh/day. Mak and Sicking have the peak occurring at roughly 5,000 veh/day for both divided and undivided highways as shown in Figure 28. When Miaou reanalyzed the Cooper data for RSAPv3 he found the peak for undivided highways occurred at 5,000 and 24,000 for divided highways. This geo- metric feature, sometimes called the “Cooper hump,” has caused considerable difficulty in using the Cooper data for benefit-cost or guideline development. At the end of the observed data, Mak and Sicking simply assumed that the rate remained the same, which accounts for the steep upward portion of the curve shown earlier in Figure 28. This upward portion of the curve is not physical since it is not a part of the observed data. All the other studies show that encroachments continue to decrease as AADT increases, a fact not apparent to Mak and Sicking. RSAPv3 generally calculates the mid-life number of encroachments and then uses that value in calculating the expected crash costs. If the mid-life ADT turns out to be on the top of the “hump” the encroachments would be overestimated for the entire life and if the mid-life ADT occurs at the bottom of the “trough” the encroachments would be underestimated. To avoid

112 Development of Safety Performance-Based Guidelines for the Roadside Design Guide this problem various approaches have been used to develop selection guidelines. For bridge rail guidelines Ray et al. used a procedure that calculates the number of encroachments at 10 equally spaced times over the life and then takes the average of these values to estimate the encroach- ments at the mid-life (Ray and Carrigan 2014b; Ray et al. 2014d). In developing pier protection guidelines, Ray et al. researchers took a different approach and set the encroachment frequency to be constant after the peak number of encroachments from the observed data was determined (Ray et al. 2018). 3.3.2.1.1.9 Recommendation. The history of encroachment studies and modeling has been discussed in the previous sections. As also discussed above, there is a new ongoing NCHRP project with the aim of collecting new, up-to-date encroachment data so the recommendations made herein may change in the near future. This, however, is not problematic since the overall risk-based procedure used in this project isolates the encroachment modeling to one discreet step. In other words, the recommended encroachment model can be changed or modified in this step and used without the need to re-analyze or change all the other portions of the procedure. This is an example of the benefit of using a single universal procedure where the individual parts can be updated independently without changing the remainder of the procedure. Unfortunately, none of the prior studies are ideal. Each has advantages and limitations. The Cooper data, old as they are, are the most recent direct observation of encroachments but, as shown in Table 25, the range of AADT conditions has its limits, and there are not many segments. On the other hand, the Carrigan data sets include a very large number of segments spread over a wide range of AADTs, but the study is an indirect one so it does not measure encroachments directly. Since the Carrigan model underpredicts with respect to the Miaou-Cooper model, it is unclear if that is because there is a real difference or if Carrigan’s estimate of crashes is simply lower than Miaou’s estimate of encroachments. Deciding between the most direct measure of encroachments (i.e., Cooper) and the best data set (i.e., Carrigan) is difficult. While the Cooper data is 40 years old it is still the most recently collected direct observation study of encroachments available, so it is recommended for continued use until a newer, better data set becomes available (from NCHRP Project 17-88, “Roadside Encroachment Database Development and Analysis”). Miaou’s reanalysis of the data for the development of RSAPv3 used the most up-to-date statistical procedures as generally used in the HSM and therefore believed to make the best of Cooper’s observed data. One change that is recommended from the Miaou-Cooper model as used in RSAPv3 is that the encroachments be held constant after the peak encroachment is attained. For developing guidelines, it does not seem prudent to decrease the encroachment frequency with increasing AADT. Likewise, the increase used in RSAP 2.0.3 and RSAPv3 at the end of Cooper’s observed data is not physical and has little basis in observations. Study (Direct) Years Segments Miles Encr. AADT Range Hutchinson Divided 1957-1963 Cooper Divided 1978 Cooper Undivided 1978 207 992 5,917 332 528 1,353 1,900 - 31,253 5,954 - 44,930 1,000 - 3,000 Study (Indirect) Years Segments Miles Crashes AADT Range Miaou Undivided 1985-1989 NR 993 159 - 17,766 Carrigan Rural Divided 2002-2010 NR NR 710 - 67,390 Carrigan Urban Divided 2002-2010 NR NR 780 - 155,340 Carrigan Rural Undivided 2002-2010 NR NR 40 - 27,540 Carrigan Urban Undivided 2002-2010 13 181 575 712 25,414 24,025 38,974 10,727 NR NR 357 - 42,836 Note: NR = Not reported in the original study. Table 25. Characteristics of encroachment study data sets.

Roadside Risk Design Methodology 113   A constant encroachment frequency seems to be the best compromise until observed data is available to determine how encroachments behave at higher AADTs. Rather than assume the rate at the end of the observed data is constant for AADTs greater than those observed, the frequency has been held constant after the peak encroachments have been reached at an AADT of 5,000 veh/day for undivided highways and 24,000 veh/day for divided highways. The model for encroachment rate and a graph of this alternative are shown in Table 26. The solid lines in Table 26 indicate the recommended values whereas the dashed lines show the equation beyond the peak encroachment value. 3.3.2.1.2 Encroachment Adjustment Factors (EAFS). The number of expected encroach- ments is also a function of the particular highway characteristics of the segment. The expected base encroachment frequency on a segment (BEFS) as a function of traffic must be further adjusted by multiplying it by the highway characteristics encroachment adjustment factors for the seg- ment (EAFS). Recall that the base encroachment model was formulated using base conditions of a straight, flat road segment with 12-foot lanes, 65 mph posted speed limit, and two-lane undivided or four-lane divided highway type. These adjustment factors allow the user to adjust the expected encroachments away from the base conditions used to represent the site conditions. This approach is very similar to that used in the HSM for calculating CMFs and has been used in roadside safety crash prediction modeling since at least the 1970s (AASHTO 2010a; Carrigan and Ray 2018a). Several research projects have established encroachment adjustment factors for the following highway characteristics: • EAFHC: Horizontal curvature. • EAFG: Grade. Table 26. Recommended encroachment frequency model (Ray et al. 2012b). Undivided Two Lane (PR Encr/mi/yr) For AADT < 5,000 veh/day: BEFUNDIV PR = � AADT 4,343 � ∙ e�0.4997−� 0.2092 ∙AADT 1,000 �� For AADT ≥ 5,000 veh/day 3 BEFUNDIV PR = 0.6667 PR encr/mi/yr Divided Four Lane (PR Encr/mi/yr) For AADT < 24,000 veh/day BEFDIV PR = � AADT 3,650 � ∙ e�−0.2104− 0.0413 ∙AADT 1,000 � For AADT ≥ 24,000 veh/day 4 BEFDIV PR = 1.9776 PR encr/mi/yr 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 PR E nc ro ac hm en ts /m i/y r Average Daily Traffic (vpd) Undivided Divided

114 Development of Safety Performance-Based Guidelines for the Roadside Design Guide • EAFLR: Encroachment side. • EAFLN: Lanes in one direction. • EAFPSL: Posted speed limit. • EAFAD: Access density. • EAFLW: Lane width (no longer recommended). • EAFTER: Terrain (no longer recommended). ∏= = = • • • • •EAF EAF EAF EAF EAF EAF EAF EAF 5S i i 1 N HC G LR LN PSL AD Full details about each of these encroachment adjustment factors are provided in the fol- lowing sections. All the EAFs are based on the statistical analysis of observed crash data, and all have been remodeled since 2012 based on relatively recent crash data. A summary table of the most up-to-date individual encroachment adjustments is provided below in Table 27. The total highway characteristics encroachment adjustment factor is simply the product of all the individual encroachment adjustment factors as shown in Equation 5. Grade: EAFG Horizontal Curve Radius:EAFHC Encroachment Side: EAFLR Pe rc en t G ra de (% ) Rural Urban D eg re e of C ur va tu re Rural Urban En cr oa ch m en t S id e A A D T R ur al D iv id ed U rb an D iv id ed U nd iv id ed D iv id ed U nd iv id ed D iv id ed U nd iv id ed D iv id ed U nd iv id ed D iv id ed -10 1.15 1.52 0.84 0.37 -25 3.11 1.00 2.07 1.00 L 1,000 0.48 0.73 -9 1.12 1.43 0.86 0.42 -20 2.13 1.00 1.63 1.00 L 5,000 0.67 0.85 -8 1.10 1.35 0.88 0.49 -15 1.46 1.00 1.28 1.00 L 10,000 0.77 0.90 -7 1.08 1.27 0.91 0.56 -10 1.00 1.00 1.00 1.00 L 20,000 0.89 0.96 -6 1.06 1.20 0.93 0.65 -5 1.00 1.00 1.00 1.00 L 30,000 0.97 0.99 -5 1.04 1.13 0.95 0.75 0 1.00 1.00 1.00 1.00 L 40,000 1.03 1.02 -4 1.02 1.06 0.98 0.87 5 1.00 1.00 1.00 1.00 L 50,000 1.07 1.04 -3 1.00 1.00 1.00 1.00 10 1.00 1.00 1.00 1.00 L 60,000 1.11 1.06 0 1.00 1.00 1.00 1.00 15 1.11 1.00 1.03 1.00 L 67,000 1.14 1.07 3 1.00 1.00 1.00 1.00 20 1.23 1.00 1.07 1.00 L 80,000 1.14 1.08 4 1.01 1.05 0.97 0.85 25 1.36 1.00 1.10 1.00 L 90,000 1.14 1.10 5 1.02 1.10 0.94 0.72 L 100,000 1.14 1.11 6 1.03 1.16 0.91 0.61 R All 1.00 1.00 7 1.04 1.22 0.89 0.51 8 1.05 1.28 0.86 0.43 9 1.06 1.34 0.83 0.37 10 1.08 1.41 0.81 0.31 Total Lanes: EAFLN Access Density: EAFAD Posted Speed Limit: EAFPSL To ta l N um be r of L an es Rural Urban M aj or A cc es s Po in ts /m i Rural Urban Po st S pe ed (m i/h r) A ll U nd iv id ed Rural Urban U nd iv id ed D iv id ed U nd iv id ed D iv id ed U nd iv id ed D iv id ed U nd iv id ed D iv id ed D iv id ed D iv id ed ≤ 2 1.00 0.83 1.00 0.89 0 1.00 1.00 1.00 1.00 ≤ 55 1.00 1.16 1.18 4 0.91 1.00 1.11 1.00 0.5 1.67 2.51 1.00 1.00 60 1.00 1.08 1.09 6 - 1.20 - 1.13 1.0 2.80 6.31 1.00 1.00 65 1.00 1.00 1.00 ≥ 8 - 1.45 - 1.27 ≥1.5 4.68 6.31 1.00 1.00 ≥ 70 1.00 0.93 0.92 Table 27. Encroachment adjustment factors (EAFs).

