Consideration of Roadside Features in the Highway Safety Manual (2022)

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26 METHOD RUN-OFF-ROAD CRASH PREDICTION MODEL FORM The functional form of the Run-off-Road (ROR) prediction model includes two crash modification functions: CMFROADWAY and a CMFROADSIDE. Equation (1) shows the general form for a ROR crash prediction model. The model predicts the number of expected crashes in a year of a particular severity that cross a particular edge of a particular segment. 𝑁𝑁𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑅𝑅𝐹𝐹𝐴𝐴𝑆𝑆 = 𝑆𝑆𝑆𝑆𝐶𝐶𝑆𝑆𝐴𝐴𝐸𝐸𝑆𝑆 ∙ 𝐶𝐶𝐶𝐶𝐶𝐶𝑅𝑅𝑂𝑂𝐴𝐴𝐴𝐴𝑅𝑅𝐴𝐴𝑆𝑆 ∙ 𝐶𝐶𝐶𝐶𝐶𝐶𝑅𝑅𝑂𝑂𝐴𝐴𝐴𝐴𝑆𝑆𝐹𝐹𝐴𝐴𝑆𝑆 (1) where: NSEVERITY = Annual number of ROR crashes of a particular severity associated with a given roadway segment edge. SPFEDGE = Safety performance function for an edge of the roadway in crashes per length of segment edge per year. CMFROADWAY = A crash modification function that adjusts for the alignment and cross- sectional features of the roadway like grade, curvature, lane width and number of lanes. CMFROADSIDE = A crash modification function that adjusts for the features of the roadside. Certain crash modification factors used within CMFROADSIDE are chosen for a particular severity. CMFROADWAY accounts for roadway characteristics that modify the likelihood of a vehicle leaving the roadway. Individual countermeasures are be multiplied together to find the aggregate CMFROADWAY as shown in equation (2). 𝐶𝐶𝐶𝐶𝐶𝐶𝑅𝑅𝑂𝑂𝐴𝐴𝐴𝐴𝑅𝑅𝐴𝐴𝑆𝑆 = �𝐶𝐶𝐶𝐶𝐶𝐶𝑝𝑝 𝑛𝑛 𝑝𝑝=1 (2) The severity of an ROR crash will depend on the object struck off the road and not the geometry of the roadway itself. CMFROADSIDE is necessarily a function that adjusts for the features of the roadside. This function predicts an increase or decrease in the frequency of various severity ROR crashes considering the portion of the roadside which is shielded and unshielded. The roadside crash modification function, CMFROADSIDE is shown in equation (3). 𝐶𝐶𝐶𝐶𝐶𝐶𝑅𝑅𝑂𝑂𝐴𝐴𝐴𝐴𝑆𝑆𝐹𝐹𝐴𝐴𝑆𝑆 = �𝛽𝛽𝑆𝑆𝑅𝑅𝐿𝐿𝐴𝐴 ∙ 𝑋𝑋𝑆𝑆𝑅𝑅𝐿𝐿𝐴𝐴 ∙�𝐶𝐶𝐶𝐶𝐶𝐶𝑗𝑗 𝑏𝑏1 𝑗𝑗=1 � + �𝛽𝛽𝑈𝑈𝑈𝑈𝑆𝑆𝑅𝑅𝐿𝐿𝐴𝐴 ∙ 𝑋𝑋𝑈𝑈𝑈𝑈𝑆𝑆𝑅𝑅𝐿𝐿𝐴𝐴 ∙�𝐶𝐶𝐶𝐶𝐶𝐶𝑏𝑏 𝑏𝑏2 𝑏𝑏=1 � (3)

27 Where: XSHLD = Proportion of the segment edge where longitudinal barriers are installed where 0 ≤ XSHLD ≤ 1. XUNSHLD = Proportion of the segment edge where there are unshielded ditches or roadside slopes and other unshielded fixed objects where 0 ≤ XUNSHLD ≤ 1. Condition: 1 = XSHLD + XUNSHLD (100% of the segment edge is accounted for). βSHLD = The probability of a longitudinal barrier crash of each severity associated with the segment edges where longitudinal barriers are installed. βUNSHLD = The probability of a non-longitudinal barrier ROR crash of each severity associated with the segment edges where there are unshielded ditches, roadside slopes or fixed objects like trees, tree lines, utility poles, bridge piers, etc. CMFj = Crash modification factors that modify the ROR crashes associated with longitudinal barriers. These CMFs account for characteristics like barrier type and barrier offset. CMFk = Crash modification factors associated with unshielded roadsides. These CMFs account for characteristics like the density of narrow fixed objects, slopes, and other unshielded objects. AVAILABLE DATA Candidate data sources for use in modeling run-off-road crashes were examined at the onset of this research. It was desirable to acquire databases which describe both the incidence of these crashes and detail the features of the roadway, roadside, median and roadside hardware which may influence the vehicle trajectory and resulting crash after departure. Data elements in the following areas were of interest: • Crash description (date, time, crash location, contributing factors) • Roadway description (e.g., lanes, alignment, surface type, grade, posted speed limit, AADT) • Roadside description (e.g., shoulder width, side slope geometry, clear zone extent) • Roadside hardware description (e.g., w-beam, concrete barrier, cable barrier, end terminal, etc.) • Object / vehicle struck (tree, pole, curb or other rollover trip mechanism or other vehicle) There are a number of ongoing and recently completed roadside data collection efforts as well are dated roadside databased. Each of these roadside-specific datasets were examined along with the available roadway-based datasets. The NCHRP Project 17-22 database is a set of 890 in-depth crash investigations of run- off road crashes collected primarily from 1997-2001 through supplemental National Automotive Sampling System (NASS) / Crashworthiness Data System (CDS) crash investigations. The dataset includes cases collected specifically in NCHRP 17-22 as well as run-off road cases collected in NCHRP 17-11 and the Federal Highway Administration (FHWA) rollover study. The Project 17-22 database contains, among other elements, reconstructed vehicle trajectories.

