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Publication Number: FHWA-RD-98-096
Date: September 1997

Modeling Intersection Crash Counts and Traffic Volume - Final Report




5.1 Distribution of interextions by traffic volumes

5.2 Total crash counts

5.3 Crashes withing and near the intersection

5.4 Crash types withing the intersection

5.5 Intersection characteristics

5.6 The length of the influence zone

5.7 Analytical modeling

5.8 Conclusion regarding the four-leg signalized intersections in California


5.1 Distribution of interextions by traffic volumes

Figures 20 and 21 show the distribution of signalized four-leg urban intersections by volume on the major and minor roads. Figure 21 shows the same distribution in a form that can be more easily compared with the figures showing smoothed crash counts and other variables. The height of the blocks is always the same, but the width of the lines indicates the number of cases in each cell. If there are no cases, no block is shown. A few isolated cases are outside the range of this diagram.




5.2 Total crash counts

Figure 22 shows total crash counts, as given in the California intersection file, moderately smoothed with a window size of (5,000 x 5,000). While the surface is somewhat "wavy," a distinct though not simple pattern is recognizable. This pattern is clearer in Figure 23 where a greater window size of (10,000 x 5,000) was used. This surface is more smooth and the waves essentially disappear.

The most obvious feature of the surface is the "ridge," at minor volumes of about 20,000 vpd, which becomes a "hump" beyond a major volume of about 50,000 vpd. Up to this ridge, crash counts increase with minor volume. Beyond it they decline and at even higher volumes they tend to rise again, but not to the height of the ridge.

Crash counts increase with increasing major volume. However, above about 20,000 vpd the increase is much slower than for lower volumes. Indeed, there is very little change for low values of the minor volume. For major volumes above 60,000 vpd, data become scarce and the pattern becomes complicated. In some areas crash counts are higher and in other areas much lower than for lower volumes.

This complicated surface cannot be represented, or even reasonably approximated, by a simple mathematical function. This contradicts the intuitive expectation that crash counts should vary in a relatively simple way with traffic volumes, and that intersection crash counts should deviate from it only because of their individual characteristics, including safety features. If these characteristics were randomly distributed over the intersections, then the shape of the relationship between crash counts and volumes should be only slightly affected. If, however, the presence of such features was related to the two volumes, then the combined effects of these features and of the pure volume effect could result in an apparent relationship between crashes and volumes that is very different from the pure crash–volumes relationship.

In our case, the complicated surface could be the result of a combination of a volume effect, and the effects of features that become increasingly common with minor volume once a value of 20,000 vpd is exceeded. There might also be intersection features that are more common for major volumes above 60,000 vpd. Another potential explanation is that different crash types may depend in different ways on the two traffic volumes. Adding several simple functions can easily result in a much more complex function.

Before we explore these possibilities, we want to make sure that the overall shape shown in Figure 23 is "real" and not an artifact of the smoothing procedure. It must be emphasized that artifacts can also be produced by analytical models. We separated the intersections into three groups. The first group included all intersections with major volumes less than or equal to 60,000 vpd and minor volumes less than or equal to 20,000 vpd, which represents most intersections. The second group included intersections where the major volume exceeded 60,000 vpd, which has no intersections with minor volume greater than 32,500 vpd. The third group included all intersections where the major volume was less than or equal to 60,000 vpd and the minor volume exceeded 20,000 vpd, which is beyond the ridge. The data for each group were smoothed separately.

Figure 24 shows the smoothed surface for the first and largest group of intersections. Comparing it with the corresponding part of Figure 23 shows that the pattern is qualitatively very similar, though quantitatively different. Note that the vertical scale is different, to accommodate the higher peak. The increase of crash counts toward the ridge is steeper, and the "hump" in Figure 23 is now a much higher "peak." It is not surprising that such features are softened in Figure 23 since this is the normal effect of smoothing. This effect is usually desired, except if there are theoretical or empirical reasons to assume that there are lines where the function is discontinuous, or has discontinuous derivatives.

