Exposure and Traffic Volume
How the literature handles traffic exposure, and what Open Road Risk can and cannot do
Exposure normalisation is the central design decision in crash-frequency modelling. A model that counts collisions without accounting for traffic volume cannot distinguish a genuinely dangerous road from a heavily used one. What follows synthesises the exposure treatment from reviewed papers and official-method sources, and maps their approaches against Open Road Risk’s data stack.
What exposure normalisation means
The canonical form for a road segment crash model places traffic exposure as a log-offset in the Poisson log-linear predictor:
\[\log(\mu_i) = \log(E_i) + \beta_0 + \sum_k \beta_k X_{ik}\]
where \(E_i = \text{AADT}_i \times L_i \times T\) is the vehicle-kilometres (or vehicle-days) of travel on segment \(i\) over period \(T\). The offset is not a coefficient to be estimated — it is a constraint that forces the model to predict a crash rate rather than a raw count. Without it, a model cannot separate roads that have more crashes because they carry more traffic from roads that have more crashes because they are inherently more dangerous.
Open Road Risk’s Stage 2 uses log(AADT × link_length_km × 365 / 1e6) as its exposure offset, which is structurally equivalent to the VMT-style offset used in Hauer et al. (2001) and directly analogous to the log(length × traffic_flow) offset used by Gilardi et al. (2022) on OS road segments in Leeds. Both papers support this exposure-offset structure for a UK segment-level crash model.
National Highways (2022) gives the same structure in official UK rate-comparison guidance: collision counts are treated as Poisson with vehicle miles as the exposure scale, \(N_i \sim \text{Poisson}(\gamma_i v_i)\). This is mathematically equivalent to a Poisson GLM with log(vehicle miles) as an offset. The guidance assumes vehicle miles are known, so it supports the offset form but does not solve Open Road Risk’s separate problem of estimated AADT uncertainty.
The AADT elasticity question
The log-offset approach implicitly assumes that crash frequency scales proportionally with traffic volume — that doubling AADT doubles expected crashes, all else equal (elasticity = 1.0). This assumption is testable and is not universally supported.
Aguero-Valverde and Jovanis (2008) fit Bayesian hierarchical models to rural Pennsylvania road segments with AADT as a free covariate (not constrained within an offset). Their estimated AADT coefficient ranges from 0.628 to 0.714 depending on model specification — well below 1.0. The paper interprets the drop from 0.714 (heterogeneity-only model) to 0.664 (spatial model) as evidence that ignoring spatial correlation biases the AADT coefficient upward via model misspecification.
Wang et al. (2009) obtain AADT elasticities of 1.2–1.9 on the M25 motorway, above the proportional assumption. They note explicitly that their elasticity is “a little high compared with some of the previous studies which reported that the elasticity ranges from 0.6–0.7” (p. 10). Pan et al. (2017) report near-unity NB coefficients on log(AADT × length) for most North American highway types, which is consistent with the offset assumption.
The evidence points in the same direction across these papers: AADT elasticity is road-type dependent and differs from the 1.0 constraint imposed by the combined offset. Motorways may have super-proportional elasticity; rural two-lane roads sub-proportional. Al-Omari (2021) finds sub-linear AADT coefficients (0.39–0.63) for dense urban road classes in Florida — well below the fixed-offset assumption.
The fixed-offset assumption (AADT and length elasticity = 1.0) is supported on average but has not been tested diagnostically in Open Road Risk. A straightforward diagnostic is to fit the Stage 2 GLM with log(AADT) and log(length) as separate free covariates and compare the estimated elasticities against 1.0. If they are materially below 1.0 for some road classes, the offset may be overweighting exposure for high-AADT links and distorting the risk percentile ranking for those classes.
Open Road Risk’s exposure data stack
The single largest gap between Open Road Risk and most papers in the literature is that Open Road Risk does not observe AADT — it estimates it.
The processed AADF table contains approximately 14,200 count points across 2015–2024, including DfT-estimated rows. Stage 1a trains on the cleaner subset of approximately 12,900 count points with at least one directly Counted observation. That direct-count training signal is equivalent to only about 0.6% as many measured locations as the 2.17 million OS Open Roads links being scored, so the remaining ~99.4% of links receive an AADT estimate from the Stage 1a machine-learning model. This introduces estimation uncertainty that most comparison papers never face: they work with either complete sensor networks (Wang 2009 on the M25, using UKHA hourly counts for all 70 segments), or nationally complete probe-based data (Roll et al. 2026, which uses INRIX commercial speed data for a second estimation tier in Oregon).
