Selection bias in post-encroachment time conflict severity modeling using a full information maximum likelihood framework

1 Introduction

Selection bias is a major and crucial problem in any data-driven study. In traffic safety, for example, studies use reported crashes or potential engagements to analyze the safety of a roadway element. An engagement is a measure on the time scale of the degree of encroachment by a vehicle in to the space of another vehicle or non-motorized mode. The measure of engagement that is often used is post encroachment time, which is observed through high-resolution video analysis in pre-defined space, such as an intersection conflict zone. Studies of post encroachment time rely on observed values (see, for example, Rahmani et al., 2026). Observed data on post encroachment time limit analysis to vehicles that have engaged in at least one interaction with another vehicle or non-motorized mode. Vehicles that are not observed to engage in interactions during the period of study may represent a portion of the driving population that may exert a different behavior under further observation. This may be because the vehicle observed in the high-resolution video in the intersection conflict zone was the only vehicle to pass through the intersection conflict zone within a consistent time window of observation (for example, 5 s or 10 s).

Limiting the use of observed engagements in a study will raise the problem of selection bias because the sample is not randomly selected and is not a representative sample of the full population. This would be particularly misleading because the vehicles observed without post encroachment may show altered engagement patterns under alternate driving conditions (higher volumes, non-random platooned arrivals, etc). The counterfactual question of what the vehicle’s post encroachment time would have been if they had been observed to engage in interactions is the critical question because it would present a full description of the true underlying propensity for engagements in terms of duration and frequency. Similar concerns have been raised in the traffic crash context, where riskier drivers can be over-represented in data that is based reported crashes (Zheng et al., 2021). Ignoring this selectivity in traffic safety will cause a biased and inaccurate estimates or predictions (Eluru and Bhat, 2007; Mannering et al., 2020; Mannering and Hensher, 1987; Oh and Shankar, 2011; Shin, and Shankar, 2013; Ventura et al., 2024; Washington et al., 2020, Zaefarian et al., 2017).

Selection bias correction methods include the well-known Heckman two-step approach (Heckman, 1979; Bendig and Hoke, 2022), where in the probability of an engagement or crash (selection equation), is factored into an observed outcome in the second stage model. The Heckman approach is useful when the second stage regression is a linear model. Its applicability to nonlinear settings is questionable since the Inverse Mills ratio measure critical to the Heckman technique provides meaningful results only in a linear regression setting. The full information maximum likelihood approach, which jointly models the selection and outcome equations, does not rely on the Inverse Mills ratio to transfer selection information to the outcome equation. Rather, it measures the degree of selection directly by including a measure of correlation between the error terms in the selection equation and the outcome equation. Correlation between the two equations arises due to unobservables that are shared between the selection process and the outcome process. For example, unobservables related to the overall driving context (for example, car following behavior in downtown environments, pedestrian crossing behavior) can influence the probability that a vehicle will engage in post encroachments as well as the severity and frequency of engagement. Despite the advantages of the joint full-information maximum likelihood approach over the two-step Heckman technique, the method’s computational complexity especially in larger datasets limits its use in model development.

This study presents a full information maximum likelihood framework to jointly model a probit-based post encroachment engagement selection equation and an ordered probit-based post-encroachment time severity outcome equation. Three levels of post encroachment engagement are used in this study: 1- interaction, 2-minor engagement, 3- critical engagement. The rest of this paper is organized as follows: a) literature review, b) data collection and empirical setting, c) full information maximum likelihood methodology, d) modeling results and marginal effects, and e) conclusions and recommendations.

2 Literature review

Many studies have raised the issue of selection bias in the traffic crash context. Ventura et al. (2024) suggest that in the study of autonomous vehicle safety, attention must be given to potential self selection among drivers choosing to drive autonomous vehicles. In the authors’ view, ignoring self selection may result in biased estimates of the parameters affecting the outcomes of a crash, since drivers who choose to operate autonomous vehicles may have high levels of risk aversion. Mannering et al. (2020) suggest that drivers who are inherently riskier than the population may be over-represented in reported crash data, thereby leading to biased and inconsistent estimates of parameters used in crash frequency and severity estimation. Islam et al. (2023) highlighted the selectivity problem in investigating crash data during the COVID-19 pandemic. They noted that people who chose to engage in social activities carried an underlying risk proneness that also predisposed them to riskier driving behavior as observed in COVID-19 crash outcomes.

(Shin and Shankar, 2013) addressed the issue of selection bias in the estimates in crash severity modeling on highway segments. They mentioned that ignoring segments with zero reported crashes can result in biased estimates of crash severity probabilities. The authors present a comparison of parameters with and without the selection bias effect in the presence of unobserved heterogeneity. Eluru and Bhat (2007) presented a joint model to account for endogenous selection in analyzing seat belt use effects on crash severity which is modeled as an ordered response in the presence of observed and unobserved heterogeneity. The authors note that drivers who choose not to wear seat belts are inherently riskier drivers with an increased likelihood of being involved severe crashes.

Recent studies on post-encroachment time engagement severity and traffic conflict measures at signalized intersections have focused on interactions between vehicle and vulnerable road users. The study by Ali et al. (2023) employed a Bayesian generalized extreme value framework to estimate crash risk in real-time from traffic conflicts measured by post encroachment time. This study used the Delta-V method to estimate the severity component in vehicle–pedestrian interactions. Alozi and Hussein (2022) proposed a modeling framework for the proactive assessment of autonomous vehicle-pedestrian conflicts using the post-encroachment time and time-to-collision. Arun et al. (2021) identified various conflict measures to quantify safety at at-grade intersections. Their study used linear regression models for time-to-collision and post-encroachment time thresholds at signalized intersections and time-to-collision thresholds at unsignalized intersections.

