Development of Analysis Methods Using Recent Data (2012)

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78 Outline of a Causal Theory of Traffic Conflicts and Collisions Gary A. Davis, John Hourdos, and Hui Xiong Department of Civil Engineering, Minnesota Traffic Observatory, University of Minnesota Using recent developments in causal analysis, a minimal model capable of rigorously representing traffic conflicts and crashes is con- structed. This model is then used to derive relationships between these types of events. The first result indicates that the magnitude of the minimum sufficient evasive action can be used to partition the space of background conditions, leading to a natural scale for ranking the severity of conflicts. The second result indicates that crashes can possibly arise from any region of the space of background conditions with the contributions of different regions of the background space to the crash population being weighted by evasive action. The third result gives a counterfactual definition of a type of conflict called a close encounter and relates the relative frequency of close encounters to that of crashes, with the evasive action determining the crash-to- conflict ratio. It is then illustrated how trajectory-based reconstruc- tion can be used to classify close encounters with respect to seriousness and to estimate the potential number of crashes in a set of close encounters. Introduction It is a commonplace observation that certain drivers or loca­ tions experience a higher frequency of traffic crashes than do other comparable entities. Hauer (1997) has identified the expected number of crashes of a stated type over a stated time interval as an appropriate measure of the safety of an entity; other things being equal, safety can be estimated by counting the crashes that have occurred. The relative rarity of crashes, however, means that even for entities and crash types of relatively high frequency, such as multiple vehicle crashes at busy urban intersections, observation periods of several years may be needed to produce estimates with acceptable statistical properties. This has led to a search for surrogate events that occur at more easily observable rates and whose frequency or nature are indicative of the fre­ quency or nature of crashes. The hypothesis that noncatastrophic events might provide information concerning the nature of catastrophic events is not new and can be found, for example, in the use of “critical incidents” in evaluating the proficiency of air traffic control­ lers and assessing pilot errors (Flanagan 1954). In traffic engi­ neering, the first use of traffic conflicts as crash surrogates is usually attributed to Perkins and Harris (1968), but during the 1970s several versions of the traffic conflict technique were developed and employed in both North America and Europe (Asmussen 1984). The Association for International Cooperation in Traffic Conflict Techniques proposed the fol­ lowing definition of traffic conflict: A traffic conflict is an observable situation in which two or more road users approach each other in space and time to such an extent that there is a risk of collision if their move­ ments remain unchanged. (Guttinger 1984, p. 18) In what follows, we will first briefly review the literature on conflicts and surrogate events, with a focus on three inter­ related issues. The first issue concerns Heinrich’s Triangle as a model for traffic safety. The second issue concerns the grad­ ing of conflicts or surrogate events with respect to seriousness or severity. The third issue concerns the relation between crash frequencies and conflict frequencies. Heinrich’s Triangle Heinrich’s Triangle is a name given to the hypothesis that events can be ranked in order of increasing severity but decreas­ ing frequency. This idea is often represented graphically with A p p e n d I x C

79 Hayward (1972) introduced the use of time to collision (TTC) as a measure of conflict severity. At a given instant, TTC is the time at which two road­using entities would collide if each persisted on its present course. Hayward illustrated how, when plotting TTC versus time, TTC would first decrease to a minimal value and then increase after one or more of the involved entities initiated evasive action. The minimum value of TTC for an encounter was taken as a measure of how close the encounter was to an actual collision. In the Swedish traffic conflict technique, TTC at the instant a road user initiates evasive action, rather than the minimum TTC over a time interval, was taken as the measure of conflict severity. In its initial implementations at an urban inter­ section, the Swedish TTC defined serious conflicts as those with TTC < 1.5 s. Using this definition, estimates of the crashes per conflict with orders of magnitude of 10-5 were reported (Hyden and Linderholm 1984). When attempting to extend this idea to rural intersections, however, it was found that vehicle speed also needed to be considered, ultimately leading Svensson and Hyden (2006) to grade severity as a function of the ratio of TTC to closing speed (CS), Severity TTC CS= ( )f ( )1 with higher values of TTC/CS indicating conflicts of lower severity. This characterization is especially interesting because TTC/CS is inversely proportional to the minimum decelera­ tion needed to bring the vehicle to a stop in the TTC interval. Svensson and Hyden also advanced the intriguing hypothesis that severe conflicts result from a mismatch between a road user’s expectations and actual events. A similar approach to grading the severity of vehicle inter­ actions has been developed in the United States to support evaluation of collision warning systems. In Smith et al. (2002), the situation between leading and following vehicles was characterized in terms of the current separation between the two vehicles, called the range (R), and the speed difference between the vehicles, called the range rate (RR). Using driver behavior data observed in driving simulators and test­track experiments, range versus range­rate curves were then used to partition the set of possible range and range­rate combina­ tions into subsets reflecting crash, near­crash, conflict, and low­risk situations. For example, the relationship R RR g= ( )( )2 2 65 2. ( ) where g denotes the gravitational acceleration, was suggested as roughly dividing the crash and near­crash conditions. In Najm et al. (2006), a TTC was defined via TTC R RR = ( )3 a pyramid divided into horizontal sections, each section rep­ resenting a class of events. The height of a section from the pyramid’s base represents