EVALUATING SAFETY PE RFORMANCE OF RURAL C OUNTY HIGHWAYS USING MIXED - EFFECTS NEGATIVE BIN OMIAL MODELS By Steven York Stapleton A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Civil Engineering Doctor of Philosophy 2021 ABSTRACT EVALUATING SAFETY PE RFORMANCE OF RURAL C OUNTY HIGHWAYS USING MIXED - EFFECTS NEGATI VE BINOMIAL MODELS By Steve n York Stapleton Safety on rural highways continues to be a serious concern in the United States. While only 20 percent of the population live in rural areas, approximately one - half of motor vehicle fatalities occur on rural roadways, resulting in a rural fatal crash rate that is approximately double that of urban areas. In many states, most rural arterial highways are owned by the state department of transportation. However, several states , including Michigan, possess a large rural county highway network. For example, nearly 75 percent of the approximately 120,000 miles of public roadways in Michigan are owned by one of the 83 county road agencies across the state. County - owned highways typically possess characteristics that differ considerably from thos e owned by the state department of transportation, which limits the usefulness of safety performance functions (SPFs) and crash modification factors (CMFs) generated based on state highways, including those found in the Highway Safety Manual (HSM) . Thus, a ssumptions made from models generated using data from state highways may not apply county highways due to differences in traffic, design, and maintenance. As a substantial proportion of rural crashes occur on county roads, identification of factors affecti ng safety performance on rural county roads is critical to support highway safety improvement programs and development of design standards. A cross - sectional safety performance analysis was performed for county highway segments and stop - controlled intersec tions throughout rural Michigan, including both federal aid and non - federal aid highways, as well as paved and unpaved road surfaces. SPFs were developed using mixed effects negative binomial regression to determine the safety effect of various design elem ents and site characteristics, including cross - sectional and geometric characteristics, which were included in the models as fixed effects. Random intercepts were incorporated into the models to account for unobserved heterogeneity between counties and bet ween individual sites. One particularly noteworthy contribution of this research was to investigate the impacts of horizontal curvature on safety performance. Curve radii data extracted from the Michigan roadway shapefile allowed for the safety performance effects of decreasing curve design speed to be assessed in an incremental manner. Horizontal curves on paved county roads with design speeds below 40 mph experienced crash occurrence that was more than four times greater than segments without substandard curvature. On unpaved roadways, such curves experienced three times greater crash occurrence compared to segments without substandard curvature. Deer - rela ted crashes, however, were shown to be fewer in frequency along horizontal curves . For stop - controlled intersections, skew angle was a variable of interest. At rural four - leg stop - controlled intersections, skew angles between 10 and 39 degrees were associ ated with increased crash frequency at intersections across all intersection classes. Skew had the greatest effect when the major road was county non - federal aid, where skew angles between 10 and 39 degrees experienced 60 percent more crashes than intersec tions without skew. Considering federal - aid intersections, the skew effect was diminished by approximately one - half. As expected, county - specific SPFs differed from models previously developed for state highways, including the SPFs included in the HSM . Ge nerally speaking, at intersections, county highways were found to experience fewer crashes per unit of traffic volume than state highways, with county non - federal aid highways showing the lowest crash occurrence. C ounty highway s egments tend to have higher crash frequency than state roads. However, this is not the case at all traffic volumes, which further shows the need for county - specific safety performance models. iv ACKNOWLEDGMENTS First and foremost , I would like to acknowledge my advisor, Dr . Timothy Gates, for all of the guidance and support he has provided throughout my education. His expertise and mentorship were critical element s in being able to reach this milestone. In addition, I would like to acknowledge my committee Dr. Mehrnaz Gha mami , Dr. Peter Savolainen , Dr. Dong Zhao , and Dr. Ali Zockaie . Their feedback throughout the writing of this dissertation has resulted in a more thorough and useful document. I would also like to thank some of the people who contributed to the data collection that made this dissertation possible. Gentjan Heqimi , Jonathan Kay, and Anthony Ingle all contributed to geographic information systems (GIS) work and developing/integrati ng datasets. Jacob Finkelman developed the process for extracting horizontal curvature data from the roadway shapefile. Others who assisted in data collection include Meghna Chakraborty, Brian Gammon, Daniel Hacker - Heck, Travis Holpuch, Alex Mullen, Moham mad Hossein (Sam) Shojaei, Jacob Swanson, and Corey Turner. v TABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ........................ vii LIST OF FIGURES ................................ ................................ ................................ ....................... xi 1. INTRODUCTION ................................ ................................ ................................ .................. 1 1.1 Problem and Kn owledge Gap ................................ ................................ ............................... 2 1.2 Research Objectives and Contributions ................................ ................................ ................ 5 1.3 Dissertation Structure ................................ ................................ ................................ ............ 7 2. LITERATURE REVIEW ................................ ................................ ................................ ....... 8 2.1 The Highway Safety Manual ................................ ................................ ................................ 8 2.1.1 Base SPF ................................ ................................ ................................ ........................ 9 2.1.2 Crash Modifica tion Factors ................................ ................................ .......................... 10 2.1.3 Calibration Factors ................................ ................................ ................................ ....... 11 2.2 Rural Highway Segment Safety Performance Characteristics ................................ ............ 19 2.2.1 Lane Width ................................ ................................ ................................ ................... 19 2.2.2 Shoulder W idth ................................ ................................ ................................ ............ 20 2.2.3 Access Points ................................ ................................ ................................ ............... 21 2.2.4 Alignment ................................ ................................ ................................ .................... 22 2.2.5 Pavement Surface ................................ ................................ ................................ ......... 23 2.2.6 Deer - Vehicle Crashes ................................ ................................ ................................ .. 24 2.3 Ru ral Intersection Safety Performance Characteristics ................................ ...................... 25 2.3.1 Turn Lane Presence ................................ ................................ ................................ ...... 26 2.3.2 Access Point Frequency ................................ ................................ ............................... 27 2.3.3 Other Geometric Factors ................................ ................................ .............................. 27 2.3. 4 Traffic - Related Factors ................................ ................................ ................................ 28 3. METHODOLOGY ................................ ................................ ................................ ............... 29 3.1 Data Collection ................................ ................................ ................................ ................... 29 3.1.1 Roadway Segmentation using the Michigan Geographic Framework ........................ 31 3.1.2 Traffic Volume Data ................................ ................................ ................................ .... 33 3.1.3 Tr affic Crash Data ................................ ................................ ................................ ........ 34 3.1.4 Additional Manual Data Collection ................................ ................................ ............. 44 3.1.5 Quality Control/Quality Assurance Verification ................................ ......................... 48 3.2 Model Calibration ................................ ................................ ................................ ............... 48 3.3 Analytical Methods ................................ ................................ ................................ ............. 49 4. SAFETY PERFORMANCE F UNCTIONS FOR COUNTY - OWNED RURAL HIGHWAY SEGMENTS ................................ ................................ ................................ ................................ . 55 4.1 Data Summary, Data Screening, and Data Diagnostics ................................ ...................... 55 4.1.1 Segment Descriptive Statistics ................................ ................................ ..................... 57 4.1.2 Data Screening ................................ ................................ ................................ ............. 60 vi 4.1.3 Data Diagnostics ................................ ................................ ................................ .......... 62 4.2 Results and Discussion ................................ ................................ ................................ ....... 70 4.2.1 Paved Federal Aid County Road Segments ................................ ................................ . 71 4.2.2 Paved Non - Federal Aid County Road Segments ................................ ......................... 77 4.2.3 Unpaved Non - Federal Aid Segments ................................ ................................ .......... 81 4.2.4 Crash Modification Factors for Rural County Segments ................................ ............. 83 4.2.5 Comparison to MDOT and Calibrated HSM SPFs ................................ ...................... 84 4.3 Summary and Conclusions ................................ ................................ ................................ . 86 5. SAFETY PERFORMANCE F UNCTIONS FOR RURAL M INOR ROAD STOP CONTROLLED INTERSECT IONS ................................ ................................ ............................ 89 5.1 Data Summary, Data Screening, and Data Diagnostics ................................ ...................... 89 5.1.1 Rural Four Leg Stop - Controlled Intersections (4ST) ................................ .................. 90 5.1.2 Rural Three - Leg Stop - Controlled Intersection s (3ST) ................................ .............. 103 5.2 Results and Discussion ................................ ................................ ................................ ..... 115 5.2.1 Four - Leg Stop - Controlled Rural Intersections (4ST) ................................ ................ 116 5.2.2 Three - Leg Stop - Controlled Rural Intersections (3ST) ................................ .............. 119 5.2.3 Crash Modification Factors Develop ed for Rural Intersections ................................ 122 5.2.4 Comparison to HSM Models ................................ ................................ ..................... 123 5.3 Summary and Conclusions ................................ ................................ ............................... 125 6. EVALUATION OF DEER C RASHES ON RURAL SEGM ENTS ................................ ... 128 6.1 Descriptive Statistics ................................ ................................ ................................ ......... 129 6.2 Results and Discussion ................................ ................................ ................................ ..... 1 33 6.3 Su mmary and Conclusions ................................ ................................ ............................... 140 7. CONCLUSIONS, AND REC OMMENDATIONS ................................ ............................ 142 7.1 Rural County - Owned Highway Segments ................................ ................................ ........ 146 7.2 Rural Minor Road Stop - Controlled Intersections ................................ ............................. 148 7.3 Recommendations for Future Work ................................ ................................ .................. 148 APPENDICES ................................ ................................ ................................ ............................ 152 Appendix A: Fixed Effects Models ................................ ................................ ........................ 153 Appendix B: Temporally Aggregated Model ................................ ................................ ......... 157 Appendix C: Mixed Effects Models without Length Offset ................................ ................... 158 Appendix D: Mixed Effects Models with M odified Curve Variables ................................ .... 161 REFERENCES ................................ ................................ ................................ ........................... 163 vii LIST OF TABLES Table 1: Rural Intersection Calibration Factors ................................ ................................ ............ 13 Table 2: Rural Segment Calibration Factors ................................ ................................ ................. 14 Table 3. Calibration Factors for HSM Models on MDOT Rural Trunkline Segments [1 6 ] ......... 16 Table 4. Calibration Factors for HSM Models on Michigan Rural County Road Segments [1 6 ] 17 Table 5. Calibration Factors for HSM Models at Michigan Rural Intersections [1 6 ] ................... 18 Table 6: Rural County Road Intersection Summary Statistics ................................ ..................... 42 Table 7: Rural County Road Segment Summary Statistics ................................ .......................... 44 Table 8: Represented Counties and Corresponding Segment Mileage (Segments) ..................... 57 Table 9: County Road Segment Summary Statistics (Federal Aid) ................................ ............. 58 Table 10: County Road Segment Summary Statistics (Paved Non - Federal Aid) ........................ 59 Table 11: County Road Segment Summary Statistics (Unpaved Non - Federal Aid) .................... 60 Table 12: Crash Severity and Crash Type Distributions for Rural Paved Federal Aid County Segments ................................ ................................ ................................ ................................ ....... 68 Table 13: Crash Severity and Crash Type Distributions for Rural Paved Non - Federal Aid County Segments ................................ ................................ ................................ ................................ ....... 69 Table 14: Crash Severity and Crash Type Distributions for Rural Unpaved Non - Federal Aid County Segments ................................ ................................ ................................ .......................... 70 Table 15: Mixed Effects Negative Binomial Model Results for Paved Federal Aid Segments ... 72 Table 16: Mixed Effects Negative Binomial Model Results for Paved Non - Federal Aid Segments ................................ ................................ ................................ ................................ ....................... 79 Table 17: Mixed Effects Negative Binomial Model Results for Unpaved Non - Federal Aid Segments ................................ ................................ ................................ ................................ ....... 82 Table 18: CMFs Developed for Rural County Segments ................................ ............................. 84 viii Table 19: Represented Counties and Intersection Count by Major Roadway Class (Four - Leg Intersections) ................................ ................................ ................................ ................................ . 93 Table 20: Descriptive Statistics for Rural 4ST Intersections (All) ................................ ............... 94 Table 21: Descriptive Statistics for Rural 4ST Intersections (Major Road MDOT) .................... 95 Table 22: Descri ptive Statistics for Rural 4ST Intersections (Major Road County FA) .............. 96 Table 23: Descriptive Statistics for Rural 4ST Intersections ( Major Road County Non - FA) ...... 97 Table 24: Crash Severity and Crash Type Distributions for Rural 4ST Intersections (All) ......... 99 Table 25: Crash Severity and Crash Type Distributions for Rural 4ST Intersections (MDOT) 100 Table 26: Crash Severity and Crash Type Distributions for Rural 4ST Intersections (County FA) ................................ ................................ ................................ ................................ ..................... 101 Table 27: Crash Severity and Crash Type Distributions for Rural 4ST Intersections (County Non - FA) ................................ ................................ ................................ ................................ ...... 102 Table 28: Represented Counties and Intersection Count by Major Roadway Class (Three - Leg Intersections) ................................ ................................ ................................ ............................... 104 Table 29: Descriptive Statistics for Rural 3ST Intersections (All) ................................ ............. 106 Table 30: Descriptive Statistics for Rural 3ST Intersect ions (MDOT) ................................ ...... 107 Table 31: Descriptive Statistics for Rural 3ST Intersections (Major Road County FA) ............ 108 Table 32: Descriptive Statistics for Rural 3ST Intersections (Major Road County Non - FA) .... 109 Table 33: Crash Severity and Crash Type Distributions for Rural 3ST Intersections (All) ....... 112 Table 34: Crash Severity and Crash Type Distributions for Rural 3ST Intersections (Major Road MDOT) ................................ ................................ ................................ ................................ ....... 113 Table 35: Crash Severity and Crash Type Distributions for Rural 3ST Intersections (Major Road County FA) ................................ ................................ ................................ ................................ . 114 Table 36: Crash Severity and Crash Type Distributions for Rural 3ST Intersections (Major Road County Non - FA) ................................ ................................ ................................ ......................... 115 Table 37: Mixed Effects Negative Binomial Model Results for 4ST Rural Intersections ......... 118 ix Table 38: Fixed Effects Negative B inomial Model Results for 4ST Rural Intersections (Major Road County FA) ................................ ................................ ................................ ........................ 119 Table 39: Fixed Effects Negative Binomial Model Results for 4ST Rural Intersections (Major Road County Non - FA) ................................ ................................ ................................ ................ 119 Table 40: Mixed Effects Negative Binomial Model Results for 3ST Rural Intersections ......... 121 Table 41: Fixed Effects Negative Binomial Model Results for 3ST Rural Intersections (Major Road County FA) ................................ ................................ ................................ ........................ 122 Table 42: Fixed Effects Negative Binomial Model Results for 3ST Rural Intersections (Major Road County Non - FA) ................................ ................................ ................................ ................ 122 Table 43: CMFs Developed for Rural Intersections ................................ ................................ ... 123 Table 4 4 : Descriptive Statistics for State Highway Segments included in Deer Crash Analysis 130 Table 4 5 : Descriptive Statistics for County Federal - Aid Segments (Deer Crashes) .................. 130 Table 4 6 : Descriptive Statistics for County Non - Federal Aid Segments (Deer Crashes) .......... 131 Table 4 7 : Descriptive Statistics for Unpaved Segments (Deer Crashes) ................................ .... 131 Table 4 8 : Mixed Effects Negative Binomial Model for Deer Crashes on State Highway Segments ................................ ................................ ................................ ................................ ..... 137 Table 4 9 : Mixed Effects Negative Binomial Model for Deer Crashes on County Federal Aid Segments ................................ ................................ ................................ ................................ ..... 138 Table 50 : Mixed Effects Negative Binomial Model for Deer Crashes on County Non - Federal Aid Segments ................................ ................................ ................................ ................................ ..... 139 Table 5 1 : Mixed Effects Negative Binomial Model for Deer Crashes on Unpaved Segments . 140 Table 5 2 : Fixed Effects Negative Binomial Model Results for Paved Federal Aid Segments .. 153 Table 5 3 : Fixed Effects Negative Binomial Model Results for Paved Non - Federal Aid Segments ................................ ................................ ................................ ................................ ..................... 154 Table 5 4 : Fixed Effects Negative Binomial Model Results for Unpaved Non - Federal Aid Segments ................................ ................................ ................................ ................................ ..... 155 Table 5 5 : Fixed Effects Negative Binomial Model Results for 3ST Rural Intersections ........... 155 Table 5 6 : Fixed Effects Negative Binomial Model Results for 4ST Rural Intersections ........... 156 x Table 5 7 : Temporally Aggregated Fixed Effects Negative Binomial Model Results for Paved Federal Aid Segments ................................ ................................ ................................ ................. 157 Table 5 8 : Mixed Effects Negative Binomial Model Results for Paved Federal Aid Segments (No Length Offset) ................................ ................................ ................................ ............................. 158 Table 5 9 : Mixed Effects Negative Binomial Model Results for Paved Non - Federal Aid Segments (No Length Offset) ................................ ................................ ................................ ...................... 159 Table 6 0 : Mixed Effects Negative Binomial Model Results for Unpaved Segments (No Length Offset) ................................ ................................ ................................ ................................ ......... 160 Table 6 1 : Mixed Effects Negative Binomi al Model Results for Paved Federal Aid Segments (Binary Curve Variables) ................................ ................................ ................................ ............ 161 Table 6 2 : Mixed Effects Negative Binomial Model Results for Paved Federal Aid Segments (Quasi - Binary Curve Variable for <40 mph Curve Design Speed ) ................................ ............ 162 xi LIST OF FIGURES Figure 1: Rural facility types for Michigan SPF development ................................ ..................... 30 Figure 2: Node identification algorithm ................................ ................................ ....................... 38 Figure 3: 3 - leg Intersection with crash search threshold ................................ .............................. 41 Figure 4: Map of rural county highway segments ................................ ................................ ........ 56 Figure 5: Annual midblock cr ashes per mile vs AADT, county segments (2011 - 2015) .............. 63 Figure 6: Lane width, shoulder width, driveway density, NFC, and curve proportions vs. AADT on paved federal aid county segments ................................ ................................ .......................... 65 Figure 7: Lane width, shoulder width, driveway density, and curve propo rtions vs. AADT on paved non - federal aid county segments ................................ ................................ ........................ 66 Figure 8: Surface width, driveway density, and curve proportions vs. AA DT on unpaved non - federal aid county segments ................................ ................................ ................................ .......... 67 Figure 9: Comparison of SPF crash results on county federal - aid segments by curve d esign speed ................................ ................................ ................................ ................................ ....................... 74 Figure 10: Comparison of SPF crash results on paved county non - federal aid segments by curve design speed ................................ ................................ ................................ ................................ .. 80 Figure 11: Comparison of SPF crash results on unpaved county non - federal aid segments by curve design speed ................................ ................................ ................................ ........................ 83 Figure 12: Comparison of SPF total crash results at base conditions ................................ ........... 86 Figure 13: Map of rural four leg stop - controlled (4ST) intersections ................................ .......... 91 Figure 14: Distribution of skew angle across 4ST intersections ................................ ................... 97 Figure 15: Annual intersection crashes vs AADT, 4ST (2011 - 2015) ................................ .......... 98 Figure 16: Map of rural three - leg stop - controlled (3ST) intersection locations ......................... 103 Figure 17: Distribut ion of skew angle across 3ST intersections ................................ ................. 110 Figure 18: Annual intersection crashes vs AADT, 3ST (2011 - 2015) ................................ ........ 111 xii Figure 19: Model results for non - deer crashes on 4ST and 3ST intersections for minor roadway AADT=500 veh/day ................................ ................................ ................................ ................... 125 Figure 20: Map of rural state highway study segments (deer crashes) ................................ ....... 132 Figure 21: Map of rural county highway study segments (deer crashes) ................................ ... 132 Figure 22: Deer crash model results under base conditions and with substandard curves ......... 135 1 1. INTRODUCTION Safety on rural highways c ontinues to be a serious concern throughout the United States. Nationwide, approximately one - half of motor vehicle fatalities occur in rural areas, although only approximately 20 percent of the U.S. population lives in rural areas. In 2018, the rural highw ay fatal crash rate (per vehicle - miles traveled) in the U.S. was approximately double that of urban areas, providing further evidence of an overrepresentation of crashes in rural areas [1] . Several factors contribute to the elevated rural crash risk, including speeds, geometry, lack of lighting, and other factors, each of which contribute to an elevated risk for lane departure crashes, including head - on, sideswipe, or run - off - road, which are among the most severe types. Begin ning with the Safe, Accountable, Flexible, Efficient Transportation Equity Act: A Legacy for Users (SAFETEA - LU) and continuing through the current transportation funding bill, states have been required to have in place a Highway Safety Improvement Program (HSIP) that - driven, strategic approach to improving highway safety on all public roads [2] . Given the prevailing focus on implementing roadway safety practices that are data - driv en, recent research has focused on gaining a more thorough understanding of how several factors affect the frequency, type, and severity of traffic crashes at specific roadway sites, such as horizontal curves and intersections. A valuable tool in this pro cess is the Highway Safety Manual ( HSM ) , published by the American Association of State Highway and Transportation Officials (AASHTO) [3] . Part C of the HSM provides a series of predictive models that can be utilized to estimat e the frequency of traffic crashes on specific road facilities as a function of traffic volumes, roadway geometry, type of traffic control, and other factors. These models, referred to as safety performance functions 2 (SPFs), are useful for estimating the s afety impacts of site - specific design alternatives or for prioritizing candidate locations for safety improvements on a network basis. As a part of this process, these SPFs can also be integrated with common decision support tools, such as Safety Analyst a nd the Interactive Highway Safety Design Model ( IHSDM ). The HSM includes separate families of SPFs to estimate annual crash occurrence for three specific roadway facility types: rural two - lane/two - way roads, rural multilane highways, and urban and suburba n arterials [3] . More recently, a supplement introduced SPFs for limited - access freeways [4] . Separate SPFs exist for intersections and road segments for the base conditions within each facility typ e, while crash modification factors (CMFs) are provided to account for deviations from the base condition of the facility type. Because the SPFs contained in the HSM were developed based on a limited sample of data collected from select states, specificall y , California, Minnesota, Texas , and Washington for the rural highway models, these functions must be calibrated or re - estimated using local data to improve their accuracy and precision [5 - 6] . A variety of states have conducted research to this end, includ ing Colorado, Florida [7] , Georgia, Illinois [8] , Kansas [9] , Michigan, North Carolina [10] , Oregon [11] , Utah, and Virginia [12] . Collectively, these studies have shown that the accuracy of the SPFs from the HSM vary considerably from state to state, a result that may be reflective of differences in geography, design practices, driver behavior, weather, crash reporting requirements, or other factors. 1.1 Problem and Knowledge Gap In many states, the majority of arterial highways in rural areas are under the jurisdiction of the state department of transportation (DOT). However, several states, including many in the Midwest and Great Lakes regions, possess a substantial rural county highway network. This 3 includes Michigan, where nearly 75 percent of the 120,000 miles of public roadways are owned by one of the 83 county road agencies across the state, with the remainder owned by the state (8 twork, it is not surprising that 60 percent of the 71,402 traffic crashes in rural areas in 2015 occurred on county - owned facilities [13] . Thus, the determination of factors affecting crashes on rural county highways , including both road segments and intersections, is important to support highway safety programs in Michigan and other states with substantial county road networks . Because the SPFs contained in the HSM were developed based on a limited sample of data collected fro m select states, direct application of the SPFs from the HSM does not tend to provide accurate results unless the models are calibrated using local data [ 5 - 12, 14] . Although the HSM provides details related to local calibration of the models, prior researc h estimating SPFs has shown that not only does the magnitude of the local curve differ from that published in the HS M , but the shape of the curves differs as well [15 - 16] . This further emphasizes the importance of developing SPFs utilizing local data, rather than simply calibrating the SPFs found within the HSM . Furthermore, the fact remains that the HSM based on data obtained from select state highways . T herefore, assumptions made on the general effect of characteristics, such as traffic volume or lane width, may not apply to low - volume, county - owned highways. In Michigan and elsewhere, county - owned highways typically have characteristics that differ considerably from those owned by the state DOT, which limits the usefulness of SPFs and CMFs generated based on data from rural state highways. Compared to state highways , the differences inherent to county roadways often include traffic characteris tics (e.g., lower traffic volumes, shorter trip lengths, greater driver familiarity, etc.), design characteristics (e.g., lower 4 design speeds, prevalen ce of unpaved/gravel surfaces, smaller curve radii, narrower lanes and shoulders, reduced sight distances , reduced clear zones, etc.), and maintenance characteristics (e.g., less aggressive snow removal, less frequent resurfacing, less frequent maintenance of traffic control devices, etc.) . Furthermore, it is also important to consider differences between th e various classes of county roadways, in particular, the distinction between those roadways that are eligible for federal funding (i.e., federal aid roadways) and those that are supported only by state and/or local funds (i.e., non - federal aid roadways). F ederal aid roadways are subject to d esign standards approved by the Federal Highway Administration (FHWA) , which ar e typically more stringent than those for non - federal aid roadways . Specifically, minimum design standards must be maintained in compliance w ith the posted speed limit for select controlling geometric elements on federal aid roadways with design speeds greater than or equal to 50 mph. These controlling geometric elements include d esign s peed, l ane w idth, sh oulder w idth, h orizontal c urve r adius, s uperelevatio n r ate, and s topping s ight d istance [17] . It is also important to note that the majority of unpaved roadways are non - federal aid. Thus , it is imperative that county roadway safety performance models account for the differences between federal aid and non - federal aid roadway designs, while also investigating differences in safety performance between these roadway types . Due to its importance as a primary controlling geometric criterion , horizontal curvature ha s been researched extensively in prior highway safety research . P revious research has found that the presence of a horizontal curve wi th a design speed at or below 55 mph on a rural highway segment contributed to 43 to 56 percent greater crash occurrence t han on segments without such curves [18 - 19] . However , these effects merely related to the presence of a horizontal curve on a segment , and do not describe the amount of curvature along the segment . 