A study by Fu et al. (2020) employed multivariate Bayesian hierarchical extreme value model rear-end crashes from four signalized intersections in the city of Surrey, British Columbia using three covariates and traffic conflict indicators. The Bayesian hierarchical structure combines traffic conflicts from different sites, incorporating several covariates and site-specific unobserved heterogeneity. Another study by Fu and Sayed, 2022 proposed a machine learning technique (Bayesian dynamic extreme value) combined with the extreme value theory approach to dynamically predict the probability of crash occurrence based on traffic conflicts. The study accounts for non-stationarity in conflict extremes through incorporating several covariates and quantitatively differentiates the real-time safety levels of a road facility with two safety indices, namely, risk of crash and return level.

Another recent study by Howlader et al. (2024) jointly modeled the post encroachment time and Delta-V to characterize right-turn and rear-end conflicts, and several covariates were used to address the time-varying heterogeneity of traffic conflict extremes and to estimate the differential crash risks at signal cycles. The results show that for both right-turn and rear-end conflicts, the corresponding conflict counts per signal cycle (post encroachment time ≤ 3.0 s for right-turn conflicts and modified time to collision ≤ 3.0 s for rear-end conflicts) are significantly related to the location parameter of the generalized extreme value distribution. The study by Nazir et al. (2024) used post encroachment time to measure vehicle–pedestrian conflicts aimed at estimating pedestrian crash risks from traffic conflicts. Results show that the sample size influences the prediction of mean crash frequencies, confidence intervals, and relative errors.

Niu et al. (2024) utilized two conflict indicators, modified time-to-collision and post-encroachment time and incorporating three cycle-related covariates. The results indicate that modified time-to-collision is more suitable for the block maxima model, while post encroachment time is better suited for the peak-over-threshold model, resulting in more accurate estimated crash frequencies (Zheng and Sayed, 2019; Zheng et al., 2019). Used time to collision and post encroachment time to analyze the interaction among left-turn vehicles and through vehicles, utilizing bivariate generalized extreme value and bivariate generalized Pareto models. The study by Zheng and Sayed (2020) applied a two-model approach that integrates different conflict indicators in a unified framework and a Bayesian hierarchical structure that combines traffic conflicts of different sites and accounts for heterogeneity in conflict extremes. The model estimation methods were applied to rear-end traffic conflicts collected at the signal cycle level from four intersections in the city of Surrey, British Columbia.

Extreme value theory has been widely used in the second approach because it is mainly designed to study the most extreme observations and extrapolate these rare or extreme engagements to crash outcomes based on the behavior of the extreme observations located in the tail of a distribution. As mentioned earlier, the objective of this study is to jointly model post encroachment time occurrence and severity while accounting for selection bias. This study goes beyond the focus on extreme value applications, as extreme value theory is designed to focus on the impact of outliers. As a result, while extreme value theory can be insightful on behavior in the tail part of the distribution, wholesome behavioral insight can be gained by including other parts of the distribution through careful analysis of selection effects. Behavioral selection is not limited to the tail; it can arise from non-random sampling of any kind, including truncated tail components of a distribution; and non-random sampling due to lack of measured values for trajectories. Therefore, the proposed framework falls within the conflict-frequency and conflict-severity modeling stream identified by Zheng et al. (2021).

Other recent intersection related studies (Ullah et, 2025; Ullah et al., 2026) employed random parameters ordered logit models with heterogeneity in means to analyze injury severities and ordered probit models coupled with bootstrap validation to support evidence-based prioritization of intersection safety improvements. Chaudhari et al. (2021) evaluated pedestrian safety at urban midblock crosswalks using traffic conflict techniques. Peesapati et al. (2013) evaluated the effectiveness of post encroachment time as a proxy for the propensity of crashes between left-turning vehicles and opposing through vehicles at four-leg signalized intersections.

3 Materials

Figure 1 illustrates post encroachment engagement points and intersection lane geometry.

The data used in this study were extracted from 360 h of video recording of vehicle traffic at the signalized intersection of 106th Avenue Northeast and Northeast 4th Street, located in Bellevue, Washington. The extraction process involved obtaining comprehensive information about the vehicle’s trajectory, such as direction, travel lane, turning radius, and measures of trajectory speed and trajectory acceleration such as minimum, average, median and maximum values. Trajectory conflict points were identified within the study intersection using travel lane centerline projections and all possible trajectory engagement paths, which resulted in a total of 39 points. Post-encroachment times less than or equal to 10 s were sampled for analysis. The published literature on trajectory analysis suggests various time scales for post encroachment time measurement, with values ranging from 5 s to 10 s .

The 10 s threshold was selected in this study to capture a broad spectrum of trajectory interactions at the studied intersection. The choice of the 10-s threshold is supported by previous research studies on engagement analysis. For example, Rahman et al. (2026) initially considered engagements using a 5-s threshold for the post encroachment time measure. However, they extended the threshold to 10-s because they discovered that a significant number of engagements showed post encroachment time values exceeding 5 s due to the conservative driving behavior of automated vehicles. Shangguan et al. (2024) detected a total of 101,364 traffic conflicts across 10 signalized intersections using a post encroachment time threshold of 10-s.

The choice of the 10-s threshold was also evaluated on the basis of theoretical and empirical knowledge of vehicle interaction and gap-acceptance behavior at intersections. For instance, the methodology developed by Gettman and Head (2003) to analyze the conflicts using the surrogate safety assessment model shows that valid post encroachment time thresholds can vary based on the characteristics of the intersection and vehicle type. The surrogate safety assessment model allows a maximum post encroachment time threshold of approximately 9.95 s.

From a behavioral standpoint, time to collision reflects a driver’s perception of imminent danger, while post encroachment time reflects drivers perception of potential for dangerous interactions. Drivers presumably accept gaps that are consistent with their intrinsic acceptance of risk based on an internal time-to-collision-based assessment, while the post encroachment times recorded in their trajectories are informative at a strategic level. As a result, it is not uncommon to observe post encroachment time values far higher than time-to-collision values. Recommended time gaps in the American Association of State Highway Transportation Officials Green Book (American Association of State Highway and Transportation Officials, 2018), based on intersection sight-distance design guidance, range from 5.5 to 12 s, depending on many intersection factors, such as traffic control, minor road design vehicle, and the desired maneuver of the minor road vehicle (Souleyrette and Hochstein, 2012). Hedlund (2015) showed in an empirical study on gap acceptance behaviors demonstrating that the probability of 10-s gap acceptance for drivers is close to 75%, whereas the acceptance gap probability at around 5 s is close to 25. These results show that larger post encroachment time values in excess of even 10 s are behaviorally inconsistent with perceived danger of engagements at intersections.