28 This dataset is available within RSAPv3. [Ray12] NCHRP 17-43, while only getting underway during this research effort, is intended to supplement the NCHRP 17-22 data indicating a strong desire among roadside safety professionals to continue to collect data such as errant vehicle trajectories. While only the NCHRP 17-22 data was available for this effort, methods which use these type of data are desirable because more data will soon be available. The Highway Safety Information System (HSIS) is a collection of roadway-based crash data. This data from nine states can be mapped to the roadway characteristics and sometime the roadside inventory. Both Ohio and Washington participate in the HSIS program and maintained a roadside inventory at the onset of this effort. While NASS/CDS provides a detailed record of approximately 5,000 crashes investigated each year, at least one of the vehicles in the crash had to have been towed from the scene to be included in NASS/CDS. [NCSA05] Further, the database includes only crashes involving cars, light trucks, vans and sport utility vehicles. Heavy vehicles and motorcycles are not included in the database. NASS/CDS does not contain many of the specialized roadside components, e.g., barrier offset or barrier type that would be desirable for evaluation of roadside hardware effectiveness. However, for several roadside safety studies, the extensive photographs taken in each NASS/CDS have been used to provide this missing information. [Gabauer09; Gabauer10] States which are not part of the HSIS program were also considered. At the onset of this research, the number of states routinely maintaining both segment and roadside inventories was very limited and did not permit the inclusion of additional states. Fortunately, by the close of this effort, these states have grown considerably in number and should provide a good amount of additional data as future research in this area develops As very briefly explained, more data sources are being developed and collected. The databases most suited for detailed analysis which would meet the statistical rigors of the HSM and which were available to this research were the HSIS data and the supplemental roadside inventories. The states of Washington and Ohio were early collectors of roadside inventory data. This research, therefore, used the Washington and Ohio roadside inventory in conjunction with the FHWA HSIS data for these states. This research also developed methods which made it possible to use the trajectory data currently available in RSAPv3. As more trajectory data becomes available, these methods will become more powerful. DATABASE COMPILATION The highway information for both the Washington and Ohio HSIS data includes geometric characteristics (i.e., horizontal and vertical curve data, lane width, shoulder width, median width, etc.) and traffic volumes for each year. Crash data is stored in separate files for each year which can be linked to the highway geometrics using the roadway name and milepost. The Ohio highway data includes heavy vehicle volumes while the Washington highway data includes the percent trucks (PT). PT was calculated for Ohio, using the traffic volumes and heavy vehicle volume so that both datasets were consistent. The highway data were used to create homogeneous segments. Each homogenous segment has consistent horizontal and vertical alignments, consistent lane and shoulder widths, and consistent traffic characteristics. After segmenting the highway data, the crash records were allocated to each segment using the route and mile post. All vehicles that ran off the road in any sequence of the crash event were included in the ROR crashes for that segment. An undivided roadway has two outside edges. A divided roadway, however, has four edges (two outside edges and two median edges). For the purposes of these discussions, the

29 increasing milepost direction will hereafter be noted as the primary direction of travel and the decreasing milepost will be the opposing direction. The outside right edge in the increasing milepost direction is therefore the primary right edge (PRE). The outside right edge in the decreasing milepost direction is the opposing right edge (ORE). Therefore, undivided and divided roadways both have a PRE and an ORE. When a median is present, the median edge in the increasing milepost direction is the primary left edge (PLE) and the median edge in the decreasing milepost direction is the opposing left edge (OLE). Many roadside safety publications reference similar nomenclature to discuss roadside encroachment directions. The primary and opposing direction of travel and the possible vehicle encroachment directions from