Guidelines for the Development and Application of Crash Modification Factors (2022)

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A-1   A P P E N D I X A Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site

A-2 A-3 Chapter 1 Introduction A-5 Chapter 2 Procedure A-5 Step 1 Identify Proposed Treatment and Subject Site A-5 Step 2 Identify Characteristics of Subject Site A-7 Step 3 Identify Existing CMFs for Proposed Treatment A-9 Step 4 Compare Subject Site Characteristics to Known CMF Influential Factors A-14 Step 5 Convert CMF to Disaggregate CMFs A-33 Step 6 Process Disaggregate CMFs A-39 Step 7 Develop Aggregate CMF Using Crash Distribution at Subject Site A-42 Result A-43 Chapter 3 Example Application of Procedure A-50 Additional Example Applications of the Step 5 Disaggregation Process A-68 Chapter 4 Supporting Research A-68 Development of Crash Modification Functions to Identify Influential Factors A-77 Evaluation of Two Estimators of Combined Average CMF A-83 References C O N T E N T S

A-3   C H A P T E R   1 Introduction Safety practitioners must often adopt CMFs developed in other geographic areas and apply those CMFs to their own intersection or road section of interest. In many cases, there are dif- ferences between the sites where the CMF was developed and the site(s) where the practitioners desire to apply the CMF. Safety practitioners need to know if a CMF developed under one set of conditions (e.g., state, roadway type, geography, and agency design practices) can be applied to a site with another set of conditions. This appendix describes a procedure for estimating the effect of a proposed treatment on a site of interest. The procedure consists of a step-by-step approach using CMFs reported in the literature, matching on major site characteristics, and adjusting them to produce a final CMF to be used at the site of interest. A major concept that underlies the procedure is the use of the crash distribution at the site of interest to produce a more reliable estimate of treatment effect at that location (relative to other locations having a different crash distribution). In other words, the CMF is made “transferable” by adjusting for differences in the crash distributions between the location where it was devel- oped and the location where the safety practitioner wishes to apply the CMF. The crash distri- bution can be described in terms of crash type, crash severity, or crash location. In this regard, the procedure uses CMFs and predicted crash frequencies for specific crash types or severities (e.g., rear-end crashes resulting in injury) to estimate the expected change in crash frequency associated with a given treatment. The procedure produces CMFs that are “locally calibrated” using the crash distribution representative of the site of interest. Importance of Disaggregate CMFs In recent years, there has been an increase in development of disaggregate CMFs, that is, CMFs that are specific to a given crash type or severity category. In some cases, the CMFs devel- oped are specific to a specific crash type and crash severity; in other cases, they may be specific to just a subset of crash types or a subset of crash severity categories. Regardless, this direction recognizes that more reliable safety estimates are obtained through the evaluation of treatment effect at a disaggregated level. In this manner, a given treatment’s overall safety effect is deter- mined by first computing its effect on each crash type and severity category, and then aggregating the results to a single estimate for the overall site. CMFs typically describe the safety effect of a treatment that is applied to both travel direc- tions on a given roadway facility. For example, CMFs have been developed for the case where a median is widened for the length of a given segment. They also describe the safety effect of a treatment that is applied to all legs of an intersection. For example, CMFs have been developed for the case where an intersection is converted from two-way stop control to signal control.

A-4 Guidelines for the Development and Application of Crash Modification Factors In this regard, the treatment is presumed to affect both travel directions for the length of the segment and all intersection legs. Hence, the CMF is accurately represented as a “spatially aggre- gate” CMF because the treatment influences the safety of the entire site. On the other hand, some treatments are only applied to one travel direction of a two-way roadway, or to just one leg of an intersection. The most accurate approach for quantifying the effect of these treatments is to quantify their effect on the treated portion of the facility. In this manner, the CMF produced would be a spatially-disaggregate CMF. It would be used in a disag- gregate safety evaluation to separately evaluate each travel direction (or each intersection leg) and then combine the results to obtain a spatially aggregate estimate of overall site safety. In summary, CMFs that are disaggregated by crash type, crash severity, or location can pro- vide more reliable insight about a treatment’s safety effect at a given site because they focus on the distribution of crashes at that location. A safety professional who is weighing the costs and benefits of a proposed treatment would ideally use a set of disaggregate CMFs to evaluate the effect on each specific crash type, severity, or location. However, many safety professionals desire to reach one total “aggregate” CMF. To this end, the procedure in this appendix provides guidelines on calculating a final aggregate CMF for the proposed treatment at the site of interest.

A-5   The procedure presented in this appendix provides a step-by-step approach to selecting and adjusting a CMF for a proposed treatment to assure its suitability for a site of interest. The pro- cedure consists of the following major steps, which are discussed individually in the following sections. They are illustrated in Figure A1. • Step 1. Identify proposed treatment and subject site • Step 2. Identify characteristics of subject site • Step 3. Identify existing CMFs for proposed treatment • Step 4. Compare subject site characteristics to known influential factors on the CMF • Step 5. Convert CMF to disaggregate CMFs • Step 6. Processing disaggregate CMFs – Step 6a. Test for homogeneity – Step 6b. Combine CMFs – Step 6c. Select the one CMF that is a best match to the subject-site characteristics or develop CMFunction • Step 7. Develop aggregate CMF using crash distribution at subject site – Step 7a. Determine the local crash distribution – Step 7b. Calculate the aggregate CMF based on the local crash distribution Step 1 Identify Proposed Treatment and Subject Site During this step, the analyst must identify the proposed treatment (countermeasure) for the subject site. The subject site is typically a single intersection or a section of road, but depending on the project, it may be on a larger scale, such as a group of intersections on a highway corridor. The treatment is defined as the action that is proposed to improve the safety of the subject site. This may be anything from low-cost treatments such as sign sheeting or reflectors to higher cost actions such as intersection reconfiguration. The treatment will need to be selected based on the specific safety issue at the subject site. Part B of the HSM provides guidelines for selecting an appropriate treatment for a site with known crash history and site conditions. Step 2 Identify Characteristics of Subject Site The analysis must identify the characteristics of the subject site before proceeding with further steps. The exact characteristics will depend on whether the site is an intersection or road segment and what is the intended treatment. Table A1 provides a list of general characteristics to identify at the subject site. C H A P T E R   2 Procedure

A-6 Guidelines for the Development and Application of Crash Modification Factors Figure A1. Procedure for selecting and adjusting a CMF for a subject site.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-7   In addition to physical and operational characteristics, it is necessary to know information about the crash history of the site for some steps of this process. In particular, the following crash information may be required. • Distribution of crash types and severities (can be obtained from crash history of site) • Empirical Bayes (EB) expected crashes before treatment installation (can be calculated from safety performance functions and observed crash history) Step 3 Identify Existing CMFs for Proposed Treatment During this step, the analyst must identify CMFs from the literature that would quantify the safety effect of the proposed treatment. If possible, the analysis should seek to identify CMFs that are high quality; are specific to crash type, severity, or location (i.e., “disaggregate”); and have known standard error values. Each of these points is discussed below. High-Quality CMFs High-quality CMFs should be used for this procedure to obtain the most reliable results. Common sources of high-quality CMFs include the Highway Safety Manual, First Edition (AASHTO 2010) and the CMF Clearinghouse (FHWA). Criteria for determining the quality of a CMF are described in the Introduction to Part D of the HSM. The Clearinghouse uses a star quality rating system that is described on its website (http://www.cmfclearinghouse.org/ sqr.cfm). According to the rating criteria used in the CMF Clearinghouse, a CMF is considered high quality if it scores highly in the following criteria. • Study Design. High-quality CMFs are developed through a statistically rigorous study design with a reference group or randomized experiment • Sample Size. High-quality CMFs are based on a large sample of data, ideally covering many years of data and many locations • Standard Error. High-quality CMFs have a small standard error compared to the CMF value • Potential Bias. High-quality CMFs are developed without undue influence from potential biases, such as confounding factors, inappropriate functional forms, or unaccounted effects from trends • Data Source. High-quality CMFs are developed using data that covers a diversity of geo- graphic areas (for wide applicability) All Sites Area type Number of lanes Traffic volume (AADT) One-way vs. two-way Speed limit Road Segments Median presence, type, and width Shoulder type and width Lane width Presence of curves and degree of curvature Roadside hazard rating Intersections Number of legs Intersection traffic control Signal phasing Table A1. Site characteristics to identify.

A-8 Guidelines for the Development and Application of Crash Modification Factors Determining whether a CMF is high quality can be done by viewing the star quality rating (if the CMF has been rated on the CMF Clearinghouse), seeing if the CMF was included in the HSM, or by considering the criteria above. CMFs Specific to Crash Type, Severity, or Location Depending on the availability of CMFs, the identified CMFs for the proposed treatment might be general CMFs that are applicable to all crash types and severities (referred to as “aggregate” CMFs) or applicable to specific crash type or severity categories (referred to as “disaggregate” CMFs). It is preferable to identify disaggregate CMFs for this process, although aggregate CMFs can be used. Disaggregate CMFs for the proposed treatment should be categorized by crash type (e.g., left- turn, head-on, run-off-road), crash severity (e.g., fatal, major injury, minor injury), and/or crash location (e.g., single direction, single intersection approach leg). • Crash type should be disaggregated into categories such that one category includes the crash types believed to be most influenced by the treatment and another category that includes all other crash types. A larger number of categories should be considered if treatment effect is believed to vary by crash type. • Crash severity should be disaggregated into two categories as a minimum: fatal-and-injury and property-damage-only (PDO). A larger number of categories (e.g., fatal vs. injury vs. PDO) may be needed when the treatment is believed to have an influence on crash severity. • Crash location should be considered if the treatment is being applied to only one of the two travel directions or to some (but not all) of the legs at an intersection. In all cases, the collective set of crash type, severity, and location categories must be inclu- sive of all crashes (i.e., the crash distribution proportions must add to 1.00). No crash category should be omitted. Practical considerations suggest that the level of disaggregation may be dictated by the CMFs in the available literature. If the CMFs are to be used with a set of disaggregate SPFs, then the CMF categories should match those of the available SPFs. For example, consider an analysis that will be based on four disaggregate SPFs that predict crashes for the following categories. • Multiple-vehicle fatal-and-injury • Single-vehicle fatal-and-injury • Multiple-vehicle property-damage-only • Single-vehicle property-damage-only) In this case, the analyst will need to identify a CMF for each of the four categories so that overall safety can be evaluated. CMFs with Known Standard Error The standard error of the CMF is a common way of indicating the statistical reliability of the CMF. Several steps in this procedure make use of the standard error of the CMF to perform calculations, so CMFs that have known standard error should be prioritized.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-9   Step 4 Compare Subject Site Characteristics to Known CMF Influential Factors In an ideal situation, the CMFs identified in Step 3 would have been developed using data from sites that are identical to the subject site. However, the reality is that the CMFs identified for the proposed treatment will often differ from the subject site on at least a few, if not many, characteristics. This step requires the user to examine the differences between the characteristics of the sites used to develop the CMF and the subject site. To reasonably assume that the CMF value is applicable to the subject site, it must match the subject site on site characteristics that are known to have a statistically significant influence on the value of the CMF. These characteristics are referred to in these guidelines as CMF influential factors. The purpose of this step is to ensure that the CMF matches on those influential factors. Later steps in this process will adjust a CMF to account for other differences between the sites where the CMF was developed and the subject sites by calibrating the CMF according to the local crash distribution. The specific influential factors will depend on what treatment is proposed. The user should determine which factors are significantly influential on the effect of the proposed treatment and ensure that the CMF(s) identified in Step 3 match the subject site on these influential factors. Tables of Known Influential Factors The project team for NCHRP Project 17-63 produced a list of influential factors for various treatments using a variety of methods, including examination of findings from past studies, tests of homogeneity on existing CMFs, and development of new knowledge from acquired data sets. The following tables provide lists of known influential factors as identified under Project 17-63. These influential factors cover 23 individual treatments across a variety of treatment categories. As seen in the tables below, some site characteristics cause a safety treatment to be more ben- eficial (lower the CMF), while others cause it to be less beneficial (raise the CMF). Additionally, some characteristics have been shown to have no effect on the magnitude of the CMF (i.e., the safety effect of the treatment is the same regardless). This knowledge on influential factors is based on research methods demonstrated in the Project 17-63 final report. For each countermeasure presented in the tables below, the influential factors are listed according to the type of CMF, since factors may differ by crash type addressed or crash severity addressed. The reader will also find a list of factors shown not to be influential. These factors were examined by the research study from which the findings were obtained but the analysis could not conclude that the factor had a significant effect on the CMF. These lists of factors are useful when deciding whether a CMF developed under one set of site conditions can be applied to another set of site conditions. If the site conditions differ on an influ- ential factor, then the CMF cannot be applied or should be applied only with extreme caution. If the site conditions differ on a factor shown not to be influential, then the CMF can be applied without hesitation. Tables A2 through A10 are grouped by general countermeasure or topic category (e.g., access management, alignment, etc.). The first column of every table describes the countermeasures, which are generally defined as actions taken to improve safety at a location and may also include infrastructure changes not made for safety but for which a safety effect can be determined. The second and third columns describe the crash severity and crash type addressed by the CMFS, since factors may differ by CMFs for the same countermeasure. The center columns contain

A-10 Guidelines for the Development and Application of Crash Modification Factors Countermeasure Crash Severity Crash Type Area Type Geographic Area of Origin Traffic Volume (AADT) Other References Increase access point density Total Total Increasing the access point density is more harmful for lower AADT compared to higher AADT Harwood et al. 2000 Modify ramp spacing Total Total Presence of auxiliary lane mitigates the harmful effect of decreasing ramp spacing Le and Porter 2012b Fatal-and- injury Total Presence of auxiliary lane mitigates the harmful effect of decreasing ramp spacing Total Multi vehicle Presence of auxiliary lane mitigates the harmful effect of decreasing ramp spacing Install TWLTL on two-lane road Total Total More beneficial in rural areas than urban areas Less beneficial for NC than for CA, IL, or AR Information on intersection and driveway density was available, but could not conclude that it was an influential factor Persaud et al. 2007 Abbreviation: TWLTL, two-way left-turn lane. Table A2. Access management related influential factors. Countermeasure Crash Severity Crash Type Speed Limit References Increase radius of horizontal curve Fatal-and-injury Total More effective on roads with higher speed limits Pratt et al. 2014 Fitzpatrick et al. 2009 Fatal-and-injury Run-off- road More effective on roads with higher speed limits Pratt et al. 2014 Fatal-and-injury Wet road More effective on roads with higher speed limits Table A3. Alignment related influential factors.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-11   information on the effect of various physical or operational characteristics. Note that these columns are not the same for every table. The last column provides the source reference for the study whose findings were used to determine the effect. Identifying Influential Factors Not Covered in This Report Tables A2 through A10 listing influential factors on CMFs do not constitute a final and exhaustive list. They were developed under the research effort for NCHRP Project 17-63 and serve two purposes: to provide a starting list of influential factors that can be used in safety practice, and to demonstrate how further influential factors can be identified. If the proposed treatment is not covered in Tables A2–A10, the user can identify influential factors for the proposed treatment on his or her own. The methods used by the 17-63 team are presented in the project final report (under Subtask 2.6a) and may be helpful in demonstrating how to identify influential factors. These methods included examining existing crash modifica- tion functions, conducting homogeneity tests, and analyzing raw data sets to identify influential factors. These methods are summarized below. Examination of Existing Crash Modification Functions Some previous studies provided information regarding influential factors, either through general statements in their results and discussion, or in the form of crash modification functions (CMFunction). For example, if a CMFunction showed that the CMF value was dependent on the level of traffic volume, then traffic volume was seen to be an influential factor on the effectiveness of paving an unpaved road. Homogeneity Test A homogeneity test (described and demonstrated later in Step 6a of this procedure) was used to compare CMFs in such a way as to determine if a significant difference in the CMFs could be attributed to a difference in a key site characteristic. The team conducted homogeneity tests for countermeasures for which there were enough disaggregate CMFs to allow for matching on all site characteristics except the one of interest. In situations where the homogeneity test concluded that the CMFs were similar enough to be combined, the team concluded that the site characteristic of interest (the characteristic that differed between the CMFs being tested) could not be considered an influential factor on the CMF. Countermeasure Crash Severity Crash Type Area Type Traffic Volume (AADT) Median References Add bike lane to urban arterial Total Total Less beneficial for higher AADT per lane More beneficial for roads with narrow medians Park et al. 2015 Install pedestrian signal Total Total Less beneficial in commercial area (possible correlation with higher ped volumes) Less beneficial for higher product of major and minor traffic volumes Sacchi et al. 2015 Fatal-and-injury Total Less beneficial in commercial area (possible correlation with higher ped volumes) Less beneficial for higher product of major and minor traffic volumes Table A4. Pedestrian and bicycle related influential factors.

A-12 Guidelines for the Development and Application of Crash Modification Factors Countermeasure Crash Severity Crash Type Area Type Geographic Area of Origin Number of Lanes / Roadway Type Traffic Volume (AADT) Shoulder Width Degree of Curvature Other References Install permanent raised pavement markers Total Nighttime More beneficial for higher AADT (for four- lane freeways) More beneficial for flatter curves Bahar et al. 2004 Install reflective pavement markings on edgelines and lane lines (freeways) Total Total More beneficial in MN compared to NC More beneficial for 6-lane freeways compared to 4-lane More beneficial at sites with lower AADT More beneficial for wider right-side shoulders More beneficial at sites with higher EB expected crashes in the before period Srinivasan, analysis of disaggregate data from 705 sites in North Carolina, Wisconsin, and Minnesota; Project 17- 63 analysis Fatal-and- Injury Total Data available, but could not conclude that it was an influential factor More beneficial for 6-lane freeways compared to 4-lane More beneficial at sites with lower AADT Data available, but could not conclude that it was an influential factor More beneficial at sites with higher EB expected crashes in the before period Total Wet road More beneficial for 6-lane freeways compared to 4-lane Data available, but could not conclude that it was an influential factor More beneficial for wider right-side shoulders More beneficial at sites with higher EB expected crashes in the before period Total Nighttime More beneficial in MN compared to NC More beneficial for 6-lane freeways compared to 4-lane More beneficial at sites with lower AADT More beneficial for wider right-side shoulders More beneficial at sites with higher EB expected crashes in the before period Total Nighttime wet road More beneficial for rural freeways More beneficial at sites with higher EB expected crashes in the before period Table A5. Delineation related influential factors.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-13   Countermeasure Crash Severity Crash Type Area Type Number of Lanes/Roadway Type Traffic Volume (AADT) Number of Intersection Legs Intersection Traffic Control References Convert signal to roundabout Total Total More beneficial in suburban areas compared to urban areas Data available, but could not conclude that it was an influential factor Less beneficial at sites with higher AADT More beneficial on 4-leg intersections compared to 3-leg Srinivasan et al. 2011 Gross et al. 2013 Srinivasan, analysis of disaggregate data using 10 states’ data from NCHRP 17-35 and other projects, Project 17-63 analysis Injury Total Data available, but could not conclude that it was an influential factor Data available, but could not conclude that it was an influential factor Data available, but could not conclude that it was an influential factor Data available, but could not conclude that it was an influential factor Srinivasan et al. 2011 Gross et al. 2013 Provide a left- turn lane on both major-road approaches Total Total Data available, but could not conclude that it was an influential factor for 4- leg stop- controlled More beneficial when volumes are more balanced than dominated by major- road volume (for urban 4- leg signalized) Data available, but could not conclude that it was an influential factor (for urban 4-leg intersections) Harwood et al. 2003* Fatal-and-injury Total Data available, but could not conclude that it was an influential factor for 4- leg stop- controlled More beneficial when volumes are more balanced than dominated by major- road volume (for urban 4- leg signalized) Data available, but could not conclude that it was an influential factor (for urban 4-leg intersections) Provide a left- turn lane on one major-road approach Total Total Data available, but could not conclude that it was an influential factor for 4- leg stop- controlled Data available, but could not conclude that it was an influential factor (for urban 4-leg signalized) Data available, but could not conclude that it was an influential factor (at stop- controlled intersections) Data available, but could not conclude that it was an influential factor (for 4-leg intersections) Fatal-and-injury Total Data available, but could not conclude that it was an influential factor for 4- leg stop- controlled More beneficial when volumes are more balanced than dominated by major- road volume (for urban 4- leg signalized) More beneficial for 3-leg vs. 4-leg (for rural stop- controlled) Data available, but could not conclude that it was an influential factor (for urban 4-leg intersections) * Influential factors identified by homogeneity tests conducted on CMFs produced by this study. Table A6. Intersection geometry related influential factors.

A-14 Guidelines for the Development and Application of Crash Modification Factors Data Analysis to Identify Influential Factors The project team conducted analyses of the various raw data sets to develop CMFunctions and identify influential factors. These included raw data that could be extracted from study reports as well as raw data sets available to the team from previous research projects. This approach requires an analyst with experience in statistical modeling and highway safety research. The research conducted in Project 17-63 to develop influential factors by analyzing raw data sets is documented in under Section A4, Supporting Research. If there is insufficient information to allow a user to follow any of these methods for identify- ing influential factors, then the user must fall back to matching on the following factors that are generally assumed to affect the CMF of most treatments. • Traffic volume • Area type • Facility type as categorized in the HSM Part C (e.g., rural two-lane, rural multilane, urban/ suburban arterial) • Type of traffic control (i.e., signalized vs. unsignalized) and number of legs, if intersection If the user were able to identify (or develop) CMFunctions for the proposed treatment in Step 3 that included terms in the function for influential factors, then there is not a need to match on the influential factors. The function itself would adjust the CMF value according to differences in these factors. Step 5 Convert CMF to Disaggregate CMFs This procedure requires the use of disaggregate CMFs. If the existing CMFs identified in Step 3 were only aggregate CMFs (i.e., disaggregate CMFs were unavailable), then this Step 5 must be followed. If disaggregate CMFs were available and identified in Step 3, skip this step, and go to Step 6. Crash Severity Crash Type Area Type Traffic Volume (AADT) Number of Intersection Legs Speed Limit Other References Convert 2- way stop control to signal control Total Total Less beneficial on higher volume roads More beneficial at 3-leg intersections compared to 4-leg intersections Less beneficial for higher speed major road (50 mph or more) More beneficial at sites with higher expected crashes per year in the before period Srinivasan and Lan 2016 Fatal- and- Injury Total Less beneficial on higher volume roads More beneficial at 3-leg intersections compared to 4-leg intersections Less beneficial for higher speed major road (50 mph or more) More beneficial at sites with higher expected crashes per year in the before period Provide flashing beacons at stop- controlled intersections Total Angle Data available, but could not conclude that it was an influential factor Srinivasan et al. 2007* * Influential factors identified by homogeneity tests conducted on CMFs produced by this study Counter- measure Table A7. Intersection traffic control related influential factors.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-15   Countermeasure Crash Severity Crash Type Geographic Area of Origin Traffic Volume (AADT) Shoulder Width Other References Decrease lane width Total Head-on, Run-off- road, Sideswipe, Single vehicle More harmful for higher AADT roads Less harmful for roads with wider shoulders Zegeer et al. 1988 Harwood et al. 2000 Le and Porter 2012 Convert traditional mainline toll plazas to hybrid mainline toll plazas Total Total Less beneficial for higher volume roads Abuzwidah et al. 2014 Fatal- and- Injury Total Less beneficial for higher volume roads Property Damage Only Total Less beneficial for higher volume roads Pave unpaved road Total Total More beneficial on higher volume roads Ksaibati et al. 2009 Install centerline and shoulder rumble strips Total Total More beneficial in MO compared to PA and KY. No difference between PA and KY. Less beneficial at sites with higher AADT Data available, but could not conclude that it was an influential factor More beneficial at sites with higher EB expected crashes in the before period Srinivasan, analysis of disaggregate data from 2000 sites in three states, Project 17-63 analysis Fatal- and- injury Total More beneficial in MO compared to PA and KY. No difference Data available, but could not conclude that it was an influential factor Data available, but could not conclude that it was an between PA and KY. influential factor Total Run-off- road Data available, but could not conclude that it was an influential factor Data available, but could not conclude that it was an influential factor Data available, but could not conclude that it was an influential factor More beneficial at sites with higher EB expected crashes in the before period Total Head-on More beneficial in MO compared to PA and KY. No difference between PA and KY. Data available, but could not conclude that it was an influential factor Data available, but could not conclude that it was an influential factor Table A8. Roadway related influential factors.