5 Furthermore, scant research exists related to the incremental effects of curve design speed on safety performance . Collectively, it is clear that further investigation is needed to provide a more comprehensive indication of the safety performance characteristics associated with horizontal curvature , in ad dition to the safety performance of other important geometric characteristics . While Michigan - specific SPFs have been previously developed , they are limited to urban and rural state - owned road segments and intersections [15 - 16] . Also, although HSM calibrat ion factors are available for Michigan county road segments and intersections, fully - specified SPFs utilizing local data have not been developed for county roadways . Furthermore, the Michigan - specific SPFs , along with those contained in the HSM , are only a pplicable to paved roads [3] , and additional research related to the safety performance of unpaved highways is also limited. Thus, there is a clear need for development of fully - specified safety performance models that are appl icable across all classes of rural county highways, including federal aid and non - federal aid roadways, while considering a broad range of geometric factors, paved and unpaved road segments , and three - leg and four - leg minor road stop - controlled intersections . 1.2 Research Objectives and Contributions The primary goal of this research was to develop a uniform, consistent approach that can be applied to estimate the safety performance of rural county road segments and intersections at the aggrega te (i.e., total crash) level . To attain this goal, a series of safety performance functions and crash modification factors were developed using data collected from across all classes of rural county roadways throughout Michigan. This included both paved an d unpaved roadways, federal aid and non - federal aid classifications, and covered both road segments and minor road stop - controlled intersections . These distinctions are important because, as previously stated, design standards are known to differ based on whether the roadway is subject to federal aid standards . 6 It was also important to consider the effects of geometric conditions on safety performance, in cluding substandard horizontal curvature , lane width, and shoulder width, as these are controlling geom etric design elements for which minimum standards must be achieved . Specifically, this research moved beyond simply considering curve presence on a segment, instead quantifying the proportion of each segment with horizontal curvature falling within a speci fic design speed range . Parameterizing the horizontal curve data in this manner also allowed for assessment of the incremental effects of curve design speed on safety performance . I ntersection skew was also an important factor to consider given the general association with crash occurrence and the relative frequency at which intersection skew occur s within the county roadway network. The study results will provide an important reference to guide states and local agencies toward making informed decisions as to planning and programming decisions for safety projects, and to provide researchers with guidance regarding future work within the realm of rural highway safety. T o achieve these aforementioned goals, t he specific objectives were as follows: 1. Review and summarize the extant literature related to SPF and CMF development and associated data collection for rural roadway segments and intersections. 2. Identify sites and collect data for the following rural segment and intersection types: a. Rural county two - lane two - way paved federal aid segments b. Rural county two - lane two - way paved non - federal aid segments c. Rural county unpaved non - federal aid segments d. Rural three - leg minor - road stop - controlled intersections e. Rural four - leg minor - road stop - controlled intersections 3. D evelop SPFs for each of the rural segment and intersection types listed above. 7 4. Develop CMFs for various design factors for each of the rural segment and intersection types listed above, including horizontal curves and intersection skew . Specifically, consi der the incremental effects of curve design speed and the curved proportion of segment on segment crash occurrence . 5. Investigate the relationship between deer crash occurrence and roadway characteristics . 6. Make comparisons and draw contrasts with existing h ighway safety research. To accomplish these objectives, county highway data, including traffic volumes, roadway characteristics, geometric characteristics, and traffic crashes were collected from across Michigan using both available datasets and manual data collection techniques . The data were subsequently analyzed utilizing mixed - effects negative binomial modeling techniques, with details provided in subsequent chapters . 1.3 Dissertation Structure This disserta tion document details the activities involved in the development of SPFs and CMFs for rural county road segments and minor road stop - controlled intersections in Michigan. The report is divided into seven chapters. Chapter 2 provides a summary of the state - of - the - art research literature. Chapter 3 describes the methods related to site selection and data collection, including details of the data sources and activities involved in database development for both rural county road segments and intersections , in a ddition to analytical methods for development of safety performance models . Chapter 4 provides the resulting SPFs and CMFs for rural, two - lane, two - way county roadway segments. Chapter 5 presents SPFs and CMFs for three - and four - leg minor road rural stop - controlled intersections along two - way two - lane county roadways. Chapter 6 presents SPFs for deer crashes along rural two - way two - lane highway segments. Conclusions and directions for future research are discussed in Chapter 7. 8 2. LITERATURE REVIEW Prior re search has explored the development of safety performance models for roadway segments and intersections and has estimated the effects of various traffic, roadway cross - sectional, geometric, and other characteristics on crashes and injuries on rural highway s . The following subsections summarize the existing research literature on these subjects. 2.1 The Highway Safety Manual S PFs are part of the core methods documented in the HSM , and the HSM incorporates many advanced analytical tools, such as the empirical Bayes (EB) method. SPFs constitute the basis for analysis in highway safety studies and key components of other types of safety analyses or evaluations. The main purpose of an SPF is to estimate the expecte d frequency of crashes given various traffic and site characteristics, such as traffic volume, segment length, and lane width . Transportation agencies and practitioners typically apply SPFs in their processes to select safety projects for funding. There ar e two general approaches described in the HSM to ensure that SPFs are appropriate to use for a particular jurisdiction: the agency or the safety analyst can either: (1) use a jurisdiction - specific SPF for the facility and crash types of interest, or (2) ca librate and use the corresponding SPF available from the HSM [3] . As defined in the HSM , an SPF has three components: (1) a base SPF, (2) CMFs and (3) a calibration factor, C. as shown in Equation 1. (1) Where , = predicted annual average crash frequency ; = predicted average crash frequency under base conditions ; = calibration factor to adjust SPF for local conditions ; and = the product of the set of applicable CMFs. 9 2.1.1 Base SPF A base SPF is a crash prediction model for a facility type that accounts for exposure to traffic flow as the only independent variable. All other variables of relevance (e.g., speed limit, number of lanes, shoulder information, etc.) are not explicitly accounted for in the base SPF because it implies a fixed value for each of these variables (i.e., they are fixed at the base conditions of the SPF). It has been argued that placing an excessive number of independent variables in the base SPF would potentially tangle the ef fects of certain variables with others [20] . The set of fixed values is referred to as the base conditions of the base SPF. These conditions may include such variables as 12 - f oo t lanes and 6 - f oo t shoulders for rural segments or no left - turn lanes for intersections. Of particular interest to this research, the generic base models for intersection SPFs (for rural or urban facilities) found in the HSM have the functional form shown in Equation 2. (2) Where, = predicted average crash frequency at base conditions, = annual average daily traffic (AADT) for the major road, = AADT for the minor road, and = est imated parameters. The base models for segment SPFs (for rural or urban facilities) found in the HSM usually have the functional form shown in Equation 3: (3) Where, = predicted average crash frequency at base conditi ons, = AADT on the segment, = segment length in miles, and = estimated parameters. Care needs to be taken when adding variables to avoid overfitting the SPF. The more complex models are often poorer predictors, only accurately predicting crashes on the segments 10 that were used to estimate its parameters, as statistical noise tends to be incorrectly included as systematic variation in crashes. To avoid this pitfall, researchers [21] suggested using backward elimination in the well - documented stepwise model selection process in statistical analysis [22] . This method identifies significant variables by a stepwise regression approach, including all variables, then eliminating each separately, to determine if each variable significantly degrades the information given by the model. 2.1.2 Crash Modification Factors The purpose of CMFs is to account for deviations from base conditions for variables known to have an impact on crash frequency, such as geometric or traffic control features. For example, if the base condition for an intersection SPF is adjacent approaches with no skew, applying this SPF to a location with one approach with a significantly skewed angle will require the application of the corresponding CMF. A CMF value above one indicates that the number of crashes is expected to increase, while a value below one means that the number of crashes is expected to go down. It is important that the applicatio n of CMFs for countermeasures be separated from the application of CMFs to adjust for base conditions. The CMFs applied to these models allow for crash estimates that distinguish between sites with various geometric or traffic control features. The HSM war ns that only the CMFs presented in Chapters 10 and 11 apply to the respective Part C predictive method as adjustments to base conditions for that facility type. Other CMFs are found in Part D, Chapter 13 for roadway segments and Chapter 14 for intersection s, and are applicable in estimating the impact of various safety countermeasures. In such cases, the expected average crash frequency of a proposed project or a project design alternative can be evaluated. 11 Chapters 10 and 11, Part C of the HSM present a se t of CMFs for rural segments (two - lane and multilane) and rural intersections. Additional CMFs can also be found in FHWA CMF Clearinghouse [23] . The CMF Clearinghouse is a web - based database of CMFs that provides supporting d ocumentation to assist users in estimating the impacts of various safety countermeasures. All CMFs are developed with an assumption that all other conditions and site characteristics remain constant, aside from the condition being represented in the CMF. F or this reason, the validity of CMFs is reliant on consistent and agreeable base conditions. The HSM documents base conditions for each of the rural segment and intersection facility types for which SPFs are developed in Chapters 10 and 11. CMFs are mainly developed from before - after and cross - sectional studies [24] . Although it is common practice to estimate the combined effect of multiple CMFs by multiplying the individual CMFs together, this practice relies on the assumption o f independence between CMFs. However, that assumption is not necessarily true in every case, and the result could be a significant overestimation or underestimation of the combined effect [25] . 2.1.3 Calibration Factors To take advantage of the value of the multiple SPFs presented in the HSM , such SPFs can be calibrated to local conditions. C alibration intends to account for the variation of crash data between different jurisdictions and for factors that were not involved i n the model. O n a project level, the development of a typical SPF can take 450 - 1,050 staff - hours, whereas calibration requires only 24 - 40 staff - hours for data collection and preparation [21] . When using an already - existing SPF taken from part C of the HSM or Safety Analyst , calibration is essential because crash frequencies fluctuate for a variety of reasons that cannot be accounted for when developing the SPF, such as climate, criteria for reporting crashes , topogr aphy, animal population, law 12 enforcement practices , vehicle characteristics, and other factors that differ between jurisdictions [21, 26 - 30] . The calibration factor is estimated using Equation 4 and is applied to the base SPF as a multiplicative scaling f actor. (4) Where, = the observed annual average crash frequency, = predicted annual average crash frequency, and n = sample size, equal to the number of sites in the calibration process. Simi larly, calibration is recommended when applying an SPF to a new jurisdiction, but a calibration between different time periods is also recommended [27, 31] . When translating SPFs across states, calibration factors are a given, but major physiographic divis ion within a state should also be considered [32] . The HSM recommends calibrating the models using data from 30 - 50 locations, which collectively possess at least 100 crashes per year. However, recent research has shown that this number of sites is insufficient for most cases [33 - 34] . Several research studies have provided further or improved guidelines to calibrate the models for local conditions [27, 35] . Considering the caveats of the calibration procedure, it is preferable to develop new predictive models if enough data are available. The use of calibration fac tors provides a standardized model to be calibrated for different jurisdictions and road conditions [36] . Calibration factors for the HSM models have been developed for rural intersections and segments in several states. The first two sections below describe studies that attempted to calibrate HSM models for rural intersections and segments. The last section covers general issues related to the calibration procedure. 13 2.1.3.1 Rural Intersections Table 1 shows the value of the calibration factor for different rural intersection models (or facilities) in Oregon [31, 37] , Florida [38] , North Carolina [39] , Maryland [40] and Misso uri [41] . As shown in Table 1 , the value of the calibration factor tends to be smaller than one, which indicates that the pre - fitted HSM models tend to overestimate the number of crashes for different types of rural inter sections for most cases documented in this table. The calibration effort in Oregon [31, 37] showed that obtaining the minor AADT flows for rural intersections is a difficult task, as these values are rarely available. To overcome this difficulty, in a more recent effort, researchers developed an AADT estimation model for minor approaches [42] . The model included land - use and demographic variables as well as the characteristics of the main highway to which the minor approach int ersects. Table 1 : Rural Intersection Calibration Factors Facility Calibration f actor Oregon Maryland Florida a N orth Carolina b Missouri Rural t wo - l ane 3 - leg, minor stop 0.31 0.16 0.8 0.57 0.77 4 - leg, minor stop 0.31 0.2 0.8 0.68 0.49 4 - leg, signalized 0.45 0.26 1.21 1.04 - Rural m ulti - l ane 3 - leg, minor stop 0.15 0.18 na na 0.28 4 - leg, minor stop 0.39 0.37 na na 0.39 4 - leg, signalized 0.15 0.11 0.37 0.49 na a For this s t ate , several yearly calibration factors were derived from 2005 to 2009. Value s derived in 2009 are reported . b Both one - and three - year period calibration factors were derived for this state. Table shows three - year factor only. Note: na = not available Calibration factors have been derived for several other types of facilities (e.g., urban intersections and segment models) in Oregon, Florida, North Carolina, Maryland , and Missouri, as well. However, this document focuses on calibration efforts documented for rural segments and intersections only. Several other states such as Utah [43] , Illinois [44] , and Alabama [45] 14 have also performed local calibration of the HSM SPFs, although rural intersections were not included in the local calibration. 2.1.3.2 Rural Segments Researchers in Kansas calibrated base model s developed using both the HSM procedure and new procedures that address specific qualities of the highway system [46] . Later, other researchers presented a revised method to develop calibration factors for five types of urban and suburban roadways with consideration of recent changes to the crash recording threshold (CRT) for property damage crashes, which occurred in Illinois in 2009 [47] . The study established a revised method to supplement and adopt a standard approach to develop calibration factors in the HSM , considering impact of the new CRT. The higher the CRT, the fewer recorded PDO crashes. Before and after the threshold change, calibration factors for four lane divided facilities were 0.68 and 0.55 respectively. Table 2 shows the value of the calibration factor for different rural segment models (or facilities) in North Carolina [39] , Oregon [37] , Florida [38] , and Illinois [44] . All the calibration factors are for all (i.e., KABCO ) crashes. Table 2 shows the value of the calibration factor varies greatly for different states, from a low of 0.36 to more than 4.0. Table 2 : Rural Segment Calibration Facto rs Facility Calibration f actor North Carolina Oregon Florida Illinois Two - lane undivided (2U) 4.04 0.74 1.05 1.58 Four - lane undivided ( 4U ) na 0.36 a na na Four - lane divided ( 4D ) na 0.78 a 0.70 a na a Referred as multilane rural highways (includes a limited number of 6 - lane segments). Note: na = not available 15 2.1.3.3 General Calibration Issues Although states usually develop one single calibration factor for the whole state, recent research on urban i ntersections in Michigan [48] showed that the value of the calibration factor could be significantly different in different regions of Michigan. To overcome this issue, the authors estimated several region - specific calibr ation factors. I n the safety literature in general, and the HSM in particular, calibration is presented as a tool to incorporate local conditions of the current jurisdiction into a model that was fitted (or developed) for another jurisdiction. However, al though calibrating the models through a scalar factor seems adequate for the overall fit of the model, there is no guarantee that same results will be achieved, even when each variable is analyzed independently (such as AADT), or by group of variables [49] . Furthermore, the application of a single scalar factor was found to be biased compared to the recently introduced Bayesian model averaging (BMA) method. T his limitation was investigated using the BMA method by carefully evaluating a series of locally developed and calibrated models [50] . Cumulative residuals (CURE) plots are often used to verify goodness of fit for the AADT variable [51] . Results from thi s study show that the bias from calibrated models is substantially larger than the BMA models. 2.1.3.4 Calibration Factors for County Roads In 2018, safety performance functions were developed for rural highways in Michigan as a part of the Michigan Depart ment of Transportation (MDOT) research program . A ddition ally, the HSM s were also calibrated using the methodology contained in the HSM . The resulting calibration factors, specific to each MDOT geographic region, are shown in Table 3 for rural sta te - owned two - lane two - way segments, Table 4 for county two - lane two - way segments, and Table 5 for rural two - lane two - way stop - controlled intersections. 16 For rural segments, calibration factors were developed for MDOT - owned (i.e., trunkline) highways, as well as county federal aid (FA) and county non - federal aid (non - FA) roadways. For intersections, calibration factors were developed for three - leg (3ST) and four - leg (4ST) rural stop - controlled intersections, and the factors presented were developed from a sample that included intersections were the major road was trunkline, county FA, and county non - FA. Calibration factors were not developed in some cases . For example, there are no rural state - owned segments in the Metro region, and so calibration was impo ssible. In other cases, calibration factors could not be developed because there were not at least 100 crashes per year on that particular class of roadway in a given region; for instance, in the Superior region, there were only 14 crashes in the county no n - federal aid sample. Table 3 . Calibration Factors for HSM Models on MDOT Rural Trunkline Segments [16] Region Count of s egments Segment m ileage N observed (Midblock c rashes 2011 - 2015) N observed (Midblock n on - d eer c rashes 2011 - 2015) N predicted ( HSM 2011 - 2015) Calibration f actor for m idblock c rashes on t angent s ections Calibration f actor for m idblock n on - d eer c rashes on t angent s ections Statewide 946 3,003 39,925 11,861 18,491 2.16 0.64 Superior 185 658 5,161 1,304 2,192 2.35 0.59 North 210 705 8,771 2,381 3,768 2.33 0.63 Grand 161 458 7,757 2,522 3,641 2.13 0.69 Bay 204 677 11,122 3,105 4,948 2.25 0.63 Southwest 99 236 3,267 1,254 1,864 1.75 0.67 University 87 269 3,847 1,295 2,078 1.85 0.62 Metro 0 0 0 0 0 Not a pplicable 17 Table 4 . Calibration Factors for HSM Models on Michigan Rural County Road Segments [16] Region Count of s egments Segment m ileage N observed (Midblock c rashes 2011 - 2015) N observed (Midblock n on - d eer c rashes 2011 - 2015) N predicted ( HSM 2011 - 2015) Calibration f actor for m idblock c rashes on t angent s ections Calibration f actor for m idblock n on - deer c rashes on t angent s ections County FA Statewide 8,318 3,558 27,661 9,858 13,078 2.12 0.75 Superior 634 303 991 304 342 2.9 0.89 North 1,496 636 4,007 1,343 1,676 2.39 0.8 Grand 2,032 845 7,103 2,586 2,704 2.63 0.96 Bay 1,085 465 3,942 1,087 1,736 2.27 0.63 Southwest 332 159 1,335 561 810 1.65 0.69 University 2,403 1,033 8,701 3,241 4,649 1.87 0.7 Metro 336 118 1,582 736 1,162 1.36 0.63 County n on - FA Statewide 2545 1293.7 3658 1330 1707 2.14 0.78 Superior 15 6.2 14 13 4 Not a pplicable North 203 76.1 198 64 120 1.65 0.53 Grand 418 212 522 239 283 1.85 0.85 Bay 321 139.4 529 190 343 1.54 0.55 Southwest 513 270.6 565 254 273 2.07 0.93 University 1061 582.7 1816 564 678 2.68 0.83 Metro 14 6.8 14 6 6 Not a pplicable Unpaved Statewide 3,054 1,436 1,474 902 541 2.73 1.67 Superior 2 3 3 2 0 Not a pplicable North 120 46 23 14 14 1.64 1 Grand 268 132 110 76 32 3.41 2.36 Bay 156 72 92 33 28 3.32 1.19 Southwest 135 67 30 17 13 2.34 1.33 University 2,056 939 965 569 349 2.76 1.63 Metro 317 177 251 191 104 2.4 1.83 18 Table 5 . Calibration Factors for HSM Models at Michigan Rural Intersections [16] Region Count of i ntersections N observed (Intersection c rashes 2011 - 2015) N predicted ( HSM 2011 - 2015) Calibration f actor for i ntersection c rashes 4ST Statewide 2,513 9,853 14,010 0.7 Superior 198 562 671 0.84 North 360 1,301 1,878 0.69 Grand 521 2,197 3,235 0.68 Bay 516 2,390 3,521 0.68 Southwest 278 1,212 1,682 0.72 University 583 1,988 2,783 0.71 Metro 57 203 239 0.85 3ST Statewide 2,297 5,395 6,376 0.85 Superior 287 583 498 1.17 North 381 1,107 1,248 0.89 Grand 388 1,030 1,182 0.87 Bay 229 691 913 0.76 Southwest 381 780 1,005 0.78 University 564 1,056 1,357 0.78 Metro 67 148 173 0.85 Upon review of the calibration factors for the various HSM models, it is evident that the accuracy of the base SPFs from the HSM for prediction of crashes in Michigan vary widely by roadway classification . These differences are reflective of several factor s, including state - specific differences (e.g., driver characteristics, road design standards, weather, etc.). The most prominent state - specific characteristic is the overabundance of animal crashes attributed to the high deer population in Michigan. Genera lly, the HSM models tend to under - predict total mid - block segment crashes, but over - predict deer - excluded mid - block crashes, although consideration must be given to the fact that a certain (albeit much lower) percentage of the HSM crash data involved anima ls. The HSM models generally tend to over - predict crashes at stop - controlled intersections. As with segments, these differences are reflective of several factors, including state - specific differences (e.g., driver characteristics, road design standards, w eather, etc.) and unobserved 19 heterogeneity between sites (e.g., vertical curvature, roadside hazard rating, etc . ). Some of these differences between the segment and intersection calibration factors may be the consequence of the method used in this study fo r distinguishing between segment and intersection crashes. These differences suggest that the accuracy of crash estimation will be improved through the development of Michigan specific SPFs . 2.2 Rural Highway Segment Safety Performance Characteristics A review of the existing research of the safety performance of rural two - lane, two - way highway segments are presented in the following subsections . 2.2.1 Lane Width Wider travel lanes on two - lane highways have been associated with reductions in single - vehicle run - off - the - road, head - on, and sideswipe type crashes [3, 52] , and the effect is most pronounced for two - lane roadways when comparing wider lines with lane widths of 9 feet or less. A case - control study revealed several interesting relationships between lane and paved surface width and crashes. The study found that increasing total pavement width was associated with a reduction in crashes; however, when evaluating the effect of different lane and shoulder widths on segments of equivalent total paved surface width, the results were less clear. The general trend, however, favored increased lane widths relative to shoulder widths [53] . Looking more specifically at minor arterials and major collectors , roadway functional classifications that are common on county highways , a study found that lane width is much more significant a factor in crash reduction t han shoulder width ; this is in contrast with principal arterials, where the study found shoulder width to be a stronger factor in crash reduction [54] . Another study found that the effect of increased lane width and reduced cra sh frequency is more pronounced on higher - volume roads [55] . 20 Not all research has found that wider lanes are less crash - prone. A recent study in rural Pennsylvania found a lower occurrence of total crashes and fatal and injury crashes at locations with narrower lane widths relative to wider lanes [19] . A nother study found that narrower lane widths were associated with reductions in same - direction crashes, and fatal and incapacitating injury crashes, but an increase in single - vehicle crashes as well as total crashes and non - incapacitating injury and property damage only (BCO) crashes [56] . Another study found that 12 - foot lanes, in particular, are the most crash prone, with lane widths both greater than and less than 12 feet showing lower crash frequency; the author noted that there were confounding factors involved, however, such as a relat ionship between lane width and speed limit [57] . Prior research in Michigan has also found a relationship between lane width and crash reduction. On rural, county - owned federal aid highways, highway segments with lane widths gr eater than 12 feet were fou nd to experience 26.3 percent fewer non - deer fatal and injury crashes relative to baseline conditions of less than 11 - foot lane widths, although lane width was not found to be significant for total non - deer crashes [18] . Other research on Michigan county - owned federal aid highways found that wider lanes are associated with a lower probability of high - severity crashes ; the same study found that traveled - way width on county non - federal aid highways is as sociated with reductions in both fatal and injury and property damage only crashes [58] . 2.2.2 Shoulder Width While the effect of lane width on crashes on this type of road segment is mixed, research has consistently found tha t wider shoulders on rural highways are associated with fewer crashes [3, 16, 52, 56] , due to the increased recovery and vehicle storage space and increased separation from roadside hazards. While the size of the effect depends on traffic volumes, the freq uency of 21 traffic crashes tends to increase as paved shoulder widths are reduced below 6 feet. Further, safety performance tends to degrade substantially as the paved shoulder width decreases below 2 feet on roadways with greater than 2,000 vehicles per day [3] . A study using data from Pennsylvania, using both case - control and cohort approaches, found that shoulder widths below 6 feet are associated with increases in crashes, while shoulder widths greater than 7 feet are associated with decreases in crashes [59] . Another study found that crashes decrease with shoulder widths of 9 feet or greater when using a case - control approach and 8 feet or greater when using a cross - sectional approach [60] . Another study found that increases in shoulder width were associated with crash decreases for interstate highways only; for state highways, there was a negative relationship, but it was not statistically significant [55] . Prior research in Michigan has found that increases in shoulder width on county paved federal aid highways are associated with a reduction in fatal and injury non - deer crashes and that wider shoulders are associated with less seve re crashes [58] . 2.2.3 Access Points Several prior studies have showed that increasing access point density leads to an increase in crash occurrence, particularly for multi - vehicle crashes [3, 61 - 62] . This is at least partiall y due to driving errors caused by intersections and/or driveways, which may result in rear - end and/or sideswipe type crashes [3] . Specifically, the NCHRP (National Cooperative Highway Research Program) Report 420 concluded that increasing the access point density from 10 to 20 per mile led to a 40 percent increase in crashes, while increasing access points to 40 per mile was associated with a doubling of crash occurrence [62] . Research from arterial roads in Oregon has found that driveway clusters are associated with higher crash frequency than isolated driveways [63] . 22 Research in Michigan has found that increasing driveway density is associated wi th increases in crashes; the highest increase was found to be for commercial driveways, while the increase in crashes was lower for industrial and residential driveways. Notably, while total crashes were similar for residential and industrial driveways, in dustrial driveways had higher rates of fatal and injury crashes than residential driveways [64] . Another study, looking at total driveway density on county federal - aid highways, found increases in crashes when driveway density was higher than 5 driveways per mile, with the greatest increases when driveway density was 15 driveways per mile or greater, although this was only significant for total non - deer crashes; it was not significant for fatal and injury crashes. Driveway densi ty was not significant with respect to crashes on non - federal aid highways [18] . 