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

The core idea builds on a framework known as integrative data analysis, or IDA, which has traditionally been used to pool information from multiple studies in order to stabilize estimates and increase sample sizes. Past work on IDA has generally treated missing data as an obstacle to be overcome before datasets can be merged, a nuisance that complicates the harmonization of measures and models. Chen’s study flips that logic on its head. Rather than viewing integration as something that must survive missing data, the new research shows that integration can actively limit the damage that missing data inflicts on the analyses researchers actually care about.

The key concept is what Chen calls auxiliary cases. These are cases drawn from other independent samples or existing studies that are not part of the researcher’s target group of interest, but that carry sufficient data on the target variables to reduce the impact of missingness on the focal analyses. In other words, if a sleep researcher is studying the relationship between daytime sleepiness and cognitive performance in a clinical sample, but many participants are missing scores on some measures, cases from a completely different study that measured the same constructs can be brought in to help. These borrowed cases do not change the research question or contaminate the target population; they simply add statistical information that stabilizes the estimation process.

The mechanism for including auxiliary cases is a multiple-group model, a structural equation modeling technique in which cases from different sources are treated as belonging to different groups within a single analysis. The auxiliary cases form their own group, and the model imposes what are known as cross-group measurement invariance constraints on the measurement portion of the model. This means that the way latent constructs are measured, including factor loadings and related measurement parameters, is forced to be the same across the target group and the auxiliary group, ensuring that the borrowed cases are genuinely measuring the same things. Crucially, however, the focal structural parameters, such as the regression paths and structural relationships that constitute the actual research question, are allowed to be freely estimated for the target group. The auxiliary group thus contributes information about the measurement apparatus without dictating the substantive findings.

To test whether this strategy actually works, Chen conducted a series of simulation studies, the standard methodological tool for evaluating statistical techniques under known conditions. The simulations compared scenarios with complete data against scenarios with missing data, varying conditions that researchers commonly face. The results were striking in one direction: the benefits of including auxiliary cases were substantially larger under missing data conditions than under complete data conditions. When data were complete, adding auxiliary cases offered only modest gains, consistent with earlier findings that integrative data analysis can stabilize estimates. But when data were missing, the borrowed cases delivered meaningful improvements in both statistical power and the efficiency of the estimates, meaning narrower standard errors and a better chance of detecting true effects that would otherwise be drowned in uncertainty.

That asymmetry makes intuitive sense once the underlying statistics are considered. Modern missing data techniques, such as full information maximum likelihood estimation, extract as much information as possible from incomplete observations, but they cannot conjure information that simply is not there. Every missing value represents lost Fisher information, and the efficiency of parameter estimates degrades accordingly. Auxiliary cases inject fresh information about the joint distribution of the target variables, effectively replenishing what the missingness drained away. Under complete data, there is little to replenish, so the auxiliary cases have little to offer. Under missingness, they arrive precisely when the analysis is most starved of information.

Yet the study is equally clear about the limits of the approach, and its warnings are as important as its promises. If the cross-group invariance constraints imposed on the measurement model are improper, that is, if the measures do not actually function equivalently across the target and auxiliary samples, the inclusion of auxiliary cases can backfire, introducing additional bias into the estimates and even reducing statistical power. The problem is compounded when inter-dataset heterogeneities, genuine differences between the datasets in populations, procedures, or contexts, are not accounted for in the analyses. Under these conditions, the borrowed cases are not contributing neutral information about the measurement apparatus; they are smuggling in assumptions that are false, and the model’s estimates absorb the error.

Notably, these risks were found to be especially pronounced when full information maximum likelihood was used as the estimation method. FIML is one of the most widely recommended approaches for handling missing data in structural equation models, prized for its efficiency and its ability to use all available information without deleting cases. But the same property that makes FIML powerful, its reliance on the model’s assumptions to fill in the informational gaps, also makes it sensitive to violations. When invariance constraints are wrong or heterogeneity is ignored, FIML propagates those errors through the entire model, and the auxiliary cases become a liability rather than an asset. The lesson for practitioners is that the auxiliary-case strategy is not a plug-and-play fix; it demands careful verification that the measures are truly comparable across datasets and that between-study differences are explicitly modeled.

To demonstrate the approach on real data rather than simulated numbers, Chen provided an empirical example based on two independent experimental datasets drawn from the National Sleep Research Resource, a publicly accessible repository supported by the National Heart, Lung, and Blood Institute. The two datasets, the Apnea Positive Pressure Long-term Efficacy Study, known as APPLES, and the Best Apnea Interventions in Research study, or BestAIR, both included the Epworth Sleepiness Scale, a widely used measure of daytime sleepiness whose psychometric properties have been extensively validated. Treating one dataset as the target sample and the other as a source of auxiliary cases, the example illustrated how the multiple-group framework operates in practice, complete with the invariance testing steps needed to justify the cross-group constraints. The analysis code for both the simulations and the empirical example has been made available through the Open Science Framework, and the datasets themselves are freely available upon request from the repository, giving other researchers a concrete template to follow.

The broader significance of this work lies in how it repositions two familiar tools on the methodological landscape. Data integration has long been framed as a way to answer bigger questions by combining evidence, and missing data handling has been framed as a way to salvage what a flawed dataset can still tell us. Chen’s study shows that these framings are not merely parallel but intertwined: the act of integrating data can itself be a missing data intervention, and the choice of which outside cases to borrow becomes a design decision with direct consequences for power, bias, and efficiency. As open data repositories grow and secondary analysis becomes ever more central to the research enterprise, the pool of potential auxiliary cases will only deepen. The approach will not suit every study, since it requires overlapping measures, defensible invariance, and honest attention to heterogeneity across sources. But for researchers staring down a dataset riddled with gaps, the message is genuinely encouraging: the remedy for missing data may already exist in someone else’s study, waiting to be borrowed, constrained carefully, and put to work.

Subject of Research: Using auxiliary cases from independent samples in multiple-group models to mitigate the impact of missing data on structural equation analyses

Article Title: Including auxiliary cases to address missing data issues through multiple-group models

Article References: Chen, P.-Y. (2026). Including auxiliary cases to address missing data issues through multiple-group models. Behavior Research Methods, 58(11), Article 304. https://doi.org/10.3758/s13428-026-03180-0

Image Credits: AI Generated

DOI: 10.3758/s13428-026-03180-0

Keywords: missing data, auxiliary cases, multiple-group models, integrative data analysis, measurement invariance, full information maximum likelihood, structural equation modeling, statistical power, data integration, simulation study, Behavior Research Methods, sleep research