the severity of the corresponding class of events, while the volume of the section represents the relative frequency for that class. For example, fatal accidents are least frequent, so they would occupy the tip of the pyra­ mid. Below fatal accidents might be injury accidents and then noninjury accidents, close calls, and so forth. Heinrich (1959) developed this hypothesis with regard to industrial accidents and conjectured that stable relative frequencies could be found between the levels of a pyramid. In traffic safety, qual­ itative versions of Heinrich’s Triangle have been invoked for conceptual illustration (e.g., Hauer 1997; Svensson and Hyden 2006), while Dingus et al. (1999) pointed out that establish­ ing predictable relationships between the levels of a Heinrich’s Triangle could justify using observations of near­crash events to evaluate the potential safety benefits of crash­avoidance technologies. Dingus et al. (2006) then provided empirical support for such a triangular relationship between conflicts, near crashes, and crashes observed in the 100­Car Naturalis­ tic Driving Study. Grading Conflict Severity Interest in grading the severity of conflicts arose relatively early in the development of traffic conflict methods, in con­ nection with difficulties encountered in establishing empirical relationships between conflict counts and observed crash fre­ quencies. For example, Guttinger (1984) and Baguley (1984) presented selected graphs showing crashes as increasing functions of conflicts, but more comprehensive studies by Cooper (1984) and Migletz et al. (1985) reported instances where crash experience appeared to increase, appeared to have no relation, or appeared to decrease as measures of con­ flicts increased. If a Heinrich’s Triangle­like relationship is accurate, then it would be expected that establishing a rela­ tionship between events in neighboring levels of the pyramid would be easier than between events at one level and events aggregated over several lower levels. This in turn means that objective criteria would be needed for assigning events to their appropriate severity levels. As an example, in a study of vehicle and pedestrian interactions, Guttinger reported a clear association between crash frequency and the frequency of serious conflicts, which were defined as follows: Serious conflict: a sudden motor reaction by a party or both of the parties involved in a traffic situation towards the other to avoid a collision, with a distance of about one metre or less between those involved. (Guttinger 1984, p. 19) Here, a serious conflict is defined using both the physical separation between the involved entities and the “sudden­ ness” of evasive action. For vehicle­to­vehicle interactions,

80 and combinations of TTC versus RR were used to define boundaries between near­crash and conflict situations. Since range rate can also be interpreted as closing speed (CS), there is a clear similarity between this approach to grading severity and that developed by Svensson and Hyden. Finally, in the 100­car study, near­crash events were defined explicitly in terms of the magnitude of a required evasive action, as Any circumstance that requires a rapid, evasive maneuver by the subject vehicle, or any other vehicle, pedestrian, cyclist, or animal to avoid a crash. A rapid, evasive maneuver is defined as a steering, braking, accelerating, or any combination of control inputs that approaches the limit of the vehicle capa­ bilities. (Dingus et al. 2006) Since, as pointed out, the ratio TTC/CS is inversely pro­ portional to a stopping deceleration, we have an additional overlap among the proposed methods for grading conflict severity. Relationship Between Crashes and Conflicts The third issue identified involves the relationship, if any, between crash frequency and conflict frequency. Early efforts focused on attempting to establish correlations between con­ flict and crash frequencies, but Hauer (1982) noted that if conflicts are treated as crash opportunities, some of which actually result in crashes, then the relationship between con­ flicts and crash frequency should take the form Expected number of crashes Number of confli= cts crash­to­conflict ratio( )× ( ) ( )4 That is, the number of conflicts is a measure of crash opportunities, while the crash­to­conflict ratio reflects the probability that a given crash opportunity results in a crash. In this case, the theoretical correlation between the random variables generating crash and conflict frequencies will depend both on the magnitude of the crash­to­conflict ratio and on the variability in the number of conflicts, and so could be low even when these quantities are known perfectly. Hauer and Garder (1986) gave a more extensive treatment of this approach, pointing out that the usefulness of Equation 4 as a predictive tool depended first on the ability to estimate the crash­to­conflict ratio with acceptable precision and second on the stability of this ratio across different entities or times. If conflicts of different degrees of severity have different probabilities of resulting in crashes, then a study that mixes together conflicts of varying severity may find it difficult to identify a stable crash­to­conflict ratio. This then leads to the problem of identifying groupings of conflicts having stable ratios. One possible solution is to interpret the crash­to­ conflict ratio as the probability that a conflict results in a crash and then allow this probability to vary continuously with a measure of conflict severity. For example, Saunier and Sayed (2008), drawing on work by Hu et al. (2004), have con­ jectured that, when conflict severity is measured using TTC, a relationship of the form P TTC TTCcrash( ) = −( ) exp ( )σ 2 5 may prove useful. An alternative approach to relating observed conflict fre­ quencies to crash likelihood is to treat crashes and conflicts as events varying continuously on some dimension, with crashes being at an extreme of this dimension. It might then be possible to apply extreme­value statistics to estimate the probability of these extreme events. This approach was ini­ tially proposed by Campbell et al. (1996), who applied it to three measures of car­following obtained from a study of drivers in instrumented vehicles: range, TTC, and available reaction time. For each of these variables, a value of zero defines the point at which a collision occurs; and for each of these variables, the fitted extreme­value distribution assigned a probability of zero to crash events. The authors pointed out that for this method to be valid, crash and noncrash events should