The consideration of autonomous vehicles is critical as well. The in-built safety features in object recognition algorithms provides for more conservative gap acceptance, and as a result extended post encroachment time engagements. The consideration of kinematic and behavioral influences strongly argues for a 10-s observation window for post encroachment time engagements in this study. The 10-s threshold also allows for heterogeneity in the analysis of severity of post encroachment time engagements than the arbitrarily set post encroachment time thresholds in the published literature.

The data set was assembled in a way that allows us to capture the count of engagements each vehicle in its trajectory. Therefore, a vehicle can have multiple post encroachment times along the length of its trajectory. In addition, if a vehicle traveled through the intersection without any engagement in the 10-s range, the trajectory was recorded with a zero count for the number of engagements (and no value for the post encroachment duration). This type of data arrangement forms the basis for selection analysis, since we do not know what the post encroachment time would have been if the vehicle had engaged in encroachments lasting less than 10 s.

Three severity levels were determined for ordered-response analysis: interaction (for engagements with post encroachment time greater than 3 s), minor engagement (greater than 2 s and less than or equal to 3 s), and critical engagement (less than or equal to 2 s). Several preliminary ordered response models were estimated to determine the number of levels that could support stable model estimation. Due to sample size limitations, the afore-mentioned severity levels supporting a viable ordered-response model were retained. This resulted in an estimation sample that contained where the resulting dataset contained 28,000 observations for interactions, 1,399 observations for minor engagements, and 786 observations for critical engagements. The raw underlying post encroachment times were analyzed for excess positive kurtosis, since high positive kurtosis would suggest the presence of extreme points. The sampled post-encroachment times showed a slight negative excess kurtosis. We sub-sampled post encroachment values at thresholds of 5 s, 7 s and 10 s, and the findings on kurtosis did not change much.

The Bellevue Vision Zero conflict-classification framework study (City of Bellevue and Together for Safer Roads, 2020) evaluated post encroachment time with an expanded category of severity. The post encroachment time thresholds used in this study are closely related to that study, which defines post encroachment time values below 2 s as critical engagements and between 2 and 3 s as minor engagements. The main difference between the adopted post encroachment time thresholds in our study and Bellevue Vision Zero conflict-classification framework is that they separated post encroachment time values above 3 s into two categories; potential conflicts (for post encroachment time values between 3 and 5 s) and interactions (for post encroachment time values between 5 and 10 s), whereas in our study we combined all post encroachment time values greater than 3 s into one single category, which is the interaction category, to facilitate severity modeling within the proposed ordered-outcome framework while preserving the commonly used 2-secondsand 3-s post encroachment severity boundaries.

Table 1 presents the descriptive statistics for key variables in the full information maximum likelihood model.

TABLE 1. Descriptive statistics for key variables in the post encroachment engagement selection-severity model.

Total number of observations in the joint selection-outcome dataset: 82,666; Total number of observations in the severity outcome dataset: 30,185.

Table 1 shows the descriptive statistics for variables found to be statistically significant in the selection-ordered probit encroachment severity model. Post encroachment severity is based on post encroachment time measurements that are transformed to indicate severity of encroachment on an increasing scale for ease of interpretation. The maximum raw post encroachment time value that was measured was 10 s, while the minimum value was 1.0 s. Values of post encroachment time ranging up to 10 s were used for the following reasons:

The transformed post encroachment measure is measured as actual post encroachment time subtracted from 10. This transformation illustrates severity on an increasing scale where measurements in the tail represent encroachments of critical engagement. For example, a value of 9 represents a raw post encroachment time of 1.0 s which is the highest observed severity of post encroachment engagement.

The next step in further defining post encroachment severity was to determine ordering that would capture discrete severity levels. The final model specification consisted of three levels, with the lowest severity level (namely, interactions) including values ranging from zero to seven; the minor engagement level including values ranging from seven to eight; and the critical engagement level including values ranging from eight to nine. Several combinations were considered, including four and five levels to conduct an initial test of viability of an ordered probit model. Due to sample size limitations and associated model stability and convergence issues, the three-categories severity model was considered further for selection effects.

Table 1 shows the descriptive statistics for the outcome variable and independent variables found to be statistically significant in the joint selection-severity model. The dataset consisted of 82,666 observations, with 30,185 observations resulting in measured post encroachment times. The remaining 52,481 observations contributed to the selection analysis since no post encroachment times were observed. The three-severity outcome variable included 92.8% interactions (28,000 observations); 4.6% minor engagements (1,399 observations), and 2.6% critical engagements (786 observations).

A critical variable of interest in the joint selection-severity model is speed skew measure. Skewness in speed is dependent on the relationship between the mean and median of speed for a given trajectory. A positive skew indicates that several instantaneous speed measurements in the trajectory are on the higher end of the observed speed scale. When such patterns exist, the ratio of average to median speed will exceed one.

Speed skew is indicative of behavioral deviations from the mean with potential to influence both post encroachment selection and the severity of post encroachment. As a policy variable, the inferred marginal effects of this variable can be valuable if the variable is found to be statistically significant. The ratio of average to median speed in the sample as a whole (82,666 observations) had a mean of 0.99, indicating a mean ratio almost close to one. The ratio appears to indicate non-skewed distribution for the dataset as a whole, while a small proportion of the data (6.36%) exhibits ratios higher than one, indicating some tail behavior in speed choice. The low proportion tail behavior resulting in observation-level right skewed speed distributions appears to suggest that the effect of speed skew (i.e., the ratio of average speed to median speed) is of interest in the joint selection-severity model. Conversely, 5.26% of trajectories have a left-skewed distribution in speed. While a vast majority of trajectories 88.32% do not have skewed speed distributions, tail behavior can potentially offer insights into both post-encroachment engagement and severity of engagement.