these directions of travel are noted as follows: primary right (PR), primary left (PL), opposing right (OR), opposing left (OL). While crashes and encroachments are different in definition, the nomenclature has been adopted here to eliminate the potential for conflicts between crash-based and encroachment-based publications and to provide consistency. Figure 9 shows these possible encroachment directions and crash edges graphically. a) Undivided Highway b) Divided Highway PR=Primary Right; PL=Primary Left; OR=Opposing Right; OL=Opposing Left PRE=Primary Right Edge; PLE=Primary Left Edge; ORE=Opposing Right Edge; OLE=Opposing Left Edge Figure 9. Possible Vehicle Crashes by Direction of Travel and Highway Type. The crash edge where a crash occurred was determined through review of the coded crash sequence of events. The coded event in the sequence of events where the vehicle exited the roadway just prior to striking an object or rolling over was used to determine the crash edge. For example, if a vehicle was traveling in the primary direction on an undivided road and exits the right side of the road, then over-corrected and crossed the centerline to exit the left side of the road where it struck a fixed object, the vehicle was counted as an opposing right edge (ORE) PL PR OLOR N PR+OL=PRE OR+PL=ORE N

30 crash because it was traveling primary direction and exited the road to the left just before hitting a fixed object. Another example would be if the vehicle traveling in the opposing direction entered the median of a divided road then crossed through the median and became involved in a head-on collision in the opposite (i.e., primary) lanes. This would be an OLE crash because the vehicle was traveling in the opposing direction and first crossed the left edge. If the direction of crash could not be determined, it was included in the dataset as unknown (Unk). Full documentation on the compiled datasets used is provided in Appendix C: Databases Documentation. Divided Roadways The difference between a primary direction crash and an opposing direction edge crash is the direction of travel and the characteristics of the roadway relative to that direction. Analysis of the data (an ultimately use of the model) can consider opposite-direction crashes as equal to primary direction crashes when the direction of travel and geometric characteristics are transformed. All segments and crashes occurring in the opposing direction of travel were transformed to occur in the primary direction of travel, thereby doubling the primary direction edge crashes. The opposing direction curve and grade signs were reversed and the crashes found to have occurred on the ORE or OLE were changed to PRE and PLE so that the direction of curvature and/or grade with respect to the direction of travel was maintain and so the data could be merged. In other words, a PRE crash is equivalent to an ORE crash since the selection of which direction is primary is arbitrary as long as the grade and curvature are recorded with respect to the direction of travel. Undivided Roadways All segments and crashes occurring in the opposing direction of travel were transformed to the primary direction of travel as described above for the divided dataset, however, there are some subtle differences. As with the divided dataset, the opposing direction curve and grade signs were reversed. The crashes which occurred in the OR or PL direction, however, were switched to have occurred in the PR and OL direction to account for vehicles that cross the center line and leave the roadway for the first time on the opposite edge. This difference represents a fundamental difference in the way undivided and divided roadways operate. An undivided roadway may have a PRE crash from a vehicle traveling in the primary direction that exits to the right or a vehicle traveling in the opposing direction that exits to the left. A vehicle which crosses over the centerline has different highway geometrics with respect to its direction of travel which in turn affects the probability of a road departure. For simplicity, these centerline crossover crashes which results in ROR crashes have been included in the total edge crashes. Each edge of an undivided highway, therefore, includes a contribution from vehicles traveling in the primary direction that depart to the right edge as well as another contribution from vehicles traveling in the opposing direction that cross the centerline and leave the road on the primary right edge. Types of ROR Crashes The development of the CMFROADSIDE included the modeling of longitudinal barrier crashes and the modeling of all other ROR crashes. Recall that any vehicle that runs off the road in any sequence of events is included in the crash dataset (ROR).