2.2.4 Alignment Horizontal curv ature is among the most critical geometric design elements related to the influence of driver behavior and crash r isk [65] . In fact, early research showed that the most significant factors in predicting crashes are degree of curve and average daily traffic ( ADT ) [66] . E arly research found that, in addition to curve flattening, widening lanes and shoulders at horizontal curves results in crash reduction ; it was less clear to what extent crashes are reduced by correcting superelevation [67] . Similar to wider shoulders and lanes being associated with reduced crashes along horizontal curves, increased sight distance, in general, is associated with crash reduction along horizontal curves [68] . Increased shoulder width is also associated with fewer crashes alon g horizontal curves involving motorcyclists, in particular [69] . One study evaluating motorcycle crashes along horizontal curves found that better pavement conditions may cause increases in crashes, suggesting users adjust beh avior to perceived risk [70] . In other behavioral factors, a naturalistic study found that driver distraction 23 plays a large role in horizontal curve crashes, with distracted drivers being three times more likely to crash than t hose who are not distracted [71] . When there is a platoon of vehicles at a [72] . Looking at curve radius, speci fically , horizontal curves with radii less than 2,600 feet tend to cause a reduction in highway running speeds below that of adjacent tangent sections, with substantial speed declines seen for curves with radii less than 800 feet [7 3] . It is generally understood that crash occurrence tends to increase as the degree of curvature and/or length of curvature increases along a rural highway segment [8, 14 - 15] . On two - lane rural highways, horizontal curves increase crash risk, particularly if operating speeds through the curve are reduced by more than 3 mph from the adjacent tangent section [6] . Any reduction in speed is associated with crash increase s , and this effect is higher with increased speed reduction (relative to speeds along tangent section) [74] . A recent analysis of state - owned rural two - lane roads in Pennsylvania found 43 percent more total c rashes and 48 percent more fatal and injury crashes at locations with curve radii less than 1 , 008 f ee t, which is the approximate radius of a curve designed for 55 mph with a superelevation of 6 percent [19] . Prior research has also showed that steeper vertical grades are associated with higher crash rates [8, 10] , especially when combined with a horizontal curve [75 - 76] . Total crash rates generally increase with the degree of vertical curva ture [8] , particularly where hidden horizontal curves, intersections, or driveways are present [65] . 2.2.5 Pavement Surface Rural unpaved roads include a wide variety of design standards, design sp eeds, and surface characteristics, which can be greatly affected by the effects of weather and heavy traffic loads. The safety of these roads may also be affected by a lack of pavement markings and insufficient 24 signage, narrow road widths, and the absence of shoulders. A limited amount of research has investigated the safety effects of paved versus unpaved surfaces for low volume roadways. Differing design standards between primary and local roadways make it difficult to compare safety performance between p aved and unpaved roadways without constraining such analyses to roadways with lower traffic volumes and lower functional classes. Nevertheless, research has found that at the lowest of volumes (e.g., less than 250 vehicles per day), little to no difference in crash occurrence between paved and unpaved roads is seen. However, at higher volumes, paved roads were found to have lower crash occurrence than unpaved roads [77] . 2.2.6 Deer - Vehicle Crashes There is currently limited conc lusive evidence regarding roadway factors or countermeasures that influence deer - vehicle crashes ( DVCs ) , which is largely due to difficulties in obtaining accurate data on deer populations and roadway crossing frequency. Many strategies to mitigate or prevent DVCs have not proven to be effective, including reflective lighting to frighten deer [78] and increased mowing frequency to reduce the roadside cover for deer [79 - 80] . The size of the deer harvest was also not found to have an impact on deer - vehicle crash rates, suggesting that hunting may not be an effective crash reduction strategy. Animal crossing warning signs have shown some evidence of reducing animal - vehicle collisions, although these findings were only supported by crash counts without accounting for differences in mileage or traffic volume between locations with sign s versus locations without signs. However, the number of signs per segment was considered [81] . A study conducted in Iowa, using deer - vehicle crash data as well as deer carcass salvage data, found that DVCs in urban areas incr eased when the speed limit was 50 mph or higher, when the adjacent land cover was grassland, and when the right shoulder was a gravel shoulder 25 (as opposed to a paved shoulder). Furthermore, deer crashes were found to be less common on two - lane roads than o n multilane [82] . There has also been research evaluating the use of odor repellant to deter deer from roadways. In Czechia, researchers, using animal carcass and crash data with a Bayesian analysis approach, found that the use of odor repellants could reduce these types of crashes by 26 to 46 percent in locations where these crashes are most common. Odor repellant was applied to wooden poles 80 cm (2.6 f ee t) tall, placed 10 m (32.8 f ee t) apart, and replenished every 3 months [83] . However, a study conducted in Ontario found that using various odor - based repellants did not have an impact on which trails wildlife chose to travel along [84] . Other studies have shown that that, while odor repellants may be effective in reducing DVCs in the short term, wildlife become habituated and therefore the treatments lose effectiveness over time [85 - 86] . A primary issue with deer crash mitigation strategies is identification of primar y deer crossing areas for installation of the treatments. Research has showed that animal crossing events can be detected with over 90 percent accuracy using a buried sensing cable along the roadside [87] . A detection system su ch as this could provide researchers with data about animal crossing locations to determine the proper locations for mitigation strategies and could also serve as an activation trigger for certain countermeasures, such as active warning devices. Other rese arch has found that roadkill data can be used to find potential hot spots [88] . 2.3 Rural Intersection Safety Performance Characteristics Prior research has explored the safety performance of rural intersections. The following paragraphs summarize the existing research literature on safety performance modeling for rural intersections, including the analytical methods specified in the HSM . Among the several types of statistical models used for SPF development, generalized linear models and negative binomial 26 models yield easy - to - interpret results and associate crash frequencies to sets of designated explanatory variables [89 - 90] . Negative binomial models are commonly used for SPFs development and have been used extensively in prior studies , including the HSM [8, 91 - 93] . Recent rural intersection SPF development in Oregon revealed the typical challenges associated with small crash sample sizes for rural intersections, as only 165 crashes occurred during a three - year period at 115 ru ral three - leg stop - controlled intersections, which represented a rate of 0.48 crashes per intersection per year. It was concluded that the lack of data and the significant costs of data collection were two major difficulties [94] . While it is widely understood that intersection crashes have a non - linear relationship with the traffic volume entering a rural stop - controlled intersection, several studies have investigated site characteristics that affect crash occurrence at both r ural three - and four - leg intersections. The effect of intersection lighting has been investigated extensively. For rural four - leg stop - controlled intersections with lighting, the HSM provides a CMF of 0.91 relative to the base condition of no lighting pres ent [3] . Research in Minnesota and California found that illuminated intersections are associated with a reduction in nighttime crash frequency of 3.6 percent and 6.5 percent, respectively [95] . Int ersection sight distance and intersection alignment have also shown to have a substantial influence on the safety of rural intersections [93] . 2.3.1 Turn Lane Presence T urn lanes generally are associated with reductions in cr ashes, relative to intersections without turn lanes, with higher crash reductions on intersections with large proportions of vehicles making turning movements. Furthermore, when additional through lanes are introduced, crash frequency tends to decrease [96] . However, right - turn lanes at three - leg intersections may increase crash likelihood, while a decrease in crashes was found when there are right - turn lanes 27 on four - legged intersections . I t was acknowledged that the presence of turn lanes is correlated with higher proportions of turning movements [93] . Another study found that right - turn lanes are associated with increased crash frequency and that left - turn lanes did not significantly re duce crashes , although this is confounded by the fact that intersections with left turn lanes are correlated with higher proportions of left - turning vehicles [97] . L eft - turn movements are associated with angle crashes [98] , which can be quite severe. One difficulty in attributing a crash effect associated with turn lane presence is that, while certain crash types ( e.g., a ngle) may increase, others (e.g. , rear - end) may decrease [56] . 2.3.2 Access Point Frequency Access point frequency also has an effect in crash frequency, with higher numbers of access points leading to increasing numbers of crashes. This is due to the increase in conflict points and the potential for veh icles turning into or out of these driveways to interfere with the [96] . Driveway density at rural intersections is particularly associated with property damage only crashes ; authors noted that, in addition to the increase in conflict points, drivers may focus attention on vehicles at driveways rather than the traffic ahead of them [99] . High driveway frequency at intersections also leads to unexp ected braking, which can cause following vehicles to rear - end the turning vehicle [97] . Commercial driveways, in particular, are prone to crashes [100] ; this is not surprising due to the high intens ity of turning movements in these locations. 2.3.3 Other Geometric Factors Other geometric factors that influence crash frequency in rural intersections include shoulder width, where increases in shoulder width are associated with crash reduction. Medians are also associated with fewer crashes when they are wider than 16 feet; when turning lanes are present , 28 medians of 5 feet or greater are associated with crash reduction. Increases in intersection skew angle are associated with increases in crash frequency [96] . Looking at skew angle, in particular, research has found that when skew angle exceeds 10 percent, susceptibility to crashes increases; this is particularly so when there is also horizontal curvature involved [101] . A study using a continuous variable for skew angle found it to be significantly positively correlated with crash frequency, with 60 - degree skew angles showing crashes increase by a factor of 1.2 [102] . A study from Ohio found that, on rural four - leg two - lane intersections, intersection angles between 60 and 55 degrees (i.e., skew angles between 30 and 35 degrees) had the highest increase in crash frequency, while the most extreme intersection angles (i. e., 20 degrees and below, corresponding with a skew angle of 70 degrees or greater) actually showed decreases in crash frequency [103] . In one study, increases in skew angle were associated with increases in fatal and injury cr ash frequency for rural two - lane two - way four - leg intersections , but was not a significant factor on the corresponding three - leg intersections [104] . 2.3.4 Traffic - Related Factors In terms of non - geometric factors, proportions of heavy vehicles or trucks during the peak hour can influence crash frequency; when trucks make up greater than 15 percent of peak hour traffic, crashes are reduced [96] . 29 3. METHODOLOGY The sections below describe the process w hereby safety performance functions for rural segments and intersections in Michigan were developed, including the data collection process, as well as the analytical method. 3.1 Data Collection To provide a better understanding of the relationship between various roadway characteristics and safety performance on rural roadways and intersections in Michigan, it was first necessary to assemble a comprehensive database of traffic crash and roadway d ata obtained for a sample of rural roadway segments and intersections across all regions of Michigan. These data were obtained from a variety of sources for the five - year period of 2011 through 2015. Details on the identification of county highway segments and collection of the relevant data are provided in the sections that follow. The correct calibration of SPFs largely depends on the quality of the data from which they are developed. SPF development requires a crash database that is comprehensive and in cludes information on specific crash location, collision type, severity, and whether the crash occurred on a segment or at an intersection, among other factors. In addition to crash data, roadway data are also collected and serve as predictor variables in the SPF models. Such factors typically relate to traffic volumes, geometry, or physical features within the right - of - way of the roadway. As a part of this study, the data wer e sought out and assembled for rural roadway segments and rural intersections fro m a diverse array of sources, including state and local agencies. Available geospatial datasets were used whenever possible, although some 30 characteristics required manual collection using satellite or street - level imagery. The aim of the data collection ta sk was to quantify relevant roadway characteristics and assemble comprehensive databases for use in SPF development for the following types of rural roadway segments and rural intersections (examples of each are displayed in Figure 1 ): a) Rural county two - la ne two - way paved federal aid segments b) Rural county two - lane two - way paved non - federal aid segments c) Rural county unpaved non - federal aid segments d) Rural three - leg minor - road stop - controlled intersections e) Rural four - leg minor - road stop - controlled intersection s County 2 - lane paved fed - aid segment County 2 - lane paved non - fed aid segment County 2 - lane unpaved segment 3 - leg minor road stop - controlled intersection 4 - leg minor road stop - controlled intersection Figure 1 : Rural f acility t ypes for Michigan SPF d evelopment 31 Data were initially collected for each of the five rural facility types from existing data sources that were available either publicly or through direct contact with MDOT. These data sources included the following databases and files Annual statewide crash database obtained from the MDOT Crash Reporting Information System (CRIS); All Roads shapefile and other relevant shapefiles based on the Michigan Geographic Framework obtained from the Michigan Geographic Data L ibrary ; Census boundary shapefiles; and . Google Earth satellite imagery was used to manually collect other data for SPF development that was not otherwise included in the existing data sets. Further details of each respective data source are provided in the following sections of this document . Up on completion of the data collection, the volume, crash, and roadway inventory data were then merged into a comprehensive dataset for each of the various roadway and intersection classes included in this analysis. 3.1.1 Roadway Segmentation using the Michi gan Geographic Framework The Michigan Geographic Framework All Roads (MGF - AR) shapefile provided the spatial basis for collection of the necessary roadway and traffic related attributes for segments and intersections. The MGF represents a digital base map for the state, consisting of all public road segments, in addition to urban boundaries, census boundaries, jurisdictional ownership, and other geographic characteristics. All roadway data collected for this study was spatially referenced based on the roadw ay linear referencing system (LRS) used in the Michigan Geographic 32 Framewor k . Updates to the framework occur annually, and version 16a, which uses 2015 data, was used in this study. The MGF - LRS subdivides the public roadway network into a series of segmen ts b ased on physical road (PR) number and begin/end mile points . The LRS allow s for data from different sources (e.g., crashes, traffic volume , other roadway characteristics) to be uniquely and independently matched to the network based on their relative roadway position. Segment begin/end mile points within the MGF - AR are based on a change in one or more primary characteristics, including pavement surface, annual average daily traffic (AADT), major junction, jurisdictional boundary, and numerous other features. Thus, the PR and mile points from the MGF - AR file effectively partition each roadway into unique homogeneous segments, which provided the roadway segmentation basis for data collection performed during this study. The roadway jurisdictional class (e.g., MDOT, county federal aid, county non - federal aid) was identified using the MGF framework classification code (FCC). U.S. Census boundaries were used to isolate rural segme nts and intersections for use in this study. Rural areas are typically defined as locations that fall outside of urban boundaries with populations greater than 5,000. However, this research sought to isolate high - speed sections of county highways where the statutory rural speed limit of 55 mph would apply. Thus, road segments falling inside any incorporated census area boundary were excluded from this sample, including small cities and villages with populations of less than 5,000 and unincorporated census d esignated places. This step was important, as speed limit signs are not required on roadways utilizing the statutory speed limit, making speed limit verification difficult. The segments and intersections were initially screened using the U.S. Census design ations found in the MGF All Roads shapefile . Only those road segments and intersections 33 falling outside of each of the following boundaries were considered rural and carried forward for further analysis A djusted census urban boundary (ACUB) minimum populat ion of 5,000 ; Urbanized area , as d esignated by the U.S. Census ; or C orporate limits of any incorporated city or village designated as partially urban by the Census. To further distinguish between rural areas and unincorporated rural communities, a shapef ile of census - designated places (CDPs) was obtained and integrated with the A ll R oads shapefile in ArcGIS. CDPs are defined as a concentration of population named by the Census Bureau for statistical purposes, exclusive of incorporated cities, towns, and v illages. For a list of CDPs and incorporated areas in Michigan, please refer to the Michigan census block maps kept by the U.S. Census Bureau . 3.1.2 Traffic Volume Data AADT volumes were obtained from two primary sources for use in this analysis . The volu me data source was dependent on the roadway federal aid classification, which are further described below. County federal aid roadway AADTs were obtained from the MDOT - maintained GIS (geographic information systems) shapefile for statewide non - trunkline fe deral aid (NTFA) roadways, entitled NTFA_Segment.shp. AADTs were obtained for either the year 2014 or 2015 for nearly the entire population of rural federal aid county roadways across all 83 counties statewide. County non - federal aid ( n on - FA) roadways AADT s, including rural collectors and local roadways, were obtained directly from the county road commission (typically from the Roadsoft 34 asset management system used by transportation agencies in Michigan) or the corresponding regional planning commission, wh ere available. Volume data for rural non - federal aid county roadways were ultimately obtained for 27 counties across all portions of the state, including: Arenac, Baraga, Barry, Charlevoix, Clinton, Dickinson, Eaton, Genesee, Grand Traverse, Gratiot, Ingha m, Iosco, Kalamazoo, Kent, Livingston, Luce, Macomb, Marquette, Mason, Mecosta, Muskegon, Oakland, Ogemaw, Roscommon, Schoolcraft, Washtenaw, and Wayne c ounties. Because the AADTs for non - federal aid county roadways were obtained directly from the county o r regional planning entity, the years for which traffic volumes were available varied from county to county. Each of the traffic volume data sets were also exported as KMZ files for access through Google Earth so that roadway inventory information could be assessed and added to a single comprehensive dataset for each facility type. Where necessary, growth factors were applied to the assembled county FA and county n on - FA annual traffic volumes to provide estimates for each of the five analysis years (2011 - factors were obtained from MDOT each year for 2011 to 2015 and were applied directly to the applicable county FA data and county n on - FA county roadway data, respectively. Growth factors Performance Monitoring System (HPMS) d atabase for the statewide county roadway network and were applied, where necessary, to the relevant n on - FA roadway volumes. 3.1.3 Traffic Crash Data The annual statewide crash databases were provided by MDOT for 2011 - 2015, which was the most recently avai lable five - year period. The crash data were provided as extracts from MDOT Crash Reporting Information System (CRIS), which is derived from the official statewide crash 35 database kept by the Criminal Justice Information Center (CJIC) of the Michigan State Police (MSP). The crash database has details of all reported public roadway crash records in the state of Michigan, sanitized of any personal information. Records in this database are kept at the crash - , vehicle - , and person - levels with a total of eight s eparate spreadsheets included in the database . For the purposes of this analysis . After joining the tw o sheets together, the information relevant to the report was exported. The relevant fields are defined below. crsh_id - unique identifier for each crash, used as the basis for linking spreadsheets date_val - contains the date the crash occurred fatl_crsh_ind - shows the crash as having at least one fatality num_injy_a - - num_injy_b - - num_injy_c - total number of people su - prop_damg_crsh_ind - shows the crash as being property damage only (PDO) crsh_typ_cd - defines the crash as single - vehicle or one of nine multiple - vehicle types mdot_area_type_cd - code provided by MDOT to differential between intersection - related and non - intersection - related crashes. spcl_crcm_deer - indicator for deer involvement in the crash ped_invl_ind - shows that a pedestrian was involved in the crash bcyl_invl_ind - shows that a bicycle was involved in the crash P R - shows the p hysical r oad on which the crash occurred MP - shows the mile point along a p hysical r oad where a crash occurred 36 detail, meaning each row in th e database represented one crash. Injury severity was defined for each crash based on the most significant injury sustained by anyone involved in the incident. Crashes involving bicycles or pedestrians were separated from vehicle - only crashes for the data analysis. From there, various aggregations of the data were performed to compute crash frequencies by injury status (i.e., fatal/injury vs. PDO) and type (i.e., single vehicle vs. multiple vehicle) on an annual basis. Deer crashes were excluded fr om the pr imary segment and intersection analyses ; deer crashes on rural highway segments were analyzed and reported separately . Since SPFs were developed separately for segment and intersection facilities, it was first necessary to filter crashes that corresponded to the proper facility type. indicated that the crash occurred on the - intersections), and were matched to the proper roadway segment based on PR ( p hysical r oad) and mile point for each segment . Intersection crashes were identified by using and were matched with each intersection by using a 0.04 - mile (211.2 f ee t) radius around the intersection node. Intersection node identification will be described later in this chapter. 3.1.3.1 Horizontal Curves Horizontal curve i nformation for each segment was obtained through an extraction process initially developed by researchers at Wayne State University and applied to all rural roadways in Michigan, including MDOT trunkline and county roadways. The extraction process estimate s the radius and length of horizontal curves based on the All Roads shapefile using tools and code written for GIS. The information includes number of curves with radii of up to 0.5 miles, length 37 of the curved part of the segment, fraction of segment lengt h that is curved, and average radii of curves up to 0.5 miles for a segment. The information was organized in cumulative categories, decreasing in order of radii, from 0.5 - mile radii to 0.088 - mile radii. The curve data were then merged with the roadway inv entory data for the respective segment. To account for segment breaks across curves, the curve data were compiled for each radius threshold in the following manner: length of the curved part of the segment, curved proportion of the segment, and the average radii of curves on the segment. After preliminary investigation, it was decided that the curved proportion of the segment was most suitable for this analysis. Curve data were then binned based on design speed increments of 5 mph (e.g., 50 to 55 mph) for Plan R - 107 - H, where a design speed of 55 mph corresponds to a curve radius of 1,008 feet, a design speed of 50 mph with a curve radius of 794 feet, a design speed of 45 mph wi th a curve radius of 614 feet, and a design speed of 40 mph with a curve radius of 464 feet, and so on [74] . For purposes of this research, 55 mph was utilized as the threshold for defining horizontal curvature . This was because any curve with a design spe ed of less than the speed limit, which was the statutory rural limit of 55 mph for the sample of county roads evaluated herein, is required to have a curve warning sign per the Manual on Uniform Traffic Control Devices (MUTCD) [75] . While it was not possib le to verify the presence of a curve warning sign at each location, 55 mph was a reasonable upper threshold as curves with design speeds falling below the statutory speed limit were deemed substandard per the MUTCD requirement . Furthermore, for federal aid county roadways with the statutory 55 mph speed limit, curves with design speeds below 40 mph would require re - alignment during a 3R or 4R (i.e., major rehabilitation or reconstruction) project unless a design exception is granted. Thus, it was deemed imp ortant to 38 assess the incremental impacts of decreasing horizontal curve design speed beginning with 55 mph and decreasing in 5 mph increments to design speeds below 40 mph . 3.1.3.2 Rural Intersection Identification and Database Assembly To identify interse developed in ArcGIS to generate nodes based on the occurrence of intersecting lines from the All_Roads.shp file. This algorithm consisted of six primary steps, which are demonstrated in Figure 2 and described in further detail in the subsequent paragraphs. Figure 2 : Node identification algorithm 39 First the full road network was obtained via the All_Roads.shp file, where each public road segment was represented by a unique line in 2 - dimensional GIS space. Points were generated at each vertex of the aggregated roadway network, where vertices have the following general properties: Vertices exist wherever a segment changes direction. Each segment contain s a beginning and ending vertex. If two segments meet together, the ending vertex of Segment 1 and beginning vertex of Segment 2 will occupy the same location in two - dimensional space. The same condition applies to three or more segments meeting together. From there, segment vertices were converted to points, and the X (longitude) and Y (latitude) coordinates were obtained for each individual point, which is repeated whenever two or more segments meet. Based on this condition, the point database then dissol ved via the concatenated XY coordinates to obtain a count of each time that the concatenated XY coordinates were repeated. This count is the number of segments meeting together at a specific spatial location. Accordingly, a potential intersection exists wh enever the count is equal to or larger than three, with the count number also being the number of legs at the intersection. To limit the node database solely to potential intersections, any point with a count of less than three was removed from the databas e. The final list is all intersections of public roadways in the state of Michigan. Following the node generation process for potential intersections, any intersection node found within an ACUB , town or village limit, or CDP were also excluded. Segment inf ormation from the All_Roads.shp file was then attached to each node for all corresponding node legs via a one - to - one spatial join with a sensitivity search radius of 5 feet. The spatial join was performed 40 to build a relationship between the node dataset an d segment dataset for purposes of joining available traffic volume data to each leg of the node. To determine the availability of traffic volume data, nodes were categorized (MDOT, county federal aid, or county non - federal aid) based on the framework class ification code (FCC) of each leg. For a node to be included in the analysis, it was necessary for both of the following conditions to be met : A t least one of the interesting roadways was county - owned ; and E ach major and minor roadway must each major and m inor roadway must have at least one leg with traffic volume data. This was only an issue for non - federal aid county roadways, as traffic volume data were available within existing statewide databases for all MDOT trunklines and county federal - aid roadways . After populating the nodes with traffic volumes for the major and minor roadways, a KMZ file was assembled for purposes of reviewing all identified nodes using Google Earth satellite imagery. Each node for which traffic volume was available for both the major and minor intersecting roadways were reviewed to verify whether nodes were properly found as a complete intersection. Nodes were excluded from further analysis if any of the following situations applied: Signalized ; Four - way stop controlled ; Not fo und at an intersection of public roadways ; Located at a roundabout ; Located at a freeway exit ramp ; Redundant or part of a larger intersection ; 41 Within 0.08 mi les (422 ft) of another node, such as at median divided intersections or ; or Merge/diverge nodes at intersections within a horizontal curve. Each crash was initially mapped in GIS (geographic information systems) space based on longitude and latitude coordinates as presented in the crash records. Crashes were associated with each node based on two primary constraints. First, eligible intersecti on crashes were each intersection for further analysis by using a 0.04 - mile (211.2 ft) radius around the intersection node, as shown in Figure 3. Figure 3 : 3 - leg Intersection with c rash s earch t hreshold 42 Table 6 provides details of the resulting data set, including a count of the number of intersections by type, as well as averages of the major AADT, minor AADT, and total annual i n the 3ST and 4ST datasets. Table 6 : Rural County Road Intersection Summary Statistics Statistic 3ST 4ST MDOT County FA County n on - FA Total MDOT County FA County n on - FA Total Number of intersections 664 1,212 421 2,297 818 1,389 306 2,513 Average major road AADT 4,715 2,033 544 2,536 4,803 2,200 619 2,855 Average minor road AADT 1,042 730 186 721 1,033 743 254 778 Average annual crashes per intersection 0.78 0.43 0.1 0.47 1.12 0.72 0.2 0.78 3.1.3.3 Rural Segment Database Assembly The county segment dataset assembly process consisted of three main parts. First, all non - trunkline rural segments were identified in the All Roads shapefile. The selection criteria for this pool excluded all state trunklines and any un - coded roadways (i.e., n ational functional classification (NFC ) is equal to 0), and included only those segments which were located outside of the ACUB a nd CDP boundaries, had a left - right rural designation, and were categorized as principal arterial, minor arterial, and general non - certified segments. AADT values were spatially matched via the developed linear referencing system (LRS) to the pool of the r ural county road segments using the PR, beginning mile point , and ending mile point values of each segment. Volumes for federal aid county roadways were matched first, due to the systemwide availability of these volumes, followed by non - federal aid county roadway volumes, where available. The latest available year of traffic volume data was used in any case where multiple years of volume data were available. In addition, because the roadway segmentation of the AADT volumes differed from the segmentation of the used framework, only those volumes 43 which were a 100 percent match with the roadway segment were applied. Segments without any AADT volumes were removed from the sample and subsequently excluded from further analysis . Following the assignment of AADT volume s to segment s , crashes occurring between 2011 and 2015 were matched to the applicable segment in an equivalent manner using the PR and MP values as presented in each crash record. A secondary criterion was implemented to include only those crashes , which represents crashes that are not associated with an interchange or intersection (i.e., midblock). Lastly, all assigned crashes were tabulated by year, type, and severity for each segment. Deer crashes were extr acted and analyzed separately from the primary analyses . Finally, the county segment database was screened to include only segments that were 0.1 miles or more in length, which is the smallest segment length recommended by the HSM to represent physical and safety conditions for the facility [3] . Table 7 provides details of the resulting data set, including a count of the number segments and segment mileage by facility type, as well as averages of the AADT, total annual segment c rashes (per mile), non - deer annual segment crashes (per mile), and deer crashes as a proportion of total segment crashes. It can be observed from Table 7 that the proportion of deer crashes ranges from 0.38 to 0.69, depending on facility type, which far ex ceeds the proportion of deer crashes (0.121) reported for the crash data from Washington state that was used to develop the two - lane two - way SPF found in the HSM . This has significant implications on the transferability of the HSM segment models for use in Michigan , and further emphasizes the need for development Michigan - specific SPFs. 44 Table 7 : Rural County Road Segment Summary Statistics Statistic County p aved FA County p aved n on - FA Unpaved Number of s egments 9 , 912 2 , 873 3 , 983 Segment m ileage 4 , 423.7 1 , 463.4 2 , 007.2 Average AADT 1 , 717 585 241 Average a nnual s egment c rashes per mile 1.49 0.56 0.24 Average a nnual n on - d eer s egment c rashes per m ile 0.58 0.22 0.15 Deer c rashes as p roportion of t otal s egment c rashes 0.61 0.61 0.38 3.1.4 Additional Manual Data Collection Although existing spatial datasets were used to the extent possible, it was also necessary to collect certain important intersection or segment attributes using manual methods. These manual data were typically us ing Google Earth, including aerial view and Street View, where available. 