Roadside Risk Design Methodology 115   While the list of encroachment adjustments given above is a good start, there are other road- way characteristics that could be added to the list of encroachment adjustment factors. There are statistical studies of high-friction surface treatments (Merritt et al. 2015), center line and edge rumble strips, shoulder type and width, delineation, and other features (AASHTO 2010a; McGee 2018a) that could be considered for addition to the list of encroachment adjustment factors. In principle, any roadway characteristic that can be shown to modify the likelihood of a vehicle leaving the traveled way could be considered for inclusion in the encroachment adjust- ment factors list. 3.3.2.1.2.1 Horizontal Curvature Encroachment Adjustment (EAFHC). Some encroach- ment adjustment factors are directionally dependent and, as such, are individually applied to each direction of travel. The horizontal curvature encroachment adjustment factor (EAFHC) is one of these. It has been presumed for some time that vehicles are more likely to encroach on the outsides of horizontal curves than on the inside but, unfortunately, there was historically little data where the direction of travel was known for use in the data analysis. Wright and Robertson’s 1975 study was “designed to identify roadway characteristics at sites where one or more vehicle occupants died when the vehicle struck a roadside object” (Wright and Robertson 1976). The study considered travel and departure direction. It was limited to crashes where the errant vehicle struck a fixed object and resulted in a fatality. If a vehicle rolled over, for example, the crash was excluded unless it involved “significant impact with an object” (Wright and Robertson 1976). The Wright and Robertson study was groundbreaking at the time since there was otherwise no information that would allow the linkage of highway alignment characteristics to crash or encroachment characteristics. The results were used for a number of years in programs such as ROADSIDE, BCAP, RSAP, and RSAPv3. Carrigan and Ray modeled the effect of the horizontal curvature on ROR crashes by direc- tion and for the full range of crash severities in NCHRP Project 17-54. In it, the EAF for horizontal curvature takes the functional form: eβi(Xi−10) where xi is the degree of curvature and βi is the coefficient (Carrigan and Ray 2018a). The values for the coefficient for each highway type and land use are shown at the bottom of Table 28. Carrigan and Ray found that the size effect for both the right and left curving divided urban and rural models was very small, and left and right curves could not be distinguished statistically, even within the large data set (i.e., rural divided data set included 124,458 segment edges and the urban divided data set included 244,050 segment edges). It was concluded that horizontal curvature on divided highways does not have the same influence that is observed on undivided highways and therefore recom- mended that a value of unity be used to represent EAFHC on curves on divided roadways in both urban and rural areas. The resulting and recommended EAFHC are tabulated as shown in Table 28 and Figure 30. 3.3.2.1.2.2 Grade Encroachment Adjustment Factor (EAFG). Another directionally depen- dent encroachment adjustment factor is the one for grade, or the EAFG. For example, if the pri- mary direction is on a negative grade the opposing direction must be on a positive grade so the adjustments for grade would be different on each side of the highway. NCHRP Report 492 inter- preted the work done by Wright and Robertson on grades and used it to adjust encroachment frequencies in RSAP version 2.0.3 (Mak and Sicking 2003; Wright and Robertson 1976). Mak and Sicking recognized the shortcomings and limitations of the Wright and Robertson study but noted, “there is no better source of information” (Mak and Sicking 2003). The interpreted values used by Mak and Sicking are shown in Table 29.

116 Development of Safety Performance-Based Guidelines for the Roadside Design Guide Miaou also developed a grade adjustment factor for rural two-lane roads in 1995 based on a much larger sample of crash data, but, unfortunately, this adjustment factor does not differ- entiate between the directions of travel (Miaou 1995). Miaou’s adjustments, shown in Table 30, tend to be somewhat larger than the Wright and Robertson adjustments. Carrigan and Ray modeled the effect of the vertical grade on ROR crashes by direction and for the full range of crash severities using a more robust and more current data set in NCHRP Project 17-54 (Carrigan and Ray 2018a). Carrigan and Ray modeled urban and rural divided and undivided data from Ohio from 2002 through 2010 and Washington from 2002 through 2007. The rural divided data set included 124,458 segment edges, and the urban divided data set included 244,050 segment edges. The rural undivided data set included 2,058,268 segment edges, and the urban undivided data set included 485,898 segment edges. Grade is continuous, therefore, Carrigan and Ray proposed the function: eβi(Xi−3) to model the effect of grade where xi is the percent of grade and βi is the coefficient. The values for the coeffi- cient for each highway type and land use are shown at the bottom of Table 31. The effect of nega- tive and positive grades in urban areas was found to be opposite the effect in rural areas for both divided and undivided roadways. In urban areas, introducing a grade reduces encroachments, and, in rural areas, it is expected to increase encroachments. The resulting and recommended EAFG are shown in Table 31 and Figure 31. Degree of Curvature Rural Urban Undivided Divided Undivided Divided -25 3.11 1.00 2.07 1.00 -20 2.13 1.00 1.63 1.00 -15 1.46 1.00 1.28 1.00 -10 1.00 1.00 1.00 1.00 -5 1.00 1.00 1.00 1.00 0 1.00 1.00 1.00 1.00 5 1.00 1.00 1.00 1.00 10 1.00 1.00 1.00 1.00 15 1.11 1.00 1.03 1.00 20 1.23 1.00 1.07 1.00 25 1.36 1.00 1.10 1.00 ßi for Curve Left 0.0756 0.0486 0.0083 0.0032 ßi for Curve Right 0.0204 0.0064 0.0112 0.0064 Table 28. Recommended EAF for horizontal curvature (Carrigan and Ray 2018a). 1.00 1.50 2.00 2.50 3.00 3.50 4.00 -25 -20 -15 -10 -5 0 5 10 15 20 25 EA F H C Degree of Curvature Rural Undivided Urban Undivided Figure 30. Recommended EAF for horizontal curvature (Carrigan and Ray 2018a). Grade (%) Adjustment <-6 2.00 -6 2.00 -2 1.00 >-2 1.00 Table 29. Wright and Robertson grade look up table (Mak and Sicking 2003). Grade (%) Adjustment ±7 7.28 ±6 6.24 ±5 5.20 ±4 4.16 ±3 3.12 ±2 2.08 ±1 1.04 0 1.00 Table 30. Miaou grade look up table (Miaou 1995).

Roadside Risk Design Methodology 117   3.3.2.1.2.3 Encroachment side encroachment adjustment (EAFLR). The Cooper data were only collected for right-edge encroachments, but these have traditionally been extended to left- edge encroachments by assuming that left-side encroachments were equally probable as right- side encroachments. Historically, there were no data available to either confirm or contradict this assumption. NCHRP Project 17-54 examined the probability of departing the road to the right versus the left for divided roadways in urban and rural environments across a wide range of traffic volumes and geometric conditions (Carrigan and Ray 2018a). Carrigan and Ray found that the long-held assumption that vehicles are equally probable to depart to the right or left was incorrect. The relationships were found to vary by traffic volume, as shown in Figure 32. The top of Figure 32 shows the rural divided highway results while the bottom shows the urban divided highway results for ROR crashes. In these figures, PR represents primarily right departures, and PL represents primarily left departures. The solid line, therefore, is the proportion of primary right-edge departures to all departures, and the dashed line is the proportion of PL edge depar- tures to all departures (Carrigan and Ray 2018a). Grade (%) Rural UrbanUndivided Divided Undivided Divided -10 1.15 1.52 0.84 0.37 -9 1.12 1.43 0.86 0.42 -8 1.10 1.35 0.88 0.49 -7 1.08 1.27 0.91 0.56 -6 1.06 1.20 0.93 0.65 -5 1.04 1.13 0.95 0.75 -4 1.02 1.06 0.98 0.87 -3 1.00 1.00 1.00 1.00 0 1.00 1.00 1.00 1.00 3 1.00 1.00 1.00 1.00 4 1.01 1.05 0.97 0.85 5 1.02 1.10 0.94 0.72 6 1.03 1.16 0.91 0.61 7 1.04 1.22 0.89 0.51 8 1.05 1.28 0.86 0.43 9 1.06 1.34 0.83 0.37 10 1.08 1.41 0.81 0.31 ßi for Up Hill 0.0104 -0.0303 0.0492 -0.1670 ßi for Down Hill 0.0194 -0.2450 0.0596 -0.1433 Table 31. Recommended EAF for vertical grade (Carrigan and Ray 2018a). 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 EA F G Grade (%) Rural Undivided Rural Divided Urban Undivided Urban Divided Figure 31. Recommended EAF for vertical grade (Carrigan and Ray 2018a).

118 Development of Safety Performance-Based Guidelines for the Roadside Design Guide The x-axis for both of these figures is the AADT. The urban data set includes a broader range of AADT values; therefore, the range in the figure is broader. Notice for both divided highway models (i.e., rural and urban), the proportion of vehicles, which crash to the right (i.e., solid line) is larger than the proportion of vehicles that crash left (i.e., dashed line) at low AADT values; however, these lines cross for both models at approximately 35,000 veh/day. This indicates that a higher proportion of vehicles crash to the left, that is the median side, in both rural and urban environments at volumes greater than 35,000  veh/day. In essence, the model indicates that in low-volume traffic conditions drivers appear more likely to encroach to the right. As the traffic volume increases, traffic in the right lanes shields vehicles in the left lanes from encroaching to the right, resulting in the shapes shown in Figure 32 (i.e., more left-hand encroachments at AADT above 35,000 veh/day). RSAPv3 assumed that left and right encroachments were equally probable in NCHRP Web- Only Document 319: Roadside Safety Analysis Program (RSAP) Update. EAFLR removes the pre- vious unsupported assumption about the probability of departing left or right and replaces it with models developed from a statically robust data set of observed crashes. The primary left- edge (PLE) and primary right-edge (PRE) models developed for rural and urban divided high- ways under NCHRP Project 17-54 take the following form for the safety performance function (SPF) (Carrigan and Ray 2017; Carrigan and Ray 2018a): = • •SPF AADT C LDIV Ai i Rural Divided Highways 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 Pr op or tio n of R O R C ra sh es AADT (in veh/day) PR/PR+PL PL/PR+PL Urban Divided Highways 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 20,000 40,000 60,000 80,000 100,000 120,000 140,000 Pr op or tio n of R O R C ra sh es AADT (in veh/day) PR/PR+PL PL/PR+PL Figure 32. Proportion of rural (top) and urban (bottom) divided highway ROR crashes by edge and AADT.

