^{1}

^{1}

^{2}

This study develops crash rate prediction models based on the premise that crash frequencies observed from adjacent paired non-weaving and weaving freeway segments are spatially correlated and therefore requires a simultaneous equation modeling approach. Simultaneous equation models for paired freeway non-weaving segments and weaving segments along with combined three freeway segments upstream and downstream were developed to investigate the relationship of crash rate with freeway characteristics. The endogenous variables have significant coefficients which indicate that unobserved variables exist on these contiguous segments, resulting in different crash rates. AADT is a variable that can show the interaction between the traffic and crashes on these contiguous segments. The results corroborate such an interaction. By comparing the simultaneous equation model and the multiple linear regression model, it is shown that more model parameters in the simultaneous models are significant than those from linear regression model. This demonstrates the existence of the correlation between the interchange and between-interchange segments. It is crucial that some variables like segment length can be identified significant in the simultaneous model, which provides a way to quantify the safety impact of freeway development.

Geometric conditions on freeways influence the occurrence of crashes. For instance, ramp spacing could be too short to provide sufficient space for drivers to negotiate with other competing drivers for maneuvers such as lane changes. Shoulders and median could have different impacts on driver behavior. When they are too wide, drivers may be encouraged to drive faster than expected which would cause speeding and the related crash. When they are too narrow, drivers may not have sufficient space to accommodate some sudden and magnificent movements, which would cause crashes.

It also has been observed that crashes generated on adjacent freeway segments exhibit spatial correlation. Either a small number of crashes in one segment corresponds to a big number of crashes in an adjacent segment or vice versus. It might be observed that crashes occurred on one segment cause congestion that migrates to the segments upstream which then cause secondary crashes. The migration to upstream segments depends on the severity of crashes downstream and the length of segment downstream. If the segments downstream are short and the crashes that happen there are severe, it would be expected to see the migration of congestion to upstream. In the other case, the congestion caused by crashes would remain local. This migration is also dependent upon whether or not there are traffic controls at the on-ramp entering the segment downstream. If there is ramp metering, traffic entering the segment containing a crash can be controlled. If traffic information is disseminated faster and taken by travelers, traffic may divert to other routes instead of passing the segment containing the crash.

Currently the models that estimate and forecast the number of crashes consider the conditions at one segment only; not including the nearby segments and their corresponding crashes. The correlation of crash frequency in space will likely invalidate the underlying assumption in current modeling, i.e., observed crash frequency is assumed independent across freeway segments. It can be imagined that the crash frequency forecast using these models will be biased and inefficient. Thus, due to the spatial adjacency nature of crash frequency observed, it is imperative that new models are developed that can consider the correlation between crashes in adjacent freeway segments.

In this study, simultaneous equations models were adopted to integrate the correlation of crashes on contiguous freeway segments. In one approach, two simultaneous equations were considered, one equation for one segment and the other for the upstream segment. In the second approach, three simultaneous equations were developed where the segments downstream and upstream of one freeway segment were considered. Three stage estimation methods are adopted to calibrate the models. The results from these models were interpreted and compared with the models where no simultaneous equation model was adopted, allowing conclusions to be made on the simultaneous equation modeling approach.

It should be noted that some studies have already developed models to address the spatial correlation of crashes on adjacent freeway segments. However, this study is the first where simultaneous equation models are employed to capture this spatial correlation. This approach is more straightforward than previous models because it does not require measures that represent the spatial proximity of the segments.

The following sections are organized as follows. The first section provides a literature review on crash frequency modeling. In the second section, the simultaneous equation modeling approach is introduced. The third section describes the data collection for model development and presents modeling results. In the last section, conclusions are made.

Researchers investigating the occurrence of crashes on freeway facilities usually use statistical models to quantify the relationship between crashes and influencing factors. Influencing factors in these studies mostly include geometric variables, traffic characteristics, as well as human factors.

The study in [

The study in [

The study in [

The study in [

The simultaneous equation model (SEM) is a structural model in which the interrelationship of variables forms a system of equations. For crash rate prediction, a system of equations can be written as follows:

where y is the vector of endogenous crash rates, Y is the vector of fitted endogenous crash rates with α estimated using instrumental variables, X is a matrix of exogenous geometric characteristics and traffic volumes with coefficients, β, and u is the error term assumed to be uncorrelated across observed values. The contemporaneous correlation across equations for the combined observations is used to obtain the covariance matrix of contemporaneous correlation error terms used in SEM estimation techniques. The covariance matrix is given as follows:

where ∑ is the covariance matrix with I, the identity matrix for the T number of observations in each equation.