The table below summarises how the reviewed papers obtain their traffic exposure.
| Paper | Traffic data source | Coverage | Missing data handling |
|---|---|---|---|
| Gilardi et al. 2022 | 2011 Census commuting OD flows, routed via shortest path | All major-road segments (derived) | No gaps — Census routing covers everything; but proxy is weak |
| Aguero-Valverde & Jovanis 2008 | Pennsylvania RMS (state road management) | All 865 rural two-lane segments | Not discussed; full coverage assumed |
| Wang et al. 2009 | UKHA hourly counts | All 70 M25 segments (near-complete) | Two segments excluded due to missing data |
| Hauer et al. 2001 | AADT assumed observed per year | Illustrative tutorial examples | Not addressed; full coverage assumed |
| Jayasinghe et al. 2019 | JICA survey counts (five cities) | Sparse sample, ~40–1500 count sites | Core problem — motivates centrality-based estimation |
| Roll et al. 2026 | HPMS observed → INRIX probe → random forest data fusion | Statewide urban intersections | Three-tier hierarchy fills gaps at each level |
| Pew et al. 2020 | UDOT entering vehicles per day | All 1,738 Utah intersections | Not discussed; full coverage assumed |
| Gao et al. 2024 | None | Not used | N/A — no traffic exposure in model |
| Open Road Risk | DfT AADF (~12,900 directly counted count points; ~0.6% of link count) → Stage 1a ML estimate | All ~2.17M links (estimated) | Stage 1a model fills the gap; uncertainty not propagated into Stage 2 |
The contrast with Gao et al. (2024) is a useful negative example. That paper uses OS-style road links in London with STATS19 crash data — superficially close to Open Road Risk — but includes no traffic exposure whatsoever. Its model cannot distinguish a high-traffic road from a high-risk road. The severity-weighted crash score it predicts is a raw exposure-unadjusted count, not a rate. This is the failure mode that Open Road Risk’s offset design is explicitly intended to avoid.
Estimated AADT as exposure: what the literature says
Jayasinghe et al. (2019) provide the closest direct reference for estimating AADT from network centrality in a sparse-count setting. Their betweenness centrality and closeness centrality model achieves random-holdout R² of 0.92–0.96 across five developing-country cities, but with critical weaknesses: very poor RMSE for low-AADT links (193% in Colombo, 412% in Phnom Penh for segments with AADT below 1000), and no spatial holdout — nearby segments share centrality structure and may inflate validation metrics.
The low-AADT failure is directly relevant to Open Road Risk, which has a long tail of minor rural and unclassified roads where AADF count coverage is near zero and Stage 1a predictions are least reliable. Huda and Al-Kaisy (2024) provide a partial mitigation: they show that for low-volume roads (≤1000 vpd), removing AADT from the model reduces R² by only 0.009. Geometry features — curvature and grade — dominate at that end of the traffic-volume distribution. This does not remove the uncertainty problem but suggests that exposure estimation error on low-AADT links may matter less to the final risk ranking than it would for high-volume roads.
Roll et al. (2026) demonstrate the three-tier data-fusion hierarchy that is conceptually the right architecture for exposure uncertainty: use observed counts first, fill with probe-based AADT second, fill with ML estimation third. Open Road Risk’s Stage 1a implements the third tier (ML estimation from network features) but has limited access to a second tier — WebTRIS provides time profiles for National Highways routes rather than AADT counts for individual links, and commercial probe data (INRIX, as used by Roll) is outside the open-data stack.
An important finding from Roll et al. (2026) on model selection: despite testing negative binomial, Poisson, XGBoost, and neural network approaches for pedestrian volume estimation, random forest was selected not because it had the best cross-validated error metric but because it was the only model that passed an application sanity check — XGBoost produced negative volume predictions, and the negative binomial produced implausibly extreme maxima in the statewide prediction. Open Road Risk should apply equivalent sanity checks to Stage 1a full-network predictions (distribution of predicted AADT by road class and rural/urban classification) rather than relying on CV metrics alone.