3.1.4.1 Intersection Data Relevant count data (e.g., number of driveways and railroad crossing presence) were collected manually using Google Earth aerial imagery based on a 211 - f oo t radius of the intersection node. The following characteristics were assessed during the manual data collection at intersections: Number of intersecting legs: Only traditional three - leg and four - leg intersections were included. Assignment of major and min or approaches: The major and minor approach legs were assigned to each intersection where the uncontrolled approach was defined as the major leg and the stop - controlled approach was defined as the minor leg . Number of stop - controlled approaches: The number of stop - controlled approaches for each 3 - leg and 4 - leg intersection was noted. Intersections for which street level imagery was not available were removed from the dataset, as it was not possible to confirm the presence of stop control on the major and mi nor approaches. This issue typically only affected intersections where the major roadway was county non - federal 45 aid, as Street View imagery was available for all MDOT roadways and many county federal aid roadways. These data were used to identify intersect ions that included stop - control on the minor approach only. Number of through traffic lanes: The number of through lanes were determined for each individual approach of the intersection. Shared use lanes (i.e., combined through/turn) were counted as a through lane . Turn lane presence: Right and left turn lanes were found based on presence of pavement markings and/or sign designations. These data were aggregated by the number of approaches with turn lanes. Tapers or widened shoulders were not considered to be turn lanes . Driveway counts: The number of driveways that were at least partially within a 211 - foot radius of the center of the intersection was counted individually for each intersection leg . Skew angle: Intersection skew angles were obtained using the heading tool in Google Earth. The HSM defines intersection skew angle as the absolute value of the deviation from an intersection angle of 90 degrees. In this definition, skew can range from zero for a perpendicular intersection and to a maximum of 89 degrees. For this study, skew was calcula ted by first measuring the smallest angle between any two legs of the intersection. The heading of each leg was measured with respect to the centerline, and the absolute difference of those two headings was then calculated. The skew angle was calculated as the absolute difference of this angle from 90 degrees. Flashing beacon presence . Lighting presence (mast - arm or single span wire with hanging light). 46 Median presence: Median divided intersections were excluded from this analysis. Curb presence: Curbs wer e considered present if they were found on any of the intersection legs within a 211 - f oo t radius of the center of the intersection. Sidewalk presence: Sidewalks were considered present if they were found on any of the intersection legs within a 211 - f oo t ra dius of the center of the intersection. Railroad crossing presence: At - grade railroad crossings that fell within a 211 - f oo t radius of the center of the intersection were identified . In addition to serving as important analytical factors for SPF and CMF dev elopment, these manually collected data were , in some cases , also used for additional screening for identification of proper study sites. For example, to provide consistency with the HSM , only cases with minor roadway stop control (i.e., one - stop leg for t hree - leg intersections and two - stop legs for four - leg intersections) were kept for further analysis . Intersections where all - way stop control existed were excluded from further analyses, as few such intersections occur on rural roadways in Michigan and, th us, were outside the scope of this research. Furthermore, intersections with high skew angles that were a part of a perpendicular intersection with a bypass curve between adjacent legs were removed from the analysis because the nature of the turning traffi c movements is not properly shown by the major and minor AADT values. This case is common in rural settings where the through movement follows a 90 - degree turn, but the tangent legs are kept as minor road approaches. 3.1.4.2 Segment Data For the county roa dway segment dataset, each segment in the KMZ file was located in Google Earth aerial imagery based on the PR and begin/end mile points from the MGF All Roads shapefile. For geometric characteristics, the Google Earth ruler tool was used to make 47 measuremen ts from the aerial imagery. It was only necessary to collect these data for the county roadways, as the data were already available within the sufficiency file or other existing spatial dataset for MDOT roadways . The following list provides details on the data that were collected manually for county roadway segments: Driveway count by type: Driveways falling within the segment boundaries were counted and classified as residential or commercial/industrial to replicate the procedure utilized by MDOT to assemb le the trunkline driveway file. Field driveways that did not lead to a structure were not included. Surface type: Surface type was classified as paved or unpaved (i.e., gravel) . Surface width: For paved roadways, the surface width (in feet) was measured f rom paved edge to paved edge. For unpaved roadways, the surface width was taken as the predominant extent of width. Traveled way width: Width in feet between edge lines (if present) on paved surfaces only. If edge lines were not present, traveled way width was equal to surface width. Lane width: Calculated as the traveled way width divided by the number of lanes. Lane width was an important safety performance characteristic to evaluate, as it is one of the controlling geometric elements that must be brought to standard during resurfacing, restoration, rehabilitation, or reconstruction projects on federal aid roadways [17] . Shoulder width: Calculated as the difference between the surface width and the traveled way width, divided by two. Similar to lane width, shoulder width is also a controlling geometric element that must be brought to standard during resurfacing, restoratio n, rehabilitation, or reconstruction projects on federal aid roadways [17] . 48 Number of lanes: Predominant number of lanes (both directions) within segment boundary. Presence of edge lines, centerlines, curbs, two - way left turn l anes, rumble strips, passing lanes, and on - street parking were each individually assessed using aerial imagery, supplemented by Street View , where present. Unobservable cases were noted. 3.1.5 Quality Control/Quality Assurance Verification In order to ensure accuracy within the data, quality assurance/quality control (QA/QC) checks were performed . The same resources used to create the initial dataset, Google Earth primarily, were used to perform the QA/QC review. This entailed a separate observer assess ing all characteristics for 5 percent of segments. Evidence of systematic errors (e.g., improper coding, inaccurate width measurements, etc.) caused all data collection for the particular observer to be repeated by a more experienced observer . 3.2 Model Ca libration The HSM presents a methodology for calibrating the models contained in the manual, and in order to evaluate the benefits of developing new safety performance functions, it was necessary to calibrate HSM models to provide a basis for comparison. Calibration was performed by estimating the number of crashes at each segment or intersection using the HSM models and comparing this estimated number to the actual number of crashes. The equation for calculating the calibration factor, C, was previously i ntroduced as Equation 4 in the literature, and is restated below for the sake of convenience. (4) 49 Where, = the observed annual average crash frequency, = predicted annual average crash frequency, and n = sample size, equal to the number of sites in the calibration process. Calculating the predicted annual average crash frequency was accomplished using the HSM Spreadsheet Tools , developed at the Texas A&M Transportation Institute and ma intained by AASHTO [105] . I t was not possible to apply the horizontal curve CMF from the HSM due to the way that the Michigan horizontal curve data were specified. Thus, only tangent segments without horizontal curvature were u tilized for calibration, and the CMF related t o horizontal curvature wa s not applied. Tangent was defined as not having any horizontal curves with radii less than 2,640 feet. Similarly, due to a lack of information, CMFs for vertical grade, roadside hazard rating, and side slopes were not applied . 3. 3 Analytical Method s In analyzing the safety performance of a given segment, there are several approaches, including: crash frequency, crash rate, and regression analysis. Crash frequency tends to be biased in f avor of prioritizing the highest volume segments for treatment, as traffic volume is positively correlated with crashes. On the other hand, using crash rate tends to be biased in favor of low - volume segments , due to the nonlinear relationship between crash es and AADT , or short segments , due to the overrepresentation of crash causal factors on such segments. For this reason, regression analysis was chosen. As crash data are comprised of non - negative integers, traditional regression techniques (e.g., ordinary least - squares) are generally not appropriate. Given the nature of such data, the Poisson distribution has been shown to provide a better fit and has been used widely to model crash frequency data. In the Poisson model, the probability of segment i experiencing y i crashes during a one - year period is given by Equation 5 : 50 ( 5 ) where P ( y i ) is probability of segment i experiencing y i crashes and i is the Poisson parameter for segment i ted number of crashes per year, E [ y i ]. Poisson models are estimated by specifying the Poisson parameter i (the expected number of crashes per period) as a function of explanatory variables . T he most common functional form is shown in Equation 6: ( 6 ) where X i is a vector of explanatory variables and is a vector of estimable parameters. A limitation of this model is the underlying assumption of the Poisson distribution that the variance is equal to the mean. As such, the model canno t handle overdispersion wherein the variance is greater than the mean. Overdispersion is common in crash data and may be caused by data clustering, unaccounted temporal correlation, model misspecification, or ultimately by the nature of the crash data, whi ch are the product of Bernoulli trials with unequal probability of events [106] . Overdispersion is generally accommodated through the use of negative binomial models (also referred to as Poisson - gamma models). The negative bin omial model is derived by rewriting the Poisson parameter for each segment as shown in Equation 7: ( 7 ) where exp( i ) is a gamma - distributed error term with mean 1 and variance . The addition of this term allows the variance to differ from the mean as . The negative binomial model is preferred over the Poisson model since the latter cannot handle overdispersion and, as such, may lead to biased parameter estimates [107] . Con sequently, the HSM recommends using the negative binomial model for the development of SPFs. 51 Poisson model. Estimation of i can be conducted through standard maximum likelihood procedures. While alternatives, such as the Conway - Maxwell model, have the advantage of accommodating both overdispersion and underdispersion (where the variance is less than the mean) [107] , the negative binomial model remains the standard in SPF development. One concern that arises when evaluating the safety of the county road system in Michigan is the occurrence of unobserved heterogeneity, defined as unknown variability in the effect of variables across the sample population . In this context, unobserved heterogeneity may be introduced when collecting data from across various counties and regions of the state, due to the inability to measure or otherwise quantify all data necessary to account for this variability . For example, design standards and maintenance practices are known to vary from county t o county, particularly for non - federal aid roads [108] . Other factors, such as weather, topograp hy, land use, and driver behavior also vary widely across the various regions of the state . If these variances are not considered , and the effects of observable variables are held the same across all observations, the estimated parameters will be biased [109] . To account for these differences, a county - specific random effect was incorporated into the analysis, whereby the intercept term is allowed to vary across counties. Furthermore, an additional site - specific random intercept term was utilized in order to acc ount for the non - independence associated with the annual replication of the data for each location within the data file. Prior research has compar ed this approach with a methodology that incorporates only one line of data per site; it was found that the tr aditional method underestimates variance, and thus may find factors to be s tatistically significant that would not be significant if site location had been controlled [110] . R ecent papers have addressed this bias by incorporati ng a site - specific 52 random effect [111] . In order to capture annual variations in crashes, the random effect approach was used; a model of county federal aid segments using fixed effects, whereby each line of data incorporates a ll five years of crashes per site can be found in Table 5 7 ( Appendix B ) . It is important to note that the model contained in Appendix B is estimating five years of crash data rather than the one - year period that all other models in this document utilize. C are needs to be taken when adding variables to avoid overfitting the SPF. More complex models are often poorer predictors, only accurately predicting crashes on the intersections that were used to estimate its parameters, as statistical noise tends to be i ncorrectly included as systematic variations in crashes. A stepwise process was used whereby factors would be removed from models to evaluate the changes in other parameter estimates and P - values. After examining the general distributions of traffic volume s and proportion of federal aid classification, it was decided that three separate series of SPFs were developed for segments (i.e., paved federal aid, paved non - federal aid, and unpaved) and six separate series of SPFs for intersections. A full list of mo dels are as follows: Rural county segments (non - deer) o Paved federal aid o Paved non - federal aid o Unpaved Rural stop - controlled intersections (non - deer) o Three - leg Combined set (major road state, county FA, and county non - FA all included) County FA 53 County non - F A o Four - leg Combined set (major road state, county FA, and county non - FA all included) County FA County non - FA Rural segments ( only deer crashes) o State o County paved federal aid o County paved non - federal aid o Unpaved With the exception of the deer - only models, o nly non - deer crashes were included in the models due to the relative frequency and unpredictability of such crashes, particularly when considering roadway design factors. It is also noted that the natural log of segment length was included as an offset in each model, with a parameter estimate fixed at one, thereby forcing the model to treat crashes as a direct one - to - one relationship with segment length. Segment length, when not constrained by an offset, tends to be very close to one [112] ; for this reason, segment is generally considered to be an offset when developing safety performance functions [113 - 115] . A set of models where segment length was not treated as an offset were developed and are included in Tables 5 8 - 60 ( Appe ndix C ) . The models in Appendix C have parameter estimates for the natural log of length that are very close to one. 54 The functional form for the mixed effects negative binomial model for prediction of annual crash frequency results is presented in Equation 8 below. ( 8 ) Where, N = estimated number of annual crashes; n = number of model parameters; L = segment length in miles; 0 = model intercept; 1 = model parameter estimate for AADT; x 2 n = additional model parameter values (for binary factors, this would equal one if present or applicable, zero if not present or applicable); and 2 n = additional model parameter estimates. The functional form for interpreting intersections model result s is presented in Equation 9 below. ( 9 ) Where, N = estimated number of annual crashes; n = number of model parameters; 0 = model intercept; 1 = model parameter estimate for maj or road AADT; 2 = model parameter estimate for minor road AADT; x 3 n = additional model parameter values (for binary factors, this would equal one if present or applicable, zero if not present or applicable); and 3 n = additional model parameter estimates. 55 4. SAFETY PERFORMANCE F UNCTIONS FOR COUNTY - OWNED RURAL HIGHWAY SEGMENTS The safety performance of rural two - way two - lane county highway segments were analyzed using the data collected from 30 counties within Mich igan , as described in Chapter 3. Due to the differences in design characteristics, traffic volumes, trip distances, driver characteristics, and other factors, separate datasets were created for federal aid and non - federal aid county highways. Non - federal a id county highways were further partitioned into paved and unpaved roadway datasets for analysis. Safety performance functions were developed for the following three categories of rural county road segments: a) Rural c ounty t wo - l ane t wo - w ay p aved f ederal a id segments ; b) Rural c ounty t wo - l ane t wo - w ay p aved n on - f ederal a id segments ; and c) Rural c ounty unpaved n on - f ederal a id segments . A series of CMFs were also developed to describe the effects of horizontal curve radius on segment safety performance. It is importa nt to note that while rural unpaved county federal aid roadways do exist, the number of such roadway within Michigan is very small and, consequently, not included in this analysis. The sections that follow will include a data summary a nd data diagnostics s ection , an explanation of the analytical methods, and results and discussion. 4.1 Data Summary , Data Screening, and Data Diagnostics Figure 4 and Table 8 display the distribution of county two - lane two - way segment study locations throughout the state of Michigan. The following counties were represented in the county road segment database: Arenac, Baraga, Barry, Charlevoix, Clinton, Dickinson, Eaton, 56 Emmet, Genesee, Grand Traverse, Gratiot , Ingham, Iosco, Kalamazoo, Kent, Keweenaw, Livingston, Luce, Macomb, Marquette, Mason, Mecosta, Monroe, Muskegon, Oakland, Ogemaw, Roscommon, Schoolcraft, Washtenaw, and Wayne. These 30 counties were utilized due to the av ailability of traffic volume data, particularly for non - federal aid county roadways. It is worth noting that each of the seven MDOT geographic regions were represented in the sample. Figure 4 : Map of r ural c ounty h ighway s egments 57 Table 8 : Represented C ounties and C orresponding S egment M ileage (Segments) County M iles County FA County n on - FA Unpaved Total Arenac 126 0 2 128 Baraga 101 0 8 108 Barry 225 4 13 243 Charlevoix 117 0 30 147 Clinton 215 99 178 493 Dickinson 101 1 12 113 Eaton 228 171 446 845 Emmet 157 0 0 157 Genesee 113 74 3 189 Grand Traverse 136 56 20 213 Gratiot 236 61 89 387 Ingham 208 309 38 554 Iosco 166 25 31 223 Kalamazoo 171 307 71 548 Kent 196 210 129 536 Keweenaw 60 0 0 60 Livingston 113 40 386 539 Luce 68 0 44 113 Macomb 29 5 117 152 Marquette 154 13 20 187 Mason 181 0 11 193 Mecosta 195 0 3 197 Monroe 161 0 0 161 Muskegon 170 10 1 180 Oakland 29 3 128 161 Ogemaw 151 6 8 165 Roscommon 99 0 2 101 Schoolcraft 83 2 63 149 Washtenaw 150 2 31 184 Wayne 23 0 6 28 Total 4,162 1,399 1,892 7,453 4.1.1 Segment Descriptive Statistics In total, 7, 453 miles of county highways were included in the analysis, of which 55.8 percent were paved federal aid segments (from 30 counties), 18. 8 percent were paved non - federal aid segments (from 19 counties), and 25. 4 percent were unpaved roadways (from 27 counties ). The full descriptive statistics for the modeled variables associated with each of the three county 58 roadway categories can be found in Table 9 (paved federal aid), Table 10 (paved non - federal aid), and Table 11 (unpaved non - federal aid). Table 9 : County Road Segment Summary Statistics (Federal Aid) Variable N (segments) Min 25th% 50th% 75th% Max Mean Std dev AADT 9,264 10 616 1,058 1,779 5,143 1,449.0 1,122.3 Segment length (mi) 9,264 0.100 0.219 0.372 0.505 8.19 0.449 0.330 Surface width (feet) 9,264 19.0 22.0 23.0 24.0 40.0 24.2 3.367 Lane width (feet) 9,264 9.5 10.5 11.0 11.0 13.0 11.0 0.685 Paved shoulder width (feet) 9,264 0.0 0.0 0.5 1.0 8.0 1.0 1.444 Driveway count 9,264 0 2 4 7 69 6.0 6.481 Driveway density (mi - 1 ) 9,264 0.0 5.0 11.3 17.8 138.7 14.5 13.612 0 - to - 4.9 driveways per mi 2,321 n/a n/a n/a n/a n/a 0.251 0.433 5 - to - 14.9 driveways per mi 3,448 n/a n/a n/a n/a n/a 0.372 0.483 15 - to - 24.9 driveways per mi 1,903 n/a n/a n/a n/a n/a 0.205 0.404 > 25 driveways per mi 1,592 n/a n/a n/a n/a n/a 0.172 0.377 Curved portion of segment > 55 mph design speed 9,264 0.000 1.000 1.000 1.000 1.000 0.969 0.144 50 - 54.9 mph design speed 9,264 0.000 0.000 0.000 0.000 1.000 0.014 0.096 45 - 49.9 mph design speed 9,264 0.000 0.000 0.000 0.000 1.000 0.010 0.077 40 - 44.9 mph design speed 9,264 0.000 0.000 0.000 0.000 1.000 0.005 0.057 <40 mph design speed 9,264 0.000 0.000 0.000 0.000 1.000 0.002 0.035 Minor arterial 803 n/a n/a n/a n/a n/a 0.087 0.281 Major collector 8,435 n/a n/a n/a n/a n/a 0.911 0.285 Minor collector 23 n/a n/a n/a n/a n/a 0.002 0.050 Local 2 n/a n/a n/a n/a n/a 0.000 0.015 Variable Five - year crash count Annual crashes per segment Min 25th% 50th% 75th% Max Mean Std dev Midblock total crashes 29,377 0 0 0 1 15 0.634 1.058 Midblock total non - deer crashes 10,862 0 0 0 0 8 0.235 0.576 Midblock fatal and injury non - deer crashes 2,956 0 0 0 0 5 0.064 0.267 Midblock property damage only non - deer crashes 7,906 0 0 0 0 7 0.171 0.472 Note: mi = miles; mph = miles per hour ; n/a = not applicable ; std dev = standard deviation ; min = minimum; max = maximum; N = number of 59 Table 10 : County Road Segment Summary Statistics (Paved Non - Federal Aid) Variable N (segments) Min 25th% 50th% 75th% Max Mean Std dev AADT 2,713 5 203 378 595 1,681 483.0 359.6 Segment length (mi) 2,713 0.100 0.257 0.490 0.636 2.012 0.515 0.301 Surface width (feet) 2,713 18.0 20.0 22.0 22.0 36.0 21.6 1.965 Lane width (feet) 2,713 9.0 10.0 10.5 11.0 13.0 10.6 0.693 Paved shoulder width (feet) 2,713 0.0 0.0 0.0 0.0 7.0 0.2 0.645 Driveway count 2,713 0 3 6 11 62 8.6 7.764 Driveway density (mi - 1 ) 2,713 0.0 8.0 14.4 21.1 108.1 17.4 13.549 0 - to - 4.9 driveways per mi 395 n/a n/a n/a n/a n/a 0.145 0.353 5 - to - 14.9 driveways per mi 1,013 n/a n/a n/a n/a n/a 0.373 0.484 15 - to - 24.9 driveways per mi 704 n/a n/a n/a n/a n/a 0.259 0.438 > 25 driveways per mi 602 n/a n/a n/a n/a n/a 0.222 0.416 Curved proportion of segment > 55 mph design speed 2,713 0.000 1.000 1.000 1.000 1.000 0.967 0.144 50 - 54.9 mph design speed 2,713 0.000 0.000 0.000 0.000 1.000 0.011 0.077 45 - 49.9 mph design speed 2,713 0.000 0.000 0.000 0.000 0.889 0.008 0.067 40 - 44.9 mph design speed 2,713 0.000 0.000 0.000 0.000 1.000 0.009 0.073 <40 mph design speed 2,713 0.000 0.000 0.000 0.000 0.785 0.005 0.044 Minor arterial 0 n/a n/a n/a n/a n/a 0.000 0.000 Major collector 14 n/a n/a n/a n/a n/a 0.005 0.073 Minor collector 508 n/a n/a n/a n/a n/a 0.187 0.390 Local 2,191 n/a n/a n/a n/a n/a 0.807 0.394 Variable Five - year crash count Annual crashes per segment Min 25th% 50th% 75th% Max Mean Std dev Midblock total crashes 3,663 0 0 0 0 7 0.270 0.621 Midblock total non - deer crashes 1,389 0 0 0 0 4 0.102 0.347 Midblock fatal and injury non - deer crashes 399 0 0 0 0 3 0.029 0.176 Midblock property damage only non - deer crashes 990 0 0 0 0 3 0.073 0.286 Note: mi = miles; mph = miles per hour; n/a = not applicable; std dev = standard deviation; min = minimum; max = maximum; N = number of 60 Table 11 : County Road Segment Summary Statistics (Unpaved Non - Federal Aid) Variable N (segments) Min 25th% 50th% 75th% Max Mean Std d ev AADT 3,747 4 78 130 203 658 172.7 132.6 Segment length (mi) 3,747 0.100 0.253 0.458 0.567 4.575 0.505 0.365 Surface width (feet) 3,747 14.0 18.0 20.0 22.0 33.0 21.0 3.526 Driveway count 3,747 0 2 4 7 50 5.9 5.944 Driveway density (mi - 1 ) 3,747 0.0 4.4 10.0 15.9 93.0 12.3 10.626 0 - to - 4.9 driveways per mi 1,013 n/a n/a n/a n/a n/a 0.270 0.444 5 - to - 14.9 driveways per mi 1,521 n/a n/a n/a n/a n/a 0.406 0.491 15 - to - 24.9 driveways per mi 773 n/a n/a n/a n/a n/a 0.206 0.405 >25 driveways per mi 440 n/a n/a n/a n/a n/a 0.118 0.322 Curved proportion of segment >55 mph design speed 3,747 0.000 1.000 1.000 1.000 1.000 0.966 0.135 50 - 54.9 mph design speed 3,747 0.000 0.000 0.000 0.000 1.000 0.008 0.057 45 - 49.9 mph design speed 3,747 0.000 0.000 0.000 0.000 1.000 0.009 0.065 40 - 44.9 mph design speed 3,747 0.000 0.000 0.000 0.000 1.000 0.007 0.057 <40 mph design speed 3,747 0.000 0.000 0.000 0.000 0.825 0.010 0.062 Minor arterial 0 n/a n/a n/a n/a n/a 0.000 0.000 Major collector 437 n/a n/a n/a n/a n/a 0.117 0.321 Minor collector 227 n/a n/a n/a n/a n/a 0.061 0.239 Local 3,083 n/a n/a n/a n/a n/a 0.823 0.382 Variable Five - year crash count Annual crashes per segment Min 25th% 50th% 75th% Max Mean Std d ev Midblock total crashes 2,080 0 0 0 0 5 0.111 0.358 Midblock total non - deer crashes 1,335 0 0 0 0 5 0.071 0.285 Midblock fatal and injury non - deer crashes 376 0 0 0 0 2 0.020 0.143 Midblock property damage only non - deer crashes 959 0 0 0 0 5 0.051 0.239 Note: mi = miles; mph = miles per hour; n/a = not applicable; std dev = standard deviation; min = minimum; max = maximum; N = number of 4.1.2 Data Screening In order to address sites with atypical characteristics, segments were constrained to those with lane widths from 10 to 13 feet on paved federal aid segments, nine to 13 feet on paved non - federal aid segments, and surface widths from 14 to 33 feet on unpav ed non - federal aid segments. Furthermore, on paved federal aid segments, those segments with paved shoulder 61 widths greater than eight feet were excluded. A thorough investigation of a representative sample of such locations found that they typically posses sed atypical features, typically involving widening at driveways, intersections, or bridges. These exceptions represented less than one percent of all segments. In addition, segments comprising of the top 5 percent of volumes were removed from this analysi s in order to focus on the lower - AADT sites where the vast majority of the data reside. These exclusions explain the difference between some of the descriptive statistics in this chapter with those in Chapter 3. In addition to the differences in data coll ection between federal aid and non - federal aid highways, previously described in Chapter 3 , the differences in funding source between federal aid and non - federal aid highways show key differences between these roadways. The most critical is the functional classification; while 91 percent of paved federal aid segments are major collectors, 81 percent of paved non - federal aid segments are local roads. Not surprisingly, these functional classifications differences are also reflected in the AADT volumes, with a 75 th percentile AADT of 1,779 vehicles per day on paved federal aid segments, but only 595 vehicles per day on paved non - federal aid segments. It is also worth noting the descriptive statistics for the crashes themselves across the roadway segment catego ries . For example, on paved federal aid highways, 6 3 percent of crashes involved deer , and on paved non - federal aid highways, 6 2 percent of crashes involved deer. In the HSM , only 12 percent of the crashes used to develop their two - way two - lane rural segme nt SPF involved animals [3] . Due to their large proportion, as well as the lack of known deer - mitigation strategies in use in Michigan, deer crashes were excluded from the primary segment models that were developed, and were re served for a separate analysis that is detailed in Chapter 6. On paved federal aid segments, 39 percent of segments experienced a crash of any kind during 62 the five - year analysis period, while only 19 percent of segments experienced a non - deer related crash . On paved non - federal aid segments, 21 percent of segments experienced a crash of any kind, while only 9.6 percent of segments experienced a non - deer related crash during the analysis period. 4.1.3 Data Diagnostics Prior to SPF development, various data d iagnostics were initially conducted to examine general trends across all locations for each facility type. This included assessment of the relationships between AADT and annual crash frequency (normalized on a per - mile basis) with scatterplots of these relationships generated for total and deer - excluded crashes for each of the three segments types, which are shown in Figure 5 . 63 a.) Total Midblock Crashes (Paved FA) b.) Deer - Excluded Midblock Crashes (Pave d FA) c.) Total Midblock Crashes (Paved Non - FA) d .) Deer - Excluded Midblock Crashes (Paved Non - FA) e.) Total Midblock Crashes ( Unpaved Non - FA) f .) Deer - Excluded Midblock Crashes ( Unp aved Non - FA) Figure 5 : Annual m idblock c rashes per m ile vs AADT, c ounty s egments (2011 - 2015) 64 Various additional factors were plotted against AADT, including lane width, paved shoulder width, driveway density, and curved segments proportions for each design speed range. These t wo - way scatter plots are displayed in Figure s 6 - 8 . Each of these factors showed correlation with AADT, which is expected , as many of these factors are design standards established based on traffic volumes. For example, design standards for lane and shoulde r width typically increase with increasing traffic volume. Not surprisingly, lane width and shoulder width (or total surface width for unpaved roads) on county road segments are found to be positively correlated with AADT. The association with AADT is stro nger for shoulder width and is strongest for total surface width on unpaved roadways. It is also worth noting that that wider lanes (i.e., 11 - and 12 - foot lanes) are found in a wider range of traffic volumes than narrower lanes (i.e., 9 - and 10 - foot lanes) . Driveway density was also found to be correlated with AADT; also, not a surprising result . Curves with the lowest design speeds are found mostly on lower - volume highways, while curves with design speeds between 45 - 49 mph and 50 - 54 mph are found along a m uch wider range of traffic volumes, including higher - volume segments. In general, these associations between the various roadway factors and AADT were strongest for the paved federal aid roadways and diminished at the lower roadway classes . 