If least squares estimators are adopted, they do not converge on the probability of being an unbiased estimator as the number of observations increases [

where _{j} = number of equation endogenous variables, K = total number of system exogenous variables, K_{j} = number of equation exogenous variables, M = total number of system endogenous variables = total number of equations,

In this study, three-stage least squares method [_{i}, the crash rate from each segment type (i.e., endogenous variables) is to regress all exogenous variables using OLS. The exogenous variables, X_{i}, are the geometric characteristics and traffic volume. The instrumental variables are used in the opposing equation for residual analysis. The residuals must be contemporaneously correlated. The second stage is calculating the covariance matrix using the residuals from the included instruments. In the third stage, the covariance matrix is used to estimate parameters using generalized least squares for the equation system in Equations (5), (6).

The results from 3SLS, as long as the disturbances are contemporaneously correlated, are consistent estimators which are asymptotically more efficient [

The data set used for modeling was collected from the freeway system in the Las Vegas area in Nevada. It includes number of lanes, shoulder width, median shoulder width, average grade change, curve radius, and segment lengths.

where N = number of crashes, V = AADT, and L = segment length in miles.

All measurements were taken using Google Earth imagery 2010. Freeway segmentations for EX-EN and EN-EX were made following the guidance in AASHTO 2001. The

Variable | Type | Unit | Description |
---|---|---|---|

CRASHRATE (EX-EN) | Endogenous | Million Vehicle Miles Traveled | Crash count per EX-EN segment taken over AADT and length in miles then multiplied by a million |

CRASRATE (EN-EX) | Million Vehicle Miles Traveled | Crash count per EN-EX segment taken over AADT and length in miles then multiplied by a million | |

TLANES | Exogenous | Number of Lanes | Number of through lanes |

SHOULDER | Feet | Measurement taken from outside lane to edge of pavement | |

MEDIAN | Feet | Measurement taken from insider lane to halfway of opposite direction inside lane edge of pavement | |

AADT | Vehicles per Day | Average annual daily traffic | |

GRADE | Percent | Average grade change for segment | |

RADIUS | Feet | Curve radius | |

LENGTH | Feet | Measurement taken from painted gore of ramp terminal to following painted gore | |

AUX | No Unit - Indicator variable | Present of auxiliary lane |

length of a segment was considered from the painted gore of the first ramp terminal to the painted gore of the next ramp and treated as base length according to HCM (2010). Freeway segments with ramp pair combinations of EX-EX and EN-EN as well as any work zone construction observed for 2010 was excluded. To relax accuracy issues when taking measurements, multiple measurements were taken and an average was recorded.

To simplify measurement, each horizontal curve observed was treated as a simple curve. Arc and chord length were recorded in Google Earth. The use of ArcGIS Curve Calculator under the COGO toolbar provided curve radii. When the same freeway segment encountered multiple curves, the shorter radius was recorded due to the stronger effect on driver comfort. This reasoning was considered for segments containing both curve radius and tangent sections. Some freeway segments shared curve radius. In this case, every segment was designated with the same curve radius measurement.

Average annual daily traffic (AADT) data was taken from the Nevada Department of Transportation. When spot volumes were not included in their traffic report, a balance approach was considered so that all freeway segments in the study had AADT values in the data set. The ramp AADT along with traffic volumes on contiguous segments were used in the balance calculations.