A-16 Guidelines for the Development and Application of Crash Modification Factors Countermeasure Crash Severity Crash Type Geographic Area of Origin Number of Lanes / Roadway Type Traffic Volume (AADT) Road Division Shoulder Type Shoulder Width References Install Safetyedge Total Total Data available, but could not conclude that it was an influential factor (between GA and IN) Data available, but could not conclude that it was an influential factor (between paved and unpaved shoulder) Graham et al. 2011* Fatal- and- Injury Total Data available, but could not conclude that it was an influential factor (between GA and IN) Data available, but could not conclude that it was an influential factor (between paved and unpaved shoulder) Total Run-off- road Data available, but could not conclude that it was an influential factor (between GA and IN) Data available, but could not conclude that it was an influential factor (between paved and unpaved shoulder) Fatal- and- injury Run-off- road Data available, but could not conclude that it was an influential factor (between GA and IN) Data available, but could not conclude that it was an influential factor (between paved and unpaved shoulder) Widen shoulder and install shoulder rumble strips Total Total More beneficial on roads with narrow Park et al. 2014 Park et al. 2015 (4–6 ft) shoulders Fatal- and- injury Total More beneficial on roads with narrow (4–6 ft) shoulders Table A9. Shoulder related influential factors.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-17   Countermeasure Crash Severity Crash Type Geographic Area of Origin Number of Lanes / Roadway Type Traffic Volume (AADT) Road Division Shoulder Type Shoulder Width References Install shoulder rumble strips Total Total Data available, but could not conclude that it was an influential factor More beneficial at higher AADT Data available, but could not conclude that it was an influential factor More beneficial on roads with narrow (4–6 ft) shoulders Park et al. 2014 Project 17-63 analysis of data from Carrasco et al. 2004 and Torbic et al. 2009 Fatal- and- injury Total More beneficial on roads with narrow (4–6 ft) shoulders Park et al. 2014 Fatal- and- injury Multiple vehicle More beneficial for PA than MN or MO Most beneficial for 2-lane roads, least beneficial for multilane highways Project 17-63 analysis of data from Torbic et al. 2009 Fatal- and- injury Single vehicle More beneficial for PA than MN or MO Most beneficial for 2-lane roads, least beneficial for multilane highways PDO Multiple vehicle More beneficial for PA than MN or MO Most beneficial for 2-lane roads, least beneficial for multilane highways PDO Single vehicle More beneficial for PA than MN or MO Most beneficial for 2-lane roads, least beneficial for multilane highways Change shoulder width Total Head-on, Run-off- road, Sideswipe, Single vehicle Higher AADT magnifies the effect of shoulder width change (widening is more beneficial, narrowing is more harmful) Harwood et al. 2000 (continued on next page) Table A9. (Continued).

A-18 Guidelines for the Development and Application of Crash Modification Factors Countermeasure Crash Severity Crash Type Geographic Area of Origin Number of Lanes / Roadway Type Traffic Volume (AADT) Road Division Shoulder Type Shoulder Width References Widen shoulder Total Total More beneficial on roads with narrow (4–6 ft) shoulders Park et al. 2014 Park et al. 2015 Fatal- and- injury Total More beneficial on roads with narrow (4–6 ft) shoulders * Influential factors identified by homogeneity tests conducted on CMFs produced by this study Table A9. (Continued). Countermeasure Crash Severity Crash Type Traffic Volume (AADT) Degree of Curvature Roadside Hazard Rating References Install new fluorescent curve signs or upgrade existing curve signs to fluorescent sheeting Total Lane departure More beneficial at sites with higher AADT More beneficial on sharper curves (radius <150 m) in Connecticut, but radius was not an influential factor in Washington. More beneficial on more hazardous roadsides (RHR >5) Srinivasan et al. 2009 Total Dark More beneficial at sites with higher AADT More beneficial on sharper curves (radius <150 m) in Connecticut, but radius was not an influential factor in Washington. More beneficial on more hazardous roadsides (RHR >5) Total Lane departure during dark More beneficial at sites with higher AADT More beneficial on sharper curves (radius <150 m) in Connecticut, but radius was not an influential factor in Washington. More beneficial on more hazardous roadsides (RHR >5) Abbreviation: RHR, roadside hazard rating. Table A10. Sign related influential factors. The aggregate CMF must be converted to a complementary set of disaggregate CMFs that will break down the effect of the treatment into its effect on each crash type or crash severity. The categories for the disaggregate CMFs will depend on the safety issue of interest (see discussion in Step 4). Converting the aggregate CMF to these specific disaggregate values will allow the safety effect to be estimated more accurately at the subject site which has a different distribution of crash types or severities than the locations from which the CMF was developed. The process for calculating the total effect of the treatment at the subject site using these disaggregated CMFs is accomplished later in Step 7. This section on Step 5 begins with a discussion of the data required for developing disag- gregate CMFs and some considerations for generalized regressions models. Following that is a listing of four sub-steps, Steps 5.1 through 5.4, that walk the user through the disaggregation procedure. The conversion to disaggregate CMFs can be done by using the procedure below or by using the CMF Regression Software developed under this project.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-19   Required Data This section describes the number of aggregate CMFs needed for the procedure, the charac- teristics of the crash distribution used in the procedure, the statistical weight associated with each CMF observation, and a technique for estimating CMFs from a function of variables. Number of Aggregate CMFs Step 5 is used to estimate disaggregate CMFs for n categories of crash type or crash sever- ity. Additional categories are needed to describe CMF value variation by spatial location if it is believed that treatment effect will vary by spatial location. For example, if two crash type categories are considered with two crash severity categories and the focus is four-leg intersections then n equals 16 (= 2 × 2 × 4). Guidelines for defining these categories are described in Step 3. A reasonable first-order approximation is to assume the treatment effect is the same for each location. With this assumption, spatial location categories are not considered when determining n. If this assumption is acceptable, the number of aggregate CMFs needed is small and the regres- sion model is kept reasonably simple. Thus, if two crash type categories are considered with two crash severity categories and the focus is four-leg intersections but the treatment effect is believed to be the same for each leg, then n equals 4 (= 2 × 2). At least n aggregate CMFs must be found in the literature to apply the procedure. Desirably, they would have the same site characteristics as the site of interest. These characteristics were identified in Step 2. However, it may not be possible to find n CMF observations that match these site characteristics. In addition, there may be other characteristics for which the collective set of aggregate CMFs show some variation. If either of these two situations occurs, then the requirement to select only CMFs with the same site characteristics may need to be relaxed to obtain the minimum CMF sample size. In this situation, an independent variable is included in the regression model to account for each difference in site characteristics. The number of these differences is denoted by the variable m. For example, if the set of aggregate CMFs found in the literature is expanded to collectively include both urban and rural area types (i.e., m = 1), then it will be necessary to find n + 1 or more aggregate CMFs so that the effect of area type can be quantified (and, if it is found to be influential, used to adjust the disaggregate CMFs to reflect the area type of the site of interest). If there are less than n + m aggregate CMFs, then it will not be possible to estimate the dis- aggregate CMFs. The analyst may reflect on whether eliminating one or more of the m char- acteristics will satisfy the sample size requirement without compromising the accuracy of the disaggregated CMFs. If there are at least n + m aggregate CMFs available from the literature, then the disaggregated CMFs can be estimated using regression analysis. In summary, the number of aggregate CMFs obtained from the literature should equal or exceed n + m where n is the number of crash type or severity categories, and m is the number of site-characteristic categories (e.g., area type) or variables (e.g., lane width) believed to influence safety among the sites represented in the set of CMFs. A larger number of observations will allow some interaction effects to be included in the regression model. Crash Distribution For each aggregate CMF observation, a representative distribution of crashes must also be available. Ideally, this distribution would be obtained from the original publication documenting the development of the aggregate CMFs. The crash distribution must represent the crash history before the treatment is applied. Once the treatment is applied, it will alter the crash distribution.

A-20 Guidelines for the Development and Application of Crash Modification Factors If the sites were selected for treatment for reasons that do not include a recent increase in crashes, then the crash distribution can be obtained from the crash history of the treatment sites. This approach requires the crash distribution to be available in the original publication or from the researchers. If the sites were selected for treatment because of a recent increase in crashes, then there may be some regression-to-the-mean artifacts in the crash history of the treated sites. One option to remove this bias is to base the crash distribution on a collective set of sites that are like the site of interest. The same period should be used to define the crash history for each site in the set of sites. Importantly, none of the sites included in this set should have the treatment present during the crash history period. This distribution would likely be obtained directly from the public agency for which the original research was conducted. A second option is to base the crash distribution on the “expected crash frequency for the after period had the site not been treated.” This expected crash frequency is computed using the method described in Appendix A to Chapter 9 of the HSM. It uses the empirical Bayes method to remove the regression-to-the-mean bias from the data. The original data may need to be acquired from the researchers to pursue this option. A third option is available for spatially aggregate CMFs. It is based on the use of traffic volume to estimate the desired proportions. This option recognizes that crash frequency is approxi- mately proportional to traffic volume when the range of volumes of interest is relatively small. Thus, the proportion of segment crashes associated with one travel direction of a two-way road- way can be estimated using the following equation. p d AADT d AADT d AADT seg i i Equation A1 _ _ _ , 1 2 = + where pseg,i = proportion of crashes associated with segment travel direction i (i = 1, 2) d_AADTi = directional (one-way) annual average daily traffic volume for travel direction i (i = 1, 2) For some applications, it may be useful to define i equal to 1 for the highest volume travel direction and 2 for the other travel direction. Similarly, the proportion of intersection crashes associated with one leg can be estimated using the following equation. p AADT AADT AADT AADT AADT int i i Equation A2, 1 2 3 4 = + + + where pint,i = proportion of crashes associated with intersection leg i (i = 1, 2, 3, 4) AADTi = annual average daily traffic volume for intersection leg i (i = 1, 2, 3, 4) For some applications, it may be useful to define i equal to 1 for the highest volume major- street leg, 2 for the other major-street leg, 3 for the highest volume minor-street leg, and 4 for the other minor-street leg. If Equation A2 is used for a three-leg intersection, then the fourth term in the denominator (i.e., AADT4) is deleted or set to zero (0). To use Equation A1 or Equation A2, the published research or the public agency would need to provide the traffic volumes for the sites studied in the original research.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-21   Standard Error and Weight The standard error is needed for each aggregate CMF observation. This statistic is used to determine the weight of the CMF observation, relative to other observations. This weight is computed using the following equation. =    2 Equation A3w CMF s i i i where wi = weight of CMF observation i CMFi = value of CMF observation i si = standard error of CMF observation i The regression analysis should be based on aggregate CMFs that are associated with a weight of 4.0 or more. CMFs with a smaller weight are less reliable because they are based on a relatively small sample size (i.e., a small number of observed crashes). As a minimum, most of the CMFs in each group should have a weight of 4.0 or more. CMF Observations from a Function The previous discussion implies that the CMFs obtained from the literature are constants. However, aggregate CMF observations can be estimated from a function of variables. The func- tion can be a CMF function, a multiple-variable safety performance function, or a safety predic- tion model. The independent variable of interest in the function is that describing the subject treatment. For example, if the function describes the effect of changing lane width, then lane width is the variable of interest. All other independent variables in the function are each set to an appropriate (i.e., representative) value for the site of interest. Just as multiple aggregate CMF constants can be found in the literature for a given treatment, multiple functions can also be found in the literature. A technique for estimating CMF observa- tions from one or more functions is described in this subsection. Initially, the range for the independent variable of interest is established. This range should be consistent with that associated with the range for the variable of interest in the function. The document that describes the development of the function is reviewed to determine the range associated with the original data that were used to calibrate the function. This range is divided into M–1 equal intervals and M values. Every combination of change from one value to another value is enumerated and then each pair of values is used with the function to create one of M2 CMF regression observations. The variable M should have a value of 3 or more, with a larger value being more appropriate for more complex, non-linear functions. For example, if a func- tion that includes a variable for “lane width” was reported to be applicable to a range of values from 10 to 12 ft, then it could be divided into 2 intervals with the following 3 values (i.e., M = 3): 10, 11, 12. The CMF observations would be based on the following 9 lane width pairs: change from 10 ft to 10 ft, change from 10 ft to 11 ft, change from 10 ft, to 12 ft, . . . , change from 12 ft to 11 ft, change from 12 ft to 12 ft. Each CMF observation i is computed using the following equation. CMF f x x x x b b b b f x x x x b b b b b a i interest n interest n interest n interest n Equation A4 , , , . . . , , , , . . . , , , , . . . , , , , . . . , : , ,a,i 1 2 1 2 ,b,i 1 2 1 2 ( ) ( )=

A-22 Guidelines for the Development and Application of Crash Modification Factors where CMFb:a,i = value of CMF observation i when variable of interest changes from b to a (i = 1 to M2) f(x,b) = function of independent variables x and regression coefficients b) xinterest, a,i = independent variable of interest having a value describing site after (or with) the change xinterest, b,i = independent variable of interest having a value describing site before (or without) the change x–j = independent variable j having a value representative of the site of interest binterest = regression coefficient associated with the independent variable of interest bj = regression coefficient associated with variable xj Equation A4 is shown to have the same function repeated in the numerator and the denom- inator. With one exception, all variables in each function are set equal to an appropriate (i.e., representative) value for the site of interest. The exception is the variable of interest whose value in the numerator and in the denominator reflects the “after” and “before” conditions, respectively. The standard error of the computed CMF is also needed. If the standard error of the regres- sion coefficient associated with the variable of interest is available, the standard error can be estimated using the following equation (Bahar et al. 2007). * 0.5 , , , . . . , , , , . . . , , , , . . . , , , , . . . , , , , . . . , , , , . . . , , , , . . . , , , , . . . , : , ,a,i 1 2 , 1 2 ,b,i 1 2 , 1 2 ,a,i 1 2 , 1 2 ,b,i 1 2 , 1 2 Equation A5 ( ) ( ) ( ) ( ) = × + σ + σ   − − σ − σ   s abs f x x x x b b b b f x x x x b b b b f x x x x b b b b f x x x x b b b b b a i interest n interest b interest n interest n interest b interest n interest n interest b interest n interest n interest b interest n where s*b:a,i = function-based estimate of standard error of CMF observation i when variable of interest changes from b to a σb,interest = standard error of regression coefficient associated with the variable of interest abs[z] = absolute value of z Equation A5 provides a reasonably accurate estimate of the standard error in most cases; however, it results in an estimate of zero when there is no change in the variable of interest. In fact, the true standard error converges to the following value as the change in the variable of interest approaches zero. 1 1 1, : , : , : , 0.5 Equation A6= + +        s CMF CMF N CMF min i b a i b a i o b a i where smin,i = minimum standard error of CMF observation i No = number of observed crashes in the database used to develop the function One final consideration in the calculation of standard error is relevant when there is more than one function identified in the literature (e.g., one function from Study A and two CMF constants from Study B, or one function from Study A and one function from Study B). In this situation, the standard error of the CMF obtained from the function should be multiplied by M. This adjustment properly weighs the CMF observations among studies.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-23   Based on the preceding discussion, the following equation is used to estimate the standard error of the regression coefficient associated with the variable of interest. This standard error is used in Equation A3 to estimate the weight associated with the CMF observation. s M larger of s sb a i b a i min i( )= × Equation A7: * ,: , : , , where sb:a,i = standard error of CMF observation i when variable of interest changes from b to a and adjusted for number of observations M = number of values selected to be representative of the range of the variable of interest If two or more functions are used, then separate values of M (i.e., M1, M2, etc.) are defined based on the individual function documentation, and Equations A4 through A7 are separately applied for each function to create the desired set of CMF observations. Considerations for Generalized Regression Models This section describes considerations when developing a regression model to estimate disag- gregate CMFs. Model Development The regression model is used to define the analytic relationship between the dependent and independent variables. The regression model is determined by the desired disaggregation categories. The aggregate CMF represents the dependent variable, the disaggregate CMF vari- ables represent the regression coefficients, and the crash distribution proportions represent the independent variables. Model for Crash Type, Severity, and Segment Location. In this subsection, the form of the model is illustrated using four crash type and severity categories. Then, variations of the model are introduced. Finally, the model is more formally described using a generalized equa- tion. The following crash type and severity categories are represented in the following regression model; the model includes one term for each of these four categories: • Multiple-vehicle fatal-and-injury • Single-vehicle fatal-and-injury • Multiple-vehicle property-damage-only • Single-vehicle property-damage-only CMF b p b p b p b p agg mv fi mv fi sv fi sv fi mv pdo mv pdo sv pdo sv pdo Equation A8 exp exp exp exp , , , , , , , , ( ) ( ) ( ) ( ) = × + × + × + × where CMFagg = aggregate CMF bi,j = regression coefficient for crash type category i (i = mv: multiple vehicle, sv: single vehicle) and crash severity category j (j = fi: fatal-and-injury, pdo: property damage only) exp(bi,j) = CMFi,j; disaggregate CMF for crash type category i and crash severity category j pi,j = proportion of crashes associated with crash type category i and crash severity category j

A-24 Guidelines for the Development and Application of Crash Modification Factors The proportions included in Equation A8 must add to 1.0 for each observation in the regres- sion database. The exponential function is used with each regression coefficient to ensure that the estimated CMF is non-negative. When Equation A8 is applied to segments, its form assumes that both travel directions are treated. If the aggregate CMFs from literature are described as applicable to overall segment crash frequency, but they represent treatments that were applied to only one travel direction, then the spatially disaggregate CMF can be estimated using the following equation (where travel direction 1 is the treated direction). exp exp exp exp 1.0 1 , , ,1 , , ,1 , , ,1 , , ,1 ,1 ,1 Equation A9 [ ] [ ] ( ) ( ) ( ) ( ) ( ) = × + × + × + × × + × − −CMF b p b p b p b p p p sp agg mv fi mv fi sv fi sv fi mv pdo mv pdo sv pdo sv pdo seg seg where pi,j,k = proportion of crashes associated with crash type category i, crash severity category j; and segment travel direction k (k = 1, 2); pseg,k = proportion of crashes associated with segment travel direction k (k = 1, 2); and all other variables are previously defined. The proportions pi,j,k in Equation A9 are based on the crash distribution for the treated travel direction (i.e., direction 1). These proportions must add to 1.0 for each observation. If the collective set of aggregate CMFs show some variation by one or more site characteristics (e.g., area type, terrain, grade, etc.), then one or more variables can be added to the regression model to quantify the effect of each characteristic on the CMF value. If the characteristic is continuous (e.g., grade), then one variable is added as a multiplicative adjustment factor. If the characteristic has m categories, then m-1 indicator variables are added to the regression model as multiplicative adjustment factors. This technique is shown in the following regression model for the case where factors are added for area type (i.e., urban or rural). CMF b p b p b p b p b I agg mv fi mv fi sv fi sv fi mv pdo mv pdo sv pdo sv pdo urban urban Equation A10 exp exp exp exp exp , , , , , , , , [ ] ( ) ( ) ( ) ( ) ( ) = × + × + × + × × × where Iurban = indicator variable for area type (= 1.0 if urban, 0.0 otherwise) burban = regression coefficient for effect of urban area type All other variables as previously defined The concepts introduced in the previous paragraphs were used to develop the following gen- eral regression model for CMFs aggregated by crash type and severity. This model can be used to estimate disaggregated CMFs when the aggregated CMFs are available in the literature. CMF c x b pagg k k k i j i j Equation A11exp exp , , i,j ∑ ∑[ ] [ ]( )= ×   × ×

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-25   where CMFagg = aggregate CMF bi,j = regression coefficient for crash type category i (i = mv: multiple vehicle, sv: single vehicle) and crash severity category j (j = fi: fatal-and-injury, pdo: property damage only) ck = regression coefficient for site characteristic k xk = variable describing site characteristic k exp(bi,j) = CMFi,j; disaggregate CMF for crash type category i and crash severity category j pi,j = proportion of crashes associated with crash type category i and crash severity category j The general regression model is applicable to any number of crash type and severity catego- ries. The exact number will depend on the treatment being considered, the types of crashes it is expected to affect (as determined in Step 4) and the availability of CMFs in the literature. The first term of the model is an optional adjustment factor to account for variation in site charac- teristics among the CMF observations. The following model can be used with the crash location distribution to estimate spatially- disaggregate CMFs for segments. CMF f b p f b psp agg a I seg a I seg Equation A12exp exp1 ,1 2 ,21 2{ } { }( ) ( )= × × + × × − with f c xa k k k Equation A13exp ∑[ ]= ×   where CMFsp-agg = spatially aggregate CMF fa = site characteristics adjustment factor bi = regression coefficient for travel direction i (i = 1, 2) exp(bi) = CMFsp,i; spatially-disaggregate CMF for travel direction i (i = 1, 2) pseg,i = proportion of crashes associated with segment travel direction i (i = 1, 2) Ii = indicator variable for treatment application to travel direction i (= 1.0 if treatment applied, 0.0 otherwise) All other variables as previously defined The two proportions in Equation A12 must add to 1.0. If the treatment effect is the same in each direction and both directions are treated, then b1 equals b2 and Equation A12 reduces to CMFsp-agg = exp(∑[ck × xk])×CMFsp, where CMFsp = CMFsp,1 = CMFsp,2. It reduces further to CMFsp-agg = CMFsp if there are no adjustment factors. If there are multiple observations of CMFsp-agg in the database, then the value of CMFsp obtained from the regression will represent an average of the observations in the database. Model for Intersection Location. The regression model can be developed to estimate spatially-disaggregate CMFs for intersection legs. The model may optionally include crash type and severity categories. The form of the regression model is initially illustrated in this subsection for all crash types and severities combined. Then, model variations are introduced. Finally, the model is more formally described using a generalized equation.