Exposure uncertainty is a first-class limitation of Open Road Risk that is not present in most comparison papers. The current Stage 2 model treats estimated AADT as observed, with no propagation of Stage 1a prediction uncertainty into the Stage 2 crash rate estimate or the risk percentile. This means that the risk percentile for a low-volume rural link combines uncertain exposure with sparse collision counts — two sources of instability that are compounded rather than separated. The Huda and Al-Kaisy (2024) finding (geometry dominates at low AADT) is partial mitigation, not a solution.
The temporal exposure problem: why AADT is a biased argument
Using AADT as the SPF exposure argument introduces a structural approximation error that is distinct from AADT estimation uncertainty. This is the argument-averaging bias first formalised by Mensah and Hauer (1998).
Within a year, traffic flow varies continuously: peak-hour motorway flows may be ten times the overnight flow. Crashes, however, are not uniformly distributed across the flow distribution — they occur at specific instantaneous conditions. A model fitted using the annual average (AADT) as its exposure argument is fitted to a smoothed version of the exposure that may systematically misrepresent the underlying flow-accident relationship.
Mensah and Hauer (1998) derive a correction factor for the exponential SPF \(\mu = \alpha q^\beta\):
\[w \approx 1 + \tfrac{1}{2}(\beta^2 - \beta) \cdot \text{CV}^2(q)\]
where \(\text{CV}(q) = \sigma(q)/\mathbb{E}(q)\) is the coefficient of variation of hourly traffic flow within a year. For the commonly observed range \(\beta \in (0.5, 0.9)\) and typical rural road \(\text{CV}(q) \approx 0.79\), the correction factor \(w\) is approximately 0.92–0.95 — meaning that fitting an SPF to AADT-based data underestimates the true SPF by 5–8% for a typical exponential specification. The bias is moderate, not catastrophic, for standard road types, but it is non-zero and direction-dependent: for \(\beta < 1\) (sub-linear elasticity), \(w < 1\) and the SPF is underestimated; for \(\beta > 1\) (super-linear), \(w > 1\).
Mensah and Hauer (1998) also identify a second problem — function averaging — that is more directly relevant to Open Road Risk’s Stage 1b work. Fitting a single annual SPF to combined daytime and nighttime data produces a composite function that does not represent a true cause-effect relationship. For any given average flow, there are more crashes at night than during the day. The composite SPF’s predicted crash count depends not only on the average flow but also on the ratio of daytime to nighttime flow, which varies by road section. A link with predominantly overnight traffic will have a different composite crash-to-flow relationship than a link with the same AADT but primarily daytime flow. This is not captured by a single annual exposure offset.
The argument-averaging correction factor \(w\) requires two inputs: a free AADT elasticity estimate \(\beta\), and an approximate \(\text{CV}(q)\) per road type derivable from Stage 1b WebTRIS time-zone fractions. The current Stage 2 GLM constrains AADT elasticity to 1.0 via the offset; \(\beta\) would need to be estimated from a separate free-elasticity diagnostic model (fitting log(AADT) and log(length) as free covariates rather than as a combined offset).
Qin et al. (2006) provide empirical evidence that the flow-crash relationship is non-linear (\(\alpha_v \neq 1\)) for all four crash types and both states tested (Michigan and Connecticut rural two-lane highways). Two findings are particularly relevant:
Opposing crash-type relationships: The flow exponent \(\alpha_v\) for single-vehicle crashes is frequently negative (crash probability decreases as hourly volume increases, consistent with congestion reducing speeds), while multi-vehicle crash types show positive \(\alpha_v\). Open Road Risk’s Stage 2 models total injury collisions — combining single-vehicle and multi-vehicle crash types in one model. The opposing flow-crash relationships for these two types partially cancel in the combined outcome, potentially producing a GLM coefficient on log(AADT) that underestimates the true relationship for both types individually. This is the empirical counterpart to Mensah and Hauer’s function-averaging-over-accident-types concern.
Time-of-day stratification changes the relationship: For at least one crash type in both states, the flow exponent \(\alpha_v\) varies significantly by time-of-day period (7am–3pm, 3pm–11pm, 11pm–7am). Fitting a single annual model collapses these into a composite function. Stage 1b time-zone profiles are the available mechanism in Open Road Risk for conditioning the model on temporal flow patterns, but they are not currently used in Stage 2.
