Safety Prediction Methodology and Analysis Tool for Freeways and Interchanges (2021)

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100 CHAPTER 5: PREDICTIVE MODEL FOR FREEWAY SEGMENTS This chapter describes the activities undertaken to calibrate and validate safety predictive models for freeway segments and for freeway speed-change lanes. Each model consists of a SPF and a family of CMFs. The SPF is derived to estimate the crash frequency for segments and speed-change lanes with specified design elements and operating conditions. The CMFs are used to adjust the SPF estimate whenever one or more elements or conditions deviate from those that are specified. The calibrated safety predictive models were used to develop a freeway safety predictive method. This method describes how to use the models to evaluate freeway safety, as may be influenced by road geometry, roadside features, traffic volume, and lane-change-related traffic maneuvers. The predictive method for freeways is documented in Appendix C. The predictive method includes predictive models for freeway segments and for speed- change lanes. The freeway segment models are used to evaluate both freeway travel directions combined. In contrast, the speed-change lane models are used to evaluate the one travel direction associated with a speed-change lane. Collectively, the predictive models for freeway segments address the following area type and lane combinations. ● rural freeway with four through lanes, ● rural freeway with six through lanes, ● rural freeway with eight through lanes, ● urban freeway with four through lanes, ● urban freeway with six through lanes, ● urban freeway with eight through lanes, and ● urban freeway with ten through lanes. The speed-change models address ramp entrances and ramp exits for freeways with the area types and lane combinations identified in the preceding list. This chapter is divided into six parts. The first part provides some background information on the topic of predictive models for freeway segments. The second part describes the theoretic development of selected CMFs. The third part describes the method used to calibrate the proposed models. The fourth part describes the calibration of the models to predict FI crash frequency. The fifth part describes the calibration of the models to predict PDO crash frequency. The sixth part provides a list of the variables defined in this chapter.

101 BACKGROUND This part of the chapter consists of three sections. The first section describes the decomposition of a freeway facility into analysis units (i.e., sites). The second section provides a brief overview of the predictive model structure. The third section reviews the highway safety data assembled for model calibration. Roadway Segments, Ramp Segments, and Ramp Terminals For analysis purposes, a freeway facility is considered to include a freeway section and, possibly, one or more interchanges. The freeway section includes a contiguous set of freeway segments and, possibly, one or more speed-change lanes. An interchange is considered to consist of a set of ramp segments, crossroad ramp terminals, and, possibly, one or more C-D road segments. These components are also referred to as “sites.” Figure 38 illustrates the sites associated with a short section of freeway near an interchange. The arrangement shown is intended to illustrate the types of sites used for freeway facility safety evaluation–it is not a typical interchange and many freeway segments are not likely to include speed-change lanes. As indicated in the HSM (Highway, 2010), road segment boundaries are typically defined by intersections or by a change in the cross section. This guidance applies to crossroad segments and to freeway segments. Specifically, freeway segment boundaries are defined by the presence of a gore point associated with a ramp. However, the length of the freeway segment is reduced by the length of the speed-change lane. This distinction stems from: (1) the lack of a clear and consistent designation of speed-change-related crashes in state DOT crash databases and (2) the presence of a speed-change lane on only one side of the freeway. For these reasons, a speed- change-related crash was identified as any crash that occurred: (1) between the marked gore point and taper points of a ramp merge or diverge area and (2) on the same side of the freeway as the merge or diverge area. This approach requires a reduction in the effective length of the freeway segment to account for the crashes re-assigned to any speed-change lanes that are located on the segment. This reduction is shown in the bottom of Figure 38 for a freeway section with one ramp entrance and one ramp exit. Safety Predictive Models The predicted average crash frequency for a freeway section is computed as the sum of the predicted average crash frequency of all sites that comprise the section. This calculation is described by Equation 10. ( ) ( ) ( )  +++= segmentsall exitsall ex entrancesall ensvmvctionse NNNNN (10) where, Nsection = predicted average crash frequency within the limits of a freeway section, crashes/yr; Nmv = predicted average multiple-vehicle non-entrance/exit crash frequency, crashes/yr; Nsv = predicted average single-vehicle non-entrance/exit crash frequency, crashes/yr; Nen = predicted average ramp-entrance-related crash frequency, crashes/yr; and Nex = predicted average ramp-exit-related crash frequency, crashes/yr.

102 PLAN VIEW COMPONENT PARTS Speed-Change Lane Speed-Change Lane Crossroad Ramp Terminal Type: ramp entrance Type: ramp exit Type: diagonal, 4-leg Seg. length = Len Seg. length = Lex 250-ft influence area Interchange Ramp Proper Interchange Ramp Proper Type: entrance ramp Type: exit ramp Seg. length = Lenr Seg. length = Lexr (note: ramp can be comprised of several segments) (note: ramp can be comprised of several segments) Freeway Segment Effective segment length, L* = Lfr - Len/2 - Lex/2 (note: freeway segment length does not include length of speed-change lanes, if these lanes are adjacent to the segment) Ramp Entrance Length, Len Ramp Exit Length, Lex SCen Fr2 Ren In C ro ss ro ad Lfr2 SCen SCex SCex In LexrLenr Ren Rex Fr1 Fr2 Rex Fr3 Lfr1 Lfr3 Fr3Fr1 Figure 38. Illustrative freeway facility analysis sites. The predicted average crash frequency for each site is computed using a predictive model. Each model represents the combination of an SPF and several CMFs. The SPF is used to estimate the average crash frequency for a generic site whose attributes are consistent with the SPF’s stated base conditions. The CMFs are used to adjust the SPF estimate when the attributes of the subject site are not consistent with the base conditions. The general form of the four safety predictive models developed in this research is shown below as Equation 11 to Equation 14. ( ) ( )kwmvmvmvspfmvmv CMFCMFCMFCMFNCN ×××××××= ...... 1,1,, (11) ( ) ( )kxsvsvsvspfsvsv CMFCMFCMFCMFNCN ×××××××= ...... 1,1,, (12)

103 ( ) ( )kyenenenspfenen CMFCMFCMFCMFNCN ×××××××= ...... 1,1,, (13) ( ) ( )kzexexexspfexex CMFCMFCMFCMFNCN ×××××××= ...... 1,1,, (14) where, Nspf, mv = predicted average multiple-vehicle non-entrance/exit crash frequency for base conditions, crashes/yr; Nspf, sv = predicted average single-vehicle non-entrance/exit crash frequency for base conditions, crashes/yr; Nspf, en = predicted average ramp-entrance-related crash frequency for base conditions, crashes/yr; Nspf, ex = predicted average ramp-exit-related crash frequency for base conditions, crashes/yr; Cmv = local calibration factor for multiple-vehicle non-entrance/exit crashes; Csv = local calibration factor for single-vehicle non-entrance/exit crashes; Cen = local calibration factor for ramp-entrance-related crashes; Cex = local calibration factor for ramp-exit-related crashes; CMFmv, 1 ... CMFmv, w = crash modification factors for multiple-vehicle crashes at a site with specific geometric design features w; CMFsv, 1 ... CMFsv, x = crash modification factors for single-vehicle crashes at a site with specific geometric design features x; CMFen, 1 ... CMFen, y = crash modification factors for ramp-entrance-related crashes at a site with specific geometric design features y; CMFex, 1 ... CMFex, z = crash modification factors for ramp-exit-related crashes at a site with specific geometric design features z; and CMF1 ... CMFk = crash modification factors for freeway segment crashes at a site with specific geometric design features k. The first term in parentheses in Equations 11 to 14 recognizes that the influence of some geometric factors is unique to each crash type. In contrast, the second term in parentheses in these equations recognizes that some geometric factors have a similar influence on all crash types. Highway Safety Database The HSIS was used as the primary source of data for model calibration and validation. The “HSIS” states California, Maine, and Washington were identified as including ramp volume data, which was of fundamental importance to all aspects of the project. These data were not available from the other HSIS states. Hence, the database assembly focused on these three states. They are called the “study states” in this report. In addition to ramp volume data, each study state database included a range of data describing the location, area type, traffic characteristics, geometry, and lane use for freeway segments. The data acquired from these databases is summarized in Table 29.

104 TABLE 29. Freeway variables from HSIS database Category Variable Description Descriptive state Source of data (CA, ME, WA) rte_nbr State route number rte_suf State route suffix county County number (established by state DOT) begmp Begin milepost (established by state DOT in CA, WA; by researchers for ME) endmp End milepost (established by state DOT in CA, WA; by researchers for ME) seg_lng Segment length, miles rodwycls Road functional classification (established by HSIS staff) rururb Area type (urban, rural) Traffic ave_adt Segment AADT averaged for a three-year period entr_to_begmp_adt AADT of ramp located at entr_begmp (see Table 30) exit_to_begmp_adt AADT of ramp located at exit_begmp (see Table 30) entr_to_endmp_adt AADT of ramp located at entr_endmp (see Table 30) exit_to_endmp_adt AADT of ramp located at exit_endmp (see Table 30) weav_aadt_inc_ent AADT of ramp entering weaving section for travel in increasing direction weav_aadt_inc_ext AADT of ramp exiting weaving section for travel in increasing direction weav_aadt_dec_ent AADT of ramp entering weaving section for travel in decreasing direction weav_aadt_dec_ext AADT of ramp exiting weaving section for travel in decreasing direction ramp_aadt AADT of ramp associated with speed-change lane 1 : repeat variable above for each of up to four speed-change lanes on segment Crash nk_mv Count of reported fatal, multiple-vehicle non-entrance/exit crashes during three-year period na_mv Count of reported incapacitating injury, multiple-vehicle non-entrance/exit crashes during three-year period nb_mv Count of reported non-incapacitating injury, multiple-vehicle non- entrance/exit crashes during three-year period nc_mv Count of reported possible-injury, multiple-vehicle non-entrance/exit crashes during three-year period no_mv Count of reported PDO, multiple-vehicle non-entrance/exit crashes during three-year period : repeat five variables above for single-vehicle non-entrance/exit crashes, crashes in ramp entrance area, and crashes in ramp exit area The data identified as “Descriptive” in Table 29 were obtained directly from the HSIS database for each study state. The data identified as “Traffic” or “Crash Data” were derived from the HSIS data. SAS software was used to manipulate the HSIS data to compute the desired variables. The ramp data identified in the table required that each ramp be visually located first (using aerial photography) using the freeway milepost reference system. As discussed in Appendix B, several of the geometry and lane use variables in the study state databases were of unknown accuracy. Also, several variables often had subtly different

105 definitions among states. Moreover, the study state databases often did not include variables that describe road-related factors known to be associated with crash frequency. To overcome these limitations, the study-state databases were enhanced using data from other sources. These variables are listed in Table 30. The collection of these data required the location of each ramp using geographic coordinates and aerial photography, based on the freeway milepost reference system in HSIS. TABLE 30. Freeway variables from supplemental data sources Category Variable Description Descriptive lat_lon_coord Latitude and longitude of begin milepost inc_lane_use Special lane use for travel in increasing milepost dec_lane_use Special lane use for travel in decreasing milepost inc_shldr_use Use of shoulder by time of day for travel in increasing milepost dec_shldr_use Use of shoulder by time of day for travel in decreasing milepost Traffic pct_hrs Proportion of hours in average day that volume exceeds 1,000 veh/h/ln pct_veh Proportion of AADT during hours where volume exceeds 1,000 veh/h/ln Roadway inc_lanes Number of lanes for travel in increasing milepost dec_lanes Number of lanes for travel in decreasing milepost inc_drop-add_lanes Number of lane drops or adds on seg. for travel in increasing milepost dec_drop-add_lanes Number of lane drops or adds on seg. for travel in decreasing milepost out_shld_meas Outside shoulder width (average of both directions) lane_meas Lane width (average for all lanes in both directions) in_meas Width of inside shoulders and non-shoulder median (= median width) in_shld_meas Inside shoulder width (average of both directions) med_width_meas Width of median measured between near edges of inside shoulder med_nontrav_meas Width of median barrier, if present Roadside or median med_type_meas Median type (1 = raised curb, 2 - barrier, 3 = depressed or unsurfaced) in_barrier_off Average offset to barrier in median, measured from face of barrier to near edge of inside shoulder in_barrier_len Total length of barrier in median out_barrier_off Average offset to barrier on roadside, measured from face of barrier to near edge of outside shoulder out_barrier_len Total length of barrier on roadside inc_clear_zone Average clear zone width for travel in increasing milepost dec_clear_zone Average clear zone width for travel in decreasing milepost Alignment nbr_curves Count of curves on segment curv_rad Radius of curve 1 curv_ang_deg Deflection angle of curve 1 curv_lgt_ft Length of curve 1 curv_begmp Begin milepost for curve 1 curv_lgt_on_seg Length of curve 1 on segment : repeat five variables above for each of up to three curves on segment

106 TABLE 30. Freeway variables from supplemental data sources (continued) Category Variable Description Speed-change lane n_sc_lanes Count of speed-change lanes adjacent to segment (in whole or part) sc_design Design for speed-change lane 1 (e.g., P=parallel, T=taper, etc.) sc_lane_use Lane use for speed-change lane 1 (e.g., N = normal, R=meter) sc_type Orientation of speed-change lane 1 (e.g., entrance/exit, left/right side) sc_mrk_begmp Freeway begin milepost at start of speed-change lane1 pavement marking sc_mrk_endmp Freeway end milepost at end of speed-change lane 1 pavement marking sc_lgt_on_seg Length of speed-change lane 1 sc_ramp_lanes Number of lanes in speed-change lane 1 at gore point sc_trav_dir Freeway travel direction adjacent to speed-change lane 1 : repeat eight variables above for each of up to four speed-change lanes on segment Weaving section inc_A_lanes Number of lanes on freeway and ramps before the weaving section inc_B_lanes Number of lanes on freeway (including auxiliary lanes) in the weaving section inc_C_lanes Number of lanes on freeway and ramps after weaving section inc_D_lanes Number of lanes on right-side entrance ramp before weaving section inc_E_lanes Number of lanes on right-side exit ramp after weaving section inc_Lw Length of weaving section (gore to gore) inc_wev_lgt_on_seg Length of weaving section on segment dec_A_lanes Number of lanes on freeway and ramps before the weaving section dec_B_lanes Number of lanes on freeway (including auxiliary lanes) in the weaving section dec_C_lanes Number of lanes on freeway and ramps after weaving section dec_D_lanes Number of lanes on right-side entrance ramp before weaving section dec_E_lanes Number of lanes on right-side exit ramp after weaving section dec_Lw Length of weaving section (gore to gore) dec_wev_lgt_on_seg Length of weaving section on segment Other ramp_exit_cnt Count of ramp exit gore points adjacent to segment ramp_ent_cnt Count of ramp entrance gore points adjacent to segment in_rumble Proportion of segment length with rumble strips on inside shoulders out_rumble Proportion of segment length with rumble strips on outside shoulders entr_begmp Milepost for ramp entrance marked gore point. Ramp is located on side of road where travel is in increasing milepost and upstream of segment begin milepost (Q ramp) exit_begmp Milepost for ramp exit marked gore point. ramp is located on side of road where travel is in decreasing milepost and downstream of segment begin milepost (R ramp) entr_endmp Milepost for ramp entrance marked gore point. Ramp is located on side of road where travel is in decreasing milepost and upstream of segment end milepost (S ramp) exit_endmp Milepost for ramp exit marked gore point. Ramp is located on side of road where travel is in increasing milepost and downstream of segment end milepost (P ramp) One source of enhanced data was the continuous traffic counting station data routinely collected by each state DOT. The freeway station nearest to each segment in the study state database was identified and used to compute the proportion of hours per day that are considered

107 to have “high volume” and the proportion of daily traffic using the segment during these high- volume hours. Aerial photography was used as a second source of enhanced data. These photographs were obtained from the Internet using Google Earth software. The data collected include the width of road cross section elements, barrier presence and location, horizontal curvature, ramp configuration, ramp entrance location, and median type. A description of the variables acquired from aerial photography is provided in Table 30. The “inc_lane_use” and “dec_lane_use” variables were used to identify the type of lane use that exists on a segment. Lane uses considered include: bus-only lane, reversible lane, truck- only lane, and HOV lane. These uses were not ranked high in the prioritization process, as documented in Chapter 3. Hence, only “normal” lane use is represented in the freeway and speed-change lane database. For similar reasons, segments with “shoulder use by time of day” are not represented in the database. CMF DEVELOPMENT This part of the chapter describes the development of three CMFs. The first section describes the development of a CMF that predicts the effect of lane-change frequency on crash frequency. The second section describes the development of a CMF that predicts the effect of high-volume conditions on crash frequency. The last section describes the development of a CMF that predicts the effect of horizontal curvature on crash frequency. Lane-Change CMF The lane-change CMF is intended to reflect the effect of lane-changing activity on crash frequency. A review of the literature (documented in Chapter 2) indicated that lane-change frequency is highest in the vicinity of ramp entrances and ramp exits. It then declines with increasing distance from the ramp. It also increases in proportion to the ramp volume. Equation 4 (in Chapter 2) was derived to describe these influences in the form of a crash modification factor. High-Volume CMF The volume-to-capacity ratio relates the demand volume to the capacity of a roadway segment. As volume nears capacity, average speed tends to decrease and headway is reduced. Logically, these changes have some influence on crash characteristics, including crash frequency, crash type distribution (i.e., single vehicle versus multiple vehicle), and crash severity distribution. Some research has been undertaken to examine the relationship between volume-to- capacity ratio and crash character. This research typically compares average hourly volume estimates with the crashes that occur during the same hour for one or more years. In this manner, the analysis is often structured by time of day. There are issues of sample size, day versus night, and autocorrelation that complicate this type of analysis.

108 A review of the literature (documented in Chapter 2) confirmed an association between time of day and crash frequency, as well as between volume-to-capacity ratio and crash frequency. It also highlights some of the aforementioned issues. Based on the findings from the literature review, it was rationalized that a “complete” safety evaluation (and models that support this evaluation) should consider all of the crashes that occur during a 24-hour period; not just those crashes that occur during a particular hour (e.g., the peak hour). Treatments that improve safety during one hour of the day may degrade safety during other hours. Moreover, given that crashes are rare events, the safety evaluation should be based on annual crash frequency; not just those crashes occurring during a peak hour, day, or month. Finally, the safety predictive models should include variables that reflect a sensitivity to the fact that high-volume conditions tend to last for only a few peak hours, and sometimes only during a peak season. Two statistics were developed to support a complete safety evaluation and provide the sensitivity to recurring high-volume conditions. These statistics were derived from a pragmatic perspective. That is, it was determined that they had to be quantifiable by practitioners that use the predictive method. The first statistic developed is the “proportion of hours in the average day that volume exceeds 1,000 veh/h/ln” (i.e., proportion of hours). The second statistic developed is the “proportion of AADT during hours where volume exceeds 1,000 veh/h/ln” (i.e., “proportion of volume”). Both statistics are defined to have a value of zero if the volume on the associated segment does not exceed the threshold value for any hour of the day. Both statistics were quantified using the continuous traffic counting station data routinely collected by each state DOT. The freeway station nearest to each segment in the study state database was identified and the hourly volume distribution for the average day acquired. This distribution was then used to compute the hourly traffic volume for each hour of the average day for each segment. The threshold value of 1,000 veh/h/ln was used to define “high-volume” conditions. It was selected after considering several values. The volume of 1,000 veh/h/ln corresponds to an average vehicle headway of 3.6 s. The freeway speed-volume relationship shown in Chapter 23 of the Highway Capacity Manual (Highway, 2000) suggests that the average speed tends to drop as flow rates increase beyond 1,000 veh/h/ln. This trend suggests that drivers are reducing speed to improve their comfort and safety as their headway gets shorter than 3.6 s. Higher threshold values were considered because they would reflect the presence of a more congested condition. However, with a threshold of 1,000 veh/h/ln, only 44 percent of the segments were found to have a non-zero value for either statistic. With a higher threshold, this percentage decreased significantly, leaving a large majority of segments with no information about the extent of traffic congestion they experience. The value of both statistics increases with an increase in the number of hours that exceed the threshold value. If the volume during each hour of the day exceeds the threshold value, then both statistics equal 1.0. In general, the proportion-of-hours statistic is large when hourly volumes are continuously high throughout the day (i.e., the hourly volume distribution is relatively flat during daylight hours). In contrast, the proportion-of-volume statistic is large when

109 0.00 0.02 0.04 0.06 0.08 1 3 5 7 9 11 13 15 17 19 21 23 Hour of Day Pr op or tio n A A D T du rin g H ou r Route 51, Sacramento Co., Calif. Route 5, Shasta Co., Calif. 0 500 1,000 1,500 2,000 1 3 5 7 9 11 13 15 17 19 21 23 Hour of Day H ou rly V ol um e, v eh /h /ln Route 51, Sacramento Co., Calif. Route 5, Shasta Co., Calif. hourly volumes are continuously high or when there are a few peak hours with an exceptionally large volume. The hourly distribution of traffic volume on two freeway segments is shown in Figure 39. Figure 39a shows the distribution as a proportion of the AADT volume. Route 51 is shown to have a “flatter” distribution during the daylight hours than Route 5. Figure 39b shows the distribution in terms of hourly volume per lane. Route 51 is shown to have 16 hours that exceed the threshold value (proportion of hours = 0.67) and 87 percent of its volume served during these hours (proportion of volume = 0.87). Route 5 has 2 hours that exceed the threshold value (proportion of hours = 0.08) and 15 percent of its volume served during these hours (proportion of volume = 0.15). a. Proportion AADT volume during hour. b. Hourly volume distribution. Figure 39. Hourly volume distribution for two freeway segments. The trend in the proportion-of-volume statistic for several hundred freeway segments is shown in Figure 40. A similar trend is found for the proportion-of-hours statistic. There are many segments with a statistic value of 0.0 because these segments do not experience an hourly volume in excess of 1,000 veh/h/ln during the average day. The trend in the data in Figure 40 indicates that the statistic value increases with increasing daily volume per lane. However, there is considerable variability in this variable value for segments with a volume in the range of 10,000 to 25,000 veh/day/ln. The variability declines for higher volumes, which indicates that these segments consistently operate at high volume for most, or all, of the day. The variability in this statistic suggests that it may contain information about segment conditions, beyond that provided by the AADT and number-of-lanes variables.