A-26 Guidelines for the Development and Application of Crash Modification Factors All models described in this subsection apply to four-leg intersections. When applied to three-leg intersections, the fourth term (i.e., with subscripts indicating leg 4) is deleted. The intersection regression model for all crash types and severities combined is represented by the following equation. CMF b p p b p p b p p b p p sp agg int int I int int I int int I int int I Equation A14 exp 1 exp 1 exp 1 exp 1 1 ,1 ,1 2 ,2 ,2 3 ,3 ,3 4 ,4 ,4 1 2 3 4 [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = × + − × × + − × × + − × × + − − where CMFsp-agg = spatially aggregate CMF bk = regression coefficient for intersection leg k (k = 1, 2, 3, 4) exp(bk) = CMFsp,k; spatially-disaggregate CMF for leg k (k = 1, 2, 3, 4) pint,k = proportion of crashes associated with intersection leg k (k = 1, 2, 3, 4) Ik = indicator variable for treatment application on intersection leg k (= 1.0 if treat- ment applied, 0.0 otherwise) The proportions included in Equation A14 must add to 1.0 for each observation in the regres- sion database. The exponential function is used with each regression coefficient to ensure that the estimated CMF is non-negative. When Equation A14 is applied to crash type or severity categories, the number of unique regression coefficients in the regression model can become quite large (which dictates a similar increase in the number of CMF observations). To eliminate this complication, it may be reason- able to assume that the treatment effect on each leg is the same. This form of the regression model is illustrated in the following equation for the case where crash severity is explicitly recognized. CMF b p b p p p b p b p p p b p b p p p b p b p p p sp agg fi int fi pdo int pdo int int I fi int fi pdo int pdo int int I fi int fi pdo int pdo int int I fi int fi pdo int pdo int int I [ ] [ ] [ ] [ ] { } ( ) { } ( ) { } ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = × + × × + − × × + × × + − × × + × × + − × × + × × + − − Equation A15 exp exp 1 exp exp 1 exp exp 1 exp exp 1 ,1, ,1, ,1 ,1 ,2, ,2, ,2 ,2 ,3, ,3, ,3 ,3 ,4, ,4, ,4 ,4 1 2 3 4 where bj = regression coefficient for crash severity category j (j = fi: fatal-and-injury, pdo: prop- erty damage only) exp(bj) = CMFsp,j; spatially disaggregate CMF for crash severity category j pint,k,j = proportion of crashes associated with intersection leg k (k = 1, 2, 3, 4) and for crash severity category j All other variables as previously defined The proportions pint,k,j in Equation A15 are based on the crash severity distribution for leg k. For each observation, these proportions must add to 1.0 for each leg. In contrast, the proportions pint,k are based on the crash location distribution for the intersection. These proportions must add to 1.0 for each observation. The concepts introduced in the previous paragraphs were used to develop Equation A16. This equation represents a general regression model for CMFs aggregated by crash type, severity, and

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-27   intersection leg. This model can be used to estimate disaggregated CMFs when the aggregated CMFs are available in the literature. The proportions pint,k,i,j in Equation A16 are based on the crash type and severity distributions for leg k. For each observation, these proportions must add to 1.0 for each leg. In contrast, the proportions pint,k are based on the crash location distribution for the intersection. These proportions must add to 1.0 for each observation. CMF f b p p p f b p p p f b p p p f b p p p sp agg a i j int i j i j int int I a i j int i j i j int int I a i j int i j i j int int I a i j int i j i j int int I Equation A16 exp 1 exp 1 exp 1 exp 1 , ,1, , , ,1 ,1 , ,2, , , ,2 ,2 , ,3, , , ,3 ,3 , ,4, , , ,4 ,4 1 2 3 4 ∑ ∑ ∑ ∑ { } ( ) { } ( ) { } ( ) { } ( ) ( ) ( ) ( ) ( ) = × × × + −       × × × × + −       × × × × + −       × × × × + −       − where CMFsp-agg = spatially aggregate CMF fa = site characteristics adjustment factor (i.e., Equation A13) bi,j = regression coefficient for crash type category i (i = mv: multiple vehicle, sv: single vehicle) and crash severity category j (j = fi: fatal-and-injury, pdo: property damage only) exp(bi,j) = CMFsp,i,,j; spatially disaggregate CMF for any leg, crash type category i, and crash severity category j pint,k,j = proportion of crashes associated with intersection leg k (k = 1, 2, 3, 4) and for crash type category i and crash severity category j pint,k = proportion of crashes associated with intersection leg k (k = 1, 2, 3, 4) Ik = indicator variable for treatment application on intersection leg k (= 1.0 if treat- ment applied, 0.0 otherwise) The general regression model is applicable to any number of crash type and severity catego- ries. The exact number will depend on the treatment being considered, the types of crashes it is expected to affect (as determined in Step 4) and the availability of CMFs in the literature. The variable fa represents the optional adjustment factor for site characteristics. Estimation Method A maximum likelihood criterion is used to quantify the regression model coefficients and variance scale parameter. The dependent variable of the regression model is the CMF, which is asymptotic to the lognormal distribution when the CMF is based on many crashes (Griffin and Flowers 1997). For this application, the log-likelihood function for the lognormal distribution is described by the following equation. LL CMF CMF v w v w v w CMFi i i i i i i Equation A17 1 2 ln ln 0.5 ln ln 2 2ln 2( ) ( )( ) ( )= − − +  +    + π +        

A-28 Guidelines for the Development and Application of Crash Modification Factors where LLi = log likelihood for observation i v = predicted variance scale parameter wi = weight of CMF observation i (from Equation A3) CMFi = value of CMF observation i CMFi = predicted value of the CMF for observation i When the weight of each CMF observation is defined using Equation A3, the variance term v in Equation A17 represents a scale parameter. Values of this parameter equal 1.0 when the vari- ance in the CMFs is explained by their standard error. Values of this parameter that exceed 1.0 indicate the presence of additional variability in the CMF observations (beyond that explained by their standard error). Each observation in the regression database represents one aggregate CMF, its standard error, and the crash type, crash severity, and crash distribution proportions corresponding to the aggregate CMF. Additional site characteristics (e.g., area type, terrain, grade, etc.) may be included if these characteristics are thought to have some influence on the treatment’s safety effect. If a site-characteristic variable is included in the database, then a value for this variable must be available for each observation. The best-fit coefficients can be determined by a variety of software tools that can adjust the model coefficients (and variance scale parameter) to maximize the log likelihood. The SAS pro- cedure NLMIXED is one such tool. The Solver feature in the Excel spreadsheet is another tool that can be used for this purpose. Model Fit Statistics The calibrated regression model can be evaluated using a chi-square analysis of the observed and predicted CMF values. The chi-square treatment and chi-square homogeneity statistics in Table A11 can be used for this purpose. The chi-square treatment is used to determine whether the treatment influences crash frequency. The chi-square homogeneity is used to determine whether the observed treatment effect varies from the predicted effect by an amount that is more than can be explained by just random variation (i.e., that some unexplained systematic influence is likely present such that the predicted effect may not accurately describe the treatment effect associated with each observation). To assess the level of treatment influence or homogeneity, the computed chi-square statistic is compared with the chi-square distribution for the specified degrees of freedom. If the computed chi-square statistic for treatment has a probability less than 0.05, then the null hypothesis that the treatment has no effect is rejected (i.e., the treatment is likely to have some effect on crash frequency). If the computed chi-square statistic for homogeneity is less than 0.05, then the null hypothesis that the CMF observations are equal is rejected (i.e., there is likely some unexplained systematic variation present). Source Chi-Square Statistic Degrees of Freedom Treatment p + 1 Homogeneity N – p – 1 Total N Note: p = number of regression coefficients in model; N = number of CMF observations; “1” for predicted variance scale parameter. Table A11. Chi-square analysis of regression model.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-29   Steps for Disaggregating CMFs This section outlines the steps within Step 5 for disaggregating CMFs (Steps 5.1 through 5.4). These steps are described using the example data for shoulder rumble strip presence that are listed in Table A12. There are two categories of crash type and two categories of severity repre- sented in this table. These categories are identified in the following list. • All crash types, all severities • All crash types, fatal-and-injury • Single-vehicle run-off-road (SV-ROR), all severities • Single-vehicle run-off-road (SV-ROR), fatal-and-injury There are 36 aggregate CMFs in Table A12 above. The procedure below demonstrates how these aggregate CMFs were used to estimate the following disaggregate CMFs. • Multiple-vehicle fatal-and-injury • Single-vehicle run-off-road fatal-and-injury • Multiple-vehicle property-damage-only • Single-vehicle run-off-road property-damage-only Step 5.1 Assemble CMFs and Crash Distribution Step 5.1 involves identifying the CMFs to be used in the disaggregation procedure. These may include some or all the CMFs identified in Step 3. The considerations for determining the number of CMFs needed is discussed in the section above titled Number of Aggregate CMFs. Also assembled in this step is the crash distribution and standard error associated with each CMF. Considerations for selecting the crash distribution are outlined in the section titled Crash Distribution. A technique for estimating CMF values (and their standard errors) from a func- tion is described in the section titled CMF Observations from a Function. For the rumble strip data, CMFs were selected for the four crash type and severity categories. All CMFs represent the application of shoulder rumble strips for both travel directions on a segment. Additional variables for area type, roadway type, and state were used to determine whether CMF values varied among these characteristics. The crash counts in Table 22 of the report by Torbic et al. (2009) were used to compute the crash distribution. These counts were from the sites used in the cross-section study (conducted in parallel to the before-after study) that did not have rumble strips. Step 5.2 Define Regression Model During Step 5.2, the considerations described in the Model Development section are used to develop the regression model. Equation A11 was selected as the regression model for the rumble strip data. This model includes one disaggregate CMF for each of the four crash type and severity categories in the previous bullet list (like that shown in Equation A8). Several different combinations of site-characteristic variables were explored in a preliminary regression analysis. The form of the model that included terms for “Presence in Minnesota or Missouri,” “Presence on Freeway,” and “Presence on Multilane Highway” was found to provide the most reasonable fit to the observed CMF values (i.e., m = 3). Step 5.3 Convert Crash Distribution and other CMF Characteristics into Observations During Step 5.3, the crash distribution assembled in Step 5.1 is used to develop the indepen- dent variables in the regression model. The counts used in computing the proportion of crashes are dictated by the crashes represented in the aggregate CMF. A CMF that applies to all crash

A-30 Guidelines for the Development and Application of Crash Modification Factors Area Type Roadway Type Median State Crash Type Severity CMF Urban Freeway Divided Pennsylvania All All 0.9862 Rural Freeway Divided Missouri All All 1.0789 Pennsylvania All All 1.0033 Multilane Divided Minnesota All All 1.1022 Missouri All All 1.2200 Pennsylvania All All 0.8671 Two-lane Undivided Minnesota All All 1.1438 Missouri All All 1.4049 Pennsylvania All All 0.7560 Urban Freeway Divided Pennsylvania All Fatal & Injury 0.8399 Rural Freeway Divided Missouri All Fatal & Injury 0.9416 Pennsylvania All Fatal & Injury 0.8739 Multilane Divided Minnesota All Fatal & Injury 0.7779 Missouri All Fatal & Injury 0.9475 Pennsylvania All Fatal & Injury 0.5988 2-lane Undivided Minnesota All Fatal & Injury 1.0513 Missouri All Fatal & Injury 0.8076 Pennsylvania All Fatal & Injury 0.8203 Urban Freeway Divided Pennsylvania SV-ROR All 0.9419 Rural Freeway Divided Missouri SV-ROR All 0.9209 Pennsylvania SV-ROR All 0.8229 Multilane Divided Minnesota SV-ROR All 1.3836 Missouri SV-ROR All 1.4478 Pennsylvania SV-ROR All 0.7454 Two-lane Undivided Minnesota SV-ROR All 1.1072 Missouri SV-ROR All 1.1687 Pennsylvania SV-ROR All 0.5641 Urban Freeway Divided Pennsylvania SV-ROR Fatal & Injury 0.9257 Rural Freeway Divided Missouri SV-ROR Fatal & Injury 0.8436 Pennsylvania SV-ROR Fatal & Injury 0.7680 Multilane Divided Minnesota SV-ROR Fatal & Injury 0.8971 Missouri SV-ROR Fatal & Injury 1.0016 Pennsylvania SV-ROR Fatal & Injury 0.8014 2-lane Undivided Minnesota SV-ROR Fatal & Injury 0.6759 Missouri SV-ROR Fatal & Injury 0.5541 Pennsylvania SV-ROR Fatal & Injury 0.6334 Abbreviation: SV-ROR, single-vehicle run-off-road crash. Table A12. CMFs of shoulder rumble strips.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-31   type and severity categories would have its proportions computed using a denominator equal to the count of all crashes, regardless of crash type or severity. The numerator of each proportion would be equal to the count of crashes associated with a specific combination of crash type and severity. In contrast, a CMF that applies to all crash types and a specific severity would have its proportions computed using a denominator equal only to the count of crashes of the specified severity. The numerator of each proportion would be equal to the count of crashes (of the speci- fied severity) for a specific crash type. This process is extended to other aggregation combina- tions. It is illustrated by example in the next paragraph. The computed proportions are shown in Columns 4 to 7 of Table A13. The proportions shown in each row are specific to the corresponding crash type and severity category shown in the table. Thus, the proportion in Column 4 of the first row represents the ratio of multiple- vehicle fatal-and-injury crashes to total crashes (i.e., all crash types and severities combined). The proportion in Column 4, Row 10 represents the ratio of multiple-vehicle fatal-and-injury crashes to fatal-and-injury crashes (all crash types). With this technique, the proportions in each row correctly add to 1.0. The aggregated CMFs are shown in the second-to-last column of the table. The reported standard error is shown in the last column of the table. All values shown in the table were obtained from the report by Torbic et al. (2009). Site Characteristics Crash Distribution Crash Type Severity CMFagg Standard Error MN or MO Free- way Multi -lane pmv,fi psv,fi pmv,pdo psv,pdo 0 1 0 0.289 0.200 0.291 0.220 All All 0.9862 0.0572 1 1 0 0.126 0.203 0.303 0.368 All All 1.0789 0.0413 0 1 0 0.166 0.317 0.198 0.319 All All 1.0033 0.1180 1 0 1 0.171 0.140 0.509 0.180 All All 1.1022 0.1468 1 0 1 0.124 0.142 0.544 0.190 All All 1.2200 0.0946 0 0 1 0.124 0.425 0.177 0.274 All All 0.8671 0.3564 1 0 0 0.239 0.148 0.447 0.167 All All 1.1438 0.0801 1 0 0 0.221 0.127 0.473 0.179 All All 1.4049 0.1800 0 0 0 0.252 0.319 0.153 0.276 All All 0.7560 0.0861 0 1 0 0.591 0.409 0.000 0.000 All FI 0.8399 0.0725 1 1 0 0.382 0.618 0.000 0.000 All FI 0.9416 0.0641 0 1 0 0.343 0.657 0.000 0.000 All FI 0.8739 0.1462 1 0 1 0.549 0.451 0.000 0.000 All FI 0.7779 0.1963 1 0 1 0.467 0.533 0.000 0.000 All FI 0.9475 0.1231 0 0 1 0.226 0.774 0.000 0.000 All FI 0.5988 0.4252 1 0 0 0.618 0.382 0.000 0.000 All FI 1.0513 0.1266 1 0 0 0.635 0.365 0.000 0.000 All FI 0.8076 0.2182 0 0 0 0.441 0.559 0.000 0.000 All FI 0.8203 0.1159 0 1 0 0.000 0.477 0.000 0.523 SV-ROR All 0.9419 0.0732 1 1 0 0.000 0.356 0.000 0.644 SV-ROR All 0.9209 0.0571 0 1 0 0.000 0.498 0.000 0.502 SV-ROR All 0.8229 0.1227 1 0 1 0.000 0.437 0.000 0.563 SV-ROR All 1.3836 0.2662 1 0 1 0.000 0.428 0.000 0.572 SV-ROR All 1.4478 0.1479 0 0 1 0.000 0.608 0.000 0.392 SV-ROR All 0.7454 0.3744 1 0 0 0.000 0.469 0.000 0.531 SV-ROR All 1.1072 0.1707 1 0 0 0.000 0.415 0.000 0.585 SV-ROR All 1.1687 0.2176 0 0 0 0.000 0.537 0.000 0.463 SV-ROR All 0.5641 0.0913 0 1 0 0.000 1.000 0.000 0.000 SV-ROR FI 0.9257 0.0993 1 1 0 0.000 1.000 0.000 0.000 SV-ROR FI 0.8436 0.0822 0 1 0 0.000 1.000 0.000 0.000 SV-ROR FI 0.7680 0.1571 1 0 1 0.000 1.000 0.000 0.000 SV-ROR FI 0.8971 0.2863 1 0 1 0.000 1.000 0.000 0.000 SV-ROR FI 1.0016 0.1584 0 0 1 0.000 1.000 0.000 0.000 SV-ROR FI 0.8014 0.5695 1 0 0 0.000 1.000 0.000 0.000 SV-ROR FI 0.6759 0.1761 1 0 0 0.000 1.000 0.000 0.000 SV-ROR FI 0.5541 0.2316 0 0 0 0.000 1.000 0.000 0.000 SV-ROR FI 0.6334 0.1335 Abbreviations: SV-ROR, single-vehicle run-off-road crash; FI, fatal or injury crash. Table A13. Shoulder rumble strip data.

A-32 Guidelines for the Development and Application of Crash Modification Factors The observations in Table A13 are organized into four groups according to crash type and severity (i.e., All/All, All/FI, SV-ROR/All, and SV-ROR/FI). Collectively, these groups represent a mixed data set where (1) each group has dependent and independent variables that are defined the same within the group but are defined differently among groups and (2) regression coefficients for the independent variables that are defined the same for all groups. Regarding point “a,” the crash distribution variables and the dependent variable are defined differently among groups (but the site-characteristic variables are defined the same for all groups). For the first group, the proportions and the CMF are based on all crashes. In contrast, for the second group, the propor- tions and the CMF are based on only fatal-and-injury crashes. The benefit of the use of a mixed data set is that it provides more observations upon which to base the estimates. The four groups are partially unrelated because of the differences in variable definition. On the other hand, they are partially related such that there is some repetition of the underlying crash counts among observations. This repetition among observations is likely to result in standard errors that are biased slightly low. However, the calibration coefficients and resulting disaggregate CMFs will not be biased. Step 5.4 – Estimate Model Coefficients During Step 5.4, the maximum likelihood approach is used to quantify the regression coef- ficients using Equation A17. The SAS procedure NLMIXED is then used to determine the coeffi- cients and their standard errors. The analysis can also be undertaken with Excel Solver to obtain the coefficient and parameter values (although Solver does not report the standard error of each value). Additional details regarding the model estimation process are provided in the section titled Estimation Method. The best-fit form of the model for the rumble strip CMFs is shown in Equation A18. CMF p p p p I I I agg p mv fi sv fi mv pdo sv pdo mn mo freeway multilane [ ]( ) ( ) ( ) ( ) ( ) = − × + − × + × + − × × × + × + × Equation A18 exp 0.120 exp 0.247 exp 0.110 exp 0.034 exp 0.111 0.0128 0.136 , , , , , , where CMFagg,p = predicted aggregate CMF Imn,mo = indicator variable for state (= 1.0 if segment in Minnesota and Missouri, 0.0 otherwise) Ifreeway = indicator variable for freeway (= 1.0 if segment on freeway, 0.0 otherwise) Imultilane = indicator variable for multilane (= 1.0 if segment on multilane highway, 0.0 otherwise) The standard errors for the coefficients in Equation A18 (in the order shown in the equation) are: 0.096, 0.072, 0.096, 0.086, 0.041, 0.047, 0.063. Those values that are underlined are associated with a coefficient that has a significance level of 0.05 or less. If the calibrated equation is used to estimate the aggregate CMF for rural two-lane highways, then Ifreeway and Imultilane are both set to 0.0. The best-fit disaggregate CMF values for two-lane highways in Pennsylvania are provided in the following list: • Multiple-vehicle fatal-and-injury 0.887 • Single-vehicle fatal-and-injury 0.781 • Multiple-vehicle property-damage-only 1.117 • Single-vehicle property-damage-only 0.966 Based on the coefficient values in Equation A18, each CMF value in the bullet list is increased by 1.3 percent when applied to freeway sites and by 14 percent for multilane highway sites. They are increased by 12 percent when applied to sites in Minnesota or Missouri.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-33   The chi-square treatment is found to have a chi-square value of 47.4 (8 degrees of freedom, = 7+1). It corresponds to a p-value of 0.0001. This value is smaller than 0.05, so the null hypoth- esis is rejected, and we can be confident that the treatment has an overall average effect on safety in the three states combined. The chi-square homogeneity is found to have a chi-square value of 37.6 (28 degrees of freedom, = 36–7–1). It corresponds to a p-value of 0.11. This value is larger than 0.05, so the null hypoth- esis cannot be rejected, and we cannot be confident that there is some remaining system atic variation that might be explained (i.e., the model may be explaining all the systematic variability in the data). Validation This section provides some empirical evidence of the validity of the model developed in the previous section (i.e., Equation A18). If this regression model can predict CMF values that are reasonably like CMF values developed using data for another state (i.e., a state whose data were not used to calibrate the model) then there is evidence that the model can accurately predict CMF values for other locations. More importantly, it would imply that the concept of “local adjustment of aggregate CMFs based on crash distribution” is valid. CMFs for shoulder rumble strip installation on two-lane rural highways in Florida that were developed by Park and Abdel-Aty (2015) are the subject of this examination. The data needed to apply the calibrated model are listed in the first eight columns of Table A14. The crash dis- tribution proportions are based on the expected crash counts for the “before” period that were reported by Park and Abdel-Aty. It is notable that the distribution values indicate that 89.1 per- cent of the crashes are fatal or injury. In contrast, the states represented in Table A13 have about 43 percent fatal or injury crashes on rural two-lane highways. For these reasons, the data in Table A14 provide a good basis for testing whether the local crash distribution can be used to estimate the aggregate CMF for one state based on disaggregate CMFs calibrated in another state. The CMF values reported by Park and Abdel-Aty are listed in Column 7. These values can be compared with those predicted using the model developed in the previous section (i.e., Equa- tion A18). The predicted CMF values are listed in the last column of the table. The CMF values in a common row are shown to be similar in magnitude, which is evidence that local adjustment based on crash distribution is viable. Step 6 Process Disaggregate CMFs If the previous steps yielded two or more disaggregate CMFs for a common crash distribution category, then this step must be followed to process those CMFs and arrive at a final CMF to be used to represent that category. As noted in Figure A1, this step is performed for each crash distribution category represented in the collective set of disaggregate CMFs. Crash Distribution Crash Type Severity CMF Standard Error Predicted CMF (Equation A18) pmv,fi psv,fi pmv,pdo psv,pdo 0.719 0.172 0.055 0.055 All All 0.83 0.07 0.886 0.807 0.193 0.000 0.000 All FI 0.84 0.08 0.867 0.000 0.759 0.000 0.241 SV-ROR All 0.75 0.14 0.826 0.000 1.000 0.000 0.000 SV-ROR FI 0.80 0.16 0.781 Abbreviations: SV-ROR, single-vehicle run-off-road crash; FI, fatal or injury crash. Table A14. Data describing the safety effect of shoulder rumble strip installation on two-lane rural highways in Florida.

A-34 Guidelines for the Development and Application of Crash Modification Factors For each crash distribution category of interest, the user must determine how many CMFs were identified for that category. If more than one CMF was identified for a particular category, Steps 6a and 6b or 6c must be applied. For example, consider a situation where a safety profes- sional using this procedure identified CMFs for the following types of crash severity in Step 3. • Fatal—1 CMF identified • Serious injury—3 CMFs identified • Minor injury—1 CMF identified • PDO—1 CMF identified The existence of more than one CMF for the “serious injury” category requires the application of Steps 6a and 6b or 6c to obtain the best estimate of treatment effect. In contrast, the CMFs for fatal, minor injury, and PDO are used directly (i.e., this step is not applied) because there is only one CMF available for each of those categories. Step 6a Test for Homogeneity When more than one study has produced CMFs for the same treatment, the values will often not be in exact agreement. Differences in CMF value can be due to random variation, system- atic differences in circumstances among the study locations, or both. Systematic differences can result from the treatment effect being influenced by the location’s road geometry, traffic control, traffic characteristics, topography, population demographics, or the driving environment. When there are multiple eligible disaggregate CMFs for a given category, then the CMFs should be tested for homogeneity. If the homogeneity test results indicate that the differences among CMF values are small, then the individual CMFs can be combined into a single CMF average value for the given category (Step 6b). If the test results indicate that the differences among CMF values are large, then further investigation is needed to determine which value (if any) is most applicable to the site of interest (Step 6c). The test for homogeneity can be carried out by following the procedure below or by using the either spreadsheet tool developed under this project. The CMF Regression Software is a more full-feature spreadsheet tool that would be used for several steps in this entire procedure. The CMF Combination Tool is a pared-down, simpler tool that is intended only for the calculations needed for Steps 6a and 6b. To apply the homogeneity test, two or more CMF values (and their associated standard error) are needed. Presumably, each CMF represents a different location. Table 1 lists two CMF values and their associated standard errors (i.e., there are two “observations”). This table is used to illustrate the test. Step 6a.1 Compute Weight In Step 6a.1, the user computes the statistical weight to be associated with each CMF value. This weight is computed using the following equation. w CMF s i i Equation A191 2 =    where wi = weight of CMF observation i CMFi = value of CMF observation i si = standard error of CMF observation i The weight calculated for each observation is shown in Column 4 of Table A15.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-35   The homogeneity test should be based on CMFs that are associated with a weight of about 4.0 or more. CMFs with a smaller weight are less reliable because they are based on a relatively small sample size (i.e., a small number of observed crashes). As a minimum, most of the CMFs being tested in each group should have a weight of 4.0 or more. Weights less than 4.0 corre- spond roughly to a 95th percentile confidence interval for the CMF that includes negative values (assuming a normal distribution). Step 6a.2 Compute Log In Step 6a.2, the user computes the natural log of the CMF value. This calculation is shown by the following equation. L CMFi Equation A20ln 1( )= where Li = natural log of CMF observation i The values calculated using Equation A20 are shown in column 5 of Table A15. Step 6a.3 Compute Average Log In Step 6a.3, the user computes the weighted average value of L. This average is computed using the following equation. L w L w i i i i i Equation A21 ∑ ∑ = where L– = weighted average value of L Equation A21 is used with the values in columns 4 and 5 of Table A15 to compute a weighted average value of −0.332 for L–. Step 6a.4 Compute Chi-Square Statistic In Step 6a.4, the user computes the chi-square value for each observation. This calculation is shown by the following equation. w L Li i i Equation A222 2( )χ = − where χi2 = chi-square value of CMF observation i Step 6a.5 Check Results In Step 6a.5, the user adds the chi-square values and compares this result with the chi-square distribution for n−1 degrees of freedom, where n is the number of CMF observations. The total Obs. (i) Crash Modification Factor (CMF) Standard Error (s) Weight (w) Natural Log of CMF (L) Chi-Square Value ( 2iχ ) 1 0.75 0.04 351.6 -0.288 0.691 2 0.62 0.06 106.8 -0.478 2.276 Total: 458.3 Total: 2.968 Table A15. Example calculation of homogeneity test.