Roadside Risk Design Methodology 119   where: SPFDIV = Frequency of ROR crashes by divided segment edge per year. AADT = Annual average daily traffic, in veh/day. L = Segment length, in miles. Ai = Regression coefficient. Ci = Constant. PLE = Primary left encroachment. PRE = Primary right encroachment. Table  32 provides the coefficients and constants recommended for use in these models. Note that the models predict ROR crashes per segment edge, not encroachments. It is common to develop encroachment adjustment factors using crash-based models due to the vast availability of crash-based data and limited availability of encroachment data. This crash-based EAF, therefore, is assumed to modify encroachments in the same proportion as crashes. The derivation of the EAF is accomplished through simplification of the ratio of the left-edge model to the right-edge model, as follows: = = =− • •EAF derivation AADT AADT AADT C C AADT C A A A A PLE PRE A i . . . . PLE CPLE L PRE CPRE L PLE PRE i The resulting urban and rural coefficients and constants are shown in Table 33 and the EAF is shown as a multiplier in Figure 33. When the EAF equals unity, there is no predicted differ- ence between right- and left-edge departure. When the EAF is greater than one, there are more predicted left-edge departures than right-edge departures. Conversely, when the EAF is less than one, there are more predicted right-edge departures than left-edge departures. The rural model development was limited to AADT values below 67,000 veh/day; therefore, the EAF is held constant at AADT values above 67,000 veh/day which results in an EAF of 1.14. The values for EAFLR shown in Table 34 result in a reduction in left encroachments at lower traffic volumes but an increase in left encroachment at higher traffic volumes. No adjustments are recommended for undivided roadways. Coefficients and Constants Rural Urban APLE 1.036E+00 6.540E-01 CPLE 2.032E-05 1.973E-03 APRE 8.308E-01 5.652E-01 CPRE 1.742E-04 4.960E-03 Table 32. SPFEDGE coefficients and constants recommended for inclusion in the HSM (Carrigan and Ray 2018a). Coefficients and Constants Rural Urban Ai 2.052E-01 8.883E-02 Ci 1.166E-01 3.978E-01 Table 33. Left departure EAFLR for divided highways.

120 Development of Safety Performance-Based Guidelines for the Roadside Design Guide 3.3.2.1.2.4 Total Number of through Lanes Encroachment Adjustment (EAFLN). The pur- pose of the multi-lane adjustment factor is to adjust the base encroachment frequency from base conditions of two-lane undivided or four-lane divided to the actual segment-specific total number of through lanes. EAFLN is applied to the whole highway segment, not by the direction of travel so the number of lanes refers to the number of lanes of the whole highway. Turning lanes, weaving lanes, and other such supplementary lanes should not be counted in the total lanes, only through travel lanes. Ray et al. documented the development of an EAF for the number of lanes based on an analysis of a 1998 and 1999 TxDOT crash database of median crashes from 52 Texas Counties (Ray et al. 2012a). The EAFLN developed using TxDOT crash data in NCHRP Web-Only Document 319 is shown in Table 35. The effect of the number of lanes on ROR crashes was investigated again by Carrigan and Ray using urban and rural divided and undivided crash and highway data from Ohio from 2002 through 2010 and Washington from 2002 through 2007 (Carrigan and Ray 2018a). The rural divided data set included 124,458 segment edges, and the rural undivided data set included 2,058,268 segment edges. The urban divided data set included 244,050 segment edges, and the urban undivided data set included 485,898 segment edges. The proportional representation of the number of lanes per segment within each data set is shown in Table 36. As would be expected, the divided roadways in the available data were dominated by four-lane roadways, and the undivided roadways are dominated by two-lane roadways. This was convenient for this effort because the encroachment data uses the same base conditions. 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 0 20,000 40,000 60,000 80,000 100,000 V EH IC LE S/ D AY Bi-directional AADT (veh/day) Rural Urban Figure 33. Urban and rural EAF for left departures on divided highways. Encroachment Side Rural Urban Undivided Divided† Undivided Divided Left 1.00 AADT0.2052 ∙ 0.1166 1.00 AADT0.0888 ∙ 0.3978 Right 1.00 1.00 1.00 1.00 † AADT > 67,000 EAF = 1.14 Table 34. Recommended EAF for encroachment side. Total Number of Lanes Undivided Divided 2 1.000 1.000 4 0.755 1.000 6+ 0.755 0.910 Table 35. Number of lanes EAFLN (Ray et al. 2012a).

Roadside Risk Design Methodology 121   The modeling results showed that the number of lanes was a highly significant (i.e., p-value < 2e-16) predictor for ROR crashes in each data set. Moreover, the results included tight confidence intervals. The results are shown in Table 37, and these are values recommended for use. 3.3.2.1.2.5 Posted Speed Limit Encroachment Adjustment (EAFPSL). The purpose of the posted speed limit adjustment factor is to adjust the BEFs from the base conditions of 65 mph to the posted speed limit on the segment of interest. Ray et al. modeled posted speed limit during the reanalysis of the Cooper data under NCHRP Project 22-27 for undivided roadways only (Ray et al. 2012b). The Cooper data included 1,353 encroachments observed on 575 undivided high- way segments. The base condition for the undivided roadway encroachment frequency model was established at 55 mph or less. It was found that, for undivided roadways with posted speed limits greater than 55 mph, an adjustment of 0.7 should be applied to the encroachment fre- quency. To coordinate with other portions of the model which were also adjusted by speed (i.e., the EFCCR65 crash severity model), a base condition for the entire encroachment probability model was set to 65 mph in RSAPv3 (Ray et al. 2012b). Posted speed limit was investigated again as a model covariate by Carrigan and Ray under NCHRP Project 17-54 when modeling urban and rural divided and undivided data from Ohio from 2002 through 2010 and Washington from 2002 through 2007 (Carrigan and Ray 2018a). The rural divided data set included 124,458 segment edges, and the rural undivided data set included 2,058,268 segment edges. The urban divided data set included 244,050 segment edges, and the urban undivided data set included 485,898 segment edges. The distribution of posted speed limits within the modeled data is shown in Table 38. Carrigan and Ray found that, for undivided roadways in both urban and rural settings, the posted speed limit had a significant, but not measurable, influence on ROR crash frequency. An increase to the posted speed limit of urban and rural divided roadways, however, indicates ROR crash frequency would decrease. There is very little data available below 55 mph (i.e., see Table 38) for divided roadways; therefore, it is recommended that the EAF be held constant below 55 mph. The results reported by Carrigan and Ray with a base condition of 55 mph have been updated to a base of 65 mph, as shown in Table 39. Total Number of Lanes Rural Urban Undivided Divided Undivided Divided ≤ 2 98.59 3.33 66.33 3.41 4 1.13 85.01 29.34 63.59 6 - 8.90 1.26 19.74 ≥ 8 - 0.09 - 6.24 Table 36. Number of lanes by proportion within Ohio and Washington NCHRP Project 17-54 data set (Carrigan and Ray 2018a). Total Number of Lanes Rural Urban Undivided Divided Undivided Divided ≤ 2 1.00 0.83 1.00 0.89 4 0.91 1.00 1.11 1.00 6 - 1.20 - 1.13 ≥ 8 - 1.45 - 1.27 Table 37. Recommended EAFLN for number of lanes (Carrigan and Ray 2018a).

122 Development of Safety Performance-Based Guidelines for the Roadside Design Guide 3.3.2.1.2.6 Access Density Encroachment Adjustment (EAFAD). The purpose of the access density adjustment factor (EAFAD) is to adjust for the number of major road and highway access points per mile in any segment. EAFAD represents the effect of these major access points on encroachment frequency based on the assumption that increased access creates more traffic con- flicts which in turn result in increased encroachments. The base condition is an access density of zero access points per mile. Cooper explained that he was able to add the access density variable to his data in 1981 through “using up-to-date route maps together with the section alignment plans prepared during the data collection” (Cooper 1981). The number of major access points along the length of section was estimated. Ray et al. modeled the influence of access density during the reanalysis of the Cooper data for NCHRP Web-Only Document 319 (Ray et al. 2012b). The Cooper data included 1,353 encroach- ments observed on 575 undivided highway segments and 528 encroachments observed on 181 divided highway segments. There was a mean of 1.2 major access points per kilometer in the undivided data set and a mean of 0.9 major access points per kilometer in the divided data set. There was not a broad range of values. Using the models developed by that research, an access density adjustment has been tabulated, as shown in Table 40. This is the best available informa- tion at this time and is recommended for use in the risk assessment of roadside designs. 3.3.2.1.2.7 Lane Width Encroachment Adjustment (EAFLW). The base condition for lane width within the remodeled Cooper data is not known since lane width was not gathered during Rural Urban Undivided Divided Undivided Divided Min. 20 25 20 20 1st Qu. 55 60 35 55 Median 55 65 40 60 Mean 52.6 63.3 40.98 57.22 3rd Qu. 55 70 50 65 Max. 65 70 70 70 Table 38. Posted speed limit within Ohio and Washington NCHRP 17-54 data set (Carrigan and Ray 2018a). Posted Speed Limit Rural Urban Undivided Divided Undivided Divided ≤ 55 1.00 1.16 1.00 1.18 60 1.00 1.08 1.00 1.09 65 1.00 1.00 1.00 1.00 ≥ 70 1.00 0.93 1.00 0.92 Table 39. Recommended EAF for posted speed limit (Carrigan and Ray 2018a). Major Access Points per Mile Rural Urban Undivided Divided Undivided Divided 0 1.00 1.00 1.00 1.00 0.5 1.67 2.51 1.00 1.00 1.0 2.80 6.31 1.00 1.00 ≥1.5 4.68 6.31 1.00 1.00 Table 40. Recommended EAF for access density (Ray et al. 2012b).

Roadside Risk Design Methodology 123   the original data collection (DeLeuw Cather and ADI Limited 1978). During the update to the Roadside Safety Analysis Program, a lane width adjustment factor was adopted from the AASHTO HSM and used in the RSAPv3 program (AASHTO 2010a; Ray et al. 2012a). Ray et al. observed at that time that “the purpose of the lane width adjustment factor is to adjust the baseline encroach- ment frequency assumption of 12-foot lanes to the appropriate average lane width for each road segment” (Ray et al. 2012b). The HSM lane width CMF available at that time was applicable to all crash types, including the ROR crashes. Carrigan and Ray have since modeled a lane width adjustment factor (EAFLW) for ROR crashes exclusively. The results of this analysis for the lane width adjustment, however, proved counterintuitive. The expected ROR crashes increased as lane width increased (Carrigan and Ray 2018a). At this time, due to the lack of knowledge about the actual lane widths in the Cooper data and the counterintuitive results available to adjust for lane width, a lane width adjustment factor should not be included. 3.3.2.1.2.8 Terrain Encroachment Adjustment (EAFTER). Ray et al. modeled the influence of terrain and found it to be a significant predictor of encroachments for undivided roadways during the reanalysis of the Cooper data under NCHRP Project 22-27, but it was not significant for divided roadways (Ray et al. 2012b). Ray et al. mistakenly characterized this variable as repre- sentative of the general area for which DeLeau Cather collected the data used by Cooper (DeLeuw Cather and ADI Limited 1978; Cooper 1980). Upon further review of the Cooper reports it appears that the terrain variable was not collected directly but rather was interpreted from the grade and curvature variables collected for each segment. Cooper explains that “a general align- ment descriptor was also given to each road section as an expression of its average alignment char- acteristics. This code . . . was a 2-digit number comprising either 1-long curves and tangents or 2-reverse curves; together with either 6-flat terrain, 7-rolling terrain or 8-mountainous terrain” (Cooper 1980). Section 3.3.2.1.2.1: Horizontal Curvature Encroachment Adjustment (EAFHC) and section 3.3.2.1.2.2: Grade Encroachment Adjustment Factor (EAFG) of this report describe the separate EAFs for horizontal curvature and vertical grade so the characteristics underlying the terrain EAF have already been used to developed EAFHC and EAFG. It is recommended that the EAFTER in NCHRP Web-Only Document 319 for terrain not be used in favor of using the separate EAFHC and EAFG discussed earlier for horizontal curvature and vertical grade. These separate encroachment adjustment factors are believed to be better representations of the influence of these individual characteristics. 3.3.2.2 Provide for Safe Recovery The second of the three FHWA countermeasures is to provide safe recovery for vehicles that leave the roadway. Stated another way, the objective of this countermeasure is to minimize the chance that a vehicle will strike a fixed object or roll over on the roadside terrain where the conditional probability of a crash given that an encroachment has occurred is given by: ∏=    = − •CRASH P THRj cj i i 1 j 1 The above equation has two terms. The first, Pcj, is the conditional probability that a vehicle will interact with a roadside feature given that the vehicle encroaches on the roadside. The likeli- hood that a vehicle will interact with a feature is a function of the size, location, and character- istics of the feature. For example, a utility pole that is placed 10 feet from the edge of travel is more likely to be struck than one placed 20 feet from the edge of travel. Similarly, a 100-foot-long vertical rock cut is more likely to be struck than a 20-foot-long rock cut. The second term, THRi, is the conditional probability that a vehicle will pass through, over, or under the roadside feature