31 A longitudinal barrier (LB) crash subset of the complete (ROR) dataset was defined as any crash where the longitudinal barrier is the first object struck off the road. In other words, if a vehicle runs of the road to the left and hits a w-beam, it is a longitudinal barrier crash. If a vehicle side-swipes another vehicle then runs off the road to the right and hits a longitudinal barrier, it is still a longitudinal barrier crash. On the other hand, if a vehicle runs of the road to the right and hits a tree then a longitudinal barrier, it is an “other” ROR crash (OC). Using these definitions for LB and OC, the P(LB) and P(OC) are mutually exclusive events. Roadside Inventories The segments used to develop the SPFs and on-road CMFs were developed from road- based datasets which include information about the frequency, severity, and location of crashes relative to the highway geometry, posted speed limit, traffic mix, etc. The longitudinal barrier inventories maintained by the states were used to determine the percentage of longitudinal barrier for each segment and subsequently the percentage of unprotected roadside (i.e., XSHLD and XUNSHLD). Unfortunately, the inventories do not include detailed information about the barriers located in the medians of divided highway. Specifically, it was not clear from the inventories if the barriers present are median barriers or single-faced barriers. Single-faced barriers struck from behind do not function as a longitudinal barrier is intended to function and single-faced barriers placed in the median may be subjected to these types of non-designed impact scenarios. SPF MODELING The state of the practice for modeling count data such as highway crashes is to fit a negative binomial model, usually with a Poisson-gamma mixture distribution. “In statistics, count data refer to observations that have only nonnegative integer values ranging from zero to some greater undetermined value.” [Hilbe14] In highway safety, zero counts of crashes are particularly important and represent areas where crashes were not observed (i.e., lower risk areas). One approach to tracking zero counts as well as the non-zero counts is to track crashes by highway segment. This approach has the added benefit to allowing the consideration of the influence of segment characteristics on crash frequency. The crash counts become the response variable and the segment characteristics such as segment length and traffic volume quantify exposure while highway type (i.e., divided or undivided), traffic mix, highway geometrics, and area type (i.e., urban or rural) become the explanatory variables which explain the occurrence of the crashes. The characteristics of segments are used to explain why each segment experiences more or fewer crashes then other segments. When robust datasets are available, models can be segregated by some of the explanatory variable (e.g., different models may be developed for area type and/or highway type). Ideally, all possible predictor variables would be known. This ideal situation remains unrealized, however, it is common to consider the known explanatory variables when developing a model to ensure the effect of the explanatory variable of interest is not misrepresented. For this research, models were developed relating the explanatory variables to the response variable using the method of maximum likelihood to quantify the magnitude of each predictors relationship. Along with each model, the fit statistics are presented. The p-value is a measure of how probable the result observed may have occurred by chance. A low p-value indicates the results are statistically significant and were unlikely to have occurred by chance (e.g., p<0.05).