65 Figure 6 : Lane width, shoulder width, driveway density, NFC, and curve proportions vs. AADT on paved federal aid county segments 66 Figure 7 : Lane width, shoulder width, driveway density, an d curve proportions vs. AADT on paved non - federal aid county segments 67 Figure 8 : Surface width, driveway density, and curve proportions vs. AADT on unpaved non - federal aid county segments Table s 12 - 14 show the crash distributions for each of the three segment types. In comparison to the default distribu tions presented in Chapter 10 of the HSM - lane two - way county segments have much lower proportions of severe crashes and much greater proportions of anim al (deer) crashes. In direct comparison to MDOT state trunkline rural two - way two - lane segments, the two - lane county segments have a higher proportion of other single - vehicle crashes, which includes fixed object collisions, across all crash severities [16] . This type 68 of crash might likely be related to the available clear zone, road hazard rating, or sideslopes, none of which were feasible for collection in this study, but would typically be reflected in the design standards for county roadways compared to MDOT trunkline highways. The over - representation of deer crashes on county segments explains why the proportion of multiple - - lane county segments is so much lower than the default distributio ns in the HSM . Table 12 : Crash Severity and Crash Type Distributions for Rural Paved Federal Aid County Segments Crash s everity l evel Count of m idblock c rashes (2011 - 2015) Percentage of t otal m idblock c rashes Fatal (Type K) 124 0.4% Incapacitating i njury (Type A) 432 1.5% Non - incapacitating i njury (Type B) 1,075 3.7% Possible i njury (Type C) 1,691 5.8% Fatal + i njury (Type K+ABC) 3,322 11.3% Property d amage o nly (Type PDO) 26,055 88.7% Single m otor v ehicle 26,827 91.3% Single m otor v ehicle ( d eer e xcluded) 8,387 28.5% Deer c rashes 18,515 63.0% Multiple v ehicle c rashes 2,460 8.4% Day c rashes 10,131 34.5% Dark c rashes 19,246 65.5% Total c rashes (5 years) 29,377 100.0% Collision t ype Percentage of f atal and i njury Percentage of p roperty d amage o nly Percentage of t otal s egment c rashes Single - vehicle crashes Collision with animal 11.0% 69.7% 63.0% Collision with bicycle 1.4% 0.0% 0.2% Collision with pedestrian 1.1% 0.0% 0.1% Overturned 19.9% 3.8% 5.6% Other single - vehicle crash 45.2% 19.8% 22.7% Total single - vehicle crash 78.7% 93.3% 91.6% Multiple - vehicle crashes Angle collision 3.0% 0.9% 1.1% Head - on collision 5.8% 0.4% 1.0% Read - end collision 7.6% 2.2% 2.8% Sideswipe collision 3.4% 2.1% 2.2% Other multiple - vehicle collision 1.6% 1.1% 1.1% Total multiple - vehicle collision 21.3% 6.7% 8.4% Total Crashes 100.0% 100.0% 100.0% 69 Table 13 : Crash Severity and Crash Type Distributions for Rural Paved Non - Federal Aid County Segments Crash s everity l evel Count of m idblock c rashes (2011 - 2015) Percentage of t otal m idblock c rashes Fatal (Type K) 23 0.6% Incapacitating i njury (Type A) 51 1.4% Non - incapacitating i njury (Type B) 153 4.2% Possible i njury (Type C) 229 6.3% Fatal + i njury (Type K+ABC) 456 12.4% Property d amage o nly (Type PDO) 3,207 87.6% Single m otor v ehicle 3,312 90.4% Single m otor v ehicle ( d eer e xcluded) 1,051 28.7% Deer c rashes 2,274 62.1% Multiple v ehicle c rashes 331 9.0% Day c rashes 1,184 32.3% Dark c rashes 2,479 67.7% Total c rashes (5 years) 3,663 100.0% Collision t ype Percentage of f atal and i njury Percentage of p roperty d amage o nly Percentage of t otal s egment c rashes Single - vehicle crashes Collision with animal 12.5% 69.1% 62.1% Collision with bicycle 0.9% 0.0% 0.1% Collision with pedestrian 3.3% 0.0% 0.4% Overturned 14.0% 2.5% 3.9% Other single - vehicle crash 54.2% 20.2% 24.4% Total single - vehicle crash 84.9% 91.8% 91.0% Multiple - vehicle crashes Angle collision 3.7% 1.3% 1.6% Head - on collision 3.9% 0.4% 0.8% Read - end collision 3.7% 2.2% 2.4% Sideswipe collision 2.2% 2.5% 2.5% Other multiple - vehicle collision 1.5% 1.7% 1.7% Total multiple - vehicle collision 15.1% 8.2% 9.0% Total Crashes 100.0% 100.0% 100.0% 70 Table 14 : Crash Severity and Crash Type Distributions for Rural Unpaved Non - Federal Aid County Segments Crash s everity l evel Count of m idblock c rashes (2011 - 2015) Percentage of t otal m idblock c rashes Fatal (Type K) 7 0.3% Incapacitating i njury (Type A) 58 2.8% Non - incapacitating i njury (Type B) 146 7.0% Possible i njury (Type C) 183 8.8% Fatal + i njury (Type K+ABC) 394 18.9% Property d amage o nly (Type PDO) 1,686 81.1% Single m otor v ehicle 1,804 86.7% Single m otor v ehicle ( d eer e xcluded) 1,063 51.1% Deer c rashes 745 35.8% Multiple v ehicle c rashes 267 12.8% Day c rashes 927 44.6% Dark c rashes 1,153 55.4% Total c rashes (5 years) 2,080 100.0% Collision t ype Percentage of f atal and i njury Percentage of p roperty d amage o nly Percentage of t otal s egment c rashes Single - vehicle crashes Collision with animal 4.6% 43.1% 35.8% Collision with bicycle 1.0% 0.1% 0.2% Collision with pedestrian 1.0% 0.0% 0.2% Overturned 24.4% 6.8% 10.1% Other single - vehicle crash 59.4% 36.5% 40.8% Total single - vehicle crash 90.4% 86.4% 87.2% Multiple - vehicle crashes Angle collision 1.8% 3.2% 2.9% Head - on collision 2.0% 1.4% 1.5% Read - end collision 1.8% 2.1% 2.0% Sideswipe collision 3.0% 5.1% 4.7% Other multiple - vehicle collision 1.0% 1.8% 1.7% Total multiple - vehicle collision 9.6% 13.6% 12.8% Total Crashes 100.0% 100.0% 100.0% 4.2 Results and Discussion The final model results for estimating non - deer crashes on county - owned rural two - lane, two - way roads on paved federal aid segments, paved non - federal aid segments, and unpaved non - federal aid segments are displayed in Table 15 , Table 16 , and Table 17 , respectively . C oefficient estimates, standard errors, and p - values are provided in each table. Due to the s mall number of 71 crashes at each location, models estimating crashes of all severity levels were developed rather than isolating specific crash types and severities . Unless otherwise noted, further discussion of crashes in this chapter should assume exclusio n of deer crashes , which are modeled separately in Chapter 6 . 4.2.1 Paved Federal Aid County Road Segments The results of the mixed effects negative binomial models for federal aid county road segments, which are presented in Table 15 , yielded several interesting results. Horizontal curvature was found to be a significant factor , and the parameter estimates were found to increase as the curve radius decreased, suggesting greater crash occurrence with decreasing curve design speeds. The effect was found to increase relatively consistently and monotonically with decreasing design speed. N sequent discussion of the horizontal curve effects will consider the value of this variable to equal 1.0, indicating a fully curved segment. H orizontal curves with design speeds from 50 - 54 .9 mph experienced approximately twice as many crashes than the base line condition of no substandard horizontal curves along the segment . For curves with a design speed below 40 mph, the increase in crashes more than doubles again , with a 4. 2 times increase in crashes relative to base condition . This is shown visually in F igure 9 , which shows the expected annual frequency of crashes per mile for curves across the varying design speed categories, compared to the base condition. Not e that base conditions were assumed for all other factors when generating these plots . 72 Table 15 : Mixed Effects Negative Binomial Model Results for Paved Federal Aid Segments Factor Description Est Exp(B) Std error P - value Intercept - 5.932 0.003 0.152 <0.001 Segment length O ffset, natural log of, miles AADT N atural log of, vehicles per day 0.681 1.976 0.020 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 0.724 2.063 0.137 <0.001 45 - 49.9 mph C urved proportion of segment 1.097 2.995 0.149 <0.001 40 - 44.9 mph C urved proportion of segment 0.999 2.715 0.225 <0.001 <40 mph C urved proportion of segment 1.432 4.186 0.327 <0.001 10 - ft lane Baseline 11 - ft lane Width in feet 0.016 1.016 0.041 0.701 12 - ft lane Width in feet 0.053 1.054 0.048 0.274 13 - ft lane Width in feet 0.131 1.140 0.088 0.139 0 to 1 ft shoulder Baseline 2 - ft shoulder Width in feet - 0.044 0.957 0.049 0.362 3 to 8 ft shoulder Width in feet 0.045 1.046 0.037 0.215 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.108 1.114 0.036 0.003 15 - to - 24.9 driveways per mile Binary indicator variable 0.173 1.189 0.040 <0.001 > 25 driveways per mile Binary indicator variable 0.267 1.306 0.043 <0.001 Site random effect 0.559 County random effect 0.263 Overdispersion parameter 0.044 AIC 49,264.3 Log likelihood - 24,615.2 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion ; mph = miles per hour; ft = foot As previously mentioned, when considering segments containing substandard horizontal curvature, 46 percent of the length of these segments, on average, does not consist of substandard curvature. Therefore, a separate analysis was conducted to determine if the extrapolations drawn from these results are valid. Only to types of segments were retained in this analysis: 1) segments which were entirely substandard within one particular design speed category; or 2) segments that entirely met standards . This was d one in order to have a truly binary dataset with respect to horizontal curve design speed. Results are presented in Table 6 1 (Appendix D), and are comparable to the results presented in this chapter for curves with design speeds between 40 - 54.9 mph. Howeve r, t here were not enough segments whose entire segment length had a design 73 speed of <40 mph for significant results to be found within this category. Therefore, a second additional analysis was performed that included segments where the proportion of the s egment containing a curve with a design speed of less than 40 mph ranged from 0.8 to 1.0. For these segments, zero to 20 percent of segment length consisted of roadway that was tangent or whose curvature met design standards; these segments did not contain horizontal curves with design speeds between 40 - 54.9 mph, as this would interfere with the binary status of those design speed categories. These results are presented in Table 6 2 (Appendix D) and are similar to those presented in this chapter. Previous r esearch on federal aid highways in Michigan showed that segments with presence of a curve with a design speed below 55 mph , as a binary factor, experienced 56 percent more total crashes and 54 percent more fatal and injury crashes than segments without suc h curves [18] . This corresponds well to the model results presented in this paper; when looking only at segments the contain substandard horizontal curvature, the average segment comprises, in terms of length, of 46 percent of the segment having no substandard curvature, 24 percent of the segment h aving a design speed between 50 - 54.9 mph, 17 percent between 45 - 49.9 mph, 9 percent between 40 - 44.9 mph, and 4 percen t of the segment having a design speed below 40 mph. When calculati ng the crash effect of such curvature ( using the model results presented in Table 15 , the formula would be ) , it corresponds to a 66 percent increase in crashes. I n Pennsylvania, a 43 percent increase in total crashes and a 48 percent increase in injury crashes was observed on 55 mph state highway segments that included a 55 - mph curve on state - owned highways [19] . The research presented here builds upon prior research in that it incl udes a continuous variable for the curved proportion of the segment . This is important as the curve effects are parameterized based 74 on the amount of curvature on the segment, rather than simply a binary indicator for curve presence . Figure 9 : Comparison of SPF crash results on county federal - aid segments by curve design speed Increasing driveway density was also associated with increasing crashes, which is not a surprising result, as increasing the number of driveways increases the number of conflict points. However, d riveway density did not appear to increase crashes to the same extent as substandard horizontal curvature. Segments with 5 to 14.9 driveways per mile had approximatel y 11 percent more crashes than segments with fewer than five driveways per mile, while segments with 25 or 75 more driveways per mile saw nearly 31 percent more crashes. Prior research has also found total crashes to increase with increasing driveway density on rural two - lane two - way roads [116] . Lane width did not have a significant effect on total crashes, although preliminary fixed effects models showed reduced crashes associated with wider lanes. Fixed effects models can be found in Appendix A. Those findings correspond somewhat with previous research associating wider lanes with fewer non - incapacitating injury and PDO crashes (BCO crashes) [56] and f ewer total crashes [3, 52] . As previously shown in the data diagnostics section, l ane width and AADT were found to be correlated with each other , which likely confounds the results of these models . Similar to lane wid th, s houlder width was not found to be a significant factor. Although the body of research has consistently found that wider shoulders result in fewer crashes [3, 19, 52, 56] , the findings of this analysis are likely confounded by the correlation between s houlder width and AADT . Furthermore, as a roadside assessment was not conducted, the impacts of roadside characteristics could not be determined. The effect of roadside character istics presents another potential confounding variable with shoulder width, with regards to the effect of fixed object crashes, whereby both shoulder width and roadside conditions will affect the ability of drivers to correct course to avoid collision, as well as visibility at driveways . Model fit can be evaluated in many ways, two of which are log likelihood and Akaike information criterion (AIC). Log likelihood is the natural logarithm of the maximum likelihood funct ion, and is a measure of how closely d likelihood is a summation of the log likelihood of individual observation s. The equation for calculating log likelihood (LL) can be found below in Equation 10 [117] : ( 10 ) 76 Where, n is the number of observations, 2 represents the variance of a disturbance term, Y is the dependent variable (i.e., crash occurrence ), and X represents a matrix of the dependent variables. L og likelihood is a useful measure for comparing the goodness - of - fit for models using the same dataset, rather than comparing models with different datasets. Log likelihood values are negative and better - fitting models are closer to zero. AIC is directly related to log lik elihood. The equation for calculating AIC is shown as Equation 11 : ( 11 ) where Q is the number of parameters an d is the log - likelihood at convergence [117] . Similar to log likelihood, values of AIC closer to zero indicate a better fit, although, unlike log likelihood, the values of AIC are positive. As indicated in the equation, AIC penalizes the use of a large number of parameters. When comparing the mixed ef fects model in Table 15 with the fixed effects model in Table 5 2 (Appendix A ) , it can be seen that both AIC and log likelihood are improved in the mixed effects model, with values being closer to zero. Compared to the fixed effects model, the mixed effects model provides more conservative estimates regarding the crash effect of parameters. The standard deviations of the site - and county - specific random effects indicate that there is more variation between sites in general than there is variation in sites be tween counties. In addition, the overdispersion parameter for the mixed effects model is lower than on the fixed effects model; this means that when applying these models using the predictive method outlined in the HSM , the expected number of crashes at an y given location will be influenced more by the model parameters than with the fixed effects model. It is worth noting that, due to the rare and random nature of crashes, there are variations from year - to - year in crash occurrence 77 at any given site. A manua l review of the highest crash sites found that, while on average, they experience high crash frequency from year - to - year, but often experience a year or two with few - to - no crashes. Care should be taken when applying models to particularly high - crash sites, as models may under - predict crashes in these locations. 4.2.2 Paved Non - Federal Aid County Road Segments Paved non - federal aid segments, displayed in Table 16 , showed somewhat similar model results to those of federal aid segments. While horizontal curves with design speeds from 45 to 54 mph did not perform significantly different from base conditions, curves with design speeds from 40 - 44 mph experienced 2. 6 times more crashes than base conditions when the curve occupied the entire segment , and curv es with design speeds below 40 mph occupying an entire segment. experienced 4. 2 times more crashes than base conditions, which is comparable to federal aid segment performance. This is shown in Figure 10 , where curves of varying design speeds are compared, with the assumption that the curve occupies the entire segment, and all other base conditions prevail . Curves with design speeds between 45 and 49 mph and between 50 and 54 mph were not found to be statistically significant, but are shown on the figure. W hile not statistically significant from base conditions, curves with design speeds of 45 - 49 mph and curves with design speeds of 50 - 54 mph show slightly higher crash frequency from base conditions; due to the similar relative risk that these two categories have (1.308 for 50 - 54.9 mph and 1.310 for 45 - 49.9 mph) they appear as a single line on the chart . Also similar to federal aid roadways, the effects of lane width and paved shoulder width was not significant. This is somewhat surprising, as increases in su rface width are typically associated with reductions in crashes for reasons previously discussed. However, given that approximately 81 percent of the included paved non - federal aid segments were classified as local 78 roads, driver familiarity and travel beha vior will likely be much different than for arterials, which have provided the basis for the majority of prior research on the safety effects of shoulders, including the HSM SPFs. Furthermore, as was the case on federal aid segments, shoulder width and AAD T are positively correlated. The body of research is unclear on the effect of lane width on safety [19, 56] . It is difficult to untangle the effect of lane width and roadside characteristics on driver speed, particula rly on rural, low - volume roads. There are other potential confounds; it is possible that the safety benefit associated with wider lanes (i.e., more space to correct course) could be counteracted by the safety detriment caused by faster operating speeds ass ociated with wider lanes. Driveway densities of 25 driveways per mile or greater were associated with a 37 percent increase in crashes larger than the 31 percent increase found on federal aid roads. It is unclear why the effect is greater on non - federal aid highways; one potential reason could be reduced visibility. Similar to on federal aid segments, the diagnostic parameters (i.e., AIC and log likelihood) in the mixed effects model (Table 16 ) show a better fit than the corresponding fixed effects mode l, located in Appendix A (Table 53 ), as well as a lower overdispersion parameter. 79 Table 16 : Mixed Effects Negative Binomial Model Results for Paved Non - Federal Aid Segments Factor Description Est Exp(B) Std error P - value Intercept - 6.848 0.001 0.310 <0.001 Segment length O ffset, natural log of, miles AADT N atural log of, vehicles per day 0.800 2.225 0.047 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 0.269 1.308 0.431 0.533 45 - 49.9 mph C urved proportion of segment 0.270 1.310 0.568 0.635 40 - 44.9 mph C urved proportion of segment 0.944 2.570 0.446 0.034 <40 mph C urved proportion of segment 1.440 4.220 0.664 0.030 11 - ft lane Baseline 12 - or 13 - ft lane Width in feet - 0.067 0.935 0.110 0.544 9 - or 10 - ft lane Width in feet 0.043 1.044 0.069 0.539 0 - ft shoulder Baseline 1 - ft shoulder Width in feet 0.059 1.061 0.088 0.500 2 - ft shoulder or wider Width in feet 0.093 1.098 0.165 0.572 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.096 1.101 0.111 0.384 15 - to - 24.9 driveways per mile Binary indicator variable 0.072 1.075 0.115 0.532 > 25 driveways per mile Binary indicator variable 0.313 1.367 0.116 0.007 Site random effect 0.638 County random effect <0.001 Overdispersion parameter 0.022 AIC 8,418.4 Log likelihood - 4,193.2 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion ; mph = miles per hour; ft = foot 80 Figure 10 : Comparison of SPF crash results on paved county non - federal aid segments by curve design speed 81 4.2.3 Unpaved Non - Federal Aid Segments Unpaved non - feder al aid segments, displayed in Table 17 , showed somewhat different model results compared to those for the other segment types. While horizontal curves with design speeds below 55 mph did show increases in crashes, the relationship was not in the same manner as on paved roads (i.e., a clear pattern of ascending crash frequency with descending curve - feel comfortable traveling in the absence of traffic or wea ther conditions, is often below 55 mph on unpaved roads for a variety of reasons, including surface quality and visibility. Curves with design speeds below 55 mph showed a clear increase in crashes relative to base conditions , but in terms of the four cate gories of substandard horizontal curvature, they did not experience significantly different results from each other. The effects of the various design speed categories of horizontal curvature are shown in Figure 11 , where it was assumed the curve occupied the whole length of the segment and all other base conditions prevailed. Similar to other models, surface width was not found to be a significant factor. The model diagnostics (i.e., AIC and log likelihood) show that the mixed effects model below in Table 17 has a better fit than the fixed effects model located in Appendix A (Table 5 4 ). 82 Table 17 : Mixed Effects Negative Binomial Model Results for Unpaved Non - Federal Aid Segments Factor Description Est Exp(B) Std error P - value Intercept - 5.781 0.003 0.668 <0.001 Segment length O ffset, natural log of, miles AADT N atural log of, vehicles per day 0.608 1.836 0.051 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 1.715 5.556 0.367 <0.001 45 - 49.9 mph C urved proportion of segment 1.384 3.992 0.311 <0.001 40 - 44.9 mph C urved proportion of segment 1.270 3.561 0.382 0.001 <40 mph C urved proportion of segment 1.283 3.606 0.335 <0.001 Surface width W idth in feet, natural log of 0.080 1.083 0.235 0.733 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.156 1.169 0.086 0.070 15 - to - 24.9 driveways per mile Binary indicator variable 0.009 1.009 0.098 0.929 > 25 driveways per mile Binary indicator variable 0.163 1.177 0.108 0.130 County random effect 0.753 Overdispersion parameter 0.025 AIC 8,944.5 Log likelihood - 4,460.2 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion ; mph = miles per hour; ft = foot 83 Figure 11 : Comparison of SPF crash results on un paved county non - federal aid segments by curve design speed 4.2.4 Crash Modification Factors for Rural County Segmen ts From these results, a set of CMFs were developed for horizontal curvature and driveway density within each funding and pavement surface condition, and these are shown in Table 18 . These CMFs are the reciprocal of the Exp(B) values , or relative risk, in the results tables. The values in the results tables describe the effect of these site characteristics when deviating from base conditions, but the reciprocal is taken to determine the opposite effect (i.e., returning to base conditions). 84 Table 18 : CMFs Developed for Rural County Segments Original c ondition CMFs County FA County n on - FA Unpaved Remarks Design speed 50 - 54.9 mph 0. 48 ns 0. 18 Final condition: design speed > 55 mph Design speed 45 - 49.9 mph 0.3 3 ns 0.2 5 Design speed 40 - 44.9 mph 0.3 7 0.3 9 0.2 8 Design speed <40 mph 0.2 4 0.2 4 0. 28 5 - to - 14.9 driveways per mile 0.92 n s ns Final condition: <5 driveways per mile 15 - to - 24.9 driveways per mile 0.86 ns ns 25 driveways per mile or greater 0.80 0.76 ns Note: ns = Not significant 4.2. 5 Comparison to MDOT and Calibrated HSM SPFs Comparisons were also made between the three county road SPFs presented in this document and those previously developed for state - owned rural two - lane highways in Michigan [16] . In addition, a comparison with the H SM is included . The HSM model presented is calibrated for each class of roadway ; a discussion of the method for calibrating HSM models for county - owned roadways can be found in Chapter s 2 and 3 . Deer crashes were not included in the state - owned models, nor were they included in the HSM calibration; this is because, as previously noted , deer crashes were not included in the models developed for county roadways presented in this chapter. T he calibration factor calculated for county federal aid roadways was 0.79, for county non - federal aid the calibration factor was 0.87, and for unpaved the value was calculated to be 1.81. Calibrated models were evaluated for goodness - of - fit using mean abs olute deviation (MAD), which is a measure of how calculated values relate with the mean value of the original dataset. The equation for calculating MAD is shown in Equation 12 below , as presented in [117] : Where, n = the number of observations, i represents an individual observation, N observed = the number of crashes observed, and N predicted = the number of crashes observed in a given model. 85 On paved segments, calibrat ed models performed well compared to Michigan - specific models when evaluating using MAD. On paved federal aid roads, the model presented in this chapter had an MAD value of 0.34 while the calibrated model had a value of 0.3 1 . On paved non - federal aid roads , both the model presented in this chapter, as well as the calibrated model, had a MAD value of 0.17. For unpaved roads, the model in this chapter had an MAD value of 0.096, while the calibrated model had a value of 0.12 . MAD is useful to understanding how models compare with the dataset used to develop the models in the aggregate. However, in comparing models, it is also useful to compare plots of model results to determine how model shape differs. F igure 12 makes the need for county - specific, rather than calibrated, models clear; the HSM model, calibrated for county federal aid highways in Michigan, is a linear model, and underpredicts crashes below AADTs of approximately 3 ,800 and overpredicts when volumes are higher . Because the calibrated models are bas ed on the summation of all crashes within the dataset, they are susceptible to bias from outlier sites. All models were set at the same base conditions, which included 12 - f oo t lanes (or equivalent total surface width), 6 - f oo t shoulders, no driveways, and n o substandard horizontal curves . E ach SPF was then plotted as a function of AADT within the general range of traffic volumes . It can be seen that the shape of each curve is different, and that the elasticity of the AADT parameter affects the shape . For ins tance, below AADTs of approximately 80, gravel roads have the highest crash occurrence, but over 80 vehicles per day, paved federal aid has the highest crash frequency. The HSM calibrated models are linear, and do not take into account the differences in e lasticity with respect to AADT that can be found in the Michigan - specific models. 86 Figure 12 : Comparison of SPF total crash results at base conditions 4.3 Summary and Conclusions The safety performance of county or local roadways is rarely investigated to the same level of detail as stat e highways. However, several states , including Michigan , possess a substantial network of rural county highways. While SPFs exist within the HSM and other resources, in Michigan and elsewhere, coun ty - owned highways typically possess characteristics that differ considerably from those under the jurisdiction of the state DOT. This limits the usefulness of SPFs generated based on state - owned rural highways, including those developed by MDOT or 87 found wi thin the HSM . Furthermore, the SPFs contained in the HSM are only applicable to paved roads, which limits application for county and local road agencies for whom unpaved roads may be a substantial portion of their network. Another concern that arises is th e differences in design and maintenance standards between counties; however, in all models, the standard deviation of the random effect for site was greater than the random effect for county, indicating that there is greater variation between sites in gene ral than there is between counties. Since greater than one - half of rural crashes in Michigan occur off of the state highway system, there is a clear need for additional guidance on how to better design county roadways for safety, given the different design and user characteristics compared to rural state highways. To that end, research was undertaken to SPFs unique to county highways. To accomplish this objective, county highway inventory data from 30 counties across Michigan were obtained and paired with t raffic crashes from 2011 - 2015. Separate SPFs were then generated for paved county federal aid roadways, paved county non - federal aid roadways, and unpaved non - federal aid highways. Not surprisingly, the county SPF results were generally different than woul d be expected on state - owned facilities. County paved federal aid roadways showed a higher crash occurrence rate than county paved non - federal aid, the calibrated HSM model for county federal roadways (at volumes below 3, 8 00 vehicles per day) , and MDOT highways. However, lane width, roadway surface width, and paved shoulder width had little to no impact on crashes across each of the three county roadway models . Increasing driveway density was found to be associated with increased crash occurrence, although these results were only significant for crashes on paved county roadways. 88 The most consistent roadway geometry related results were associated with the presence of curve s with a design speed at or below 55 mph . This relationship was most clear on county federal aid segments, with lower curve radii associated with higher crash frequencies . On paved federal aid and unpaved non - federal aid highways, curves with substandard design speeds (i.e., less than 55 mph) resulted in higher crash frequency than base conditions , while on paved non - federal aid highways, only curves with design speeds below 45 mph performed significantly different from base conditions. The results of this research can be utilized towards various safety programs within Michigan and beyond. This includes performing network screening and crash prediction estimates to support local agency safety programs, in addition to providing similar support for the High Risk Rural Roads program. 89 5. SAFETY PERFORMANCE F UNCTIONS FOR RURAL MINOR ROAD STOP CONTROLLED INTERSECT IONS Rural two - way two - lane intersections were analyzed on county - owned roads in Michigan . Both three - leg (3ST) and four - leg (4ST) minor road stop - controlled intersections were analyzed, and intersections where the major road was state - owned , county federal aid, and county non - federal aid were included. Intersections with state - owned highways were included only if the minor cross - road was a county road . This was deemed important in order to fully capture the safety performance imp acts of stop - controlled intersections that included county roadways . However, it is important to note that when the major road was under state jurisdiction, the entire intersection was under the jurisdiction of MDOT rather than the county road commission, and subject to different design standards . In order to account for this, separate models were developed for each jurisdictional class . The sections that follow will include a data summary, an explanation of the analytical method, and results and discussion. 5.1 Data Summary, Data Screening, and Data Diagnostics The subsections below summarize the descriptive statistics for 3ST and 4ST intersections with stop control on the minor roadway . It should be noted that the free - flowing roadway was always designated as the major roadway, while the minor roadway was stop - controlled. For that reason, the minor AADT was greater than the major AADT in a small number of cases. The final dataset included a total of 5, 659 rural stop - controlled intersections, of wh ich there were slightly more 4ST intersections than 3ST . All 83 counties in Michigan were represented . The focus of this analysis was the evaluation of intersections with county - owned highways; sites where the 90 were included because they intersect with county highways, which are stop - controlled. 5.1.1 Rural Four Leg Stop - Controlled Intersections (4ST) Table 19 displays the number intersections from each county, by each of the three funding and jurisdictional cla sses. Considering intersections where the major road is under MDOT jurisdiction, road was county federal aid, 82 counties were included , and 16 counties had at least one major road non - federal aid intersection included in the sample. T he small sample of counties for non - federal aid roadways was due to the limited availability of traffic volume data for these roadways and gave further impetus to development of models based on ma jor road jurisdictional classification A map of the location of the 4ST intersections included is displayed in Figure 13 . 91 Figure 13 : Map of r ural f our l eg s top - c ontrolled (4ST) i ntersections Table 20 provides summary statistics (i.e., mean, minimum, maximum, standard deviation) for all relevant variables of interest considered during 4ST SPF development. Table 21 shows the same information for intersections whose major road is under MDOT jurisdiction, Table 22 for coun ty federal aid, and Table 23 for county non - federal aid. Approximately 57 percent of intersections were county federal aid, 33 percent were under MDOT jurisdiction, and county non - federal aid jurisdiction made up the remainder. Approximately 42 percent of intersections had street lighting present. Driveway within 211 feet (0.04 miles) of 4ST intersections were relatively sparse , with a mean of 2.7 per intersection , which is indicative of the fact that only rural areas were included in the sample . The majori ty of crashes (73 percent) were property damage only. Forty - five percent of intersections experienced at least one crash of 92 any kind, while 38 percent of intersections experienced at least one non - deer related crash during the 5 - year analysis period . The mean skew angle for the entire sample was 5. 7 degrees . The skew data was also categorized into a series of binary variables for analytical purposes, as follows: 0 degrees, 1 to 9 degrees, 10 to 39 degrees, greater than or equal to 40 degrees . These ca tegorization ranges were formulated based on the similarity of parameter estimates obtained from preliminary modeling efforts . Categorization of the skew variable in this manner allowed for improved model fit . A histogram showing the frequency of various s kew angle categories can be seen in Figure 14 , which shows that the vast majority of intersections have a skew angle less than five degrees, and very few intersections have skew angles of 40 degrees or more . 