124 Development of Safety Performance-Based Guidelines for the Roadside Design Guide and continue its trajectory on the roadside. Each of these terms and their constituent parts are discussed in the following sections. 3.3.2.2.1 Conditional Probability of Reaching Lateral Offset Y [Py(Yj)]. The probability of a collision given an encroachment, Pcj, depends on the cumulative probability distributions of the lateral and longitudinal extents of encroachment so each of these terms will be discussed. The maximum lateral extent of encroachment distribution [i.e., Py(Yj)] has been obtained in several studies of encroachments over the years. Mak first used the maximum lateral extent col- lected by Cooper when developing RSAP 2.0.3. Subsequently, he reconstructed 890 passenger- vehicle trajectories as a part of NCHRP Report 665 (Mak and Sicking 2003; Mak et al. 2010). Gabler assembled updated data from NHTSA’s NASS CDS system for NCHRP Project 17-43, “Long-Term Roadside Crash Data Collection Program.” NCHRP Project 17-43 has collected data on 1,125 cases from the 2012 through 2015 National Automotive Sampling System (NASS) Crashworthy Data System (CDS). Each of these data sets has one serious limitation with respect to determining the cumulative probability distribution of the maximum lateral extent of encroachment: each case was collected because a collision occurred, or the vehicle rolled over. The vehicle would likely have continued farther had the trajectory not been interrupted by a fixed object collision or rollover. The lateral extent of encroachment from these earlier studies, therefore, has resulted in an underestimate of the cumulative lateral extent distribution. Ray et al. tried to alleviate this problem in RSAPv3 by extrapolating the trajectories in a straight line from the last point of the trajectory until the energy in the trajectory was expended (Ray et al. 2012b). The result is shown in Figure 34 where the solid line represents the reconstructed trajectories that were interrupted by collisions and rollovers, and the dashed line represents extrapolated trajectories. The result is that the cumulative distribution is shifted upward and to the right. Johnson and Gabler used a similar technique to investigate guardrail runout lengths using the NCHRP Project 17-43 data (Johnson et al. 2015). Carrigan and Ray attempted to solve this problem another way by using an innovative tech- nique borrowed from the medical statistics literature (Carrigan and Ray 2019). Survival analysis is a method for analyzing the amount of time until an event occurs. The event might be death 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 10 20 30 40 50 60 70 80 90 100 Pr ob ab ili ty o f L at er al E xt en t Y j Offset from Travelway, Y (ft) NCHRP 17-22 (Reconstructed) NCHRP 17-22 (Extrapolated) Figure 34. Lateral extent of encroachment from Report 665 as reconstructed (solid line) and extrapolated (dashed line) (Mak and Sicking 2003; Ray et al. 2012b).

Roadside Risk Design Methodology 125   from lung cancer, AIDS in a medical setting, or the time when a mechanical component fails in a machine design setting. In the analysis of roadside vehicle trajectories, the “event” is reaching the end of the sloped terrain. In trajectory analysis, time is measured in units of distance so survival analysis techniques can be used to estimate the probability of a vehicle reaching a certain lateral extent on a variety of roadside and median terrains. If 100 vehicles leave the roadway on a particular 30-foot-wide slope, the proportion of them that reach a lateral extent of 30 feet is considered to have “survived” the slope. If a vehicle returns to the roadway, rolls over, or stops on the slope the trajectory did not survive the slope. These other events (i.e., rolling over, returning to the roadway, and stopping on the slope) are referred to as competing risk, and their probabilities can also be modeled using survival analysis techniques. Carrigan obtained 43,200 CarSim trajectory simulations from the NCHRP Research Report 911 research team (Sheikh et al. 2019). For NCHRP Research Report 911 these simula- tions were performed using the following parameter variations: • Five slopes (i.e., −10:1, −6:1, −4:1, −3:1, and −2:1). • Four foreslope widths (i.e., 8, 16, 32, and 105 ft). • Six encroachment speeds (i.e., 25 to 75 in 10-mph increments). • Six encroachment angles (5 to 30 in 5-degree increments). • Five driver steering and braking inputs. • Four vehicle types (i.e., pickup, small car, mid-size passenger car, and SUV). These conditions resulted in 43,200 CarSim trajectories. The research for NCHRP Web-Only Document 296 has generated another 57,600 trajectories involving trajectories traversing ditches, but the effect of ditches on lateral extent was not included in Carrigan’s analysis due to resource constraints (Bullard 2020). The technique developed by Carrigan could prove a useful way to model a wider variety of terrain configurations than would be possible to model using crash and roadside inventory data. Carrigan found that the parameters most highly correlated to the maximum lateral extent of encroachment as measured by the Spearman correlation coefficient were the encroachment angle (0.5513), driver input (−0.5149), speed (0.4111), and slope (0.1663). These parameters, excepting driver input, were used to generate Kaplan-Meir cumulative probability distributions of the survival proportion for measured maximum lateral extents of encroachment and compet- ing risks (Pintilie 2006). Full details of the model development and analysis are presented in a paper (Carrigan and Ray 2019). Simulated data support the survival analysis model; therefore, the resulting model was adjusted to account for field-measured encroachment conditions. The bulleted list of parameter variations listed above does not represent the actual distribution of encroachment speeds and angles in the field. The mean encroachment speed and angle for NCHRP Research Report 911’s data were 50 mph and 17.5 degrees, respectively. The mean encroachment speed and angle for the NCHRP Project 17-43, “Long-Term Roadside Crash Data Collection Program” data, on the other hand, were 48.6 mph and 13.8 degrees. Mak reported very similar average encroach- ment conditions (i.e., 49.3 mph and 16.9 degrees) in NCHRP Report 665 (Mak et al. 2010). The NCHRP 17-43 data represents the most current data available and were used to scale the survival curve such that it had the same mean encroachment conditions as the NCHRP 17-43 data set. The results of the analysis, shown in Table 41, are a cumulative probability distribution of the maximum lateral extent of encroachment. The solid line represents the cumulative distribution of the actual 43,200 CarSim simulations adjusted to the observed mean speed, and the dotted line is a regression fit to these data. This cumulative probability distribution is used in this method to estimate the conditional probability that a vehicle trajectory will reach a particular lateral offset from the road and the conditional probability that a vehicle will roll over on the slope.

126 Development of Safety Performance-Based Guidelines for the Roadside Design Guide Knowing the conditional probability of the trajectory reaching the specified lateral offset is used to estimate the probability of striking a roadside feature at that offset. For example, if a vertical rock cut is located 20 feet laterally from the edge of the travel lanes, the probability that an encroaching vehicle will strike that rock cut is 0.67 as shown in Table 41. The probability of an encroaching vehicle striking an obstacle located 30 feet from the edge of the traveled way is 0.57. As expected, the probability of striking a feature decreases exponentially as the feature is located farther from the traveled way. 3.3.2.2.2 Conditional Probability of Reaching Longitudinal Position X [Px(Xj)]. The same 43,200 CarSim trajectory simulations from NCHRP Research Report 911 that were used by Carrigan to characterize the maximum lateral extent of encroachment were used in this project to examine the maximum extent of longitudinal encroachments (Carrigan and Ray 2019; Sheikh et al. 2019). The longitudinal extents of encroachments from earlier studies share the same limi- tation as the lateral extent: the trajectories are terminated by a collision or a rollover. The longitu- dinal extents of trajectories reconstructed under NCHRP Report 665 and NCHRP Project 17-43 would likely have been longer if they had not been stopped as a result of the collision or rollover. As for the lateral extent data, the 43,200 NCHRP Research Report 911 CarSim trajectory simu- lations were allowed to continue until the vehicle came to rest or returned to the road without a collision. Those trajectories with a vehicle rollover are accounted for as competing risks. The same parametric variations of the trajectories and modeling techniques discussed in the last Table 41. Probability of an encroachment reaching a feature at lateral offset Y [Py(Yj)]. Lateral Extent (ft) Py(Yj) LateralExtent (ft) Py(Yj) LateralExtent (ft) Py(Yj) 1 0.9761 13 0.7376 45 0.4063 2 0.9431 14 0.7277 50 0.3622 3 0.9090 15 0.7191 55 0.3254 4 0.8844 16 0.7105 60 0.2887 5 0.8650 17 0.7008 65 0.2531 6 0.8394 18 0.6910 70 0.2307 7 0.8267 19 0.6825 75 0.2115 8 0.8089 20 0.6741 80 0.1918 9 0.7912 25 0.6238 85 0.1752 10 0.7737 30 0.5699 90 0.1624 11 0.7612 35 0.5082 95 0.1515 12 0.7488 40 0.4603 100 0.1416 y = 0.9888e-0.02x R² = 0.9953 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 10 20 30 40 50 60 70 80 90 100 Pr ob ab ili ty o f L at er al E xt en t Y , P Y (Y j) Offset from Travelway, Y (ft)

Roadside Risk Design Methodology 127   section for the lateral extent were used for the longitudinal extent. Like the lateral extent trajec- tories, the NCHRP Research Report 911 trajectories were adjusted to match the mean encroach- ment conditions of the NCHRP Project 17-43 data. e cumulative probability density function for the longitudinal extent of encroachment for NCHRP Research Report 911’s CarSim simulations is shown with the dashed line in Figure 35. is line represents the probability of reaching a longitudinal extent X if all the trajectories start from the same upstream position. For example, if all 43,200 trajectories departed from the same location on the roadway (i.e., X = 0) then about 30% of them (12,960) would cross a line perpendicular to the road 400 feet downstream, and about 4.5% (1,944) would cross another perpendicular line 1,000 feet downstream. In considering the probability of a collision, however, encroachments could leave from any point on the road upstream of the roadside feature. To determine the probability of all possible encroachments that depart upstream of the roadside feature, the sum of all the probabilities must be calculated. The probability of striking a downstream object is zero if the object is farther than the longest trajectory in the bundle of analyzed trajectories so trajectories origi- nating farther than this point need not be considered. Assume the longest trajectory in the bundle of trajectories is 1,000 feet. e area of concern, therefore, starts 1,000 feet upstream of the roadside feature. is point is dened as X = 0, and the location of the roadside feature is at X = 1,000. Figure 35 shows that 4.5% of the trajectories that leave from X = 0 will reach X = 1,000. If the same bundle of trajectories is then shied down- stream such that they depart from X = 200, 5.5% will travel 800 feet to X = 1,000. If the same bundle of trajectories is again shied downstream further such that they depart from X = 800, 73% will travel 200 feet to X = 1,000. e probability that one of the trajectories originating at X = 0, 200, and 800 reaching the location X = 1,000 feet is the sum of the probabilities divided by the number of starting points or (4.5 + 5.5 + 73)/3 = 27.7%. e probability of a trajectory depart- ing anywhere 1,000 feet upstream of a roadside feature would, therefore, be the integral of the cumulative density function with bounds at X = 0 and 1,000. Summing the probabilities at each one-foot increment and dividing by the distance (i.e., 1,000) is a good approximation for the integral as follows: ∑∫ ( ) ( )( ) = ≤ ≈ ≤ = = P X 1 X P X X dx 1 X P X Xx x j x j x 0 X x 0 X 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 200 400 600 800 1000 Longitudinal Extent (ft) Pr ob ab ili ty o f L on gi tu di na l E xt en t X j Figure 35. Probability of an encroachment reaching a feature at longitudinal offset X [Px(Xj)].