32 A higher p-value only indicates that the results have not proven the null hypothesis false, not that the null hypothesis is true. The p-value cannot be relied on alone. The pseudo R2 statistic was determined for each model. The pseudo- R2 is not interpreted the same way as the coefficient of determination is for an ordinary least squares regression. A low value of the pseudo R2 can indicate lack of fit while higher values carry no such indication. There is no definition of a low value. The Akaike Information Criterion (AIC) fit statistic provides comparative information, with lower values indicating a better fitting model then the model it is compared to. Bayesian Information Criterion (BIC) is interpreted the same way. Both are calculated from the likelihood function. [Hilbe07] Negative binomial model parameters are estimated using maximum likelihood, where the parameters of the probability distribution that characterize the data are estimated. The log of the likelihood function is used to determine which parameters make the model most likely to be the case when the data is considered. Through an iterative process, the derivative of the log likelihood function is taken and set to zero to estimate the parameters. When the difference between iterative values is less than a specified tolerance (i.e., 10 ^-6), the iterations stop and the values are at the maximum likelihood estimated values. The log likelihood (LL) is also reported with the models, however, is only useful when calculating other fit statistics (e.g., AIC and BIC). Any measurement has uncertainty, which should be communicated. This uncertainty in statistical analysis can be conveyed through noting the standard error or the confidence interval along with the measurements. The standard error is a measure of how much the estimate could change within the model. The 95% confidence interval is essentially the same type of statistic as standard error; the 95% confidence interval limits indicates that the analysist is 95% confident the true value of the coefficient is within the stated range. It is important to note the 95% confidence interval is equal to twice the standard error for normally distributed error. Negative binomial models are assumed to have normally distributed error. The negative binomial regression models were developed using the COUNT package available in R. [COUNT16, R17] CMF MODELING A variety of procedures are available for the development of CMFs. The Federal Highway Administration released technical report FHWA-SA-10-032, titled “A Guide to Developing Quality Crash Modification Factors” (see http://www.cmfclearinghouse.org). This document addresses the various issues identified to date regarding the creation and assessment of CMFs. Since CMFs are statistical estimates of the effect of treatments on the predicted number of crashes, methodologies to develop these quantities allow for statistical estimation of their associated standard error. A comparison group approach is sometimes used for the development of CMFs. In this approach the analyst selects two groups with comparable trends in the “before” period, applies the treatment under study to one of the groups, and compares the “after” periods via sample odds ratio. The Empirical Bayes (EB) method estimates the expected number of crashes from a weighted average of two sources of information: the observed (historic) number of crashes at the particular site or sites, and the expected number of crashes based on a known trend in similar sites. The expected number of crashes is typically determined from a SPF for the specific facility type.

33 The Full Bayes (FB) method is a modeling approach comparable to Generalized Linear Regression. The FB approach develops a long term distribution as the expected value, as opposed to the EB which uses a point estimate. Similar to the before-after version, the cross-sectional studies approach directly compares two groups of different sites with different treatment levels. Since other factors may also help to explain performance, the underlying assumption is that the differences as identified using this method are, in fact, caused by the treatment. The case-control study approach uses two samples determined from a selection process based on the outcome (crash/no crash) and then classified by the prior condition (treatment/no treatment). One potential issue with case-control studies is the opportunity for a flawed definition of cases and controls. A cohort study approach begins with the observation of two groups of sites (i.e., with and without treatment) and updates the status as crashes occur. The relative risk is then estimated for the two groups. Meta-analysis refers to the procedure of combining the CMFs and the corresponding standard errors from different studies and using it to arrive at a global estimate with the associated global standard error. The quality of the input CMFs and the associated study design should be considered. Additionally, there is the risk of incurring “in-publication bias” which is to say that unpublished works are not included in the meta-analysis. The expert panel and Delphi methods develop CMFs through iterative consultation with a group of practitioners as they revise and select the best CMFs, in their opinion, from a set of studies. Each of the selection iterations will reduce the differences in opinion until consensus or near consensus is achieved. The Delphi method differs in that it tries to minimize the influence that the panelists may be exerting on each other. Two assumptions underlie the use of CMFs. First, methods for deriving CMFs are based on an assumption that CMFs account for the independent effects of the various treatments. A second underlying assumption is that the data used to develop CMFs is representative of a wide range of sites. Unfortunately, many countermeasure applications only occur at locations where crash mitigation is needed and so this assumption requires careful attention. Attention should be given to the policy and procedures surrounding construction of the countermeasure to ensure that, by the very construction, the site has not been biased. CMFROADWAY The state of the practice in highway safety for developing CMFs from cross-sectional models is to estimate the variable coefficients using a negative binomial regression model. The negative binomial regression model is assumed to take the following form: ln (𝑁𝑁) = �𝛽𝛽𝑝𝑝𝑋𝑋𝑝𝑝 + ln(𝑆𝑆𝑆𝑆𝐶𝐶𝑆𝑆𝐴𝐴𝐸𝐸𝑆𝑆) + 𝜀𝜀 where: N = Expected number of ROR crashes per segment edge per year; ßi = Regression coefficients; Xi = Explanatory variables; and SPFEDGE = Safety Performance Function form used to predict edge crash frequency. ɛ = Gamma-distributed error term.