93 Table 19 : Represented Counties and Intersection Count by Major Roadway Class ( Four - Leg Intersections) County N umber of sites (major road) County N umber of sites (major road) State County FA County n on - FA Total State County FA County n on - FA Total Alco na 7 13 0 20 Lake 15 21 0 36 Alger 15 2 0 17 Lapeer 13 27 0 40 Allegan 18 39 0 57 Leelanau 11 4 0 15 Alpena 8 13 0 21 Lenawee 20 19 1 40 Antrim 8 10 0 18 Livingston 1 54 49 104 Arenac 10 16 0 26 Luce 9 2 0 11 Baraga 8 3 0 11 Mackinac 13 5 0 18 Barry 24 23 0 47 Macomb 1 31 8 40 Bay 7 36 0 43 Manistee 15 13 0 28 Benzie 8 7 0 15 Marquette 11 3 0 14 Berrien 18 25 0 43 Mason 8 22 0 30 Branch 5 32 0 37 Mecosta 16 14 0 30 Calhoun 15 14 1 30 Menominee 20 23 0 43 Cass 22 16 0 38 Midland 3 14 0 17 Charlevoix 10 6 0 16 Missaukee 9 11 0 20 Cheboygan 12 21 0 33 Monroe 6 22 0 28 Chippewa 20 23 0 43 Montcalm 23 33 0 56 Clare 8 8 0 16 Montmorency 7 6 0 13 Clinton 19 97 37 153 Muskegon 6 25 0 31 Crawford 6 11 0 17 Newaygo 25 30 0 55 Delta 10 5 0 15 Oakland 0 18 8 26 Dickinson 6 1 0 7 Oceana 13 19 0 32 Eaton 32 68 58 158 Ogemaw 11 11 0 22 Emmet 7 10 0 17 Ontonagon 8 1 0 9 Genesee 13 34 6 53 Osceola 17 24 0 41 Gladwin 12 15 0 27 Oscoda 9 8 0 17 Gogebic 10 2 0 12 Otsego 3 7 0 10 Grand Traverse 19 22 9 50 Ottawa 2 20 0 22 Gratiot 32 44 15 91 Presque Isle 17 9 0 26 Hillsdale 25 26 0 51 Roscommon 2 5 0 7 Houghton 6 9 0 15 Saginaw 22 47 0 69 Huron 25 29 1 55 St. Clair 6 31 0 37 Ingham 12 49 12 73 St. Joseph 17 16 0 33 Ionia 10 27 0 37 Sanilac 27 27 0 54 Iosco 14 23 12 49 Schoolcraft 9 3 0 12 Iron 15 2 0 17 Shiawassee 11 38 0 49 Isabella 3 38 0 41 Tuscola 24 27 0 51 Jackson 10 23 1 34 Van Buren 17 16 0 33 Kalamazoo 4 84 51 139 Washtenaw 6 29 0 35 Kalkaska 15 13 0 28 Wayne 0 3 0 3 Kent 27 88 51 166 Wexford 17 10 0 27 Keweenaw 3 0 0 3 TOTAL 1,028 1,775 3,20 3,123 94 Table 20 : Descriptive Statistics for Rural 4ST Intersections (All) Variable N (sites) Min 25th% 50th% 75th% Max Mean Std dev AADT - major roadway 3,123 57 1,064 2,251 3,824 21,414 3,049.3 2,691.4 AADT - minor roadway 3,123 2 288 638 1,090 10,009 981.2 1,056.5 Lighting provided 1,305 n/a n/a n/a n/a n/a 0.418 0.493 Overhead beacon provided 468 n/a n/a n/a n/a n/a 0.150 0.357 Skew angle 3,123 0.0 0.0 0.0 0.0 79.3 5.663 12.500 Skew 0 degrees 2,408 n/a n/a n/a n/a n/a 0.771 0.420 Skew 1 to 9 degrees 121 n/a n/a n/a n/a n/a 0.039 0.193 Skew 10 to 39 degrees 475 n/a n/a n/a n/a n/a 0.152 0.359 Skew > 40 degrees 119 n/a n/a n/a n/a n/a 0.038 0.191 Number of through lanes (major) 3,123 1 1 1 1 2 1.031 0.174 Number of through lanes (minor) 3,123 0 1 1 1 2 1.002 0.047 Number of right turn lanes 3,123 0 0 0 0 4 0.192 0.609 Number of left turn lanes 3,123 0 0 0 0 4 0.214 0.744 Railroad crossing within 211 feet of intersection 49 n/a n/a n/a n/a n/a 0.016 0.124 Driveway count 3,123 0 1 2 3 18 2.683 2.950 MDOT major roadway 1,028 n/a n/a n/a n/a n/a 0.329 0.470 County FA major roadway 1,775 n/a n/a n/a n/a n/a 0.568 0.495 County n on - FA major roadway 320 n/a n/a n/a n/a n/a 0.102 0.303 Variable Five - year crash count Annual crashes per intersection Min 25th% 50th% 75th% Max Mean Std dev Midblock total crashes 12,898 0 0 0 1 13 0.826 1.267 Midblock total non - deer crashes 10,671 0 0 0 1 13 0.683 1.184 Midblock fatal and injury non - deer crashes 3,585 0 0 0 0 6 0.230 0.565 Midblock property damage only non - deer crashes 9,313 0 0 0 1 11 0.596 0.993 Note : n/a = not applicable; std dev = standard deviation; min = minimum; max = maximum; N = number of 95 Table 21 : Descriptive Statistics for Rural 4ST Intersections (M ajor Road MDOT ) Variable N (sites) Min 25th% 50th% 75th% Max Mean Std dev AADT - major roadway 1,028 431 2,844 4,400 5,872 21,414 4,990.7 3,107.9 AADT - minor roadway 1,028 2 441 898 1,445 9,914 1,258.9 1,212.0 Lighting provided 622 n/a n/a n/a n/a n/a 0.605 0.489 Overhead beacon provided 221 n/a n/a n/a n/a n/a 0.215 0.411 Skew angle 1,028 0.0 0.0 0.0 8.7 75.4 8.902 15.128 Skew 0 degrees 678 n/a n/a n/a n/a n/a 0.660 0.474 Skew 1 to 9 degrees 52 n/a n/a n/a n/a n/a 0.051 0.219 Skew 10 to 39 degrees 224 n/a n/a n/a n/a n/a 0.218 0.413 Skew > 40 degrees 74 n/a n/a n/a n/a n/a 0.072 0.258 Number of through lanes (major) 1,028 1 1 1 1 2 1.077 0.266 Number of through lanes (minor) 1,028 1 1 1 1 2 1.001 0.031 Number of right turn lanes 1,028 0 0 0 0 4 0.483 0.895 Number of left turn lanes 1,028 0 0 0 0 4 0.470 1.006 Railroad crossing within 211 feet of intersection 26 n/a n/a n/a n/a n/a 0.025 0.157 Driveway count 1,028 0 1 2 4 17 3.207 3.446 Variable Five - year crash count Annual crashes per intersection Min 25th% 50th% 75th% Max Mean Std dev Midblock total crashes 6,228 0 0 1 2 13 1.212 1.543 Midblock total non - deer crashes 5,139 0 0 0 1 13 1.000 1.461 Midblock fatal and injury non - deer crashes 1,614 0 0 0 0 6 0.314 0.669 Midblock property damage only non - deer crashes 4,614 0 0 1 1 11 0.898 1.222 Note: n/a = not applicable; std dev = standard deviation; min = minimum; max = maximum; N = number of 96 Table 22 : Descriptive Statistics for Rural 4ST Intersections ( Major Road County FA) Variable N (sites) Min 25th% 50th% 75th% Max Mean Std dev AADT - major roadway 1,775 68 1,013 1,815 2,756 12,191 2,360.5 1,837.3 AADT - minor roadway 1,775 10 336 632 1,054 10,009 949.4 981.3 Lighting provided 662 n/a n/a n/a n/a n/a 0.373 0.484 Overhead beacon provided 244 n/a n/a n/a n/a n/a 0.137 0.344 Skew angle 1,775 0.0 0.0 0.0 0.0 79.3 4.358 10.965 Skew 0 degrees 1,444 n/a n/a n/a n/a n/a 0.814 0.390 Skew 1 to 9 degrees 62 n/a n/a n/a n/a n/a 0.035 0.184 Skew 10 to 39 degrees 226 n/a n/a n/a n/a n/a 0.127 0.333 Skew > 40 degrees 43 n/a n/a n/a n/a n/a 0.024 0.154 Number of through lanes (major) 1,775 1 1 1 1 2 1.011 0.103 Number of through lanes (minor) 1,775 0 1 1 1 2 1.002 0.058 Number of right turn lanes 1,775 0 0 0 0 4 0.059 0.339 Number of left turn lanes 1,775 0 0 0 0 4 0.103 0.572 Railroad crossing within 211 feet of intersection 22 n/a n/a n/a n/a n/a 0.012 0.111 Driveway count 1,775 0 1 2 3 18 2.596 2.760 Variable Five - year crash count Annual crashes per intersection Min 25th% 50th% 75th% Max Mean Std dev Midblock total crashes 6,357 0 0 0 1 10 0.716 1.112 Midblock total non - deer crashes 5,287 0 0 0 1 10 0.596 1.040 Midblock fatal and injury non - deer crashes 1,883 0 0 0 0 6 0.212 0.532 Midblock property damage only non - deer crashes 4,474 0 0 0 1 8 0.504 0.861 Note: n/a = not applicable; std dev = standard deviation; min = minimum; max = maximum; N = number of 97 Table 23 : Descriptive Statistics for Rural 4ST Intersections ( Major Road County Non - FA) Variable N (sites) Min 25th% 50th% 75th% Max Mean Std dev AADT - major roadway 320 57 230 438 756 5,255 633.2 604.3 AADT - minor roadway 320 18 98 180 271 2,030 265.2 262.7 Lighting provided 21 n/a n/a n/a n/a n/a 0.066 0.248 Overhead beacon provided 3 n/a n/a n/a n/a n/a 0.009 0.096 Skew angle 320 0.0 0.0 0.0 0.0 55.0 2.491 8.351 Skew 0 degrees 286 n/a n/a n/a n/a n/a 0.894 0.308 Skew 1 to 9 degrees 7 n/a n/a n/a n/a n/a 0.022 0.146 Skew 10 to 39 degrees 25 n/a n/a n/a n/a n/a 0.078 0.268 Skew > 40 degrees 2 n/a n/a n/a n/a n/a 0.006 0.079 Number of through lanes (major) 320 1 1 1 1 1 1.000 0.000 Number of through lanes (minor) 320 1 1 1 1 1 1.000 0.000 Number of right turn lanes 0 0 0 0 0 0 0.000 0.000 Number of left turn lanes 1 0 0 0 0 2 0.006 0.112 Railroad crossing within 211 feet of intersection 1 n/a n/a n/a n/a n/a 0.003 0.056 Driveway count 320 0 0 1 2 8 1.484 1.465 Variable Five - year crash count Annual crashes per intersection Min 25th% 50th% 75th% Max Mean Std dev Midblock total crashes 313 0 0 0 0 4 0.196 0.483 Midblock total non - deer crashes 245 0 0 0 0 4 0.153 0.428 Midblock fatal and injury non - deer crashes 88 0 0 0 0 2 0.055 0.239 Midblock property damage only non - deer crashes 225 0 0 0 0 3 0.141 0.398 Note: n/a = not applicable; std dev = standard deviation; min = minimum; max = maximum; N = number of Figure 14 : Distribution of s kew angle across 4ST intersections 98 5.1.1.1 Data Diagnostics Prior to SPF development, various data diagnostics were initially conducted to examine general trends across all locations for each facility type. This included assessment of the relationships between AADT and annual crash frequency , with scatterplots of these relationships generated for total and deer - ex cluded crashes for 4ST intersections, which are shown in Figure 15 . Crash severity and crash type distributions were also reported and analyzed . a.) Total intersection crashes ( 4 ST) b.) Deer - excluded intersection crashes ( 4 ST) Figure 15 : Annual intersection crashes vs AADT, 4 ST (2011 - 2015) Table s 24 - 27 show the crash severity and crash type distributions for rural four - leg intersections . It can be observed that MDOT intersections (Table 25) have a lower proportion of crashes involving fatalities and/or injuries compared to the other jurisdictions (Tables 26 - 27) . In addition, 4ST rural Michigan intersections (Table 24) experience a lower proportion of fatal/injury crashes than the default di stributions presented in Chapter 10 of the HSM [3] . Within the crash type distribution , angle collisions comprised a far greater proportion of intersection crashes for intersections under county jurisdictions compared to MDOT i ntersections. A potential explanation for this situation is the available intersection sight distance at MDOT 99 intersections as compared to the county road system. This could manifest either in horizontal sight triangles clear of obstructions or vertical si ght distance along the approaches. Table 24 : Crash Severity and Crash Type Distributions for Rural 4 ST Intersections (All) Crash s everity l evel, c ollision t ype, or l ight c ondition Count of i ntersection c rashes (2011 - 2015) Percent of t otal i ntersection c rashes Fatal (Type K) 133 1.03% Incapacitating i njury (Type A) 486 3.77% Other i njury (Type B+C) 2,966 23.00% Fatal + i njury (Type K+ABC) 3,585 27.80% Property d amage o nly (Type PDO) 9,313 72.20% Single m otor v ehicle 4,066 31.52% Single m otor v ehicle ( d eer e xcluded) 1,862 14.44% Deer c rashes 2,227 17.27% Multiple v ehicle c rashes 8,778 68.06% Day c rashes 8,545 66.25% Dark c rashes 4,353 33.75% Total n on - d eer c rashes (5 years) 10,671 82.73% Total c rashes (5 years) 12,898 100.00% Collision t ype Percent of FI i ntersection c rashes Percent of PDO i ntersection c rashes Percent of t otal i ntersection cr ashes Single - vehicle crashes Collision with deer 0.98% 23.29% 17.27% Collision with bicycle 0.45% 0.02% 0.14% Collision with pedestrian 0.81% 0.08% 0.28% Other single - vehicle crash 12.05% 15.93% 14.68% Total single - vehicle crash 13.03% 39.22% 31.94% Multiple - vehicle crashes Angle collision 60.42% 27.67% 36.77% Head - on collision 1.67% 0.59% 0.89% Read - end collision 10.49% 14.76% 13.58% Sideswipe collision 1.95% 14.76% 4.02% Other multiple - vehicle collision 12.44% 2.99% 12.79% Total multiple - vehicle collision 86.97% 60.78% 68.06% Total c rashes 100.00% 100.00% 100.00% 100 Table 25 : Crash Severity and Crash Type Distributions for Rural 4ST Intersections ( MDOT ) Crash s everity l evel, c ollision t ype, or l ight c ondition Count of i ntersection c rashes (2011 - 2015) Percent of t otal i ntersection c rashes Fatal (Type K) 53 0.85% Incapacitating i njury (Type A) 220 3.53% Other injury (Type B+C) 1,341 21.53% Fatal + i njury (Type K+ABC) 1,614 25.92% Property d amage o nly (Type PDO) 4,614 74.08% Single m otor v ehicle 1,865 29.95% Single m otor v ehicle ( d eer e xcluded) 783 12.57% Deer c rashes 1,089 17.49% Multiple v ehicle c rashes 4,331 69.54% Day c rashes 4,160 66.80% Dark c rashes 2,068 33.20% Total n on - d eer c rashes (5 years) 5,139 82.51% Total c rashes (5 years) 6,228 100.00% Collision t ype Percent of FI i ntersection c rashes Percent of PDO i ntersection c rashes Percent of t otal i ntersection cr ashes Single - vehicle crashes Collision with deer 0.93% 23.13% 17.49% Collision with bicycle 0.56% 0.04% 0.18% Collision with pedestrian 1.05% 0.09% 0.34% Other single - vehicle crash 10.59% 13.96% 12.97% Total single - vehicle crash 11.52% 37.08% 30.46% Multiple - vehicle crashes Angle collision 53.04% 24.82% 32.13% Head - on collision 1.86% 0.59% 0.92% Read - end collision 14.31% 17.23% 16.47% Sideswipe collision 2.48% 17.23% 4.91% Other multiple - vehicle collision 16.79% 3.06% 15.11% Total multiple - vehicle collision 88.48% 62.92% 69.54% Total c rashes 100.00% 100.00% 100.00% 101 Table 26 : Crash Severity and Crash Type Distributions for Rural 4 ST Intersections (County FA) Crash s everity l evel, c ollision t ype, or l ight c ondition Count of i ntersection c rashes (2011 - 2015) Percent of t otal i ntersection c rashes Fatal (Type K) 76 1.20% Incapacitating i njury (Type A) 259 4.07% Other injury (Type B+C) 1,548 24.35% Fatal + i njury (Type K+ABC) 1,883 29.62% Property d amage o nly (Type PDO) 4,474 70.38% Single m otor v ehicle 2,064 32.47% Single m otor v ehicle ( d eer e xcluded) 1,009 15.87% Deer c rashes 1,070 16.83% Multiple v ehicle c rashes 4,272 67.20% Day c rashes 4,191 65.93% Dark c rashes 2,166 34.07% Total n on - d eer c rashes (5 years) 5,287 83.17% Total c rashes (5 years) 6,357 100.00% Collision t ype Percent of FI i ntersection c rashes Percent of PDO i ntersection c rashes Percent of t otal i ntersection cr ashes Single - vehicle crashes Collision with deer 1.06% 23.13% 16.83% Collision with bicycle 0.37% 0.00% 0.11% Collision with pedestrian 0.58% 0.07% 0.22% Other single - vehicle crash 13.17% 17.48% 15.97% Total single - vehicle crash 14.23% 40.61% 32.80% Multiple - vehicle crashes Angle collision 65.91% 30.38% 40.90% Head - on collision 1.59% 0.56% 0.87% Read - end collision 7.54% 12.72% 11.18% Sideswipe collision 1.54% 12.72% 3.30% Other multiple - vehicle collision 9.19% 3.02% 10.95% Total multiple - vehicle collision 85.77% 59.39% 67.20% Total c rashes 100.00% 100.00% 100.00% 102 Table 27 : Crash Severity and Crash Type Distributions for Rural 4 ST Intersections (County Non - FA) Crash s everity l evel, c ollision t ype, or l ight c ondition Count of i ntersection c rashes (2011 - 2015) Percent of t otal i ntersection c rashes Fatal (Type K) 4 1.28% Incapacitating i njury (Type A) 7 2.24% Other injury (Type B+C) 77 24.60% Fatal + i njury (Type K+ABC) 88 28.12% Property d amage o nly (Type PDO) 225 71.88% Single m otor v ehicle 137 43.77% Single m otor v ehicle ( d eer e xcluded) 70 22.36% Deer c rashes 68 21.73% Multiple v ehicle c rashes 175 55.91% Day c rashes 194 61.98% Dark c rashes 119 38.02% Total n on - d eer c rashes (5 years) 245 78.27% Total c rashes (5 years) 313 100.00% Collision t ype Percent of FI i ntersection c rashes Percent of PDO i ntersection c rashes Percent of t otal i ntersection cr ashes Single - vehicle crashes Collision with deer 0.00% 29.78% 21.73% Collision with bicycle 0.00% 0.00% 0.00% Collision with pedestrian 1.14% 0.00% 0.32% Other single - vehicle crash 14.77% 25.78% 22.36% Total single - vehicle crash 14.77% 55.56% 44.09% Multiple - vehicle crashes Angle collision 78.41% 32.44% 45.37% Head - on collision 0.00% 1.33% 0.96% Read - end collision 3.41% 4.89% 4.47% Sideswipe collision 1.14% 4.89% 0.96% Other multiple - vehicle collision 2.27% 0.89% 4.15% Total multiple - vehicle collision 85.23% 44.44% 55.91% Total c rashes 100.00% 100.00% 100.00% 103 5.1.2 Rural Three - Leg Stop - Controlled Intersections (3ST) Table 28 displays the number of intersections included from each county, by each of the three intersections where t represented for county federal aid major road intersections, while 15 counties had at least one major road non - federal aid intersection included in the sample. The small sample of counties f or non - federal aid roadways was due to the limited availability of traffic volume data for these roadways. The lack of statewide coverage for non - federal aid intersections further emphasized the importance of developing separate models across the three jur isdictional classes of roadways. A map of the location of the 3ST intersections included is displayed in Figure 16 . Figure 16 : Map of r ural t hree - l eg stop - c ontrolled (3ST) i ntersection l ocations 104 Table 28 : Represented Counties and Intersection Count by Major Roadway Class ( Three - Leg Intersections) County Number of sites (major road) County Number of sites (major road) State County FA County non - FA Total State County FA County non - FA Total Alcona 8 8 0 16 Lake 4 11 0 15 Alger 9 8 0 17 Lapeer 2 12 1 15 Allegan 6 26 0 32 Leelanau 12 9 0 21 Alpena 7 5 0 12 Lenawee 14 18 0 32 Antrim 19 18 0 37 Livingston 5 64 104 173 Arenac 3 3 0 6 Luce 6 6 0 12 Baraga 13 9 0 22 Mackinac 17 14 0 31 Barry 22 20 0 42 Macomb 4 12 8 24 Bay 1 4 0 5 Manistee 9 10 0 19 Benzie 22 7 0 29 Marquette 23 12 1 36 Berrien 7 6 0 13 Mason 5 11 0 16 Branch 12 19 0 31 Mecosta 6 10 0 16 Calhoun 6 17 3 26 Menominee 15 20 0 35 Cass 13 10 0 23 Midland 3 11 0 14 Charlevoix 13 12 0 25 Missaukee 7 10 0 17 Cheboygan 12 13 0 25 Monroe 4 16 0 20 Chippewa 17 8 0 25 Montcalm 5 19 0 24 Clare 4 5 0 9 Montmorency 11 6 0 17 Clinton 2 55 25 82 Muskegon 1 12 0 13 Crawford 5 9 0 14 Newaygo 13 24 0 37 Delta 17 13 0 30 Oakland 0 39 4 43 Dickinson 7 4 0 11 Oceana 5 24 0 29 Eaton 21 44 68 133 Ogemaw 2 13 0 15 Emmet 6 13 0 19 Ontonagon 14 3 0 17 Genesee 14 12 3 29 Osceola 7 12 0 19 Gladwin 15 3 0 18 Oscoda 6 6 0 12 Gogebic 10 7 0 17 Otsego 4 6 0 10 Grand Traverse 20 27 8 55 Ottawa 1 8 0 9 Gratiot 1 41 7 49 Presque Isle 12 8 0 20 Hillsdale 14 10 0 24 Roscommon 3 12 0 15 Houghton 8 7 0 15 Saginaw 5 11 0 16 Huron 17 3 0 20 St. Clair 4 22 0 26 Ingham 4 52 11 67 St. Joseph 13 18 0 31 Ionia 11 10 0 21 Sanilac 7 8 0 15 Iosco 19 28 5 52 Schoolcraft 23 8 0 31 Iron 14 10 0 24 Shiawassee 6 18 0 24 Isabella 0 7 0 7 Tuscola 7 2 0 9 Jackson 16 25 0 41 Van Buren 5 15 0 20 Kalamazoo 20 97 135 252 Washtenaw 5 17 0 22 Kalkaska 6 11 0 17 Wayne 0 4 0 4 Kent 23 78 47 148 Wexford 8 6 0 14 Keweenaw 6 2 0 8 Total 773 1,333 430 2,536 105 Table 29 provides summary statistics for all relevant variables of interest considered during 3ST SPF development. Table 30 shows the same information for intersections whose major road is under MDOT jurisdiction , Table 31 for county federal aid, and Table 32 for county non - federal aid. More than 52 percent of intersections were county federal aid, 31 percent were under the ju risdiction of MDOT, and the remainder were county non - federal aid jurisdictions. Relative to 4ST intersections, a lower proportion of 3ST intersections were lit, with a round 34 percent of intersections having lighting present. Driveway counts were also sli ghtly lower for 3ST, with a mean of 1.9 per intersection. The majority of crashes (75 percent) were property damage only. Thirty - one percent of intersections experienced any kind of crash, while 25 percent of intersections experienced a non - deer related cr ash . Interestingly, compared to 4 ST intersections, skew was more common at 3 ST intersections, with 6 7 percent of intersections possessing no skew compared to 7 7 percent of 4 ST intersections . The skew was also more extreme at 3 ST intersections, as 7.2 percent of intersections possessed skew greater than 40 degrees, compared to only 3.8 percent of 4 ST intersections . The average skew as also higher at 3 ST when compared to 4 ST intersections ( 9.00 degrees vs. 5.66 degrees). A histogram showing the frequenc y of various skew angle categories can be seen in Figure 17 , which shows that the vast majority of intersections have a skew angle less than five degrees, and very few intersections have skew angles of 40 degrees or more. 106 Table 29 : Descriptive Statistics for Rural 3ST Intersections (All) Variable N (sites) Min 25th% 50th% 75th% Max Mean Std d ev AADT - major roadway 2,536 26 771 1,740 3,140 32,006 2,651.3 2,822.0 AADT - minor roadway 2,536 4 167 456 871 8,480 803.7 991.6 Lighting provided 863 n/a n/a n/a n/a n/a 0.340 0.474 Overhead beacon provided 90 n/a n/a n/a n/a n/a 0.035 0.185 Skew angle 2,536 0.0 0.0 0.0 8.0 80.0 9.003 16.238 Skew 0 degrees 1,669 n/a n/a n/a n/a n/a 0.658 0.474 Skew 1 to 9 degrees 151 n/a n/a n/a n/a n/a 0.060 0.237 Skew 10 to 39 degrees 534 n/a n/a n/a n/a n/a 0.211 0.408 Skew > 40 degrees 182 n/a n/a n/a n/a n/a 0.072 0.258 Number of through lanes (major) 2,536 1 1 1 1 2 1.030 0.169 Number of through lanes (minor) 2,536 0 1 1 1 1 0.948 0.223 Number of right turn lanes 2,536 0 0 0 0 2 0.123 0.396 Number of left turn lanes 2,536 0 0 0 0 3 0.104 0.380 Railroad crossing within 211 feet of intersection 47 n/a n/a n/a n/a n/a 0.019 0.135 Driveway count 2,536 0 0 1 2 13 1.936 2.079 MDOT major roadway 773 n/a n/a n/a n/a n/a 0.305 0.460 County FA major roadway 1,333 n/a n/a n/a n/a n/a 0.526 0.499 County n on - FA major roadway 430 n/a n/a n/a n/a n/a 0.170 0.375 Variable Five - year crash count Annual crashes per intersection Min 25th% 50th% 75th% Max Mean Std dev Midblock total crashes 6,178 0 0 0 1 14 0.487 0.924 Midblock total non - deer crashes 4,663 0 0 0 0 13 0.368 0.811 Midblock fatal and injury non - deer crashes 1,188 0 0 0 0 5 0.094 0.334 Midblock property damage only non - deer crashes 3,475 0 0 0 0 12 0.274 0.682 107 Table 30 : Descriptive Statistics for Rural 3ST Intersections ( MDOT ) Variable N (sites) Min 25th% 50th% 75th% Max Mean Std dev AADT - major roadway 773 177 2,229 4,100 5,787 32,006 4,815.4 3,592.1 AADT - minor roadway 773 10 314 752 1,327 8,200 1,187.4 1,267.0 Lighting provided 431 n/a n/a n/a n/a n/a 0.558 0.497 Overhead beacon provided 64 n/a n/a n/a n/a n/a 0.083 0.276 Skew angle 773 0.0 0.0 0.0 18.0 80.0 13.026 18.217 Skew 0 degrees 466 n/a n/a n/a n/a n/a 0.603 0.489 Skew 1 to 9 degrees 50 n/a n/a n/a n/a n/a 0.065 0.246 Skew 10 to 39 degrees 189 n/a n/a n/a n/a n/a 0.245 0.430 Skew > 40 degrees 68 n/a n/a n/a n/a n/a 0.088 0.283 Number of through lanes (major) 773 1 1 1 1 2 1.089 0.285 Number of through lanes (minor) 773 0 1 1 1 1 0.868 0.338 Number of right turn lanes 773 0 0 0 0 2 0.320 0.606 Number of left turn lanes 773 0 0 0 0 3 0.263 0.573 Railroad crossing within 211 feet of intersection 19 n/a n/a n/a n/a n/a 0.025 0.155 Driveway count 773 0 0 1 3 12 2.025 2.271 Variable Five - year crash count Annual crashes per intersection Min 25th% 50th% 75th% Max Mean Std dev Midblock total crashes 3,098 0 0 0 1 14 0.802 1.223 Midblock total non - deer crashes 2,364 0 0 0 1 13 0.612 1.109 Midblock fatal and injury non - deer crashes 609 0 0 0 0 5 0.158 0.439 Midblock property damage only non - deer crashes 1,755 0 0 0 0 12 0.454 0.931 108 Table 31 : Descriptive Statistics for Rural 3ST Intersections (Major Road County FA) Variable N (sites) Min 25th% 50th% 75th% Max Mean Std dev AADT - major roadway 1,333 87 867 1,583 2,510 15,947 2,073.9 1,729.3 AADT - minor roadway 1,333 5 207 508 877 8,480 778.7 845.9 Lighting provided 416 n/a n/a n/a n/a n/a 0.312 0.463 Overhead beacon provided 26 n/a n/a n/a n/a n/a 0.020 0.138 Skew angle 1,333 0.0 0.0 0.0 5.5 75.0 8.298 15.701 Skew 0 degrees 900 n/a n/a n/a n/a n/a 0.675 0.468 Skew 1 to 9 degrees 88 n/a n/a n/a n/a n/a 0.066 0.248 Skew 10 to 39 degrees 257 n/a n/a n/a n/a n/a 0.193 0.395 Skew > 40 degrees 88 n/a n/a n/a n/a n/a 0.066 0.248 Number of through lanes (major) 1,333 1 1 1 1 2 1.004 0.061 Number of through lanes (minor) 1,333 0 1 1 1 1 0.977 0.148 Number of right turn lanes 1,333 0 0 0 0 2 0.049 0.229 Number of left turn lanes 1,333 0 0 0 0 3 0.046 0.248 Railroad crossing within 211 feet of intersection 25 n/a n/a n/a n/a n/a 0.019 0.136 Driveway count 1,333 0 1 2 2 13 2.017 2.103 Variable Five - year crash count Annual crashes per intersection Min 25th% 50th% 75th% Max Mean Std dev Midblock total crashes 2,862 0 0 0 1 7 0.429 0.782 Midblock total non - deer crashes 2,138 0 0 0 0 7 0.321 0.670 Midblock fatal and injury non - deer crashes 541 0 0 0 0 3 0.081 0.300 Midblock property damage only non - deer crashes 1,597 0 0 0 0 7 0.240 0.571 109 Table 32 : Descriptive Statistics for Rural 3ST Intersections ( Major Road County Non - FA) Variable N (sites) Min 25th% 50th% 75th% Max Mean Std dev AADT - major roadway 430 26 177 346 560 9,871 550.8 801.9 AADT - minor roadway 430 4 67 113 193 2,436 191.4 234.2 Lighting provided 16 n/a n/a n/a n/a n/a 0.037 0.189 Overhead beacon provided 0 n/a n/a n/a n/a n/a 0.000 0.000 Skew angle 430 0.0 0.0 0.0 0.0 65.0 3.956 11.761 Skew 0 degrees 369 n/a n/a n/a n/a n/a 0.858 0.349 Skew 1 to 9 degrees 9 n/a n/a n/a n/a n/a 0.021 0.143 Skew 10 to 39 degrees 35 n/a n/a n/a n/a n/a 0.081 0.274 Skew > 40 degrees 17 n/a n/a n/a n/a n/a 0.040 0.195 Number of through lanes (major) 430 1 1 1 1 2 1.002 0.048 Number of through lanes (minor) 430 0 1 1 1 1 0.998 0.048 Number of right turn lanes 430 0 0 0 0 1 0.002 0.048 Number of left turn lanes 430 0 0 0 0 1 0.002 0.048 Railroad crossing within 211 feet of intersection 3 n/a n/a n/a n/a n/a 0.007 0.083 Driveway count 430 0 0 1 2 8 1.523 1.519 Variable Five - year crash count Annual crashes per intersection Min 25th% 50th% 75th% Max Mean Std dev Midblock total crashes 218 0 0 0 0 3 0.101 0.340 Midblock total non - deer crashes 161 0 0 0 0 3 0.075 0.290 Midblock fatal and injury non - deer crashes 38 0 0 0 0 1 0.018 0.132 Midblock property damage only non - deer crashes 123 0 0 0 0 2 0.057 0.259 110 Figure 17 : Distribution of s kew angle across 3 ST intersections 5.1.2.1 Data Diagnostics Prior to SPF development, various data diagnostics were initially conducted to examine general trends across all locations for each facility type. This included assessment of the relationships between AADT and annual crash frequency with scatterplots of these relationships generated for total and deer - exc luded crashes for 3ST intersections, which are shown in Figure 18 . Crash severity and crash type distributions were also reported and analyzed . 111 a.) Total intersection crashes (3ST) b.) Deer - excluded intersection crashes (3ST) Figure 18 : Annual intersection crashes vs AADT, 3ST (2011 - 2015) Table s 33 - 36 show the crash severity and crash type distributions for rural three leg intersections . In comparison to the default distributions presented in Chapter 10 of the HSM [3] , (Table 33) . In consideration of crash types, a relatively high proportion of single vehicle crashes involved deer (approximately 25 percent) , likely contrib uting to the lower severity compared to the HSM . Angle and rear - end collisions are the most prevalent specific categories of multiple - vehicle crashes at 3ST intersections in Michigan, which is consistent with the default distributions in the HSM . Angle cra shes make up 11 percent of crashes at 3ST intersections compared with 37 percent of crashes at 4ST intersections. The proportion of crashes occurring in dark conditions is notably higher than the default distribution in the HSM [3] , again, likely due to the high proportion of deer crashes. Compared with 4ST intersections, crashes at 3ST intersections tend to be less severe, with 80.4 percent of crashes being PDO at 3ST intersections, compared with 72.2 percent at 4ST. In additi on, the proportion of multiple - vehicle crashes is much lower at 3ST intersections, with only 43.4 percent 112 being multiple - vehicle, compared with 68.06 percent at 4ST intersections, likely reflecting the reduced number of conflict points at 3ST. Table 33 : Crash Severity and Crash Type Distributions for Rural 3ST Intersections (All) Crash s everity l evel, c ollision t ype, or l ight c ondition Count of i ntersection c rashes (2011 - 2015) Percent of t otal i ntersection c rashes Fatal (Type K) 26 0.42% Incapacitating i njury (Type A) 163 2.64% Other injury (Type B+C) 1,022 16.54% Fatal + i njury (Type K+ABC) 1,211 19.60% Property d amage o nly (Type PDO) 4,967 80.40% Single m otor v ehicle 3,472 56.20% Single m otor v ehicle ( d eer e xcluded) 1,972 31.92% Deer c rashes 1,515 24.52% Multiple v ehicle c rashes 2,681 43.40% Day c rashes 3,350 54.22% Dark c rashes 2,828 45.78% Total n on - d eer c rashes (5 years) 4,663 75.48% Total c rashes (5 years) 6,178 100.00% Collision t ype Percent of FI i ntersection c rashes Percent of PDO i ntersection c rashes Percent of t otal i ntersection cr ashes Single - vehicle crashes Collision with deer 1.90% 29.74% 24.52% Collision with bicycle 0.66% 0.10% 0.21% Collision with pedestrian 0.74% 0.06% 0.19% Other single - vehicle crash 41.29% 30.14% 32.08% Total single - vehicle crash 43.19% 59.88% 56.60% Multiple - vehicle crashes Angle collision 18.83% 9.32% 11.18% Head - on collision 3.39% 0.79% 1.29% Read - end collision 14.04% 14.46% 14.37% Sideswipe collision 2.56% 14.46% 3.32% Other multiple - vehicle collision 18.00% 1.11% 13.22% Total multiple - vehicle collision 56.81% 40.12% 43.40% Total c rashes 100.00% 100.00% 100.00% 113 Table 34 : Crash Severity and Crash Type Distributions for Rural 3ST Intersections (Major Road MDOT ) Crash s everity l evel, c ollision t ype, or l ight c ondition Count of i ntersection c rashes (2011 - 2015) Percent of t otal i ntersection c rashes Fatal (Type K) 12 0.39% Incapacitating i njury (Type A) 82 2.65% Other injury (Type B+C) 531 17.14% Fatal + i njury (Type K+ABC) 625 20.17% Property d amage o nly (Type PDO) 2,473 79.83% Single m otor v ehicle 1,477 47.68% Single m otor v ehicle ( d eer e xcluded) 746 24.08% Deer c rashes 734 23.69% Multiple v ehicle c rashes 1,607 51.87% Day c rashes 1,819 58.72% Dark c rashes 1,279 41.28% Total n on - d eer c rashes (5 years) 2,364 76.31% Total c rashes (5 years) 3,098 100.00% Collision t ype Percent of FI i ntersection c rashes Percent of PDO i ntersection c rashes Percent of t otal i ntersection cr ashes Single - vehicle crashes Collision with deer 2.56% 28.91% 23.69% Collision with bicycle 0.80% 0.12% 0.26% Collision with pedestrian 0.64% 0.08% 0.19% Other single - vehicle crash 30.72% 22.97% 24.44% Total single - vehicle crash 33.28% 51.88% 48.13% Multiple - vehicle crashes Angle collision 18.72% 9.46% 11.33% Head - on collision 3.52% 0.73% 1.29% Read - end collision 19.68% 20.06% 19.98% Sideswipe collision 3.36% 20.06% 3.91% Other multiple - vehicle collision 21.44% - 2.18% 15.36% Total multiple - vehicle collision 66.72% 48.12% 51.87% Total c rashes 100.00% 100.00% 100.00% 114 Table 35 : Crash Severity and Crash Type Distributions for Rural 3ST Intersections (Major Road County FA) Crash s everity l evel, c ollision t ype, or l ight c ondition Count of i ntersection c rashes (2011 - 2015) Percent of t otal i ntersection c rashes Fatal (Type K) 12 0.42% Incapacitating i njury (Type A) 77 2.69% Other injury (Type B+C) 459 16.04% Fatal + i njury (Type K+ABC) 548 19.15% Property d amage o nly (Type PDO) 2,314 80.85% Single m otor v ehicle 1,836 64.15% Single m otor v ehicle ( d eer e xcluded) 1,124 39.27% Deer c rashes 724 25.30% Multiple v ehicle c rashes 1,016 35.50% Day c rashes 1,425 49.79% Dark c rashes 1,437 50.21% Total n on - d eer c rashes (5 years) 2,138 74.70% Total c rashes (5 years) 2,862 100.00% Collision t ype Percent of FI i ntersection c rashes Percent of PDO i ntersection c rashes Percent of t otal i ntersection cr ashes Single - vehicle crashes Collision with deer 1.28% 30.47% 25.30% Collision with bicycle 0.36% 0.09% 0.14% Collision with pedestrian 0.91% 0.04% 0.21% Other single - vehicle crash 51.82% 36.73% 39.20% Total single - vehicle crash 53.10% 67.20% 64.50% Multiple - vehicle crashes Angle collision 19.16% 9.08% 11.01% Head - on collision 3.28% 0.86% 1.33% Read - end collision 8.21% 9.12% 8.94% Sideswipe collision 1.46% 9.12% 2.73% Other multiple - vehicle collision 14.78% 4.62% 11.50% Total multiple - vehicle collision 46.90% 32.80% 35.50% Total c rashes 100.00% 100.00% 100.00% 115 Table 36 : Crash Severity and Crash Type Distributions for Rural 3ST Intersections ( Major Road County Non - FA) Crash s everity l evel, c ollision t ype, or l ight c ondition Count of i ntersection c rashes (2011 - 2015) Percent of t otal i ntersection c rashes Fatal (Type K) 2 0.92% Incapacitating i njury (Type A) 4 1.83% Other injury (Type B+C) 32 14.68% Fatal + i njury (Type K+ABC) 38 17.43% Property d amage o nly (Type PDO) 180 82.57% Single m otor v ehicle 159 72.94% Single m otor v ehicle ( d eer e xcluded) 102 46.79% Deer c rashes 57 26.15% Multiple v ehicle c rashes 58 26.61% Day c rashes 106 48.62% Dark c rashes 112 51.38% Total n on - d eer c rashes (5 years) 161 73.85% Total c rashes (5 years) 218 100.00% Collision t ype Percent of FI i ntersection c rashes Percent of PDO i ntersection c rashes Percent of t otal i ntersection cr ashes Single - vehicle crashes Collision with deer 0.00% 31.67% 26.15% Collision with bicycle 2.63% 0.00% 0.46% Collision with pedestrian 0.00% 0.00% 0.00% Other single - vehicle crash 63.16% 43.89% 47.25% Total single - vehicle crash 63.16% 75.56% 73.39% Multiple - vehicle crashes Angle collision 15.79% 10.56% 11.47% Head - on collision 2.63% 0.56% 0.92% Read - end collision 5.26% 6.11% 5.96% Sideswipe collision 5.26% 6.11% 2.75% Other multiple - vehicle collision 7.89% 1.11% 5.50% Total multiple - vehicle collision 36.84% 24.44% 26.61% Total c rashes 100.00% 100.00% 100.00% 5.2 Results and Discussion The sections below wi ll present and explain the model results for both 4ST and 3ST type intersections. Coefficient estimates, standard errors, and p - values are provided in each table. Due to the small number of crashes at each location, models estimating crashes of all severity levels were developed rather than isolating specific crash types and severities. Unless otherwise noted, further discussion of crashes in this chapter should assume exclusion of deer crash es. 116 5.2.1 Four - Leg Stop - Controlled Rural Intersections (4ST) The model results for annual crash occurrence at four - leg stop - controlled rural intersections of all jurisdictional classifications are summarized in Table 37 . Four - leg stop - controlled rural inte rsections with skew angles between 10 degrees and 39 degrees were found to have 2 8 percent greater crash frequency relative to intersections with no intersection skew . On the other hand, skew angles greater than or equal to 40 degrees or less than 10 degrees were not found to be significantly different from those with no intersection skew . Intersection skew is associated with Minor skew (<1 0 degrees) does not appear to impact impacted safety performance . Safety performance is also not impacted by extreme skew, although this is likely at least somewhat due to small sample size . However, it may also be due to drivers proceeding with increased caution at intersections with such extreme skew . Further exploration into these effects is warranted in future work . Turning to the intersection jurisdiction factors, intersections where a state highway was the major jurisdiction were found to have the hig hest crash frequency, while county non - federal aid were found to have the lowest crash frequency. The presence of a railroad crossing within the 26 percent. The presence of a railr oad crossing creates an opportunity for rear - end crashes, as does the presence of any traffic control device that compels drivers to stop or yield. All other factors were not found to have a significant effect on crash occurrence, including lighting , drive way county, and the presence of left turn lanes, which in many cases , was due to small sample sizes. Notably, a