110 0.0 0.2 0.4 0.6 0.8 1.0 0 10,000 20,000 30,000 40,000 Average Daily Traffic Demand per Lane, veh/day/ln Pr op or tio n A A D T du rin g H ig h- Vo lu m e H ou rs Pv = 1 - e[1.45 - 0.124 x AADT/lanes/1000] Figure 40. Proportion-of-volume statistic for freeway segments in three states. The CMF used to represent the effect of high-volume conditions is described using the following equation. hvhv Pb hv eCMF = (15) where, CMFhv = high-volume crash modification factor; bhv = calibration coefficient; and Phv = proportion of AADT during hours where volume exceeds 1,000 veh/h/ln. The CMF is shown using the proportion-of-volume statistic. A similar CMF can be derived using the proportion-of-hours statistic. However, only one of these CMFs is used in the model given that they are highly correlated. Horizontal Curve CMF The horizontal curve CMF is defined by the following equation. seg curveseg hc N NN CMF + = (16) where, CMFhc = horizontal curve crash modification factor; Nseg = predicted average crash frequency on segment (regardless of curvature), crashes/yr; and Ncurve = predicted average additional crashes due to curvature; crashes/yr. The following equations are rationalized to represent the components of Equation 16. ic b seg CMFLAADTaN = (17)

111 0.0 1.0 2.0 3.0 4.0 0.00 0.10 0.20 0.30 0.40 Side Friction Demand C ra sh R at e, c ra sh es /m vm 85th% tangent speed = 65 mph Fatal + Injury + PDO Fatal + Injury idc b curve CMFfIAADTaN = (18) ( ) e R Vf cd −= 2.32 47.1 2 (19) where, CMFi = crash modification factor for element i (i = lane width, shoulder width, etc.); Lc = length of horizontal curve (= Ic × R / 5280 / 57.3), mi; Ic = curve deflection angle, degrees; fd = side friction demand factor; Vc = average curve speed, mi/h; e = superelevation rate, ft/ft; and R = curve radius, ft. Equation 18 does not include curve length Lc as a measure of exposure. Rather, curve deflection angle Ic is rationalized as the appropriate measure of exposure for crashes that occur because the segment is curved. Just as segment length L represents the product of forward velocity and travel time, deflection angle represents the product of angular velocity and travel time. Side friction demand is included in Equation 18 as being logically correlated with crash frequency. As side friction demand reaches the roll threshold for tall vehicles, the propensity for a roll-over crash increases. Also, as side friction demand reaches the limit of available friction, the propensity for sliding off the road increases. The relationship between crash rate and side friction demand derived by Bonneson et al. (2007b) is shown in Figure 41. Figure 41. Relationship between crash rate and side friction demand.

112 Equations 16 through 19 can be combined to yield the following horizontal curve CMF. ( ) 2 2 0 2.32 47.10.1 R VbCMF chc += (20) where, b0 = calibration coefficient. The contribution of the superelevation term (i.e., e/100) in Equation 19 is small and is removed for mathematical convenience. This CMF is compared in Figure 16 with similar CMFs for several highway types. The speed variable in Equation 20 is not available in the study state databases. However, the typical freeway speed limit is similar among the three states. In rural areas, it is 70, 65, and 70 mi/h for California, Maine, and Washington, respectively. In urban areas, it is 65, 55, and 60 mi/h in California, Maine, and Washington, respectively. METHODOLOGY This part of the chapter describes the methodology used to calibrate the freeway segment and speed-change lane safety predictive models. It is divided into three sections. The first section describes several supplemental variables used to calibrate the predictive models. The second section describes several analytic relationships used to calibrate non-homogeneous segments. The last section provides an overview of the approach used to calibrate the predictive models. Supplemental Variables As noted in a previous part of this chapter, several variables in the database were obtained from aerial photographs of the freeway segments and speed-change lanes represented in the study state databases. Of these variables, some of the more complex ones are defined in this section. Ramp Entrance/Exit Length and Type Ideally, the defined location of a speed-change lane for use in a predictive model would coincide with that used in the Green Book (Policy, 2004). However, crash location attributes in most state DOT databases are not sufficiently refined as to allow the accurate identification of crashes associated with the speed-change lane definitions in the Green Book, as shown in Figure 11. More importantly, these databases do not explicitly identify speed-change-related crashes. Given the challenges in identifying crashes associated with speed-change lane operation, a revised definition of speed-change lane location was developed. The revised definition was dictated primarily by the desire to identify all speed-change-related crashes. To this end, the entrance ramp speed-change lane is considered to coincide with the “ramp entrance length” shown in Figure 11, and include all lanes associated with the freeway and ramp for a common travel direction. Similarly, the exit ramp speed-change lane is considered to coincide with the “ramp exit length” shown in Figure 11, and include all lanes associated with the freeway and

113 ramp for a common travel direction. These definitions are consistent with the ramp entrance and exit sites identified in Figure 38. By applying the aforementioned definition, a speed-change-related crash is identified as a crash that is located between the taper point and the gore point and on the same side of the freeway as the ramp. This definition is likely to identify most speed-change-related crashes, but it may also identify some freeway crashes that are not associated with the speed-change lane. For this reason, some factors that influence crash frequency on freeway segments (e.g., lane width) may also influence crashes in the speed-change lane. Weaving Length and Type Entrance-exit ramp pairs that form a weaving section were identified in the database. In all cases, these ramp pairs were on the right-hand side of the freeway. The type of weaving section was also identified using the definitions in Chapter 13 of the Highway Capacity Manual (Highway, 2000). Figure 42 illustrates the three weaving section types defined in this manual. There are 17 mi of Type A weaving section in the database and 19 mi of Type B weaving section in the database. In contrast, there are only 1.3 mi of Type C weaving section in the database. There were no two-sided Type C weaving sections in the database (this configuration is not shown in Figure 42). a. Type A Weaving Sections. b. Type B Weaving Sections. c. Type C Weaving Sections. Figure 42. Weaving section types.

114 Lwev = w eaving section length 2' Lwev 2' It is generally recognized that the length of the weaving section has an important influence on the operation of the freeway segment. This influence relates to the degree to which the weaving activity is concentrated along the freeway. In recognition of a possible correlation between weaving section concentration and crash frequency, the length of the weaving section was also included in the database. The convention used to measure this length is shown in Figure 43. Figure 43. Weaving section length measurement. Horizontal Clearance Distance The clear zone distance was measured from the outside edge of the traveled way to either the nearest continuous obstruction (e.g., fence line, utility poles, etc.) or the near edge of the frontage road traveled way, whichever is closer to the freeway. If a barrier was present on the roadside for only a portion of the segment, the clear zone distance was measured only for that portion of the segment without the barrier. Longitudinal Barriers Longitudinal barriers (i.e., cable barrier, concrete barrier, guardrail, or bridge rail) were noted when present on a segment. Attributes used to quantify barrier presence include barrier offset, length, and width. Barrier offset, length, and width were each measured separately for inside and outside locations (e.g., inside barrier offset, outside barrier offset, etc.). Barrier offset represents a lateral distance measured from the near edge of the shoulder to the face of the barrier (i.e., it does not include the width of the shoulder). Barrier length represents the length of lane paralleled by a barrier; it is a total for both travel directions. For example, if the outside barrier extends for the length of the roadway on both sides of the roadway, then the outside barrier length equals twice the segment length. Median barrier width represents either the physical width of the barrier if only one barrier is used, or the lateral distance between barrier “faces” if two parallel barriers are provided in the median area. A barrier face is the side of the barrier that is exposed to vehicle traffic. Analytic Relationships for Non-Homogeneous Segments In most instances, the segments in the study state databases were not homogeneous in terms of the variables of interest. One or more geometric elements were often found to start or

115 end at some point along the length of a segment. When this occurs, the length of the segment associated with the element’s initial condition and the length associated with its changed condition were recorded in the database. Geometric elements that were sometimes only partially located on a segment are identified in the following list. ● horizontal curve presence, ● weaving section presence, ● ramp entrance presence, ● ramp exit presence, ● rumble strip presence on inside shoulder, ● rumble strip presence on outside shoulder, ● median barrier presence, and ● roadside barrier presence. CMFs are typically developed for application to homogeneous segments. Thus, they do not include variables that allow them to be modified for application to segments on which they only partially apply. However, the following equation was used to convert a CMF for homogeneous segments into one that could be used for non-homogeneous segments. ( ) iiLiLaggi CMFPPCMF ,,| 0.10.1 +−= (21) where, CMFi|agg = aggregated CMF for element i; PL, i = proportion of the segment length with element i; and CMFi = crash modification factor for element i. To illustrate the use of Equation 21, it can be combined with Equation 20 to compute the aggregate CMF for segments with combined curve and tangent portions. The result is shown in the following equation. ( ) cagghc PR VbCMF 2 2 0| 2.32 47.10.1 += (22) where, CMFhc|agg = aggregated horizontal curve CMF for a segment with both tangent and curved portions; Pc = proportion of the segment length with curvature; and CMFhc = horizontal curve crash modification factor. Modeling Approach Combined Regression Models The calibration activity used SAS that employs maximum-likelihood methods and a negative binomial distribution of crash frequency. Four models were calibrated. The form of each model is shown in the following equations. ( ) ( )kwmvmvmvspfmv CMFCMFCMFCMFNN ××××××= ...... 1,1,, (23) ( ) ( )kxsvsvsvspfsv CMFCMFCMFCMFNN ××××××= ...... 1,1,, (24)

116 ( ) ( )kyenenenspfen CMFCMFCMFCMFNN ××××××= ...... 1,1,, (25) ( ) ( )kzexexexspfex CMFCMFCMFCMFNN ××××××= ...... 1,1,, (26) The SPFs associated with these models are defined using the following equations. ( ) )000,1/ln(,,,,, 1,0,5.05.0 AADTbbisegexisegenmvspf mvmveLLLN + −−= (27) ( ) )000,1/ln(,,,,, 1,0,5.05.0 AADTbbisegexisegensvspf svsveLLLN + −−= (28) ( ) )000,2/ln(,,, 1,0, AADTbbisegenenspf eneneLN += (29) ( ) )000,2/ln(,,, 1,0, AADTbbisegexexspf exexeLN += (30) where, L = length of segment, mi; Len, seg, i = length of ramp entrance i on segment, mi; Lex, seg, i = length of ramp exit i on segment, mi; AADT = AADT volume on segment, veh/day; and bj, i = calibration coefficients for model j ( j = mv, sv, en, ex), i = 0, 1. The second term of Equations 23 to 26 recognizes that the influence of some geometric factors is unique to each crash type. In contrast, the third term of these equations recognizes that some geometric factors have a similar influence on all crash types. The use of common CMFs in multiple models required the use of a combined-model approach. With this approach, the regression analysis evaluated all four models simultaneously and used the total log-likelihood statistic for all four models to determine the best-fit calibration coefficients. A simulation analysis was undertaken to determine if this type of regression would bias the calibration coefficients or their standard error. The results of this analysis indicated that: (1) the coefficients were not biased and (2) that the standard error of those coefficients associated with a variable were not biased. The regression analysis is described in more detail in the next part of this chapter. Cross-Sectional Database The database is described as cross-sectional (as opposed to panel). It represents a common three-year study period for all observations. Study duration in “years” is represented as an offset variable in the regression model. One reason for using cross-sectional data for model calibration relates to the accuracy of the AADT values in most highway safety databases. Examination of the AADT volume in the assembled database (and examination of associated database and state DOT documentation) indicated that segment AADT volume is frequently extrapolated by the state DOT from partial- year counts taken at temporary count stations located several miles from the subject segment.

117 Thus, there are accuracy implications associated with this temporal and spatial extrapolation. Moreover, State DOT practice when a current count is not available for a segment is sometimes to adjust the AADT volume from the last year it was counted (which could be several years previous); sometimes it is to leave the variable as missing. In fact, it is common for a segment’s AADT volume to be missing for one or more years. In these cases, the researchers for this project have had to estimate a value using the AADT volume trends for adjacent years and adjacent segments. Thus, averaging each segment’s AADT volume over years minimizes the variability in AADT volume which, based on the aforementioned observations, is considered largely random. More generally, cross-sectional data provide a more robust predictive model than panel data when the year-to-year variability in the independent variables is largely random. A second reason for using cross-sectional data for model calibration is to minimize the problems associated with over-representation of segments or intersections with zero crashes. Statistical methods have been developed to improve the fit of a model to this “zero-inflated” data. However, Lord et al. (2007) indicate that when these methods have been applied to highway crash data, they have (1) an inherent tendency to over-fit the data, (2) a theoretic explanation of dual state highway safety that is problematic, and (3) the potential to obfuscate the interpretation of predictive model trend and coefficient meaning. Thus, summing each segment’s crashes over years minimizes the proportion of segments or intersections with zero crashes in the database and precludes the need for a dual state distribution. It was assumed that segment crash frequency is Poisson distributed, and that the distribution of the mean crash frequency for a group of similar segments is gamma distributed. In this manner, the distribution of crashes for a group of similar segments can be described by the negative binomial distribution. The variance of this distribution is described by the following equation. ( ) LK NyNyXV 2 ][ += (31) where, V[X] = crash frequency variance for a group of similar locations, crashes2; N = predicted average crash frequency, crashes/yr; X = reported crash count for y years, crashes; y = time interval during which X crashes were reported, yr; and K = inverse dispersion parameter (= 1/k, where k = overdispersion parameter), mi-1. Prediction of PDO Crash Frequency Experience with regression-based calibration of SPFs and CMFs using total crashes and using only FI crashes indicates that the calibration coefficients often vary among model types for common variables. Some of this variation is likely due to the fact that geometric elements often have a different effect on FI crashes than on PDO crashes. As a result, the search for correlation and possible causation is challenged when using total crash data to build total crash prediction models because total crashes combine FI and PDO crashes. The presence of barrier is one example of a geometric component that has a different effect on FI crash frequency than it has on PDO crash frequency. These observations suggest that PDO-based models are preferable to total- crash models.

118 It is widely-recognized that PDO crash counts vary widely on a regional basis due to significant variation in the reporting threshold. This issue was discussed in Chapter 4 where it was noted that there was wide variation in the representation of PDO crashes in the study-state databases. When crash frequency varies systematically from county to county, district to district, and state to state because of formal and informal differences in the reporting threshold, the use of PDO crash data to build PDO crash prediction models is problematic. This observation suggests that PDO-based and total-crash models are likely to include regional biases and added uncertainty due to variation in reporting thresholds. Based on these issues, the following model-building process was developed. It was rationalized that (1) FI crash data are likely to provide the most accurate insight into regression model structure and factors influencing safety and (2) PDO-based models are preferred to total- crash models. However, the development of PDO regression models is problematic because of under-reporting. Therefore, the FI regression model structure was developed first and then used as a “starting point” for the development of the PDO regression model. To minimize the influence of reporting threshold variability among counties, districts, and states, the “county” and “state” variables were treated as both fixed and random effects in separate versions of the PDO regression model. The Hausman test (Hausman, 1978) was used to determine when the fixed-effect treatment was appropriate. The fixed-effect treatment of county and state required that the PDO regression model include an indicator variable for each county and state combination represented in the database. Regardless of whether a fixed- or random-effect model is used, significant county and state differences often emerged through this process. As a result, some geometric variables that were significant in the FI model were less significant in the PDO model. Specifically, the standard error was increased for those geometric variables that varied more among counties than within counties. Unfortunately, it is not known whether the among-county variation is due to differences in reporting threshold (as may be informally applied at different levels within a state) or because of differences in geometry. This approach often resulted in the PDO model having fewer geometric variables than the FI model. However, the benefit of this approach is that the remaining calibration coefficients in the PDO model are less likely to be biased by differences in the reporting threshold. MODEL CALIBRATION FOR FI CRASHES This part of the chapter describes the calibration and validation of the combined freeway segment and speed-change lane predictive models based on FI crashes. The first section identifies the data used for model calibration. The second section describes the structure of each of the four predictive models, as used in the regression analysis. The third section summarizes statistical analysis methods used for model calibration. The fourth section describes the regression statistics for each of the calibrated models. The fifth section describes a validation of the calibrated models. The sixth section describes the proposed predictive models and calibrated CMFs. The seventh section provides a sensitivity analysis of the predictive models over a range of traffic demands. The last section describes some techniques for extending the models to address atypical freeway conditions.