result as random outcomes from the same underlying probability distribution, and that if crash events and normal­ driving events come from different populations, then the standard assumptions of extreme­event statistics may not be satisfied. This is an important point, and it will be helpful to give it a more formal restatement. Suppose driving events can be divided into two subsets, “normal driving” and “close calls.” Let y denote a numerical variable characterizing these events, and let yc denote a critical value for y such that when y < yc, a crash occurs. Then the probability of a crash is given by the appropriate mixture of the two sets of conditions P P y yc Pcrash normal driving normal drivi( ) = <( ) ng close call close call ( ) + <( ) ( )P y yc P ( )6 Given an adequate sample of normal driving conditions, standard extreme­value methods could be used to approxi­ mate P(y < yc |normal driving), and given a sample of close calls, extreme­value methods could be used to approximate P(y < yc |close call). However, if P(y < yc |normal driving) ≠ P(y < yc |close call), extreme­value methods applied to obser­ vations of normal driving conditions will generally not be sufficient to estimate crash probabilities.

81 how these formal results might be applied to actual crash and near­crash events. Causal Model of Crashes and Conflicts Our starting point is Pearl’s (2000) notion of a causal model, which consists of a set of exogenous variables, a set of endo g­ enous variables and, for each endogenous variable, a struc­ tural equation describing how that variable responds to changes in other model variables. A causal model can be rep­ resented qualitatively using a directed graph, with the nodes of the graph representing variables and directed arrows indi­ cating direct causal dependencies. Figure C.1 displays the simple graphical model that underlies our treatment. The node u, which may be vector­valued, denotes variables describ­ ing background conditions. The node x denotes the variable describing evasive or avoidance action, and the node y is a crash­related outcome. Associated with y is a critical value yc such that if y(u,x) < yc, then a crash results. In what follows, we will assume that the evasive action x takes on values from a discrete set X = {x1,x2, . . . , xn}, with x1 < x2 < . . . < xn, and the background conditions take one value from a denumerable set U. Since continuous ranges of pos­ sible evasive action values or background conditions can be approximated arbitrarily well by discrete sets, this is a rela­ tively weak constraint. It will, however, allow us to derive results using elementary mathematics and avoid some of the technicalities that arise when treating general sets X. To complete the model specification, we need a probability distribution over the values taken on the background vari­ ables, denoted by P(u), and a structural equation describing how evasive actions are selected. However, the results we will develop require only the weaker condition that we have a conditional probability distribution for the evasive action, denoted by P(x |u). The conditional independence structure implied by the graphical model in Figure C.1 then means that the full joint distribution for our model factors into one with the form P y x u P y x u P x u P u, , , ( )( ) = ( ) ( ) ( ) 7 The nature of u and x and the functional form for y(u,x) will, of course, vary depending on the type of crash under Analytic results relevant to this issue have been given by Al­Hussaini and El­Adll (2004). One of the key features of extreme­value statistics is that the distribution functions of extreme values from random samples tend to converge, after suitable normalization, to one of only three limiting distribu­ tions. This means that if it is possible to identify which of these three distributions is appropriate for a given problem, then predictions of extreme events can be accomplished even when detailed knowledge of the underlying probability model is lacking. Al­Hussaini and El­Adll (2004) show that the clas­ sic convergence results do not necessarily apply to data gen­ erated by finite mixtures of distributions, such as Equation 6. In this case, the limiting distribution for the extreme values can depend both on the type of mixture components and on their relative weights. One approach to solving this problem would be to identify an observable variable that could serve as a proxy for the dif­ ferent driving conditions and then allow one’s extreme­value model to depend on this. Apparently independently of the aforementioned work, Songchitruksa and Tarko (2006) applied extreme­event methods to observations of postencroach­ ment time (PET) obtained from video recordings of vehicle encounters at intersections. By allowing their extreme­value distributions to vary with measures of traffic flow, the authors were able to generate predicted crash frequencies that were, at least for some intersections, consistent with observed crash frequencies. To summarize, there has been a long­standing interest in using surrogate events such as traffic conflicts as a proxy for crashes, and a long­standing belief that it should be possible to relate the expected number of crashes to a measure of con­ flict frequency. Initial difficulties in establishing stable crash­ to­conflict relationships led to an interest in grading the severity of conflicts, with the idea being that severe conflicts would be more reflective of crashes. Currently in the litera­ ture there are descriptions of several methods for assessing severity of a noncrash event, which rely on measures of the kinematic variables, and show some overlap. One point to emphasize is that the working definitions of observable conflicts cited previously all have counterfactual components. That is, a conflict is defined as a noncrash event where, had things been different, a crash would have resulted. Counterfactuals are somewhat difficult to address rigorously (Lewis 1973), but recently Pearl (2000) has described formal tools that allow one to assess probabilities assigned to both factual and counterfactual statements. In what follows, we will use some of these tools to give a rigorous counterfactual definition of traffic conflict and then use that definition to derive a relationship between the relative frequency of crashes and that of conflicts of differing severity, where it turns out that the crash­to­conflict ratios are governed by drivers’ abil­ ity to achieve successful evasive action. We will then illustrate u x y Figure C.1. Simple graphical model of crash events depicting conditional independence structure.