Figures 25 and 26 show the smoothed surfaces for the other two groups of intersections. Figure 27 shows the smoothed surface for all intersections with the part for the first group "cut out" to make the comparison of the second and third groups to all intersections easier. The overall shape of the surfaces for both groups is very similar to that for all intersections, but there are differences in the details. For instance, the surface for the third group shows a steeper increase for high values of the major volume than does the smoothed surface for all intersections.




5.3 Crashes withing and near the intersection

The crash data file shows whether a crash occurred within the curb lines of the intersection, on a major approach leg, or on a minor approach leg.

Figure 28 shows the smoothed surface for crashes within the intersection. Comparison with Figure 23, representing total crashes, shows that crashes within the intersection proper are only a relatively small proportion of all crashes. The comparison also shows that the relationship between crash counts and volumes appears much weaker. There are more "local" patterns. One is that intersections with both volumes around 40,000 to 50,000 vpd have many more crashes than most other intersections, and that there are a few intersections with very high crash counts at major volumes of about 77,500 vpd and minor volumes of about 25,000 vpd. Furthermore, the hump at a major volume of 60,000 vpd and a minor volume of 20,000 vpd is more pronounced. This suggests that individual intersection characteristics have a strong influence on crash counts.

Figure 29 shows the smoothed surface of crashes on the major approaches. Comparison with Figure 23 shows that these crashes account for a high percentage of all crashes, and that the shape of the surface is very similar to the surface for all crashes.

Figures 30, 31, and 32 also show the same surface as Figure 29, but cut off at major volumes of 20,000, 40,000, and 60,000 vpd, respectively. These cuts show the nature of the ridge clearly. For low major volumes the ridge is near a minor volume of 15,000 vpd and is a soft maximum. For roads with higher major volumes the ridge occurs at minor volumes of about 20,000 vpd and is much sharper.

Figure 33 shows smoothed crash counts on the minor approaches. There is a clear pattern for major volume up to 60,000 vpd and minor volume up to 20,000 vpd, which consists of a nearly plane surface, increasing in a roughly proportional rate with minor volume, and with relatively little or no increase with major volume. The cross – sections shown in Figures 34, 35, and 36 clearly show the initial nearly proportional increase with minor volume. For higher major road volumes, however, the nearly proportional increase stops abruptly and reverses.




5.4 Crash types withing the intersection

Crashes that occurred within the curb lines of an intersection were further classified. Those involving a single vehicle were excluded, because they do not directly result from the intersection of two traffic streams. Crashes involving pedestrians, bicycles, and animals also were excluded. Although some of these crashes result from crossings of the road at an intersection, and thus are typical intersection crashes, their study requires additional exposure measures for pedestrians, bicyclists, and even animals.

We distinguished the following crash types indicated in the crash data file:

  • rear–end collision
  • angle collision
  • right–turn collision
  • left–turn collision (or U–turn)

and combined head–on, sideswipe, and other multiple–vehicle collision into "other" collisions.

Figure 37 shows the smoothed surface for left–turn collisions within the intersection. This surface is very similar to that for all crashes within the intersection, shown in Figure 28, but the dependence on the major volume appears to be slightly weaker.

Figure 38 shows the smoothed surface for right–turn collisions. The pattern also is similar to that of all crashes though the isolated "peaks" at the right side, outstanding in the surfaces for all within–intersection and left–turn crashes, are missing.

The smoothed surface for rear–end collisions, Figure 39, shows an unusually simple pattern, consisting of a roughly linear increase with the major volume up to about 60,000 vpd, and a general tendency to increase with minor volume, albeit with some waves.

Figure 40 shows the smoothed surface for angle collisions. There is no clear pattern, except for the hump similar to the one seen for all within–intersection crashes, and the "spike" nearby, which also appears for all within–intersection crashes and left–turn crashes.

The smoothed surface for "other" collisions, shown in Figure 41, is the only surface that has a ridge, similar to the surface for all intersection crashes, but in a slightly different position.