119 Calibration Data The data collection process consisted of a series of activities that culminated in the assembly of a highway safety database suitable for the development of a comprehensive safety prediction methodology for freeways and speed-change lanes. These activities are described Chapter 4. Crash data were identified for each segment and speed-change lane using the most recently available data from the HSIS. Three years of crash data were identified for each segment. The analysis period is 2005, 2006, and 2007 for the California and Washington segments. It is 2004, 2005, and 2006 for the Maine segments. The AADT volume for each year was merged into the assembled database. A total of 8,381 FI crashes and 18,045 PDO crashes are represented in the database. Additional information about the database is provided in Chapter 4. Model Development This section describes the proposed predictive models and the methods used to calibrate them. The following regression model form was used to facilitate the analysis of the combined models. ( ) aggmbiswlwexexenensvsvmvmvj CMFCMFCMFININININN |×××+++= (32) with, ( ) ( )   ≥ < = − ftWIfe ftWIfe CMF l b l Wb lw lw llw 13: 13: 0.1 12 (33) ( )6−= isis Wbisw eCMF (34) ( ) icbbar Wbibibaggmb ePPCMF /| 0.10.1 +−= (35) where, Nj = predicted average crash frequency for model j (j = mv if Imv = 1.0; j = sv if Isv = 1.0; j = en if Ien = 1.0; j = ex if Iex = 1.0); crashes/yr; Nmv = predicted average multiple-vehicle non-entrance/exit crash frequency, crashes/yr; Nsv = predicted average single-vehicle non-entrance/exit crash frequency, crashes/yr; Nen = predicted average ramp-entrance-related crash frequency, crashes/yr; Nex = predicted average ramp-exit-related crash frequency, crashes/yr. Imv = crash indicator variable (= 1.0 if multiple-vehicle non-entrance/exit crash data, 0.0 otherwise); Isv = crash indicator variable (= 1.0 if single-vehicle non-entrance/exit crash data, 0.0 otherwise); Ien = crash indicator variable (= 1.0 if ramp-entrance-related crash data, 0.0 otherwise); Iex = crash indicator variable (= 1.0 if ramp-exit-related crash data, 0.0 otherwise); CMFlw = lane width crash modification factor; Wl = lane width, ft; CMFisw = inside shoulder width crash modification factor; Wis = inside shoulder width, ft; CMFmb|agg = aggregated median barrier crash modification factor;

120 Pib = proportion of segment length with a barrier present in the median (i.e., inside); Wicb = distance from edge of inside shoulder to barrier face, ft; bi = calibration coefficient for condition i (see Table 31); and other variables are previously defined. The final form of the regression model is described here, before the discussion of regression analysis results. However, this form reflects the findings from several preliminary regression analyses where alternative model forms were examined. The form that is described represents that which provided the best fit to the data, while also having coefficient values that are logical and constructs that are theoretically defensible and properly bounded. Equation 32 combines four “component” models. The multiple-vehicle and single-vehicle models apply to basic freeway segments. The two ramp-related models apply to freeway segments with a speed-change lane. The regression model form of each model is described in the following subsections. The CMFs in Equation 32 apply to all component models. The other CMFs that were developed were found to provide a better fit to the data when applied to just one or two of the component models. Freeway Segment - Multiple-Vehicle Non-Entrance/Exit Crash Frequency ( ) ( )hvmvagghcmvaggmwmvagglcmv mvspfmvspfmvspfmvspfmv CMFCMFCMFCMF ININININN ,|,|,|, 1010,,88,,66,,44,, ×××× +++= (36) with, ( ) ruralruralmvmvmv IbAADTbbisegexisegenmvspf eLLLN ,1,4, )000,1/ln(,,,,4,, 5.05.0 ++ −−= (37) ( ) ruralruralmvmvmv IbAADTbbisegexisegenmvspf eLLLN ,1,6, )000,1/ln(,,,,6,, 5.05.0 ++ −−= (38) ( ) ruralruralmvmvmv IbAADTbbisegexisegenmvspf eLLLN ,1,8, )000,1/ln(,,,,8,, 5.05.0 ++ −−= (39) ( ) ruralruralmvmvmv IbAADTbbisegexisegenmvspf eLLLN ,1,10, )000,1/ln(,,,,10,, 5.05.0 ++ −−= (40) ( ) ( )declcdecwevinclcincwevagglcmv ffffCMF ,,,,|, 5.05.0 += (41) ( ) incwevwev LbincwevBincwevBincwev ePPf ,/,,, 0.10.1 +−= (42) ( ) decwevwev LbdecwevBdecwevBdecwev ePPf ,/,,, 0.10.1 +−= (43) [ ] [ ]       −+×         −+= − +− − +− Lb x AADTbXb Lb x AADTbXb inclc x extevextex x entbventbx e Lb e e Lb ef 0.10.1 0.10.1 )000,1/ln( )000,1/ln( , ,, ,, (44)

121 [ ] [ ]       −+×         −+= − +− − +− Lb x AADTbXb Lb x AADTbXb declc x extbvextbx x enteventex e Lb e e Lb ef 0.10.1 0.10.1 )000,1/ln( )000,1/ln( , ,, ,, (45) ( ) ( ) ( )482482|, ,,0.1 −−− +−= icbmwmvismmwmv WbibWWbibaggmwmv ePePCMF (46) ic i i crmvagghcmv PR bCMF , 3 1 2 ,|, 730,50.1  =       += (47) hvhvmv Pb hvmv eCMF ,, = (48) where, Nspf, mv, n = predicted average multiple-vehicle non-entrance/exit crash frequency for number of through lanes n (n = 4, 6, 8, 10); crashes/yr; In = cross section indicator variable (= 1.0 if cross section has n lanes, 0.0 otherwise); CMFmv, lc|agg = aggregated lane change crash modification factor for multiple-vehicle crashes; CMFmv, mw|agg = aggregated median-width crash modification factor for multiple-vehicle crashes; CMFmv, hc|agg = aggregated horizontal curve crash modification factor for multiple-vehicle crashes; CMFmv, hv = high-volume crash modification factor for multiple-vehicle crashes; Irural = area type indicator variable (= 1.0 if area is rural, 0.0 if it is urban); fwev, inc = weaving section adjustment factor for travel in increasing milepost direction; fwev, dec = weaving section adjustment factor for travel in decreasing milepost direction; flc, inc = lane change adjustment factor for travel in increasing milepost direction; flc, dec = lane change adjustment factor for travel in decreasing milepost direction; PwevB, inc = proportion of segment length within a Type B weaving section for travel in increasing milepost direction; PwevB, dec = proportion of segment length within a Type B weaving section for travel in decreasing milepost direction; Lwev, inc = weaving section length for travel in increasing milepost direction (may extend beyond segment boundaries), mi; Lwev, dec = weaving section length for travel in decreasing milepost direction (may extend beyond segment boundaries), mi; Xb, ent = distance from segment begin milepost to nearest upstream entrance ramp gore point, for travel in increasing milepost direction, mi; Xb, ext = distance from segment begin milepost to nearest downstream exit ramp gore point, for travel in decreasing milepost direction, mi; Xe, ent = distance from segment end milepost to nearest upstream entrance ramp gore point, for travel in decreasing milepost direction, mi; Xe, ext = distance from segment end milepost to nearest downstream exit ramp gore point, for travel in increasing milepost direction, mi; AADTb, ent = AADT volume of entrance ramp located at distance Xb, ent, veh/day; AADTb, ext = AADT volume of exit ramp located at distance Xb, ext, veh/day;

122 AADTe, ent = AADT volume of entrance ramp located at distance Xe, ent, veh/day; AADTe, ext = AADT volume of exit ramp located at distance Xe, ext, veh/day; Wm = median width (measured from near edges of traveled way in both travel directions), ft; Ri = radius of curve i, ft; Pc,i = proportion of segment length with curve i; Phv = proportion of AADT during hours where volume exceeds 1,000 veh/h/ln; bi = calibration coefficient for condition i (see Table 31) and other variables are previously defined. The lane change CMF, as described in Equations 41 to 45, is a directional CMF because its terms are derived to apply to specific travel directions along the segment. The constant “0.5” in Equation 41 is used to compute an average of the CMF value for each direction. The weaving section adjustment factor provides a sensitivity to weaving section type. The preliminary regression analysis indicated that the Type B weaving sections were correlated with crash frequency, so the CMF is derived to include this sensitivity. The constant “48” in Equation 46 represents a base median width of 60 ft and a base inside shoulder width of 6 ft (i.e., 48 = 60 - 2×6). Freeway Segment - Single-Vehicle Non-Entrance/Exit Crash Frequency ( ) ( ) ( )aggrssvhvsvagghcsv aggobsvaggocsvaggmwsvaggoswsv nbAADTbb isegexisegensv CMFCMFCMF CMFCMFCMFCMF eLLLN svsvsv |,,|, |,|,|,|, )000,1/ln( ,,,, 2,1,0,5.05.0 ××× ×××× −−= ++ (49) with, ( ) ( ) ( ) ( )10,10,|, ,tan,0.1 −−  +−= scursss WbicWbicaggoswsv ePePCMF (50) ( ) ( ) ( )482482|, ,,0.1 −−− +−= icbmwsvismmwsv WbibWWbibaggmvsv ePePCMF (51) ( ) ( ) ( )2020|, 0.1 −−− +−= ocbocshcoc WbobWWbobaggocsv ePePCMF (52) ( ) ocbbar Wbobobaggobsv ePPCMF /|, 0.10.1 +−= (53) ic i i crsvagghcsv PR bCMF , 3 1 2 ,|, 730,50.1  =       += (54) hvhvsv Pb hvsv eCMF ,, = (55) ( ) ( ) curicicaggrssv fPfPCMF  +−= ,tan,|, 0.1 (56) ( ) ( )tan,tan, 0.1]0.1[5.00.1]0.1[5.0tan rsrs bororbirir ePPePPf +−++−= (57)

123 ( ) ( )currscurrs bororbirircur ePPePPf ,, 0.1]0.1[5.00.1]0.1[5.0 +−++−= (58) where, n = number of through lanes on segment (n = 4, 6, 8, 10); CMFsv, osw|agg = aggregated outside shoulder width crash modification factor for single-vehicle crashes; CMFsv, mw|agg = aggregated median-width crash modification factor for single-vehicle crashes; CMFsv, oc|agg = aggregated outside clearance crash modification factor for single-vehicle crashes; CMFsv, ob|agg = aggregated outside barrier crash modification factor for single-vehicle crashes; CMFsv, hc|agg = aggregated horizontal curve crash modification factor for single-vehicle crashes; CMFsv, hv = high-volume crash modification factor for single-vehicle crashes; CMFsv, rs|agg = aggregated shoulder rumble strip crash modification factor for single-vehicle crashes; Ws = outside shoulder width, ft; Whc = clear zone width, ft; Pob = proportion of segment length with a barrier present on the roadside (i.e., outside); Wocb = distance from edge of outside shoulder to barrier face, ft; ftan = factor for rumble strip presence on tangent portions of the segment; fcur = factor for rumble strip presence on curved portions of the segment; Pir = proportion of segment length with rumble strips present on the inside shoulders; Por = proportion of segment length with rumble strips present on the outside shoulders; bi = calibration coefficient for condition i (see Table 31) and other variables are previously defined. The shoulder rumble strip CMF, as described in Equations 56 to 58, is sensitive to rumble strip location in terms of their use on the outside shoulders, inside shoulders, or both. The constant “0.5” in Equations 57 and 58 is used to compute an average of the CMF value for both locations. The constant “20” in Equation 52 represents a base clear zone width of 30 ft and a base outside shoulder width of 10 ft (i.e., 20 = 30 - 10). Speed-Change Lane - Ramp Entrance Crash Frequency ( ) ( )aggenagghvmvagghcmvaggmwmv IbnbAADTbb isegenen CMFCMFCMFCMF eLN ruralruralenenenen ||,|,|, )000,2/ln( ,, ,2,1,0, ×××× = +++ (59) with, ( ) ( )  ++= isegen AADTbLbIb isegen aggen L eL CMF iradtenienlenenileftleft ,, )000,1/ln(/ ,, | ,,,,, (60) where, CMFen|agg = aggregated ramp entrance crash modification factor; Ileft,i = ramp side indicator variable for ramp i (= 1.0 if entrance or exit is on left side of through lanes, 0.0 if it is on right side); Len, i = length of ramp entrance for ramp i (may extend beyond segment boundaries), mi; AADTr, i = AADT volume of ramp i, veh/day; bi = calibration coefficient for condition i (see Table 31) and other variables are previously defined.

124 This model is derived to predict the total number of ramp-entrance-related crashes on a segment with one or more ramp entrances. This form is dictated by the non-homogeneous character of most freeway segments in the database. Specifically, many segments include only a portion of a speed-change lane. The calibrated version of this model can be algebraically re- written such that it can be applied separately to individual ramp entrance speed-change lanes. Speed-Change Lane - Ramp Exit Crash Frequency ( ) ( )aggenagghvmvagghcmvaggmwmv AADTbb isegexex CMFCMFCMFCMF eLN exex ||,|,|, )000,2/ln( ,, 1,0, ×××× = + (61) with, ( ) ( )  += isegex LbIb isegex aggex L eL CMF iexlenexileftleft ,, / ,, | ,,, (62) where, CMFex|agg = aggregated ramp exit crash modification factor; Lex, i = length of ramp exit for ramp i (may extend beyond segment boundaries), mi; bi = calibration coefficient for condition i (see Table 31); and other variables are previously defined. This model is derived to predict the total number of ramp-exit-related crashes on a segment with one or more ramp exits. This form is dictated by the non-homogeneous character of most freeway segments in the database. Specifically, many segments include only a portion of a speed-change lane. The calibrated version of this model can be algebraically re-written such that it can be applied separately to individual ramp exit speed-change lanes. Barrier Variable Calculations Two key variables that are needed for the evaluation of barrier presence are the inside barrier distance Wicb and the outside barrier distance Wocb. As indicated in Equations 35 and 53, this distance is included as a divisor in the exponential term. This relationship implies that the correlation between distance and crash frequency is an inverse one (i.e., crash frequency decreases with increasing distance). When multiple sections of barrier exist along the segment, a length-weighted average of the reciprocal of the individual distances is needed to properly reflect this inverse relationship. The length used to weight the average is the barrier length. Additional key variables include the proportion of segment length with a barrier present in the median Pib and the proportion of segment length with a barrier present on the roadside Pob. Equations for calculating these proportions and the aforementioned distances are described in the following paragraphs. For segments with a continuous barrier centered in the median (i.e., symmetric median barrier), the following equations are used to estimate Wicb and Pib.

125 ( )ibism iib isiinoff iib icb WWW LL WW L LW −− − + − =  25.0 2 2 , ,, , (63) 0.1=ibP (64) where, Lib, i = length of lane paralleled by inside barrier i (include both travel directions), mi; Wib = inside barrier width (measured from barrier face to barrier face), ft; and Woff, in, i = horizontal clearance from the edge of the traveled way to the face of inside barrier i, ft. The first summation term “∑” in Equation 63 applies to short lengths of barrier in the median. It indicates that the ratio of barrier length Lib to clearance distance Woff, in, i - Wis should be computed for each individual length of barrier that is found in the median along the segment (e.g., a barrier protecting a sign support). The continuous median barrier is not included in this summation. For segments with a continuous barrier adjacent to one roadbed (i.e., asymmetric median barrier), the following equations should be used to estimate Wicb and Pib. nearibism iib isiinoff iib isnear icb WWWW LL WW L WW L LW −−− − + − + − =  2 2 , ,, , (65) 0.1=ibP (66) where, Wnear = “near” horizontal clearance from the edge of the traveled way to the continuous median barrier (measure for both travel directions and use the smaller distance), ft. Similar to the previous guidance, the first summation term “∑” in Equation 65 applies to short lengths of barrier in the median. The ratio of barrier length Lib to the distance Woff,in, i - Wis should be computed for each individual length of barrier that is found in the median along the segment. The continuous median barrier is not included in this summation. For segments with a depressed median and some short sections of barrier in the median (e.g., bridge rail), the following equations should be used to estimate Wicb and Pib.   − = isiinoff iib iib icb WW L L W ,, , , (67) L L P iibib 2 ,= (68)

126 For segments with depressed medians without a continuous barrier or short sections of barrier in the median, the following equation should be used to estimate Pib. 0.0=ibP (69) As suggested by Equation 35, the calculation of Wicb is not required when Pib = 0.0. For segments with barrier on the roadside, the following equations should be used to estimate Wocb and Pob.   − = siooff iob iob ocb WW L L W ,, , , (70) L L P iobob 2 ,= (71) where, Lob, i = length of lane paralleled by outside barrier i (include both travel directions), mi; and Woff, o, i = horizontal clearance from the edge of the traveled way to the face of outside barrier i, ft. For segments without barrier on the roadside, the following equation should be used to estimate Pob. 0.0=obP (72) As suggested by Equation 53, the calculation of Wocb is not required when Pob = 0.0. Statistical Analysis Methods The nonlinear regression procedure (NLMIXED) in the SAS software was used to estimate the proposed model coefficients. This procedure was used because the proposed predictive model is both nonlinear and discontinuous. The log-likelihood function for the negative binomial distribution was used to determine the best-fit model coefficients. Equation 31 was used to define the variance function for all models. The variable L in this equation was adjusted to account for the presence of one or more speed-change lanes on a segment. The procedure was set up to estimate model coefficients based on maximum-likelihood methods. Several statistics were used to assess model fit to the data. One measure of model fit is the Pearson χ2 statistic. This statistic is calculated using the following equation. ( ) = −= n i i ii XV NyX 1 2 2 ][ χ (73) where, n = number of observations.

127 This statistic follows the χ2 distribution with n-p degrees of freedom, where n is the number of observations (i.e., segments) and p is the number of model variables (McCullagh and Nelder, 1983). This statistic is asymptotic to the χ2 distribution for larger sample sizes. The root mean square error se is a useful statistic for describing the precision of the model estimate. It represents the standard deviation of the estimate when each independent variable is at its mean value. This statistic is computed using the following equation. ( ) pn NyX y s n i ii e − − =  =1 2 0.1 (74) where, se = root mean square error of the model estimate, crashes/yr. The scale parameter φ is used to assess the amount of variation in the observed data, relative to the specified distribution. This statistic is calculated by dividing Equation 73 by the quantity n-p. A scale parameter near 1.0 indicates that the assumed distribution of the dependent variable is approximately equivalent to that found in the data (i.e., negative binomial). Another measure of model fit is the coefficient of determination R2. This statistic is commonly used for normally distributed data. However, it has some useful interpretation when applied to data from other distributions when computed in the following manner (Kvalseth, 1985). It is computed using the following equation. SST SSER += 0.12 (75) with, ( ) = −= n i ii NyXSSE 1 2 (76) ( ) = −= n i i XXSST 1 2. (77) where, X. = average crash frequency for all n observations. The last measure of model fit is the dispersion-parameter-based coefficient of determination Rk2. This statistic was developed by Miaou (1996) for use with data that exhibit a negative binomial distribution. It is computed using the following equation. null k k kR −= 0.12 (78) where, knull = overdispersion parameter based on the variance in the observed crash frequency. The null overdispersion parameter knull represents the dispersion in the reported crash frequency, relative to the overall average crash frequency for all segments. This parameter can be

128 obtained using a null model formulation (i.e., a model with no independent variables but with the same error distribution, link function, and offset in years y). Model Calibration The predictive model calibration process was based on regression analysis using combined models, as discussed in the section titled Modeling Approach. With this approach, the component models and CMFs (represented by Equations 32 to 58) are calibrated using a database of common sites. This approach is needed because several CMFs were common to two or more component models. The database assembled for calibration included four replications of the original database. The dependent variable in the first replication was set equal to the multiple-vehicle crashes. The dependent variable in the second replication was set equal to the single-vehicle crashes. That for the third replication was set equal to the ramp-entrance-related crashes and that for the fourth replication was set equal to the ramp-exit-related crashes. The predicted crash frequency from each of the four component models was computed for each segment. The four values were then totaled for each segment and compared with the total reported crash frequency for the segment. The difference between the two totals was then summed for all segments. This sum was found to be very small (i.e., less than 0.5 percent of the total reported crash frequency), so it was concluded that there was no bias in the component models in terms of their ability to predict total crash frequency. The models were calibrated using the California and Washington data. The Maine data were reserved for model validation. The discussion in this section focuses on the findings from the model calibration. The findings from model validation are provided in the next section. The results of the regression model calibration are presented in Table 31. The Pearson χ2 statistic for the model is 4,069, and the degrees of freedom are 4036 (= n − p = 4,076 −40). As this statistic is less than χ2 0.05, 4036 (= 4,185), the hypothesis that the model fits the data cannot be rejected. The t-statistic for each coefficient is listed in the last column of Table 31. These statistics describe a test of the hypothesis that the coefficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 2.0 indicate that the hypothesis can be rejected with the probability of error in this conclusion being less than 0.05. For those few variables where the absolute value of the t-statistic is smaller than 2.0, it was decided that the variable was important to the model and its trend was found to be intuitive and, where available, consistent with previous research findings (even if the specific value was not known with a great deal of certainty as applied to this database).