A-36 Guidelines for the Development and Application of Crash Modification Factors chi-square value is 2.968. Relating this value to the chi-square distribution with for 1 degree of freedom (= 2−1) yields a probability of 0.085. There is an 8.5 percent chance that CMF value differences this large could exist and still be due only to random variation. Guidelines The application of the homogeneity test is typically used to determine whether there are location-specific differences underlying a set of CMF values. A chi-square value associated with a probability (α) of 0.05 is often used for this test (Griffin and Flowers 1997). When the computed chi-square value is large such that it is associated with a probability that is smaller than 0.05, the null hypothesis is rejected, and the conclusion is that the variability in the CMF values cannot be explained by random variation alone. The implication is that the values are also very likely to include some systematic variation, such that they should not be combined. For the example in Table A15, the computed value of 0.085 exceeds the suggested threshold of 0.05, so it is concluded that the CMF values can be combined to obtain an overall average CMF. For a given set of CMF observations, the homogeneity test is more likely to indicate that the CMFs can be combined when the average number of crashes per CMF observation is small. For this reason, the confidence interval of the overall average CMF should also be checked to con- firm that the range of likely CMF values is acceptable for the intended application. This check is described in the next section. Step 6b Combine CMFs If the differences between CMFs are small, the CMFs should be combined to create a single value. The process for combining CMFs is presented in this step. It can also be accomplished using either spreadsheet tool developed under this project. This procedure assumes that CMF values (and their respective standard errors) for two or more locations are available from the literature and that the values are unbiased. Several meth- ods for estimating a CMF and its standard error are available in the literature, depending on the study design (e.g., Hauer 1997). The procedure is also based on the premise that more detailed data (e.g., crash counts) are not available for the collective set of locations. There are more robust procedures available for computing an overall average CMF when detailed data (e.g., crash data) are available for the collective set of locations (e.g., Hauer 1997). These procedures should be used to compute the overall average CMF when detailed data are available. Step 6b.1 Compute Standard Error The standard error of the overall average CMF is computed using the following equation. s e LCMF L se Equation A23= × with L w se i i Equation A241 0.5 ∑ =     where sCMF = standard error of CMF Lse = standard error for L –

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-37   Using the data in Table A15, the value of eL – is computed as 0.717. The value of sCMF is computed as 0.034 (= 0.717/[351.6+106.8]0.5). Step 6b.2 Compute Overall Average CMF The following equation can be used to estimate the overall average CMF. CMF e fL c Equation A25= × with f w L L w c i i i i i Equation A26exp 0.574 2∑ ∑ ( ) = −          where CMF = overall average CMF fc = correction factor n = number of observations Equation A25 requires a correction factor fc to remove the bias introduced by the log trans- form. The factor provides reliable results when the weight w of a majority (desirably all) of the CMF observations being combined has a value of about 4.0 or more. The derivation of the correction factor is described in Section A4, Supporting Research, under Evaluation of Two Estimators of Combined Average CMF. Using the data in Table A15, the value of eL – is computed as 0.717. The correction factor fc is computed as 1.0037 (= exp[0.574 × 2.968 / 458.3]). The value of CMF is computed as 0.720 (= 0.717 × 1.0037). Step 6b.3 Compute Confidence Interval The confidence interval for the overall average CMF is computed using the following equation. CI CMF CMFp u p l p Equation A27: ,, ,( ) with CMF CMF eu p z Lse Equation A28, = × ( )× CMF CMF el p z Lse Equation A29, = × ( )− × where CIp = pth percentile confidence interval of the overall average CMF CMFu,p = upper limit of the pth percentile confidence interval CMFl,p = lower limit of the pth percentile confidence interval z = standard normal variable associated with percentile p (z = 1.96 for 95% confidence interval) The 95th percentile confidence interval for the data in Table A15 is computed as 0.657 to 0.789. Step 6b.4 Check Results The confidence interval of the overall average CMF should be checked to confirm that the range of likely CMF values is acceptable for the intended application.

A-38 Guidelines for the Development and Application of Crash Modification Factors Criterion 1. Implementation. When the treatment is to be implemented at a given location, the confidence interval should exclude values of 1.0 or larger. When the interval includes values greater than 1.0, the implication is that there is a possibility that a location’s expected crash frequency may increase following implementation. Criterion 2. Prediction. When the treatment effect is being quantified for purposes of predict- ing crash frequency, then the range ratio (as a proportion of the overall average CMF) should be less than 0.40. This ratio is computed using the following equation. R CMF CMF CMF p u p l p Equation A30, ,( )= − where Rp = range ratio based on the pth percentile confidence interval The range ratio for the data in Table A15 is computed as 0.183 (= [0.789 – 0.657]/0.720). This value is less than 0.40 so the CMF value is known with reasonable accuracy for predictive applications. Guidelines If the confidence interval check does not satisfy the criterion corresponding to the intended application, then overall average CMF may not be known with sufficient accuracy to form the basis for sound decisions. More generally, the homogeneity test and the confidence interval check should both be satisfied before the combined average CMF can be applied with reason- able confidence. Increasing the number of observations is one means of satisfying this check. However, increas- ing the number of observations may make it more difficult to satisfy the homogeneity test if the individual CMFs are not similar. In other words, the overall average CMF is meaningful when there is (1) enough observations to provide an estimate that is useful for the intended applica- tion and (2) sufficient similarity among individual CMF values to confirm that a combined CMF describes treatment effect for a wide range of locations. Step 6c Select the One CMF That Is a Best Match to the Subject-Site Characteristics or Develop CMFunction If the differences between CMFs are large enough so that they cannot be combined, the user has two options: • Select the CMF that is the best match to the subject site characteristics. If sufficient infor- mation is available for each of the identified CMFs, the user should select the CMF which most closely matches with the subject site. This would involve examining the details of the CMFs to determine the characteristics of the sites at which they were developed as well as the geographic area of the study. See Table A1 for a list of site characteristics to consider in the comparison. • Develop a CMFunction. The differences between the identified CMFs may represent the effect due to a site characteristic (e.g., shoulder width, turn lane presence, etc.) which can be quantitatively accounted for by developing a CMFunction. The function would be able to incorporate all identified CMF values and use a variable to account for the differences in site conditions. Guidelines on developing CMFunctions is presented in Appendix C. It should be noted that developing a CMFunction requires statistical modeling experience.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-39   Step 7 Develop Aggregate CMF Using Crash Distribution at Subject Site The previous step resulted in a final set of disaggregate CMFs for the treatment of interest. If the user needs to arrive at an aggregate (total crash) CMF for the treatment of interest, Step 7 must be performed. The disaggregate CMFs are aggregated together using the local crash distribution of the subject site to arrive at the aggregate CMF. This step consists of two parts, determining the local crash distribution of the subject site (Step 7a) and calculating the aggregate CMF (Step 7b). If the user does not need to arrive at an aggregate CMF for the treatment, then Step 7 does not need to be performed. Step 7a Determine the Local Crash Distribution During this step, a representative local crash distribution is developed for the site of interest. It should have the same categories as represented in the disaggregate CMFs. The crash distribution must represent the crash history before the treatment is applied. Once the treatment is applied, it will alter the crash distribution. If the site was selected for treatment for reasons that do not include a recent increase in crashes, then the crash distribution can be obtained from the crash history of the site. The dis- tribution should reflect a three- to five-year average to insure reasonable stability in the com- puted proportions. There should be no major changes in the site characteristics during the years included in the crash history. The crash history should include the most recent years and include as many years as needed (up to five years) such that it includes more than 100 crashes. If the site of interest was selected for treatment because of a recent increase in crashes or if its crash history includes less than 100 crashes, then the crash distribution should be based on the crash history of a collective set of sites that are like the site of interest. The same period should be used to define the crash history for each site in the set of sites. Importantly, none of the sites included in this set should have the treatment present during the crash history period. For spatially-disaggregate CMFs, the local crash distribution can be estimated using traffic volume if crash data are not readily available. This approach recognizes that crash frequency is approximately proportional to traffic volume when the range of volumes of interest is relatively small. Thus, the proportion of segment crashes associated with one travel direction of a two-way roadway can be estimated using the following equation. p d AADT d AADT d AADT seg i i Equation A31 _ _ _ , 1 2 = + where pseg,i = proportion of crashes associated with segment travel direction i (i = 1, 2) d_AADTi = directional (one-way) annual average daily traffic volume for travel direction i (i = 1, 2) For some applications, it may be useful to define i equal to 1 for the highest volume travel direction and 2 for the other travel direction. Similarly, the proportion of intersection crashes associated with one leg can be estimated using the following equation. p AADT AADT AADT AADT AADT int i i Equation A32, 1 2 3 4 = + + +

A-40 Guidelines for the Development and Application of Crash Modification Factors where pint,i = proportion of crashes associated with intersection leg i (i = 1, 2, 3, 4) AADTi = annual average daily traffic volume for intersection leg i (i = 1, 2, 3, 4) For some applications, it may be useful to define i equal to 1 for the highest volume major- street leg, 2 for the other major-street leg, 3 for the highest volume minor-street leg, and 4 for the other minor-street leg. If Equation A32 is used for a three-leg intersection, then the fourth term in the denominator (i.e., AADT4) is deleted or set to zero (0). Step 7b Calculate the Aggregate CMF Based on the Local Crash Distribution During this step, the aggregate CMF is computed using the disaggregate CMFs and the local crash distribution. This step can be carried out using the CMF Regression Software spreadsheet tool. Three example cases are shown below to illustrate this step. Case 1 Variation Represented by Crash Type and Severity Distribution If the analysis involves CMFs that differ by crash type distribution, severity distribution, or both, then the following equation is used to compute the aggregate CMF. This equation follows from Equation A32 but allows for any number of crash categories. CMF CMF pagg i ii n Equation A331∑ ( )= ×= where CMFagg = aggregate CMF CMFi = disaggregate CMF for category i (i = 1, 2, 3, . . . , n) pi = proportion of crashes associated with category i (i = 1, 2, 3, . . . , n) The proportions included in Equation A33 must add to 1.0. Example: Consider a segment for which it is known that the proportion of fatal-and-injury (FI) crashes is 0.30 and the proportion of property-damage-only (PDO) crashes is 0.70. A specific segment treatment is known to reduce FI crashes by 60 percent and PDO crashes by 10 percent. Using Equation A33, the aggregate CMF is computed as 0.75 [= (0.40 × 0.30) + (0.90 × 0.70)]. Case 2 Variation Represented by Crash Location Distribution Intersection Case. For intersection analysis where treatment application varies by leg, the following equation can be used to compute the spatially aggregate CMF. This CMF would be applied to overall intersection crash frequency. 1 1 1 1 ,1 ,1 ,1 ,2 ,2 ,2 ,3 ,3 ,3 ,4 ,4 ,4 1 2 3 4 Equation A34 [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( ) = × + − × × + − × × + − × × + − −CMF CMF p p CMF p p CMF p p CMF p p sp agg sp int int I sp int int I sp int int I sp int int I where CMFsp-agg = spatially aggregate CMF (CMF for total intersection crashes when not all legs are treated) CMFsp,i = spatially-disaggregate CMF for intersection leg i (i = 1, 2, 3, 4) pint,i = proportion of crashes associated with intersection leg i (i = 1, 2, 3, 4) Ii = indicator variable for treatment presence on intersection leg i (= 1.0 if treatment present, 0.0 otherwise)

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-41   The proportions included in Equation A34 must add to 1.0. If Equation A34 is used for a three-leg intersection, then the fourth term in brackets (i.e., with subscripts indicating leg 4) is deleted. The form of Equation A34 is different from Equation A33, since it must account for the interaction between the addition of a bay on one leg and the frequency of crashes within the intersection (but not on the treated leg). This interaction is described in the text associated with Table 6 in the main body of this report. Example: Consider a four-leg urban signalized intersection for which it is known that 25 per- cent of the crashes occur on each leg. One leg has a left-turn bay added. None of the other legs have a left-turn bay. The addition of a left-turn bay to one leg is known to reduce crashes associated with the leg by 40 percent. Using Equation A34, the spatially aggregate CMF is computed as 0.9 {= [(0.60 × 0.25) + (1 – 0.25)]1.0 × [ . . . ]0.0 × [ . . . ]0.0 × [ . . . ]0.0}. The first term of Equation A34 has a value other than 0.0 because only one leg is treated. The terms associated with the other three legs have an indicator variable value of 0.0 which makes the corresponding term equal to 1.0. Segment Case. For segment analysis where treatment application can vary by travel direc- tion, the following equation can be used to compute the spatially aggregate CMF. This CMF would be applied to overall segment crash frequency. CMF CMF p CMF psp agg sp seg sp seg Equation A35,1 ,1 ,2 ,2( ) ( )= × + ×− where CMFsp-agg = spatially aggregate CMF (CMF for total segment crashes when not all directions are treated CMFsp,i = spatially-disaggregate CMF for travel direction i (i = 1, 2) pseg,i = proportion of crashes associated with segment travel direction i (i = 1, 2) The two proportions in Equation A35 must add to 1.0. If a treatment is not applied in one travel direction, then the CMFsp value for that direction is 1.0. Example: Consider a road segment where a treatment is applied to only one travel direction. It is known that the proportion of traffic traveling in the treated direction is 0.55 and that the treatment can reduce crashes by 10 percent in the treated travel direction. Using Equation A35, the spatially aggregate CMF is computed as 0.945 [= (0.90 × 0.55) + (1.0 × 0.45)]. Case 3 Variation Represented by Crash Type, Severity, and Location Distribution If the evaluation involves CMFs that differ by crash type, severity, and location, then the concepts described for Cases 1 and 2 in the preceding paragraphs are used in combination. The CMFs used for this case will need to be disaggregated by the selected crash type, severity, and location categories. Consider an intersection for which a left-turn bay is being added to one leg. Disaggregate CMFs are available for the following categories: FI crashes for single-leg treatment, PDO crashes for single-leg treatment. As a first step, the procedure for Case 2 would be applied to determine the spatially aggregate CMF for FI crashes. It would be repeated to determine the spatially aggregate CMF for PDO crashes. Then, the procedure for Case 1 would be applied to deter- mine the aggregated CMF (as aggregated across the two severity categories being considered: FI and PDO).

A-42 Guidelines for the Development and Application of Crash Modification Factors Result At the conclusion of the process, the user will have a CMF or set of CMFs that can be used to estimate the effect of the proposed treatment. If the user desired an aggregate (total crash) CMF and followed Step 7, the CMF will be appropriate for estimating the effect of the proposed treat- ment on the expected number of total crashes at the subject site. After following this procedure, this estimated effect will be calibrated to the subject site according to its distribution of crash type, severity, or treatment location.

A-43   A city engineer is seeking to improve the safety of an intersection in her city. The inter section is currently stop-controlled, and her diagnosis of the intersection crash patterns shows that installing a traffic signal would be an appropriate countermeasure. She needs to conduct a quan- titative benefit-cost analysis to show the expected benefit from installing the traffic signal. To do this, she will need to determine the most appropriate CMFs to use in the analysis to estimate the benefit due to crash reductions. To identify these CMFs, she uses the step-by-step process presented in this appendix. Each step is demonstrated below. Step 1 Identify Proposed Treatment and Subject Site The proposed treatment is the installation of a standard traffic signal. The subject site is the stop-controlled intersection under analysis by the engineer. Step 2 Identify Characteristics of Subject Site The subject site is a four-leg intersection with stop control on the minor road, four legs, a speed limit of 45 mph on the major road, and an average volume on the major road of 7,000 vehicles per day. The engineer uses SPFs to calculate that the expected crashes at this site before signal installation are 2.3 crashes per year. Step 3 Identify Existing CMFs for Proposed Treatment The engineer consults CMF resources for information on the effect of installing a traffic signal. She identifies CMFs from three studies. Study 1 provided CMFs for each level of crash severity (i.e., K, A, B, C, and O). Studies 2 and 3 produced CMFs only for fatal and A-level injury crashes (i.e., K and A). All studies used data from urban, stop-controlled, 4-leg intersections. All studies were con- ducted in a state other than the subject site. The CMFs identified from these studies address all crash types and are as follows. C H A P T E R   3 Example Application of Procedure Study 1 Crash Severity CMF Standard Error K 0.58 0.12 A 0.68 0.05 B 0.80 0.11 C 0.83 0.08 O 1.15 0.05

A-44 Guidelines for the Development and Application of Crash Modification Factors Step 4 Compare Subject Site Characteristics to Known CMF Influential Factors The engineer consults Table A7 and sees that traffic volume, number of intersection legs, speed limit, and frequency of expected crashes before treatment are influential factors on the effect of a traffic signal installation. To use the CMFs identified in Step 3, the characteristics of the subject site must match the data of the CMF on these influential factors. The engineer reviews information about each of the studies and the intersections that were used in each study. Table A16 shows the comparison between the characteristics of the engineer’s subject site and the characteristics (or range of characteristics) for the sites used in each of the three studies under consideration. Both Study 1 and Study 2 were conducted using sites with characteristics that match the sub- ject site on the influential factors. However, Study 3 used three-leg intersections, which does not match the subject site (four legs). The engineer concludes that only the CMFs identified from Studies 1 and 2 are eligible to use in analyzing the subject site. Step 5 Convert CMF to Disaggregate CMFs Step 5 is not necessary, since the engineer was able to identify CMFs that were already dis- aggregated (the identified CMFs address each level of crash severity). Step 6 Process Disaggregate CMFs Step 6 must be done for each level of crash severity—K, A, B, C, and O. For the B, C, and O severities, only one CMF was identified (from Study 1), so no processing is needed for these CMFs. However, for the fatal and A-injury severities, the engineer identified two CMFs for each severity (one from each study). Thus, she must conduct steps 6a through 6c to determine how to use the fatal and A-injury severity CMFs. Study 2 Crash Severity CMF Standard Error K 0.45 0.27 A 0.80 0.03 Study 3 Crash Severity CMF Standard Error K 0.55 0.21 A 0.68 0.12 Influential Factor Subject Site Characteristics CMFs from Study 1 CMFs from Study 2 CMFs from Study 3 Characteristic Match Characteristic Match Characteristic Match Traffic volume 7,000 vehicles per day on major road 4,000 to 15,000 vehicles per day on major road Yes 6,000 to 11,500 vehicles per day on major road Yes 5,000 to 21,000 vehicles per day on major road Yes Number of legs 4 4 Yes 4 Yes 3 No Speed limit 45 mph 35 to 55 mph Yes 45 mph Yes 45 to 55 mph Yes Frequency of expected crashes before treatment 2.3 crashes per year 0.6 to 7.5 crashes per year Yes 0.2 to 9 crashes per year Yes 0.2 to 9 crashes per year Yes Table A16. Comparison of subject site characteristics to CMF characteristics.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-45   If the homogeneity test in Step 6a indicates that the differences among CMF values are small, then the individual CMFs can be combined into a single CMF average value for the given category (Step 6b). If the test results indicate that the differences among CMF values are large, then further investigation is needed to determine which value (if any) is most applicable to the site of interest (Step 6c). These steps will be repeated for both sets of CMFs (fatal and A-injury severities). Step 6a (for Fatal Crash CMFs)—Test for Homogeneity Step 6a.1 Compute Weight w CMF s i i i =    2 where wi = weight of CMF observation i CMFi = value of CMF observation i si = standard error of CMF observation i The weight calculated for each CMF observation is shown in Column 4 of Table A17. Step 6a.2 Compute Log. The second step is to compute the natural log of the CMF value. The natural log values of the CMFs are shown in column 5 of Table A17. Step 6a.3 Compute Average Log. The third step is to compute the weighted average value of L. This average is computed using the following equation. L w L w i i i i i ∑ ∑ = where L– = weighted average value of L Using the values in columns 4 and 5 of Table A17, the weighted average value L– is computed as -0.572. Step 6a.4 Compute Chi-Square Statistic. The fourth step is to compute the chi-square value for each observation. This calculation is shown by the following equation. w L Li i i2 2( )χ = − where χi2 = chi-square value of CMF observation i The chi-square values of the CMFs are shown in Column 6 of Table A17. Obs. (i) Crash Modification Factor (CMF) Standard Error (s) Weight (w) Natural Log of CMF (L) Chi-Square Value ( 2 iχ ) 1 0.58 0.12 23.4 -0.545 0.017 2 0.45 0.27 2.8 -0.799 0.143 Total: 26.1 0.16 Table A17. Values for homogeneity test calculations for fatal crash CMFs.

A-46 Guidelines for the Development and Application of Crash Modification Factors Step 6a.5 Check Results. The last step is to add the chi-square values and compare this result with the chi-square distribution for n − 1 degrees of freedom, where n is the number of CMF observations. The total chi-square value is 0.16. Relating this value to the chi-square dis- tribution with for 1 degree of freedom (= 2 − 1) yields a probability of 0.685. There is a 68.5 per- cent chance that CMF value differences this large could exist and still be due only to random variation. This value of 0.685 exceeds the suggested threshold of 0.05, so it is concluded that the two CMF values can be combined to obtain an overall average CMF (Step 6b). Step 6b (for Fatal Crash CMFs)—Combine CMFs Given that there is not a statistically significant difference between the two CMFs, they can be combined to produce a single CMF to be used for estimating the change in fatal crashes using the steps below. Step 6b.1 Compute Standard Error. The standard error of the overall average CMF is com- puted as follows. s e LCMF L se= × with L w se i i 1 0.5 ∑ =     where sCMF = standard error of CMF Lse = standard error for L – Using the data in Table A17, the value of eL – is computed as e−0.572 = 0.564. The value of Lse is cal- culated as (23.4 + 2.8)0.5 = 0.196. The value of sCMF is computed as 0.564 ∗ 0.196 = 0.110. Step 6b.2 Compute Overall Average CMF. The overall average CMF is calculated as follows. CMF e fL c= × with f w L L w c i i i i i exp 0.574 2∑ ∑ ( ) = −          where CMF = overall average CMF fc = correction factor n = number of observations As previously calculated, the value of eL – is 0.564. The correction factor fc is computed as 1.0035. The overall average CMF is computed as 0.564 × 1.0035 = 0.57.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-47   Step 6b.3 Compute Confidence Interval. The engineer calculates a confidence interval for the combined CMF using 95% confidence (z-value of 1.96). The upper limit of the CMF confi- dence interval is calculated as follows. CMF CMF eu p z Lse 0.830.57 e, 1.96 0.196= × = × =( ) ( )× × The lower limit of the CMF confidence interval is calculated as follows. CMF CMF el p z Lse 0.390.57 e, 1.96 0.196= × = × =( ) ( )− × − × Step 6b.4 Check Results. Criterion 1. Implementation. The confidence interval of the com- bined CMF is 0.39 to 0.83. This range does not include 1.0, so it is reasonably assumed that the effect of the countermeasure will reduce fatal crashes (i.e., very unlikely that the countermeasure would increase fatal crashes). Criterion 2. Prediction. Although the countermeasure is expected to reduce fatal crashes, the combined CMF has a relatively large standard error, so it may not be a reliable basis for decision- making. To use this combined CMF, the engineer must consider the decision that would be made if the true CMF value were as low as 0.39, and the decision that would be made if the true CMF value were as high as 0.83. If the decisions reached are not the same, then the combined CMF should not be used. The conclusion for the two fatal CMFs under consideration is that they are combined to pro- duce an overall average CMF of 0.57. This combined CMF will be used in the engineer’s analysis to predict the change in fatal crashes due to the traffic signal. The engineer now repeats Steps 6a through 6c for the two A-injury CMFs under consideration. Step 6a (for A-Injury CMFs)—Test for Homogeneity Step 6a.1 Compute Weight w CMF s i i i =    2 where wi = weight of CMF observation i CMFi = value of CMF observation i si = standard error of CMF observation i The weight calculated for each CMF observation is shown in Column 4 of Table A18. Step 6a.2 Compute Log. The second step is to compute the natural log of the CMF value. The natural log values of the CMFs are shown in Column 5 of Table A18. Obs. (i) Crash Modification Factor (CMF) Standard Error (s) Weight (w) Natural Log of CMF (L) Chi-Square Value ( 2 iχ ) 1 0.68 0.05 185.0 -0.386 3.077 2 0.80 0.03 711.1 -0.223 0.80 Total: 896.1 3.877 Table A18. Values for homogeneity test calculations for A-injury CMFs.