82 The distance separating the major approach from the col­ lision point at this time is y u x y v t r t y v t a t r, ˆ , ˆ ˆ ˆ( ) = − > − + −( ) 2 2 1 2 1 2 2 1 2 1 2 2 2 , ˆ ( ) r t≤    1 11 A collision is taken to occur when y(u,x) < 0. In what follows, we will assume that the function y(u,x) is monotonic in x, in the sense that, for each u, x1 > x2 implies that y(u,x1) > y(u,x2). That is, for a given set of background conditions, increasing the magnitude of the evasive action does not produce a crash where one would not have occurred otherwise. It is straightforward to verify that, given reason­ able constraints of the background conditions, this monoto­ nicity condition is satisfied by both the above models. Model properties The first consequence of our model is that set X, together with the monotonic structural equation y(u,x), can be used to partition the set of background conditions. That is, the sets U1 = {u: y(u,xj) ≥ yc, j = 1, . . , n} U2 = {u: y(u,x1) < yc & y(u,xj) ≥ yc, j = 2, . . , n} . . . Uj = {u: y(u,xi) < yc, i = 1, . . , j - 1 & y(u,xi) ≥ yc, i = j, . . , n} . . . Un+1 = {u: y(u,xi) < yc, j = 1, . . , xn} form a partition of the set of background conditions. A proof of this claim is given in the appendix to this paper. One way to interpret the sets U1, . . , Un+1 is as follows. Define the function xmin(u) = the smallest value xj such that y(u,xj) ≥ yc. For each possible background condition u, xmin(u) gives the lowest value of the evasive action that prevents a crash. U1 is then the set of conditions where a crash never occurs, Uj is the set of all conditions for which xmin(u) = xj, j = 1, . . . n, and Un+1 is the set of conditions where a crash is unavoidable. The back­ ground conditions belonging to a given set in our partition share the same value for xmin, so each set in our partition is identified by its characteristic minimum successful evasive action. If xmin(u) is taken as a measure of the severity of an event, then we have a partition of events with respect to severity. The second consequence is that probability of a crash is a mixture of the crash probabilities from the different sets in our partition. consideration. For example, in Brill’s (1972) simple two­ vehicle rear­ending collision model, the initial speed and braking deceleration of the leading vehicle can be denoted by v1 and a1, respectively, while v2 and a2 denote the initial speed and braking deceleration of the following vehicle, and h2 and r2 denote the following headway and reaction time of the fol­ lowing driver. A collision occurs when the stopping distance available to the following driver is less than that needed to stop without colliding with the lead vehicle. That is, when v h v a v r v a2 2 1 2 1 2 2 2 2 22 2 0 8+ − +     < ( ) If we take the following driver’s deceleration as the avoid­ ance action, then the variables (v1, a1, v2, r2, h2) would be com­ ponents of u, the evasive action x would be a2, and the outcome function would be y u x v h v a r v v a , ( )( ) = + − −2 2 1 2 1 2 2 2 2 22 2 9 Placing a probability distribution over the values taken on by the model’s exogenous variables produces what Pearl (2000) calls a probabilistic causal model. This distribution, together with the structural equations for the endogenous variables, is then sufficient to compute probabilities assigned to both factual and counterfactual events. As another example, consider the interaction of major and minor approach vehicles at a two­way­stop controlled inter­ section, where the minor approach vehicle crosses in front of the major approach vehicle. Let: y2 = initial distance of major approach vehicle from colli­ sion point v2 = initial speed of major approach vehicle r2 = reaction time of major approach driver a2 = deceleration of major approach driver y1 = initial distance of minor approach vehicle from colli­ sion point v1 = initial speed of minor approach vehicle a1 = acceleration of minor approach driver Here, the deceleration of the major approach driver is taken as the evasive action and the remaining variables describe background conditions. The time needed by the minor approach driver to clear the collision point is ˆ ( )t v a y a1 1 2 1 1 1 2 10= +