129 TABLE 31. Freeway FI model statistical description–combined model–two states Model Statistics Value R2: 0.66 Scale parameter φ: 1.01 Pearson χ2: 4,069 (χ20.05, 4036 = 4,185) Observations no: 1,644 seg. (7,379 injury or fatal crashes in 3 years) 584 ramp entrances, 462 ramp exits Calibrated Coefficient Values Variable Inferred Effect of... Value Std. Dev. t-statistic bmv, cr Horizontal curvature on 2+ veh. crashes 0.0187 0.0069 2.7 bsv, cr Horizontal curvature on 1 veh. crashes 0.0847 0.0162 5.2 blw Lane width -0.0147 0.0374 -0.4 bs, tan Outside shoulder width on 1 veh. crashes, tangents -0.0618 0.0175 -3.5 bs, cur Outside shoulder width on 1 veh. crashes, curves -0.0850 0.0249 -3.4 bis Inside shoulder width -0.0161 0.0062 -2.6 brs, tan Shoulder rumble strip on 1 veh. crashes, tangents -0.168 0.0782 -2.1 brs, cur Shoulder rumble strip on 1 veh. crashes, curves 0.293 0.1185 2.5 bmv, mw Median width on 2+ veh. crashes -0.00335 0.0009 -3.6 bsv, mw Median width on 1 veh. crashes 0.00042 0.0007 0.6 bbar Barrier presence 0.1023 0.0496 2.1 boc Outside clearance on 1 veh. crashes -0.00197 0.0054 -0.4 bwev Type B weaving section presence on 2+ veh. crashes 0.179 0.0549 3.3 bv Ramp AADT on lane-change-related crashes -0.294 0.1507 -2.0 bx Distance from ramp on lane-change-related crashes 13.876 4.8850 2.8 bleft Left side entrance or exit on speed-change lane crashes 0.487 0.5610 0.9 ben, adt Ramp AADT on ramp-entrance-related crashes 0.202 0.0754 2.7 ben, len Ramp entrance length on related crashes 0.0283 0.0150 1.9 bex, len Ramp exit length on related crashes 0.0103 0.0108 1.0 bmv, hv High-volume conditions on 2+ veh. crashes 0.238 0.1515 1.6 bsv, hv High-volume conditions on 1 veh. crashes -0.0804 0.1179 -0.7 bmv, 4 4 lanes on 2+ vehicle crashes in urban areas -6.000 0.536 -11.2 bmv, 6 6 lanes on 2+vehicle crashes in urban areas -6.103 0.600 -10.2 bmv, 8 8 lanes on 2+ vehicle crashes in urban areas -6.192 0.649 -9.5 bmv, 10 10 lanes on 2+ vehicle crashes in urban areas -6.416 0.692 -9.3 bmv, 1 AADT on 2+ vehicle crashes 1.620 0.141 11.5 bmv, rural Added effect of rural area type on 2+ veh. crashes -0.582 0.062 -9.3 bsv, 0 1 veh. crashes -2.499 0.248 -10.1 bsv, 2 Number of lanes on 1 veh. crashes 0.0328 0.023 1.4 bsv, 1 AADT on 1 veh. crashes 0.700 0.080 8.7 ben, 0 Ramp-entrance crashes in urban areas -3.331 0.556 -6.0 ben, 2 Number of lanes on ramp-ent. crashes in urban areas -0.142 0.052 -2.7 ben, 1 AADT on ramp-entrance crashes 1.248 0.202 6.2 ben, rural Added effect of rural area type on ramp-ent. crashes -0.191 0.154 -1.2 bex, 0 Ramp exit crashes -3.017 0.626 -4.8 bex, 1 AADT on ramp exit crashes 1.010 0.157 6.4

130 The findings from an examination of the coefficient values on the corresponding CMF or SPF predictions are documented in a subsequent section. In general, the sign and magnitude of the calibration coefficients in Table 31 are logical and consistent with previous research findings. An indicator variable for the state of California was included in an initial version of the regression model. The coefficient for this variable was very small and not statistically significant. This finding is evidence that the combined-model form is able to explain differences in crash occurrence among the two states. Model Validation Model validation was a two-step process. The first step required using the calibrated models to predict the crash frequency for sites from a third state (i.e., Maine). The objective of this step was to demonstrate the robustness of the model structure and its transferability to another state. The second step required comparing the calibrated CMFs with similar CMFs reported in the literature, where such information was available. The objective of this step was to demonstrate that the calibrated CMFs were consistent with previous research findings. The findings from the first step of the validation process are described in this section. Those from the second step are described in the next section. The first step of the validation process consisted of several tasks. The first task was to quantify the local calibration factor for each of the four models (i.e., Cmv, Csv, Cen, Cex), which would be the first step for any agency using the HSM methodology. This produced a “re- calibrated” set of four models (i.e., the models with the coefficients from Table 31 plus the local calibration factors). The local calibration factor values for the Maine data are provided in the list below: ● Calibration factor for multiple-vehicle non-entrance/exit crashes, Cmv = 1.27 ● Calibration factor for single-vehicle non-entrance/exit crashes, Csv = 1.08 ● Calibration factor for ramp-entrance-related crashes, Cen = 0.89 ● Calibration factor for ramp-exit-related crashes, Cex = 0.95 The second task was to apply the re-calibrated models to the Maine data to compute the predicted average crash frequency for each segment or speed-change lane (i.e., Nmv, Nsv, Nen, Nex). The predicted crash frequency was then compared to the reported crash frequency for each site. The third task was to compute the fit statistics and assess the robustness of the calibrated model. These statistics are listed in Table 32. The Pearson χ2 statistic for each component model, and for the overall model, is less than χ20.05 so the hypothesis that the model fits the validation data cannot be rejected.

131 TABLE 32. Freeway model validation statistics Component Model R 2 Rk2 Scale Parameter φ Pearson χ2 Deg. of Freedom χ20.05, n - 1 Multiple-veh. non-entrance/exit 0.57 0.89 1.14 228.4 201 235.1 Single-veh. non-entrance/exit 0.35 0.92 1.11 224.9 202 236.2 Ramp-entrance-related 0.28 0.97 1.20 68.3 57 75.6 Ramp-exit-related 0.41 0.70 0.60 38.3 64 83.7 Overall: 0.45 1.06 559.8 527 581.5 The findings from this validation step indicate that the trends in the Maine data are not significantly different from those in the California and Washington data. These findings also suggest that the model structure is transferable to other states (when locally calibrated) for the prediction of FI crash frequency. Based on these findings, the data for the three states were combined and used in a second regression model calibration. The larger sample size associated with the combined database reduced the standard error of several calibration coefficients. Bared and Zhang (2007) also used this approach in their development of predictive models for urban freeways. Combined Model The data from the three study states were combined and the predictive models were calibrated a second time using the combined data. The calibration coefficients for the four models are described in the next subsection. The subsequent four subsections describe the fit of each component model. The fit statistics were separately computed using the calibrated component model and an analysis of its residuals. Aggregate Model The results of the regression model calibration are presented in Table 33. The Pearson χ2 statistic for the model is 4,574, and the degrees of freedom are 4,568 (= n − p = 4,604 −36). As this statistic is less than χ2 0.05, 4568 (= 4,726), the hypothesis that the model fits the data cannot be rejected. Several segments were removed as a result of outlier analysis such that the calibration database included only 8,038 of the 8,381 crashes identified in Chapter 4. The t-statistic for each coefficient is listed in the last column of Table 33. These statistics have generally increased, relative to their counterparts in Table 31, as a result of the increased sample size. With a few exceptions, these statistics have an absolute value that is larger than 2.0, which indicates that the null hypothesis can be rejected with the probability of error in this conclusion being less than 0.05. For those few variables where the absolute value of the t- statistic is smaller than 2.0, it was decided that the variable was important to the model and its trend was found to be intuitive and, where available, consistent with previous research findings (even if the specific value was not known with a great deal of certainty as applied to this database). This consistency is demonstrated in a subsequent section.

132 TABLE 33. Freeway FI model statistical description–combined model–three states Model Statistics Value R2: 0.65 Scale parameter φ: 1.00 Pearson χ2: 4,574 (χ20.05, 4568 = 4,726) Observations no: 1,850 seg. (8,038 injury or fatal crashes in 3 years) 642 ramp entrances, 527 ramp exits Calibrated Coefficient Values Variable Inferred Effect of... Value Std. Dev. t-statistic bmv, cr Horizontal curvature on 2+ veh. crashes 0.0172 0.0064 2.7 bsv, cr Horizontal curvature on 1 veh. crashes 0.0719 0.0144 5.0 blw Lane width -0.0376 0.0358 -1.1 bs, tan Outside shoulder width on 1 veh. crashes, tangents -0.0647 0.0163 -4.0 bs, cur Outside shoulder width on 1 veh. crashes, curves -0.0897 0.0248 -3.6 bis Inside shoulder width -0.0172 0.0060 -2.9 brs, tan Shoulder rumble strip on 1 veh. crashes, tangents -0.209 0.0724 -2.9 brs, cur Shoulder rumble strip on 1 veh. crashes, curves 0.274 0.1039 2.6 bmv, mw Median width on 2+ veh. crashes -0.00302 0.0009 -3.5 bsv, mw Median width on 1 veh. crashes 0.00102 0.0006 1.6 bbar Barrier presence 0.131 0.0478 2.7 boc Outside clearance on 1 veh. crashes -0.00451 0.0052 -0.9 bwev Type B weaving section presence on 2+ veh. crashes 0.175 0.0519 3.4 bv Ramp AADT on lane-change-related crashes -0.272 0.1421 -1.9 bx Distance from ramp on lane-change-related crashes 12.561 3.5232 3.6 bleft Left side entrance or exit on speed-change lane crashes 0.594 0.5440 1.1 ben, adt Ramp AADT on ramp-entrance-related crashes 0.198 0.0730 2.7 ben, len Ramp entrance length on related crashes 0.0318 0.0147 2.2 bex, len Ramp exit length on related crashes 0.0116 0.0095 1.2 bmv, hv High-volume conditions on 2+ veh. crashes 0.350 0.1350 2.6 bsv, hv High-volume conditions on 1 veh. crashes -0.0675 0.1135 -0.6 bmv, 4 4 lanes on 2+ vehicle crashes in urban areas -5.470 0.426 -12.9 bmv, 6 6 lanes on 2+vehicle crashes in urban areas -5.587 0.484 -11.5 bmv, 8 8 lanes on 2+ vehicle crashes in urban areas -5.635 0.525 -10.7 bmv, 10 10 lanes on 2+ vehicle crashes in urban areas -5.842 0.562 -10.4 bmv, 1 AADT on 2+ vehicle crashes 1.492 0.115 13.0 bmv, rural Added effect of rural area type on 2+ veh. crashes -0.505 0.058 -8.7 bsv, 0 1 veh. crashes -2.266 0.219 -10.4 bsv, 2 Number of lanes on 1 veh. crashes 0.0351 0.022 1.6 bsv, 1 AADT on 1 veh. crashes 0.646 0.071 9.1 ben, 0 Ramp-entrance crashes in urban areas -3.194 0.500 -6.4 ben, 2 Number of lanes on ramp-ent. crashes in urban areas -0.130 0.050 -2.6 ben, 1 AADT on ramp-entrance crashes 1.173 0.187 6.3 ben, rural Added effect of rural area type on ramp-ent. crashes -0.180 0.146 -1.2 bex, 0 Ramp exit crashes -2.679 0.488 -5.5 bex, 1 AADT on ramp exit crashes 0.903 0.132 6.8

133 Indicator variables were included for the states of California and Maine in the regression model. The coefficient for each variable was very small and not statistically significant. This finding is evidence that the combined-model form is able to explain differences in crash occurrence among states. Model for Predicting Multiple-Vehicle Non-Ramp-Related Crash Frequency The results of the multiple-vehicle model calibration are presented in Table 34. The Pearson χ2 statistic for the model is 1,659, and the degrees of freedom are 1,577 (= n − p = 1,591 −14). As this statistic is less than χ2 0.05, 1577 (= 1,670), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.66. An alternative measure of model fit that is better suited to the negative binomial distribution is Rk2. The Rk2 for the calibrated model is 0.92. TABLE 34. Freeway FI model statistical description–multiple-vehicle model–three states Model Statistics Value R2 (Rk2): 0.66 (0.92) Scale parameter φ: 1.05 Pearson χ2: 1,659 (χ20.05, 1577 = 1,670) Inverse dispersion parameter K: 17.6 mi-1 Observations no: 1,591 segments (4,392 injury or fatal crashes in 3 years) Standard deviation se: ±1.07 crashes/yr The inverse dispersion parameter is relatively large when compared to that for other models reported in the literature. This trend implies that there is less unexplained site-to-site variability in the predicted mean crash frequency for groups of similar sites. There are several reasons for this trend. Those reasons that tend to reduce the overall variability in the database include: ● The data is cross-sectional such that each independent variable value is averaged for each site over the analysis period. In contrast, with panel data, each independent variable value is measured for each site for each year in the analysis period. Thus, cross-sectional data inherently have less variability than panel data when the year-to-year variability in the independent variables is largely random (Lord and Park, 2008). Of particular note is the random variability in the AADT data found in most highway safety databases, as discussed previously in the section titled Modeling Approach. ● The manual collection of geometric variables using aerial photographs for this project was found to significantly reduce the variability of these variables, relative to that found in the equivalent variables in highway safety databases obtained from state agencies. For example, the standard deviation of lane width in one state database is 2.0 ft while that in the manually assembled data for the same segments is 0.6 ft. ● The researchers that assembled the database for this project intentionally removed exceptionally short and exceptionally long segments for reasons of statistical control. This approach results in less variability in the database.

134 0 5 10 15 20 25 30 35 40 0 10 20 30 40 50 Predicted Injury + Fatal Crash Frequency, cr/3 yrs R ep or te d C ra sh F re qu en cy , cr /3 y rs Each data point represents an average of 10 sites. 1 1 An additional reason for the relatively large inverse dispersion parameter value stems from this project’s development of a “full”model (i.e., one with multiple variables). This type of model inherently explains more variability than a simple model (e.g., an AADT-to-a-power model), and results in a larger inverse dispersion parameter (Mitra and Washington, 2007). The coefficients in Table 33 were combined with Equations 37 to 40 to obtain the calibrated SPFs for multiple-vehicle non-entrance/exit crashes. The form of each model is described in the following equations. ( ) ruralIAADTisegexisegenmvspf eLLLN 505.0)000,1/ln(492.1470.5,,,,4,, 5.05.0 −+− −−= (79) ( ) ruralIAADTisegexisegenmvspf eLLLN 505.0)000,1/ln(492.1587.5,,,,6,, 5.05.0 −+− −−= (80) ( ) ruralIAADTisegexisegenmvspf eLLLN 505.0)000,1/ln(492.1635.5,,,,8,, 5.05.0 −+− −−= (81) ( ) ruralIAADTisegexisegenmvspf eLLLN 505.0)000,1/ln(492.1842.5,,,,10,, 5.05.0 −+− −−= (82) The calibrated CMFs used with these SPFs are described in a subsequent section. The fit of the calibrated models is shown in Figure 44. This figure compares the predicted and reported crash frequency in the calibration database. The trend line shown represents a “y = x” line. A data point would lie on this line if its predicted and reported crash frequency were equal. The data points shown represent the reported multiple-vehicle non-entrance/exit crash frequency for the segments used to calibrate the corresponding component model. Figure 44. Predicted vs. reported multiple-vehicle freeway FI crashes. Each data point shown in Figure 44 represents the average predicted and average reported crash frequency for a group of 10 segments. The data were sorted by predicted crash frequency

135 to form groups of segments with similar crash frequency. The purpose of this grouping was to reduce the number of data points shown in the figure and, thereby, to facilitate an examination of trends in the data. The individual segment observations were used for model calibration. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency for segments experiencing up to 40 multiple-vehicle crashes in a three- year period. Model for Predicting Single-Vehicle Non-Ramp-Related Crash Frequency The results of the single-vehicle model calibration are presented in Table 35. The Pearson χ2 statistic for the model is 1,836, and the degrees of freedom are 1,837 (= n − p = 1,850 −13). As this statistic is less than χ2 0.05, 1837 (= 1,938), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.65. The Rk2 for the calibrated model is 0.93. TABLE 35. Freeway FI model statistical description–single-vehicle model–three states Model Statistics Value R2 (Rk2): 0.65 (0.93) Scale parameter φ: 0.99 Pearson χ2: 1,836 (χ20.05, 1837 = 1,938) Inverse dispersion parameter K: 30.1 mi-1 Observations no: 1,850 segments (2,762 injury or fatal crashes in 3 years) Standard deviation se: ±0.51 crashes/yr The inverse dispersion parameter is relatively large when compared to that for other models reported in the literature. The reasons for this trend were identified in the discussion associated with Table 34. The coefficients in Table 33 were combined with Equation 49 to obtain the calibrated SPF for single-vehicle non-entrance/exit crashes. The form of this model is described in the following equation. ( ) nAADTisegexisegensvspf eLLLN 0351.0)000,1/ln(646.0266.2,,,,, 5.05.0 ++− −−= (83) The calibrated CMFs used with this SPF are described in a subsequent section. The fit of the calibrated model is shown in Figure 45. This figure compares the predicted and reported crash frequency in the calibration database. Each data point shown represents the average predicted and average reported crash frequency for a group of 10 segments. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency for segments experiencing up to 15 single-vehicle crashes in a three-year period.

136 0 2 4 6 8 10 12 14 16 0 5 10 15 20 Predicted Injury + Fatal Crash Frequency, cr/3 yrs R ep or te d C ra sh F re qu en cy , cr /3 y rs Each data point represents an average of 10 sites. 1 1 Figure 45. Predicted vs. reported single-vehicle freeway FI crashes. Model for Predicting Ramp-Entrance-Related Crash Frequency The results of the ramp-entrance-related model calibration are presented in Table 36. The Pearson χ2 statistic for the model is 597, and the degrees of freedom are 630 (= n − p = 642 −12). As this statistic is less than χ2 0.05, 630 (= 690), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.50. The Rk2 for the calibrated model is 0.96. TABLE 36. Freeway FI model statistical description–ramp entrance model–three states Model Statistics Value R2 (Rk2): 0.50 (0.96) Scale parameter φ: 0.93 Pearson χ2: 597 (χ20.05, 630 = 690) Inverse dispersion parameter K: 26.1 mi-1 Observations no: 642 ramp entrances (624 injury or fatal crashes in 3 years) Standard deviation se: ±0.47 crashes/yr The coefficients in Table 33 were combined with Equation 59 to obtain the calibrated SPF for ramp-entrance-related crashes. The form of this model is described in the following equation. ( ) ruralInAADTenenspf eLN 180.0130.0)000,2/ln(173.1194.3, −−+−= (84) The calibrated CMFs used with this SPF are described in a subsequent section. This SPF is applied to a ramp entrance speed-change lane, as shown in Figure 38. The “segment” length is equal to the ramp entrance length Len, which is measured using the gore and taper points identified in Figure 11.