128 Development of Safety Performance-Based Guidelines for the Roadside Design Guide The solid line in Figure 35 represents the probability of a vehicle reaching a downstream location given that the encroachment could have happened anywhere upstream of the feature. Knowing the conditional probability of the trajectory reaching the specified downstream loca- tion is used to estimate the probability of striking a roadside feature at that location. For example, for a vehicle leaving the roadway anywhere within 1,000 feet upstream of the utility pole, the probability of the vehicle reaching that longitudinal position is 0.35. This does not mean the vehicle will strike the utility pole, only that it will reach that longitudinal position. Determining whether the vehicle strikes the pole will require using both the longitudinal and lateral offset distributions as will be discussed in the next section. Figure 35 shows that the likelihood of reaching a particular longitudinal position decays exponentially as the feature is located farther from the point of encroachment as expected. 3.3.2.2.3 Conditional Probability of Interacting with a Roadside Feature (Pcj). Now that the cumulative probability distributions for both the lateral and longitudinal extents of encroach- ment are known, the probability of colliding with a roadside feature at some specific location can be determined. Figure 36 shows a generic roadside feature with an arbitrary location. WFj is the lateral offset to the traffic face of the feature, WBj is the lateral offset to the back of the feature, and Lj is the length along the road of the feature. The cumulative probability distributions for the lateral extent and longitudinal extent are overlaid on a two-lane roadway. As illustrated by Figure 36, some vehicles will not travel far enough longitudinally to reach the roadside feature, and others will not travel far enough laterally. Some trajectories will pass between the feature and the roadside such that a collision does not occur. Other trajectories may pass behind the feature, and a few will strike the feature. The roadside feature has two potential impact faces: the traffic face and the upstream face. The probability of a collision with the upstream face is the product of the probability of reaching the downstream location of the feature and reaching a lateral offset between WFj and WBj (i.e., [Px (LTMax)(Py (WFj) − Py (WBj)]). Similarly, the probability of striking the traffic face at a location WFj from the traveled way is Py (WFj). Each of these terms must be proportioned by its charac- teristic length and the length of the segment since encroachments are predicted on the entire segment. The probability of a collision with a feature is, therefore, given by: ( ) ( )(( )=     +     − ( )• S P L L P L L P L P W P W 6cj S y W TMax x TMax y Fj y Bj j Fj where LTMax is the largest longitudinal extent in the trajectory bundle (e.g., 1,101.40 feet for the NCHRP 17-55 trajectories which was rounded down to LTMax = 1,000 ft) and the other terms are as defined earlier in Equation 2. This expression can be specified for three particular types of features: (1) continuous features, (2) point features, and (3) area features. Continuous Features: As Py (WFj) − Py (WBj) → 0 the second term of Equation 6 is diminished such that the feature is essentially a zero-width line. Such features can be idealized as a line. Such continuous linear features include roadside bar- riers, median barriers, bridge railings, and other long but narrow objects. Some special features like median crossovers, penetrating through a bridge railing or crossing into the workspace of a workzone, are included in this group. Any object that has a length much greater than its width and can be reasonably considered to be a line fits into this group. ( )=     •P L L P Wcj S y Fj j

Roadside Risk Design Methodology 129   Point Features: If the length along the road, Lj, is very small (e.g., one foot) in compari- son to the segment length, the rst term of Equation 6 approaches zero. Such features are essentially a point in space. Point features include xed objects like trees, utility poles, trac signal supports, and luminaire sup- ports. Barrier end treatments and crash cushions are also considered point features. Any object that has both a small length and width and is reasonably considered a point ts into this group. ( ) ( )(( )=     +     − ( )•P L L P L L P L P W P Wcj S y W TMax S x TMax y Fj y Bj j Fj Area Features: Area features have appreciable widths and lengths in comparison to the segment length so all terms of Equation 6 are needed. ese features Figure 36. Probability of collision with a roadside feature as a function of the lateral and longitudinal extents of encroachment.

130 Development of Safety Performance-Based Guidelines for the Roadside Design Guide include terrain-related features like foreslopes, backslopes, and ditches as well as buildings. Features that have a substantial width and length t into this group. ( ) ( )(( )=     +     − ( )•P L L P L L P L P W P Wcj S y W TMax S x TMax y Fj y Bj j Fj e trajectories from NCHRP Project 17-55 are recommended for use in this procedure so LTMax = 1,000  and PX(LTMAX) = 0.3508 in the above equations. e conditional probability of striking a roadside feature knowing its shape (i.e., continuous line, point, or area) and longitudinal and lateral position can be calculated using the equations presented above and the distributions presented earlier in Table 44 and Figure 35. 3.3.2.2.4 Reduce Secondary Impacts (THRj). THRj is a variable that represents the pro- portion of vehicles that pass through, over, or under a feature and continue on to potentially interact with another roadside feature. For example, a vehicle may strike a guardrail shielding a non-traversable slope. ere is a small chance that the vehicle will cross over the barrier line by vaulting over the barrier, under riding it, or penetrating it. If the vehicle does cross the barrier line it will then interact with the slope where it may roll over or strike another xed object on the slope. Similarly, a vehicle may travel on a roadside slope, interact with, and penetrate a median barrier, travel on another roadside slope, then enter the opposing lanes where another vehicle may strike it. e proportion that passes through for each category of roadside feature (i.e., the rst and second slope and the median barrier) is dependent on variables that are unique to the specic type of feature. e variable THRj represents the conditional probability that a vehicle will pass through, over, or under the feature and continue its trajectory. Figure 37 provides a hypothetical example that illustrates the use of the THRj variable for a v-ditched median with a median barrier located nearer the northbound travel lanes. Assum- ing 100 vehicles encroach into the median from the southbound direction, Figure 37 indicates that 11 returned to the roadway, 10 came to a stop on the foreslope, four rolled over on the foreslope, and 75 passed completely across the foreslope, crossed the ditch line, and entered the backslope. e vehicles that returned to the road or stopped on the foreslope (i.e., 11 + 10 = 21) Figure 37. Example of the use of the THRj variable.

Roadside Risk Design Methodology 131   are predicted by the lateral extent of encroachment variable Py(Yj) discussed in Section 3.3.2.2.1: Conditional Probability of Reaching Lateral Offset Y [Py(Yj)]. Of the 75 + 4 = 79 vehicles that completed an interaction with the slope, 75 passed through so the value of THRFORESLOPE, in this case, is 75/79 = 0.94. Seventy-five vehicles entered the backslope and four stopped somewhere on the backslope, three rolled over on the back slope, and 61 + 7 = 68 struck the median barrier. THRBACKSLOPE is, therefore, (61 + 7)/(61 + 7 + 3) = 0.96. Of the 68 vehicles that struck the median barrier, 61 were contained and redirected and seven penetrated so THRBARRIER, in this case, is 7/(61 + 7) = 0.1. Finally, of the seven vehicles that entered the opposing lanes, three were struck by vehicles traveling northbound and four were not, so the value of THREOL = 4/(3 + 4) = 0.57. For longitudinal barriers, THRj is the proportion of vehicles that penetrate through, over, or under the barrier. The proportion is dependent on the test level of the barrier and the percentage of trucks in the traffic stream. SUTs have higher centers of gravity than passenger vehicles, so they can roll over or vault a barrier that may retain and redirect a passenger vehicle. Tractor-trailer trucks are even taller, so they will roll over a barrier even more frequently than single-unit trucks. Carrigan developed a lengthy and comprehensive list for the values of THRBAR as a function of highway type, barrier type, land use, and percentage of trucks based on observational crash data and crash testing research (Carrigan and Ray 2022). The observational data, however, was very limited in terms of identifying when longitudinal barriers were penetrated so the complicated model was considered more detailed than the data could support. Instead, a simple model was developed based on the test level of the barrier based on the following assumptions: TL-2 All trucks penetrate, roll over, or vault over a TL-2 barrier. Passenger vehicles are contained when posted speed limits are less than or equal to 45 mph. TL-3 Longitudinal barriers contain all passenger vehicles, but all trucks breach the barrier. TL-4 Longitudinal barriers contain passenger vehicles and single-unit trucks (SUTs), but all tractor-trailer and tractor-tanker trucks (TTs) breach the barrier. TL-5 Longitudinal barriers contain all vehicles. With these assumptions the following simple model for THRBAR, which shows the percentage of trucks (PT) in the traffic stream as a number (i.e., not decimal), was developed and is recommended: =     • THR A PT 100BAR As stated by Carrigan, “it should be recognized that there are no assurances that all crashes of any type will be contained nor that all crashes of any type will not be contained, however, these assumptions were necessary to differentiate between different test levels of barriers” (Carrigan and Ray 2022). The values recommended for THRBAR are shown in Table 42. Percent age of Trucks (%) Test Level Coefficient A 0 5 10 15 20 25 30 50 2 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.50 3 1.00 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.50 4 0.75 0.00 0.04 0.08 0.11 0.15 0.19 0.23 0.38 5 0.000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Table 42. Encroachments passing through, over, or under barriers (THRBAR) as a function of the percentage of trucks (Carrigan and Ray 2022).