The average trajectory length was observed to be approximately 90 feet. Trajectory length is an exposure measure which needs to be controlled for in developing the selection-severity model. It is therefore an essential variable. Turning radius comprised approximately three percent of the dataset. Approach leg median presence comprised 40% of the dataset, which indicates that separation of traffic flow effects will be of interest in the selection-severity model. Lane availability in the dataset appears to indicate lack of symmetry in approach and departure cross sections. Asymmetric cross sections will affect post encroachment time due to changes in driver expectations, and therefore, the difference between the number of lanes in the departure leg and the approach leg is of interest in the selection-severity model.

The width of the departure leg crosswalk is of interest particularly due to the presence of turning movements, with an average value of 9.75 feet. In association with the departure leg crosswalk width variable, approach leg crosswalk length is of interest due to the presence of asymmetry in crosswalk length across approaches. Approach leg crosswalk length was on average approximately 60 feet (indicating roughly a five-lane cross section). Average lane width difference between departure and approach legs is of interest since mean value is approximately one foot, while the maximum is 5 feet. The significant variation in lane width indicates asymmetry due to departure legs being constrained to the typical two-lane cross section.

Approach legs on the other hand are more likely to support dedicated turning movements while total roadway width remains approximately the same. As a result, the asymmetry resulting from added lane capacity on the approach leg versus the added lane width on the departure leg if of interest in the selection-severity model. The availability of departure leg right shoulder is of interest in the selection-severity model since there appears to be asymmetry in departure leg right shoulder width in (average of 0.96 feet and a maximum of 6 feet) the receiving legs across the four intersection legs. Right shoulder availability is likely to influence turning movement speeds and drivers’ expectations in terms of accepted gaps in the opposing traffic stream. The shared lane indicator comprised approximately 46% of the dataset. Share lane use has a composite effect on turning movement speeds and associated post encroachment time, and is therefore an important variable for consideration in the selection-severity model.

Land use variables such as the presence of high rise buildings at the intersection comprise approximately 11 percent of the dataset. The presence of high rise buildings is likely to influence vehicle-pedestrian interactions, right turn movements on the red signal, and unobserved effects arising from perception of available sight distance. The weekday and Saturday indicators comprised 87 percent of the dataset. In addition, the daytime and evening peak indicators comprised 60% of the dataset. Temporal indicators suggest that speed choice and driver behavior relating to gap acceptance can vary during a whole week of observation as this dataset has afforded. In order to control temporal effects, the day of the week and time of day indicators are of interest in the selection-severity model. Research on temporal stability of parameters in crash analysis (see, for example, Venkataraman et al., 2016; Mannering 2018)) appears to suggest time effects can potentially arise in the underlying safety behavior of vehicles at the trajectory and post encroachment levels.

4 Methodology

The presented full information maximum likelihood (FIML) in this study jointly models the probit and ordered probit models (De Luca and Perotti, 2011) with the account for the correlation between the two error terms. The probit model (selection equation) is structured in Equation 1 as follows,

Z n * = β 1 ´ Χ 1 + u n (1)

The ordered probit model can be presented in Equation 2 below as follows, where Z n * is the unobserved (latent) variable indicating the probability of a trajectory n being engaged in an engagement; Χ 1 is a vector of explanatory variables affecting the probability of a trajectory 1 being engaged in an engagement; β 1 is a vector of estimated parameters corresponding to the explanatory variables affecting the selection; u n is the error term which assumed to follow a normal distribution u n ∼ N 0 , 1. The observed severity categories and threshold structure are defined in Equation 3 above, where Y n * is the latent variable represents the underlaying severity, Χ 2 is a vector of explanatory variables affecting engagement severity for trajectory 2, β 2 is a vector of estimated parameters (coefficients) corresponding to the explanatory variables affecting the engagement severity, ε n is the error term, ε n ∼ N 0 , 1, and µ 0 , µ 1 are the cut points (thresholds). Assuming u n and ε n are distributed bivariate normal with mean zero and variance matrix 1 ρ ρ 1, the full information log-likelihood in Equation 4 below is given by:where S is the set of observations for which y j is observed, Φ 2 · is the cumulative bivariate normal distribution function in the observed outcome equation, Φ · is the standard cumulative normal in the selection equation part (when observation is not selected), µ 0 and µ 1 are the upper and lower limits in the ordered probit model, respectively, X 1 and X 2 are covariates from the selection and outcome equations respectively, and ρ is the correlation term. In the maximum likelihood estimation, ρ is not directly estimated. Directly estimated is a ⁡ tanh ⁡ ρ in Equation 5 below:

Y n * = β 2 ´ Χ 2 + ε n (2)

Y n = 0 1 2 i f i f i f Y n * ≤ µ 0 µ 0 < Y n * ≤ µ 1 Y n * > µ 1 (3)

ln ⁡ L = ∑ j ∉ S ln Φ − Z j γ + ∑ h = 1 Η ∑ j ∈ S y j = v h ln Φ 2 Z j γ , µ 0 − x j β , − ρ − Φ 2 Z j γ , µ 1 − x j β , − ρ (4)

a ⁡ tanh ⁡ ρ = 1 2 ln 1 + ρ 1 − ρ (5)

If ρ = 0 , the log likelihood for the ordered probit sample-selection model is equal to the sum of the ordered probit model for the outcome y and the selection model. We can perform a likelihood-ratio test by comparing the log likelihood of the full model with the sum of the log likelihoods for the ordered probit and selection models.