subsequent analysis of nighttime crashes found lighting to remain insignificant. Similarly, a follow up analysis of left - turn head - on collisions 117 found the presence of left - turn lanes to remain insignificant, although this is likely at least partially due to the very small sample of intersections possessing left - turn lanes. To account for differences in roadway, driver, and trip characteristics between the three jurisdictional and funding classes, and to focus on county - owned roadways, separate models were generated for cases where the major intersecti ng road way was county fe deral aid , in addition to a separate model for county non - federal aid. Due to the small number of intersections with crashes , fixed effects analysis was used rather than mixed effects . Results for county federal aid intersections, shown in Table 38 , were very similar to the mixed effects model for all jurisdictions , with the same factors being significant, and with While the mixed effects model estimates a 2 8 percent increase in crashes at intersections when t he skew angle is between 10 and 39 degrees, the county federal aid - specific fixed effects model estimates a 30 percent increase. The county non - federal aid model ( Table 39 ) shows a n even stronger effect, with a 60 percent greater crash occurrence when skew angle is between 10 and 3 9 degrees compared to intersections with no skew . Intersections with skew angles of 40 degrees or more did not demonstrate a significant difference from intersections with no skew in any analysis , nor did intersections with skew a ngles between 1 and 9 degrees . Again, the lack of measurable effects of extreme skew are likely at least some what due to small sample size, although drivers may also be proceeding with increased caution at such locations . Further exploration into these eff ects is warranted in future work . Model diagnostics (i.e., AIC and log likelihood) show that the mixed effects model in Table 37 is more accurate in predicting crashes that the fixed effects model located in Appendix A ( Table 56 ). Mixed effects models, i.e., models that incorporate random effects in addition to 118 fixed effects, tend to be more accurate than fixed effects models. As previously discussed, fixed effects models generally include one line of data per year which introduces a bias ; each line of data is assumed to be independent, but multiple observations at the same site are not truly independent from each other. The random effects incorporated in this model address this bias through the site - specific random effect, in addit ion to a county - specific random effect to address both the differences in maintenance and design practices between county road commissions as well as weather and population differences , among others . Table 37 : Mixed Effects Negative Binomial Model Results for 4 ST Rural Intersections Factor Description Est Exp(B) Std error P - value Intercept - 7.499 0.001 0.215 <0.001 Major road AADT N atural log of, vehicles per day 0.411 1.508 0.026 <0.001 Minor road AADT N atural log of, vehicles per day 0.551 1.735 0.021 <0.001 Railroad c ross ing P resent within 211 feet 0.228 1.256 0.124 0.067 Skew 0 degrees D eviation from 90 degrees baseline Skew 1 to 9 degrees D eviation from 90 degrees 0.115 1.121 0.087 0.186 Skew 10 to 39 degrees D eviation from 90 degrees 0.244 1.276 0.046 <0.001 Skew > 40 degrees D eviation from 90 degrees - 0.015 0.985 0.086 0.862 State h ighway M ajor road jurisdiction baseline County FA M ajor road jurisdiction - 0.139 0.870 0.041 0.001 County n on - FA M ajor road jurisdiction - 0.343 0.710 0.101 0.001 County random effect 0.129 Site random effect 0.631 Overdispersion parameter 0.0393 AIC 30,151.20 Log - likelihood - 15,063.6 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion 119 Table 38 : F ixed Effects Negative Binomial Model Results for 4ST Rural Intersections ( Major Road County FA) Factor Description Est Exp(B) Std error P - value Intercept - 7.548 0.001 0.188 <0.001 Major road AADT N atural log of, vehicles per day 0.432 1.540 0.027 <0.001 Minor road AADT N atural log of, vehicles per day 0.548 1.730 0.021 <0.001 Railroad crossing P resent within 211 feet 0.139 1.149 0.136 0.308 Skew 0 degrees Deviation from 90 degrees baseline Skew 1 to 9 degrees Deviation from 90 degrees 0.018 1.018 0.089 0.839 Skew 10 to 39 degrees Deviation from 90 degrees 0.265 1.304 0.046 <0.001 Skew > 40 degrees Deviation from 90 degrees 0.096 1.100 0.102 0.350 Overdispersion parameter 0.845 AIC 16,963.00 Log - likelihood - 16,947.3 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion Table 39 : Fixed Effects Negative Binomial Model Results for 4ST Rural Intersections ( Major Road County Non - FA) Factor Description Est Exp(B) Std error P - value Intercept - 7.641 0.000 0.574 <0.001 Major road AADT N atural log of, vehicles per day 0.534 1.706 0.108 <0.001 Minor road AADT N atural log of, vehicles per day 0.409 1.505 0.100 <0.001 Railroad cross ing P resent within 211 feet 1.450 4.263 0.445 0.001 Skew 0 degrees Deviation from 90 degrees baseline Skew 1 to 9 degrees Deviation from 90 degrees 0.534 1.706 0.350 0.127 Skew 10 to 39 degrees Deviation from 90 degrees 0.468 1.597 0.200 0.019 Skew > 40 degrees Deviation from 90 degrees - 20.490 0.000 21370.00 0.999 Overdispersion parameter 0.186 AIC 1,317.5 Log - likelihood - 650.7 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion 5.2.2 Three - Leg Stop - Controlled Rural Intersections (3ST) The e model results for deer - excluded crashes occurring at three - leg stop - controlled rural intersections are summarized in Table 40 . Intersection skew angle did not show a significant difference betw een intersections with no skew and those with skew angles between one and 39 degrees . On the other hand, there was a negative correlation between crash reduction and skew 120 angles of 40 degrees or greater ; a prior study found that skew angles of 70 degrees o r higher were associated with crash reduction [103] . It is important to note that the vast majority of sites had skew angles of zero , and very few sites had skew angles of 40 degrees or greater . The cause of this decrease is un clear; however, there are some potential explanations that may be rooted in driver behavi or, such as drivers being more cautious at intersections with extreme skew. Similar to four - leg intersections, county non - federal aid intersections experienced lower c rash occurrence than MDOT or county federal aid intersections. All other factors, including lighting, left turn lanes, and driveway counts, were not found to significantly affect crash occurrence at rural 3ST intersections. Similar to 4ST intersections, t o account for differences in roadway, driver, and trip characteristics between the three jurisdictional and funding classes, separate models w ere generated for cases where the major intersecti ng roadway was county federal aid or county non - federal aid. Due to the small number of crashes at each location and the small number of sites , fixed effects analysis was used rather than mixed effects . Results for county federal aid intersections, shown in Table 41 , were quite similar to th ose in the mixed effects model, with the T he mixed effects model estimates a 28 percent decrease in crashes and the county federal aid - specific model estimate s a 2 4 per cent de crease in crashes at intersections when the skew angle is 40 degrees or greater , while the county non - federal aid - specific fixed effects model ( Table 42 ) does not find this factor to be significant . No models found a significant difference in crash frequency when skew angle was between 1 and 39 degrees. Model diagnostics (i.e., AIC and log likelihood) show that the mixed effects model below in Table 40 is more accurate in predicting crashes that the fixed effects model located in 121 Appendix A ( Table 55 ). Mixed effects models, i.e., models that incorporate random effects in addition to fixed effects, tend to be more accurate than fixed effects models. As previously discussed, fixed effects models generally include one line of data per year which introduc es a bias; each line of data is assumed to be independent, but multiple observations at the same site are not truly independent from each other. The random effects incorporated in this model address this bias through the site - specific random effect, in add ition to a county - specific random effect to address both the differences in maintenance and design practices between county road commissions as well as weather and population differences. Table 40 : Mixed Effects Negative Binomial Mo del Results for 3ST Rural Intersections Factor Description Est Exp(B) Std error P - value Intercept - 7.117 0.001 0.263 <0.001 Major road AADT N atural log of, vehicles per day 0.345 1.413 0.032 <0.001 Minor road AADT N atural log of, vehicles per day 0.530 1.698 0.024 <0.001 Skew 0 degrees Deviation from 90 degrees baseline Skew 1 to 9 degrees Deviation from 90 degrees - 0.034 0.967 0.093 0.714 Skew 10 to 39 degrees Deviation from 90 degrees - 0.042 0.959 0.055 0.449 Skew > 40 degrees Deviation from 90 degrees - 0.327 0.721 0.090 <0.001 State h ighway M ajor road jurisdiction baseline County FA M ajor road jurisdiction - 0.245 0.783 0.052 <0.001 County n on - FA M ajor road jurisdiction - 0.635 0.530 0.117 <0.001 County random effect 0.133 Site random effect 0.658 Overdispersion parameter 0.00657 AIC 17,089.10 Log - likelihood - 8,533.6 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion 122 Table 41 : F ixed Effects Negative Binomial Model Results for 3ST Rural Intersections ( Major Road County FA) Factor Description Est Exp(B) Std error P - value Intercept - 7.243 0.001 0.261 <0.001 Major road AADT N atural log of, vehicles per day 0.376 1.456 0.037 <0.001 Minor road AADT N atural log of, vehicles per day 0.508 1.662 0.027 <0.001 Skew 0 degrees Deviation from 90 degrees baseline Skew 1 to 9 degrees Deviation from 90 degrees 0.059 1.061 0.094 0.527 Skew 10 to 39 degrees Deviation from 90 degrees 0.070 1.072 0.059 0.236 Skew > 40 degrees Deviation from 90 degrees - 0.274 0.760 0.105 0.009 Overdispersion parameter 0.393 AIC 8,962.60 Log - likelihood - 4,474.3 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion Table 42 : Fixed Effects Negative Binomial Model Results for 3ST Rural Intersections ( Major Road County Non - FA) Factor Description Est Exp(B) Std error P - value Intercept - 7.537 0.001 0.607 <0.001 Major road AADT N atural log of, vehicles per day 0.310 1.363 0.116 0.007 Minor road AADT N atural log of, vehicles per day 0.576 1.778 0.112 <0.001 Skew 0 degrees Deviation from 90 degrees baseline Skew 1 to 9 degrees Deviation from 90 degrees 0.744 2.105 0.435 0.087 Skew 10 to 39 degrees Deviation from 90 degrees 0.193 1.213 0.237 0.415 Skew > 40 degrees Deviation from 90 degrees - 0.938 0.391 0.590 0.112 Overdispersion parameter 0.497 AIC 1,086.6 Log - likelihood - 536.3 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion 5.2.3 Crash Modification Factors Developed for Rural Intersections From these results, a set of CMFs were developed for correcting intersection skew angle within each funding and jurisdictional category , and these are shown i n Table 43 . These CMFs are the reciprocal of the Exp(B) values in the results tables. The values in the results tables describe the effect of these site characteristics when deviating from base conditions, but the reciprocal is taken to determine the oppos ite effect (i.e., returning to base conditions). 123 Table 43 : CMFs Developed for Rural Intersections Original c ondition CMFs for the following major road classification MDOT County FA County n on - FA Remarks 4ST intersections Skew 1 to 9 degrees ns ns ns Final condition: Skew 10 to 39 degrees 0.78 0.77 0.63 s kew 0 degrees Skew > 40 degrees ns ns ns 3ST intersections Skew 1 to 9 degrees ns ns ns Final condition: Skew 10 to 39 degrees ns ns ns s kew 0 degrees Skew > 40 degrees 1.39 1.32 ns Note: ns = not significant 5.2. 4 Comparison to HSM Models A graphical representation of the Michigan - specific rural 4ST and 3ST model results, as presented in this chapter, are shown in Figure 19 . The respective HSM base models were also included in the figures for comparison purposes . The HSM models have been calibrated to major road county federal aid data using the methodology presented in Chapter s 2 and 3 . The calibration factor for three - leg intersections was found to be 0.66 and the for four - leg intersections it was found to be 0.59 . Only lower volumes are shown in Figure 19 order to emphasize the role that model shape plays in prediction, and how shape c an vary based on jurisdiction. A minor road volume of 500 vehicles per day was selected as the median value for minor road AADT was 456 and 638 vehicles per day at 3ST and 4ST intersections , respectively . Calibrated models were evaluated for goodness - of - f it using mean absolute deviation. On paved segments, calibrated models performed well compared to Michigan - specific models when evaluating using MAD. At three - leg intersections where the major road was county federal aid , the model presented in this chapte r had an MAD value of 0. 21 while the calibrated model had a value of 0.3 8, indicating a better fit for the Michigan - specific model . At four - leg intersections where the major road was county federal aid, the model presented in this chapter had an MAD 124 value of 0.24 while the calibrated model had a value of 0.77, indicating an even greater improvement in accuracy. For 3ST intersections, the calibrated HSM - predicts crashes at lower major roadway volumes, but begins to over - predict crashes when the major roadway AADT exceeds approximately 3 , 0 00 veh icles per day. The calibrated HSM model over - predicts crashes at major roadway volumes that exceed approximately for 1,000 vehicles per day at 4ST intersections. At higher volumes (i.e., 2,000 v ehicles p er d ay ), the calibrated HSM - prediction of 4ST crashes increases to 27 percent at county federal aid intersections . This is not surprising, as c a libration of the HSM HSM generally overpredicts crash occurrence . For example, in North Carolina, a calibration factor of 0.68 was assigned, while in Oregon, the HSM was found to overpredict by an even greater degree, with a calibration factor of 0.31 assigned [118] . Another analysis found that stop - controlled intersections in North America (both 3ST and 4ST) should be assigned a calibration factor of 0.56 [119] . 125 Figure 19 : Model results for non - deer crashes on 4ST and 3ST intersections for minor roadway AADT= 5 00 veh/day 5.3 Summary and Conclusions This study involved the estimation of SPF s for low - volume rural stop - controlled intersections in Michigan. I n order to create a robust sample of intersections within this volume range, both state represe nted in the 5, 659 - intersection sample. A robust sample of roadway characteristic data, including traffic crashes, traffic volumes, roadway classification, geometry, cross - sectional features, and other site characteristics were collected for the period of 2 011 - 2015. After the data were assembled for the rural intersection sample, a series of SPFs were developed to estimate annual crash occurrence on three - leg (3ST) and four - leg (4ST) intersections that included intersections of all funding and jurisdictional classes (i.e., MDOT, county federal aid, and county non - federal aid) . The models were specified considering factors 126 such as driveway density, presence of lighting, turn lane presence, and inter section skew, in addition to volume. To account for the unobserved heterogeneity associated with differing design standards and other county - to - county differences, random effects negative binomial models with a county - specific random effect were utilized. Furthermore, a site - specific random effect was used to account for the lack of independence among the five data points each intersection provided. In addition, due to the fact that models developed for different funding or jurisdictional classes are expect analyses were performed for intersections where the major road was county federal aid and county non - federal aid. Due to a small sample size, fixed effects analyses were used fo r these subsets. fixed effects models in terms of which factors were significant and in which direction the results trended in, the shape of each model was different. The mixed effects negative binomial analysis found that of the aforementioned factors, skew angles of between 10 and 39 degrees led to significantly greater crash occurrence for 4ST intersections. Intersections with skew angles in this category comprised of o nly approximately 15 percent of intersections. Other factors were found to have little impact on crash occurrence, even when considering only targeted crash types, although this is likely a result of small crash sample sizes. Comparison of the Michigan - spe cific models to the uncalibrated HSM base models showed that the HSM 3ST model under - predicts crashes at lower major roadway volumes, but begins to over - predict crashes when the major roadway AADT exceeds 3 , 0 00 veh icles per day. Compared to the Michigan - specific 4ST models, the HSM over - predicts crashes when AADT exceeds 1,000 vehicles per day . 127 The rural intersection models developed herein will be of use to transportation professionals, as there is a limited amount of rese arch on safety performance at low - volume, rural stop - controlled intersections. Particularly noteworthy is the inclusion of county - owned intersections, including those on minor collectors and local roadways, as these facilities tend to have design and maint enance characteristics, travel patterns, and driver types that vary greatly from state - owned facilities. Ultimately, the results of this study provide a number of methodological tools that will allow for proactive safety planning activities, including netw ork screening and identification of high - risk sites. 128 6. EVALUATION OF DEER C RASHES ON R URAL SEGMENTS Deer - vehicle crashes continue to be a problem in the United States, with 1.2 million such crashes occurring annually . Such crashes are a particular issue on two - lane rural highways in Michigan, accounting for more than 60 percent of all crashes . Such a high proportion of deer vehicle crashes limits the transferability of existing safety models, including those found in the HSM , that are often based on data from states with considerably lower proportions of deer crashes . Furthermore, deer crashes also introduce unwanted bias when modeling the relationships between crash occurrence and geometry or other roadway related factors . As a result, the primary safety performance functions developed in this study for county road segments and intersections , presented in C hapters 4 and 5, categorically excluded deer crashes from these models in order to improve the prediction capabilities of the roadway related factors However, there remains a clear need for further research on the impacts that roadway characteristics have on deer crash occurrence across the primary classes of rural roadways, including both state and county two - lane highway segments . A cross - sectional an alysis of deer crashes was performed using the 2011 - 2015 crash data sample described in Chapter 3 . This analysis also included state - owned rural two - lane highways, for which roadway data were obtained from the MDOT sufficiency file, which serves as the pri mary roadway inventory file for the MDOT rural highway network . The data were analyzed across four categories of rural two - lane roadways, including: state highways, federal aid county roadways, non - federal aid county roadways , and unpaved county roadways . Mixed effects negative binomial regression models utilizing spatial (i.e., county) and temporal (i.e., crash year) random effects were generated separately for each of the rural two - lane roadway types . The following sections detail 129 the descriptive statisti cs, analytical method, results and discussion of deer - related crashes on county - owned highway segments. 6.1 Descriptive Statistics A total of 17,285 segments comprising of 12,746 miles of rural two - lane roadway were analyzed. 42 percent of miles analyzed w ere state highways, 35 percent were paved county federal aid, 11 percent were paved county non - federal aid, and 12 percent were unpaved. The location of state highway segments is shown in Figure 20 and county segments in Figure 21 . Table s 44 - 4 7 show summar y statistics for state, county federal aid, county non - federal aid, and unpaved rural roadway segments. Particularly noteworthy is the proportion of deer crashes to total crashes, which is above 0.6 in all categories, with the exception of unpaved roads. A ADT values range from an average of more than 4,000 vehicles per day, with a standard deviation of more than 3,000 vehicles, on state highway segments, to an average of 217 vehicles per day on unpaved segments. All sites across all jurisdictional categorie s were two - lane roads. In order to have a clear understanding of the effect of lane width, segments were constrained to those with lane widths between 9 ft and 13 ft for all analyses, which accounts for 100 percent of state - owned segments, 99.3 percent of county federal - aid segments, 99.4 percent of county non - federal - aid segments, and 84.4 percent of unpaved segments. Lane widths were rounded to the nearest foot (i.e., 10.5 ft was rounded to 11 ft), as suggested by the HSM . Maps of the state and county hig hway segments utilized in the deer crash analysis are provided in the figures that follow . 130 Table 44 : Descriptive Statistics for State Highway Segments included in Deer Crash Analysis Statistic Mean St d ev Min Max Segment length (mi) 3.439 2.798 0.105 21.743 AADT 4,382.15 3,016.97 23 23,481.00 Number of lanes 2 0 2 2 Lane width (ft) 11.633 0.5 10 12 Paved shoulder width (ft) 4.771 2.521 0 12 Driveway density (driveways/mi) 14.773 9.956 0 85.053 Substandard curves/mi 0.076 0.443 0 9.524 Public deer licenses per sq uare mi le 1.575 1.57 0 5.717 Private deer licenses per sq uare mi le 7.656 5.901 0 22.133 Superior Region 0.225 North Region 0.262 Grand Region 0.108 Bay Region 0.161 Southwest Region 0.124 University Region 0.106 Metro Region 0.013 Total annual midblock crashes/mi 2.705 2.277 0.000 28.571 Midblock annual deer crashes/mi 1.781 1.760 0.000 20.161 Midblock annual FI deer crashes/mi 0.034 0.141 0.000 4.032 Percent deer crashes 65.8% Number of segments 1,556 Note: ft = feet, mi = miles, std dev = standard deviation; min = minimum; max = maximum Table 45 : Descriptive Statistics for County Federal - Aid Segments ( Deer Crashes ) Statistic Mean St dev Min Max Segment length (mi) 0.447 0.329 0.100 8.188 AADT 1,721 1,678 10 12,781 Number of lanes 2 0 2 2 Lane width (ft) 11.0 0.697 9.0 13.0 Paved shoulder width (ft) 1.1 1.551 0.0 10.0 Driveway density (driveways/mi) 14.922 13.895 0.000 138.686 Substandard curves/mi 0.210 1.043 0.000 15.625 Public deer licenses/ sq mi (5 - y ea r average) 1.020 0.902 0.000 3.419 Private deer licenses/ sq mi (5 - y ea r average) 12.626 8.053 0.000 25.640 Total annual midblock crashes/mi 1.590 2.803 0.000 42.017 Midblock deer crashes/mi 0.960 2.085 0.000 33.613 Percent deer crashes 60.4% Number of segments 9,847 Note: ft = feet, mi = miles, std dev = standard deviation; min = minimum; max = maximum 131 Table 46 : Descriptive Statistics for County Non - Federal Aid Segments ( Deer Crashes ) Statistic Mean St dev Min Max Segment length (mi) 0.51 0.299 0.1 2.012 AADT 586.358 636.213 5 12,628 Number of lanes 2 0 2 2 Lane width (ft) 10.585 0.689 9 13 Paved shoulder width (ft) 0.249 0.683 0 8 Driveway density (driveways/mi) 17.626 13.694 0 108.108 Substandard curves/mi 0.242 1.138 0 13 Public deer licenses/ sq mi (5 - y ea r average) 1.089 0.768 0 3 Private deer licenses/ sq mi (5 - y ea r average) 15.794 6.258 0 25.64 Total annual midblock crashes/mi 0.588 1.494 0 23 Midblock deer crashes/mi 0.355 1.121 0 19 Percent deer crashes 60.4% Number of segments 2,856 Note: ft = feet, mi = miles, std dev = standard deviation; min = minimum; max = maximum Table 47 : Descriptive Statistics for Unpaved Segments ( Deer Crashes ) Statistic Mean St dev Min Max Segment length (mi) 0.508 0.365 0.1 4.575 AADT 216.639 332.945 7 6,298 Number of lanes 2 0 2 2 Lane width (ft) 10.604 1.201 9 13 Paved shoulder width (ft) 0 0 0 0 Driveway density (driveways/mi) 12.766 10.894 0 93 Substandard curves/mi 0.258 1.148 0 15 Public deer licenses/ sq mi (5 - y ea r average) 1.618 1.121 0 3 Private deer licenses/ sq mi (5 - y ea r average) 17.52 6.409 0 25.64 Total annual midblock crashes/mi 0.254 0.945 0 18 Midblock deer crashes/mi 0.091 0.542 0 9 Percent deer crashes 35.8% Number of segments 3,026 Note: ft = feet, mi = miles, std dev = standard deviation; min = minimum; max = maximum 132 Figure 20 : Map of rural state highway study segments (deer crashes) Figure 21 : Map of rural county highway study segments (deer crashes) 133 6.2 Results and Discussion The mixed effects negative binomial regression models yielded several interesting results. The full model results for state highways, county federal aid, county non - federal aid, and unpaved segments are presented in Table 48 , Table 49 , Table 50 , and Table 51 respectively. In all cases, factors potentially related to speed were found to be a significant predictor variable for deer crashes. This is an intuitive finding, as faster speeds give drivers less time to react to deer, increasing the likelihood of col lision. For example, wider lanes were associated with increased crashes in the case of county federal aid, county non - federal aid, and unpaved segments, although this relationship was not significant on county - non - federal aid segments. In particular, on county federal aid segments, a 12 - f oo t lane is associated with 24 percent more deer crashes than a 10 - f oo t lane, while on unpaved segments, it is associated with an 8 percent increase. Readers should be aware that the average lane width for county segments (both federal aid and non - federal aid) was approximately 11 f ee t, and are referred to Table s 44 - 47 for full descriptive statistics. These results were consistent on state highways as well, where 12 - f oo t lanes were associated with 14 percent greater fatal and injury crashes relative to 11 - f oo t lanes. The effect of lane width was not significant for 10 - f oo t lanes, or property damage only (PDO) crashes on state highway segments. While lane width was not significant for county non - federal - aid segments, the res ults still suggested increased crashes with wider lanes, consistent with all other segment categories. These findings are consistent with the notion that wider lanes are associated with faster speeds [120] , perhaps contributing to greater crash occurrence, particularly those involving injury. On the other hand, wider shoulders were associated with fewer PDO crashes, perhaps due to the increased separation between the roadside and traveled way along with the additional 134 recovery area when evasive maneuvers are necessary for collision avoidance. Some research supports the notion that wider cross - sections, which wider shoulders imply, are associated with decreased deer activity [82] . Analysis of segment design - hour level of service (LOS) provided perhaps another inventory file includes design hour level of service ratings for all state highway segments, which w as subsequently included in the state highway model. Relative to LOS A, all other levels of service showed a significant decrease in deer crashes, including decreases in fatal and injury crashes. With PDO crashes, where all categories of level - of - service w ere statistically significant, each decline in level - of - service was associated with a further decline in deer crashes. The last factor likely related to speed was the number of curves designed below 55 mph (the statutory speed limit on all study segments was 55 mph), which was also associated with a decrease in deer crashes on paved roads across all jurisdictional categories. However, number of curves was not a significant factor on unpaved roads. This is shown graphically in Figure 22 , which shows the est imated number of annual crashes for each jurisdictional category (under the following conditions: no paved shoulder, 12 - ft lanes, no driveways, no substandard curves) compared with the same road classification with one substandard curve. This could be due to drivers generally traveling more slowly on unpaved roads, providing additional reaction time. This hypothesis is supported by the lack of deer crashes in relation to total crashes on unpaved roads (36 percent) compared to state highways, county federal aid, and county non - federal aid roadways (66 percent, 60 percent, and 60 percent, respectively). 135 Figure 22 : Deer c rash m odel r esults u nder b ase c onditions and with s ubstandard c urves Driveway density was also a significant factor in deer crashes across all categories, although this result differed between state highway and county segments. On state highway segments, driveway density was associated with an increase in deer crashes. Anec dotally, hunters will modify and manipulate trails and access roads to direct deer to them [121 - 122] , as wildlife will often use human - made paths. However, this trend did not hold for county segments, 136 where driveway density was associated with a small decl ine in deer crashes, perhaps due to the increased human presence associated with greater driveway density. Further research is needed to more completely investigate the relationship between driveway density and deer crashes. Lastly, the number of antlerle ss deer licenses offered by the Michigan Department of Natural Resources was included as a variable. The purpose was to serve as a surrogate for deer management practices, as antlerless deer licenses are the only type of deer hunting license offered in Mic higan with geographic restrictions in order to incentivize hunting in specific areas. The results were inconsistent in terms of significance and sign, and very small in magnitude. This could indicate several things, one of which is that incentivizing deer hunting in specific locations may not have an influence on crashes. However, there are several limitations certainly leading to the statistical uncertainty that readers should be aware of. This variable measures the density of antlerless deer licenses avai lable in the county or DMU of a given road segment. This is geographically imprecise, and some DMUs span several counties, making this even less precise. There is also a lack of a licensing system that considers geography with respect to bucks with antlers , and a lack of estimates of the total deer population by county or region. 