Dutta and Fontaine (2020) provide empirical validation of these theoretical concerns at scale. Testing AADT-based, average-hourly, average 15-minute, and raw-hourly SPF models on 110 rural and 80 urban Virginia freeway segments (2011–2017), they find that average-hourly volume models outperform AADT-based models on all out-of-sample validation measures:
| Comparison | MAD improvement | MAPE improvement | MSPE improvement |
|---|---|---|---|
| Urban total crashes: average-hourly vs AADT | 20% | 22% | 38% |
| Rural total crashes: average-hourly vs AADT | 11% | 33% | 29% |
Raw hourly data — the highest temporal resolution — performed worse than AADT because 23% of raw hourly observations failed quality checks, introducing noise. Average-hourly data (using available observations smoothed to a typical hourly profile) gave the best results. This is directly relevant to Open Road Risk’s WebTRIS data situation: Stage 1b already builds smoothed time-zone profiles, which is the correct approach for sparse and imperfect data.
Sung et al. (2024) confirm the same pattern on a larger scale — 1,095 Korean national highway segments with complete VDS sensor coverage — finding that modified temporal SPFs outperform AADT-based models consistently. However, their validation uses a random 8:2 split without spatial holdout, and the training data is balanced to a 1:1 crash/non-crash ratio, which inflates apparent model performance. The directional finding (temporal disaggregation helps) is credible; the specific magnitude of improvement is not directly transferable.
The Dutta and Fontaine (2020) improvements (20–38% in MSPE) are from Virginia freeways with direct sensor coverage on every segment. Open Road Risk must estimate time-zone fractions via Stage 1b for most links rather than observing them. The improvements are therefore an upper bound on what temporal conditioning could achieve in Open Road Risk — the actual gain will be smaller, particularly for minor roads where Stage 1b coverage is thinnest. The experiment with core_overnight_ratio as a Stage 2 feature (+0.004 R²) is more directly representative of the achievable gain in the current pipeline.
The Empirical Bayes connection
Hauer et al. (2001) provide the canonical mathematical connection between the exposure model (SPF) and the EB shrinkage step. The EB weight formula requires:
\[w = \frac{1}{1 + \eta/\phi}\]
where \(\eta = \mu \times L \times Y\) is the SPF-predicted expected count over the observation period (incorporating AADT, length, and years) and \(\phi\) is the overdispersion parameter per unit length estimated from negative binomial regression. When AADT is estimated rather than observed, \(\eta\) carries estimation uncertainty — but the EB procedure at least ensures that for links with very uncertain AADT (typically low-volume rural roads), the EB estimate shrinks heavily toward the SPF mean rather than being dominated by the raw observed count. For a link with near-zero expected crashes (\(\eta\) small), \(w \to 1\) and the EB estimate is almost entirely determined by the prior.
This means that exposure uncertainty on minor roads, while real, is partially absorbed by the EB shrinkage mechanism: a link whose AADT is poorly estimated will have a poorly estimated \(\eta\), but the EB estimator will respond by weighting the SPF prior heavily — which is the correct conservative response in the absence of reliable evidence.
The full EB procedure (Hauer et al. 2001, equation 7) accommodates year-specific AADT changes by summing \(\mu_\text{year}\) across years. Open Road Risk’s Stage 1a produces a year-specific AADT estimate per link per year, which directly supports this extension.
Stage 1a validation: what the literature suggests
Two papers speak directly to how Stage 1a AADT estimation should be validated and what its failure modes are.
Jayasinghe et al. (2019) show that a learning-curve diagnostic — measuring validation error as a function of the number of calibration count points — is a simple, useful way to understand how performance degrades with sparse coverage. Around 40 calibration observations produced RMSE below 30% in their five case cities. Open Road Risk has approximately 12,900 directly counted DfT AADF count points in the current Stage 1a training signal, which is dense by comparison, but the points are highly skewed toward major roads. The learning curve for minor road AADT estimation would look quite different if assessed separately.
Roll et al. (2026) show that CV metrics alone are insufficient: application sanity checks (checking the full-network prediction distribution by road class and AADT band) detected implausible outputs in two of the four models tested, failures that the CV metrics did not reveal. Open Road Risk should compare the distribution of Stage 1a predicted AADT by road class and rural/urban classification against observed AADF values and known traffic-volume relationships.