34 A negative binomial regression model of the edge crashes was estimated for both divided and undivided highways. The model relates the explanatory variables to the response variable using the method of maximum likelihood to quantify the magnitude of the relationship. The analysis was conducted until the model only included explanatory variables with reasonable representation in the dataset and an acceptable p-value in the final cross-sectional model. CMFs were explored for a variety of geometric features. CMFs which take the form of single point estimates and the form of a function were explored to find estimates which best represent the data. A variety of mathematical manipulations of the estimated model coefficients can then be used to solve for the CMFs. Once the coefficients are determined, each CMF needs to be formulated such that under the base conditions the CMF value is unity. This is accomplished algebraically as follows: 𝐶𝐶𝐶𝐶𝐶𝐶𝑝𝑝 = 𝑁𝑁𝑝𝑝 𝑁𝑁𝑡𝑡 = 𝑒𝑒𝛽𝛽𝑖𝑖𝑋𝑋𝑖𝑖 ∙ 𝑆𝑆𝑆𝑆𝐶𝐶𝑆𝑆𝐴𝐴𝐸𝐸𝑆𝑆 ∙ 𝑒𝑒𝜀𝜀 𝑒𝑒𝛽𝛽𝑜𝑜𝑋𝑋𝑜𝑜 ∙ 𝑆𝑆𝑆𝑆𝐶𝐶𝑆𝑆𝐴𝐴𝐸𝐸𝑆𝑆 ∙ 𝑒𝑒𝜀𝜀 𝐶𝐶𝐶𝐶𝐶𝐶𝑝𝑝 = 𝑒𝑒𝛽𝛽𝑖𝑖𝑋𝑋𝑖𝑖 𝑒𝑒𝛽𝛽0𝑋𝑋𝑜𝑜 𝐶𝐶𝐶𝐶𝐶𝐶𝑝𝑝 = 𝑒𝑒(𝛽𝛽𝑖𝑖𝑋𝑋𝑖𝑖−𝛽𝛽0𝑋𝑋𝑜𝑜) where: CMFi = Crash modification factor i for non-base condition roadway feature (e.g., lane width, shoulder width, etc.). Ni and No = Observed number of ROR crashes/year/segment edge for condition i. ßi = Regression coefficient from model for non-base condition variable i. Xo, Xi = Independent characteristics under consideration. SPFEDGE = Safety performance function for ROR segment edges ɛ = Gamma-distributed error term. The Transportation Research Circular E-C142 documents the inclusion process used to filter CMFs to ensure only reliable information is included in the HSM. “CMFs to be considered for future editions of the HSM will be a least as stable as the CMFs found in the first edition.” [Bahar10] CMFs are considered for inclusion in the HSM if the value does not change by more than 50 percent (P<0.5) of the value which is currently published and the standard error is equal to or less than 0.1 (SN≤0.1). The 95% confidence interval is shown for all the CMFs. Recall the 95% confidence interval equals twice the standard error. CMFROADSIDE Recall that any vehicle that runs off the road in any sequence of events is included in the crash dataset (ROR). The longitudinal barrier crash type (LB) subset of the complete ROR dataset was defined as any crash where a longitudinal barrier is the first object struck off the road. In other words, if a vehicle runs off the road to the left and hits a w-beam, it is a longitudinal barrier crash. If a vehicle side-swipes another vehicle then runs off the road to the right and hits a longitudinal barrier, it is still a longitudinal barrier crash. On the other hand, if a vehicle runs of the road to the right and hits a tree then a longitudinal barrier, it is an “other” ROR crash (OC). Using these definitions for LB and OC, the crashes are mutually exclusive events. Thus, the same vehicle cannot be counted as a LB and an OC crash, but only as an LB or