137 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 2.0 4.0 6.0 8.0 Predicted Injury + Fatal Crash Frequency, cr/3 yrs R ep or te d C ra sh F re qu en cy , cr /3 y rs Each data point represents an average of 10 sites. 1 1 The fit of the calibrated model is shown in Figure 46. This figure compares the predicted and reported crash frequency in the calibration database. Each data point shown represents the average predicted and average reported crash frequency for a group of 10 segments. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency for segments experiencing up to 8.0 ramp-entrance-related crashes in a three- year period. Figure 46. Predicted vs. reported ramp-entrance-related FI crashes. Model for Predicting Ramp-Exit-Related Crash Frequency The results of the ramp-exit-related model calibration are presented in Table 37. The Pearson χ2 statistic for the model is 481, and the degrees of freedom are 518 (= n − p = 527 −9). As this statistic is less than χ2 0.05, 518 (= 572), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.41. The Rk2 for the calibrated model is 0.95. The inverse dispersion parameter for this model is dimensionless because crash frequency variance was found to be insensitive to ramp exit length (i.e., the variable L was removed from Equation 31). TABLE 37. Freeway FI model statistical description–ramp exit model–three states Model Statistics Value R2 (Rk2): 0.41 (0.95) Scale parameter φ: 0.91 Pearson χ2: 481 (χ20.05, 518 = 572) Inverse dispersion parameter K: 1.78 Observations no: 527 ramp exits (624 injury or fatal crashes in 3 years) Standard deviation se: ±0.47 crashes/yr

138 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 1.0 2.0 3.0 4.0 5.0 Predicted Injury + Fatal Crash Frequency, cr/3 yrs R ep or te d C ra sh F re qu en cy , cr /3 y rs Each data point represents an average of 10 sites. 1 1 The coefficients in Table 33 were combined with Equation 61 to obtain the calibrated SPF for ramp-exit-related crashes. The form of this model is described in the following equation. ( ) )000,2/ln(903.0679.2, AADTexexspf eLN +−= (85) The calibrated CMFs used with this SPF are described in a subsequent section. This SPF is applied to a ramp exit speed-change lane, as shown in Figure 38. The “segment” length is equal to the ramp exit length Lex, which is measured using the gore and taper points identified in Figure 11. The fit of the calibrated models is shown in Figure 47. This figure compares the predicted and reported crash frequency in the calibration database. Each data point shown represents the average predicted and average reported crash frequency for a group of 10 segments. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency for segments experiencing up to 4.0 ramp-exit-related crashes in a three-year period. Figure 47. Predicted vs. reported ramp-exit-related FI crashes. Calibrated CMFs Several CMFs were calibrated in conjunction with the SPFs. All of them were calibrated using FI crash data. Collectively, they describe the relationship between various geometric factors and crash frequency. These CMFs are described in this section and, where possible, compared with the findings from previous research as means of model validation. Many of the CMFs found in the literature are typically derived from (and applied to) the combination of multiple-vehicle and single-vehicle crashes. That is, one CMF is used to indicate the influence of a specified geometric feature on total crashes. In contrast, the models developed for this project include several CMFs that are calibrated for a specific crash type. In these

139 instances, Equation 86 is used to facilitate a comparison of the CMFs reported in the literature with those developed for this project. Specifically, this equation is used to convert the CMFs developed for a specific crash type to one that applies to total crashes. ( ) imvmvisvmvaggi CMFPCMFPCMF ,,| 0.1 +−= (86) where, CMFi|agg = aggregated CMF for element i; Pmv = proportion of multiple-vehicle crashes; and CMFj, i = crash modification factor for element i and crash type j (j = mv, sv). The proportion of multiple-vehicle crashes used in this equation is obtained from Table 38. The data in this table were obtained from the study state databases. TABLE 38. Distribution of FI crashes on freeways Area Type Number of Through Lanes Multiple-Vehicle Non-Entrance/Exit FI Crashes (MV) Single-Vehicle Non- Entrance/Exit FI Crashes Proportion MV Crashes Rural 4 326 573 0.363 6 432 434 0.499 8 222 269 0.452 Urban 4 456 316 0.591 6 947 567 0.625 8 1,382 393 0.779 10 888 235 0.791 Horizontal Curve CMF. The calibrated horizontal curve CMF has two forms, depending on which component model is being used. The CMF for multiple-vehicle non- entrance/exit crashes, ramp-entrance-related crashes, and ramp exit related crashes is described using the following equation. ic m i i agghcmv PR CMF , 1 2 |, 730,50172.00.1  =       += (87) The CMF for single-vehicle crashes is described using the following equation. ic m i i agghcsv PR CMF , 1 2 |, 730,50719.00.1  =       += (88) These two CMFs are derived to be applicable to a segment that has a mixture of uncurved and curved lengths. The variable Pc, i is computed as the ratio of the length of curve i on the segment to the length of the segment. For example, consider a segment that is 0.5 mi long and a curve that is 0.2 mi long. If one-half of the curve is on the segment, then Pc, i = 0.20 (= 0.1/0.5). In fact, this proportion is the same regardless of the curve’s length (provided that it is 0.1 mi or longer and 0.1 mi of this curve is located on the segment).

140 1.0 1.1 1.2 1.3 1.4 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000 Curve Radius, ft C ra sh M od ifi ca tio n Fa ct or . Two-Lane Highway Harwood et al. (2000) Def. Angle = 20 deg. 60 mi/h, Bonneson and Pratt (2009) 55 mi/h Rural 4-lane, proposed Urban 4-lane, proposed Rural 6-lane, proposed Urban 6-lane, proposed The combined horizontal curve CMF is shown in Figure 48 using a series of thick, solid trend lines. Equation 86 was used to create these trend lines. They represent different combinations of area type and through lanes through their association with the different proportions of multiple-vehicle crashes in Table 38. The radii used to calibrate this CMF range from 1,500 to 12,000 ft. The base condition for this CMF is an uncurved (i.e., tangent) segment. Figure 48. Calibrated freeway horizontal curve CMF for FI crashes. Also shown in Figure 48 are CMFs developed by other researchers. Thin lines are used to differentiate these CMFs from those developed for this research project. The correlation between curve radius and crash frequency is consistent among all CMFs. The sensitivity to speed found in the model by Bonneson and Pratt (2009) suggests that the CMFs developed for this project reflect speeds of 60 to 65 mi/h. Speed data were not available in the study state databases. Lane Width CMF. The lane width CMF is described using the following equation. ( )    ≥ < = −− ftWIf ftWIfe CMF l l W lw l 13:963.0 13:120376.0 (89) The lane width used in this CMF is an average for all through lanes on the segment. The CMF is discontinuous, breaking at a lane width of 13 ft. Widths of 13 ft to 14 ft were not found to have a correlation with crash frequency. Collectively, the crash data for segments with lane widths in excess of 13 ft tended to be insensitive to lane width. These segments represent about 4.5 percent of the freeway segments evaluated. The lane width CMF is shown in Figure 49 using a thick, solid trend line. The lane widths used to calibrate this CMF range from 10.5 to 14 ft. The base condition for this CMF is a 12-ft lane width.

141 0.90 0.95 1.00 1.05 1.10 1.15 10 11 12 13 14 Lane Width, ft C ra sh M od ifi ca tio n Fa ct or Urban, Bonneson and Pratt (2009) Rural, Bonneson and Pratt (2009) proposed - Rural multilane divided highway (Highway, 2010) Figure 49. Calibrated freeway lane width CMF for FI crashes. Also shown in Figure 49 are CMFs developed by other researchers. Thin lines or data points are used to differentiate these CMFs from those developed for this research project. The correlation between lane width and crash frequency is consistent among all CMFs. Outside Shoulder Width CMF. The outside shoulder width CMF is described using the following equation. ( ) ( ) ( ) ( )100897.0,100647.0,|, 0.1 −−−−  +−= ss WicWicaggoswsv ePePCMF (90) This CMF is applicable to single-vehicle crashes. The regression analysis indicated that the outside shoulder width had an insignificant correlation with multiple-vehicle crashes. The shoulder width used in this CMF is an average for both directions of travel. The variable Pc,i is computed as the ratio of the length of curve i on the segment to the length of the segment. The combined outside shoulder width CMF for uncurved segments is shown in Figure 50 using a series of thick, solid trend lines. Equation 86 was used to create this CMF (with CMFmv,osw = 1.0). It represents the likely correlation between outside shoulder width and total crash frequency. The trend lines represent different combinations of area type and through lanes through their association with the different proportions of multiple-vehicle crashes in Table 38. The shoulder widths used to calibrate this CMF range from 6 to 14 ft. The base condition for this CMF is a 10-ft shoulder width.

142 0.9 1.0 1.1 1.2 6 7 8 9 10 11 12 Outside Shoulder Width, ft C ra sh M od ifi ca tio n Fa ct or . Rural Access Controlled Highway Urban Access Controlled Highway (Harkey et al., 2008) - Rural multilane divided highway (Highway, 2010) Rural 4-lane (no curve), proposed Urban (no curve), proposed Rural 6-lane (no curve), proposed Figure 50. Calibrated freeway outside shoulder width CMF for FI crashes. Also shown in Figure 50 are CMFs developed by other researchers. Thin lines or data points are used to differentiate these CMFs from those developed for this research project. The correlation between outside shoulder width and crash frequency is consistent among all CMFs. Inside Shoulder Width CMF. The inside shoulder width CMF is described using the following equation. ( )60172.0 −−= isWisw eCMF (91) The shoulder width used in this CMF is an average for both directions of travel. The inside shoulder width CMF is shown in Figure 51 using a thick, solid trend line. The shoulder widths used to calibrate this CMF range from 2 to 11 ft. The base condition for this CMF is a 6-ft shoulder width. Also shown in Figure 51 are CMFs developed by other researchers. Thin lines are used to differentiate these CMFs from those developed for this research project. The correlation between inside shoulder width and crash frequency is consistent among all CMFs.

143 0.8 0.9 1.0 1.1 1.2 2 4 6 8 10 Inside Shoulder Width, ft C ra sh M od ifi ca tio n Fa ct or . Principal Arterial Highways Milton and Mannering (1998) Rural Freeway (Hadi et al., 1995) Rural, Bonneson and Pratt (2009) Urban Proposed Figure 51. Calibrated freeway inside shoulder width CMF for FI crashes. Median Width CMF. The calibrated median width CMF has two forms, depending on which component model is being used. The CMF for multiple-vehicle non-entrance/exit crashes, ramp-entrance-related crashes, and ramp exit related crashes is described using the following equation. ( ) ( ) ( )48200302.048200302.0|, 0.1 −−−−− +−= icbism WibWWibaggmwmv ePePCMF (92) The CMF for single-vehicle crashes is described using the following equation. ( ) ( ) ( )48200102.048200102.0|, 0.1 −−− +−= icbism WibWWibaggmvsv ePePCMF (93) The median width used in either CMF is an average for the segment. These two CMFs are derived to be applicable to a segment that has median barrier present along some portion of the segment. Guidance for computing the variables Pib and Wicb was provided previously in the subsection titled Barrier Variable Calculations. The sign of the calibration coefficients in Equation 92 indicates that multiple-vehicle crash frequency decreases with an increase in median width. However, the coefficients in Equation 93 indicate that single-vehicle crash frequency increases slightly with an increase in median width. This latter trend reflects the fact an errant vehicle is more likely to have a single- vehicle crash with a wide median and a multiple-vehicle crash with a narrow median. Considering both single-vehicle and multiple-vehicle crashes together, the combined CMF indicates that the total number of FI crashes decreases with an increase in median width. The combined median-width CMF is shown in Figure 52 using the thick, solid trend line labeled “No barrier, proposed.” Equation 86 was used to create this CMF. The trend line shown represents an urban freeway. The slope of the line is slightly flatter for a rural freeway. The median widths used to calibrate this CMF range from 9 to 140 ft. The base condition for this

144 0.90 0.95 1.00 1.05 1.10 1.15 1.20 20 30 40 50 60 70 80 Median Width, ft C ra sh M od ifi ca tio n Fa ct or . Barrier in center of median, proposed - No barrier, rural multilane divided highway (Highway, 2010) (adjusted for 60-ft base median width) No barrier, Bonneson and Pratt (2009) No barrier, proposed Barrier in center of median, Bonneson and Pratt (2009) CMF is a 60-ft median width and an inside shoulder width of 6.0 ft. The trend lines labeled “Barrier in center...” are discussed with the next CMF. Figure 52. Calibrated freeway median-width CMF for FI crashes. Also shown in Figure 52 are CMFs developed by other researchers. Thin lines or data points are used to differentiate these CMFs from those developed for this research project. The correlation between median width and crash frequency is consistent among all CMFs. Median Barrier CMF. The median barrier CMF is described using the following equation. ( ) icbWibibaggmb ePPCMF /131.0| 0.10.1 +−= (94) Guidance for computing the variables Pib and Wicb was provided previously in the subsection titled Barrier Variable Calculations. The variable Wicb (representing the distance from the edge of inside shoulder to median barrier face) ranges in value from 1.0 to 17 ft in the database. The base condition for this CMF is no barrier. This median barrier CMF is shown in Figure 52. The relevant trend lines are labeled “Barrier in center...”. The values shown represent the median barrier CMF multiplied by the median-width CMF because both CMFs are sensitive to the variable Wicb. The trend line shown is for urban freeways. It drops slightly for rural freeways. The calibration coefficient in Table 33 for this CMF is 0.131. However, research by Bonneson and Pratt (2008) using Texas data found a coefficient value of 0.890 for an identical CMF. There is a significant difference between these two coefficients, where the smaller value suggests that barrier presence produces a smaller CMF value. This difference can also be seen in Figure 52 by comparing the two trend lines associated with barrier in the center of the median. The trend line attributed to Bonneson and Pratt (2009) is based on rigid and semi-rigid barrier on

145 Texas freeways. In contrast, the other barrier-related trend line is based on a mixture of rigid, semi-rigid, and cable barrier types in the study states. It is possible that some of the difference between the two trend lines reflects differences in barrier design between Texas and the study states. Shoulder Rumble Strip CMF. The shoulder rumble strip CMF is described using the following equation. ( ) ( ) curicicaggrssv fPfPCMF  +−= ,tan,|, 0.1 (95) ( ) ( )811.00.1]0.1[5.0811.00.1]0.1[5.0tan ororirir PPPPf +−++−= (96) ( ) ( )32.10.1]0.1[5.032.10.1]0.1[5.0 ororirircur PPPPf +−++−= (97) This CMF is applicable to single-vehicle crashes. The regression analysis indicated that shoulder rumble strip presence had an insignificant correlation with multiple-vehicle crashes. The proportion Pir represents the proportion of the segment length with rumble strips present on the inside shoulders. It is computed by summing the length of roadway with rumble strips on the inside shoulder in both travel directions and dividing by twice the segment length. The proportion Por represents the proportion of the segment length with rumble strips present on the outside shoulders. It is computed by summing the length of roadway with rumble strips on the outside shoulder in both travel directions and dividing by twice the segment length. The constant “0.811” in Equation 96 represents the calibration coefficient after conversion. It corresponds to a CMF value of 0.811 for FI single-vehicle crashes on uncurved (i.e., tangent) road segments when shoulder rumble strips are continuously present. This value compares favorably with the CMF value of 0.84 recommended by Torbic et al. (2009) for crashes of the same type and severity on rural freeways. The constant “1.32” in Equation 97 represents the calibration coefficient after conversion. It corresponds to a CMF value of 1.32 for FI single-vehicle crashes on curved road segments when shoulder rumble strips are continuously present. It suggests that there are 32 percent more crashes on curved road segments when rumble strips are present. A review of the literature on the safety effect of shoulder rumble strips on curves did not reveal any evidence that could support or refute this finding. In related research, Torbic et al. (2009) examined the safety effect of centerline rumble strips on rural two-lane highways in three states. They found that centerline rumble strips increased total crashes 3.5 percent on curved segments (although this result was not statistically significant). Their examination of FI crash frequency for the same curves showed a range of results—crashes were reduced by 36.7 percent in one state but increased 9.8 percent in another state. Outside Clearance CMF. The calibrated outside clearance CMF is described using the following equation. ( ) ( ) ( )2000451.02000451.0|, 0.1 −−−−− +−= ocbshc WobWWobaggocsv ePePCMF (98) This CMF is applicable to single-vehicle crashes. The regression analysis indicated that outside clearance had an insignificant correlation with multiple-vehicle crashes. The clear zone

146 1.00 1.02 1.04 1.06 1.08 1.10 10 15 20 25 30 Clear Zone Width, ft C ra sh M od ifi ca tio n Fa ct or . Rural 4-lane, Bonneson and Pratt (2009) Rural 6-lane 10-ft outside shoulder width Rural 4-lane, proposed Rural 6-lane, proposed Roadside has barrier 2-ft from shoulder edge for 100 percent of segment, proposed 1.00 1.02 1.04 1.06 1.08 1.10 10 15 20 25 30 Clear Zone Width, ft C ra sh M od ifi ca tio n Fa ct or . Urban, Bonneson and Pratt (2009) 10-ft outside shoulder width Roadside has barrier 2-ft from shoulder edge for 100 percent of segment, proposed Urban, proposed width used in this CMF is an average for both directions of travel. This CMF is derived to be applicable to a segment that has roadside barrier present along some portion of the segment. Guidance for computing the variables Pob and Wocb was provided previously in the subsection titled Barrier Variable Calculations. The combined outside clearance CMF is shown in Figure 53 using a series of thick, solid trend lines. Equation 86 was used to create this CMF (with CMFmv, oc = 1.0). The trend lines represent different combinations of area type and through lanes through their association with the different proportions of multiple-vehicle crashes in Table 38. The clear zone widths used to calibrate this CMF range from 0 to 30 ft. The base condition for this CMF is a 30-ft clear zone and a 10-ft outside shoulder width. The trend lines labeled “Roadside has barrier...” are discussed with the next CMF. a. Rural freeway segments. b. Urban freeway segments. Figure 53. Calibrated freeway outside clearance CMF for FI crashes. Also shown in Figure 53 are CMFs developed by other researchers. Thin lines are used to differentiate these CMFs from those developed for this research project. The correlation between clear zone width and crash frequency is consistent among all CMFs. Outside Barrier CMF. The calibrated outside barrier CMF is described using the following equation. ( ) ocbWobobaggobsv ePPCMF /131.0|, 0.10.1 +−= (99) This CMF is applicable to single-vehicle crashes. The regression analysis indicated that outside barrier presence had an insignificant correlation with multiple-vehicle crashes. Guidance for computing the variables Pob and Wocb was provided previously in the subsection titled Barrier Variable Calculations. The variable Wocb (representing the distance from the edge of outside shoulder to median barrier face) ranges in value from 1.0 to 17 ft in the database. The base condition for this CMF is no barrier.

147 The combined outside barrier CMF is shown in Figure 53. The relevant trend lines are labeled “Roadside has barrier...”. Equation 86 was used to create this CMF (with CMFmv, ob = 1.0). The values shown represent the outside barrier CMF multiplied by the outside clearance CMF because both are sensitive to the variable Wocb. The trend lines represent different combinations of area type and through lanes; however, the sensitivity to these influences is very small. Lane Change CMF. The calibrated lane change CMF is described using the following equations. ( ) ( )declcdecwevinclcincwevagglcmv ffffCMF ,,,,|, 5.05.0 += (100) ( ) incwevLincwevBincwevBincwev ePPf ,/175.0,,, 0.10.1 +−= (101) ( ) decwevLdecwevBdecwevBdecwev ePPf ,/175.0,,, 0.10.1 +−= (102) [ ] [ ]       −+×         −+= − −− − −− L AADTbX L AADTX inclc e L e e L ef extevexte entbentb 56.12 )000,1/ln(272.056.12 56.12 )000,1/ln(272.056.12 , 0.1 56.12 0.1 0.1 56.12 0.1 ,, ,, (103) [ ] [ ]       −+×         −+= − −− − −− L AADTX L AADTX declc e L e e L ef extbextb enteente 56.12 )000,1/ln(272.056.12 56.12 )000,1/ln(272.056.12 , 0.1 56.12 0.1 0.1 56.12 0.1 ,, ,, (104) The variables for weaving section length (i.e., Lwev, inc, Lwev, dec) in Equations 101 and 102 are intended to reflect the degree to which the weaving activity is concentrated along the freeway. This variable has negligible correlation with segment length L. The variables PwevB, inc and PwevB, dec in Equations 101 and 102, respectively, are computed as the ratio of the length of the weaving section on the segment to the length of the segment. If the segment is wholly located in the weaving section, then this variable is equal to 1.0. The calibration coefficient in these two equations indicates that lane change CMF value will increase if the segment is in a Type B weaving section. The amount of this increase is inversely related to the length of the weaving section. This CMF consists of several component equations but only requires a few input variables. These variables describe the distance to (and volume of) the four nearest ramps to the subject segment. Two of the ramps of interest are on side of the freeway with travel in the increasing milepost direction. One ramp on this side of the freeway is upstream of the segment and one ramp is downstream of the segment. Similarly, one ramp on the other side of the freeway is upstream of the segment and one ramp is downstream. Only those entrance ramps that

148 1.0 1.1 1.2 1.3 0.05 0.15 0.25 0.35 0.45 Distance from Gore (x), mi C ra sh M od ifi ca tio n Fa ct or Rural 6-lane freeway Ramp AADT = 6,000 veh/day Segment Length = 0.1 mi Average CMF for 0.5-mi road section = 1.05 x contribute volume to the subject segment are of interest. Hence a downstream entrance ramp is not of interest. For similar reasons, an upstream exit ramp is not of interest. If the segment is in a Type B weaving section, then the length of the weaving section is also an input. To illustrate this CMF, consider a 0.5-mi section of rural six-lane freeway. It consists of five segments that are each 0.1 mi in length. There is an interchange at one end of the section. The distance to the next interchange is sufficiently large that its ramp traffic has no influence on segment lane change activity. No weaving section is present. Under this scenario, fwev, inc equals 1.0, fwev, dec equals 1.0, the second term of Equation 103 equals 1.0, and the first term of Equation 104 equals 1.0. The CMF for each of the five segments is plotted in Figure 54. The example 0.5-mi freeway section is shown in plan view in the upper left corner of this figure. Figure 54. Lane change CMF as a function of distance from ramp gore. The CMF is 1.18 for the segment that starts at the ramp gore and extends 0.1 mi (i.e., segment center is x = 0.05 mi). The CMF value for the next segment is 1.05. CMF values continue to decrease for each subsequent segment. This decline in CMF value reflects the decreasing number of lane changes with increasing distance from the ramp gore. The overall average CMF for all five segments is 1.05. Consider a 0.5-mi section of rural six-lane freeway that is bounded on each end by an interchange. It consists of five segments that are each 0.1 mi in length. A Type A weaving section exists in each travel direction. Under this scenario, fwev, inc equals 1.0 and fwev, dec equals 1.0. The CMF for each of the five segments is plotted in Figure 55. The example 0.5-mi freeway section is shown in the upper left corner of this figure.

