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Using Bayesian Networks to Identify the Causal Effect of Speeding in Individual Vehicle/Pedestrian Collisions

On roads showing significant violations of posted speed limits, one measure of the safety effect of speeding is the difference between the road's actual accident count and the count that would have occurred if the posted speed limit had been strictly obeyed. An estimate of this accident reduction can be had by computing the probability that speeding was a necessary condition for each of set of accidents. This is an instance of assessing individual probabilities of causation, which is generally not possible absent prior knowledge of causal structure. For traffic accidents such prior knowledge is often available and this paper illustrates how, for a commonly occurring class of vehicle/pedestrian accidents, approaches to uncertainty and causal analyses appearing in the accident reconstruction literature can be unified using Bayesian networks. Measured skidmarks, pedestrian throw distances, and pedestrian injury severity are treated as evidence, and using the Gibbs Sampling routine BUGS, the posterior probability distribution over exogenous variables, such as the vehicle's initial speed, location, and driver reaction time, is computed. This posterior distribution is then used to compute the"probability of necessity"for speeding.


INTRODUCTION
Selecting a speed limit involves balancing the mobility of motorists against the external costs that mobility imposes on other road users. In setting speed limits, current traffic engineering practice gives preference to the 85th percentile of the vehicle speed distribution, and so for streets where the 85th percentile speed exceeds the current posted speed limit it can be argued that the posted speed limit is too low.
Where this is the case, one measure of the external cost imposed by motorists' choice of speed is the number of vehicle/pedestrian accidents that would have been prevented had the prevailing speed limit been obeyed. This measure is especially important for local and residential streets, where pedestrians are naturally to be expected. To predict the accident reduction resulting from a safety countermeasure, traffic engineers usually apply an externally estimated accident reduction factor (RF) to an observed or predicted accident count. RFs are in turn generally estimated as where rB and rA respectively denote accident rates estimated before and after the application of the countermeasure.
Unfortunately, external estimates from studies that are well enough designed to support causal interpretations tend to be rare (Davis 2000), and for pedestrian accidents on local streets even questionable estimates are not available (TRB 1998). Interpreting the accident rates as probabilities, an RF estimated from a well-designed study can then be interpreted as what Pearl calls a "probability of necessity," computed assuming minimal knowledge of how accidents actually occur (2000, p. 292). By representing traffic accidents using structural models, an alternative estimate of the effect of speeding, specific to the accident history of a given area, can be had by computing the probability that speeding was necessary for the occurrence of each accident in that area.
To show this, we can begin with Rubin's potential response model (Holland 1986

DAVIS UAI2001
Assuming that enforcement of the speed limit causes Issues concerning individual accident causation frequently arise in legal contexts, where a jury must assess "cause in fact" (Robertson 1997), and so it is not surprising that over the past 60 years the discipline of traffic accident reconstruction has developed primarily to assist the legal system in resolving such issues. The definition of cause generally accepted in accident reconstruction work has been given by Baker ( 1975), who first defines a causal factor as" ...  Balke and Pearl (1994) have also shown how technical difficulties arising from the need to describe or store a posterior distribution can be circumvented by applying updating methods to a DAG model that has been augmented to include nodes representing counterfactual outcomes.

STRUCTURAL MODEL
We will consider pedestrian/vehicle collisions consistent with the following scenario, illustrated in Figure  follows that the impact speed is given by The theoretical length of the skidmark is computed as the difference between the total braking distance and the distance traversed during the transition phase, theoretic skidmark = v2 /(2fg)-(vt5-jgr; /2) Garrot and Guenther ( 1982), describing results from a series of controlled braking tests, reported that the standard deviation of measured skidmarks tended to increase as the initial speed increased, and that the average coefficient of variation for the difference between the measured and theoretical skid lengths was approximately equal to 0.11. In the reconstructions described below the measured skidmark, denoted by sf, was assumed to be a lognormal random Specifically, the range for fwas [0.45,1.0), and was taken from Fricke (1990, p. 62-14), where the lower bound corresponds to a dry, travel-polished asphalt pavement and the upper bound to a dry, new concrete pavement. The range for the perception/reaction time, lr, was [0.5 seconds , 2.5 seconds], which brackets the values obtained by Fambro et al. (1998) in surprise braking tests, and the midpoint of which ( 1.5 seconds) equals a commonly chosen nominal value (Stewart-Morris 1995).
For the braking transient time, Neptune et al. (1995) reported values ranging between 0.1 and 0.35 seconds for a well-tuned braking system, while Reed and Keskin (1989) reported values in the range of 0.4-0.5 seconds, so the chosen range was [0.1 seconds, 0.5 seconds]. The strategy for the initial distance and initial speed was to use ranges wide enough so that no reasonable possibility was excluded a priori. The range for v was [5 meters/second, 50 meters/second], and the range for x was [0 meters, 200 meters). Finally, uncertainty in the parameters for the throw distance and injury severity models was incorporated by treating the actual parameter values as normal random variables, with means and standard deviations equal to the estimates presented earlier. In Figure   2 the nodes labeledp1 and p1 represent these parameters.

APPLICATION
To estimate the accident reduction due to speed limit The application of this approach will be illustrated using data collected by Kloeden et al. (1997), where accident reconstruction methods were used to estimate the speeds of accident-involved vehicles as part of a study seeking to relate speed to accident risk. All the investigated accidents occurred on roads with a 60 kmlh speed limit, but a sample of iterations with every I Oth iteration being saved for analysis.
Three separate Gibbs sampling chains were generated from different initial values and random number seeds, convergence was checked using the Gelman and Rubin test, and sample size was checked using the Raftery and Lewis test, as implemented in CODA (Best et al. 1995).

CONCLUSION
Investigation and reconstruction of traffic accidents is often done to support criminal and civil legal proceedings and, to a lesser extent, to support traffic safety research. In all cases though, the ultimate objective is to use the evidence from an accident to identify or exclude possible causal factors, as rationally and objectively as is possible. Unlike more standard problems in causal inference, the underlying causal structure of a reconstruction problem can often be identified,