83 Here the notation P(Uj) has been used as shorthand for P(u∈Uj). This result can then be used to eliminate P(Uj) in Equation 13, giving the probability of a crash in terms of close encounters and evasive actions P y yc P y yc x x y yc P x x j x x j n j j <( ) = ≥ = <( ) ≤ = = + − ∑ & & 1 2 1 − ( ) =( )     1 14 U P x x U j j j ( ) For background events in set Uj, a crash occurs if x ≤ xj-1, while a close encounter occurs if x = xj. The ratio P(x ≤ xj-1 |Uj)/P(x = xj |Uj) can thus be interpreted as a crash­ to­conflict ratio, and Equation 14 states that crashes can result from close encounters of varying severity, with possibly different crash­to­conflict ratios. Equation 14 is a generaliza­ tion of the relationship proposed in Equation 4, with an explicit representation of the crash­to­conflict ratio. To summarize, Equation 12 states that, for this model, the probability of crash can be expressed in terms of the proba­ bility distribution for background conditions together with knowledge of how evasive actions are chosen. Equation 14 states that the probability distributions for the background conditions can be dispensed with if it is possible to find prob­ ability distributions for a type of traffic conflict we call a close encounter. In either case, though, it is the evasive action model that determines the relative contribution of a region in the space of background conditions to the population of crashes. Applying the Results Since the definition of a close encounter has both factual and counterfactual components, determining whether an observed event is a close encounter requires conducting a counter­ factual test. This is most easily done when one has at hand the structural equation y(u,x) and plausible estimates of the val­ ues for the event’s background variables. For the simple rear­ ending collision model described, Davis and Swenson (2006) have discussed how Bayes estimates of the background condi­ tions and the evasive action can be obtained by fitting a model to a vehicle’s trajectory data extracted from a video recording of the event. Given posterior distributions characterizing the uncertainty regarding the event’s background conditions, it is then straightforward to compute the probability of a crash as a function of different values of the evasive action, using Monte Carlo simulation. Figure C.3 shows plots of the crash probability versus braking deceleration for two noncolli­ sion events observed on Interstate 94 by (the predecessor of) the Minnesota Traffic Observatory. For the encounter P P y yc P x x u U P u Uj j n j jcrash( ) = <( ) = ≤( ∈ ) ∈(− = +∑ 1 2 1 ) ( )12 (A proof of this claim is given in the appendix to this paper.) This relationship states that, in principle, crashes can result from most regions of the space of background conditions, and it is the evasive action model that converts the possibility of crash into whether a crash actually occurs. Other things being equal, those regions where it is difficult to achieve a successful evasive action should contribute proportionally more to the set of crashes. Our third result concerns the relationship between con­ flicts and crashes. As noted, working definitions of conflict and near crash have a counterfactual component: an observed conflict or near crash is a noncrash event that, had things been different, could have been a crash. A rigorous definition of conflict will thus require a method for formally stating this counterfactual condition, and one of the strengths of Pearl’s theory is that it provides just such a method. Pearl (2000) has used this formal language to explicate different ideas regard­ ing causal effects and to identify conditions that allow causal effects to be estimated from nonexperimental studies. Davis (2002) has argued that the concepts of causal effect used in statistical safety studies, crash simulation studies, and crash reconstruction can be treated as instances of what Pearl calls probability of necessity. Following Pearl’s treatment, the notation yx=x0 will stand for the value taken on by the variable y when the variable x is set, counterfactually, to the value x0. Probabilities regarding counterfactual claims are then evaluated using the modified graphical model depicted in Figure C.2. In order to sidestep the surplus meaning attached to notions like conflict and near crash, we will define a close encounter as an event with y ≥ yc (crash does not occur), x = xj (evasive action of level xj was observed), but yx=xj-1 < yc (had the next weakest evasive action xj-1 been employed, a crash would have occurred.) It can then be shown that P CE P y yc x x y yc P x x U P U j x x j j j j ( ) = ≥ = <( ) = = )( ( = − & & 1 ) ( )13 (A proof of this claim is given in the appendix to this paper.) u x=x0 y Figure C.2. Modified graphical model of crash for assessing counterfactual event x 5 x0.