These surfaces were smoothed with the same windows as those shown in the earlier figures, so that potential similarities could be recognized. However, because of the smaller number of cases, the surfaces are more "wavy." Therefore, in Figures 42, 43, 44, 45, and 46 we also show the surfaces for the various crash types, smoothed with a larger window size of (15,000 x 10,000). The general patterns are more easily recognizable in this set of figures.

As seen in Figure 42, the number of left–turn crashes at low volumes on the minor road is practically constant and independent of the major volume. That the surface shows two collisions for no traffic is a consequence of smoothing, which extrapolates to this point. The same can occur with an analytical model if it does not force the number of crashes to be zero for major and minor volumes equal to zero. For low major volumes, left–turn crashes vary very little with minor volume. For larger values of the major volume they increase only slowly with minor volume, except for high values of the minor volume, where they increase rapidly.

The smoothed surface for right–turn collisions in Figure 43 shows a soft ridge at a minor volume of about 12,500 vpd, and a relatively simple surface, except for intersections with middle values for both the major and minor volumes.

The smoothed surface for rear–end collisions in Figure 44 is nearly a plane up to major volumes of about 60,000 vpd, and minor volumes of about 20,000 vpd. The surface shows that rear–end crashes decline rapidly for larger values of the major volume. For larger values of the minor volume they level off and even decline somewhat if the minor volume becomes even larger.

The smoothed surface for angle collisions in Figure 45 is smooth, but not simple.

The familiar ridge is evident in the smoothed surface for "other" collisions shown in Figure 46. The ridge is located at a minor volume of 20,000 vpd for low values of the major volume, and at a minor volume of 25,000 vpd for high values of the major volume. Up to the ridge, the increase with minor volume is roughly linear.

The patterns that appear in the proportions of the different types of crashes may be missed if only the smoothed surfaces of crash counts are examined. Since the proportions of crash type showed greater variations than crash counts, a larger smoothing window size of (15,000 x 15,000) was used. Figures 47, 48, 49, 50, and 51 show the smoothed surfaces for proportions of different crash types. The smoothed surface for the proportion of left–turn collisions in Figure 47 initially declines with increasing volumes, but changes relatively little when volumes are greater, except when the volumes on the major and minor roads are in the middle range. The surface for the proportion of right–turn collisions increases rapidly with both volumes, but levels off later and shows relatively little change, except at the highest volumes on the major road. The smoothed surface for the proportion of rear–end collisions shows an initial increase with both volumes. For higher volumes on the major road there is relatively little change, and there is a slight decline with increasing volume on the minor road. There is a decrease in the proportion of rear–end collisions at the highest volumes on the major road. Figure 50 shows that the proportion of angle crashes is fairly constant, although there is a slight increase when both the major and minor volumes are in the middle range. The smoothed surface for the proportion of "other" crashes shown in Figure 51 also increases initially with both volumes, but then remains fairly constant, except for a group of intersections with the highest volumes on the major road.

Care must be taken when trying to interpret such figures. For instance, left–turn collisions usually occur with oncoming vehicles on the same road. Therefore, it is expected that their occurrence depends primarily on the volume of the road the vehicles are traveling on and not on the volume of the crossroad. However, the same holds for vehicles on the crossroads. Thus, the total number of left–turn collisions is the sum of the left turns on both roads, each separately determined by its own volume. Similar arguments hold for rear–end collisions. However, the occurrence of angle and right–turn collisions is expected to be determined by an interaction of both volumes.




5.5 Intersection characteristics

Intersection characteristics, especially those reflecting crash countermeasures are expected to have an effect on the occurrence of crashes. Then, intersections with a certain, safety related feature would have fewer crashes than otherwise comparable intersections without this feature or characteristic. Comparing otherwise comparable intersections with and without the feature should allow the effect of the feature on crashes to be estimated.