149 1.0 1.1 1.2 1.3 0.05 0.15 0.25 0.35 0.45 Distance from Left-Side Gore (x), mi C ra sh M od ifi ca tio n Fa ct or Rural 6-lane freeway Ramp AADT = 6,000 veh/day Segment Length = 0.1 mi Average CMF for 0.5-mi road section = 1.10 x Figure 55. Lane change CMF for segments between a pair of interchanges. As before, the CMF is 1.18 for the segment that starts at the ramp gore on the left and extends 0.1 mi (i.e., segment center is x = 0.05 mi). The CMF value for the next segment is 1.06. The CMF continues to decrease until the middle segment is reached and then starts to increase as the segments get closer to the next interchange. The overall average CMF for all five segments is 1.10. The lane-change CMF was applied to other weaving section lengths and the average CMF was computed for each length. This process was undertaken to facilitate a comparison of the lane-change CMF with two weaving-section CMFs shown previously in Figure 26. The results of this process are plotted in Figure 56. The trends shown indicate relatively good agreement between the lane-change CMF and these other CMFs. Weaving sections associated with a full cloverleaf interchange typically have a length of 0.10 to 0.18 miles. Figure 56 suggests that weaving sections with a length in this range will be associated with a relatively high CMF value and high crash rate. In fact, interchange design practice in the last few decades has been to relocate the weaving section associated with a full cloverleaf interchange to a collector-distributor roadway to improve freeway safety and operation. The lane-change CMF is applicable to any segment in the vicinity of one or more ramps. It is equally applicable to segments in a weaving section and segments in a non-weaving section (i.e., segments between an entrance ramp and an exit ramp where both ramps have a speed- change lane).

150 1.0 1.1 1.2 1.3 1.4 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Weaving Section Length, mi A ve ra ge C ra sh M od ifi ca tio n Fa ct or Bonneson and Pratt (2008) Full cloverleaf, Cirillo (1970) Ramp AADT = 6,000 veh/day Urban, proposed Rural 6-lane, proposed Rural 4-lane, proposed Figure 56. Average CMF value for FI crashes as a function of weaving section length. The two component models for predicting speed-change-related crash frequency (i.e., Equations 84 and 85) are not used when evaluating a weaving section because the ramps that form the weaving section do not have a speed-change lane. As a result, the predicted crash frequency for the set of segments that comprise a weaving section will tend to be smaller than that predicted for a similar set of segments located in a non-weaving section. This generalization will always be true for the Type A and Type C weaving sections. It may or may not hold for the Type B weaving section, depending on the length of the weaving section. The calibration coefficient associated with the ramp AADT term in Equations 103 and 104 is negative which is counterintuitive at first glance. It indicates that the lane change CMF is larger for segments associated with lower volume ramps. This trend may be explained by the fact that high-volume ramp flows tend to dominate the traffic stream such that a large portion of the traffic stream is changing lanes and all drivers are more aware of these maneuvers. Regardless, the entering ramp volumes are also included in the segment AADT volume and the coefficient associated with the segment AADT variable in the SPF is positive and relatively large. As a result, when all relevant SPFs and CMFs are combined, the predicted average crash frequency for a freeway segment increases with an increase in the AADT volume of nearby ramps. This trend is logical and intuitive. Ramp Entrance CMF. The ramp entrance CMF is described using the following equation. )000,1/ln(198.0/0318.0594.0 renleft AADTLI en eCMF ++= (105) This CMF is applied to a ramp entrance speed-change lane, as shown in Figure 38. The “segment” length is equal to the ramp entrance length Len, which is measured using the gore and taper points identified in Figure 11. This CMF applies only to the side of the freeway with the subject speed-change lane.

151 0.8 1.0 1.2 1.4 1.6 1.8 300 500 700 900 Ramp Entrance Length, ft C ra sh M od ifi ca tio n Fa ct or . HSM (Highway, 2010), base length = 950 ft Sarhan et al. (2006) Bonneson and Pratt (2008) Ramp AADT = 6,000 veh/day, proposed Ramp AADT = 1,000 veh/day, proposed The variable for ramp entrance length Len in Equation 105 is intended to reflect the degree to which the lane-changing activity is concentrated along the ramp entrance. This variable has negligible correlation with segment length L. The indicator variable for ramp side Ileft is associated with a positive calibration coefficient. It suggests that a ramp entrance on the left side of the through lanes is associated with an 81 percent increase in crashes, relative to one on the right side. This finding is consistent with that of Moon and Hummer (2009), and with that from the re-analysis of the data reported by Lundy (1966), as documented in Chapter 2. The ramp entrance CMF for right-side ramps is shown in Figure 57 using a thick, solid trend line. It has been adjusted using Equation 106 to facilitate a comparison with the CMFs reported by other researchers because they apply to entire segments (i.e., both sides of the freeway). The ramp entrance lengths used to calibrate this CMF range from 0.07 to 0.22 mi (370 to 1,200 ft). eni CMFCMF 5.05.0 += (106) Figure 57. Calibrated freeway ramp entrance CMF for right-side ramps and FI crashes. Also shown in Figure 57 are CMFs developed by other researchers. Thin lines are used to differentiate these CMFs from those developed for this research project. The correlation between ramp entrance length and crash frequency is consistent among all CMFs. Ramp Exit CMF. The ramp exit CMF is described using the following equation. exleft LI ex eCMF /0116.0594.0 += (107) This CMF is applied to a ramp exit speed-change lane, as shown in Figure 38. The “segment” length is equal to the ramp exit length Lex, which is measured using the gore and taper

152 0.8 1.0 1.2 1.4 1.6 1.8 200 400 600 800 1,000 Ramp Exit Length, ft C ra sh M od ifi ca tio n Fa ct or . HSM (Highway, 2010), base length = 690 ft Sarhan et al. (2006) Bauer and Harwood (1998) Proposed points identified in Figure 11. This CMF applies only to the side of the freeway with the subject speed-change lane. The variable for ramp exit length Lex in Equation 107 is intended to reflect the degree to which the lane-changing activity is concentrated along the ramp exit. This variable has negligible correlation with segment length L. The interpretation of the indicator variable for ramp side Ileft is provided with the previous CMF discussion. The ramp exit CMF for right-side ramps is shown in Figure 58 using a thick, solid trend line. It has been adjusted using Equation 106 to facilitate a comparison with the CMFs reported by other researchers, which apply to entire segments (i.e., both sides of the freeway). The ramp exit lengths used to calibrate this CMF range from 0.03 to 0.21 mi (160 to 1,100 ft). Figure 58. Calibrated freeway ramp exit CMF for right-side ramps and FI crashes. Also shown in Figure 58 are CMFs developed by other researchers. Thin lines are used to differentiate these CMFs from those developed for this research project. The correlation between ramp exit length and crash frequency is similar, but not consistent among CMFs. The calibrated CMF value is smaller than that obtained from the other CMFs for lengths less than 600 ft. High-Volume CMF. The calibrated high-volume CMF has two forms, depending on which component model is being used. The CMF for multiple-vehicle non-entrance/exit crashes, ramp-entrance-related crashes, and ramp exit related crashes is described using the following equation. hvP hvmv eCMF 350.0 , = (108) The CMF for single-vehicle crashes is described using the following equation. hvP hvsv eCMF 0675.0 , −= (109)

153 1.00 1.05 1.10 1.15 1.20 0.0 0.2 0.4 0.6 0.8 1.0 Proportion AADT during High-Volume Hours C ra sh M od ifi ca tio n Fa ct or Rural 6-lane, proposed Urban, proposed Rural 4-lane proposed Multiple-vehicle crashes 0.94 0.96 0.98 1.00 0.0 0.2 0.4 0.6 0.8 1.0 Proportion AADT during High-Volume Hours C ra sh M od ifi ca tio n Fa ct or Rural 6-lane, proposed Urban, proposed Rural 4-lane proposed Single-vehicle crashes The proportion of AADT during hours where volume exceeds 1,000 veh/h/ln Phv is computed using the average hourly volume distribution associated with the subject segment. This distribution will typically be computed using the data obtained from the nearest continuous traffic counting station (on a freeway of similar character). The variable Phv is positively correlated with the volume-to-capacity ratio experienced by the segment on an hourly basis. The high-volume CMF is shown in Figure 59 using a thick, solid trend line. The trend lines represent different combinations of area type and through lanes through their association with the different proportions of multiple-vehicle crashes in Table 38. The base condition for this CMF is a proportion Phv equal to 0.0. The trends shown in both figures are consistent with those developed by Lord et al. (2005) for multiple-vehicle and single-vehicle crashes as a function of volume-to-capacity ratio. a. Multiple-vehicle crashes. b. Single-vehicle crashes. Figure 59. Calibrated freeway high-volume CMF for FI crashes. Sensitivity Analysis The relationship between crash frequency and traffic demand, as obtained from the combined calibrated models, is illustrated in Figure 60 for a 1-mile freeway segment with two ramp entrances, two ramp exits, no curvature, and no barrier. The individual component models are illustrated in Figures 60a, 60b, 60c, and 60d. The sum of the individual component crash frequencies is illustrated in Figure 61. The length of the trend lines in Figures 60 and 61 reflect the range of AADT volume in the data. The trends in Figure 61 are comparable to those in Figure 15e and 15f. The trend lines shown in Figure 61 indicate that urban freeways have about 20 to 30 percent more crashes than rural freeways. By comparison, the crash rates listed in Table 19 indicate that urban freeways have 50 to 250 percent more crashes than rural freeways. It is likely that this latter trend reflects the influence of barrier length, ramp entrances, ramp exits, and weaving section length. As shown in Table 18, these influences are more prevalent on urban freeway segments. In contrast, these influences have been explicitly quantified in the proposed

154 0 3 6 9 12 15 0 50 100 150 200 250 Average Daily Traffic Demand (1000s), veh/day FI M ul tip le -V eh ic le C ra sh Fr eq ue nc y, c ra sh es /y r 1.0-mile segment length, no barrier 4 lanes 6 8 Rural Freeway Urban Freeway 6 10 8 0 1 2 3 4 5 0 50 100 150 200 250 Average Daily Traffic Demand (1000s), veh/day FI S in gl e- Ve hi cl e C ra sh Fr eq ue nc y, c ra sh es /y r 1.0-mile segment length, no barrier 4 lanes 6 8 Rural and Urban Freeway 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 40 80 120 Directional Average Daily Traffic (1000s), veh/day FI R am p En tra nc e C ra sh Fr eq ue nc y, c ra sh es /y r 4 lanes 8 10 6 Ramp AADT = 0.1 x Directional AADT Ramp entrance length = 700 ft Rural Freeway Urban Freeway 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 40 80 120 Directional Average Daily Traffic (1000s), veh/day FI R am p Ex it C ra sh F re qu en cy , cr as he s/ yr 4 lanes Ramp exit length = 350 ft Rural and Urban Freeway 8 10 6 model such that they do not influence the trends shown in Figure 61. The proposed model provides a more accurate indication of differences between freeway segments in rural versus urban areas, when the segments have the same barrier proportion, ramp entrance length, and weaving section length. a. Multiple-vehicle crashes. b. Single-vehicle crashes. c. Ramp-entrance-related crashes. d. Ramp-exit-related crashes. Figure 60. Freeway FI model components.

155 0 5 10 15 20 25 0 50 100 150 200 250 Average Daily Traffic Demand (1000s), veh/day To ta l F I C ra sh F re qu en cy , cr as he s/ yr 1.0-mile segment length, no barrier 2 entrance ramps, 2 exit ramps 4 lanes 8 Rural Freeway Urban Freeway 6 10 4, 6, 8 lanes Figure 61. Freeway FI model. The trend lines shown in Figure 61 also indicate that crash frequency is lower on freeways with many lanes than it is on freeways with few lanes. In fact, the models indicate that an urban six-lane freeway segment has about 7 percent fewer crashes than an urban four-lane segment and that a rural six-lane segment has about 3 percent fewer crashes than an urban four- lane segment. These trends are counter to those found when comparing the crash rates in Table 19, which indicate that crash rate is higher on freeways with many lanes. It is likely that these trends in crash rate reflect the fact that the proportion of barrier along a freeway segment typically increases (and the lateral clearance decreases) with an increase in the number of lanes. The proposed predictive models account for the influence of barrier presence and lateral clearance and, therefore, provide a more accurate indication of the relationship between number of lanes and crash frequency. Model Extensions This section describes two extensions to the predictive models described in the previous sections. The models were developed to be applicable to a wide range of geometric conditions. However, limitations of the data did not allow model calibration for some situations. These situations are addressed in this section as model extensions. AADT Volume Varies by Direction The speed-change models and several of the CMFs are sensitive to geometric elements and ramp volumes that exist in only one travel direction on the segment (e.g., weaving section). These features make the safety evaluation more adaptive to segment geometry on a directional basis and the results more accurate.

156 One situation that could not be addressed in the SPF development was a segment with an unbalanced daily traffic demand. The AADT volumes in the study state database represent both travel directions and it is commonly assumed that the daily directional distribution is balanced (i.e., the same in each direction). If the daily directional distribution is known, then the calibrated SPFs are revised as shown in the following equations. ( ) ( ) ( ))000,1/ln(492.1)000,1/ln(492.1505.0470.5 ,,,,24,, 21 5.05.0 468.10.1 406.1 AADTPAADTPI isegexisegen p mvspf eee LLLN rural +×× −−        + = −− σ (110) ( ) ( ) ( ))000,1/ln(646.0)000,1/ln(646.00351.0266.2 ,,,,2, 21 5.05.0 457.00.1 782.0 AADTPAADTPn isegexisegen p svspf eee LLLN +×× −−        − = +− σ (111) ( ) rurali InAADTPenenspf eLN 180.0130.0)000,1/ln(173.1194.3, −−+−= (112) ( ) )000,1/ln(903.0679.2, AADTPexexspf ieLN +−= (113) with, ( )212 5.0 Pp −=σ (114) where, P1 = proportion of AADT in travel direction 1; P2 = proportion of AADT in travel direction 2 (= 1.0 - P1); σp = standard deviation of Pi; and Pi = proportion of AADT in travel direction i that corresponds to subject speed-change lane. The first term in Equations 110 and 111 represents a correction factor for using directional traffic demand in the SPF. Equation 110 is the SPF for multiple-vehicle crashes on four-lane freeway segments. The other multiple-vehicle-crash SPFs can be similarly converted using the same correction factor. It is noted that both correction factors have a value in the range of 0.97 to 1.02 for most conditions. The AADT divisor in Equations 112 and 113 is “1,000,” rather than the “2,000” used in the calibrated versions of these equations. The value “2,000” was intentionally used for calibration because it corresponds to a value of Pi equal to 0.5. Number of Lanes Varies by Direction There are few freeway segments with an unequal number of lanes in opposing directions from the standpoint of statistically valid sample sizes. However, there are a sufficient number of these segments encountered during a safety evaluation that an extension is needed to provide reasonable estimates of average crash frequency. If a freeway segment has X total lanes that represent Y lanes in one direction and Z lanes in the opposite direction (i.e., X = Y + Z) and Y is not equal to Z, then it recommended that the

157 segment be evaluated twice. One evaluation would be conducted where the number of lanes is equal to 2Y and one evaluation would be conducted where the number of lanes is equal to 2Z. All other inputs to the models would be unchanged between evaluations. The two estimates of average crash frequency obtained in this manner are then averaged to obtain the best estimate of the predicted average crash frequency for the subject segment. MODEL CALIBRATION FOR PDO CRASHES This part of the chapter describes the calibration of the combined freeway segment and speed-change lane predictive models based on PDO crashes. The methodology used to calibrate the models is described in the part titled Methodology. The calibration data, model development, and statistical analysis methods are described in the part titled Model Calibration for FI Crashes. An initial regression analysis was undertaken with county and state variable combinations treated as fixed effects and as random effects. The Hausman test was performed using the covariance matrix to determine whether the fixed-effect model was appropriate (Hausman, 1978). The null hypothesis is that the regression coefficients from the two model treatments are consistent. This hypothesis was rejected (p = 0.0001) indicating that the coefficients are different (i.e., inconsistent) among the two treatments. In this case, it is concluded that the regression coefficient values are influenced by county so a fixed-effect treatment is needed to remove the county effect. Model Calibration The results of the regression model calibration are presented in Table 39. The Pearson χ2 statistic for the model is 4,888, and the degrees of freedom are 4,557 (= n − p = 4,587 −30). This statistic is greater than the χ2 0.05, 4577 (= 4,715) so the hypothesis that the model fits the data is rejected by this test. Several segments were removed as a result of outlier analysis such that the calibration database included only 17,265 of the 18,045 crashes identified in Chapter 4. Closer examination of the data indicates that a small number of sites with a length less than 0.10 mi and many crashes were causing the lack of fit. A test of the scaled deviance was also conducted. This statistic is also chi-square distributed but is less sensitive to sites with short length and many crashes. The scaled deviance for this model is 4,532, which is less than 4,715 so the hypothesis that the model fits the data cannot be rejected by this test. An examination of the other fit statistics in Table 39 indicates the model provides a relatively good fit, and supports the conclusion reached by the scaled deviance test. The t-statistic for each coefficient is listed in the last column of Table 39.These statistics describe a test of the hypothesis that the coefficient value is equal to 0.0. Those t-statistics with an absolute value that is larger than 2.0 indicate that the hypothesis can be rejected with the probability of error in this conclusion being less than 0.05. For those few variables where the absolute value of the t-statistic is smaller than 2.0, it was decided that the variable was important to the model and its trend was found to be intuitive and, where available, consistent with previous research findings (even if the specific value was not known with a great deal of certainty as applied to this database).

