The problem is deceptively simple. Studies of post-encroachment time typically rely on observed values extracted from video analysis, which limits the sample to vehicles that engaged in at least one interaction with another vehicle or vulnerable road user. Vehicles that pass through the intersection without any observable engagement are excluded from the analysis. Yet these non-engaging vehicles may represent a fundamentally different portion of the driving population—one whose behavior under higher traffic volumes or platooned arrivals could differ markedly from those that did engage. The counterfactual question—what would the post-encroachment time have been for a vehicle that never engaged?—is the critical one, because it would present a full description of the true underlying propensity for dangerous interactions in terms of both duration and frequency.

This is not a new concern in traffic safety research. Prior studies have raised the alarm about selection bias in crash severity modeling, noting that riskier drivers can be over-represented in reported crash data, leading to biased and inconsistent parameter estimates. Similar selectivity problems have been documented during the COVID-19 pandemic, where people who chose to engage in social activities carried an underlying risk proneness that also predisposed them to riskier driving. But the Texas Tech team, led by Karam Daoud, Abdallah Alrawashdeh, and Narayan Venkataraman, is among the first to tackle the issue in the context of post-encroachment time engagements at signalized intersections using a full-information maximum likelihood framework.

The conventional tool for correcting selection bias is the Heckman two-step approach, where the probability of an engagement or crash is factored into a second-stage model. But the Heckman technique relies on the Inverse Mills ratio, which provides meaningful results only in a linear regression setting. Its applicability to nonlinear settings—such as the ordered probit models needed for severity analysis—is questionable. The full-information maximum likelihood approach, by contrast, jointly models the selection and outcome equations, measuring the degree of selection directly through a correlation parameter between the error terms of the two equations. This correlation arises from unobservables shared between the selection process and the outcome process—factors like car-following behavior in downtown environments or pedestrian crossing tendencies that influence both the probability of engagement and its severity.

The computational complexity of the full-information maximum likelihood approach has historically limited its use, especially in larger datasets. The Texas Tech researchers overcame this by carefully scaling variables and re-parameterizing the correlation parameter on the real line, which supported stable and smooth convergence without the need to switch optimization techniques mid-estimation. They transformed the correlation parameter through the inverse hyperbolic tangent function, expanding the parameter space to enable smoother likelihood behavior. The result was a convergent joint model applied to 82,666 observations extracted from the Bellevue intersection, with 30,185 observations yielding measured post-encroachment times and the remaining 52,481 contributing to the selection analysis.

The severity outcome was modeled as an ordered probit with three levels: interaction (post-encroachment time greater than 3 seconds), minor engagement (greater than 2 seconds and less than or equal to 3 seconds), and critical engagement (less than or equal to 2 seconds). These thresholds align closely with the Bellevue Vision Zero conflict-classification framework, which defines post-encroachment time values below 2 seconds as critical and between 2 and 3 seconds as minor. The dataset contained 28,000 observations for interactions, 1,399 for minor engagements, and 786 for critical engagements—a stark imbalance that reflects the reality that most engagements at signalized intersections are low-severity interactions rather than near-crashes.

The joint model converged at a log-likelihood of −54,056.836, compared to −54,061.188 for the ordered probit model without selection. The likelihood-ratio test produced a chi-squared value of 8.704, demonstrating statistically significant rejection of the null hypothesis that there is no correlation due to unobservables. The correlation parameter itself was estimated at −0.174 with a t-statistic of −2.78, confirming that shared unobservables—hypothesized to include driver tendencies unique to the downtown context such as interactions with parked vehicles, non-motorized traffic, gap acceptance behavior, and perceived turning movement safety—exert a real influence on both engagement occurrence and severity.

When the researchers compared coefficients from the joint model to those from the naïve ordered probit without selection, the bias was unmistakable. Standard errors in the outcome equation were on average 14 percent higher under selection. Coefficient bias averaged 5.8 percent, but the largest distortions appeared for the speed ratio variable—measuring the ratio of average to median trajectory speed as a proxy for speed skewness—and the daytime indicator, with changes of 35 percent and 73 percent respectively. The speed ratio effect was over-estimated in magnitude in the presence of selection, while the daytime variable was under-estimated. Signs remained unchanged, but the true direction and magnitude of effects by severity level could only be recovered through a comprehensive analysis of marginal effects.

The marginal effects analysis revealed three types of behavioral insight: direct effects from variables in the severity equation, indirect effects from variables in the selection equation, and full effects combining both. The speed skew measure exerted noteworthy full marginal effects on the probability of the lowest severity level, as did lane width difference and lane count difference between departure and approach legs. The lane count difference variable—the difference between the number of lanes on the departure leg and the approach leg—had the greatest magnitude effect, contributing a −0.449 decrease in the probability of interaction engagement while increasing the probabilities of minor and critical engagements. This suggests that lane balance between approach and departure legs can yield lowered encroachment severity likelihoods, while dedicated turning movement lanes or parking-induced lane reductions may inadvertently escalate engagement severities through merging maneuvers.

The policy implications are concrete. Traffic control measures that mitigate right-skewed speed distributions can yield beneficial effects in terms of lowered post-encroachment times. Narrower approach lane widths to accommodate higher turning movement capacity may support lower encroachment severities. Approach leg medians, sharper turning radii, and departure leg right shoulders all exerted measurable effects on engagement probabilities across the severity spectrum. The researchers recommend extending the 10-second measurement technique to multiple intersections to capture richer behavioral insights across downtown, urban, suburban, and rural contexts, particularly for variables related to traffic operations and control, capacity recapture policies such as elimination of approach leg parking, and lane balance decisions. The study focused on a single intersection, so findings capture within-site variations and selection effects—but the framework itself offers a replicable template for any traffic safety analysis that relies on observed engagements rather than the full population of vehicles passing through a conflict zone.

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

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

Article References: Daoud, K., Alrawashdeh, A., & Venkataraman, N. (2026). Selection bias in post-encroachment time conflict severity modeling using a full information maximum likelihood framework. Frontiers in Future Transportation, 7, Article 1893114. https://doi.org/10.3389/ffutr.2026.1893114

Image Credits: AI Generated

DOI: 10.3389/ffutr.2026.1893114

Keywords: traffic safety, selection bias, post-encroachment time, full-information maximum likelihood, ordered probit, signalized intersections, trajectory analysis, Vision Zero, Selection, bias, post-encroachment, time