Freeway segments were paired initially by EX-EN segment and the following (or downstream) EN-EX (see

Three segment clusters were formed by adding the upstream EN-EX to the paired segments (see

It can be seen from

Variable | EX-EN Upstream | EN-EX Downstream | ||||||
---|---|---|---|---|---|---|---|---|

Mean | Min | Max | Std. Dev | Mean | Min | Max | Std. Dev | |

CHASHRATE | 0.262 | 0 | 2.655 | 0.463 | 0.165 | 0 | 1.569 | 0.268 |

TLANES | 3.23 | 2 | 5 | 0.73 | 3.30 | 2 | 6 | 0.84 |

SHOULDER | 11.98 | 3.54 | 18.20 | 2.40 | 11.88 | 5.30 | 20.56 | 2.46 |

MEDIAN | 15.77 | 3.07 | 45.78 | 10.23 | 15.27 | 4.02 | 45.96 | 10.23 |

AADT | 122,151 | 25,500 | 291,600 | 56,368 | 136,230 | 33,000 | 298,100 | 63,313 |

GRADE | −0.09 | −3.50 | 3.10 | 1.41 | −0.03 | −3.80 | 3.20 | 1.20 |

RADIUS | 1448 | 0 | 11,088 | 2746 | 3244 | 0 | 10,768 | 3230 |

LENGTH | 3175 | 797 | 5930 | 877 | 3877 | 897 | 14,119 | 2163 |

AUX | - | - | - | - | 0.63 | 0 | 1 | 0.49 |

than that of EN-EX segments. These two segments have similar number of lanes, should width, median width, AADT. Their grades and radii are quite different.

The model was estimated adopting three-stage least squares method using R software and the results for the paired segments is indicated in

It is obvious from

Variable | EX-EN | EN-EX Downstream | ||||||
---|---|---|---|---|---|---|---|---|

Coefficient | Standard Error | t-statistic | p-value | Coefficient | Standard Error | t-statistic | p-value | |

Constant | 1.215 | 0.267 | 4.550 | 0.000 | 0.480 | 0.177 | 2.718 | 0.009 |

CRASHRATE (EN-EX) | −0.359 | 0.146 | −2.463 | 0.017 | - | - | - | - |

SHOULDER | −0.066 | 0.015 | −4.325 | 0.000 | −0.018 | 0.010 | −1.790 | 0.079 |

MEDIAN | −0.020 | 0.009 | −2.105 | 0.039 | ||||

AADT | 0.000001 | 0.0000005 | 2.429 | 0.018 | −0.0000006 | 0.0000004 | −1.744 | 0.086 |

RADIUS | −0.00004 | 0.000018 | −2.382 | 0.020 | −0.00002 | 0.00001 | −1.979 | 0.053 |

RMSE | 0.291 | |||||||

McElroy’s R^{2} | 0.605 |

In the paired segment model, shoulder width, median shoulder width, AADT and curve radius were found significant in the estimated EX-EN model equation. Any increase in shoulder width and median width would decrease crash rate, which is reasonable. Widening these areas of the freeway would decrease the chance of vehicle collision when drivers make abnormal driving actions.

In the EN-EX model equation, the shoulder width and curve radius have the same impact on crash rate as on the EX-EN equation. However, the median shoulder width was not found significant at the 95% level.

The SEM model for three sequential freeway segments, up-stream EN-EX, EX-EN and down-stream EN-EX are indicated in

From

Variable | EN-EX Upstream | EX-EN | EN-EX Downstream | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Coefficient | Std. Error | t-statistic | p-value | Coefficient | Std. Error | t-statistic | p-value | Coefficient | Std. Error | t-statistic | p-value | |

Constant | 3.126 | 0.540 | 5.79 | 0.00 | 1.607 | 0.321 | 5.01 | 0.00 | 2.043 | 0.548 | 3.73 | 0.00 |

CRASHRATE (EN-EX) | - | - | - | - | −0.146 | 0.085 | −1.71 | 0.087 | - | - | - | - |

TLANES | ||||||||||||

SHOULDER | −0.128 | 0.033 | 0.03 | 0.00 | −0.062 | 0.015 | −4.19 | 0.00 | −0.049 | 0.028 | −1.73 | 0.084 |

MEDIAN | −0.076 | 0.016 | −4.87 | 0.00 | −0.040 | 0.011 | −3.61 | 0.00 | −0.060 | 0.019 | −3.25 | 0.001 |

AADT | 0.000002 | 0.0000007 | 3.03 | 0.002 | ||||||||

GRADE | 0.127 | 0.0723 | 1.75 | 0.080 | ||||||||

RADIUS | −0.00006 | 0.000019 | −3.20 | 0.001 | ||||||||

LENGTH | −0.00008 | 0.00003 | −2.75 | 0.006 | ||||||||

RMSE | 0.368 | 0.239 | 0.395 | |||||||||

McElro’s R^{2} | 0.545 |

variables are held. This result is reasonable and very important because the distance between interchanges should not be continuously decreasing. In Las Vegas, many interchanges have been built in recent years by adding interchanges between two existing interchanges, shortening the distance between interchanges. This result on segment length provides a way to quantify the impact of interchange space on traffic safety. This result cannot be obtained from other models.