158 TABLE 39. Freeway PDO model statistical description–combined model –three states Model Statistics Value R2: 0.66 Scale parameter φ: 1.07 Pearson χ2: 4,888 (χ20.05, 4557 = 4,715) Observations no: 1,848 segments (17,265 PDO crashes in 3 years) 637 ramp entrances, 513 ramp exits Calibrated Coefficient Values Variable Inferred Effect of... Value Std. Dev. t-statistic bmv, cr Horizontal curvature on 2+ veh. crashes 0.0340 0.0071 4.8 bsv, cr Horizontal curvature on 1 veh. crashes 0.0626 0.0116 5.4 bs, cur Outside shoulder width on 1 veh. crashes, curves -0.0840 0.0224 -3.7 bis Inside shoulder width -0.0153 0.0056 -2.7 brs, cur Shoulder rumble strip on 1 veh. crashes, curves 0.186 0.0934 2.0 bmv, mw Median width on 2+ veh. crashes -0.00291 0.0007 -4.3 bsv, mw Median width on 1 veh. crashes -0.00289 0.0006 -4.7 bbar Barrier presence 0.169 0.0424 4.0 bwev Type B weaving section presence on 2+ veh. crashes 0.123 0.0503 2.5 bv Ramp AADT on lane-change-related crashes -0.283 0.1551 -1.8 bx Distance from ramp on lane-change-related crashes 13.461 3.5160 3.8 bleft Left side entrance or exit on speed-change lane crashes 0.824 0.4888 1.7 ben, len Ramp entrance length on related crashes 0.0252 0.0130 1.9 bmv, hv High-volume conditions on 2+ veh. crashes 0.283 0.1218 2.3 bsv, hv High-volume conditions on 1 veh. crashes -0.611 0.1142 -5.3 bmv, 4 4 lanes on 2+ vehicle crashes in urban areas -6.355 0.369 -17.2 bmv, 6 6 lanes on 2+vehicle crashes in urban areas -6.616 0.415 -15.9 bmv, 8 8 lanes on 2+ vehicle crashes in urban areas -6.804 0.451 -15.1 bmv, 10 10 lanes on 2+ vehicle crashes in urban areas -7.067 0.478 -14.8 bmv, 1 AADT on 2+ vehicle crashes 1.936 0.101 19.2 bmv, rural Added effect of rural area type on 2+ veh. crashes -0.332 0.050 -6.7 bsv, 0 1 veh. crashes -1.955 0.204 -9.6 bsv, 2 Number of lanes on 1 veh. crashes -0.0193 0.022 -0.9 bsv, 1 AADT on 1 veh. crashes 0.876 0.073 11.9 ben, 0 Ramp-entrance crashes in urban areas -2.180 0.386 -5.6 ben, 2 Number of lanes on ramp-ent. crashes in urban areas -0.101 0.041 -2.5 ben, 1 AADT on ramp-entrance crashes 1.215 0.140 8.7 ben, rural Added effect of rural area type on ramp-ent. crashes -0.0989 0.108 -0.9 bex, 0 Ramp exit crashes -1.575 0.381 -4.1 bex, 1 AADT on ramp exit crashes 0.932 0.107 8.7 The coefficients for 43 county indicator variables are not shown in Table 39 because their individual significance is not directly relevant to model fit assessment or its application. However, it is recognized that the “intercept” variables in Table 39 (i.e., bmv, 4, bmv, 6, bmv, 8, bmv, 10, bsv, 0, ben, 0, bex, 0) correspond to only one state and county combination. Desirably, the intercept

159 would represent an average value for all states and counties in the database. To this end, the predicted crash frequencies from the model described by Table 39 were submitted to a second regression analysis using Equation 115. 0 ,, c bI i i ee NyX icoico = (115) where, Xi = reported crash count for y years in county i, crashes; y = time interval during which X crashes were reported, yr; Ni = predicted average crash frequency for county i, crashes/yr; Ico, i = county indicator variable (= 1.0 if county i, 0.0 otherwise); bco, i = county i regression coefficient; and co = regression coefficient. The regression coefficient co was determined to be -0.193 for multiple-vehicle crashes, -0.203 for single-vehicle crashes, -0.212 for ramp-entrance-related crashes, and -0.223 for ramp- exit-related crashes. Each of these values is added to the appropriate intercept variables to compute an average intercept value for the overall database. This addition is shown in each of the next four sections. Model for Predicting Multiple-Vehicle Non-Ramp-Related Crash Frequency The results of the multiple-vehicle model calibration are presented in Table 40. The Pearson χ2 statistic for the model is 1,743, the scaled deviance is 1,580, and the degrees of freedom are 1,576 (= n − p = 1,589 −13). The Pearson χ2 statistic is greater than χ2 0.05, 1576 (= 1,669) but the scaled deviance is less than 1,669. For reasons cited previously with regard to the full model, the hypothesis that the model fits the data is not rejected. The R2 for the model is 0.66. An alternative measure of model fit that is better suited to the negative binomial distribution is Rk2. The Rk2 for the calibrated model is 0.93. TABLE 40. Freeway PDO model statistical description–multiple-vehicle model–three states Model Statistics Value R2 (Rk2): 0.66 (0.93) Scale parameter φ: 1.10 Pearson χ2: 1,743 (χ20.05, 1576 = 1,669) Inverse dispersion parameter K: 18.8 mi-1 Observations no: 1,589 segments (9,960 PDO crashes in 3 years) Standard deviation se: ±2.59 crashes/yr The coefficients in Table 39 were combined with Equations 37 to 40 to obtain the calibrated SPFs for multiple-vehicle non-entrance/exit crashes. The form of each model is described in the following equations. ( ) ruralIAADTisegexisegenmvspf eLLLN 332.0)000,1/ln(936.1193.0355.6,,,,4,, 5.05.0 −+−− −−= (116)

160 0 5 10 15 20 25 30 35 40 0 10 20 30 40 50 Predicted PDO Crash Frequency, cr/3 yrs R ep or te d C ra sh F re qu en cy , cr /3 y rs Each data point represents an average of 10 sites. 1 1 ( ) ruralIAADTisegexisegenmvspf eLLLN 332.0)000,1/ln(936.1193.0616.6,,,,6,, 5.05.0 −+−− −−= (117) ( ) ruralIAADTisegexisegenmvspf eLLLN 332.0)000,1/ln(936.1193.0804.6,,,,8,, 5.05.0 −+−− −−= (118) ( ) ruralIAADTisegexisegenmvspf eLLLN 332.0)000,1/ln(936.1193.0067.7,,,,10,, 5.05.0 −+−− −−= (119) The calibrated CMFs used with these SPFs are described in a subsequent section. The fit of the calibrated models is shown in Figure 62. This figure compares the predicted and reported crash frequency in the calibration database. The trend line shown represents a “y = x” line. A data point would lie on this line if its predicted and reported crash frequency were equal. The data points shown represent the reported multiple-vehicle non-entrance/exit crash frequency for the segments used to calibrate the corresponding component model. Figure 62. Predicted vs. reported multiple-vehicle freeway PDO crashes. Each data point shown in Figure 62 represents the average predicted and average reported crash frequency for a group of 10 segments. The data were sorted by predicted crash frequency to form groups of segments with similar crash frequency. The purpose of this grouping was to reduce the number of data points shown in the figure and, thereby, to facilitate an examination of trends in the data. The individual segment observations were used for model calibration. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency for segments experiencing up to 50 multiple-vehicle crashes in a three- year period. Model for Predicting Single-Vehicle Non-Ramp-Related Crash Frequency The results of the single-vehicle model calibration are presented in Table 41. The Pearson χ2 statistic for the model is 2,036, the scaled deviance is 1,927, and the degrees of freedom are

161 1,838 (= n − p = 1,848 −10). The Pearson χ2 is greater than χ2 0.05, 1838 (= 1,939) but the scaled deviance is less than 1,939. For reasons cited previously with regard to the full model, the hypothesis that the model fits the data is not rejected. The R2 for the model is 0.65. The Rk2 for the calibrated model is 0.89. TABLE 41. Freeway PDO model statistical description–single-vehicle model–three states Model Statistics Value R2 (Rk2): 0.65 (0.89) Scale parameter φ: 1.10 Pearson χ2: 2,036 (χ20.05, 1838 = 1,939) Inverse dispersion parameter K: 20.7 mi-1 Observations no: 1,848 segments (5,372 PDO crashes in 3 years) Standard deviation se: ±0.88 crashes/yr The coefficients in Table 39 were combined with Equation 49 to obtain the calibrated SPF for single-vehicle non-entrance/exit crashes. The form of this model is described in the following equation. ( ) nAADTisegexisegensvspf eLLLN 0193.0)000,1/ln(876.0203.0955.1,,,,, 5.05.0 −+−− −−= (120) The calibrated CMFs used with this SPF are described in a subsequent section. The fit of the calibrated model is shown in Figure 63. This figure compares the predicted and reported crash frequency in the calibration database. Each data point shown represents the average predicted and average reported crash frequency for a group of 10 segments. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency for segments experiencing up to 20 single-vehicle crashes in a three-year period.

162 0 5 10 15 20 25 0 5 10 15 20 25 30 Predicted PDO Crash Frequency, cr/3 yrs R ep or te d C ra sh F re qu en cy , cr /3 y rs Each data point represents an average of 10 sites. 1 1 Figure 63. Predicted vs. reported single-vehicle freeway PDO crashes. Model for Predicting Ramp-Entrance-Related Crash Frequency The results of the ramp-entrance-related model calibration are presented in Table 42. The Pearson χ2 statistic for the model is 623, and the degrees of freedom are 627 (= n − p = 637 −10). As this statistic is less than χ2 0.05, 627 (= 686), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.57. The Rk2 for the calibrated model is 0.96. TABLE 42. Freeway PDO model statistical description–ramp entrance model–three states Model Statistics Value R2 (Rk2): 0.57 (0.96) Scale parameter φ: 0.93 Pearson χ2: 623 (χ20.05, 627 = 686) Inverse dispersion parameter K: 24.8 mi-1 Observations no: 637 ramp entrances (1,369 PDO crashes in 3 years) Standard deviation se: ±0.96 crashes/yr The coefficients in Table 39 were combined with Equation 59 to obtain the calibrated SPF for ramp-entrance-related crashes. The form of this model is described in the following equation. ( ) ruralInAADTenenspf eLN 0989.0101.0)000,2/ln(215.1212.0180.2, −−+−−= (121) The calibrated CMFs used with this SPF are described in a subsequent section. This SPF is applied to a ramp entrance speed-change lane, as shown in Figure 38. The “segment” length is equal to the ramp entrance length Len, which is measured using the gore and taper points identified in Figure 11.

163 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 0.0 2.0 4.0 6.0 8.0 10.0 Predicted PDO Crash Frequency, cr/3 yrs R ep or te d C ra sh F re qu en cy , cr /3 y rs Each data point represents an average of 10 sites. 1 1 The fit of the calibrated model is shown in Figure 64. This figure compares the predicted and reported crash frequency in the calibration database. Each data point shown represents the average predicted and average reported crash frequency for a group of 10 segments. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency for segments experiencing up to 9.0 ramp-entrance-related crashes in a three- year period. Figure 64. Predicted vs. reported ramp-entrance-related PDO crashes. Model for Predicting Ramp-Exit-Related Crash Frequency The results of the ramp-exit-related model calibration are presented in Table 43. The Pearson χ2 statistic for the model is 487, and the degrees of freedom are 504 (= n − p = 513 −9). As this statistic is less than χ2 0.05, 504 (= 557), the hypothesis that the model fits the data cannot be rejected. The R2 for the model is 0.53. The Rk2 for the calibrated model is 0.95. The inverse dispersion parameter for this model is dimensionless because crash frequency variance was found to be insensitive to ramp exit length. TABLE 43. Freeway PDO model statistical description–ramp exit model–three states Model Statistics Value R2 (Rk2): 0.53 (0.95) Scale parameter φ: 0.95 Pearson χ2: 487 (χ20.05, 504 = 557) Inverse dispersion parameter K: 1.58 Observations no: 513 ramp exits (564 PDO crashes in 3 years) Standard deviation se: ±0.58 crashes/yr

164 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 1.0 2.0 3.0 4.0 5.0 Predicted PDO Crash Frequency, cr/3 yrs R ep or te d C ra sh F re qu en cy , cr /3 y rs Each data point represents an average of 10 sites. 1 1 The coefficients in Table 39 were combined with Equation 61 to obtain the calibrated SPF for ramp-exit-related crashes. The form of this model is described in the following equation. ( ) )000,2/ln(932.0223.0575.1, AADTexexspf eLN +−−= (122) The calibrated CMFs used with this SPF are described in a subsequent section. This SPF is applied to a ramp exit speed-change lane, as shown in Figure 38. The “segment” length is equal to the ramp exit length Lex, which is measured using the gore and taper points identified in Figure 11. The fit of the calibrated models is shown in Figure 65. This figure compares the predicted and reported crash frequency in the calibration database. Each data point shown represents the average predicted and average reported crash frequency for a group of 10 segments. In general, the data shown in the figure indicate that the model provides an unbiased estimate of expected crash frequency for segments experiencing up to 4.0 ramp-exit-related crashes in a three-year period. Figure 65. Predicted vs. reported ramp-exit-related PDO crashes. Calibrated CMFs Several CMFs were calibrated in conjunction with the SPFs. All of them were calibrated using PDO crash data. Collectively, they describe the relationship between various geometric factors and PDO crash frequency. Many of the CMFs found in the literature are typically derived from (and applied to) the combination of multiple-vehicle and single-vehicle crashes. That is, one CMF is used to indicate the influence of a specified geometric factor on total crashes. In contrast, the models developed for this research project include several CMFs that are calibrated for a specific crash type. In

165 these instances, Equation 86 is used to convert the CMFs developed for this project into equivalent total-crash CMFs for the purpose of illustrating the overall trend. The proportion of multiple-vehicle crashes used in this equation is obtained from Table 44. The data in this table were obtained from the study state databases. TABLE 44. Distribution of PDO crashes on freeways Area Type Number of Through Lanes Multiple-Vehicle Non-Entrance/Exit PDO Crashes (MV) Single-Vehicle Non- Entrance/Exit PDO Crashes Proportion MV Crashes Rural 4 628 1,162 0.351 6 926 1,131 0.450 8 388 346 0.529 Urban 4 906 662 0.578 6 2,280 1,110 0.673 8 2,868 635 0.819 10 2,198 326 0.871 Horizontal Curve CMF. The calibrated horizontal curve CMF has two forms, depending on which component model is being used. The CMF for multiple-vehicle non- entrance/exit crashes, ramp-entrance-related crashes, and ramp exit related crashes is described using the following equation. ic m i i agghcmv PR CMF , 1 2 |, 730,50340.00.1  =       += (123) The CMF for single-vehicle crashes is described using the following equation. ic m i i agghcsv PR CMF , 1 2 |, 730,50626.00.1  =       += (124) These two CMFs are derived to be applicable to a segment that has a mixture of uncurved and curved lengths. The variable Pc, i is computed as the ratio of the length of curve i on the segment to the length of the segment. The combined horizontal curve CMF is shown in Figure 66 using a series of thick, solid trend lines. Equation 86 was used to create these trend lines. They represent different combinations of area type and through lanes through their association with the different proportions of multiple-vehicle crashes in Table 44. The radii used to calibrate this CMF range from 1,500 to 12,000 ft. The base condition for this CMF is an uncurved (i.e., tangent) segment.

166 1.0 1.1 1.2 1.3 1.4 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000 Curve Radius, ft C ra sh M od ifi ca tio n Fa ct or . Rural 4-lane Urban 4-lane Rural 6-lane Urban 6-lane Figure 66. Calibrated freeway horizontal curve CMF for PDO crashes. Outside Shoulder Width CMF. The outside shoulder width CMF is described using the following equation. ( ) ( ) ( )100840.0,,|, 0.10.1 −− +−= sWicicaggoswsv ePPCMF (125) This CMF is applicable to single-vehicle crashes. The regression analysis indicated that the outside shoulder width had an insignificant correlation with multiple-vehicle crashes. The shoulder width used in this CMF is an average for both directions of travel. The variable Pc,i is computed as the ratio of the length of curve i on the segment to the length of the segment. The CMF value is 1.0 when applied to a segment that is straight (i.e., no curves). The combined outside shoulder width CMF for curved segments is shown in Figure 67. Equation 86 was used to create this CMF (with CMFmv, osw = 1.0). It represents the likely correlation between outside shoulder width and total crash frequency. The trend lines represent different combinations of area type and through lanes through their association with the different proportions of multiple-vehicle crashes in Table 44. The shoulder widths used to calibrate this CMF range from 6 to 14 ft. The base condition for this CMF is a 10-ft shoulder width.

167 0.9 1.0 1.1 1.2 6 7 8 9 10 11 12 Outside Shoulder Width, ft C ra sh M od ifi ca tio n Fa ct or . Rural 4-lane (curve) Urban 4-lane (curve) Rural 6-lane (curve) Urban 6-lane (curve) 0.8 0.9 1.0 1.1 1.2 2 4 6 8 10 Inside Shoulder Width, ft C ra sh M od ifi ca tio n Fa ct or . Figure 67. Calibrated freeway outside shoulder width CMF for PDO crashes. Inside Shoulder Width CMF. The inside shoulder width CMF is described using the following equation. ( )60153.0 −−= isWisw eCMF (126) The shoulder width used in this CMF is an average for both directions of travel. The inside shoulder width CMF is shown in Figure 68. The shoulder widths used to calibrate this CMF range from 2 to 11 ft. The base condition for this CMF is a 6-ft shoulder width. Figure 68. Calibrated freeway inside shoulder width CMF for PDO crashes.

168 0.90 0.95 1.00 1.05 1.10 1.15 1.20 20 30 40 50 60 70 80 Median Width, ft C ra sh M od ifi ca tio n Fa ct or . Barrier in center of median No barrier Median-Width CMF. The calibrated median-width CMF has two forms, depending on which component model is being used. The CMF for multiple-vehicle non-entrance/exit crashes, ramp-entrance-related crashes, and ramp exit related crashes is described using the following equation. ( ) ( ) ( )48200291.048200291.0|, 0.1 −−−−− +−= icbism WibWWibaggmwmv ePePCMF (127) The CMF for single-vehicle crashes is described using the following equation. ( ) ( ) ( )48200289.048200289.0|, 0.1 −−−−− +−= icbism WibWWibaggmvsv ePePCMF (128) The median width used in either CMF is an average for the segment. These two CMFs are derived to be applicable to a segment that has median barrier present along some portion of the segment. Guidance for computing the variables Pib and Wicb was provided previously in the subsection titled Barrier Variable Calculations. The combined median-width CMF is shown in Figure 69 using the line labeled “No barrier.” Equation 86 was used to create this CMF. The trend line shown represents an urban freeway. The slope of the line is slightly flatter for a rural freeway. The median widths used to calibrate this CMF range from 9 to 140 ft. The base condition for this CMF is a 60-ft median width and an inside shoulder width of 6.0 ft. The trend line labeled “Barrier in center...” is discussed with the next CMF. Figure 69. Calibrated freeway median-width CMF for PDO crashes. Median Barrier CMF. The median barrier CMF is described using the following equation. ( ) icbWibibaggmb ePPCMF /169.0| 0.10.1 +−= (129)

169 Guidance for computing the variables Pib and Wicb was provided previously in the subsection titled Barrier Variable Calculations. The variable Wicb (representing the distance from the edge of inside shoulder to median barrier face) ranges in value from 1.0 to 17 ft in the database. The base condition for this CMF is no barrier. This median barrier CMF is shown in Figure 69 using the line labeled “Barrier in center...”. The values shown represent the median barrier CMF multiplied by the median-width CMF because both CMFs are sensitive to the variable Wicb. The trend line shown is for urban freeways. It drops slightly for rural freeways. Shoulder Rumble Strip CMF. The shoulder rumble strip CMF is described using the following equation. ( ) ( ) curicicaggrssv fPPCMF  +−= ,,|, 0.10.1 (130) ( ) ( )20.10.1]0.1[5.020.10.1]0.1[5.0 ororirircur PPPPf +−++−= (131) This CMF is applicable to single-vehicle crashes. The regression analysis indicated that shoulder rumble strip presence had an insignificant correlation with multiple-vehicle crashes. The proportion Pir represents the proportion of the segment length with rumble strips present on the inside shoulders. It is computed by summing the length of roadway with rumble strips on the inside shoulder in both travel directions and dividing by twice the segment length. The proportion Por represents the proportion of the segment length with rumble strips present on the outside shoulders. It is computed by summing the length of roadway with rumble strips on the outside shoulder in both travel directions and dividing by twice the segment length. The constant “1.20” in Equation 131 represents the calibration coefficient after conversion. It corresponds to a CMF value of 1.20 for PDO crashes on curved road segments when shoulder rumble strips are continuously present. It suggests that there are 20 percent more crashes on curved road segments when rumble strips are present. A review of the literature on the safety effect of shoulder rumble strips on curves did not reveal any evidence that could support or refute this finding. In related research, Torbic et al. (2009) examined the safety effect of centerline rumble strips on rural two-lane highways in three states. They found that centerline rumble strips increased total crashes 3.5 percent on curved segments (although this result was not statistically significant). Outside Barrier CMF. The calibrated outside barrier CMF is described using the following equation. ( ) ocbWobobaggobsv ePPCMF /169.0|, 0.10.1 +−= (132) This CMF is applicable to single-vehicle crashes. The regression analysis indicated that outside barrier presence had an insignificant correlation with multiple-vehicle crashes. Guidance for computing the variables Pob and Wocb was provided previously in the subsection titled Barrier Variable Calculations. The variable Wocb (representing the distance from the edge of outside shoulder to median barrier face) ranges in value from 1.0 to 17 ft in the database. The base condition for this CMF is no barrier.