137 Table 48 : Mixed Effects Negative Binomial Model for Deer Crashes on State Highway Segments Factor Fatal and i njury c rashes Property d amage o nly c rashes Est Std error Sig Est Std error Sig Intercept - 7.14 0.644 <0.001 - 3.022 0.177 <0.001 Segment length (ln[mi]) 1 1 Volume (ln[AADT]) 0.456 0.081 <0.001 0.405 0.021 <0.001 Antlerless deer license quota (per square mile within county) Public land - 0.032 0.017 0.071 - 0.005 0.005 0.306 Private land - 0.001 0.097 0.989 - 0.004 0.025 0.885 Lane width 12 ft baseline 10 ft - 0.194 0.488 0.692 - 0.156 0.115 0.175 11 ft - 0.133 0.08 0.098 - 0.016 0.021 0.448 Paved shoulder width <4 ft baseline >4 ft 0.043 0.076 0.568 - 0.048 0.02 0.014 Level of service A baseline B - 0.241 0.102 0.018 - 0.108 0.026 <0.001 C - 0.193 0.112 0.086 - 0.131 0.03 <0.001 D - 0.215 0.171 0.209 - 0.318 0.046 <0.001 E - 0.544 0.444 0.22 - 0.299 0.102 0.003 Driveway density (mile - 1 ) <5 driveways baseline >5 driveways 0.309 0.132 0.019 0.426 0.032 <0.001 Number of substandard curves - 0.059 0.055 0.285 - 0.081 0.013 <0.001 Random effects MDOT region 0.114 0.134 Year 0.085 0.061 Overdispersion parameter 0.115 0.34 AIC 5,275.00 37,731.40 Log - likelihood - 2,621.50 - 18,849.70 Note: ft = feet, mi = miles, std = standard; sig = statistical significance 138 Table 49 : Mixed Effects Negative Binomial Model for Deer Crashes on County Federal Aid Segments Factor Est. Std error Sig Intercept - 4.054 0.253 <0.001 Segment length (ln[mi]) 1.000 Volume (ln[AADT]) 0.395 0.011 <0.001 Public - 0.018 0.013 0.158 Private 0.017 0.002 <0.001 Driveway density (per mi) - 0.007 0.001 <0.001 Lane width 9 ft Baseline 10 ft 0.894 0.237 <0.001 11 ft 1.044 0.237 <0.001 12 ft 1.112 0.237 <0.001 13 ft 1.349 0.246 <0.001 Number of substandard curves - 0.396 0.032 <0.001 Random effects MDOT region 0.157 Year 0.039 Overdispersion parameter 0.572 AIC 75,921 Log likelihood - 37,947 Note: ft = feet, mi = miles, std = standard; sig = statistical significance 139 Table 50 : Mixed Effects Negative Binomial Model for Deer Crashes on County Non - Federal Aid Segments Factor Est. Std error Sig Intercept - 5.381 0.289 <0.001 Segment length (ln[mi]) 1.000 Volume (ln[AADT]) 0.663 0.028 <0.001 Public land 0.073 0.037 0.047 Private land 0.003 0.008 0.688 Driveway density (per mile) - 0.015 0.002 <0.001 Lane width 9 ft Baseline 10 ft 0.124 0.212 0.558 11 ft 0.313 0.213 0.142 12 ft 0.346 0.220 0.116 13 ft 0.152 0.407 0.709 Number of substandard curves - 0.416 0.085 <0.001 Random effects MDOT region 0.257 Year 0.048 Overdispersion parameter 0.665 AIC 12,650 Log - likelihood - 6,312 Note: ft = feet, mi = miles, std = standard; sig = statistical significance 140 Table 51 : Mixed Effects Negative Binomial Model for Deer Crashes on Unpaved Segments Factor Est St d error Sig Intercept - 5.259 0.391 <0.001 Segment length (ln[mi]) 1.000 Volume (ln[AADT]) 0.456 0.054 <0.001 Public land - 0.091 0.059 0.122 Private land 0.015 0.014 0.310 Driveway density (per mile) - 0.016 0.005 0.001 Lane width 9 ft Baseline 10 ft 0.312 0.138 0.024 11 ft 0.402 0.160 0.012 12 ft 0.392 0.159 0.014 13 ft 0.794 0.198 <0.001 Number of substandard curves - 0.054 0.069 0.433 Random effects MDOT region 0.579 Year 0.116 Overdispersion parameter 0.636 AIC 5,076 Log - likelihood - 2,525 Note: ft = feet, mi = miles, std = standard; sig = statistical significance 6.3 Summary and Conclusions The primary objective of this research was to determine relationships between deer crashes and roadway characteristics across all classes of two - lane rural roadways in Michigan, including both pave d and unpaved roadway surfaces. To accomplish this objective, highway data, including traffic volumes, roadway characteristics, and traffic crashes, were collected on state - owned rural roads statewide and on county - owned rural roads within a 30 - county samp le, and subsequently analyzed using mixed effect negative binomial modeling techniques. The results showed that factors likely to be speed - related, including lane width and horizontal curvature, had a significant effect on vehicle deer crashes occurring on most categories of rural two - lane two - way roadway segments in the state of Michigan, although these factors did not have as much of an effect on unpaved roads, which see fewer DVCs and lower 141 travel speeds Wider lanes were associated with a greater occurre nce of deer crashes, perhaps due to higher prevailing travel speeds. Conversely, more curves with design speeds lower than the statutory speed limit were associated with fewer deer crashes, perhaps due to lower travel speeds on curved segments. Wider shoul ders, which afford greater separation between the travel lanes and the roadside, were found to significantly reduce deer crash occurrence, furthering the hypothesis that wider clear zones are associated with a decrease in deer activity. Unfortunately, the concentration of hunting licenses, a potentially useful predictor for deer crashes, did not appear to have a consistent influence on vehicle - deer crashes. Policymakers and practitioners can use this information in several ways. Primarily, decision - makers should be aware of the impact speed - related geometric features have on deer crashes, particularly as the state continues its trend of raising speed limits on highways and freeways in rural areas. For instance, the conventional wisdom is that wider lanes ar e safer, but this may not be the case in locations with high deer populations due to higher travel speeds and subsequent reduced reaction times. Adding paved shoulders and widening the clear - zone may also help mitigate deer - vehicle crashes in problem areas . Further research needs to be conducted to determine a more precise relationship between vehicle speeds and/or speed limit policy and vehicle - deer crashes, as well as the relationship between the roadside conditions and these crashes 142 7. CONCLUSIONS , AND R ECOMMENDATIONS Since 2005, federal highway funding bills in the U.S. have required states to have data - driven strategic highway safety plans [123] . In most states , considerable attention is given within these plans to wards addressing rural highway safety issues, which remain a considerable problem in many parts of the country . The Highway Safety Manual assists towards that end, by providing models for estimating crash occurrence, as well as crash modification f actors when p arameters differ from base conditions. But w hile safety performance models for rural highway segments and intersections exist within the HSM and other literature sources , these models were typically developed using data from state - owned highways, which lim its the transferability to secondary classes of rural highways, including those owned and maintained by county road agencies. While Michigan - specific SPFs have been previously been developed , they were limited to urban and rural state - owned road segments and intersections [15 - 16] . Also, although HSM for county road segments and intersections, fully - specified SPFs utilizing local data have not been developed f or county roadways . Prior research has shown that, when evaluating highways of the same functional class, improvements in the predictive capabilities will generally be achieved if SPFs are developed using local data rather than calibrating HSM SPFs, due to the variability in the parameter estimates between the HSM and state - specific models . Rural county highways typically possess traffic, driver, and geometric characteristics that differ considerably from rural state highways . However, the safety performanc e of rural county roadways is rarely investigated to the same level of detail as that for state highways . This is an 143 important gap, as many states, particularly those in the Midwest and Great Lakes regions, possess a substantial network of rural county - own ed highways . Thus, determination of how various roadway and traffic related factors affect crashes on rural county highways , including both road segments and intersections, was the primary aim of this research . The findings would serve to support development of guidance for roadway designs and highway safety programs unique to rural county roadways and other rural secondary road networks . As a part of this research, it was also important to consider differences between the various classes of county roadways, in particular, the distinction between federal aid and non - federal aid roadways . Federal aid roadways are subject to design standards approved by the FHWA, which are typically more stringent than those for non - federal aid roadways. S pecifically, minimum design standards must be maintained in compliance with the posted speed limit for select controlling geometric elements, most notably horizontal and vertical curvature, on high speed federal aid roadways . Furthermore, nearly all available safety p erformance models are only applicable to paved roads, which further limits the applicability of these models for use by county or other local road agencies, which often maintain a substantial network of unpaved (gravel) roads . Thus, it was imperative that the county roadway safety performance models account for the differences between federal aid and non - federal aid roadway designs, while also investigating differences in safety performance between these roadway types . It was also important to provide a mor e detailed investigation into the safety performance impacts of various roadway geometric characteristics, most notably horizontal curvature . While prior research has investigated the safety performance effects related to the presence of a horizontal curve on a segment, these models did not account for the amount curvature along the segment . Furthermore, there was little prior research available related to the incremental effects 144 of curve design speed on safety performance . The safety performance effects of other design attributes, including intersection skew, were also taken into consideration. To address these gaps, research was undertaken to investigate the safety performance characteristics of rural county highways . This included development of a series of safety performance functions for rural county highway segments and stop - controlled intersections . A series of fully - specified safety performance models were developed across all classes of rural county highways, including federal aid and non - federal ai d roadways, while considering a broad range of geometric factors, paved and unpaved road segments, and 3 - leg and 4 - leg stop - controlled intersections . Specifically, safety performance functions were developed for the following roadway facility types using d ata collected from across Michigan: a. Rural county two - lane two - way paved federal aid segments b. Rural county two - lane two - way paved non - federal aid segments c. Rural county unpaved non - federal aid segments d. Rural three - leg minor - road stop - controlled intersections e. Rural four - leg minor - road stop - controlled intersections CMFs were also developed for various design factors for each of the rural segment and intersection types listed above, most notably, horizontal curves and intersection skew . Specific consideration w as given to the incremental effects of curve design speed and the curved proportion of segment on segment crash occurrence, as these two aspects had not been fully researched in prior safety performance models . The r esults of this research serve to provide an important reference to guide states and local agencies toward making informed decisions as to planning and programming decisions for safety projects and roadway design standards , and to 145 provide researchers with guidance regarding future work within the realm of safety performance on rural secondary roadways . In general, the resulting county SPF results were generally different than prior models developed for similar state - owned highways, although some similarities were observed . It was determined that the jurisdiction and surface type of a roadway also affected model shape, which further demonstrated the need for county - specific SPFs. The SPFs and CMFs developed in this dissertation will provide additional tools for highway engineers to make safety - rela ted design decisions on county roadways, as opposed to the common method of calibrating or otherwise applying SPFs and CMFs developed for state highways to county roadways. Each of the models were developed utilized negative binomial regression, and, where appropriately, also included one or more random intercept terms, thereby resulting in mixed - effects models. Negative binomial regression is generally used in developing SPF s, and is the technique that was used to develop most of the models contained in the HSM [3] . However, one problem that arises when using a fixed effects model is that each observation is assumed to be independent from other ob servations. However, this is not the case, as each site or segment has five observations (from five years of annual crash counts), which are not truly independent from each other. For this reason, site - specific random effects were incorporated. In addition , this research addressed county highways, with multiple sites or segments analyzed within a given county. Because each county road commission maintains its own practices regarding construction, maintenance, and/or design for non - federal aid roadways, in a ddition to climatic, geographic, and driver related differences, there exists unobserved county - to - county heterogeneity that cannot be easily quantified by fixed factors, prompting the inclusion of a county - level random effect. Very little existing researc h has involved the use of county - specific 146 random effects, which reflects the general lack of research on the safety performance of county highways. 7.1 Rural County - Owned Highway Segments Some key findings concerning county - owned segments include the effec t horizontal curvature has on crash frequency, particularly on federal aid segments. In general, lower curve design speed is associated with greater crash occurrence . The relationship between crash frequency and substandard curvature (i.e., horizontal curv es that have design speeds below the statutory speed limit of 55 mph) was present across all curve design speed categories on paved federal - aid highways, which tend to be major collectors, and the magnitude of this increase monotonically increases with dec reasing design speed. For paved non - federal aid county highways, for which construction and maintenance is funded solely by state and/or local dollars, the increase in crashes was only significant for curves with design speeds of less than 45 mph . This fin ding is important, as such extreme horizontal curvature is more likely to be encountered on non - federal aid roadways compared to higher classes of rural highways . The CMFs presented in this dissertation provide an opportunity for designers to make educated decisions concerning horizontal curve correction during reconstruction and rehabilitation projects, and also opens the opportunity for local agencies to receive safety funding that require appropriate CMFs to justify spending. In particular, on paved high ways, correcting horizontal curves with design speeds of less than 40 mph could reduce crashes by more than fourfold. On all three classes of roadway, there was a significant increase in crashes, relative to base conditions, when curve radius was lower tha n 40 miles per hour. This is significant, because during roadway reconstruction or rehabilitation of a federal - aid roadway, horizontal curves with 147 design speeds 15 miles per hour or lower than the posted speed limit or overall roadway design speed (e.g., 4 0 mph for a 55 - mph posted speed limit), must either be re - aligned or granted a design exception from FHWA [124] . Interestingly, the deer - specific analysis from Chapter 6 showed the opposite results, where the presence of horiz ontal curvature is associated with reduced crash frequency, likely due to the reduced travel speeds of motorists at horizontal curve locations. This research also confirms prior research demonstrating a higher crash rate associated with higher access point frequency. This was especially notable for paved segments with 25 driveways per mile or more; on county federal aid highways, where drivers may be less familiar with their surroundings due to trip characteristics (non - federal aid highways tend to be local roads while federal aid tend to be collectors), crash occurrence also increased when there were between 5 and 25 driveways per mile. This provides additional evidence pointing to the need to consolidate driveways on federal aid highways, in particular. T his research demonstrates the importance of using SPFs developed specifically for county - owned roadways, and between funding categories. For rural highway segments, it was shown that the shape of each function is quite different; although there are overarc hing trends concerning which category of roadway (i.e., state, county federal aid, county non - federal aid, and unpaved) experience the most or fewest crashes, these patterns do not hold at all traffic volumes. For instance, unpaved roads experience the hig hest crash frequency at low AADTs but the lowest crash frequency at more moderate AADTs. Similarly, MDOT roadways experience fewer crashes than paved non - federal aid segments at lower volumes, but more crashes at higher volumes. If calibration were used, r ather than developing new SPFs, this could lead to under - and over - prediction of crashes. 148 7.2 Rural Minor Road Stop - Controlled Intersections At rural stop - controlled intersections, the most significant geometric factor that influenced crash frequency was i ntersection skew. Intersections with skew angles between 1 and 9 degrees did not experience significant differences in crashes compared to intersections with no skew. However, at four leg stop - controlled intersections, sites with skew angles between 10 and 39 degrees experienced significantly more crashes than those with no skew, with model results estimating an increase in crashes of 28 percent. On four - leg intersections where the major road is county non - federal aid, the increase in crashes was even great er, with a 60 percent increase in crashes. In contrast, the models developed for three - leg stop controlled intersections did not show a significant increase in crashes when skew angle was between 10 and 39 degrees. Similar to rural segments, the safety pe rformance of rural intersections varied depending on site type. Intersections with the major roadway under state jurisdiction experienced the highest crash occurrence, while county federal aid experienced the least. There were also significant differences between the models developed here and the models presented in the HSM , with the HSM generally overpredicting. As with county segments, model shape varied depending on the site type. 7.3 Recommendations for Future Work The research described in this disser tation has implications for future research. For rural county highway segments, horizontal curvature was one of the factors evaluated, and CMFs for various design speeds were developed. However, the methods for collecting and subsequently structuring the d ata was focused on describing the attributes of the segments themselves, rather than curves, specifically. This is best demonstrated by the fact that segmentation was provided by the uently integrated 149 into the dataset. A relevant future analysis would begin by identifying all horizontal curves within a state, or a subset of the state, in order to investigate the safety effects of whether the curve was isolated or as a part of a series of successive curves. Thus, such an analysis must include characteristics such as the length of the tangent leading into the curve (or between successive curves), as it is expected that a compound curve (S - curve) or any series of horizontal curves would ha ve different safety performance than an isolated curve. Furthermore, although the curved proportion of the segment was included as a safety performance factor, the length of the curve itself was not considered in this research, again, owing to the nature o f how the data were collected. Another key question related to horizontal curvature that arises from this research is to investigate the radius at which a horizontal curve begins to possess safety performance that is equivalent to a tangent segment. In other words, determining the minimum radius at which curvature no longer impacts safety performance . While this research compared curves with design speeds below 55 mph with segments without such substandard curves, future research should also include hori zontal curves of with design speeds above 55 mph. This dissertation did demonstrate that on non - federal highways, horizontal curves with design speeds between 45 mph and 55 mph did not perform significantly different from base conditions, but it is importa nt to remember that curves with design speeds of 55 mph or greater were included in the base condition . It was not possible to separate curves with higher design speeds from pure tangent sections based on the way that the data were collected for this study . Another area that was beyond the scope of this research is the effect of speed transition zones, i.e., reduced speed limits as vehicles lead into build - up areas. This research focused on highway segments with speed limits of 55 miles per hour, the statu tory speed limit in Michigan 150 for rural county roads. However, in order to develop guidance with respect to when speed transitions are warranted, the manner in which they are implemented (e.g., how many 10 mph in advance of the built - up area they should be With respect to rural intersections, this research presents a comprehensive analysis of three - and four - leg rural intersections of all jurisdictio nal classes, with a focus on intersection skew angle. This research confirmed a previous piece of research that found that, while moderate skew angles are associated with crash increases, extreme skew angles can be associated with crash reductions relative to intersections with little - to - no skew. However, a causal explanation for this counter - intuitive result has not been determined. While there are several potential explanations, such as drivers taking more care at intersections they perceive to be dangero us, the model results themselves do not indicate the cause of this effect. Future research to evaluate skew could involve the use of the SHRP - 2 naturalistic driving experiment to determine how drivers behave at intersections of varying skew angles. There are other approaches that do not involve direct observation of human subjects. For instance, researchers could evaluate skew along with other factors, such as the percentage of left - turning versus right - turning traffic, and correlate this to the angle at w hich most vehicles are turning (i.e., are most vehicles making a turn greater than or less than 90 degrees). Turning movement data were not available when this research was being completed. Other aspects of the intersection zone, which was defined in this research as being within a 211 - foot radius of the center of the intersection, can be explored further. For instance, skew was explored in this paper; however, correcting intersection skew requires the introduction of horizontal curvature in advance of the intersection on the leg that is stop - controlled. The effect of 151 curvature within the intersection influence zone is an important area that needs to be researched further, as it can be used to develop guidelines on when, and how, to correct intersections wit h nonzero skew angles. Lastly, while this research focused on traditional three - and four - leg intersections, there are other types of intersections whose safety performance should be quantified. For instance, intersections with five or more legs without a traffic signal are uncommon, but do exist, and it is useful to know how their safety can be improved. More common atypical intersection configuration which can be researched further include so - where the free - - leg intersection where stop - controlled legs are separated from each other by some lateral distance. 152 APPENDICES 153 Appendix A: Fixed Eff ects Models Table 52 : Fixed Effects Negative Binomial Model Results for Paved Federal Aid Segments Factor Description Est Exp(B) Std error P - value Intercept - 6.006 0.002 0.108 <0.001 Segment length O ffset, natural log of, miles AADT N atural log of, vehicles per day 0.730 2.076 0.015 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 0.776 2.174 0.114 <0.001 45 - 49.9 mph C urved proportion of segment 1.029 2.798 0.126 <0.001 40 - 44.9 mph C urved proportion of segment 1.033 2.810 0.191 <0.001 <40 mph C urved proportion of segment 1.613 5.015 0.260 <0.001 10 - ft lane Baseline 11 - ft lane Width in feet - 0.063 0.939 0.032 0.046 12 - ft lane Width in feet - 0.056 0.945 0.036 0.113 13 - ft lane Width in feet - 0.143 0.866 0.068 0.036 0 to 1 ft shoulder Baseline 2 - ft shoulder Width in feet - 0.088 0.916 0.038 0.020 3 to 8 ft shoulder Width in feet - 0.014 0.986 0.026 0.590 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.150 1.162 0.030 <0.001 15 - to - 24.9 driveways per mile Binary indicator variable 0.224 1.251 0.033 <0.001 > 25 driveways per mile Binary indicator variable 0.308 1.360 0.035 <0.001 Overdispersion parameter 0.172 AIC 50,205.7 Log likelihood - 25,087.9 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion ; mph = miles per hour; ft = foot 154 Table 53 : Fixed Effects Negative Binomial Model Results for Paved Non - Federal Aid Segments Factor Description Estimate Exp(B) Std error P - value Intercept - 6.683 0.001 0.278 <0.001 Segment length O ffset, natural log of, miles AADT N atural log of, vehicles per day 0.804 2.234 0.042 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 0.287 1.332 0.377 0.447 45 - 49.9 mph C urved proportion of segment 0.288 1.334 0.517 0.577 40 - 44.9 mph C urved proportion of segment 0.916 2.498 0.393 0.020 <40 mph C urved proportion of segment 1.534 4.638 0.563 0.006 11 - ft lane Baseline 12 - or 13 - ft lane Width in feet - 0.077 0.926 0.097 0.428 9 - or 10 - ft lane Width in feet 0.068 1.070 0.061 0.268 0 - ft shoulder Baseline 1 - ft shoulder Width in feet 0.058 1.059 0.075 0.442 2 - ft shoulder or wider Width in feet 0.062 1.064 0.144 0.667 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.094 1.099 0.100 0.344 15 - to - 24.9 driveways per mile Binary indicator variable 0.076 1.079 0.104 0.462 > 25 driveways per mile Binary indicator variable 0.316 1.371 0.103 0.002 Overdispersion parameter 0.082 AIC 8,465.6 Log likelihood - 4,218.8 155 Table 54 : F ixed Effects Negative Binomial Model Results for Unpaved Non - Federal Aid Segments Factor Description Est Exp(B) Std error P - value Intercept - 7.322 0.001 0.539 <0.001 Segment length O ffset, natural log of, miles AADT N atural log of, vehicles per day 0.634 1.885 0.047 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 1.596 4.935 0.377 <0.001 45 - 49.9 mph C urved proportion of segment 1.593 4.920 0.313 <0.001 40 - 44.9 mph C urved proportion of segment 1.051 2.860 0.407 0.010 <40 mph C urved proportion of segment 1.563 4.772 0.343 <0.001 Surface width W idth in feet, natural log of 0.528 1.695 0.199 0.008 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.566 1.762 0.082 <0.001 15 - to - 24.9 driveways per mile Binary indicator variable 0.557 1.745 0.091 <0.001 > 25 driveways per mile Binary indicator variable 0.752 2.122 0.099 <0.001 Overdispersion parameter 0.048 AIC 9,110.8 Log likelihood - 4,544.4 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion ; mph = miles per hour; ft = foot Table 55 : F ixed Effects Negative Binomial Model Results for 3 ST Rural Intersections Factor Description Est Exp(B) Std error Sig Intercept - 6.831 0.001 0.191 <0.001 Major road AADT N atural log of, vehicles per day 0.336 1.399 0.023 <0.001 Minor road AADT N atural log of, vehicles per day 0.535 1.707 0.018 <0.001 Skew 0 degrees Deviation from 90 degrees baseline Skew 1 to 9 degrees Deviation from 90 degrees - 0.056 0.945 0.068 0.411 Skew 10 to 39 degrees Deviation from 90 degrees - 0.063 0.939 0.041 0.126 Skew > 40 degrees Deviation from 90 degrees - 0.339 0.712 0.068 <0.001 State h ighway M ajor road jurisdiction baseline County FA M ajor road jurisdiction - 0.254 0.776 0.037 <0.001 County n on - FA M ajor road jurisdiction - 0.616 0.540 0.09353 <0.001 Overdispersion parameter 0.515 AIC 17,581.00 Log - likelihood - 8,781.7 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion 156 Table 56 : F ixed Effects Negative Binomial Model Results for 4ST Rural Intersections Factor Description Est Exp(B) Std error Sig Intercept - 7.060 0.001 0.143 <0.001 Major road AADT N atural log of, vehicles per day 0.375 1.455 0.018 <0.001 Minor road AADT N atural log of, vehicles per day 0.557 1.746 0.015 <0.001 Railroad cross ing P resent within 211 feet 0.238 1.268 0.081 0.003 Skew 0 degrees Deviation from 90 degrees baseline Skew 1 to 9 degrees Deviation from 90 degrees 0.075 1.078 0.061 0.221 Skew 10 to 39 degrees Deviation from 90 degrees 0.263 1.301 0.031 <0.001 Skew > 40 degrees Deviation from 90 degrees - 0.029 0.972 0.060 0.632 State h ighway M ajor road jurisdiction baseline County FA M ajor road jurisdiction - 0.112 0.894 0.027 <0.001 County n on - FA M ajor road jurisdiction - 0.319 0.727 0.0759 <0.001 Overdispersion parameter 0.505 AIC 31,295.00 Log - likelihood - 15,637.4 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion 157 Appendix B: Temporally Aggregated Model Table 57 : Temporally Aggregated Fixed Effects Negative Binomial Model Results for Paved Federal Aid Segments Factor Description Est Exp(B) Std error P - value Intercept - 4.858 0.008 0.135 <0.001 Segment length O ffset, natural log of, miles AADT N atural log of, vehicles per day 0.443 1.558 0.019 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 0.581 1.787 0.143 <0.001 45 - 49.9 mph C urved proportion of segment 0.863 2.370 0.148 <0.001 40 - 44.9 mph C urved proportion of segment 0.393 1.482 0.243 0.105 <40 mph C urved proportion of segment 1.207 3.343 0.340 <0.001 10 - ft lane Baseline 11 - ft lane Width in feet - 0.096 0.908 0.042 0.021 12 - ft lane Width in feet - 0.157 0.855 0.047 0.001 13 - ft lane Width in feet - 0.096 0.909 0.088 0.273 0 to 1 ft shoulder Baseline 2 - ft shoulder Width in feet - 0.058 0.944 0.053 0.277 3 to 8 ft shoulder Width in feet - 0.133 0.876 0.037 <0.001 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.345 1.411 0.039 <0.001 15 - to - 24.9 driveways per mile Binary indicator variable 0.428 1.534 0.044 <0.001 > 25 driveways per mile Binary indicator variable 0.326 1.386 0.047 <0.001 Overdispersion parameter 1.400 AIC 27,636.7 Log likelihood - 13,803.4 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion ; mph = miles per hour; ft = foot 158 Appendix C: Mixed Effects Models without Length Offset Table 58 : M ixed Effects Negative Binomial Model Results for Paved Federal Aid Segments (No Length Offset) Factor Description Est Exp(B) Std error P - value Intercept - 5.949 0.003 0.151 <0.001 Segment length N atural log of, miles 0.936 2.550 0.020 <0.001 AADT N atural log of, vehicles per day 0.677 1.968 0.020 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 0.699 2.011 0.136 <0.001 45 - 49.9 mph C urved proportion of segment 1.061 2.888 0.149 <0.001 40 - 44.9 mph C urved proportion of segment 0.966 2.626 0.224 <0.001 <40 mph C urved proportion of segment 1.409 4.091 0.324 <0.001 10 - ft lane Baseline 11 - ft lane Width in feet 0.014 1.014 0.041 0.729 12 - ft lane Width in feet 0.048 1.050 0.048 0.314 13 - ft lane Width in feet 0.123 1.131 0.088 0.161 0 to 1 ft shoulder Baseline 2 - ft shoulder Width in feet - 0.048 0.953 0.048 0.323 3 to 8 ft shoulder Width in feet 0.042 1.043 0.036 0.250 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.119 1.126 0.036 0.001 15 - to - 24.9 driveways per mile Binary indicator variable 0.178 1.195 0.040 <0.001 > 25 driveways per mile Binary indicator variable 0.262 1.299 0.043 <0.001 Site random effect 0.555 County random effect 0.254 Overdispersion parameter 0.045 AIC 49,256.7 Log likelihood - 24,610.3 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion ; mph = miles per hour; ft = foot 159 Table 59 : Mixed Effects Negative Binomial Model Results for Paved Non - Federal Aid Segments (No Length Offset) Factor Description Est Exp(B) Std error P - value Intercept - 6.848 0.001 0.310 <0.001 Segment length N atural log of, miles 1.017 2.764 0.059 <0.001 AADT N atural log of, vehicles per day 0.802 2.229 0.047 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 0.275 1.317 0.432 0.524 45 - 49.9 mph C urved proportion of segment 0.279 1.322 0.569 0.624 40 - 44.9 mph C urved proportion of segment 0.959 2.610 0.450 0.033 <40 mph C urved proportion of segment 1.442 4.230 0.665 0.030 11 - ft lane Baseline 12 - or 13 - ft lane Width in feet - 0.066 0.936 0.110 0.548 9 - or 10 - ft lane Width in feet 0.043 1.044 0.070 0.539 0 - ft shoulder Baseline 1 - ft shoulder Width in feet 0.060 1.062 0.088 0.495 2 - ft shoulder or wider Width in feet 0.096 1.101 0.166 0.561 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.092 1.096 0.112 0.410 15 - to - 24.9 driveways per mile Binary indicator variable 0.068 1.070 0.117 0.560 > 25 driveways per mile Binary indicator variable 0.311 1.365 0.116 0.007 Site random effect 0.638 County random effect <0.001 Overdispersion parameter 0.022 AIC 8,420.4 Log likelihood - 4,193.2 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion ; mph = miles per hour; ft = foot 160 Table 60 : Mixed Effects Negative Binomial Model Results for Unp aved Se gments (No Length Offset) Factor Description Est Exp(B) Std error P - value Intercept - 5.792 0.003 0.667 <0.001 Segment length N atural log of, miles 0.944 2.571 0.051 <0.001 AADT N atural log of, vehicles per day 0.601 1.823 0.052 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 1.721 5.592 0.365 <0.001 45 - 49.9 mph C urved proportion of segment 1.357 3.886 0.310 <0.001 40 - 44.9 mph C urved proportion of segment 1.244 3.469 0.380 0.001 <40 mph C urved proportion of segment 1.278 3.590 0.334 <0.001 Surface width W idth in feet, natural log of 0.087 1.091 0.235 0.711 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.168 1.182 0.087 0.053 15 - to - 24.9 driveways per mile Binary indicator variable 0.019 1.019 0.099 0.846 > 25 driveways per mile Binary indicator variable 0.167 1.181 0.108 0.122 County random effect 0.732 Overdispersion parameter 0.026 AIC 8,945.3 Log likelihood - 4,459.7 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion ; mph = miles per hour; ft = foot 161 Append i x D: Mixed Effects Models with M odified Curve Variable s Table 61 : Mixed Effects Negative Binomial Model Results for Paved Federal Aid Segments (Binary Curve Variables) Factor Description Est Exp(B) Std error P - value Intercept - 5.947 0.003 0.156 <0.001 Segment length O ffset, natural log of, miles AADT N atural log of, vehicles per day 0.681 1.976 0.021 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 0.614 1.848 0.223 0.006 45 - 49.9 mph C urved proportion of segment 1.149 3.154 0.218 <0.001 40 - 44.9 mph C urved proportion of segment 0.932 2.539 0.383 0.015 <40 mph C urved proportion of segment - 12.272 0.000 593.110 0.983 10 - ft lane Baseline 11 - ft lane Width in feet 0.027 1.027 0.043 0.537 12 - ft lane Width in feet 0.059 1.061 0.050 0.240 13 - ft lane Width in feet 0.133 1.142 0.091 0.145 0 to 1 ft shoulder Baseline 2 - ft shoulder Width in feet - 0.055 0.946 0.050 0.271 3 to 8 ft shoulder Width in feet 0.048 1.049 0.038 0.200 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.113 1.120 0.037 0.002 15 - to - 24.9 driveways per mile Binary indicator variable 0.194 1.214 0.041 <0.001 > 25 driveways per mile Binary indicator variable 0.291 1.338 0.044 <0.001 Site random effect 0.558 County random effect 0.239 Overdispersion parameter 0.040 AIC 46,161.7 Log likelihood - 23,063.9 Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion ; mph = miles per hour; ft = foot 162 Table 62 : Mixed Effects Negative Binomial Model Results for Paved Federal Aid Segments (Quasi - Binary Curve Variable for <40 mph Curve Design Speed ) Factor Description Est Exp(B) Std error P - value Intercept - 5.951 0.003 0.156 <0.001 Segment length O ffset, natural log of, miles AADT N atural log of, vehicles per day 0.681 1.976 0.021 <0.001 Horizontal curve design speed >55 mph B aseline 50 - 54.9 mph C urved proportion of segment 0.614 1.848 0.223 0.006 45 - 49.9 mph C urved proportion of segment 1.149 3.155 0.218 <0.001 40 - 44.9 mph C urved proportion of segment 0.931 2.537 0.383 0.015 <40 mph * Curved proportion of segment 1.080 2.944 0.500 0.031 10 - ft lane Baseline 11 - ft lane Width in feet 0.026 1.026 0.043 0.542 12 - ft lane Width in feet 0.059 1.061 0.050 0.240 13 - ft lane Width in feet 0.133 1.142 0.091 0.145 0 to 1 ft shoulder Baseline 2 - ft shoulder Width in feet - 0.055 0.947 0.050 0.275 3 to 8 ft shoulder Width in feet 0.049 1.051 0.038 0.189 0 - to - 4.9 driveways per mile Baseline 5 - to - 14.9 driveways per mile Binary indicator variable 0.114 1.121 0.037 0.002 15 - to - 24.9 driveways per mile Binary indicator variable 0.194 1.214 0.041 <0.001 > 25 driveways per mile Binary indicator variable 0.292 1.339 0.044 <0.001 Site random effect 0.558 County random effect 0.240 Overdispersion parameter 0.040 AIC 46,189.9 Log likelihood - 23,077.9 *The quasi - curve variable only applies to <40 mph curve design speed segments, all other curve variables are binary Note: Est = parameter estimate, Std = standard, AIC = Akaike information criterion ; 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