5 Model estimation results

The joint selection-severity model estimated through full information maximum likelihood converged without stability issues, with normal exit from iterations. Table 2 shows the results of the modeling process. Two sets of results are shown–results from the joint selection-severity model, and for bias comparison, results from the severity model without selection. The convergent likelihood for the joint model was determined to −54,056.836, while the ordered probit model without selection converged at −54,061.188. The effect of the correlation parameter for an extra degree of freedom in the joint model resulted in a chi-squared value of 8.704 demonstrating that the joint model resulted in a statistically significant rejection of the null hypothesis that there is no correlation due to unobservables in the model. This is also confirmed by the statistical significance of the correlation parameter whose value was determined to be −0.17369 with a t-statistic of −2.78.

TABLE 2. Full information maximum likelihood post encroachment severity selection–outcome model**.

Ordered probit coefficients and t-statistics for ordered probit without selection shown in parentheses.

Eight variables were found to be significant in the outcome equation (ordered severity component). In addition to speed ratio, high-rise building presence and daytime indicators were the non-geometric effects determined to be statistically significant. Geometric effects included approach leg crosswalk length, lane width difference between departure and approach legs, lane count difference between departure and approach legs, approach leg median presence indicator and departure leg right shoulder width. The significance of all of the variables exceeded the 99 percent confidence level. The threshold parameter was determined to be 0.519 at the 99 percent confidence level.

In the selection equation, 13 variables were found to be statistically significant, including kinematic, land use, geometric, movement type, temporal factors. In addition to speed ratio and trajectory length, high-rise building presence, weekday, Saturday, daytime and evening peak indicators were the non-geometric effects determined to be statistically significant. The day of the week indicator that served as the baseline was the Sunday indicator. Similarly, the time-of-day indicator that served as the baseline was the evening non-peak indicator.

Geometric effects in the selection equation included departure leg crosswalk width, approach leg crosswalk length, lane width difference between departure and approach legs, lane count difference between departure and approach legs, approach leg median presence indicator and turning radius indicator (less than 25-foot radius). Turning movement effects from shared lanes were found to be statistically significant. This indicator shows that shared lane effects whether they are shared left-through or shared through-right exert the same statistical effect. All variables in the selection equation were found to be statistically significant at the 99 percent level.

A comment on the likelihood behavior of the joint model bears mention. The full information maximum likelihood estimator was found to be highly sensitive to the inclusion of trajectory level speed variations such as the ratio of average trajectory speed to median trajectory speed, much more so than direct speed measures such average, median or maximum speed. This was reflected in the observation that the difference in likelihood between the restricted model (correlation parameter constrained to be equal to zero) and the joint selection-severity model was much smaller, indicating significant reduction in unobserved effects that were shared across the two equations.

In addition, the speed ratio variable reflects the effect of scaling in the independent variable. If the original speed measures are used, notwithstanding the fact that the variable is not as statistically informative, scaling the average speed measure by the median speed measure supports stable likelihood behavior and convergence that is not premature. This also helps avoid the need for switching optimization techniques such as the Newton-Raphson and quasi-Newton methods. The lack of need for technique switching also ensures that one can estimate the joint likelihood in a reasonably sized dataset (86,666 observations) such as the one used in this study using a single quasi-Newton method that relies on a rank-two matrix updating scheme. In cases where additional behavioral insight is sought through heterogeneous specifications for the threshold parameter involving threshold specific constants and speed as an additional variable, convergence issues can arise due to scaling problems. The solutions in these cases are not readily available. Popular techniques such as threshold constraints may not resolve the poor convergence issues. The end result of poor convergence is that the joint severity-severity model iterations can terminate in a likelihood that is worse than the restricted ordered model likelihood that assumes zero correlation.

A viable solution to resolve this problem is one that provides for integration of richer behavioral insight through a re-scaled speed parameter directly in the severity equation without additional alterations to the threshold parameter. Further, estimation of the correlation parameter can be problematic if direct estimation of ρ is involved. Since ρ is bounded by [−1,1], transforming the parameter space through the inverse hyperbolic tangent allows for expansion of the parameter space to − ∞ , ∞ which enables smoother likelihood behavior. The relationship between atanh ⁡ ρ and ρ is discussed in the methodology section. This is the approach that was used in this study.

Regarding trajectory level speed variations, average speed and median speed contribute to the skewness of the speed distribution. While the mean of this ratio in the whole sample is close to one, nearly 8.8% at the observation level showed ratios exceeding one standard deviation from the mean, indicating right skewed speed choice behavior was prevalent in the sample. Since the speed measurements are determined at a high resolution level, values exceeding the median speed reflect near-instantaneous speeds within the length of the trajectory as opposed to corridor level speed choice.

5.1 Selection bias and error-term correlation

The second observation on the full information maximum likelihood estimator that bears mention is the increase in standard errors in the outcome equation under selection effect compared to the ordered probit model standard errors without selection. Standard errors on average were 14 percent higher under selection. In association with the increase in standard errors, coefficients in the outcome equation were biased to varying degrees. Coefficient bias in the outcome equation on average was determined to be 5.8%, while the largest biases were identified for the speed ration and daytime variables (35% and 73% respective changes in coefficient values).

The largely biased parameters are worth further discussion. The speed ratio effect measuring skewness in speed appears to be over-estimated in magnitude in the presence of selection. The daytime variable appears to be under-estimated in magnitude in the presence of selection. The signs of both variables remain unchanged under selection; however, the true direction of their effects and magnitudes by severity level will be recovered only through a comprehensive analysis of the marginal effects of all variables. This discussion follows in a separate section.

The significance of the correlation parameter bears mention from a behavioral standpoint. Six out of eight variables in the severity model were also found to be statistically significant in the selection model. In addition to these variables, seven additional variables were found to be statistically significant in the selection model. The seven additional variables captured the temporal and geometric effects in the selection model. This shows that after controlling for all plausible effects due to temporal, land use, geometric, and kinematic factors, unobservables remain in both error terms.

Potential contributors to unobservables in the error terms include driver tendencies in the downtown context unique to the intersection such as interactions with parked vehicles, nonmotorized traffic, gap acceptance behavior and perceived level of turning movement safety. One can hypothesize that these unobservables would influence both the selection and severity occurrence processes from a behavioral standpoint. The statistically significant correlation parameter and the significant chi-squared statistic from the likelihood ratio computation confirms this expectation.