Both papers confirm that low-AADT links are the hardest to estimate well. Jayasinghe et al. (2019) report RMSE of 193–412% in the lowest AADT category. Huda and Al-Kaisy (2024) find that AADT explains almost nothing in the crash-risk model for links with fewer than 1000 vehicles per day — geometry dominates at that end. This does not mean Stage 1a performance on low-volume links is unimportant, but it contextualises what is at stake: for minor rural roads, exposure uncertainty matters less to the final risk ranking than the quality of the geometric and network features.
For a consolidated view of how the findings on this page map to the current pipeline state, open gaps, and recommended diagnostic actions, see the Literature–Pipeline Alignment page.
References
| ID | Citation |
|---|---|
| LIT-001/002 | Aguero-Valverde, J. & Jovanis, P.P. (2008). Analysis of road crash frequency with spatial models. TRB Annual Meeting; Transportation Research Record 2061, 55–63. |
| LIT-012/013/014 | Gilardi, A., Mateu, J., Borgoni, R. & Lovelace, R. (2022). Multivariate hierarchical analysis of car crashes data considering a spatial network lattice. Journal of the Royal Statistical Society Series A, 185(3), 1150–1177. DOI: 10.1111/rssa.12823 |
| LIT-015 | Hauer, E., Harwood, D.W., Council, F.M. & Griffith, M.S. (2001). Estimating safety by the empirical Bayes method: a tutorial. National SPF Summit, Chicago. |
| LIT-016/042 | Huda, K.T. & Al-Kaisy, A. (2024). Network screening on low-volume roads using risk factors. Future Transportation, 4(1). DOI: 10.3390/futuretransp4010013 |
| LIT-017/043 | Jayasinghe, A., Sano, K., Abenayake, C. & Mahanama, P.K.S. (2019). A novel approach to model traffic on road segments of large-scale urban road networks. MethodsX. DOI: 10.1016/j.mex.2019.04.024 |
| LIT-025/037 | Pan, G., Fu, L. & Thakali, L. (2017). Development of a global road safety performance function using deep neural networks. International Journal of Transportation Science and Technology, 6(3), 159–173. DOI: 10.1016/j.ijtst.2017.07.004 |
| LIT-032 | Pew, T., Warr, R.L., Schultz, G.G. & Heaton, M. (2020). Justification for considering zero-inflated models in crash frequency analysis. Transportation Research Interdisciplinary Perspectives, 8, 100249. DOI: 10.1016/j.trip.2020.100249 |
| LIT-028/045 | Roll, J., Anderson, J. & McNeil, N. (2026). Developing a pedestrian safety performance function for Oregon. FHWA-OR-RD-26-06. |
| LIT-029 | Wang, C., Quddus, M.A. & Ison, S.G. (2009). Impact of traffic congestion on road safety: a spatial analysis of the M25 motorway in England. Accident Analysis & Prevention. |
| LIT-034 | Gao, X. et al. (2024). Uncertainty-aware probabilistic graph neural networks for road-level traffic crash prediction. arXiv:2309.05072v4. |
| LIT-035 | Balawi, M. & Tenekeci, G. (2024). Time series traffic collision analysis of London hotspots. Heliyon. DOI: 10.1016/j.heliyon.2024.e25710 |
| LIT-049 | Mensah, A. & Hauer, E. (1998). Two problems of averaging arising in the estimation of the relationship between accidents and traffic flow. Transportation Research Record 1635, Paper No. 98-0232. |
| LIT-050 | Qin, X., Ivan, J.N., Ravishanker, N., Liu, J. & Tepas, D. (2006). Bayesian estimation of hourly exposure functions by crash type and time of day. Accident Analysis and Prevention, 38(6), 1071–1080. DOI: 10.1016/j.aap.2006.04.012 |
| LIT-051 | Dutta, N. & Fontaine, M.D. (2020). Improving freeway crash prediction models using disaggregate flow state information. VTRC 20-R15, Virginia DOT. |
| LIT-055 | National Highways (2022). Statistical methods for comparing road traffic collision and casualty rates: proposed approach. National Highways PR81/22. |
| LIT-052 | Sung, Y., Kim, S., Park, J. & Wang, L. (2024). Development of modified temporal safety performance function considering various time flows. Journal of Advanced Transportation. DOI: 10.1155/2024/7970454 |