The common way to overcome this problem is to use all available intersections to develop an analytic model. Such a model would have the crash counts as the dependent variable, the traffic volume or continuous independent variables, and other intersection characteristics, including the presence or absence of certain countermeasures, or categorical independent variables. Because such a model is based on much greater case numbers than those available for "matched" comparisons, the standard error of the coefficient, including those reflecting countermeasures, tend to be lower.

However, even this may not resolve the problem of lacking comparable intersections. For instance, many features reducing the crash risk will also improve traffic flow. Therefore, they are more likely to be installed at high–volume intersections. This means that there is a correlation between the independent variables volumes and presence of the feature. If the correlation is high enough, no statistical technique can separate the two effects. Furthermore, the estimate effect of a feature depends completely on the assumed mathematical form of the relation with the volumes. There is no theoretical basis for the mathematical form of this function. Those mathematical functions used are arbitrary, though often plausible.

Therefore, we explored whether the unexpected complicated relation between crashes and traffic volumes could be explained by the presence of intersection characteristics related to the two traffic volumes.

The California intersection file contains a number of intersection characteristics that may have an effect on crashes. These include:

  • design speed
  • traffic control type
  • lighting type
  • major road left–turn channelization
  • major road right–turn channelization
  • major road traffic regulation
  • major road number of lanes
  • crossroad left–turn channelization
  • crossroad right–turn channelization
  • crossroad traffic regulations
  • crossroad number of lanes
  • median on major road
  • information on shoulders
  • details on the median
  • road bed dimensions


We selected the following characteristics that are most likely to have an effect on crashes:

  • design speed
  • traffic control type: 2–phase signal or multi phase signal
  • major road left–turn channelization
  • major road provision for free right turn
  • minor road left turn channelization
  • minor road provisions for free right turn
  • major road number of lanes
  • minor road number of lanes
  • median on major road

We also considered using the presence of turn restrictions, distinguishing intersections where left turns were not permitted or were restricted during peak hours. However, only 13 intersections had such restrictions on the main road, and nine intersections had such restrictions on the crossroads.

Figure 52 shows the average design speed in relation to the two volumes. The design speeds are mostly between 80 and 100 km/h and no clear pattern is seen. It should be noted that this is the design speed and not the speed limit or actual travel speed. Thus, no conclusions on relations between speed and traffic volume and indirectly on relations between design speed and crashes can be drawn from our data.

Figure 53 shows that the proportion of intersections with multi–phase signals is very high with no simple pattern. The figure clearly shows that there is no consistent increase in multi–phase signals with traffic volumes.

The proportion of intersections with left–turn channelization on the main road is shown in Figure 54. It is very high and shows only minor variations with volume.

The corresponding proportion of intersections with channelization on the minor road is shown in Figure 55. The proportion varies only slightly with volume of the major road, but strongly with the volume of the minor road. This variation is roughly linear up to minor volumes of around 20,000 to 25,000 vpd. Beyond that it levels off and even reverses in one part of the diagram.

This surface bears some similarity to the smoothed surface of crash counts on the minor approaches shown in Figure 33. Both crashes and the presence of left–turn lanes increase with increasing volume. Thus, while left–turn lanes may possibly reduce collisions within the intersection, they may increase crashes on approaches. This is well known to traffic engineers. However, the pattern in Figure 37 gives no indication that left–turn crashes on any approach are inversely related to left–turn channelization on the minor approach.

Figures 56 and 57 show the proportions of intersections with provisions for free right turns on the major and minor roads, respectively. There is some similarity between the surfaces, especially since free right turns become rarer with increasing volume on the minor road. However, there are local deviations from these patterns. One of these is a slight local "hump" near major volumes of 60,000 vpd and minor volumes of 20,000 vpd, which appears in several graphs.

These two surfaces show no similarity to any of the crash surfaces. Figure 58 shows the number of lanes on the major road. As expected, it increases from an average of three lanes for the lowest–volume roads to six lanes for the highest. There is relatively little variation with the volume of the minor road, except for an area where both roads have middle–range volumes.

