Byung-Jung Park

Application of finite mixture models for vehicle crash data analysis

Developing sound or reliable statistical models for analyzing motor vehicle crashes is very important in highway safety studies. However, a significant difficulty associated with the model development is related to the fact that crash data often exhibit over-dispersion. Sources of dispersion can be varied and are usually unknown to the transportation analysts. These sources could potentially affect the development of negative binomial (NB) regression models, which are often the model of choice in highway safety. To help in this endeavor, this paper documents an alternative formulation that could be used for capturing heterogeneity in crash count models through the use of finite mixture regression models. The finite mixtures of Poisson or NB regression models are especially useful where count data were drawn from heterogeneous populations. These models can help determine sub-populations or groups in the data among others. To evaluate these models, Poisson and NB mixture models were estimated using data collected in Toronto, Ontario. These models were compared to standard NB regression model estimated using the same data. The results of this study show that the dataset seemed to be generated from two distinct sub-populations, each having different regression coefficients and degrees of over-dispersion. Although over-dispersion in crash data can be dealt with in a variety of ways, the mixture model can help provide the nature of the over-dispersion in the data. It is therefore recommended that transportation safety analysts use this type of model before the traditional NB model, especially when the data are suspected to belong to different groups.

Bias properties of Bayesian statistics in finite mixture of negative binomial regression models in crash data analysis

Factors that cause heterogeneity in crash data are often unknown to researchers and failure to accommodate such heterogeneity in statistical models can undermine the validity of empirical results. A recently proposed finite mixture for the negative binomial regression model has shown a potential advantage in addressing the unobserved heterogeneity as well as providing useful information about features of the population under study. Despite its usefulness, however, no study has been found to examine the performance of this finite mixture under various conditions of sample sizes and sample-mean values that are common in crash data analysis. This study investigated the bias associated with the Bayesian summary statistics (posterior mean and median) of dispersion parameters in the two-component finite mixture of negative binomial regression models. A simulation study was conducted using various sample sizes under different sample-mean values. Two prior specifications (non-informative and weakly-informative) on the dispersion parameter were also compared. The results showed that the posterior mean using the non-informative prior exhibited a high bias for the dispersion parameter and should be avoided when the dataset contains less than 2,000 observations (even for high sample-mean values). The posterior median showed much better bias properties, particularly at small sample sizes and small sample means. However, as the sample size increases, the posterior median using the non-informative prior also began to exhibit an upward-bias trend. In such cases, the posterior mean or median with the weakly-informative prior provided smaller bias. Based on simulation results, guidelines about the choice of priors and the summary statistics to use are presented for different sample sizes and sample-mean values.

Accident Modification Factors for Medians on Freeways and Multilane Rural Highways in Texas

With the growing demand for safer streets and highways, state and national transportation agencies are investigating the relationships between roadway characteristics and crashes. The objective of this study was to develop accident modification factors (AMFs) for median characteristics on urban and rural freeways and on rural multilane highways. Data available for use in the evaluation included 458 mi of with-barrier segments (primarily urban, with some rural sites), 359 mi of urban without-barrier segments, and 436 mi of rural without-barrier segments. A series of negative binomial regression models was used to determine the effects of independent variables on crashes. Variables considered in developing the base models included average daily traffic, left-shoulder width, barrier offset, median (with shoulder) width, and pole density. Crashes were examined in relation to median crashes for 5 years (1997 to 2001). An AMF represents the change in safety when a particular geometric design element changes in size from one value to another. In this project, the AMFs were estimated directly from the coefficients of the models. This approach for AMF development assumes that (a) each AMF is independent because the model parameters are assumed to be independent, and (b) the change in crash frequency is exponential. AMF equations were developed for urban and rural medians with rigid barriers, urban medians without barriers, and rural medians without barriers.