132 Development of Safety Performance-Based Guidelines for the Roadside Design Guide For terrain features like foreslopes, backslopes, and ditch bottoms, the proportion of vehicles that pass through the feature is determined by predicting the proportion of rollover crashes that occur when the encroachment enters the slope until departing the slope. A statistical model for this was a byproduct of the survival analysis performed by Carrigan and discussed in Section 3.3.2.2.1: Conditional Probability of Reaching Lateral Offset Y [Py(Yj)] (Carrigan and Ray 2019). The THRFORESLOPE is the proportion of vehicles that travel all the way across the slope feature without rolling over. The value (1 − THRFORESLOPE), then, is the proportion of encroachments that entered the slope but roll over before leaving it. The values of THRFORESLOPE for a selection of lateral distances and slope values are shown in Table 43 based on recent research (Carrigan and Ray 2022). Tables like Table 43 are needed for backslopes and ditch type and width but have not yet been developed. As expected, the proportion of trajectories that “survive” the slope without rolling over decreases as the slope increases. For example, 96% of trajectories will survive a 50-foot-wide −10:1 slope compared with 86% on a 50-foot-wide −2:1 slope. A trajectory on the −2:1 slope is 3.5 times more likely to result in a rollover than a trajectory on a −10:1 slope [i.e., (1 − 0.86)/ (1 − 0.96) = 3.5]. Vehicles in the opposing lanes of traffic are a median-related feature that should be consid- ered when assessing the need for median barriers. The probability of passing across the oppos- ing lanes without striking another vehicle is a function of the traffic volume in the opposing lanes. If the traffic volume is very light, a vehicle that enters the opposing lanes is unlikely to be struck or strike a vehicle in the opposing lanes whereas if the traffic volume is high, it is more likely a vehicle will be present that may strike or be struck by the encroaching vehicle. NCHRP Research Report 996 developed a statistical model of cross-median crashes given an Lateral Extent THRFORESLOPE (ft) -12:1 or flatter -10:1 -6:1 -4:1 -3:1 -2:1 or steeper 0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995 15 0.9996 0.9992 0.9993 0.9998 0.9997 0.9985 20 0.9981 0.9963 0.9962 0.9957 0.9966 0.9948 25 0.9961 0.9921 0.9911 0.9885 0.9887 0.9835 30 0.9938 0.9876 0.9851 0.9811 0.9782 0.9659 35 0.9902 0.9804 0.9784 0.9712 0.9643 0.9356 40 0.9877 0.9755 0.9731 0.9640 0.9516 0.9092 45 0.9843 0.9687 0.9639 0.9557 0.9381 0.8813 50 0.9819 0.9638 0.9567 0.9446 0.9252 0.8577 55 0.9790 0.9579 0.9507 0.9382 0.9139 0.8320 60 0.9772 0.9543 0.9451 0.9298 0.9018 0.8073 65 0.9743 0.9487 0.9384 0.9181 0.8852 0.7832 70 0.9714 0.9428 0.9330 0.9113 0.8757 0.7670 75 0.9708 0.9416 0.9296 0.9058 0.8638 0.7514 80 0.9697 0.9393 0.9264 0.8976 0.8550 0.7392 85 0.9670 0.9340 0.9227 0.8903 0.8453 0.7267 90 0.9654 0.9307 0.9168 0.8846 0.8377 0.7186 95 0.9648 0.9295 0.9139 0.8805 0.8323 0.7068 100 0.9633 0.9266 0.9104 0.8756 0.8275 0.7001 Table 43. Encroachments passing all the way through a foreslope (THRFORESLOPE).

Roadside Risk Design Methodology 133   encroachment (Carrigan and Ray 2022; Carrigan and Ray 2018). The probability of a cross- median crash (CMC) and cross-median event (CME) was shown to be: ( ) ≈ + −    P CMC CME 1 1 e 52,800 AADT 20,000 THREOL, however, is the proportion of vehicle not striking a vehicle in the opposing lanes so values from the above equation are subtracted from one. These recommended values are tabu- lated in Table 44. For fixed roadside objects like trees, utility poles, and bridge piers, THRFO is generally taken to be zero since typical passenger vehicles are unlikely to break away such features and continue a trajectory. At this time there is no data to estimate the passing through probability, or THR, values for objects like large and small breakaway signs, slipbase and transformer base lumi- naires, and other breakaway features although a value of unity may be sufficient. 3.3.2.3 Reduce Crash Severity The third of the three FHWA countermeasures is to reduce the crash severity for vehicles that leave the roadway and interact with a roadside feature. The crash severity term is given by: ( )= − δ     • •SEV P 1 THR PSL 65j SEV j j s 3 3j 3.3.2.3.1 Conditional Probability of a Fatal or Serious Injury Crash (PSEVj). PSEVj is the conditional probability that the outcome of interest (e.g., a fatal or serious injury crash) will be observed given that an interaction with feature j occurs. The method for calculating PSEVj was described by Ray et al. for different roadside barriers based on an earlier method for estimating crash severity using observed crash data that includes estimating the proportion of unreported crashes (Ray et al. 2014c; Ray et al. 2018). The results for some roadside and median features are shown in Table 45. Cells in Table 45 with “RN” indicate areas where research is needed to determine the values. The longitudinal barrier options listed in the top rows of Table 45 are generic in the sense that they apply to any barrier in that category. Listing barriers by categories is preferred at this AADT THREOL AADT THREOL AADT THREOL AADT THREOL 1,000 0.9302 13,000 0.8797 25,000 0.8006 37,000 0.6878 2,000 0.9269 14,000 0.8744 26,000 0.7925 38,000 0.6770 3,000 0.9234 15,000 0.8688 27,000 0.7841 39,000 0.6660 4,000 0.9198 16,000 0.8629 28,000 0.7756 40,000 0.6548 5,000 0.9161 17,000 0.8569 29,000 0.7667 41,000 0.6434 6,000 0.9121 18,000 0.8507 30,000 0.7577 42,000 0.6318 7,000 0.9080 19,000 0.8442 31,000 0.7484 43,000 0.6201 8,000 0.9038 20,000 0.8375 32,000 0.7389 44,000 0.6083 9,000 0.8993 21,000 0.8306 33,000 0.7291 45,000 0.5963 10,000 0.8947 22,000 0.8235 34,000 0.7191 >46,000 0.6000 11,000 0.8899 23,000 0.8161 35,000 0.7089 12,000 0.8849 24,000 0.8085 36,000 0.6985 Table 44. Proportion of vehicles passing across the opposing lanes without striking an opposing vehicle given that the vehicle enters the opposing lanes (THREOL).

134 Development of Safety Performance-Based Guidelines for the Roadside Design Guide time due to a lack of data for specific barrier types. When ISPE results begin to provide quality in-service information on particular hardware features, they could be added to a list like Table 45. The specificity of the barrier types is only limited by the availability of ISPE studies. While KA crashes are recommended for evaluations in this method, four additional outcomes are listed in Table 45: the proportion of fatal crashes (K), serious and fatal crashes (KA), fatal, serious and minor injury crashes (KAB), and fatal and injury crashes (KABC also sometimes called F + I). Some highway agencies prefer evaluating these other severities, and this same methodology can be used by just using the desired severity outcome from Table 45. Each of these outcomes is a crash severity outcome and they are listed at a base posted speed of 65 mph. Equation 2 includes a term (i.e., PSL3s/653), which can adjust the crash severity up or down based on the posted speed limit as will be discussed in Section 3.3.2.3.3: Posted Speed Limit Affects (PSL). While the procedure outlined in Equation 2 allows the users to define the outcome they are interested in, the RDG and MASH are generally concerned with minimizing fatal and serious injury crashes (i.e., KA) so usually the outcome of interest is KA65 as listed in Table 45. Table 45 also includes the terrain features foreslope, backslope, and ditch bottom. Some vehicles when interacting with terrain features will roll over. Rollover has been assumed to have the same crash severity regardless of the value of the slope or whether it occurs on the foreslope or backslope or is caused by interacting with the ditch bottom. Essentially, it is assumed at this time that a rollover has the same crash severity regardless of the cause of the rollover although this could be examined and updated pending the availability of sufficiently detailed crash data. Feature K65 KA65 KAB65 KABC65 Reference Longitudinal Barriers Cable Barrier 0.0009 0.0050 0.0297 0.0849 Strong-Post W-Beam Barrier 0.0015 0.0094 0.0422 0.0977 Weak-Post W-Beam Barrier 0.0006 0.0091 0.0321 0.1187 Closed Faced Concrete Barriers 0.0021 0.0159 0.0810 0.1667 Guardrail Terminals RN 0.0500 RN RN (Ray and Carrigan 2018) Crash Cushions RN RN RN RN Terrain Features Foreslope Rollover 0.0142 0.0589 0.3138 0.4836 Backslope Rollover 0.0142 0.0589 0.3138 0.4836 Ditch Bottom Rollover 0.0142 0.0589 0.3138 0.4836 Fixed Objects Trees and Utility Poles 0.0142 0.0589 0.3138 0.4836 Bridge Piers 0.0278 0.0656 0.1729 0.2444 (Ray et al. 2018) Other Users Crash in Opposing Lanes 0.0098 0.0451 0.1290 0.1938 Crash in Work Zone RN RN RN RN Crash with Pedestrian/Cyclist RN RN RN RN Enter the following from above: Waterbody 0.0049 0.0343 0.1421 0.2254 Minor Transportation Facility RN RN RN RN Major Transportation Facility RN RN RN RN Low-Risk Environment RN 0.0589 RN RN Medium-Risk Environment RN 0.4737 RN RN High-Risk Environment RN 1.0000 RN RN Note: RN = research needed in this area to determine the values. (Carrigan and Ray 2022) (Carrigan and Ray 2022) (Carrigan and Ray 2022) (Carrigan and Ray 2022) (Carrigan and Ray 2022) (Carrigan and Ray 2022) (Carrigan and Ray 2022) (Ray and Carrigan 2014b) (Ray and Carrigan 2014b) (Ray and Carrigan 2014b) Table 45. Outcomes for selected roadside and median features (PSEVj).