A-48 Guidelines for the Development and Application of Crash Modification Factors Step 6a.3 Compute Average Log. The third step is to compute the weighted average value of L. This average is computed using the following equation. L w L w i i i i i ∑ ∑ = where L– = weighted average value of L Using the values in columns 4 and 5 of Table A17, the weighted average value L– is computed as -0.257. Step 6a.4 Compute Chi-Square Statistic. The fourth step is to compute the chi-square value for each observation. This calculation is shown by the following equation. w L Li i i2 2( )χ = − where Xi2 = chi-square value of CMF observation i The chi-square values of the CMFs are shown in Column 6 of Table A18. Step 6a.5 Check Results. The last step is to add the chi-square values and compare this result with the chi-square distribution for n − 1 degrees of freedom, where n is the number of CMF observations. The total chi-square value is 3.877. Relating this value to the chi-square dis- tribution with for 1 degree of freedom (= 2 − 1) yields a probability of 0.049. There is a 4.9 per- cent chance that CMF value differences this large could exist and still be due only to random variation. This value of 0.049 is below the suggested threshold of 0.05, so it is concluded that the two CMF values cannot be combined to obtain an overall average CMF. Therefore, the engineer must follow step 6c. Step 6c (for A-Injury CMFs)—Select the One CMF That Is a Best Match to the Subject-Site Characteristics or Develop CMFunction Given that the A-injury CMFs from Study 1 (0.68) and Study 2 (0.80) could not be reliably combined into an overall average CMF, the engineer must select the one that is the best match to the subject site or develop a CMFunction. The engineer does not have sufficient staff expertise or resources to develop a CMFunction, so she investigates the details of each study to determine which one is a better match to the subject site. She has already matched both studies to her sub- ject site on the influential factors of traffic volume, number of legs, speed limit, and expected crashes, so she examines other details about the sites used in each study. She sees that Study 1 used intersections with a range of speed limits on the major road (35 to 55 mph). While this does include the condition of her subject site (45 mph), she sees that all the intersections in Study 2 had a speed limit of 45 mph on the major road. Additionally, the geo- graphic area of Study 2 is close to her jurisdiction, whereas Study 1 was conducted farther away, in an area that has different topography, weather, and driving environment. For both reasons, she decides that the results of Study 2 would be more appropriate for her subject site than Study 1. Thus, she selects the CMF of 0.80 to use for the A-injury severity.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-49   At the end of Step 6, she has identified or calculated the following disaggregate CMFs for each severity level. • Fatal 0.57 • A-injury 0.80 • B-injury 0.80 • C-injury 0.83 • PDO 1.15 The engineer can end the procedure here and apply these CMFs in the benefit-cost analysis. However, if she required a total (aggregate) CMF for the traffic signal installation, she would complete Step 7. Step 7 Develop Aggregate CMF Using Crash Distribution at Subject Site The disaggregate CMFs are aggregated together using the local crash distribution of the sub- ject site to arrive at the aggregate CMF. This step consists of two parts, determining the local crash distribution of the subject site (Step 7a) and calculating the aggregate CMF (Step 7b). Step 7a Determine the Local Crash Distribution The engineer reviews five years of crash data at the subject site to determine the crash distri- bution by severity. However, the crash history was not sufficient to compile a reliable sample (i.e., 100 crashes) on which to base the severity distribution. Thus, she reviews crash data from a group of similar stop-controlled intersections in the same area to achieve a total of 100 crashes. Based on this compiled crash data, she develops the severity distribution shown in Table A19. Step 7b Calculate the Aggregate CMF Based on the Local Crash Distribution The aggregate CMF is calculated based on the CMFs of the individual severity levels and the proportions of those severities in the crash distribution. The general equation is. CMF CMF pagg i ii n 1∑ ( )= ×= where CMFagg = aggregate CMF CMFi = disaggregate CMF for category i (i = 1, 2, 3, . . . , n) pi = proportion of crashes associated with category i (i = 1, 2, 3, . . . , n) The engineer calculates the aggregate CMF as follows: CMFagg = (0.57 ∗ 0.013) + (0.80 ∗ 0.017) + (0.80 ∗ 0.15) + (0.83 ∗ 0.318) + (1.15 ∗ 0.502) = 0.98 This aggregate CMF of 0.98 indicates that the overall number of crashes would be reduced by only 2 percent. Although the reductions in severe crashes would be substantial (CMFs of 0.57 and 0.80), the subsequent increase in property-damage-only crashes (CMF of 1.15) and the high proportion of property-damage-only crashes (0.502) cause the overall frequency of crashes at the intersection to remain fairly unchanged. The primary safety benefit of the traffic signal is seen in the reduction of the severity of crashes. Severity Proportion K 0.013 A 0.017 B 0.15 C 0.318 O 0.502 Table A19. Severity distribution at example intersections.

A-50 Guidelines for the Development and Application of Crash Modification Factors Result The engineer has identified a set of CMFs for the traffic signal that are specific to each level of severity. She can now use these CMFs to calculate the anticipated change in for each category of crash severity and use the resulting benefit in the benefit-cost analysis. Additional Example Applications of the Step 5 Disaggregation Process The example application above did not involve any calculations in Step 5, because the engi- neer in the example had identified CMFs that were already disaggregated (specific to each sever- ity level). If the engineer had only identified aggregate (total crash) CMFs, she would have had to follow Step 5 to calculate disaggregate CMFs. For the reader’s benefit, below are some example applications of the procedure in Step 5 for disaggregating aggregate CMFs. The procedure is demonstrated through the following four additional example applications. • Conversion from Signalized Intersection to Roundabout • Installation of Permanent Raised Pavement Markings • Use of Narrow Lanes and Shoulder to Add HOV Lane • Change in Median Width The results from each example application reflect the disaggregation of the CMF observations obtained from the referenced studies. Additional CMF observations from other studies could be pooled with the observations identified in this paper to produce a larger sample upon which to base the disaggregated CMFs. Conversion from Signalized Intersection to Roundabout This section describes the disaggregation of aggregate CMFs that quantify the safety effect of converting a signalized intersection to a roundabout. The aggregate CMF observations were obtained from reports by Russo et al. (2014) and Persaud et al. (2001). The CMF observations were reported at the total crash (i.e., all severities) level and at the fatal-and-severe-injury (i.e., KAB) level. CMFs for the KAB level and the possible-injury-and-property-damage-only (i.e., CO) level were estimated using the disaggregation procedure. Assemble CMFs The first step of the disaggregation procedure requires the assembly of CMFs from the lit- erature. Research by Gross et al. (2013) indicated that the effectiveness of conversion (from signal) to roundabout decreased as the total daily entering traffic volume increased. In fact, they found that the CMF for total crashes exceeded 1.0 for daily entering volumes greater than about 14,400 to 17,500 vehicles per day. For this reason, the literature search was limited to studies where the researchers reported the AADT (and CMF) for each conversion site. A few published reports were found that satisfied this criterion; however, only two studies were con- sistent in the severity categories for which the CMFs were derived. The reported CMFs for these two studies are listed in Table A20 and Table A21 for total crashes (all severities) and fatal-and- severe-injury crashes, respectively. The data in Table A20 describe the characteristics of the 26 intersections studied. The data col- lectively represent sites in three states (i.e., Michigan, Florida, and Colorado). The CMFs listed in column 7 range from 0.339 to 6.796. This large range suggests that the conversion effect on

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-51   safety can vary widely from site to site. This observation is supported by a homogeneity test that indicates that the null hypothesis (i.e., that all sites have the same CMF value) can be rejected at a significance level of 0.0001. The data in Table A21 describe the change in fatal-and-severe-injury (i.e., KAB) crash fre- quency at the same 26 intersections listed in Table A20. The CMFs range from 0.086 to 3.53. This range is large, like the finding for the total crash CMFs. The homogeneity test was not conducted for these data because the weight (= [CMF/standard error]2) for almost all observations is less than 4.0. An observation with a value smaller than about 4.0 is associated with a small number of crashes and may not yield reliable test results. These guidelines regarding the statistical weight of a CMF are discussed in Step 3 in the main body of this appendix. Define Regression Model The second step of the disaggregation procedure is to define the regression model. The regres- sion model chosen includes one disaggregate CMF for fatal-and-severe-injury crashes and one Site ID1 Number Legs Circulating Lanes AADT2, veh/day Crash Severity Distribution3 CMF Standard Error KAB CO 1 4 1 14,756 0.035 0.965 0.364 0.161 2 4 1 22,360 0.075 0.925 2.396 0.710 3 4 1 11,451 0.028 0.972 0.508 0.252 4 4 1 26,055 0.071 0.929 0.873 0.263 5 4 1 17,854 0.060 0.940 0.434 0.156 6 3 1 13,806 0.054 0.946 0.842 0.361 7 3 2 33,010 0.045 0.955 0.673 0.143 8 3 2 23,000 0.061 0.939 1.105 0.248 9 4 2 30,490 0.091 0.909 1.141 0.280 10 4 3 27,155 0.065 0.935 2.707 0.488 11 4 2 28,710 0.068 0.932 1.372 0.227 12 4 3 36,670 0.039 0.961 2.611 0.309 13 4 3 33,330 0.052 0.948 3.484 0.510 14 4 3 36,160 0.038 0.962 2.405 0.289 15 4 2 11,300 0.023 0.977 1.579 0.360 16 4 3 53,139 0.065 0.935 2.723 0.353 174 4 2 31,680 0.082 0.918 6.796 1.346 18 4 2 19,210 0.033 0.967 1.179 0.312 19 4 1 31,498 0.089 0.911 1.344 0.407 20 3 1 22,259 0.040 0.960 0.897 0.254 21 4 1 18,408 0.047 0.953 0.548 0.212 22 4 1 17,911 0.032 0.968 0.399 0.132 A 4 2 22,030 0.108 0.892 0.866 0.179 B 4 2 18,475 0.076 0.924 0.417 0.140 C 4 2 18,795 0.102 0.898 0.339 0.092 D 4 1 5,322 0.271 0.729 2.088 0.907 Notes: 1. Sites 1 to 22 reported by Russo et al. (2014). Sites A to D reported by Persaud et al. (2001). 2. AADT: annual average daily traffic volume entering the signalized intersection before conversion to a roundabout. 3. Crash severity distribution based on reported “expected crash frequency for the after period if conversion does not occur.” 4. Site 17 data is an outlier and was removed from analysis. Table A20. Reported safety effect of conversion from signalized intersection to roundabout: CMFs for total crashes.

A-52 Guidelines for the Development and Application of Crash Modification Factors disaggregate CMF for possible-injury-and-property-damage-only crashes (i.e., n = 2). Several different combinations of site-characteristic variables were explored in a preliminary regression analysis. The form of the model that included terms for “number of legs,” “number of circulat- ing lanes,” and “AADT per circulating lane” was found to provide the most reasonable fit to the observed CMF values (i.e., m = 3). The final model is described by the following equation. CMF CMF p CMF pagg KAB KAB CO CO Equation A36= × + × with CMF b b N b N b AADT N CMF b b N b N b AADT N KAB KAB legs legs lanes lanes AADT lanes CO CO legs legs lanes lanes AADT lanes exp ln 0.001 exp ln 0.001 [ ] [ ] ( ) ( ) = + × + × + × = + × + × + × Site ID1 Number Legs Circulating Lanes AADT2, veh/day Crash Severity Distribution3 CMF Standard Error KAB CO 1 4 1 14,756 1.00 0.00 0.712 1.031 2 4 1 22,360 1.00 0.00 1.314 1.379 3 4 1 11,451 1.00 0.00 2.970 3.114 44 4 1 26,055 1.00 0.00 3.530 1.792 5 4 1 17,854 1.00 0.00 0.232 0.336 6 3 1 13,806 1.00 0.00 0.702 1.015 7 3 2 33,010 1.00 0.00 1.607 0.829 8 3 2 23,000 1.00 0.00 1.162 0.600 9 4 2 30,490 1.00 0.00 0.178 0.256 10 4 3 27,155 1.00 0.00 1.207 0.565 11 4 2 28,710 1.00 0.00 0.673 0.385 12 4 3 36,670 1.00 0.00 1.276 0.638 13 4 3 33,330 1.00 0.00 1.155 0.597 14 4 3 36,160 1.00 0.00 0.937 0.531 15 4 2 11,300 1.00 0.00 1.699 1.295 16 4 3 53,139 1.00 0.00 0.683 0.509 17 4 2 31,680 1.00 0.00 0.679 0.703 18 4 2 19,210 1.00 0.00 0.679 0.984 19 4 1 31,498 1.00 0.00 1.662 1.287 20 3 1 22,259 1.00 0.00 1.010 1.054 21 4 1 18,408 1.00 0.00 1.117 1.161 22 4 1 17,911 1.00 0.00 2.384 1.569 A 4 2 22,030 1.00 0.00 0.168 0.177 B 4 2 18,475 1.00 0.00 0.183 0.271 C 4 2 18,795 1.00 0.00 0.086 0.124 D 4 1 5,322 1.00 0.00 2.010 1.395 Notes: 1. Sites 1 to 22 reported by Russo et al. (2014). Sites A to D reported by Persaud et al. (2001). 2. AADT: annual average daily traffic volume entering the signalized intersection before conversion to a roundabout. 3. Crash severity distribution based on reported “expected crash frequency for the after period if conversion does not occur.” 4. Site 4 data is an outlier because it has an exceptionally large CMF value and a large weight (=3.9), so it was removed from the analysis. Table A21. Reported safety effect of conversion from signalized intersection to roundabout: CMFs for fatal and severe injury crashes.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-53   where CMFagg = aggregate CMF CMFKAB = disaggregate CMF for fatal-and-severe-injury (i.e., KAB) crashes CMFCO = disaggregate CMF for possible-injury-and-property-damage-only (i.e., CO) crashes bKAB = regression coefficient for KAB crashes bCO = regression coefficient for CO crashes pKAB = proportion of KAB crashes pCO = proportion of CO crashes bk = regression coefficient for site characteristic k Nlegs = number of roundabout legs Nlanes = number of circulating lanes AADT = annual average daily traffic volume entering the signalized intersection before con- version to roundabout, veh/d Convert Crash Distribution and Other CMF Characteristics into Observations The third step of the disaggregation procedure is to determine the proportion of crashes for the KAB and CO severity levels associated with each CMF observation. For the CMF observa- tions associated with the KAB severity, the proportion pKAB is equal to 1.0 and the proportion pCO is equal to 0.0. For the CMF observations associated with total crashes (all severities), the proportion pKAB is computed using the reported expected crash frequencies (i.e., the EB-based expected crash fre- quencies reported by Russo et al., 2014; and by Persaud et al., 2001). Specifically, the proportion pKAB is computed using the “expected crash frequency for the after period if conversion does not occur.” The numerator of this ratio is the expected KAB crash frequency, and the denominator is the expected total crash frequency. The proportion pCO is computed as 1.0 – pKAB. The result- ing crash distribution proportions are listed in Columns 5 and 6 of Table A20 and Table A21. Estimate Model Coefficients The fourth step of the disaggregation procedure is to estimate the regression coefficients using Equation A36. The maximum likelihood criterion was used with a lognormal distribution for the residuals (more details of the analysis procedure are provided in A.2., Step 5, in the section titled Estimation Method). The SAS procedure NLMIXED was used to determine the coeffi- cients and their standard errors. The analysis can also be undertaken with Excel Solver to obtain the coefficient and parameter values (although Solver does not report the standard error of each value). Many of the KAB CMFs were associated with a weight w less than 3.0. These CMFs were removed from the data set to produce more reliable results. The best-fit form of the model is shown in the following equation. CMF CMF p CMF pagg KAB KAB CO CO Equation A37= × + × with CMF N N AADT N CMF N N AADT N KAB legs lanes lanes CO legs lanes lanes exp 2.326 0.195 0.594 0.244 ln 0.001 exp 2.235 0.195 0.594 0.244 ln 0.001 [ ] [ ] ( ) ( ) = − + × + × + × = − + × + × + × The coefficients and their standard errors (in parentheses) are: −2.326 (1.17), 0.195 (0.237), 0.594 (0.109), 0.244 (0.236), and −2.235 (1.16). All underlined values are associated with a sig- nificance level of 0.06 or less.

A-54 Guidelines for the Development and Application of Crash Modification Factors The chi-square treatment is found to have a chi-square value of 277 (6 degrees of freedom). It corresponds to a p-value of 0.0001. This value is smaller than 0.05, so the null hypothesis can be rejected, and we can be reasonably confident that the treatment influences safety at the sites studied. The chi-square homogeneity is found to have a chi-square value of 103 (26 degrees of freedom, = 32 – 5 – 1). It corresponds to a p-value of 0.0001. This value is smaller than 0.05, so the null hypothesis can be rejected, and we can be reasonably confident that there is some remaining systematic variation that is unexplained by the model. Additional site-characteristic variables would be needed to determine whether some additional variability can be explained by the model. Table A22 illustrates the magnitude of the disaggregated CMF values obtained from the cali- brated equation. The values listed apply to the number-of-legs, circulating lanes, and entering AADT inputs shown—other CMF values would be obtained for other inputs. In general, the three- leg roundabouts have CMF values that are smaller than those for four-leg roundabouts. Round- abouts with one circulating lane have CMF values that are smaller than those for roundabouts with two circulating lanes. The CMFs for KAB crashes are smaller than those for CO crashes. Figure A2 illustrates the change in CMF values associated with the change in lane volume. The trends in the figure apply to four-leg roundabouts. The CMFs for three-leg roundabouts are about 20 percent smaller. The trends in the figure suggest that the CMF value for PDO and C-injury crash frequency increases with an increase in traffic volume, which is consistent with the finding of Gross et al. (2013). Number Legs Circulating Lanes Entering AADT, veh/day CMFKAB CMFCO 3 1 10,000 0.558 0.611 2 20,000 1.010 1.106 4 1 10,000 0.678 0.743 2 20,000 1.228 1.345 Table A22. Predicted disaggregate CMF values for conversion from signalized intersection to roundabout. Figure A2. CMF for conversion of signalized intersection to roundabout based on traffic volume.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-55   Installation of Permanent Raised Pavement Markings This section describes the disaggregation of aggregate CMFs that quantify the safety effect of installing permanent raised pavement markings (PRPMs) on two-lane highways and four-lane freeways. The aggregate CMF observations were obtained from the report by Bahar et al. (2004). The CMF observations were reported for two time-of-day levels, and two severity levels. These combinations are identified in the following list. • All times of day, all severities, non-intersection • All times of day, fatal-and-injury, non-intersection • Daytime crashes, all severities, non-intersection • Daytime crashes, fatal-and-injury, non-intersection • Nighttime crashes, all severities, non-intersection • Nighttime crashes, fatal-and-injury, non-intersection CMFs for the following levels were estimated using the disaggregation procedure. • Daytime crashes, fatal-and-injury (FI), non-intersection • Daytime crashes, property-damage-only (PDO), non-intersection • Nighttime crashes, fatal-and-injury, non-intersection • Nighttime crashes, property-damage-only, non-intersection Assemble CMFs The first step of the disaggregation procedure requires the assembly of CMFs from the litera- ture. Research by Bahar et al. (2004) indicated that PRPMs installed on low-volume roadways (i.e., about 5,000 veh/day or less) can be associated with an increase in crash frequency. The only report found in the literature that reported the AADT of the roadways studied was that by Bahar et al. The AADTs and CMFs from their report are listed in Table A23. The data in Table A23 describe the characteristics of the two-lane highway segments and four- lane freeway segments studied. There are 42 CMF observations in the table, representing five states. Each group of three rows (for four-lane freeways) or four rows (for two-lane highways) corresponds to a specific combination of crash time-of-day and severity. Within each group, the CMF values somewhat widely; often having some values below 1.0 and other values above 1.0. A homogeneity test of state-to-state variability within each group of three or four rows supports the finding of wide CMF variability. The test results indicate that the null hypothesis (i.e., that all states have the same CMF value) can be rejected at a significance level of 0.028. An alternative interpretation of this finding is that the CMF values may vary with AADT, as noted by Bahar et al. (2004). Define Regression Model The second step of the disaggregation procedure is to define the regression model. The regres- sion model chosen includes one disaggregate CMF for each of the following levels: daytime FI crashes, daytime PDO crashes, nighttime FI crashes, and nighttime PDO crashes (i.e., n = 4). Several different combinations of site-characteristic variables were explored in a preliminary regression analysis. Characteristics considered included “state,” “number of through lanes,” and “percent-shoulder-rumble-strips.” The form of the model that included a separate AADT term for daytime and for nighttime periods was found to provide the most reasonable fit to the observed CMF values (i.e., m = 2). The final model is described by the following equation. CMF CMF p CMF p CMF p CMF p agg day fi day fi day pdo day pdo night fi night fi night pdo night pdo Equation A38 , , , , , , , , = × + × + × + ×

A-56 Guidelines for the Development and Application of Crash Modification Factors with CMF b b AADT CMF b b AADT CMF b b AADT CMF b b AADT day fi day fi AADT day day pdo day pdo AADT day night fi night fi AADT night night pdo night pdo AADT night exp ln exp ln exp ln exp ln , , , , , , , , , , , , ( ) ( ) ( ) ( ) [ ] [ ] [ ] [ ] = + × = + × = + × = + × where CMFagg = aggregate CMF CMFday,fi = disaggregate CMF for daytime FI crashes CMFday,pdo = disaggregate CMF for daytime PDO crashes CMFnight,fi = disaggregate CMF for nighttime FI crashes CMFnight,pdo = disaggregate CMF for nighttime PDO crashes Facility State AADT, veh/day Time of Day Severity CMF Standard Error 2-lane highway Illinois 2,850 All All 1.091 0.035 New Jersey 10,944 All All 1.032 0.027 New York 9,140 All All 0.905 0.034 Pennsylvania 5,486 All All 0.980 0.030 Illinois 2,850 All Fatal & Injury 1.071 0.065 New Jersey 10,944 All Fatal & Injury 0.955 0.038 New York 9,140 All Fatal & Injury 1.020 0.057 Pennsylvania 5,486 All Fatal & Injury 1.017 0.068 Illinois 2,850 Daytime All 1.179 0.051 New Jersey 10,944 Daytime All 1.047 0.034 New York 9,140 Daytime All 1.003 0.048 Pennsylvania 5,486 Daytime All 0.963 0.038 Illinois 2,850 Daytime Fatal & Injury 1.080 0.086 New Jersey 10,944 Daytime Fatal & Injury 0.976 0.044 New York 9,140 Daytime Fatal & Injury 1.074 0.072 Pennsylvania 5,486 Daytime Fatal & Injury 0.978 0.086 Illinois 2,850 Nighttime All 1.001 0.045 New Jersey 10,944 Nighttime All 0.991 0.040 New York 9,140 Nighttime All 0.873 0.052 Pennsylvania 5,486 Nighttime All 1.039 0.048 Illinois 2,850 Nighttime Fatal & Injury 1.106 0.091 New Jersey 10,944 Nighttime Fatal & Injury 0.899 0.058 New York 9,140 Nighttime Fatal & Injury 1.000 0.097 Pennsylvania 5,486 Nighttime Fatal & Injury 1.074 0.110 4-lane freeway Missouri 14,007 All All 0.979 0.012 New York 15,390 All All 1.031 0.074 Penn. 24,995 All All 0.943 0.019 Missouri 14,007 All Fatal & Injury 0.946 0.021 New York 15,390 All Fatal & Injury 1.179 0.141 Pennsylvania 24,995 All Fatal & Injury 1.000 0.047 Missouri 14,007 Daytime All 0.979 0.015 New York 15,390 Daytime All 1.046 0.100 Pennsylvania 24,995 Daytime All 0.935 0.024 Missouri 14,007 Daytime Fatal & Injury 0.938 0.026 New York 15,390 Daytime Fatal & Injury 1.195 0.183 Pennsylvania 24,995 Daytime Fatal & Injury 1.023 0.062 Missouri 14,007 Nighttime All 0.991 0.020 New York 15,390 Nighttime All 0.900 0.090 Pennsylvania 24,995 Nighttime All 0.960 0.028 Missouri 14,007 Nighttime Fatal & Injury 0.975 0.035 New York 15,390 Nighttime Fatal & Injury 0.951 0.171 Pennsylvania 24,995 Nighttime Fatal & Injury 0.988 0.070 Table A23. Reported safety effect of installing permanent raised pavement markings.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-57   bday,fi = regression coefficient for daytime FI crashes bday,pdo = regression coefficient for daytime PDO crashes bnight,fi = regression coefficient for nighttime FI crashes bnight,pdo = regression coefficient for nighttime PDO crashes bAADT,day = regression coefficient for AADT adjustment to daytime crashes bAADT,night = regression coefficient for AADT adjustment to nighttime crashes pday,fi = proportion of daytime FI crashes pday,pdo = proportion of daytime PDO crashes pnight,fi = proportion of nighttime FI crashes pnight,pdo = proportion of nighttime PDO crashes AADT = annual average daily traffic volume (two-way total), veh/d Convert Crash Distribution and Other CMF Characteristics into Observations The third step of the disaggregation procedure is to determine the proportion of crashes for the time-of-day and severity categories associated with each CMF observation. The “expected crash frequency for the after period if installation does not occur” was used to compute the desired proportions. This expected value was reported by Bahar et al. (2004) for the categories identified in the following list. • All times of day, all severities, non-intersection • All times of day, fatal-and-injury, non-intersection • Daytime crashes, all severities, non-intersection • Daytime crashes, fatal-and-injury, non-intersection • Nighttime crashes, all severities, non-intersection • Nighttime crashes, fatal-and-injury, non-intersection Estimates of the expected PDO crash frequency were computed from the reported categories. The corresponding crash distribution proportions are listed in Table A24. Estimate Model Coefficients The fourth step of the disaggregation procedure is to estimate the regression coefficients using Equation A38. The SAS procedure NLMIXED was used to determine the coefficients and their standard errors. The analysis can also be undertaken with Excel Solver to obtain the coefficient and parameter values (although Solver does not report the standard error of each value). The best-fit form of the model is shown in the following equation. CMF CMF p CMF p CMF p CMF p agg day fi day fi day pdo day pdo night fi night fi night pdo night pdo Equation A39 , , , , , , , , = × + × + × + × with CMF AADT CMF AADT CMF AADT CMF AADT day fi day pdo night fi night pdo ( ) ( ) ( ) ( ) [ ] [ ] [ ] [ ] = − × = − × = − × = − × exp 0.637 0.0699 ln exp 0.650 0.0699 ln exp 0.261 0.0293 ln exp 0.253 0.0293 ln , , , , The coefficients and their standard errors (in parentheses) are 0.637 (0.156), 0.650 (0.159), 0.261 (0.178), 0.253 (0.180), −0.0699 (0.0166), and −0.0293 (0.0190). All coefficient values are associated with a significance level of 0.17 or less.