Evaluating the Effects of Freeway Design Elements on Safety

Increased emphasis has been placed on improving the explicit role of highway safety in making decisions on highway planning, design, and operations. This end can be achieved by quantifying the safety effects of geometric design elements for various highway facilities. The objective of this study was to investigate the safety effects of two important design elements for freeways: ramp density and horizontal curve. Data available for use in the evaluation included 324.2 centerline mi of freeways collected in Texas. Five years (1997–2001) of freeway crashes were examined. Negative binomial regression models were used to estimate the effects of independent variables on crashes. The final model for evaluation revealed that crashes on freeway segments were associated with average daily traffic, on-ramp density, degree of curvature, median width with inside shoulder, the number of lanes (for urban freeways), and whether the freeway is in an urban or rural area. Off-ramp density was not statistically significant in the model. Furthermore, the effect of on-ramp density on freeway crashes was significant for horizontal curves but not for tangent sections and indicates that freeway designers should eliminate or minimize the number of on-ramps within the horizontal curves. The statistical modeling results were geared into the development of accident modification factors for on-ramp density and horizontal curves that can be used for safety prediction of freeways.

Horizontal Curve Accident Modification Factor with Consideration of Driveway Density on Rural Four-Lane Highways in Texas

Agencies are seeking a better understanding of those roadway or roadside features that affect safety. The objectives of this study were to develop a horizontal curve accident modification factor (AMF) for rural four-lane divided and undivided highways and to determine if the effect of driveway density is different for horizontal curves as compared to tangent sections. Data available for use in the evaluation included 194.8 centerline km (121 centerline mi) of rural four-lane highways. Negative binomial regression models were used to determine the effects of independent variables on crashes. Variables considered in developing the base models included driveway density, lane width, outside shoulder width, median width (which included inside shoulder width), median type, degree of curve, segment length, and average daily traffic. Five years (1997–2001) of driveway and segment crashes were examined. An AMF for horizontal curves was identified. Reviewing the findings with respect to driveway density revealed that the effect of driveway density is different for horizontal curves and tangents; however, the differences were relatively minor.

Adjustment for Maximum Likelihood Estimate of Negative Binomial Dispersion Parameter

The negative binomial (NB) (or Poisson–gamma) model has been used extensively by highway safety analysts because it can accommodate the overdispersion often exhibited in crash data. However, it has been reported in the literature that the maximum likelihood estimate of the dispersion parameter of NB models can be significantly affected when the data are characterized by small sample size and low sample mean. Given the important roles of the dispersion parameter in various types of highway safety analyses, there is a need to determine whether the bias could be potentially corrected or minimized. The objectives of this study are to explore whether a systematic relationship exists between the estimated and true dispersion parameters, determine the bias as a function of the sample size and sample mean, and develop a procedure for correcting the bias caused by these two conditions. For this purpose, simulated data were used to derive the relationship under the various combinations of sample mean, dispersion parameter, and sample size, which encompass all simulation conditions performed in previous research. The dispersion parameter was estimated by using the maximum likelihood method. The results confirmed previous studies and developed a reasonable relationship between the estimated and true dispersion parameters for reducing the bias. Details for the application of the correction procedure were also provided by using the crash data collected at 458 three-leg unsignalized intersections in California. Finally, the study provided several discussion points for further work.

Finite mixture modeling for vehicle crash data with application to hotspot identification

The application of finite mixture regression models has recently gained an interest from highway safety researchers because of its considerable potential for addressing unobserved heterogeneity. Finite mixture models assume that the observations of a sample arise from two or more unobserved components with unknown proportions. Both fixed and varying weight parameter models have been shown to be useful for explaining the heterogeneity and the nature of the dispersion in crash data. Given the superior performance of the finite mixture model, this study, using observed and simulated data, investigated the relative performance of the finite mixture model and the traditional negative binomial (NB) model in terms of hotspot identification. For the observed data, rural multilane segment crash data for divided highways in California and Texas were used. The results showed that the difference measured by the percentage deviation in ranking orders was relatively small for this dataset. Nevertheless, the ranking results from the finite mixture model were considered more reliable than the NB model because of the better model specification. This finding was also supported by the simulation study which produced a high number of false positives and negatives when a mis-specified model was used for hotspot identification. Regarding an optimal threshold value for identifying hotspots, another simulation analysis indicated that there is a discrepancy between false discovery (increasing) and false negative rates (decreasing). Since the costs associated with false positives and false negatives are different, it is suggested that the selected optimal threshold value should be decided by considering the trade-offs between these two costs so that unnecessary expenses are minimized.