Roadside Risk Design Methodology 135   The bottom portion of Table 45 (i.e., Other Users) includes some examples of features that potentially put other users at risk. As an example, a crash in the opposing lanes of traffic is another important outcome for cross-median crashes that was investigated by Carrigan (Carrigan and Ray 2022; Carrigan and Ray 2018b). The outcome for entering a work zone, pedestrian area, or bicycle area has not been documented, but could be developed from crash data. Table 45 also gives the crash severity values for several types of fixed objects: trees, utility poles, and bridge piers. Not surprisingly, bridge piers appear to be the single most hazardous object listed in Table 45. As might be expected, all the fixed object and terrain features have expected crash severities that are at least an order of magnitude greater than the longitudinal barrier crash severities. Analysis of the data produced a wide range of values, but the confidence intervals all overlapped so it was decided to use a single composite value for all fixed objects. The conditional probability of an outcome given a particular crash severity (e.g., KA65 crashes) can be developed for a variety of roadside and median features based on the availability of crash data and/or ISPE results. The method proposed by Ray and Carrigan can be used to define the conditional probability PSEV j of any type of feature as long as there is sufficient observed crash data to perform an analysis (Ray et al. 2018). The features listed above in Table 45 should be added to or modified in the future such that a wider range and specificity of roadside and median features are available to the designer. 3.3.2.3.2 Crash Severity Interaction Term. The crash severity interaction term, δj, in Equation 2 accounts for whether the outcome associated with the particular feature occurs for all vehicles interacting with the feature (δj = 0) or just those that do not pass through the feature (δj = 1). For example, occupants of vehicles that strike a longitudinal barrier are likely to experi- ence similar outcomes regardless of whether the barrier contains and redirects the vehicle, or the vehicle penetrates through, vaults over, and goes beneath the barrier. On the other hand, if a vehicle does not roll over while interacting with a foreslope, the vehicle occupants will not expe- rience any harm associated with the foreslope feature. The value of δj, therefore, is as follows: δj = 0 For all longitudinal barriers, breakaway devices, crash cushions, guardrail terminals, and fixed objects because the potential for the outcome will be the same for all vehicles that interact with the hardware whether they pass through or not. δj = 1 For all-terrain features and other geometric features where the harm is only associated with those vehicles that do not make it through the feature. 3.3.2.3.3 Posted Speed Limit Affects (PSL). The relationship between speed and crash severity, in general, has been well established by many researchers (Bowie and Walz 1993; Nilsson 1982; O’Day and Flora 1982; Stuster et al. 1998). As stated in a 1998 FHWA synthesis on speed effects and crash severity: The relationship between vehicle speed and crash severity is unequivocal and based on the laws of physics. The kinetic energy of a moving vehicle is a function of its mass and velocity squared.. . . . Generally, the more kinetic energy to be dissipated in a collision the greater the potential for injury to vehicle occupants. Because kinetic energy is determined by the square of the vehicle’s speed rather than by speed alone, the probability of injury and the severity of injuries that occur in a crash, increase expo- nentially with vehicle speed (Stuster et al. 1998). The linkage, therefore, between speed and severity has been made both statistically and based on the physics of vehicle crashes. Göran Nilsson showed that for all types of crashes the number of injury crashes increases as a square of the ratio of velocities, to the third power for severe injury crashes and to the fourth power for fatal crashes (Nilsson 1982). Similar results have been obtained in the United States by Noble Bowie, Hans Joksch, and James O’Day to name several (Bowie and Walz 1993; Joksch 1993; O’Day and Flora1982). Nilsson showed that the ratio of

136 Development of Safety Performance-Based Guidelines for the Roadside Design Guide injury crashes prior to a change in average travel speed to those after is proportional to the ratio of speed squared. For example, Nilsson’s conclusion that severe injury crashes are propor- tional to speed to the third power would indicate that a particular rural two-lane road that experiences 10 severe injury crashes/mile/year when the average travel speed is 55 mph would experience on average 5.5 severe injury crashes/mile/year if the average travel speed were reduced to 45 mph, that is [5.5/10] = [55/45]3. If this exemplar road has utility poles, one would expect that part of the reason for a decrease in severe crashes is the reduced speed of utility pole crashes. Since Nilsson has shown that injury crashes increase in severity as a function of the velocity to some power, it would appear to be a reasonable assumption that a better crash severity model would be obtained if the crash severity model were a function of some power of the velocity. Ray et al. used this same principle to develop a procedure using observed crash data across a range of posted speeds to normalize crash severity of different roadside features to a base posted speed of 65 mph as discussed earlier in Section 3.3.2.3.1: Conditional Probability of a Fatal or Serious Injury Crash (PSEVj) (Ray et al. 2014c; Ray et al. 2018). This same approach is used in this method to scale the 65 mph crash severities (i.e., KA65) from Table 45 using the PSL term in Equation 2. 3.3.3 Segment Risk and Making Design Decisions The segment risk (i.e., OUTCOMES) is the total risk of observing a fatal or serious injury crash (i.e., KA crash) on the evaluated roadway segment. The segment risk is simply the summation of all the feature risks on the road segment as follows: ∑ ( ) = = − = − = OUTCOME OUTCOME OR OUTCOME OUTCOME 1 RR OUTCOME S j j 1 N SAlt Null SAlt SNull Alt Null SNull The first step in the roadside design process is for a highway agency to establish a specific, numeric goal as shown in Figure 38. This is a policy decision for each individual highway agency. The decision of one agency need not be the same as another since local priorities, conditions, and constraints will dictate which approach is best suited to local conditions. Individual designs are evaluated in the subsequent steps to determine whether a roadside safety treatment is likely to improve the safety of the segment by reducing the risk. Once the segment risk is calculated, it is compared to the goal. If the goal is met, the design is acceptable and should be considered for implementation. If the goal is not met, the agency should make changes in the design and re-analyze the new design to see if it meets the goal. If no design alternatives meet the goal, the alternative with the least risk is likely the best alternative. It is possible in some situations that the least-risk alternative is to leave the site alone. While the goal of roadside design presented in Table 7 involves minimizing the frequency of KA ROR crashes, there are a variety of methods that can be used to make design decisions based on accomplishing that goal. Highway agencies can use the following five decision-making methods to make decisions and assign priorities: 1. The segment risk is less than the absolute risk goal (i.e., OUTCOMEAlt < OUTCOMENull). 2. The relative risk on the segment is less than one (i.e., <1 OUTCOME OUTCOME Alt Null ).

Roadside Risk Design Methodology 137   3. The benefit-cost ratio is greater than the minimum acceptable benefit-cost ratio (i.e., BCRAlt/Null > BCRMIN). 4. The incremental cost-effectiveness ratio (ICER) is greater than the minimum acceptable cost- effectiveness ratio (i.e., ICER > ICERMIN). 5. The internal rate of return is greater than the minimum acceptable rate of return (i.e., IRRAlt > IRRMIN). The Null alternative in the list above represents the unaltered roadside whereas Alt represents a roadside design alternative meant to reduce the risk of a KA crash. The risk-based decision methods (i.e., items 1 and 2) establish an absolute or relative risk threshold independent of the cost of implementing the alternative whereas the economic measures (i.e., items 3 through 5) attempt to find the best use of highway agency resources. Each method has its advantages and disadvantages as will be discussed in the following sections. Figure 38. Flow chart of proposed safety performance-based roadside design guidelines.

138 Development of Safety Performance-Based Guidelines for the Roadside Design Guide 3.3.3.1 Absolute Risk There are two common ways to present crash rates: rates based on a unit length and rates based on traffic. It would be equally appropriate to measure KA crashes per mile per year and KA crashes per hundred MVMT, or 100 MVMT. Rates using MVMT tend to direct more resources to higher- volume roadways whereas rates using length tend to direct resources to lower volume roads. A common compromise is to have differing rates to represent performance measures or goals for different classes or types of roads (e.g., functional classes, highway types, etc.). While AASHTO TCRS and/or FHWA might establish national goals, each user agency should also establish its own goals based on the safety performance of its own roads. The following sections calculate basic fatal plus serious injury crash rates for several states on a per volume and per mile basis. Table 46 summarizes the ROR crash rates by roadway edge by both volume and length for Illinois, Maine, North Carolina, Ohio, and Washington State. Complete details and tables of results are provided in Appendix C. While the KA ROR crash rate on a per mile basis varies from a low of 0.0087 on undivided rural roads in North Carolina to a high of 0.0903 for urban divided highways in Ohio, the average rate for all states examined and all roadway types was found to be 0.0325 KA crashes/edge-mile/year. 3.3.3.2 Relative Risk The goal 0.0325 KA crashes/edge-mile/year is an absolute risk goal in that it is based on an explicit number of expected crashes. Another way to set goals is using a relative scale. For example, if a row of utility poles is close to the edge of the roadway, the choice is to move them farther from the traveled way, shield them with guardrails, or leave them unshielded. If the poles cannot be moved, the shield or unshielded alternatives have safety advantages. On the one hand, a collision with a guardrail is presumably less risky than a collision with a utility pole. On the other hand, the guardrail is continuous and is more likely to be struck since it is longer and closer to the road than the line of utility poles. In this case, a relative performance measure can KA Crashes/100 MVMT State Years R ur al U nd iv id ed U rb an U nd iv id ed A ll U nd iv id ed R ur al D iv id ed U rb an D iv id ed A ll D iv id ed A ll H ig hw ay Ty pe s Illinois 2010-2014 - - - - - - 1.9258 Maine† 2010-2018 - - 1.3267 - - - 2.6494 North Carolina 2010-2015 2.4140 0.9677 1.6237 0.1446 0.1092 0.1188 1.5655 Ohio 2010-2015 3.6457 1.1935 2.1566 0.3174 0.2964 0.3016 2.2889 Washington 2010-2017 1.0364 0.2601 0.5764 0.0983 0.0561 0.0662 0.4833 Average 2.3654 0.8071 1.4208 0.1868 0.1539 0.1622 1.7826 KA Crashes/edge-mile/year State Years R ur al U nd iv id ed U rb an U nd iv id ed A ll U nd iv id ed R ur al D iv id ed U rb an D iv id ed A ll D iv id ed A ll H ig hw ay Ty pe s Illinois 2010-2014 - - - - - - 0.0073 Maine† 2010-2018 - - 0.0083 - - 0.0083 0.0166 North Carolina 2010-2015 0.0087 0.0129 0.0096 0.0134 0.0206 0.0096 0.0105 Ohio 2010-2015 0.0452 0.0821 0.0528 0.0348 0.0903 0.0528 0.1118 Washington 2010-2017 0.0135 0.0372 0.0161 0.0094 0.0263 0.0161 0.0161 Average 0.0225 0.0441 0.0217 0.0192 0.0458 0.0217 0.0325 † The Maine data available did not distinguish between urban and rural. Also, with the exception of 367 miles of interstate, all roads were undivided. It was not possible to separate out the divided highway crashes, so they are included in the undivided roads since the mileage is small. Table 46. Fatal and serious injury ROR crash rates for Illinois, Maine, North Carolina, Ohio, and Washington State.

Roadside Risk Design Methodology 139   be used. A guardrail should only be installed to shield the utility poles if the guardrail alterna- tive is less risky than leaving the poles unshielded. Many design questions in roadside safety are best addressed using a relative goal (e.g., installing median barriers, shielding fixed objects, etc.). Risk is the proportion of poor outcomes to all outcomes, and the relative risk (i.e., risk ratio) is the risk of one type of treatment (e.g., install a roadside appurtenance to shield a feature) divided by the risk for a different treatment (e.g., leaving the roadside feature unshielded) (Bonita et al. 2006; Higgins and Green 2011; Irwig et al. 2008). Risk (R) and relative risk (RR) are defined as follows: = = R FREQ FREQ RR R R i Poor Outcomes AllOutcomes 1 2 1 2 The objective is to minimize serious and fatal injury crashes (i.e., KA crashes) so the risk is the proportion of KA crashes divided by crashes of all severities with the same feature (i.e., KABCO). Relative risk can also be assessed by dividing the risk for one type of roadside treat- ment (e.g., installing a w-beam) by the risk of another type (e.g., cable). Risk, when defined as above, is a very useful quantity of measure for roadside safety. Unlike 40 years ago when researchers often had few alternatives for observable measures of risk, today all states collect and maintain extensive databases of crashes that occur on their roadways, and most conform to the minimum standards in the MMUCC (NHTSA and GHSA 2017). Current interest from the states, AASHTO, and the FHWA in performing roadside safety in-service performance evalu- ations will accelerate the availability of good quality crash, roadside inventory, and traffic data. While highway agencies and AASHTO’s TCRS are certainly encouraged to establish their own specific goals, the roadside design goals used in developing guidance for this project are shown in Table 47. Although the focus of these procedures is a risk-based design, economic analysis can be used as a follow-on step to further prioritize projects based on the effective use of highway agency funds. The first step in any such analysis is to estimate the reduction in KA ROR crashes as discussed previously. Economic analysis can optionally be used next to further evaluate projects. Three typical types of economic analyses can be easily based on the results of a risk assessment: • Benefit-cost. • Cost-effectiveness. • Internal rate of return. Each of these techniques will be discussed in the following sections. 3.3.3.3 Benefit-Cost One of the most common techniques for maximizing value used in many technical fields is benefit-cost analysis (Newman 1977). In the context of roadside safety, the benefit is usually considered to be the reduction in societal costs associated with roadside crashes, and the costs Table 47. Recommended roadside design risk performance goals. Absolute Goal Divided and undivided highways should be designed such that the expected number of fatal and serious injury crashes is less than or equal to 0.0325 KA ROR crashes/edge-mile/year. Relative Goal Where a decision is needed to deploy or not deploy a roadside safety treatment, the performance goal must be that the treatment poses less risk than the untreated condition of the site.