A point of note pertaining to the behavioral assumptions in the construction of the joint model for model identification. The assumption of normality allows for a straightforward construction of the joint likelihood. The important consideration for identification arises along three fronts: a) exclusion restrictions; b) non-linearity of underlying distribution functions; and c) inclusion of continuous variables in the specifications. The extant literature in econometrics had shed significant light on these issues (see, for example, Meng and Schmidt, 1985; Lee, 1985; Keane, 1992; Manski, 1988; De Luca and Perotti, 2011). The exclusion restriction is the commonly applied line of reasoning for model identification. The restriction rule suggests that at least one variable in the selection equation should not be present in the outcome equation. The model estimated in this study far exceeds this restriction, with seven variables that are present in the selection equation but not in the severity outcome equation.

The non-linearity assumption for the underlying distribution assumptions has been debated in the literature, primarily from a weak instrument standpoint, since the assumptions call for functional form restrictions. Further, the third line of reasoning relating to the inclusion of continuous variables in the selection equation in particular deserved particular mention. Continuous variables such as speed ratio (albeit on the positive real line), trajectory length, pseudo-continuous effects such as lane width and lane count differences between the departure and approach legs and departure length crosswalk width provide richer support for the X vector, thereby diluting potential for confounders that can compromise model identification. In the outcome equation, speed ratio amplifies the support for the outcome X vector in addition to pseudo-continuous effects such as lane width and lane count differences between the departure and approach legs, departure leg right shoulder width and approach leg crosswalk length.

It should be noted that the exposure variable, trajectory length was found to be statistically significant in the selection equation but not in the severity equation. Trajectory length appears to lose statistical significance once all potential confounders relating to land use, temporal, geometric and kinematic effects are controlled for in the outcome equation. From a behavioral standpoint, this is an important finding. For example, temporal factors influence how aggressively drivers move through the intersection including maneuvers that can allow for smaller gap acceptance. Geometric effects can exert a similar influence by constraining the turning movement envelope. Land use effects exert behavioral influence through their impact on drivers’ perception of available sight distance.

5.2 Marginal effects

5.2.1 Severity specific marginal effects

A full set of behavioral insights is made possible through a comprehensive marginal effects analysis of the variables found to be statistically significant in the model. Marginal effects are shown in Table 3. Table 3 shows marginal effects with and without selection. We discuss marginal effects under selection in detail below. Marginal effects bias is discussed for the appropriate variables. All marginal effects appear to exert the expected direction in effects, while certain magnitudes of certain effects deserve special attention. An important point to mention here relates to the computation of marginal effects in the joint selection-severity model.

TABLE 3. Full information maximum likelihood post encroachment severity specific direct, indirect, and full marginal effects in selection–outcome model.

Three types of marginal effects are computed for each severity level. The first type is the direct marginal effect which arises from the change in the severity level probability due to a unit change in the X variable in the severity equation. Direct effects are computed for speed ratio, high-rise building presence, daytime indicator, approach leg crosswalk length, lane width difference between departure and approach legs, lane count difference between departure and approach legs, approach leg median presence indicator and departure leg right shoulder width.

The second type of marginal effect arises from the effect of variables in the selection equation which result in indirect effects on the severity outcome probability. In combination, the final type of marginal effects are computed, i.e., full marginal effects.

Full marginal effects are the sum of direct and indirect effects. For variables that appear in both equations, the full marginal effects can be computed, whereas for variables that appear in the outcome equation alone, direct marginal effects can be computed, and for variables that appear in the selection equation alone, indirect marginal effects can be computed. It should be noted that while direct marginal effects will end in a net zero change across the severity spectrum, indirect marginal effects do not follow this constraint, Their effects can be unidirectional.

Marginal effects were also evaluated for statistical significance. The standard errors for the marginal effects were estimated using the delta method. The delta method stipulates that the standard error of a marginal effect can be computed using the Jacobian of the marginal effects vector using Taylor’s series approximation. The resulting variance estimator is given by V ϑ ^ = ∂ ϑ ^ ∂ ϑ ^ ´ N − 1 C ^ ∂ ϑ ^ ∂ ϑ ^ ´ ´ as noted in Cameron and Trivedi (2005), where ϑ ^ is the marginal effects vector, the first and last terms on the right hand side being Jacobians, N the sample size and C being the original variance matrix for the estimated parameters. The bootstrap estimator, although not estimated in this study, is a viable alternative for consistent estimation of the standard error of the marginal effects. Using the delta method, with the exception of the high-rise building indicator’s full marginal effect in the high-severity level (critical engagement), all marginal effects were statistically significant at the 99 percent confidence level.

In the absence of selection, the marginal effects for speed ratio, daytime indicator, lane count difference variable, departure leg right shoulder width and approach leg median presence variables were noticeably biased in the naïve ordered model. While the marginal effects were consistent in sign, bias magnitudes ranged from 23 percent to as high as 61 percent since the effects of selection are not accounted for.

5.2.2 Variables with full marginal effects

Ratio of average speed to median speed: The full marginal effect of a one unit increase in the speed ratio variable is a large decrease in the probability of low severity (interaction engagement) while resulting in much smaller decreases in the probabilities of the minor and critical engagement outcomes. This finding suggests that a higher speed ratio results in a stronger right skew in the speed distribution either due to a decrease in the median speed or an increase in the average speed. From a driving behavior perspective, this suggests that higher severity probabilities decrease at a substantially lower rate compared to the lowest severity level (interactions).

High rise building indicator: This indicator is a dummy variable. The baseline is the absence of a high rise building on the trajectory approach. The addition of a high rise building contributes to a full marginal effect that results in a substantial increase in the probability of interactions, while resulting in much lower increases in the probability of minor and critical engagements. As mentioned in previous sections of this paper, the presence of a high rise building raises the perception of limited sight distance for turning movements as well as the perception that the intersection conflict zone is a contested space due to interactions with pedestrian traffic generated by the high rise land use. Perceptions of contested driving space and limited sight distance likely result in cautious driving behaviors which result in escalations in low severity engagements the most.