Figure 59, showing the number of lanes on the minor approach, reveals the complementary pattern with little variation with the volume of the major road and a roughly linear increase with minor road volume. These is no similarity to any crash pattern.

Figure 60, showing the presence of a median on the main road, yields an interesting pattern. The main feature is that the presence of medians increases rapidly with volumes on the main road up to about 30,000 vpd. Beyond 30,000 vpd, medians are nearly always present. However, for high values a "valley"' appears near a minor volume of about 20,000 vpd. This is more recognizable in Figures 61 through 63. That means that at these volumes a median is less common than at other volumes. One might speculate that this could allow relatively more collisions on the main approach, accounting for the "ridge" that appears for crashes on this approach. However, it would not explain the less sharp ridge that appears for crashes on the minor approach except if presence of a median on the major road is related to presence of a median on the minor road.

To determine whether a high number of crashes on the relatively many intersections with undivided main approaches creates the ridge and/or hump, we separated intersections by divided and undivided main approach. We had to use the term "main" approach, rather than the "major," because division was given for only one approach. Thus, these surfaces are not strictly comparable with the other surfaces in this study. Figure 64 shows the smoothed crash counts for the major approaches for divided intersections. Comparing this figure with Figure 29 shows that the overall patterns, with the ridge and to some extent the hump, are similar. Figure 65 shows the smoothed surface of crash counts for undivided highways. There are indeed a few intersections with very high crash counts in the area with main volumes of 50,000 to 55,000 vpd and crossroad volumes of about 25,000 vpd. However, they are already beyond the ridge and do not substantially contribute to it. Consideration of these two figures and Figure 33 indicates that division of the major highway, or lack of it, cannot explain the ridge.

Comparing Figure 64 with Figure 65 shows another interesting feature. In the area of low volumes where there are sufficient numbers of intersections with divided and undivided main approaches, the surface for undivided approaches tends to be below that for divided approaches. This contradicts the intuitive expectation that divided highways should have fewer crashes than undivided highways. Other factors must have a stronger effect than the separation of traffic streams, and the divisions may have been made in response to high crash counts without reducing them to the level of those intersections where the approach remained undivided.




5.6 The length of the influence zone

Crash counts on intersection approaches depend not only on intersection characteristics and traffic – flows, but also on the length of the "influence zones." These zones are defined by convention or judgment and are not a specific function of other intersection characteristics. Therefore, they have to be considered when studying crashes on intersection approaches, as well as intersection crashes that include those within the influence zones.

The California data base always had values for the length of an influence zone for the main approaches. The value for the length of influence zone on the crossroad approaches was usually zero and only had positive values for a few cases. Fifty–three percent of the intersections have influence zones of 75 meters. The next most frequent value is 45 meters for 8 percent of the zones, followed by 60 meters and 30 meters for 4 percent of the zones. The maximum length of the influence zone is 350 feet and the minimum is 7 feet. Nine percent of the influence zones are less than 100 feet. While one would not expect the crash count to be proportional to the length of the influence zone, one would expect it to increase with the length. Therefore, if the length of the influence zone is related to traffic volumes, which is not implausible, a relationship between crash counts and traffic volumes is expected. Indeed, the length of the influence zone determines the "exposure" on the approaches.

A simple relationship between the length of the influence zone and traffic volumes should be recognizable in Figure 66, which shows the smoothed average length of the influence zone on the main approach in relation to the two traffic volumes. There could be more subtle relations, e.g., in some areas of the diagram the lengths might be equal for all intersections, whereas in other areas the average could still be the same, but the individual values vary widely.

The surface shows no overall trend and an appreciable variation. Of special interest is the slight local hump that appears in the area where we have previously found a hump with a ridge. However, the ridge appears for crossroad volumes of 15,000 vpd, whereas at 20,000 vpd there is a valley, where previously there was a ridge.