84 This ratio will tend to be large for subsets where the minimum successful evasive action comes from the right­hand tail of the distribution of evasive actions. That is, other things being equal, crashes should tend to have extreme values of the eva­ sive action, and this is what Figure C.4 shows. This gives us a possible interpretation of the empirical findings that the fre­ quency of serious conflicts tends to be more reliably associated with the frequency of crashes. The results in Figure C.4 also suggest a means for approx­ imating the expected number of crashes from a set of non­ crash events, where detailed knowledge of the evasive action model may not be necessary. If the limited data shown in Figure C.4 are typical of rear­ending events, the space of back­ ground conditions can be roughly divided into two subsets, one where crash occurrence tends to be negligible, character­ ized by P(x < xmin(u)) ≈ 0, and one subset from which crashes tend to be generated—that is, where P(x < xmin(u)) > 0. If, in addition, the evasive action model can be taken as roughly constant in this crash­producing region, then an estimate of the number of crashes in a set of noncrash events can be computed as the sum of the probabilities that each of these events could have been a crash. That is, if P(x = xj |u = ui) = P(x = xj), then P P y yc P x x P x xj j jcrash( ) = <( ) = >( ) =( )∑ min ( )16 Figure C.5 shows the two relationships depicted in Fig­ ure C.3, along with a proposed probability distribution for between Vehicles 1 and 2, braking decelerations on the part of Driver 2 below about 7 ft/s2 would probably have resulted in a collision, while for the encounter between Vehicles 5 and 6, braking decelerations by Driver 6 below about 15 ft/s2 would probably have resulted in a collision. The results shown in Figure C.3 make no reference to the actual braking decelerations used and so by themselves are not sufficient to determine if the event qualifies as a close encounter. Figure C.4, however, displays point estimates of actual and minimal braking decelerations for vehicles involved in three freeway rear­ending crash events, also taken from Davis and Swenson (2006), and including the two events illustrated in Figure C.3. The three rightmost points on Figure C.4 represent decelerations for colliding vehicles, while the points in the cluster to the left represent, with one exception, decelerations of successful stops before colliding. The first interesting observation from Figure C.4 is that the noncolliding drivers appeared to select decelerations close to the minima needed to stop without colliding. Since for each noncolliding vehicle a relatively small decrease in deceleration would have led, other things being equal, to a collision, each of the successful stops arguably qualifies as a close encounter. The second interesting observation is that the decelerations used by colliding drivers tend to be substantially greater than those used in the observed close encounters. Referring to Equation 14, the crash­to­conflict ratio for near­crash events for subset Uj takes the form P x x U P x x Uj j j j≤( ) =( )−1 15( ) 0 5 10 15 20 25 0 0.5 1 Vehicles 1 & 2 Vehicles 5 & 6 feet/sec2 Pr ob ab ili ty o f c ra sh 1.1 0 p12 p56 221 a Figure C.3. Crash probability versus braking deceleration for two noncrash events. The event involving Vehicles 1 and 2 is arguably less severe than the event involving Vehicles 5 and 6.

85 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 crash 1 crash 2 crash 3 slope=1 line 1.0 0 A2g 1 A10g 1 A13g 1 Aeq 1 1.00 Amin2g 1 Amin10g 1, Amin13g 1, Aeq 2, Figure C.4. Minimum versus observed decelerations (in g units) for vehicles stopping on a congested freeway. The x axis is minimum deceleration in g units, and the y axis is observed deceleration in g units. The three rightmost points are from colliding vehicles. 0 5 10 15 20 25 0 0.5 1 Vehicles 1 & 2 Vehicles 5 & 6 Evasive pdf feet/sec2 Pr ob ab ili ty o f c ra sh 1.1 0 p12 p56 pa 251 a Figure C.5. Crash probability versus deceleration curves for two noncrash events, together with a proposed probability density for decelerations.