35 an OC crash. Therefore, 𝑆𝑆(𝐿𝐿𝐿𝐿 ∪ 𝑂𝑂𝐶𝐶) = 𝑆𝑆(𝐿𝐿𝐿𝐿) + 𝑆𝑆(𝑂𝑂𝐶𝐶). The P(LB) and P(OC) was determined individually and added together to find the P(ROR). Recall this dataset is only ROR crashes, therefore the P(ROR) of any severity in this dataset is equal to unity and the P(LB) and P(OC) are some fraction of one. This relationship is the basis for CMFROADSIDE (i.e., CMFROADSIDE = P(ROR)). It was assumed that the data could be modeled using a proportional odds model. This assumption was checked using a multinomial logit model. The proportional odds models were fit to each dataset (i.e., urban and rural; divided and undivided) using the MASS package available in R. [Venables02; R17] The multinomial logit models were then fit to each dataset using the nnet package in R. [Venables02; R17] The chi-square distribution was determined and the models were compared using the resulting p-value. When the p-value was high, the proportional odds model was considered to fit the data as well as the multinomial logit model. In these cases, the proportional odds model was favored over the more complicated multinomial logit model. The odds of an event occurring are simply the probability of an event (P) occurring divided by the probability of the event not occurring (1-P). [Hilbe11] 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 = 𝑂𝑂𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒𝑂𝑂𝑂𝑂𝑒𝑒𝑂𝑂 𝑓𝑓𝑎𝑎𝑖𝑖𝑖𝑖𝑠𝑠𝑖𝑖𝑒𝑒𝑂𝑂 = 𝑆𝑆 1 − 𝑆𝑆 For example, if the probability of snow in November is 0.25, the odds of having snow in November are 0.25 1−0.25 = 0.3333 = 1/3. A person gambling on snow in November would say there is a 1 in 3 chance it will snow in November or a 3 to 1 chance it will not snow in November. The limited range of probability (i.e., 0 to 1) presents a problem when used directly in regression modeling, therefore the odds (i.e., P 1−P ) are used instead. Log odds (i.e., logit) provide a more suitable variable for regression modeling because a line is fit using the maximum likelihood method. Using the definitions for ROR, LB, and OC, the relationships used to determine CMFROADSIDE can be conceptualized as follows for any level of severity i, where βi is the slope and ε is the intercept: 𝑖𝑖𝑙𝑙 � P(LB𝑆𝑆𝑆𝑆𝑆𝑆𝑝𝑝) 1 − P(LB𝑆𝑆𝑆𝑆𝑆𝑆𝑝𝑝) � = 𝑖𝑖𝑙𝑙𝑙𝑙𝑖𝑖𝑙𝑙[𝑆𝑆(𝐿𝐿𝐿𝐿𝑆𝑆𝑆𝑆𝑆𝑆𝑝𝑝)] = ε + βiXi Extending these relationships to the proportional log odds model where we are modeling the probability of being in one category or lower, we can explicitly determine the probability of an LB or OC crash of any severity (i.e., K, A, B, C, or O). In this model, there is only one slope, but multiple intercepts (i.e., K, A, B, C, O) with SEVj levels. For example, the model will find the probability of an “O” crash or above, a “C” crash or above, and so forth. [Ford15] 𝑖𝑖𝑙𝑙𝑙𝑙𝑖𝑖𝑙𝑙[𝑆𝑆(𝐿𝐿𝐿𝐿𝑆𝑆𝑆𝑆𝑆𝑆𝑝𝑝 ≤ 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆)] = ε𝑆𝑆𝑆𝑆𝑆𝑆𝑗𝑗 + βX Where X=1 for OC and 0 for LB. The probability of having a crash of a particularly severity, for example the probability of having a “B” level crash, can be found by subtraction: P(LBSEVi=B) = P(LBSEVi≤B) - P(LBSEVi≤A).