Day time indicator: This indicator is a dummy variable. The baseline is the time of day that includes night time driving. The full marginal effect of a change in this variable is that trajectory engagements that occur during the day time are expected to result in engagement probabilities across the spectrum, with the greatest increase at the lowest severity level. This is consistent with the expectation that day time interactions occur more frequently, while night time engagements occur with lesser frequency but potentially higher severity.

Approach leg median presence indicator: This indicator is a dummy variable. The baseline is the absence of a median. The full marginal effect of a change in this variable is that trajectory engagement probabilities are expected to decrease across the spectrum of severities with the strongest decrease in the probability of minor interactions. The inclusion of a median indicates that lane widths will be decreased for a given roadway cross section, which in turn results in lowered speed of engagements. However, the inclusion of a median can result in lane offsetting and more cautious behavior as drivers maneuver to avoid contact with the raised median. A comparison of the full marginal effect magnitudes across the severities indicates that statistically they are not dissimilar based on an evaluation of their 99 percent confidence intervals which overlap substantially.

Lane width difference between departure and approach legs: This indicator is a pseudo-continuous variable, since lane width changes typically occur in 0.5-foot increments. The full marginal effect of this variable is an expected increase in the probability of interactions and decreases in the probability of minor and critical engagements. Lane differences can increase due to an increase in departure leg lane widths or a decrease in approach leg lane widths. Increased departure leg lane widths as expected will result in lower minor and critical engagements, while increases interaction engagement can occur with decreased approach leg lane widths due to perceptions of narrower lanes and resulting caution in driver behavior.

Lane count difference between departure and approach legs: This indicator is a pseudo-continuous variable, since lane count changes typically occur in full lane additions. The lane count difference between departure and approach legs variable appears to have the greatest magnitude effect. A one unit increase in the difference contributes to −0.449 decrease in the probability of interaction engagement while increasing the probabilities of the minor and critical engagement probabilities. Asymmetry in lane count availability emerges as a critical policy variable in the assessment of post encroachment severity, since lane balance appears to result in counter productive effects on post engagement severity.

The lane difference measure is computed as the difference between the number of lanes in the departure leg and the approach leg. The finding on this variable raises two critical insights. The dedicated turning movement lane scenario results in lane count differences. This suggests that while dedicated turning movement lanes are added, separation of turning movements also contributes to post encroachment processes that can potentially escalate engagement severities. The second lane difference scenario is one that arises when departure legs are reduced by a lane due to parking needs. In this instance as well, a one unit change contributes to escalation in post encroachment severities due to vehicle merging maneuvers through the intersection conflict zone.

5.2.3 Variables with direct marginal effects alone

Departure leg right shoulder width: A one unit increase in the departure leg right shoulder width variable decreases the probability of low severity (interaction engagement) while increasing the probabilities of the minor and critical engagement outcomes. This is consistent with expectations. The provision of a traversable shoulder on the departure leg incentives more aggressive and less cautious departure speeds.

Approach leg crosswalk length: A one unit increase in the approach leg crosswalk length variable on the other hand decreases the probability of low severity (interaction engagement) while increasing the probabilities of the minor and critical engagement outcomes. This is also consistent with expectations. The provision of a longer crosswalk occurs when lane widths are expanded or additional lanes are provided, or in the unusual case, the crosswalk is re-oriented to be oblique due to offset cross street alignments. The first two design scenarios incentivize more aggressive driving while disincentivizing driver caution. The third scenario which involves the provision of an oblique and longer crosswalk creates an asymmetry in approach and departure conflict area identifications, thereby resulting escalating minor and critical engagement probabilities.

5.2.4 Variables with indirect marginal effects only

Weekday and Saturday indicators: The baseline is the Sunday indicator. The indirect marginal effect computation suggests that relative to a Sunday trajectories on a weekday or Saturday will experience increased probabilities across the severity spectrum, with interaction probabilities increasing the most. It should be noted here that the weekday marginal effect was found to be statistically different from the Saturday effect. This evaluation was through an examination of the 99 percent confidence intervals which do not overlap. Increased severity probabilities on weekday and Saturday are consistent with expectations. Increased driver activity during these periods are likely to increase temporal exposure across the severity spectrums.

Shared lane indicator: The baseline is the dedicated lane indicator. Relative to the dedicated lane baseline, the provision of a shared lane as expected will result in increased engagement probabilities across the severity spectrum, with probability increases in interactions being the largest. Shared lane trajectories involve disruptions due to turning movements, lane changes due to drivers choosing to move into a dedicated through lane and the lack of protected left turns, depending on the lane of the trajectory.

Sharp turning radius indicator: This is an indicator variable that takes a value of one if the turning radius on the trajectory is less than or equal to 25 feet. The baseline is a trajectory with turning radii that exceed 25 feet. The marginal effect resulting from the provision of a sharper turning radius is an expected decrease in engagement probabilities across the severity spectrum. This is consistent with expectations since turning radii less than 25 feet are sufficiently sharp to disincentivize aggressive turning movements. In addition, the provision of a smaller radius improves the field of vision for both the approaching vehicle and a pedestrian who might enter the conflict zone. The largest expected decrease in engagement probabilities due to a sharper turning radius will occur in interactions.

Evening peak indicator: This is an indicator variable with the baseline being night time indicator. Relative to the night time baseline, the severity engagement probabilities are expected to increase across the spectrum, with probabilities increases in interactions being the largest. The evening peak effect is comprised of driving behavior in the commuting population. As a result, one can expect the effects of congestion to escalate engagement probabilities albeit at lowered speeds.

Departure leg crosswalk width: This is a continuous variable. A one-unit increase in this variable is expected to increase engagement probabilities across the severity spectrum, with the largest increase expected to occur in interactions. The provision of wider departure leg crosswalks increases the conflict zone area and the spatial envelope surrounding the intersection conflict points.