170 1.00 1.02 1.04 1.06 1.08 10 15 20 25 30 Distance from Edge of Traveled Way to Barrier, ft C ra sh M od ifi ca tio n Fa ct or Rural, 4 Lanes Rural, 6 Lanes Urban 4 Lanes 10-ft Outside Shoulder Width Barrier or Bridge Rail for 100% of Segment Urban 6 Lanes The combined outside barrier CMF is shown in Figure 70. Equation 86 was used to create this CMF (with CMFmv, ob = 1.0). It represents the likely correlation between outside barrier offset distance and crash frequency for total crashes. The trend lines represent different combinations of area type and through lanes through their association with the different proportions of multiple-vehicle crashes in Table 44. Figure 70. Calibrated freeway outside barrier CMF for PDO crashes. Lane Change CMF. The calibrated lane change CMF is described using the following equations. ( ) ( )declcdecwevinclcincwevagglcmv ffffCMF ,,,,|, 5.05.0 += (133) ( ) incwevLincwevBincwevBincwev ePPf ,/123.0,,, 0.10.1 +−= (134) ( ) decwevLdecwevBdecwevBdecwev ePPf ,/123.0,,, 0.10.1 +−= (135) [ ] [ ]       −+×         −+= − −− − −− L AADTbX L AADTX inclc e L e e L ef extevexte entbentb 46.13 )000,1/ln(283.046.13 46.13 )000,1/ln(283.046.13 , 0.1 46.13 0.1 0.1 46.13 0.1 ,, ,, (136) [ ] [ ]       −+×         −+= − −− − −− L AADTX L AADTX declc e L e e L ef extbextb enteente 46.13 )000,1/ln(283.046.13 46.13 )000,1/ln(283.046.13 , 0.1 46.13 0.1 0.1 46.13 0.1 ,, ,, (137)

171 The variables for weaving section length (i.e., Lwev, inc, Lwev, dec) in Equations 134 and 135 are intended to reflect the degree to which the weaving activity is concentrated along the freeway. This variable has negligible correlation with segment length L. The variables PwevB, inc and PwevB, dec in Equations 134 and 135, respectively, are computed as the ratio of the length of the weaving section on the segment to the length of the segment. If the segment is wholly located in the weaving section, then this variable is equal to 1.0. The calibration coefficient in these two equations indicates that lane change CMF value will increase if the segment is in a Type B weaving section. The amount of this increase is inversely related to the length of the weaving section. This CMF consists of several component equations but only requires a few input variables. These variables describe the distance to (and volume of) the four nearest ramps to the subject segment. Two of the ramps of interest are on side of the freeway with travel in the increasing milepost direction. One ramp on this side of the freeway is upstream of the segment and one ramp is downstream of the segment. Similarly, one ramp on the other side of the freeway is upstream of the segment and one ramp is downstream. Only those entrance ramps that contribute volume to the subject segment are of interest. Hence a downstream entrance ramp is not of interest. For similar reasons, an upstream exit ramp is not of interest. If the segment is in a Type B weaving section, then the length of the weaving section is also an input. The lane-change CMF was applied to a range of weaving section lengths and the average CMF was computed for each length. The results of this process are plotted in Figure 71. The lane-change CMF is applicable to any segment in the vicinity of one or more ramps. It is equally applicable to segments in a weaving section and segments in a non-weaving section (i.e., segments between an entrance ramp and an exit ramp where both ramps have a speed- change lane). The two component models for predicting speed-change-related crash frequency (i.e., Equations 121 and 122) are not used when evaluating a weaving section because the ramps that form the weaving section do not have a speed-change lane. As a result, the predicted crash frequency for the set of segments that comprise a weaving section will tend to be smaller than that predicted for a similar set of segments located in a non-weaving section. This generalization will always be true for the Type A and Type C weaving sections. It may or may not hold for the Type B weaving section, depending on the length of the weaving section.

172 1.0 1.1 1.2 1.3 1.4 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Weaving Section Length, mi A ve ra ge C ra sh M od ifi ca tio n Fa ct or Ramp AADT = 6,000 veh/day Urban, 6-lane Rural 6-lane Rural 4-lane Urban, 4-lane Figure 71. Average CMF value for PDO crashes as a function of weaving section length. The calibration coefficient associated with the ramp AADT term in Equations 136 and 137 is negative which is counterintuitive at first glance. It indicates that the lane change CMF is larger for segments associated with lower volume ramps. This trend may be explained by the fact that high-volume ramp flows tend to dominate the traffic stream such that a large portion of the traffic stream is changing lanes and all drivers are more aware of these maneuvers. Regardless, the entering ramp volumes are also included in the segment AADT volume and the coefficient associated with the segment AADT variable in the SPF is positive and relatively large. As a result, when all relevant SPFs and CMFs are combined, the predicted average crash frequency for a freeway segment increases with an increase in the AADT volume of nearby ramps. This trend is logical and intuitive. Ramp Entrance CMF. The ramp entrance CMF is described using the following equation. enleft LI en eCMF /0252.0824.0 += (138) This CMF is applied to a ramp entrance speed-change lane, as shown in Figure 38. The “segment” length is equal to the ramp entrance length Len, which is measured using the gore and taper points identified in Figure 11. This CMF applies only to the side of the freeway with the subject speed-change lane. The variable for ramp entrance length Len in Equation 138 is intended to reflect the degree to which the lane-changing activity is concentrated along the ramp entrance. This variable has negligible correlation with segment length L. The indicator variable for ramp side Ileft is associated with a positive calibration coefficient. It suggests that a ramp entrance on the left side of the through lanes is associated with a 128 percent increase in crashes, relative to one on the right side. This finding is consistent

173 1.00 1.05 1.10 1.15 1.20 1.25 1.30 300 500 700 900 Ramp Entrance Length, ft C ra sh M od ifi ca tio n Fa ct or . with that of Moon and Hummer (2009), and with that from the re-analysis of the data reported by Lundy (1966), as documented in Chapter 2. The ramp entrance CMF for right-side ramps is shown in Figure 72. It has been adjusted using Equation 106 to convert it into a CMF that is applicable to the entire segment (i.e., both sides of the freeway). The ramp entrance lengths used to calibrate this CMF range from 0.07 to 0.22 mi (370 to 1,200 ft). Figure 72. Calibrated freeway ramp entrance CMF for right-side ramps and PDO crashes. Ramp Exit CMF. The ramp exit CMF is described using the following equation. leftI ex eCMF 824.0= (139) This CMF is applied to a ramp exit speed-change lane, as shown in Figure 38. The “segment” length is equal to the ramp exit length Lex, which is measured using the gore and taper points identified in Figure 11. This CMF applies only to the side of the freeway with the subject speed-change lane. The interpretation of the indicator variable for ramp side Ileft is provided with the previous CMF discussion. High-Volume CMF. The calibrated high-volume CMF has two forms, depending on which component model is being used. The CMF for multiple-vehicle non-entrance/exit crashes, ramp-entrance-related crashes, and ramp exit related crashes is described using the following equation. hvP hvmv eCMF 283.0 , = (140)

174 1.00 1.05 1.10 1.15 1.20 0.0 0.2 0.4 0.6 0.8 1.0 Proportion AADT during High-Volume Hours C ra sh M od ifi ca tio n Fa ct or Rural 6-lane Urban 6-lane Rural 4-lane Multiple-vehicle crashes Urban 4-lane 0.80 0.85 0.90 0.95 1.00 0.0 0.2 0.4 0.6 0.8 1.0 Proportion AADT during High-Volume Hours C ra sh M od ifi ca tio n Fa ct or Rural 6-lane Urban 6-lane Rural 4-lane Single-vehicle crashes Urban 4-lane The CMF for single-vehicle crashes is described using the following equation. hvP hvsv eCMF 611.0 , −= (141) The proportion of AADT during hours where volume exceeds 1,000 veh/h/ln Phv is computed using the average hourly volume distribution associated with the subject segment. This distribution will typically be computed using the data obtained from the nearest continuous traffic counting station (on a freeway of similar character). The variable Phv is positively correlated with the volume-to-capacity ratio experienced by the segment on an hourly basis. The high-volume CMF is shown in Figure 73. The trend lines represent different combinations of area type and through lanes through their association with the different proportions of multiple-vehicle crashes in Table 44. The base condition for this CMF is a proportion Phv equal to 0.0. The trends shown in both figures are consistent with those developed by Lord et al. (2005) for multiple-vehicle and single-vehicle crashes as a function of volume-to- capacity ratio. a. Multiple-vehicle crashes. b. Single-vehicle crashes. Figure 73. Calibrated freeway high-volume CMF for PDO crashes. Sensitivity Analysis The relationship between crash frequency and traffic demand, as obtained from the combined calibrated models, is illustrated in Figure 74 for a 1-mile freeway segment with two ramp entrances, two ramp exits, no curvature, and no barrier. The individual component models are illustrated in Figures 74a, 74b, 74c, and 74d. The sum of the individual component crash frequencies is illustrated in Figure 75. The length of the trend lines in Figures 74 and 75 reflect the range of AADT volume in the data.

175 0 10 20 30 40 50 0 50 100 150 200 250 Average Daily Traffic Demand (1000s), veh/day PD O M ul tip le -V eh ic le C ra sh Fr eq ue nc y, c ra sh es /y r 1.0-mile segment length, no barrier 4 lanes 6 8 Rural Freeway Urban Freeway 6 10 8 0 2 4 6 8 10 0 50 100 150 200 250 Average Daily Traffic Demand (1000s), veh/day PD O S in gl e- Ve hi cl e C ra sh Fr eq ue nc y, c ra sh es /y r 1.0-mile segment length, no barrier 4 lanes 6 8 Rural and Urban Freeway 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 40 80 120 Directional Average Daily Traffic (1000s), veh/day PD O R am p En tra nc e C ra sh Fr eq ue nc y, c ra sh es /y r 4 lanes 8 10 6 Ramp entrance length = 700 ft Rural Freeway Urban Freeway 0.0 0.5 1.0 1.5 0 40 80 120 Directional Average Daily Traffic (1000s), veh/day PD O R am p Ex it C ra sh Fr eq ue nc y, c ra sh es /y r 4 lanes Rural and Urban Freeway 8 10 6 a. Multiple-vehicle crashes. b. Single-vehicle crashes. c. Ramp-entrance-related crashes. d. Ramp-exit-related crashes. Figure 74. Freeway PDO model components. The trend lines shown in Figure 75 indicate that urban freeways have about 15 percent more crashes than rural freeways. By comparison, the crash rates listed in Table 19 indicate that urban freeways have 50 to 250 percent more crashes than rural freeways. It is likely that this latter trend reflects the influence of barrier length, ramp entrances, ramp exits, and weaving section length. As shown in Table 18, these influences are more prevalent on urban freeway segments. In contrast, these influences have been explicitly quantified in the proposed model such that they do not influence the trends shown in Figure 75. The proposed model provides a more accurate indication of differences between freeway segments in rural versus urban areas, when the segments have the same barrier proportion, ramp entrance length, and weaving section length.

176 0 10 20 30 40 50 60 70 0 50 100 150 200 250 Average Daily Traffic Demand (1000s), veh/day To ta l P D O C ra sh F re qu en cy , cr as he s/ yr 1.0-mile segment length, no barrier 2 entrance ramps, 2 exit ramps 4 lanes 8 Rural Freeway Urban Freeway 6 10 4, 6, 8 lanes Figure 75. Freeway PDO model. The trend lines shown in Figure 75 also indicate that crash frequency is lower on freeways with many lanes than it is on freeways with few lanes. In fact, the models indicate that an urban six-lane freeway segment has about 15 percent fewer crashes than an urban four-lane segment and that a rural six-lane segment has about 10 percent fewer crashes than an urban four- lane segment. These trends are counter to those found when comparing the crash rates in Table 19, which indicate that crash rate is higher on freeways with many lanes. It is likely that these trends in crash rate reflect the fact that the proportion of barrier along a freeway segment typically increases (and the lateral clearance decreases) with an increase in the number of lanes. The proposed predictive models account for the influence of barrier presence and lateral clearance and, therefore, provide a more accurate indication of the relationship between number of lanes and crash frequency. NOMENCLATURE σp = standard deviation of Pi; AADT = AADT volume on segment, veh/day; AADTb, ent = AADT volume of entrance ramp located at distance Xb,ent, veh/day; AADTb, ext = AADT volume of exit ramp located at distance Xb,ext, veh/day; AADTe, ent = AADT volume of entrance ramp located at distance Xe,ent, veh/day; AADTe, ext = AADT volume of exit ramp located at distance Xe,ext, veh/day; AADTr = AADT volume of ramp, veh/day; AADTr, i = AADT volume of ramp i, veh/day; b0 = calibration coefficient; bi = calibration coefficient for condition i bj,i = calibration coefficients for model j (j = mv, sv, en, ex), i = 0, 1; Cen = local calibration factor for ramp-entrance-related crashes; Cex = local calibration factor for ramp-exit-related crashes;

177 CMF1 ... CMFk = crash modification factors for freeway segment crashes at a site with specific geometric design features k; CMFen = ramp entrance crash modification factor; CMFen|agg = aggregated ramp entrance crash modification factor; CMFen,1 ... CMFen, y = crash modification factors for ramp-entrance-related crashes at a site with specific geometric design features y; CMFex, 1 ... CMFex, z = crash modification factors for ramp-exit-related crashes at a site with specific geometric design features z; CMFex|agg = aggregated ramp exit crash modification factor; CMFhc = horizontal curve crash modification factor; CMFhc|agg = aggregated horizontal curve CMF for a segment with both tangent and curved portions; CMFhv = high-volume crash modification factor; CMFi = crash modification factor for element i; CMFi|agg = aggregated CMF for element i; CMFisw = inside shoulder width crash modification factor; CMFj, i = crash modification factor for element i and crash type j (j = mv, sv); CMFlc = lane change crash modification factor; CMFlw = lane width crash modification factor; CMFmb|agg = aggregated median barrier crash modification factor; CMFmv, 1 ... CMFmv, w = crash modification factors for multiple-vehicle crashes at a site with specific geometric design features w; CMFmv, hc|agg = aggregated horizontal curve crash modification factor for multiple-vehicle crashes; CMFmv, hv = high-volume crash modification factor for multiple-vehicle crashes; CMFmv, lc|agg = aggregated lane change crash modification factor for multiple-vehicle crashes; CMFmv, mw|agg = aggregated median-width crash modification factor for multiple-vehicle crashes; CMFsv, 1 ... CMFsv, x = crash modification factors for single-vehicle crashes at a site with specific geometric design features x; CMFsv, hc|agg = aggregated horizontal curve crash modification factor for single-vehicle crashes; CMFsv, hv = high-volume crash modification factor for single-vehicle crashes; CMFsv, mw|agg = aggregated median-width crash modification factor for single-vehicle crashes; CMFsv, ob|agg = aggregated outside barrier crash modification factor for single-vehicle crashes; CMFsv, oc|agg = aggregated outside clearance crash modification factor for single-vehicle crashes; CMFsv, osw|agg = aggregated outside shoulder width crash modification factor for single- vehicle crashes; CMFsv, rs|agg = aggregated shoulder rumble strip crash modification factor for single-vehicle crashes; Cmv = local calibration factor for multiple-vehicle non-entrance/exit crashes; Csv = local calibration factor for single-vehicle non-entrance/exit crashes; e = superelevation rate, ft/ft; fcur = factor for rumble strip presence on curved portions of the segment; fd = side friction demand factor; flc, dec = lane change adjustment factor for travel in decreasing milepost direction; flc, inc = lane change adjustment factor for travel in increasing milepost direction;

178 ftan = factor for rumble strip presence on tangent portions of the segment; fwev, dec = weaving section adjustment factor for travel in decreasing milepost direction; fwev, inc = weaving section adjustment factor for travel in increasing milepost direction; Ic = curve deflection angle, degrees; Ien = crash indicator variable (= 1.0 if ramp-entrance-related crash data, 0.0 otherwise); Iex = crash indicator variable (= 1.0 if ramp-exit-related crash data, 0.0 otherwise); Ileft,i = ramp side indicator variable for ramp i (= 1.0 if entrance or exit is on left side of through lanes, 0.0 if it is on right side); Imv = crash indicator variable (= 1.0 if multiple-vehicle non-entrance/exit crash data, 0.0 otherwise); In = cross section indicator variable (= 1.0 if cross section has n lanes, 0.0 otherwise); Irural = area type indicator variable (= 1.0 if area is rural, 0.0 if it is urban); Isv = crash indicator variable (= 1.0 if single-vehicle non-entrance/exit crash data, 0.0 otherwise); k = overdispersion parameter, mi; K = inverse dispersion parameter (= 1/k), mi-1; knull = overdispersion parameter based on the variance in the observed crash frequency; L = length of segment, mi; Lc = length of horizontal curve (= Ic × R / 5280 / 57.3), mi; Len, i = length of ramp entrance for ramp i (may extend beyond segment boundaries), mi; Len, seg, i = length of ramp entrance i on segment, mi; Lex, i = length of ramp exit for ramp i (may extend beyond segment boundaries), mi; Lex, seg, i = length of ramp exit i on segment, mi; Lib, i = length of lane paralleled by inside barrier i (include both travel directions), mi; Lob, i = length of lane paralleled by outside barrier i (include both travel directions), mi; and Lwev, dec = weaving section length for travel in decreasing milepost direction (may extend beyond segment boundaries), mi; Lwev, inc = weaving section length for travel in increasing milepost direction (may extend beyond segment boundaries), mi; n = number of observations; n = number of through lanes on segment; N = predicted average crash frequency, crashes/yr; Ncurve = predicted average additional crashes due to curvature; crashes/yr; Nen = predicted average ramp-entrance-related crash frequency, crashes/yr; Nex = predicted average ramp-exit-related crash frequency, crashes/yr; Nj = predicted average crash frequency for model j (j = mv, sv, en, ex); crashes/yr; Nmv = predicted average multiple-vehicle non-entrance/exit crash frequency, crashes/yr; Nsection = predicted average crash frequency within the limits of a freeway section, crashes/yr; Nseg = predicted average crash frequency on segment (regardless of curvature), crashes/yr; Nspf, en = predicted average ramp-entrance-related crash frequency for base conditions, crashes/yr; Nspf, ex = predicted average ramp-exit-related crash frequency for base conditions, crashes/yr; Nspf, mv, n = predicted average multiple-vehicle non-entrance/exit crash frequency for base conditions for number of through lanes n (n = 4, 6, 8, 10); crashes/yr;

179 Nspf, mv = predicted average multiple-vehicle non-entrance/exit crash frequency for base conditions, crashes/yr; Nspf, sv = predicted average single-vehicle non-entrance/exit crash frequency for base conditions, crashes/yr; Nsv = predicted average single-vehicle non-entrance/exit crash frequency, crashes/yr; P1 = proportion of AADT in travel direction 1; P2 = proportion of AADT in travel direction 2 (= 1.0 - P1); Pc = proportion of the segment length with curvature; Pc,i = proportion of segment length with curve i; Phv = proportion of AADT during hours where volume exceeds 1,000 veh/h/ln; Pi = proportion of AADT in travel direction i that corresponds to subject speed-change lane; Pib = proportion of segment length with a barrier present in the median (i.e., inside); Pir = proportion of segment length with rumble strips present on the inside shoulders; PL, i = proportion of the segment length with element i; Pmv = proportion of multiple-vehicle crashes; Pob = proportion of segment length with a barrier present on the roadside (i.e., outside); Por = proportion of segment length with rumble strips present on the outside shoulders; PwevB, dec = proportion of segment length within a Type B weaving section for travel in decreasing milepost direction; PwevB, inc = proportion of segment length within a Type B weaving section for travel in increasing milepost direction; R = curve radius, ft; Ri = radius of curve i, ft; se = root mean square error of the model estimate, crashes/yr; Vc = average curve speed, mi/h; V[X] = crash frequency variance for a group of similar locations, crashes2; Whc = clear zone width, ft; Wib = inside barrier width (measured from barrier face to barrier face), ft; Wicb = distance from edge of inside shoulder to barrier face, ft; Wis = inside shoulder width, ft; Wl = lane width, ft; Wm = median width (measured from near edges of traveled way in both travel directions), ft; Wnear = “near” horizontal clearance from the edge of the traveled way to the continuous median barrier (measure for both travel directions and use the smaller distance), ft; Wocb = distance from edge of outside shoulder to barrier face, ft; Woff, in, i = horizontal clearance from the edge of the traveled way to the face of inside barrier i, ft. Woff, o, i = horizontal clearance from the edge of the traveled way to the face of outside barrier i, ft. Ws = outside shoulder width, ft; X = reported crash count for y years, crashes; X. = average crash frequency for all n observations; Xb, ent = distance from segment begin milepost to nearest upstream entrance ramp gore point, for travel in increasing milepost direction, mi; Xb, ext = distance from segment begin milepost to nearest downstream exit ramp gore point, for travel in decreasing milepost direction, mi;

180 xb= distance from ramp gore to start of segment, mi; Xe, ent = distance from segment end milepost to nearest upstream entrance ramp gore point, for travel in decreasing milepost direction, mi; Xe, ext = distance from segment end milepost to nearest downstream exit ramp gore point, for travel in increasing milepost direction, mi; xe= distance from ramp gore to end of segment (xe > xb), mi; y = time interval during which X crashes were reported, yr;