Because Figure 66 is based on main and cross roads, and the previous figures were based on major and minor roads, they are not strictly comparable. Therefore, Figure 67 shows crashes on the main approach by volumes of main and cross roads. It corresponds to Figure 29. While there are some differences in detail, the overall pattern is the same.

Thus, we conclude that the peculiar surface for crashes on the main (and also the major) approaches cannot be explained by the length of the influence zone as a scale factor.




5.7 Analytical modeling

We also developed a simple analytical model. Its purpose is not to compete with very detailed analytical models, such as those developed by Bauer and Harwood, but to see to what extent simple analytical models can approximate the empirical representation of the data by smoothing techniques.

One important point to keep in mind is that there is no theoretical basis for a specific mathematical form. Commonly used models are based on very simple considerations of plausibility and mathematical convenience. A commonly used model is

z=a*xb yc (5-1)

The current state – of – the – art is to specify (5–1) as the model for the expected value of a Poisson distribution for the crash counts, and then get a maximum–likelihood estimate of the parameters. A negative binomial distribution with an additional parameter could also be used. We used a mathematically simpler process to obtain practically the same result. We assumed (5–1) as the model for the crash counts, but weighted each observation with the inverse of the modeled crash count, which is the variance of a Poisson distributed variable. The estimates were obtained using the SAS procedures NLIN repeatedly, re–weighting each time. Though not statistically rigorous, these estimates are usually close to true maximum likelihood estimates. Considering the only modest overall fit of the resulting model, a more sophisticated approach would not have been justified.

The result was:

z = 2.60*x0.522y0.219 (5-2)
(0.38) (0.042) (0.017)  

with standard error shown in parentheses.

The correlation between b and c is –0.20 and between a and c it is –0.02. These values are negligible. The correlation between a and b is –0.97, which is large, as is usual if the averages of the variables are far from zero. This means that despite their low relative standard errors, a and b can be varied considerably, as long as it is done according to the correlation, without affecting the model fit too much.

Figure 68 shows the surface resulting from the model, and Figure 69 shows the same surface but cut at x = 20,000 vpd and y = 62,500 vpd. Comparison with Figures 23 and 27 shows that the analytical model is only a rough approximation of the smoothed surface. In some areas the trend of the analytical model contravenes the trend of the data as reflected by the smoothed surface.

Figures 70 and 71 show this even more clearly. However, they also show that our overall smoothed surfaces still are not satisfactory representations of the crash patterns. If separate surfaces are constructed for three separate blocks of data, bounded by the apparent ridges (x £ 60,000 , y £ 20,000, x > 60,000, y > 20,000), they do not meet at the boundaries. This means that more sophisticated smoothing techniques have to be used that distinguish "real" ridges from "noise,"' and avoid "smoothing out" of such ridges. Such techniques are not yet available for routine work.




5.8 Conclusion regarding the four-leg signalized intersections in California

We have found that crashes at four–leg signalized urban intersections in California have complicated relationships with the volumes of the two roads, which cannot be expressed by mathematical functions of the form z = a*xb*yc, which is equivalent to a log–linear model. It is still possible that such a model is tenable, if it is expanded to include certain intersection or traffic characteristics. However, the intersection characteristics available in the data file show no patterns that can explain the deviation of the actual crash patterns from the analytical models. Traffic characteristics beyond the volumes of the two roads are not available in the file. We tried to infer some traffic characteristics by separating certain classes of crashes. Again, these classes of crashes showed no patterns likely to provide an explanation for the deviations. Of course, patterns of specific crash classes represent a combination of risk patterns and specific exposure patterns. Thus, it could be possible that very specific exposure measures exist that can explain the deviations.

Another important finding is that crashes on the major and minor approaches showed very different patterns in relation to the volumes, and that crashes within the intersection itself showed yet another different pattern. Different crash types within the intersection also showed different patterns.

While it might be possible to develop useable analytical model for specific crash types with the use of exposure measures specific to the type of crashes, it is very unlikely that a simple analytical model can be found that adequately represents the sum of many different crash types, the proportions of which vary across the intersections.






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