140 Development of Safety Performance-Based Guidelines for the Roadside Design Guide are the construction, maintenance, and repair costs expended by the highway agency to achieve that benefit. Since benefits are defined as the reduction in the societal cost of crashes, estimating the number and severity of crashes is at the heart of the benefit-cost method in roadside safety. When comparing design alternatives, an average annual crash cost is calculated by estimat- ing the number and severity of crashes for the considered alternative and the existing condi- tion (i.e., the null alternative) and then converting these to estimate the social costs using the willingness-to-pay concept. These crash costs are then annualized over the project life at some predefined rate of return. Any direct highway agency costs (i.e., initial installation, annual main- tenance, and periodic repairs) are likewise annualized, and the BCR is calculated. The BCR is calculated as follows: ( ) = +       = − +       • • • • • • BCR OR C VSL DC AP MC BCR 1 RR OUTCOME C VSL DC AP MC Alt Null KA Alt i,n Alt Alt Null Null KA Alt i,n Alt where: BCR = Benefit-cost ratio of the design alternative being considered with respect to the null alternative. ORAlt/Null = The outcome reduction (OR), which is the estimated difference in the annual frequency of fatal and serious injury for the design alternative being considered and the existing null alternative. ORAlt/Null = OUTCOMENull − OUTCOMEAlt ORAlt/Null = (1 − RRAlt/Null) OUTCOMENull VSL = The value of statistical life (VSL) in dollars based on the U.S. DOT recom- mendation or the agency value for a fatal crash (Monje 2016). CKA = A unitless coefficient that transforms the VSL to the average cost of a KA crash. APi,n = The capital recovery factor as a function of the interest rate, i, and service life, n, where ( ) ( ) = + + −       i i i n nAP P 1 1 1i,n (Newman 1977). DCAlt = The direct cost of constructing and maintaining the design alternative being considered over the service life of the alternative. The direct cost of the null alternative is presumed to be zero since it is the existing situation. MCAlt = The annual maintenance cost of the median barrier. RRAlt/Null = The relative risk of the considered alternative with respect to the null alter- native where RRAlt/Null = OUTCOMENull / OUTCOMEAlt. OUTCOMENull = The frequency of fatal and serious injury ROR crashes for the existing roadside (i.e., the null alternative). i = Highway agency specified rate of return as a decimal percent. n = Considered alternative projected life in years. A BCR greater than one means that the investment is less than the benefit obtained. A value of one is the minimum BCR where the project should be considered. Most highway agencies expect BCR values between two and four to maximize the benefit of scarce agency resources (AASHTO 2010b). The expected frequency of KA outcomes for the roadside design alternatives under consid- eration can be estimated using the method outlined in this section and the difference calcu- lated (ORAlt/Null). The VSL is roughly equivalent to the fatal crash cost. The VSL is periodically defined by the U.S. DOT for use in policy analyses (Monje 2016). The 2020 VSL is estimated to

Roadside Risk Design Methodology 141   be $12.3 million based on the published 2016 update procedure (Monje 2016). Many highway agencies establish their own local values for either VSL or the fatal crash cost, and these could be used as well. CKA is a coefficient that transforms the VSL into the average cost of a KA crash. Based on data from Miller et al., the value of CKA should be taken as 0.33 unless there are spe- cific local data used to calibrate it further. Although other methods can be used, the annualized cost method is recommended because the societal benefits are an annually recurring benefit as are direct maintenance costs. Other techniques like present worth or future worth can be used and will result in the same answer. Usually, highway agencies will determine an appropriate value for the rate of return (i) and design life (n) to be used in economic analyses. Generally, the design life should be on the order of 25 to 30 years for typical roadside hardware, and rates of return between 2% and 4% are typical for the rate of return. The next step is to calculate the direct costs. The direct cost of the null alternative is assumed to be zero since the null alternative is the already-existing roadside design. The annualized construction cost of the alternative roadside design being considered is added to the annual maintenance cost to determine the highway agency direct cost (DCAlt). The BCR is then easily calculated using the equation above. As shown above, the BCR can be calculated based on the difference in outcomes (ORAlt/Null) or the relative risk (RRAlt/Null = OUTCOMEAlt/OUTCOMENull) depending on which is most convenient. 3.3.3.4 Cost-Effectiveness Cost-effectiveness analysis is very similar to benefit-cost analysis, but instead of monetizing benefits, the outcome itself (i.e., the annual reduction in KA crashes) is used. The annualized cost of the roadside design improvement divided by the annual reduction in the number of KA crashes would be the incremental cost-effectiveness ratio. The ICER is defined as follows (Newman 1976): ( )= + = + − • • ICER DC AP MC OR DC AP MC 1 RR OUTCOMEi j Alt i,n Alt Alt Null Alt i,n Alt Alt Null Null where: ICERi/j = Incremental cost-effectiveness ratio of alternative j with respect to alter- native i. DCAlt = The cost of the direct construction cost for the considered alternative. The direct cost of the null alternative is assumed to be zero since it is the exist- ing condition. APi,n = The capital recovery factor as a function of the interest rate, i, and service life, n, where ( ) ( ) = + + −       i i i n nAP P 1 1 1i,n (Newman 1977). MCAlt = Annualized maintenance cost, the considered alternative. The mainte- nance cost of the null alternative is assumed to be zero since it is the exist- ing condition. ORAlt/Null = The outcome reduction, which is the estimated difference in the annual frequency of fatal and serious for the considered roadside design (Alt alter- native) and the existing roadside design (Null alternative). RRAlt/Null = The relative risk of the considered alternative with respect to the null alternative. OUTCOMENull = The frequency of fatal and serious injury ROR crashes for the existing roadside (i.e., the null alternative). i = Highway agency specified rate of return as a decimal percent. n = Considered alternative projected life in years.

142 Development of Safety Performance-Based Guidelines for the Roadside Design Guide Like benefit-cost analysis, ICER can be calculated based on the difference in outcomes (ORAlt/Null) or the relative risk (RRAlt/Null = OUTCOMEAlt/OUTCOMENull) depending on which is most convenient. Similarly, present-worth, future worth, and annual cost analyses could all be used with the same results, but annual cost analysis is used herein since the reduction in KA crashes is an annual value. One of the advantages of the ICER method of economic analysis is that it does not require the user agency to monetize fatal and serious injury crashes. Better alternatives have lower ICER values and can be chosen on that basis alone. The ICER can be viewed as a priority rank for various projects, with the higher values representing a high priority and better use of funding. The ICER can also be understood as the annual cost over the life of the design of avoiding one KA crash. 3.3.3.5 Internal Rate of Return The internal rate of return (IRR) is the discount or interest rate that makes all net present- worth values of the alternative zero (Barreto and Howland 2006). The IRR is the interest rate where the annual benefit of avoiding fatal and serious injury crashes resulting from the con- sidered alternative roadside design is exactly equal to the cost of implementing the considered alternative roadside design. In other words, the present worth of the annual benefit of fatal and serious injury crashes avoided over the life of the project exactly equals the total construction and maintenance costs of the considered alternative over the life of the project. ∑ ∑ ( ) ( ) ( ) + = + + = − + = = • • • • • • i n i n DC MC PA VSL C OR 1 IRR DC MC PA VSL C 1 IRR OUTCOME 1 IRR Alt Alt i,n KA Alt Null i 0 Alt Alt i,n KA Alt Null Null i 0 where: IRR = The highway agency internal rate of return that results in the present worth of the alternative being exactly 0. DCAlt = The cost of the direct construction cost for the considered alternative. The direct cost of the null alternative is assumed to be zero since it is the existing condition. MCAlt = Annualized maintenance cost, the considered alternative. The maintenance cost of the null alternative is assumed to be zero since it is the existing condition. PAi,n = The series present worth factor as a function of the interest rate, i, and service life, n, where ( ) ( ) = + − +       i i i n nPA A 1 1 1i,n (Newman 1977). VSL = The value of statistical life (VSL) in dollars based on the U.S. DOT recom- mendation or the agency value for a fatal crash (Monje 2016). CKA = A unitless coefficient that transforms the VSL to the average cost of a KA crash. RRAlt/Null = The relative risk of the considered alternative with respect to the null alternative. OUTCOMENull = The frequency of fatal and serious injury ROR crashes for the existing roadside (i.e., the null alternative). i = Highway agency specified rate of return as a decimal percent. n = Considered alternative projected life in years. The IRR is the rate of return where the societal benefits accrued over the life of the considered roadside design alternative exactly equal the cost installing and maintaining that alternative.

Roadside Risk Design Methodology 143   Larger IRR values indicate a higher rate of return and are, therefore, a better use of scarce high- way agency funds than projects with lower values of IRR. Until recently, the United States GDP growth has been about 2% annually. The most optimistic estimates of GDP growth for the next several years are just under 4% annually. A highway improvement project should have an IRR greater than the actual GDP growth and probably should be above the optimistic GDP growth values. If the IRR is less than about 4%, the replacement project is probably not a good way to spend scarce highway agency resources. The IRR can also be viewed as a measure of priority. Projects with the highest IRR should have the highest priority since they generate the largest societal benefit for the funds expended. A replacement project with an IRR = 8 would be a much better use of highway agency funds than another project with an IRR = 6 for example. Projects with negative IRR values would be a poor use of funds in all economic conditions. 3.4 Updating and Maintaining the Data One of the significant advantages to the method outlined in Chapter 3 is that it is modular. For example, when NCHRP Project 17-88, “Roadside Encroachment Database Development and Analysis” develops new encroachment information, the new encroachment models can be used to replace those outlined in Section 3.3.2.1.1: Base Encroachment Frequency, and shown in Table 26 without having to change the rest of the procedure. Similarly, if a research study develops new crash severities for rollovers on backslopes, the new data can be added to outline Section 3.3.2.3: Reduce Crash Severity, and Table 45 without changing the rest of the process. In this way, the method can be updated, improved, and extended as new research becomes available.