A-58 Guidelines for the Development and Application of Crash Modification Factors The chi-square treatment is found to have a chi-square value of 40.6 (6 degrees of freedom). It corresponds to a p-value of 0.0001. This value is smaller than 0.05, so the null hypothesis can be rejected, and we can be reasonably confident that the treatment influences safety at the sites studied. The chi-square homogeneity is found to have a chi-square value of 48.3 (35 degrees of freedom, = 42 – 6 – 1). It corresponds to a p-value of 0.067. This value is larger than 0.05, so the null hypothesis cannot be rejected, and we cannot be confident that there is some remaining sys- tematic variation that might be explained (i.e., the model may be explaining all the systematic variability in the data). The data used for this analysis represent the segment data aggregated for each state and facility type combination. An analysis of the individual CMFs for each roadway segment along with information about the segment’s characteristics (e.g., AADT, horizontal curvature) would be needed to determine whether some additional factors influence CMF value at the segment level. Facility State Time of Day Severity Crash Distribution pday,fi pday,pdo pnight,fi pnight,pdo 2-lane highway Illinois All All 0.149 0.333 0.135 0.383 New Jersey All All 0.260 0.398 0.115 0.227 New York All All 0.230 0.336 0.111 0.324 Pennsylvania All All 0.109 0.504 0.073 0.315 Illinois All Fatal & Injury 0.524 0.0 0.476 0.0 New Jersey All Fatal & Injury 0.694 0.0 0.306 0.0 New York All Fatal & Injury 0.675 0.0 0.325 0.0 Pennsylvania All Fatal & Injury 0.599 0.0 0.401 0.0 Illinois Daytime All 0.309 0.691 0.0 0.0 New Jersey Daytime All 0.395 0.605 0.0 0.0 New York Daytime All 0.407 0.593 0.0 0.0 Pennsylvania Daytime All 0.177 0.823 0.0 0.0 Illinois Daytime Fatal & Injury 1.0 0.0 0.0 0.0 New Jersey Daytime Fatal & Injury 1.0 0.0 0.0 0.0 New York Daytime Fatal & Injury 1.0 0.0 0.0 0.0 Pennsylvania Daytime Fatal & Injury 1.0 0.0 0.0 0.0 Illinois Nighttime All 0.0 0.0 0.261 0.739 New Jersey Nighttime All 0.0 0.0 0.336 0.664 New York Nighttime All 0.0 0.0 0.255 0.745 Pennsylvania Nighttime All 0.0 0.0 0.187 0.813 Illinois Nighttime Fatal & Injury 0.0 0.0 1.0 0.0 New Jersey Nighttime Fatal & Injury 0.0 0.0 1.0 0.0 New York Nighttime Fatal & Injury 0.0 0.0 1.0 0.0 Pennsylvania Nighttime Fatal & Injury 0.0 0.0 1.0 0.0 4-lane freeway Missouri All All 0.205 0.445 0.101 0.249 New York All All 0.134 0.358 0.110 0.398 Pennsylvania All All 0.074 0.524 0.055 0.347 Missouri All Fatal & Injury 0.671 0.0 0.329 0.0 New York All Fatal & Injury 0.548 0.0 0.452 0.0 Pennsylvania All Fatal & Injury 0.575 0.0 0.425 0.0 Missouri Daytime All 0.316 0.684 0.0 0.0 New York Daytime All 0.272 0.728 0.0 0.0 Pennsylvania Daytime All 0.124 0.876 0.0 0.0 Missouri Daytime Fatal & Injury 1.0 0.0 0.0 0.0 New York Daytime Fatal & Injury 1.0 0.0 0.0 0.0 Pennsylvania Daytime Fatal & Injury 1.0 0.0 0.0 0.0 Missouri Nighttime All 0.0 0.0 0.288 0.712 New York Nighttime All 0.0 0.0 0.217 0.783 Pennsylvania Nighttime All 0.0 0.0 0.136 0.864 Missouri Nighttime Fatal & Injury 0.0 0.0 1.0 0.0 New York Nighttime Fatal & Injury 0.0 0.0 1.0 0.0 Pennsylvania Nighttime Fatal & Injury 0.0 0.0 1.0 0.0 Note: Crash distribution based on reported “expected crash frequency for the after period if installation does not occur.” Table A24. Reported crash distribution on two-lane highways and four-lane freeways.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-59   Table A25 illustrates the magnitude of CMF values obtained from the calibrated equation. The values listed apply to two-lane highways and four-lane freeways—other CMF values would be obtained for other facility types. In general, the PRPMs tend to increase crash frequency for AADTs of 5,000 veh/day or less. This finding is consistent with that of Bahar et al. (2004) in their examination of the data. CMF values decrease with increasing AADT. It is also noted that there is a larger variation in the daytime CMF values over the range of AADTs listed (relative to the nighttime CMF values). Use of Narrow Lanes and Shoulder to Add HOV Lane This section describes the disaggregation of aggregate CMFs that quantify the safety effect of narrowing lanes and shoulders to add a high-occupancy vehicle (HOV) lane on a section of urban freeway. For each direction of travel, the number of general-purpose lanes is unchanged following the addition of the HOV lane (i.e., one lane is added, and it is operated as an HOV lane). The aggregate CMF observations were obtained from a journal article by Bauer et al. (2004). The CMF observations were reported for the following three severity levels: fatal-and-injury, fatal-and-injury- plus property-damage-only tow-away, and total crashes (all severities). CMFs for the following levels were estimated using the disaggregation procedure: fatal-and-injury and property-damage-only. Assemble CMFs The first step of the disaggregation procedure requires the assembly of CMFs from the lit- erature. The CMFs reported by Bauer et al. (2004) were the only CMFs found that included information about the crash distribution and the changes in lane and shoulder width that were implemented to accommodate the installation of an HOV lane. The reported CMFs are listed in Table A26. The data in Table A26 describe the characteristics of the freeway segments studied. All seg- ments are in southern California. There are 21 CMF observations in the table. Each direction of travel was separately evaluated on a given segment. The data in Table A26 describe the char- acteristics for the subject direction of travel as well as the CMF corresponding to the change in crash history for the subject direction. The CMF values in Table A26 are greater than 1.0 in most instances. This finding indicates that crash frequency increased after the conversion. A homogeneity test of the seven CMFs associated with total crashes (all severities) indicates that the null hypothesis (i.e., all “lane” and “bundle” combinations have the same CMF value) can be rejected at a significance level of 0.01. This finding implies that the safety effect varies among bundles due to systematic differences in the bundles. AADT, veh/day CMFday,fi CMFday,pdo CMFnight,fi CMFnight,pdo CMFagg 1,000 1.167 1.182 1.060 1.052 1.113 2,500 1.094 1.108 1.032 1.024 1.063 5,000 1.042 1.056 1.011 1.003 1.028 15,000 0.965 0.978 0.979 0.972 0.974 25,000 0.931 0.943 0.964 0.957 0.950 50,000 0.887 0.899 0.945 0.938 0.918 Note: Aggregated CMF (CMFagg) values are based on following proportions: pday,fi = 0.15; pday,pdo = 0.33; pnight,fi = 0.14; pnight,pdo = 0.38. Table A25. Predicted disaggregate CMF values for installation of PRPMs.

A-60 Guidelines for the Development and Application of Crash Modification Factors Bauer et al. speculated that the addition of an HOV lane (as opposed to a general-purpose lane) may contribute to the observed increase in crash frequency because of the large speed differential typically created between the HOV lane and the general-purpose lanes. They also speculated that the added lane may relocate the traffic bottleneck to a downstream unconverted segment such that bottleneck queues (and related crashes) may extend backward into the con- verted segments. Bauer et al. noted that Bundle H was associated with an 11.6 percent decrease in total crashes (all severities) (CMF = 0.884). For this bundle, only the inside shoulder width was reduced (it was reduced by 6.0 ft). Relative to the other bundles, this bundle underwent the smallest reduction in lane and shoulder width, and the cross section was widened the most (i.e., by 6.0 ft) to accom- modate the new HOV lane. The inside shoulder width CMF in Chapter 18 of the HSM indicates that a 6.0 ft decrease in inside shoulder width corresponds to an average 11 percent increase in fatal-and-injury crash frequency (CMF = 1.11). Combining this value with the reported CMF of 0.884, suggests that the addition of an HOV lane corresponds to a CMF value of about 0.80 (= 0.884/1.11). In contrast, the reference-site SPFs documented in Table 4 of the report by Bauer et al. suggest that a ten-lane freeway has 19 percent fewer crashes than a four-lane freeway. In other words, the addition of one general-purpose lane in each direction corresponds to an equivalent CMF of 0.81. In total, these findings suggest the following CMF values (when there is no change in shoulder width). • Add lane CMF = 0.81 • Operate new lane as HOV CMF = 0.99 • Add new lane and operate as HOV CMF = 0.80 (= 0.81 × 0.99) Severity Before Characteristics After Characteristics CMF Std. Error Lanes1,2 (bundle) Lane Width, ft Inside Shldr. Width, ft Outside Shldr. Width, ft Inside Barrier Offset, ft Lane Width, ft Inside Shldr. Width, ft Outside Shldr. Width, ft Inside Barrier Offset, ft Fatal-and- Injury 4 (B) 12 14 9 14 12 2 9 2 1.199 0.136 4 (C) 12 9 10 14 12 2 9 2 1.096 0.073 4 (D) 12 12 10 12 11.3 2 10 2 1.184 0.078 4 (G) 12 10 10 30 11.4 2 8.5 2 0.949 0.194 5 (A) 12 14 10 14 12 2 10 2 1.151 0.169 5 (F) 12 12 10 123 11.6 2 10 2 1.359 0.162 5 (H) 12 8 10 13.5 12 2 10 2 0.912 0.104 Fatal-and- Injury Plus PDO- Tow- Away 4 (B) 12 14 9 14 12 2 9 2 1.246 0.117 4 (C) 12 9 10 14 12 2 9 2 1.038 0.057 4 (D) 12 12 10 12 11.3 2 10 2 1.220 0.070 4 (G) 12 10 10 30 11.4 2 8.5 2 1.110 0.196 5 (A) 12 14 10 14 12 2 10 2 1.293 0.161 5 (F) 12 12 10 123 11.6 2 10 2 1.298 0.133 5 (H) 12 8 10 13.5 12 2 10 2 0.790 0.078 All Severities 4 (B) 12 14 9 14 12 2 9 2 1.255 0.088 4 (C) 12 9 10 14 12 2 9 2 1.081 0.046 4 (D) 12 12 10 12 11.3 2 10 2 1.174 0.047 4 (G) 12 10 10 30 11.4 2 8.5 2 1.104 0.142 5 (A) 12 14 10 14 12 2 10 2 1.119 0.103 5 (F) 12 12 10 123 11.6 2 10 2 1.293 0.101 5 (H) 12 8 10 13.5 12 2 10 2 0.885 0.069 Bundle letter in parentheses corresponds to bundle type in Table 2 of the article by Bauer et al. (2004). 1 Bundle letters correspond to descriptions provided in Table 2 of the article by Bauer et al. (2004). 2 Number of through lanes in the subject direction of travel. Lane count in the “after” period is increased by one HOV lane. 3 Unspecified in Table 2 by Bauer et al. (2004). Estimated to equal the inside shoulder width. Table A26. Reported safety effect of lane and shoulder reductions combined with HOV installation.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-61   The preceding analysis indicates that operating a lane as an HOV lane reduced crashes 1.0 percent (CMF = 0.99) on the segments in Bundle H. In contrast, the computed CMFs for “operating a lane as an HOV lane” for the other bundles indicate a CMF range of 1.13 to 1.35. In summary, after accounting for the effect of the change in shoulder width, the CMFs in Table A26 for Bundle H imply that operating a lane as an HOV lane reduces crash frequency, while all the other bundles imply that HOV operation increases crash frequency. A review of the literature indicates that there is little information on the safety effect of converting a general-purpose lane to an HOV lane. However, the information that is available suggests that (a) HOV lanes are not installed to provide safety benefits and (b) some HOV design and operation features can increase crash frequency, but they rarely reduce crash frequency (relative to a segment comprised only of general-purpose lanes). Define Regression Model The second step of the disaggregation procedure is to define the regression model. The regression model chosen includes one disaggregate CMF for each of the following levels: fatal- and-injury, property-damage-only crashes (i.e., n = 2). Several alternative model forms were explored in a preliminary regression analysis. Characteristics considered included “change in lane width”, “change in inside shoulder width”, “change in outside shoulder width”, and “change in inside barrier offset”. The form of the model that included terms for “change in lane width” and “change in inside shoulder width” was found to provide the most reasonable fit to the observed CMF values (i.e., m = 2). The model form found to provide the best fit to the observed CMF values is described by the following equation. CMF CMF p CMF p pagg fi fi pdo pdo tow pdo nontow( )= × + × + Equation A40, , with CMF b b w w b w w CMF b b w w b w w fi HOV fi lw l a l b is is a is b pdo HOV pdo lw l a l b is is a is b exp exp , , , , , , , , , , { } { }( ) { } { }( ) = + × − + × − = + × − + × − where CMFagg = aggregate CMF CMFHOV,fi = disaggregate CMF for addition of HOV lane, FI crashes CMFHOV,pdo = disaggregate CMF for addition of HOV lane, PDO crashes bHOV,fi = regression coefficient for addition of HOV lane, FI crashes bHOV,pdo = regression coefficient for addition of HOV lane, PDO crashes blw = regression coefficient for lane width bis = regression coefficient for inside shoulder width pfi = proportion of FI crashes ppdo,tow = proportion of PDO-tow-away crashes ppdo,nontow = proportion of PDO-nontow-away crashes; wl,a = average lane width after addition of HOV lane, ft wl,b = average lane width before addition of HOV lane, ft wis,a = inside shoulder width after addition of HOV lane, ft wis,b = inside shoulder width before addition of HOV lane, ft An analysis of the inside shoulder width and outside shoulder width data indicated that they were highly correlated. As a result, only one variable was included in the model. The “change in inside shoulder width” variable was selected for inclusion because it provided the best fit to the data.

A-62 Guidelines for the Development and Application of Crash Modification Factors Based on the findings regarding Bundle H (discussed in the previous section) and a preliminary examination of the data, it was concluded that the CMFs for Bundle H are likely to be influenced by other, unknown factors. This possibility was acknowledged by Bauer et al. in the discussion of findings for Bundle H. As a result, the data for Bundle H were removed from the database. Convert Crash Distribution and Other CMF Characteristics into Observations The third step of the disaggregation procedure is to determine the proportion of crashes for the severity categories associated with each CMF observation. The observed crash frequency for the two-year period prior to HOV lane installation was used to compute the desired propor- tions. These crash frequencies were reported by Bauer et al. (2004) for the following categories: fatal, injury, property-damage-only, tow-away, and property-damage-only, no tow-away crashes. Ideally, the crash distribution for each of the seven unique combinations of “lanes” and “bundle” would be used for this analysis. However, this level of disaggregation in the observed crashes was not available in the article by Bauer et al. Instead, they reported the observed crashes only for the two “lanes” categories. The computed crash distribution proportions are listed in Table A27. Estimate Model Coefficients The fourth step of the disaggregation procedure is to estimate the regression coefficients using Equation A40. The SAS procedure NLMIXED was used to determine the coefficients and their standard errors. The analysis can also be undertaken with Excel Solver to obtain the coefficient and parameter values (although Solver does not report the standard error of each value). The best-fit form of the model is shown in the following equation. CMF CMF p CMF p pagg fi fi pdo pdo tow pdo nontow( )= × + × + Equation A41, , with CMF w w w w CMF w w w w fi l a l b is a is b pdo l a l b is a is b exp 0.120 0.0402 0.0283 exp 0.136 0.0402 0.0283 , , , , , , , , { } { }( ) { } { }( ) = − − × − − × − = − − × − − × − The coefficients and their standard errors (in parentheses) are: −0.120 (0.056), −0.136 (0.059), −0.0402 (0.033), and −0.0283 (0.0059). The p-values for the coefficients are 0.047, 0.035, 0.24, and 0.0002, respectively. The p-value for the “change in lane width” coefficient is relatively large (i.e., 0.24) and suggests a high degree of uncertainty is attached to the coefficient value. How- ever, the lane width variable is retained in the model because the coefficient value is consistent in magnitude and sign with the coefficient in the lane width CMF in Chapter 18 of the HSM (i.e., −0.0376). Severity Lanes1 Crash Distribution2 pfi ppdo,tow ppdo,nontow Fatal-and-Injury 4 1.0 0.0 0.0 5 1.0 0.0 0.0 Fatal and Injury Plus PDO-Tow-Away 4 0.760 0.240 0.0 5 0.775 0.225 0.0 All Severities 4 0.357 0.113 0.531 5 0.373 0.108 0.518 1 Number of through lanes in the subject direction of travel before the HOV lane was installed. 2 Based on observed crashes in two-year period before the HOV lane was installed (Table 3, Bauer et al. 2004). Table A27. Reported crash distribution on urban freeways.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-63   Given the positive correlation between the changes in inside shoulder width and outside shoulder width in the data, it is rationalized that variables wis,a and wis,b in Equation A41 could be replaced with ws,a and ws,b and that these new variables could be used to describe the total change in both the inside and outside shoulder widths. The chi-square treatment is found to have a chi-square value of 85.8 (5 degrees of freedom). It corresponds to a p-value of 0.0001. This value is smaller than 0.05, so the null hypothesis can be rejected, and we can be reasonably confident that the treatment influences safety at the sites studied. The chi-square homogeneity is found to have a chi-square value of 6.6 (13 degrees of freedom, = 18 – 4 – 1). It corresponds to a p-value of 0.92. This value is larger than 0.05, so the null hypoth- esis cannot be rejected, and we cannot be confident that there is some remaining systematic variation that might be explained (i.e., the model may be explaining all the systematic variability in the data). Table A28 illustrates the magnitude of the disaggregated CMF values obtained from the calibrated equation. The values listed apply to the change in lane and shoulder width values shown—other CMF values would be obtained for other inputs. In general, the addition of a lane and its operation as an HOV lane can result in a CMF value less than 1.0 if the changes to lane and shoulder width are small. However, if the changes to lane or shoulder width are large, then the CMF value can be larger than 1.0. It is likely that the safety benefit is derived from the addition of the traffic lane. Whether the operation of this lane as an HOV lane has an influence on safety cannot be determined from this database (because “add lane” and “operate as an HOV lane” are not independent in the data). Change in Median Width This section describes the disaggregation of aggregate CMFs that quantify the safety effect of changing median width on a highway. The aggregate CMF observations were obtained from the report by Harkey et al. (2008). The CMF observations are based on data for a mixture of urban and rural divided highway segments, some of which had full access control and others that had partial or no access control. All segments had “traversable” medians, which were defined to be medians having no continuous median barrier. The median width was defined to be the distance from edge-of-travel-way to edge-of-traveled-way for the two roadbeds (i.e., the median width includes the width of the inside shoulders). The CMF observations were reported for cross median crashes and for total crashes (all types). CMFs were estimated for cross median crashes and non-cross median crashes using the disaggregation procedure. Assemble CMFs The first step of the disaggregation procedure requires the assembly of CMFs from the literature. The CMFs that were found were based on functions derived from multiple-variable Change in Average Lane Width, ft Change in Shoulder Width, ft CMFFI CMFPDO CMFagg 0.0 0.0 0.887 0.873 0.878 -4.0 0.993 0.978 0.983 -8.0 1.112 1.095 1.101 -0.5 0.0 0.905 0.891 0.896 -4.0 1.013 0.997 1.003 -8.0 1.134 1.117 1.123 Note: Aggregate CMF (CMFagg) values are based on following proportions: pfi = 0.36; ppdo = 0.64 Table A28. Predicted disaggregate CMF values for the addition of an HOV lane to a freeway segment.