86 In this paper, we postulated a minimal theoretical structure that incorporates crashes, near crashes, and conflicts; defined rigorously what we mean by a surrogate event; and then derived a relationship between crash propensity and the sur­ rogate. Our treatment produced a natural, one­dimensional method for grading the severity of noncrash events in terms of their minimum successful evasive actions. Our treat­ ment also produced a generalization of Hauer’s proposed relationship between expected crashes and conflicts, where the crash­to­conflict ratio is governed by the evasive action model, suggesting that a better understanding of crash­to­ conflict ratios could be had through a better understanding of evasive action. Finally, combining our treatment with some preliminary empirical work on rear­ending conflicts and crashes led to a suggested alternative method for estimat­ ing the expected number of crashes from a set of conflicts, where each conflict is assigned a probability of (counter­ factually) being a crash. It may occasionally happen that a theoretical treatment can substitute for empirical work, but more often the best a theoretical treatment can offer is an interpretation of past work and a guide for future work. For situations where crash outcome is a monotonic function of an evasive action, our treatment indicated that converting a set of conflict events to an estimate of crash tendency requires (1) counterfactual testing to correctly classify the conflict events, and (2) under­ standing how evasive actions are selected over a range of con­ flict severities, so that correct crash­to­conflict ratios can be determined. These requirements, in turn, suggest directions for future work. First, one important focus would be on developing defensible models of how drivers select evasive actions as functions of background conditions. Second, we indicated previous work where counterfactual testing was accomplished through trajectory­based reconstruction of rear­ending events, based on fairly simple algebraic trajectory models. This initial work should be extended to support more complicated models, possibly based on differential equations. Third, it may happen that the structural features of a particu­ lar problem allow Steps 1 and 2 to be simplified. We illustrated one such simplification where detailed knowledge of evasive action mechanisms was replaced by an empirically derived probability density, and highlighted the assumptions needed to justify this simplification. We suspect that the majority, if not all, of proposed traffic conflict techniques can be regarded as simplifications of Steps 1 and 2, but where the simplifying assumptions are to greater or lesser degrees unclear. If this is the case, then the credibility and usefulness of economical traffic conflict techniques could be enhanced substantially by unpacking these assumptions and determining the condi­ tions under which they are valid. emergency braking decelerations. This proposed distribution is based on a normal distribution, with mean and variance taken from the surprise braking tests reported by Fambro et al. (1997). Since for monotonic relationships y(x,u) < yc only if xmin(u) > x, the function giving crash probability ver­ sus braking deceleration is equivalent to the function giving probability of failing to achieve the minimum safe decelera­ tion versus braking deceleration. “Integrating” this function with respect to the braking deceleration distribution then gives the probability that the event could have been a crash had the following driver’s deceleration been similar to that observed by Fambro et al. (1997). Table C.1 displays these counterfactual crash probabilities computed for five leader– follower pairs of noncolliding vehicles observed on Inter­ state 94 using the Fambro et al. (1997) statistics. Pairs 1–2 and 5–6 are the same, as shown in Figures C.3 and C.5. The expected number of crashes in this set, which is simply the sum of these probabilities, would be 0.142. discussion Traffic collisions tend to be relatively rare but of nontrivial financial and personal consequence when they happen. It is not surprising then that substantial effort has been invested in finding useful surrogates for collisions, which are easier to observe but that still provide information on the frequency and nature of collisions. Our review of the literature on traffic conflicts and near crashes identified two important trends. The first is that conflicts can be ranked with respect to sever­ ity and that the more severe conflicts tend to show reliable associations with crash experience. The second is that work­ ing definitions of observed conflicts contain a counterfactual component, where a conflict is an event in which, had some action not occurred, a crash would have occurred. Table C.1. Crash Probabilities Computed for Five Leader–Follower Pairs of Noncolliding Vehicles Vehicle Number Estimated Follower Deceleration (ft/s2)a Leader Follower Actual Minimum P(crash) 1 2 6.5 (.06) 6.2 (.06) 0 2 3 12.6 (.99) 11.4 (.66) 0 3 4 14.2 (.51) 12.8 (.43) 0 4 5 16.0 (.91) 14.4 (.63) .004 5 6 17.3 (1.6) 17.1 (1.5) .138 Sum .142 a Standard deviations for deceleration estimate are given in parentheses.