Finite mixture modeling approach for developing crash modification factors in highway safety analysis

This study aimed to investigate the relative performance of two models (negative binomial (NB) model and two-component finite mixture of negative binomial models (FMNB-2)) in terms of developing crash modification factors (CMFs). Crash data on rural multilane divided highways in California and Texas were modeled with the two models, and crash modification functions (CMFunctions) were derived. The resultant CMFunction estimated from the FMNB-2 model showed several good properties over that from the NB model. First, the safety effect of a covariate was better reflected by the CMFunction developed using the FMNB-2 model, since the model takes into account the differential responsiveness of crash frequency to the covariate. Second, the CMFunction derived from the FMNB-2 model is able to capture nonlinear relationships between covariate and safety. Finally, following the same concept as those for NB models, the combined CMFs of multiple treatments were estimated using the FMNB-2 model. The results indicated that they are not the simple multiplicative of single ones (i.e., their safety effects are not independent under FMNB-2 models). Adjustment Factors (AFs) were then developed. It is revealed that current Highway Safety Manuals method could over- or under-estimate the combined CMFs under particular combination of covariates. Safety analysts are encouraged to consider using the FMNB-2 models for developing CMFs and AFs.

Safety Effectiveness of Super 2 Highways in Texas

The objective of this study was to evaluate the safety effectiveness of Super 2 highways in Texas. A before–after study was performed with the empirical Bayes (EB) method, which was superior to other methods because it could address the regression-to-the-mean bias. On the basis of potential study sites identified in seven districts (Paris, Childress, Corpus Christi, Austin, Wichita Falls, Yoakum, and Bryan) in Texas, four reference groups were considered by imposition of different restrictions. Negative binomial regression models were then used to develop safety performance functions for each reference group. From the model selection process, the most restricted reference group was selected for the final analysis. For roadway inventory and crash history data, 12 years (1997 to 2001 and 2003 to 2009) of data for Texas were examined. The analysis used fatal (K), incapacitating injury (A), nonincapacitating injury (B), and minor injury (C) crashes. Property-damage-only crashes were not included. The EB analyses were carried out on five corridors with about 53 centerline miles. The results showed that the installation of Super 2 highways led to statistically significant reductions in the incidence of crashes of 35% for crashes on segments only (KABC) and 42% for crashes on segments and at intersections (KABC) on the study corridors. These findings were consistent with those of previous studies of the safety of Super 2 corridors that showed improvements in safety with installation of passing lanes, even when traffic volumes were higher than those considered under previous guidance in Texas.

Super 2 highways in Texas: operational and safety characteristics

As traffic volumes increase in both urban and rural areas, so do demands on the highway network. Specifically, as rural traffic volumes rise in Texas, the pressure on the states network of two-lane highways rises accordingly. Previous research in Texas demonstrated that periodic passing lanes can improve operations on two-lane highways with average daily traffic lower than 5,000 vehicles. These highways, called Super 2 highways, can provide many of the benefits of a four-lane alignment at a lower cost. A recent project expanded on that research to develop design guidelines for passing lanes on two-lane highways with higher volumes. Researchers investigated the effects of volume, terrain, and heavy vehicles on traffic flow and safety. This paper discusses findings from field observations and crash analysis of existing Super 2 highway corridors in Texas and computer modeling of traffic conditions on a simulated Super 2 corridor. Results indicate that passing lanes provide added benefit at higher traffic volumes by reducing crashes, delay, and percent time spent following. Empirical Bayes analysis of crash data reveals a 35% reduction in expected nonintersection crashes with injuries. Simulation results indicate that most passing activity takes place within the first mile of the passing lane, so additional passing lanes can offer greater benefit than longer passing lanes. Whether new passing lanes are added or existing lanes are lengthened, the incremental benefit diminishes as additional length is provided and the highway more closely resembles a four-lane alignment.