A-64 Guidelines for the Development and Application of Crash Modification Factors regression models. The functions reported by Harkey et al. (2008) were complete in terms of the crash distribution characteristics needed. Separate functions were developed by Harkey et al. for the following characteristics. • Total crashes (all types), cross median crashes • Urban, rural • Full access control, no or partial access control • Four lanes, five or more lanes (only for urban with full access control) Each function predicts expected crash frequency and was developed as a multiple-variable regression model. A total of ten functions were developed. Each function has the following equation formulation. N b b AADT b AADT b L b w b w b I b I b I p o lAADT AADT L mw m is is ds jct jct no con no con Equation A42 exp ln 10000 10000 ln 55 [ ( ) ( )= + × + × + × + × + × + × + × + ×> − − where Np = predicted crash frequency, crashes/yr bo = regression coefficient for intercept blAADT = regression coefficient for natural log of AADT bAADT = regression coefficient for AADT bL = regression coefficient for segment length bmw = regression coefficient for median width bis = regression coefficient for inside shoulder width bds = regression coefficient for design speed bjct = regression coefficient for segments near a ramp or intersection bno-con = regression coefficient for segments with no access control AADT = annual average daily traffic volume (two-way total), veh/d L = segment length, miles wm = median width, ft wis = inside shoulder width, ft I>55 = indicator variable for design speed (= 1.0 if design speed is 60 mi/h or more, 0.0 otherwise) Ijct = indicator variable for influence of ramp or intersection (= 1.0 if segment is within 0.30 miles of a ramp or 250 ft of an intersection, 0.0 otherwise) Ino-con = indicator variable for no access control (= 1.0 if there is no access control, 0.0 otherwise) The regression coefficients for each function are listed in Table A29. The coefficients are based on data for divided highways in California. The database includes ten years of data for the years 1993 to 2002. The database includes segments for 1,993 miles of roadway, of which about 33 per- cent were urban. The CMF for median width is computed using Equation A42 in the following equation. CMF N f w x b b N f w x b b b a i p a i m i mw i p b i m i mw i Equation A43 , all other , all other , all other , all other : , , , ,a,i , , ,b,i ( ) ( )= = = where CMFb:a i = value of CMF observation i when variable of interest changes from b to a (i = 1 to M2) f(x,b) = function of independent variables x and regression coefficients b)

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-65   wm,a,i = median width for “after” condition of observation i, ft wm,b,i = median width for “before” condition of observation i, ft x–j = independent variable j having a value representative of the site of interest bj = regression coefficient associated with variable xj To compute the CMF observations, the median width variable for the “before” and the “after” conditions is varied over the range of 10 to 90 ft, reflecting the range in the calibration database assembled by Harkey et al. (2008). An interval of 20 ft was established for this range resulting in the following median widths being considered: 10, 30, 50, 70, and 90 ft (i.e., M = 5). CMFs were then computed for all M2 combinations of median width change (e.g., before median width = 10, after median width = 10; before median width = 10, after median width = 30; etc.) for each of the ten prediction models described in Table A29. This process resulted in 250 CMF observations (= 52 × 10) in the database. The formulation of Equation A42 is unique such that when used in Equation A43, some can- cellation occurs and the CMF reduces to CMFb:a,i = exp[bmw (wm,a,i – wm,b,i)]. Either form of the CMF equation (i.e., Equation A43 or the reduced equation) can be used to produce the same CMF estimates. More generally, the use of Equation A43 makes the approach sufficiently general that it can be used with any crash frequency prediction model formulation. The procedure described in Equation A5 to Equation A7 was used to compute the standard error of each CMF observation. Define Regression Model The second step of the disaggregation procedure is to define the regression model. The regression model chosen includes one disaggregate CMF for cross median crashes and one disaggregate CMF for non-cross median crashes (i.e., n = 2). Several different combinations of site-characteristic variables were explored in a preliminary regression analysis. The form of the model that included terms for “area type” and “lanes” was found to provide the most reasonable fit to the observed CMF values. The influence of “area type” was notably different for the cross Area Type Access Lanes Crash Type1 (count) Regression Coefficients2 b0 blAADT bAADT bL bmw (σmw) bis bds bjct bno-con Rural Full 4 Total (33009) 1.139 0.836 0.107 0.952 -0.00357 (0.00040) -0.0380 0.0 -0.405 0.0 Cross (1961) -0.684 0.691 0.0 0.971 -0.01537 (0.0014) -0.0390 0.0 -0.129 0.0 Urban Full 4 Total (35690) 1.482 0.779 0.119 0.929 -0.00547 (0.00062) -0.0271 0.0 -0.713 0.0 Cross (1554) -0.885 0.783 0.0 0.937 -0.0112 (0.0022) -0.0157 0.0 -0.580 0.0 Rural No or partial 4 Total (13255) 2.049 0.615 0.144 0.712 -0.00461 (0.00080) -0.0780 0.0 -1.084 -0.132 Cross (1593) 1.564 0.780 0.0 0.825 -0.01695 (0.00200) -0.134 0.0 -1.676 -0.166 Urban No or partial 4 Total (28185) 1.515 0.987 -0.0751 0.556 -0.00533 (0.00090) 0.0395 0.118 -1.433 -0.0519 Cross (3438) - 0.0494 1.619 -0.444 0.461 -0.0134 (0.00205) 0.0248 0.0 -1.873 0.136 Urban Full 5 or more Total (43385) 1.988 0.945 0.0142 0.829 -0.00744 (0.00063) -0.0747 0.0 -0.647 0.0 Cross (1507) -0.133 1.428 -0.127 1.094 -0.01151 (0.00234) -0.0931 0.0 -0.749 0.0 1 Count, the count of crashes in the database used to calibrate the regression model. 2 σmw, standard error of the regression coefficient for median width. Table A29. Reported regression coefficients for highway safety prediction models.

A-66 Guidelines for the Development and Application of Crash Modification Factors median vs. the non-cross median crashes so “area type” was represented using two regression coefficients (i.e., m = 3). The final model is described by the following equation. CMF CMF p CMF pagg cr cr ncr ncr Equation A44= × + × with CMF b b I b I w w CMF b b I b I w w cr cr area cr urban lanes m a m b ncr ncr area ncr urban lanes m a m b exp exp 4 , , 4 , , [ ] [ ] ( ) ( ) { } { } = + × + × × − = + × + × × − − > − > where CMFagg = aggregate CMF CMFcr = disaggregate CMF for cross median crashes CMFncr = disaggregate CMF for non-cross median crashes bcr = regression coefficient for cross median crashes bncr = regression coefficient for non-cross median crashes barea-cr = regression coefficient for area type and cross median crashes barea-ncr = regression coefficient for area type and non-cross median crashes blanes = regression coefficient for number of through lanes pcr = proportion of cross median crashes pncr = proportion of non-cross median crashes wm,a = median width after change in width, ft wm,b = median width before change in width, ft; Iurban = indicator variable for area type (= 1.0 if urban, 0.0 otherwise) Ilanes = indicator variable for number of through lanes (= 1.0 if 5 or more, 0.0 otherwise) Convert Crash Distribution and Other CMF Characteristics into Observations The third step of the disaggregation procedure is to determine the proportion of crashes for the crash type categories associated with each CMF observation. The number of crashes of each crash type in the calibration database assembled by Harkey et al. (2008) was used for this purpose. They reported the number of cross median crashes and the number of total crashes (all types). The number of non-cross median crashes was computed by subtraction of cross median crashes from total crashes. The corresponding crash distribution proportions are listed in Table A30. Estimate Model Coefficients The fourth step of the disaggregation procedure is to estimate the regression coefficients using Equation A44. The SAS procedure NLMIXED was used to determine the coefficients and their Area Type Access Lanes Crash Type Crash Distribution pcross pnon-cross Rural Full 4 Total 0.0594 0.9406 Cross 1.000 0.000 Urban Full 4 Total 0.0435 0.9565 Cross 1.000 0.000 Rural No or partial 4 Total 0.1202 0.8798 Cross 1.000 0.000 Urban No or partial 4 Total 0.1220 0.8780 Cross 1.000 0.000 Urban Full 5 or more Total 0.0347 0.9653 Cross 1.000 0.000 Notes: Crash type and distribution based on reported crashes in the calibration database assembled by Harkey et al. (2008). Table A30. Reported crash distribution on divided highways.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-67   standard errors. The analysis can also be undertaken with Excel Solver to obtain the coefficient and parameter values (although Solver does not report the standard error of each value). The best-fit form of the model is shown in the following equation. CMF CMF p CMF pagg cr cr ncr ncr Equation A45= × + × with CMF I I w w CMF I I w w cr urban m a m b ncr urban m a m b exp 0.01580 0.00431 0.00200 exp 0.00282 0.00226 0.00200 4 , , 4 , , [ ] [ ] ( ) ( ) { } { } = − + × − × × − = − − × − × × − > > The coefficients and their standard errors (in parentheses) are: −0.0158 (0.00014), −0.00282 (0.00005), 0.00431 (0.0002), −0.00226 (0.00008), and −0.00200 (0.0009). All values have a sig- nificance level of 0.05 or less. The chi-square treatment is found to have a chi-square value of 466 (6 degrees of freedom). It corresponds to a p-value of 0.0001. This value is smaller than 0.05, so the null hypothesis can be rejected, and we can be reasonably confident that the treatment influences safety at the sites studied. The chi-square homogeneity is found to have a chi-square value of 2.4 (244 degrees of freedom, = 250 – 5 – 1). It corresponds to a p-value of 0.9999. This value is larger than 0.05, so the null hypothesis cannot be rejected, and we cannot be confident that there is some remaining sys- tematic variation that might be explained (i.e., the model may be explaining all the systematic variability in the data). Table A31 illustrates the magnitude of the disaggregated CMF values obtained from the cali- brated equation. The values listed apply to the changes in median width shown—other CMF values would be obtained for other inputs. The CMF values listed indicate that widening the median will reduce crash frequency. The reduction is larger for cross median crashes and for larger median width increases. The values shown are very consistent with the CMF values reported by Harkey et al. (2008, Tables G-13 and G-14). The values shown in the last three rows of Table A31 represent an extrapolation of the trends in the data using Equation A45 because data for rural highways with five or more lanes were not collected by Harkey et al. As a result, some added caution should be taken when using these CMFs. Area Type Through Lanes Median Width After Change1, ft CMFcross CMFother CMFagg2 Urban 4 10 1.000 1.000 1.000 30 0.795 0.903 0.895 50 0.631 0.816 0.801 5 or more 10 1.000 1.000 1.000 30 0.763 0.868 0.860 50 0.583 0.753 0.740 Rural 4 10 1.000 1.000 1.000 30 0.729 0.945 0.928 50 0.531 0.893 0.864 5 or more 10 1.000 1.000 1.000 30 0.700 0.908 0.891 50 0.491 0.825 0.798 1 CMFs were calculated using 10 ft as the median width before the change occurs. 2 Aggregate CMF based on following proportions: pcross = 0.08; pother = 0.92 Table A31. Predicted disaggregate CMF values for changing median width on a divided highway.

A-68 This chapter provides details on the research conducted in Project 17-63 to develop and support portions of this appendix’s step-by-step procedure for estimating the effect of a proposed treatment at a subject site. The supporting research in this chapter is presented as two major parts. • Development of Crash Modification Functions to Identify Influential Factors—This research produced knowledge on the influential factors on CMFs, which fed into the tables of known influential factors in Step 4 of the procedure. • Evaluation of Two Estimators of Combined Average CMF—This research was used to derive the procedure for combining CMFs in Step 6b of the procedure. Development of Crash Modification Functions to Identify Influential Factors One of the efforts under Project 17-63 was to identify influential factors that can be used by practitioners to determine the transferability of CMFs. One approach for developing the list of influential factors is to estimate crash modification functions. This section provides the method- ology that was used for estimating the crash modification functions and results of this effort. The knowledge gained in these analyses was used in the development of the list of influential factors in Table A2 through Table A10. This section also documents the work to develop CMFunctions using three data sets from ongoing NCHRP and FHWA projects. The three data sets address the following countermeasures. • Signal to roundabout conversion • Centerline and shoulder rumble strips • Wet pavement markings on freeways Methodology In this effort, CMFunctions were estimated using data from empirical Bayes before-after studies. With empirical Bayes before-after studies, the equations for the CMF and the standard error of the CMF are the following (Hauer 1997): CMFi i i Equation A46* = λ π CMF Vari i i i i Equation A47 1 2 ( )= λ π + π π C H A P T E R   4 Supporting Research

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-69   where CMFi* = the biased estimate of the CMF for a particular site i CMFi = the unbiased estimate of the CMF λi = the actual number of crashes in the after period πi = the expected number of crashes in the after period had the treatment not been implemented The unbiased estimate is different from the biased estimate because the expected value of the ratio of two random numbers is different from the ratio of their expected values [i.e., if A and B are two random numbers, E(A/B) ≠ E(A)/E(B)]. Var represents the variance of these parameters. The intent is to estimate the CMF as a function of site characteristics. Equation A48CMF f site characteristics( )= With this mind, Equation A46 and Equation A47 can be rewritten as the following: CMF f site characteristicsi i i Equation A49* ( )= λ π = CMF Var f site characteristicsi i i i i Equation A50 1 2 ( )( )= λ µ + π π     = The traditional approach for estimating CMFunctions includes the use of the CMF value as the dependent variable and site/treatment characteristics as independent variables. This CMFunction could then be estimated as a regression equation. Based on a recent study by Elvik (2015), variance of the CMF needs to be considered in this estimation. The inverse of the vari- ance is typically introduced as a weight in a weighted regression model. In other words, for an observation (or site) whose CMF is CMFi with a variance of Var(CMFi), the weight will be 1/Var(CMFi). For linear regression, this would be appropriate. Some recent studies have recommended the use of a different model form such as a lognormal model that would ensure the predicted CMF from a CMFunction would always be greater than zero. In the case of the lognormal model, it can be shown that the appropriate weight for a weighted lognormal regression model would instead be [CMFi/Var(CMFi)] (this is because, based on Equation A48, the Var(CMFi) is not independent of CMFi, i.e., lower CMFs values would tend to have lower variances as well). For either the normal regression or lognormal regression models with weights, reliable esti- mates of CMFs and their variances are needed. To have reliable estimates of these parameters, sites with similar characteristics are often combined. However, this aggregation can lead to loss of useful information. To overcome this problem, an alternative approach was used. This approach involves rewriting Equation A49 as follows. f site characteristicsi i Equation A51( )λ = π × Written in the form of Equation A51, this model can be estimated as a count data model with λ as the dependent variable and π as an offset. One issue with this approach is that the offset is not a fixed value but is estimated as part of the EB procedure with a known variance. There has been some limited research on the implications of errors/variance in the independent variables, but further research is needed, possibly using simulation.

A-70 Guidelines for the Development and Application of Crash Modification Factors In this case, Equation A51 was estimated as a negative binomial regression model. The func- tional form for the negative binomial regression model was the typical log-linear form (investi- gation of other forms could be a topic for future research). With this form, the model equation was the following: i i X X X X Equation A52exp . . . . .1 1 2 2 3 3 4 4λ = π { }α+β ∗ +β +β +β + where α is the intercept Xs are the independent variables (site characteristics) βs are the coefficients of the independent variables Negative binomial regression models were estimated using PROC GENMOD in SAS. Cumu- lative residuals (CURE) plots with the predicted (fitted) value in the x axis were developed as a goodness of fit measure. Models were estimated using backward elimination, and the statisti- cally significant variables were retained at each step. Signal to Roundabout Conversion The data set containing data on the conversion of signalized intersections to roundabouts was based on the work done in NCHRP Project 17-35 and more recent efforts. Data from ten states are available in this data set, covering a total of 39 intersections that were converted. Table A32 provides the breakdown by state. CMFs were estimated for total and fatal and injury crashes. For each site, information was available on area type, total intersection AADT before the implementation of the roundabouts, and number of legs. Number of lanes in the roundabout was not available for six of the intersec- tions, and hence, was not used in this analysis. CMFunctions for Total Crashes The following variable was significant in total crashes: Total intersection AADT/10000 (denoted as total_AADT_before_s in the output) (_s is included in the variable name to indi- cate that it was scaled by dividing by 10000); higher AADT values were associated with higher CMFs. As a group, the effect of ‘state’ was marginally significant (based on the type 3 statistics) (see table below). However, this is a small data set with very few sites from each state. So, these results are probably not conclusive regarding effect of state. When the state factor was removed, AADT continued to be significant (see Table A35). State Intersections Converted CO 3 FL 1 IN 1 MD 2 MI 16 NC 2 NY 11 SC 1 VT 1 WA 1 Table A32. Signals converted to roundabouts by state.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-71   Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 -0.7474 0.7190 -2.1566 0.6618 1.08 0.2986 State CO 1 -1.5127 0.6713 -2.8284 -0.1970 5.08 0.0242 State FL 1 0.9984 0.9121 -0.7892 2.7861 1.20 0.2737 State IN 1 0.0083 0.8360 -1.6303 1.6469 0.00 0.9921 State MD 1 -0.7587 0.7142 -2.1585 0.6411 1.13 0.2881 State MI 1 -0.4236 0.5901 -1.5802 0.7330 0.52 0.4728 State NC 1 0.2277 0.7601 -1.2620 1.7174 0.09 0.7645 State NY 1 -0.0098 0.6213 -1.2276 1.2079 0.00 0.9874 State SC 1 -1.6153 0.8827 -3.3453 0.1148 3.35 0.0673 State VT 1 -0.1100 0.8268 -1.7304 1.5105 0.02 0.8942 State WA 0 0.0000 0.0000 0.0000 0.0000 . . total_AADT_before_s 1 0.4886 0.1554 0.1841 0.7931 9.89 0.0017 Dispersion 1 0.2898 0.0866 0.1613 0.5206 Note: The negative binomial dispersion parameter was estimated by maximum likelihood. Table A33. CMFunction results for signal to roundabout conversions (total crashes). LR Statistics for Type 3 Analysis Source DF ChiSq Pr > ChiSq State 9 16.87 0.0508 total_AADT_before_s 1 9.11 0.0025 Table A34. Regression statistics for CMFunction results for signal to roundabout conversions (total crashes). Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 -0.5730 0.3146 -1.1896 0.0436 3.32 0.0685 total_AADT_before_s 1 0.2838 0.1314 0.0264 0.5413 4.67 0.0307 Dispersion 1 0.5070 0.1299 0.3068 0.8377 Table A35. CMFunction results for signal to roundabout conversions (total crashes, no state factor).

A-72 Guidelines for the Development and Application of Crash Modification Factors CMFunctions for Fatal and Injury Crashes Table A36 shows the function for fatal and injury crashes. Lg_total_AADT_before_s repre- sents ln(total intersection AADT/10000). The log AADT parameter was only marginally signifi- cant. The state factor was not significant in this model. Centerline and Shoulder Rumble Strip Installation This data set contained data on the installation of centerline and shoulder rumble strips from more than 2000 ‘sites’ from KY, MO, and PA. Within this, KY has the fewest observations with data for 27 sites. This treatment was implemented on two-lane roads in these states. The fol- lowing independent variables were available: state, average shoulder width, AADT, and the EB expected number of crashes per mile in the first year of the before period. The last variable serves as a proxy for the crash propensity of a particular location—it is a useful variable since some important site characteristics (e.g., alignment and roadside hazards) are not available in this data set. Since the EB expected number of crashes per mile and AADT are likely correlated, in addition to the models shown here, additional models were also estimated with including only one of the two variables. These additional models provided conclusions that were consistent with the models provided below. CMFunction for Total Crashes The final model for total crashes is shown below. aadt_b_s denotes the average AADT in the before period/10000. lg_E_k1_mile represents the natural log of the EB expected number of crashes per mile in the first year of the before period. Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 -2.1579 0.5794 -3.2935 -1.0223 13.87 0.0002 total_AADT_before_s 1 1.0501 0.4543 0.1597 1.9405 5.34 0.0208 lg_total_AADT_before_s 1 -1.6231 0.8521 -3.2931 0.0469 3.63 0.0568 Dispersion 1 0.3451 0.2876 0.0674 1.7669 Table A36. CMFunction results for signal to roundabout conversions (fatal and injury crashes). Analysis Of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 -0.2070 0.0760 -0.3560 -0.0581 7.42 0.0064 State KY 1 0.1046 0.1460 -0.1815 0.3907 0.51 0.4738 State MO 1 -0.3989 0.0744 -0.5447 -0.2531 28.74 <.0001 State PA 0 0.0000 0.0000 0.0000 0.0000 . . aadt_b_s 1 0.3255 0.0862 0.1565 0.4945 14.25 0.0002 lg_E_k1_mile 1 -0.2359 0.0525 -0.3388 -0.1329 20.18 <.0001 Dispersion 1 0.3867 0.0584 0.2877 0.5199 Note: The negative binomial dispersion parameter was estimated by maximum likelihood. Table A37. CMFunction results for centerline and shoulder rumble strips (total crashes).

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-73   The CMF associated with Missouri is significantly lower compared to that associated Pennsylvania after accounting for AADT and the expected crashes in the before period. As seen with the round- about treatment, higher AADTs are associated with higher CMFs. Higher expected crashes in the before period are associated with lower CMFs. CMFunction for Fatal and Injury Crashes The only significant factor in fatal and injury crashes was state. Again, Missouri was associated with a lower CMF compared to Pennsylvania. CMFunctions for Run-Off-Road Crashes The only variable that was significant for run-off-road crashes was the natural log of the EB expected number of crashes per mile in the first year of the before period. As for total crashes, higher expected crashes are associated with lower CMFs. CMFunctions for Head-On Crashes The results for head-on crashes were like the ones obtained for fatal and injury crashes. Only state was significant. Wet Reflective Pavement Markings on Freeways This data set has data for 705 sites from NC, WI, and MN. However, the treatment was not the same in all the sites: the markers could be on edgelines, edgelines and lanelines, or lanelines. The “edgelines and lanelines” treatment included data from NC and MN (for 250 sites), and CMFunctions were estimated for this treatment. The other two treatments were only imple- mented in one state, and hence, could not be used for determining whether CMFs differ from LR Statistics for Type 3 Analysis Source DF ChiSq Pr > ChiSq State 2 33.81 < .0001 aadt_b_s 1 14.27 0.0002 lg_E_k1_mile 1 19.54 < .0001 Table A38. Regression statistics for CMFunction results for centerline and shoulder rumble strips (total crashes). Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 0.0167 0.0656 -0.1119 0.1453 0.06 0.7996 State KY 1 -0.0799 0.1615 -0.3964 0.2365 0.25 0.6206 State MO 1 -0.6047 0.0968 -0.7944 -0.4150 39.03 <.0001 State PA 0 0.0000 0.0000 0.0000 0.0000 . . Dispersion 1 0.3247 0.0809 0.1993 0.5290 Note: The negative binomial dispersion parameter was estimated by maximum likelihood. Table A39. CMFunction results for centerline and shoulder rumble strips (fatal and injury crashes). LR Statistics for Type 3 Analysis Source DF ChiSq Pr > ChiSq State 2 41.02 <.0001 Table A40. Regression statistics for CMFunction results for centerline and shoulder rumble strips (fatal and injury crashes).

A-74 Guidelines for the Development and Application of Crash Modification Factors one state to the other. The following site characteristics were available: AADT, expected crashes per mile in the first year of the before period, number of lanes, right-side shoulder width, left side shoulder width, and area type. Since EB expected number of crashes per mile and AADT are likely correlated, in addition to the models shown here, additional models were also estimated with including only one of the two variables. These additional models provided conclusions that were consistent with the models provided below. CMFunction for Total Crashes For the total crashes model, the offset (expected crashes in the after period) was also included as a weight in the form of a weight statement in PROC GENMOD because it provided better results (based on AIC and BIC) compared to not including a weight. The final model is shown in Table A43. The state factor is significant. The other significant variables are number of lanes, AADT, expected crashes per mile, and right shoulder width. Higher AADT values and 4 lanes (com- pared to 6 lanes) are associated with higher CMFs; this is somewhat contradicting, but it may be simply due to the correlation between number of lanes and AADT. As seen in the previous results for the other treatments, higher expected crashes are associated with lower CMFs. Also, wider right shoulders are associated with lower CMFs. CMFunction for Fatal and Injury Crashes State and shoulder width are not significant for fatal and injury crashes. Otherwise, the results are quite like those of the total crash model. CMFunction for Run-Off-Road Crashes The results for run-off-road crashes are unexpected and strange with the respect to the state factor. The coefficient of 2.7272 for MN indicates that the CMF for MN is about 15 times higher (exp(2.7272)) compared to NC. Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 -0.4294 0.0991 -0.6236 -0.2352 18.79 < .0001 lg_E_k1_mile 1 -0.1881 0.0752 -0.3355 -0.0407 6.25 0.0124 Dispersion 1 0.6319 0.1514 0.3951 1.0107 Table A41. CMFunction results for centerline and shoulder rumble strips (run-off-road crashes). Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 0.0228 0.2093 -0.3875 0.4331 0.01 0.9134 State KY 1 -0.4397 0.4247 -1.2720 0.3926 1.07 0.3005 State MO 1 -0.6920 0.2967 -1.2735 -0.1106 5.44 0.0197 State PA 0 0.0000 0.0000 0.0000 0.0000 . . Dispersion 1 0.9771 0.7228 0.2293 4.1647 Table A42. CMFunction results for centerline and shoulder rumble strips (head-on crashes).