Trajectory length: This is a continuous variable. A one-unit increase in this variable is expected to increase engagement probabilities across the severity spectrum due to increased exposure, with the largest increase expected to occur in interactions.

5.3 Model predictive performance

Table 4 shows the precision, recall, and F1-score values across each individual post encroachment severity level.

TABLE 4. Class-specific performance measures of the joint model.

As shown in Tables 3, 4, the overall accuracy of the full information maximum likelihood model was 91.57%, while the class-specific performance measures showed notable differences across each individual post encroachment severity category. The model shows excellent precision, recall, and F1-score values for the interaction post encroachment time engagement category, but on the other hand, the model exhibited lower precision, recall, and F1-score values for the other two categories. This finding reflects the strong ability of the full information maximum likelihood model to identify the dominant category more than the less frequent categories. This finding is largely attributable to the category imbalance in the used dataset, where the post encroachment time interaction category greatly outnumbers minor and critical post encroachment time engagement categories. It should be noted that the main goal of the presented framework is to examine and study the behavioral relationships in engagement occurrence and severity when applying engagement-based data and using post encroachment tme as the outcome measure, with the account for selection bias rather than to enhance the model classification performance or to use it as a predictive screening tool.

6 Conclusion and recommendations

The first major conclusion from this study is that trajectory analysis that allows for an expanded post encroachment time window (10 s) can yield potential insights into selection effects that may otherwise be ignored in shorter post encroachment time windows. There is certainly support for focused investigation of critical post encroachments from an extreme value theoretical viewpoint; we suggest that complementary analysis that includes selection effects as a parallel focus be conducted as well. The second major conclusion is that ordering of post encroachment in increments of one second intervals up to 3 seconds (with the lowest encroachment severity category being greater than 3 s and less than 10 s) appear to support viable, stable selection models at a high-resolution level for larger datasets. Likelihood stability during full information maximum likelihood estimation is a crucial issue. Scaling variables and re-parameterizing the correlation parameter on the real line appear to support stable and smooth convergence properties, without the need to resort to technique switching during the optimization process.

In terms of behavioral insights, the full information maximum likelihood approach enables the computation of full marginal effects, direct marginal effects and indirect marginal effects on the outcome of interest, which in this study was the severity of post encroachment engagement. The main behavioral variable, speed skew was determined to be significant in both the outcome equation and the selection equation. This finding is consistent with the expectation that speed choice over the length of a vehicle’s trajectory would appear to influence the likelihood of post encroachment engagement as well as the likelihood of post encroachment severity.

It was also determined that encroachment exposure (measured in this study as trajectory length) appears to influence the probability of engagement, but not the probability of encroachment severity. It was determined that when land use, temporal, geometric and speed skew factors that would confound inference on this important behavioral variable were accounted for, the statistical significance of trajectory length was weak. Trajectory length therefore appeared as an indirect effect on post encroachment severity.

Several variables that were not significant in the outcome equation were found to be significant in the selection equation, thereby providing strong support for model identification. In particular, temporal factors exerted statistically significant effects. This finding supports the expectation that temporal variations in driver behavior during the week mitigate the risk of spurious selection by capturing unobservables that emanate due to time of day, and day of week variations in driver behavior.

Policy insights in terms of magnitude of marginal effects were generated for three variables–the speed skew measure used in this study as the ratio of the average speed to median speed, lane width difference and resulting asymmetry between approach and departure legs, and lane count difference between approach and departure legs. All three variables exerted noteworthy full marginal effects on the probability of lowest severity of encroachment compared to other variables in the model. The full marginal effect of the three variables appears to weaken across higher severities, which can be attributed to sample sizes for the minor and critical engagement categories. The totality of the findings on the three variables suggests the following policy implications: a) traffic control measures that mitigate right skewed speed distributions can yield beneficial effects in terms of lowered post encroachment times; b) lane width asymmetry and lane count asymmetry should be investigated further. In this study the lane width difference variable seems to suggest that narrower lane widths in approach legs to accommodate higher turning movement capacity may support lower encroachment severities, whereas lane count differences would exert the opposite effect. The lane count difference variable appears to suggest that lane departure-approach leg through movement lane balance can yield lowered encroachment severity likelihoods.

Recommendations from this study pertain to the following streams of thought: this study focused on a single intersection and therefore, findings from this study capture within-site variations and selection effects. While we recognize that data collection at the resolution used in this study is resource intensive for a multi-site investigation, the expectation is that richer behavioral insights can be gained by extending the 10-s measurement technique to multiple intersections. It is noted that the study intersection was located in a downtown context. Richer insights that aim to capture downtown versus urban, suburban and rural contexts can provide further support for comprehensive design policy, particularly for variables related to traffic operations and control, capacity recapture policies such as elimination of approach leg parking, narrower approach lane widths, as well as lane approach and departure lane balance decisions.

A note on threshold behavior in trajectory analysis bears mention. With the limited perspective from this study, we observed that threshold heterogeneity is highly sensitive to the inclusion to behaviorally meaningful variables such as speed range. In addition, confounding factors that may potentially influence inference such as driving context variables do not appear to statistically stabilize likelihood behavior when included in threshold heterogeneity specifications. Two popular versions of threshold heterogeneity are available for investigation. The first involves specifying thresholds as an exponential function of a fixed constant and level specific parameters. The second technique involves specifying thresholds as an exponential function of a level specific constant and level specific parameters. The nature of the high-resolution variation in trajectory encroachment values appears to create flat likelihoods regardless of the heterogeneous specification that is used. This aspect merits further investigation in follow-up studies. Since the summation of the terms inside the exponent is not guaranteed to be near zero, linear approximations may not work as effectively as demonstrated by Pudney and Shields (2000).

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Footnotes

Edited by: Juneyoung Park, Hanyang University, Republic of Korea

Reviewed by: Irfan Ullah, University of Wyoming, United States

Tengfeng Lin, Korea Advanced Institute of Science and Technology (KAIST), Republic of Korea