The EX-EN estimated model equation exhibited an additional variable of average percent grade change compared to the previous paired segment model. The positive coefficient indicates that if average grade is increased by 1% then crash rate is increased by 0.127 when holding all other variables including those in the other equations. This observation is very revealing because it is clearly indicated that an interchange should be built on flater ground. Usually freeways are underpass. With high grade on interchange, the driving condition would not be favorable.

Variable | EX-EN Upstream | EN-EX Downstream | ||||||
---|---|---|---|---|---|---|---|---|

MLR | 3SLS | MLR | 3SLS | |||||

Coefficient | Standard Error | Coefficient | Standard Error | Coefficient | Standard Error | Coefficient | Standard Error | |

Constant | 1.095 | 0.256 | 1.228 | 0.288 | 0.469 | 0.176 | 0.484 | 0.173 |

SHOULDER | −0.071 | 0.014 | −0.057 | 0.016 | 0.023 | 0.011 | −0.017 | 0.010 |

MEDIAN | −0.028 | 0.010 | ||||||

AADT | 1.27E−06 | 4.85E−07 | 1.13E−06 | 4.86E−07 | −6.50E−07 | 3.50E−07 | ||

RADIUS | −4,47E−05 | 1.73E−05 | −4.26E−05 | 1.75E−05 | −2.16E−05 | 1.05E−05 | ||

R^{2} | 0.508 | 0.453 | 0.158 | 0.208 | ||||

Adjust R^{2} | 0.454 | 0.380 | 0.061 | 0.087 |

with the use of instrumental variables has provided more information for freeway segments. SEM usage for this study proves the interrelatedness of connecting EX-EN and EN-EX segments.

The freeway segment at an interchange and those between interchanges are different, one with traffic exiting off the ramp, and the other with traffic entering from the ramp. They are correlated when they are contiguous which implies that some traffic would be continuous through both of them. When a crash happens downstream on the between- interchange segment, congestion would be formed and move upstream, which would influence the traffic conditions on the interchange segment. These observations make it imperative to look at the occurrence of crashes on these two contiguous segments simultaneously, which calls for employing simultaneous equation models estimating crash or crash rate on these freeway segments.

In this study, SEM models for paired freeway interchange segment EX-EN and downstream between-interchange segment EN-EX and those with combined three segment upstream EN-EX, EX-EN and downstream EN-EX have been developed. The endogenous variables have significant coefficients which indicate that unobserved variables exist on these contiguous segments, resulting in different crash rate. AADT is a variable that can show the interaction between the traffic and crashes on these segments. The results clearly present such an interaction. By comparing the SEM model and the MLR model, it is shown that more model parameters in the SEM models are significant than those from MLR. This further demonstrates the existence of the correlation between the interchange and between-interchange segments. It is important that some variables like segment length can be identified as significant in the SEM model, providing a way to quantify the safety impact of freeway development.

The Highway Safety Manual in [

The impact of using simultaneous equation model in forecasting crashes should also be investigated. Current crash forecast models do not consider the correlation of the crashes on contiguous freeway segments. This practice would cause crashes to be predicted unreliably. This unreliability should be quantified by comparing the SEM and non-SEM models.

Some factors like segment length are shown to be statistically significant, which actually cannot be identified using other models. The impact of these factors on safety should be further investigated to determine the linearity of this impact. Real cases in Las Vegas, Nevada can be used in quantifying the impact of intensifying freeway network on safety. Other variables such as shoulder width, AADT and crash rate on other contiguous segments can also be further investigated.

Ramos, A., Teng, H.L. and Fu, Y.Y. (2016) Estimating Crash Rate of Freeway Segments Using Simultaneous Equation Model. Journal of Trans- portation Technologies, 6, 327-338. http://dx.doi.org/10.4236/jtts.2016.65029