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-75   Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 0.3189 0.2266 -0.1253 0.7631 1.98 0.1593 State MN 1 -0.6692 0.1746 -1.0114 -0.3270 14.69 0.0001 State NC 0 0.0000 0.0000 0.0000 0.0000 . . no_lanes 4 1 0.4276 0.1038 0.2242 0.6310 16.97 <.0001 no_lanes 6 0 0.0000 0.0000 0.0000 0.0000 . . AADT_b_s 1 0.1709 0.0173 0.1371 0.2048 98.03 <.0001 lg_E_k1_mile 1 -0.3839 0.0417 -0.4656 -0.3022 84.90 <.0001 rshldwid 1 -0.0675 0.0200 -0.1067 -0.0283 11.37 0.0007 Dispersion 1 1.3075 0.2056 0.9607 1.7794 Note: The negative binomial dispersion parameter was estimated by maximum likelihood. Table A43. CMFunction results for wet reflective pavement markings (total crashes). Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 -0.9506 0.3069 -1.5522 -0.3490 9.59 0.0020 no_lanes 4 1 0.4607 0.2349 0.0003 0.9211 3.85 0.0499 no_lanes 6 0 0.0000 0.0000 0.0000 0.0000 . . AADT_b_s 1 0.1542 0.0328 0.0899 0.2184 22.13 <.0001 lg_E_k1_mile 1 -0.2776 0.0663 -0.4076 -0.1476 17.52 <.0001 Dispersion 1 0.1894 0.0721 0.0898 0.3996 Table A44. CMFunction results for wet reflective pavement markings (fatal and injury crashes). Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 -2.7669 1.3343 -5.3821 -0.1517 4.30 0.0381 State MN 1 2.7272 1.0101 0.7474 4.7070 7.29 0.0069 State NC 0 0.0000 0.0000 0.0000 0.0000 . . no_lanes 4 1 1.1764 0.4359 0.3219 2.0308 7.28 0.0070 no_lanes 6 0 0.0000 0.0000 0.0000 0.0000 . . AADT_b_s 1 0.3410 0.0635 0.2165 0.4655 28.83 <.0001 lshldwid 1 0.2725 0.1338 0.0103 0.5348 4.15 0.0417 rshldwid 1 -0.2641 0.1183 -0.4959 -0.0323 4.99 0.0256 Dispersion 1 0.4877 0.1900 0.2273 1.0464 Table A45. CMFunction results for wet reflective pavement markings (run-off-road crashes).

A-76 Guidelines for the Development and Application of Crash Modification Factors Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 0.6334 0.4567 -0.2616 1.5285 1.92 0.1654 State MN 1 -0.8196 0.2653 -1.3396 -0.2996 9.54 0.0020 State NC 0 0.0000 0.0000 0.0000 0.0000 . . no_lanes 4 1 0.4017 0.1871 0.0350 0.7684 4.61 0.0318 no_lanes 6 0 0.0000 0.0000 0.0000 0.0000 . . urbrur 0 1 -0.3487 0.1802 -0.7019 0.0044 3.75 0.0529 urbrur 1 0 0.0000 0.0000 0.0000 0.0000 . . AADT_b_s 1 0.1054 0.0362 0.0344 0.1763 8.48 0.0036 lg_E_k1_mile 1 -0.3058 0.0677 -0.4385 -0.1732 20.42 <.0001 rshldwid 1 -0.0937 0.0328 -0.1580 -0.0295 8.17 0.0043 Dispersion 1 0.5754 0.1641 0.3290 1.0064 Note: The negative binomial dispersion parameter was estimated by maximum likelihood. Table A46. CMFunction results for wet reflective pavement markings (night crashes). Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 -1.1505 0.5210 -2.1716 -0.1293 4.88 0.0272 no_lanes 4 1 0.7807 0.2140 0.3614 1.2001 13.31 0.0003 no_lanes 6 0 0.0000 0.0000 0.0000 0.0000 . . AADT_b_s 1 0.4143 0.0917 0.2345 0.5941 20.40 <.0001 lg_AADT_b_s 1 -1.2198 0.4173 -2.0377 -0.4019 8.54 0.0035 lg_E_k1_mile 1 -0.3462 0.0678 -0.4790 -0.2134 26.09 <.0001 lshldwid 1 0.2008 0.0445 0.1135 0.2881 20.32 <.0001 rshldwid 1 -0.1877 0.0441 -0.2742 -0.1012 18.09 <.0001 Dispersion 1 0.6118 0.1589 0.3677 1.0180 Table A47. CMFunction results for wet reflective pavement markings (wet crashes). CMFunction for Night Crashes The results for night crashes are very similar to those of total crashes, except that the area type (rural vs. urban) is marginally significant (like the total crash model, this model also included a weight). CMFunction for Wet Crashes The state factor is not significant for wet crashes. The effect of AADT is a bit complicated (it is not monotonically increasing or decreasing). Since the signs for the two shoulder width terms are in opposite direction, the effect of left shoulder width may be opposite to that of outside shoulder width.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-77   CMFunction for Wet Night Crashes For wet night crashes, only the expected crashes per mile in the first year of the before period and area type are significant for this crash type. Evaluation of Two Estimators of Combined Average CMF This section describes the findings from an evaluation of two alternative estimators of the combined average CMF. The results from this evaluation were used to derive the procedure for combining CMFs that is presented in Step 6b. The combined average CMF is used to describe the overall safety effect of a given treatment based on either its application to several locations or the published findings from several inde- pendent studies of this treatment. In each case, there is an estimate available of the treatment effectiveness (i.e., a CMF) and the standard error of this CMF. The discussion herein is based on the premise that CMF values (and their respective stan- dard errors) for two or more locations are available from the literature. It is also based on the premise that more detailed data (e.g., crash counts) are not available for the collective set of locations. If more detailed data are available, then there are robust procedures available for estimating the combined average CMF (e.g., Elvik 1995; Hauer 1997, Chapter 10). Two Estimators of the Combined Average CMF Estimator A Bahar et al. (2007) offer the following equation for computing the combined average CMF. CMF CMF s s v i ii ii Equation A531 2 2 ∑ ∑ = where CMFv = overall average CMF based on variance weighting CMFi = value of CMF observation i si = standard error of CMF observation i Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Wald ChiSq Pr > ChiSq Intercept 1 1.0946 0.2240 0.6555 1.5337 23.87 < .0001 urbrur 0 1 -1.1083 0.2130 -1.5257 -0.6908 27.07 < .0001 urbrur 1 0 0.0000 0.0000 0.0000 0.0000 . . E_k1_mile 1 -0.7554 0.2440 -1.2336 -0.2773 9.59 0.0020 Dispersion 1 0.3736 0.1721 0.1515 0.9215 Table A48. CMFunction results for wet reflective pavement markings (wet night crashes).

A-78 Guidelines for the Development and Application of Crash Modification Factors The standard error of the combined CMF is computed using the following equation. s s CMF v ii Equation A5411, 2∑ = where sCMF v, = overall average standard error of CMFv based on variance weighting Estimator B Griffin and Flowers (1997) offer the following equation for computing the combined average CMF. = Equation A55CMF eL L with L w L w i i i i i Equation A56 ∑ ∑ = and w CMF s i i i Equation A57 2 =    where CMFL = overall average CMF based on log transform L– = weighted average value of L Li = natural log of CMF observation i (= ln[CMFi]) wi = weight of CMF observation i CMFi = value of CMF observation i si = standard error of CMF observation i The standard error of the combined CMF is computed using the following equation. s e LCMF L se Equation A58= × and L w se i i Equation A591 0.5 ∑ =     where sCMF = standard error of CMF Lse = standard error for L – Hauer (1992) used simulation to evaluate Equation A55 and two other estimators. However, he did not evaluate Equation A53. He found that Equation A55 tended to produce a biased estimate of safety effect. The bias ranged from −0.08 to +0.1, with the value dependent on the number of crashes occurring in the “before” period and the CMF value.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-79   Comparing Performance A simulation program was constructed to evaluate the two estimators. The program replicated the results from 1,000 before-after studies (i.e., 1,000 observations). An average number of crash counts in the “before” period was specified, as was the true value of the combined average CMF. For each observation, the number of crashes in the before period was generated from a Poisson distribution. The mean crash count for the before period was specified. Similarly, the number of crashes in the after period was generated from a Poisson distribution. The mean crash count for the after period was computed as the product of the mean count for the before period and the true value of the combined average CMF. The before period and the after period had the same duration. For a given set of 1,000 observations, the method described by Hauer (1997) was used to compute the CMF for each observation and its standard error. The following equations were used for this purpose. CMF N N fi a i b i c H Equation A60, , ,= × and s N N N N fi a i b i a i b i c H Equation A61 1 1, , , , ,= × + × with f N c H b i Equation A621 1 1 , , = + where CMFi = value of CMF observation i Na,i = number of crashes for the after period Nb,i = number of crashes for the before period si = standard error of CMF observation i fc,H = correction factor for the Hauer method In Equation A61, the first term is represented by the ratio of crash counts in the after and before periods (i.e., Na,i/Nb,i). The use of this ratio is consistent with Equation 6.4 of the book by Hauer (1997). Equation A61 is derived by the method of statistical differentials, as described on page 70 of the book by Hauer (1997). This point about the first term is noted here because example applications in the literature often incorrectly use CMFi as the first term in Equa- tion A61 (this use incorrectly “double counts” the effect of the correction factor and under- estimates the standard error). Several combinations of “true crash count in the before period” and “true value of the combined average CMF” were considered for the evaluation. Specifically, the crash count had values of 10 and 20. The combined average CMF had values of 0.8, 1.0, and 1.2. These varia- tions resulted in six unique combinations that were evaluated. For each combination, there were five replications of 1,000 observations simulated. This resulted in 30 (= 6 × 5) unique simulation runs (each with 1,000 observations). For each simulation run, the 1,000 estimates of the CMF and its standard error were com- puted using Equation A60 and Equation A61, respectively. Estimator A was then used with the

A-80 Guidelines for the Development and Application of Crash Modification Factors CMFs to compute an estimate of the overall average CMF. Finally, Estimator B was used with the CMFs to compute an estimate of the overall average CMF. For each combination, the error was computed for each replication and the five error values averaged to produce an average error. The error was computed as the predicted CMF minus the true CMF. The results of the analysis are shown in Table A49. The results in Table A49 indicate that both estimators consistently underestimate the true combined average CMF. The estimate from Estimator A is 20 to 30 percent below the true value. That from Estimator B is 5 to 10 percent below the true value. The error approaches zero as the crash count increases. The last column of Table A49 summarizes the average error when Equation A63 is used. This equation includes an adjustment factor that removes most of the bias associated with Estima- tor B. The development of this equation and adjustment factor is described in the next section. Adjusting for Bias This section describes the development of the adjustment factor included in Equation A63. This factor is intended to remove the bias associated with Estimator B, as described in the previ- ous section. Data Set 1 Simulation data were used to develop the adjustment factor. The program replicated the results from 1,000 before-after studies (i.e., 1,000 observations). An average number of crash counts in the “before” period was specified, as was the true value of the combined average CMF. For each observation, the number of crashes in the before period was generated from a Poisson distribution. The mean crash count for the before period was specified. Similarly, the number of crashes in the after period was generated from a Poisson distribution. The mean crash count for the after period was computed as the product of the mean count for the before period and the true value of the combined average CMF. The before period and the after period had the same duration. For a given set of 1,000 observations, the method described by Hauer (1997) was used to compute the CMF for each observation and its standard error. Equation A60 to Equation A62 were used for this purpose. Several combinations of “true crash count in the before period” and “true value of the com- bined average CMF” were considered for the evaluation. Specifically, the crash count had values of 10, 20, 30, and 100. The combined average CMF had values of 0.5, 0.8, 1.0, 1.2, and 2.0. These variations resulted in 20 unique combinations that were evaluated. For each combination, there were five replications of 1,000 observations simulated. This resulted in 100 (= 20 × 5) unique simulation runs (each with 1,000 observations). True Crash Count in Before Period (crash/observation) True Value of Combined Average CMF Average Error by Estimator Estimator A Estimator B Equation A63 10 0.8 -0.276 -0.067 0.023 1.0 -0.316 -0.095 0.002 1.2 -0.358 -0.123 -0.019 20 0.8 -0.150 -0.035 0.014 1.0 -0.179 -0.049 0.005 1.2 -0.202 -0.063 -0.005 Table A49. Comparison of two estimators of overall average CMF.

Procedure for Estimating the Effect of a Proposed Treatment at a Subject Site A-81   Data Set 2 The evaluation of Estimators A and B indicated that the amount of bias for a given estimator was influenced by the standard error associated with each observation. These standard errors are included in the weighted average technique associated with each estimator. The magnitude of the standard error is a function of the “CMF calculation method” (e.g., naïve before-after, before-after with comparison group, EB, EB with comparison group). For this reason, a second set of 100 unique simulation runs were completed to include comparison group data. Comparison group data were included because they are associated with a relatively large standard error. For the comparison group data, the number of crashes in the before period at the treatment site was set to be the same as that for the before and after periods at the comparison sites. This approach was used to reduce the variability in the results such that fewer simulation runs would be needed, while allowing an exploration of the effect of “method” on the amount of bias in Estimator B. The before periods and the after periods for the treatment and comparison groups had the same duration. For a given set of 1,000 observations, the method described by Hauer (1997) for comparison groups was used to compute the CMF for each observation and its standard error. Results Data sets 1 and 2 were combined into one database with 200 data points. The dependent vari- able was the true CMF value, as computed using the following equation. CMF e ft L c t Equation A63,= × where CMF st = true CMF fc,t = correction factor based on true CMF When a CMF is transformed by the natural log function, the transformed value is asymptotic to the normal distribution when the CMF is based on many crashes (Griffin and Flowers, 1997). It follows that CMF data are asymptotic to the lognormal distribution and the correction factor fc,t has a value of 0.5 σ2ln(CMF). This relationship was used to determine the best form for the adjust- ment factor. The following equation form was derived to provide the best fit to the data. f b w L L w c o i i i i i Equation A64exp 2∑ ∑ ( ) ′= −          where fc′ = predicted correction factor bo = regression coefficient The statistics describing the calibrated model are summarized in Table A50. The SAS pro- cedure NLMIXED was used to automate a maximum likelihood regression analysis of the true and computed CMFs. A log-likelihood function for the lognormal distribution was used in the analysis. The coefficient of determination R2 of 0.99 indicates a good fit to the data. The t-statistic for each coefficient is sufficiently large as to indicate that the coefficient is statistically significant. Figure A3 compares the predicted and true CMFs for the 200 data points. The line shown is not the line of best fit. Rather, it an “x = y” line such that the data point would lie on this line if the predicted value equaled the true value. In general, the predicted CMF value is in good agree- ment with the true CMF value for the range of CMFs evaluated.

A-82 Guidelines for the Development and Application of Crash Modification Factors Model Statistics Value R2: 0.99 Root mean square error: 0.042 Observations no: 200 data points (1,000 obs. per data point) Calibrated Coefficient Values Variable Value Std. Err. t-statistic b0 Intercept 0.574 0.0265 21.6 Table A50. Model statistical description. 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 2.5 Tr ue C M F Predicted CMF 1 1 Figure A3. Comparison of predicted and true correction factor values.

A-83   American Association of State Highway and Transportation Officials (AASHTO). 2010. Highway Safety Manual, First Edition. Abuzwidah, M., M. Abdel-Aty, and M. M. Ahmed. 2014. Safety Evaluation of Hybrid Mainline Toll Plazas. Paper presented at the 93rd meeting of the Transportation Research Board, Washington, D.C. Bahar, G., M. Parkhill, E. Hauer, F. Council, B. Persaud, C. Zegeer, R. Elvik, A. Smiley, and B. Scott. 2007. NCHRP Project 17-27: Prepare Parts I and II of the Highway Safety Manual. iTRANS Consulting, Ltd. (Appendix A). Bahar, G., C. Mollett, B. Persaud, C. Lyon, A. Smiley, T. Smahel, and H. McGee. 2004. NCHRP Report 518: Safety Evaluation of Permanent Raised Pavement Markers. Transportation Research Board, Washington, D.C. Bauer, K., D. Harwood, W. Hughes, and K. Richard. 2004. “Safety Effects of Narrow Lanes and Shoulder-Use Lanes to Increase Capacity of Urban Freeways.” Transportation Research Record 1897. Transportation Research Board, Washington, D.C., pp. 71–80. Bonneson, J., S. Geedipally, M. Pratt, and D. Lord. 2012. “Safety Prediction Methodology and Analysis Tool for Freeways and Interchanges.” NCHRP Project No. 17-45. Transportation Research Board, Washington, D.C. Bonneson, J. and M. Pratt. 2008. Calibration Factors Handbook: Safety Prediction Models Calibrated with Texas Highway System Data. FHWA/TX-08/0-4703-5. Texas Transportation Institute, College Station, Texas. Carrasco, O., J. McFadden, and P. Chandhok. 2004. “Evaluation of the Effectiveness of Shoulder Rumble Strips on Rural Multi-lane Divided Highways In Minnesota.” Washington D.C., 83rd Transportation Research Board Annual Meeting. Elvik, R. 2015. Methodological guidelines for developing accident modification functions. Accident Analysis and Prevention, Vol. 80, pp. 26–36. Elvik, R. 1995. “Meta Analysis of Evaluations of Public Lighting as Accident Countermeasure.” Transportation Research Record 1485. Transportation Research Board, Washington, D.C., pp. 112–123 Federal Highway Administration (FHWA), Crash Modification Factors Clearinghouse, www.cmfclearinghouse.org. Accessed January 2017. Fitzpatrick, K., D. Lord, and B. J. Park. 2009. “Horizontal Curve Accident Modification Factor with Consideration of Driveway Density on Rural, Four-Lane Highways in Texas.” Transportation Research Board 88th Annual Meeting Compendium of Papers CD-ROM. Washington, D.C. Fleiss, J. L. 1973. Statistical Methods for Rates and Proportions. John Wiley & Sons, New York. Graham, J. L., K. R. Richard, M. K. O’Laughlin, and D. W. Harwood. 2011. “Safety Evaluation of the Safety Edge Treatment” Report No. FHWA-HRT-11-024, Federal Highway Administration, Washington, DC. Griffin, L. and R. Flowers. 1997. A Discussion of Six Procedures for Evaluating Highway Safety Projects. Report No. FHWA-RD-99-040. Texas Transportation Institute, College Station, Texas. Gross, F., C. Lyon, B. Persaud, and R. Srinivasan. 2013. ”Safety Effectiveness of Converting Signalized Inter- sections to Roundabouts.” Accident Analysis and Prevention, Vol. 50, pp. 234–241. Harkey, D., R. Srinivasan, J. Baek, F. Council, K. Eccles, N. Lefler, F. Gross, B. Persaud, C. Lyon, E. Hauer, and J.  Bonneson. 2008. NCHRP Report 617: Accident Modification Factors for Traffic Engineering and ITS Improvements. Transportation Research Board, Washington, D.C. Harwood, D. W., K. M. Bauer, I. B. Potts, D. J. Torbic, K. R. Richard, E. R. Rabbani, E. Hauer, L. Elefteriadou, and M. S. Griffith. 2003. “Safety Effectiveness of Intersection Left- and Right-Turn Lanes.” Washington, D.C., 82nd Transportation Research Board Annual Meeting. Harwood, D. W., F. M. Council, E. Hauer, W. E. Hughes, and A. Vogt. 2000. “Prediction of the Expected Safety Performance of Rural Two-Lane Highways.” FHWA-RD-99-207, McLean, Va., Federal Highway Administration. Hauer, E. 1997. Observational Before-After Studies in Road Safety. Elsevier Science Inc., Tarrytown, New York. References

A-84 Guidelines for the Development and Application of Crash Modification Factors Hauer, E. 1992. “A Note on Three Estimators of Safety Effect.” Traffic Engineering & Control. June 1992, pp. 388–393. Ksaibati, K., C. Zhong, and B. Evans. 2009. “WRRSP: Wyoming Rural Road Safety Program.” Report No. FHWA- WY-09/06F, Cheyenne, Wy., Wyoming Department of Transportation. Le, T. and R. Porter. 2012. Safety Effects of Cross Section Design on Urban and Suburban Roads. Presented at the 92nd Annual Meeting of the Transportation Research Board, Washington, D.C. Le, T. Q., and R. J. Porter. 2012. “Safety Evaluation of Geometric Design Criteria 3 for Entrance-Exit Ramp Spacing and Auxiliary Lane Use.” Presented at the Transportation Research Board 91st Annual Meeting, Paper No. 12-2153, January 22–26, Washington, DC. Park, J., and M. Abdel-Aty. 2015. Development of adjustment functions to assess combined safety effects of multiple treatments on rural two-lane roadways. Accident Analysis and Prevention, Vol. 75, pp. 310–319. Park, J., M. Abdel-Aty, and C. Lee. 2014. Exploration and comparison of crash modification factors for multiple treatments on rural multilane roadways. Accident Analysis and Prevention, Vol. 70, pp. 167–177. Park, J., M. Abdel-Aty, J. Lee, and C. Lee. 2015. Developing crash modification functions to assess safety effects of adding bike lanes for urban arterials with different roadway and socio-economic characteristics, Accident Analysis and Prevention, Vol. 74, pp. 179–191. Persaud, B., C. Lyon, K. Eccles, N. Lefler, D. Carter, and R. Amjadi. Safety Evaluation of Installing Center Two-Way Left-Turn Lanes on Two-Lane Roads. Federal Highway Administration, FHWA Report No. FHWA-HRT-08-042. (December 2007) Persaud, B., R. Retting, P. Gardner, and D. Lord. 2001. “Safety Effect of Roundabout Conversions in the United States.” Transportation Research Record 1751. Transportation Research Board, Washington, D.C., pp. 1–8. Pratt, M. P., S. R. Geedipally, A. M. Pike, P. J. Carlson, A. M. Celoza, and D. Lord. “Evaluating the Need for Surface Treatments to Reduce Crash Frequency on Horizontal Curves.” Report No. FHWA/TX-14/0- 6714-1. Texas Department of Transportation. Austin, Texas. (May 2014) Russo, B., A. Davis, S. Frazier, P. Savolainen, and T. Gates. 2014. “Evidence of Mixed Safety Performance among Roundabouts Converted from Signalized Intersections.” Paper No. 14-0046. Presented at the annual meeting of the Transportation Research Board, Washington, D.C. Sacchi, E., T. Sayed, and A. Osama. 2015. Developing crash modification functions for pedestrian signal improve- ment. Accident Analysis and Prevention, Vol. 83, pp. 47–56. Srinivasan, R. and B. Lan. Estimation of Crash Modification Functions Using Site-Level Information from Results of Empirical Bayes Before-After Evaluations, Final Report, Southeastern Transportation Center, USDOT GRANT DTRT13-G-UTC34, January 2016 Srinivasan, R., J. Baek, D. Carter, B. Persaud, C. Lyon, K. Eccles, F. Gross, N. Lefler. 2009. “Safety Evaluation of Improved Curve Delineation.” Report No. FHWA-HRT-09-045, Federal Highway Administration, Washington, D.C. Srinivasan, R., J. Baek, S. Smith, C. Sundstrom, D. Carter, C. Lyon, B. Persaud, F. Gross, K. Eccles, A. Hamidi, and N. Lefler. 2011. NCHRP Research Report 705: Evaluation of Safety Strategies at Signalized Intersections, Washington, D.C., Transportation Research Board. Srinivasan, R., D. Carter, K. Eccles, B. Persaud, N. Lefler, C. Lyon, and R. Amjadi. “Safety Evaluation of Flash- ing Beacons at STOP-Controlled Intersections,” Federal Highway Administration, FHWA-HRT-08-044, Washington, D.C., December 2007. Torbic, D. J., J. M. Hutton, C. D. Bokenkroger, K. M. Bauer, D. W. Harwood, D. K. Gilmore, D. K. Dunn, J. J. Ronchetto, E. T. Donnell, H. J. Sommer III, P. Garvey, B. Persaud, and C. Lyon. 2009. NCHRP Report 641: Guidance for the Design and Application of Shoulder and Centerline Rumble Strips, Transportation Research Board, Washington D.C. Woolf, G. 1955. “On Estimating the Relationship between Blood Group and Disease.” Annals of Human Genetics. Vol., 19, pp. 251–253. Zegeer, C. V., D. W. Reinfurt, W. W. Hunter, J. Hummer, R. Stewart, and L. Herf. 1988. “Accident Effects of Side- slope and Other Roadside Features on Two-Lane Roads.” In Transportation Research Record 1195. Transpor- tation Research Board, Washington, D.C., pp. 33–47.