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88 P y yc y u x P x x u P u y u i j j n i j i i<( ) = ( ) =( ) ( ) = = ∑∑ ˆ , ˆ , 1 x P x x u P u y u x P j j j n u U j ( ) =( ) ( )   + ( ) =∈ ∑∑ 11 ˆ , x x u P u y u x P x j j n u U j =( ) ( )   + + ( ) =∈ ∑∑ 12 . . . ˆ , =( ) ( )  =∈ ∑∑+ x u P ujj n u Un 11 But ˆ ,y u x P x x u P u P xj j j n u U ( ) =( ) ( )   = ( ) ==∈ ∑∑ 11 0 x u P ujj n u U ( ) ( )   = =∈ ∑∑ 11 0 and for k = 2, . . , n + 1 ˆ ,y u x P x x u P u P x j j j n u Uk ( ) =( ) ( )   = ( ) = =∈ ∑∑ 1 1 x u P u P x x u P u P u U k k 1 11 0 ( ) ( )+( + ( ) =( ) ( ) + ( ) ∈ − ∑ . . . x x u P u P x x u P u P x x u k n k =( ) ( )+ + ( ) =( ) ( )) = ≤( − . . . 0 1 ) ( ) = ≤ ∈( ) ∈( ) ∈ − ∑ P u P x x u U P u U u U k k k k 1 Substituting appropriately then gives us the desired result. Proposition 3. P(y ≥ yc & x = xj & yx=xj-1 < yc) = P(x = xj|u∈Uj) P(u∈Uj) Proof: P y yc x x y yc y u x y u x j x x i j i j ≥ = <( ) = − ( )( ) = − & & ˆ , ˆ , 1 1 j i j i i i j i j i P x x u P u y u x P x x u − − ( ) =( ) ( ) = ( ) = ∑ ∑ 1 1 ˆ , ( ) ( ) − ( ) ( ) =( ) ∑ − P u y u x y u x P x x u P u i i j i i j j i i ˆ , ˆ , 1 ( ) Now, ˆ , ˆ , y u x P x x u P u y u x i j i j i i i j u U − − ∈ ( ) =( ) ( ) = ( ) ∑ 1 1 1 ∑ ∑=( ) ( ) ( ) = − ∈ + P x x u P u y u x P x x j i i i j u Un + . . . + ˆ , 1 1 j i iu P u( ) ( ) Referring to the partition of U defined above, for any i = 1,2, . . , n ˆ , , , , , y u x if u U k i if u U k i i k k ( ) = ∈ ≥ + ∈ ≤    1 1 0 Appendix to “Outline of a Causal Theory of Traffic Conflicts and Collisions” Derivations of Main Results Let U denote the set of possible values for the background variables u and X denote the set of possible values for the eva­ sive action x. For simplicity, we will assume that U is count­ able, and X is finite, X x x x x x xn n= { } < < <1 2 1 2, , , . . .. . , with Proposition 1. Consider the sets U1 = {u∈U : y(u,xj) ≥ yc, j = 1, . . , n} U2 = {u∈U: y(u,x1) < yc, y(u,xj) ≥ yc, j = 2, . . , n} . . . Uj = {u∈U : y(u,xi) < yc, i = 1, . . , j - 1 & y(u,xi) ≥ yc, i = j, . . , n} . . . Un+1 = {u∈U : y(u,xj) < yc, j = 1, . . , n} The sets U1, U2, . . , Un+1 form a partition of U. Proof: We need to show that U1∪U2∪ . . . Un+1 = U, and Ui∩Uj = f whenever i ≠ j. Let u∈Uj. Then u∈U. Now let u∈U. Then y(u,x1), y(u,x2), . . y(u,xn) are all well defined. If y(u,x1) ≥ yc, u∈U1, while if y(u,xn) < yc, then u∈Un+1. Now, suppose y(u,x1) < yc and y(u,xn) ≥ yc. By the monotonicity property of y(u,x), there exists value k such that y(u, x) < yc for all x < xk and y(u,x) ≥ yc for all x ≥ xk. Hence, u∈Uk, and we have shown that U1∪U2∪ . . . Un+1 = U. Next, suppose i ≠ j and there exists u belonging to both Ui and Uj. If i < j, u∈Ui implies y(u,xi) ≥ yc but u∈Uj implies y(u,xi) < yc, a contradiction. The case i > j is handled similarly, giving us Ui∩Uj = f when i ≠ j. Proposition 2. P(y < yc) = j n = +∑ 2 1 P(x ≤ xj-1|u∈Uj)P(u∈Uj) Proof: Let a crash indicator function be defined as ˆ , , , , , y u x if y u x yc if y u x yc i j i j i j ( ) = ( ) <( ) ≥   1 0 Then, P y yc y u x P x x u P u y u i j j n i j i i<( ) = ( ) =( ) ( ) = = ∑∑ ˆ , ˆ , 1 x P x x u P u y u x P j j j n u U j ( ) =( ) ( )   + ( ) =∈ ∑∑ 11 ˆ , x x u P u y u x P x j j n u U j =( ) ( )   + + ( ) =∈ ∑∑ 12 . . . ˆ , =( ) ( )  =∈ ∑∑+ x u P ujj n u Un 11

89 and since ˆ , ˆ , , , , , y u x y u x if u U k j if u U k j j k k − ( ) ( ) = ∈ ≥ + ∈ 1 1 1 0 ≤    j we get ˆ , ˆ ,y u x y u x P x x u P u P x x u i j i i j j i i j − ( ) ( ) =( ) ( ) = = ∑ 1 ∈( ) ∈( )+ = ∈( ) ∈( + + + + U P u U P x x u U P u U j j j n n 1 1 1 1 . . . + ) Putting these together we have P y yc x x y yc P x x u U P u U j x x j j j j ≥ = <( ) = = ∈( ) ∈( )+ = − & & 1 . . .+( = ∈( ) ∈( )) − = ∈( ) + + + P x x u U P u U P x x u U j n n j j 1 1 1 P u U P x x u U P u U P x j j n n ∈( )+( + = ∈( ) ∈( )) = = + + + 1 1 1 . . . x u U P u Uj j j∈( ) ∈( ) This then implies ˆ ,y u x P x x u P u P x x u U P u U i j u U j k k k ( ) =( ) ( ) = =( ∈ ) ∈( ∈ ∑ ) ≥ + ≤    , , if k i if k i 1 0 so that ˆ ,y u x P x x u P u P x x u U P u Ui j j i i j j j−( ) =( ) ( )= = ∈( ) ∈(1 )+ + = ∈( ) ∈( ) ∑ + + . . . i j n nP x x u U P u U1 1 Similarly, ˆ , ˆ , ˆ , y u x y u x P x x u P u y u x i j i j j i i i i ( ) ( ) =( ) ( ) = − ∑ 1 j j u U j i i i y u x P x x u P u y u − ∈ ( ) ( ) =( ) ( ) +∑ 1 1 ˆ , ˆ , . . . + x y u x P x x u P uj j u U j i i n − ∈ ( ) ( ) =( ) ( ) + ∑ 1 1 ˆ ,