Safety Performance Functions and Suitable Models

Safety Performance Functions and Suitable Models

Define the prediction

A Safety Performance Function (SPF) estimates crash frequency for a defined facility and outcome as a function of explanatory variables, often traffic volume and segment length. Some models include additional characteristics, while Highway Safety Manual predictive procedures can separately apply CMFs and local calibration to base-condition models. An SPF is not a site's observed crash count or a guaranteed forecast of the exact next-year count.

Predicted frequency describes the modeled mean for comparable conditions. Observed frequency is the recorded realization during the study period. Expected frequency using Empirical Bayes combines model information with site history under the chosen method. Keep these terms distinct when interpreting a screening measure or evaluation.

FHWA safety performance function resources support model selection and interpretation. The model's documentation is essential: an equation without its facility, outcome, units, and limitations is not ready for application.

Read the components and units

A general illustrative log-link form can be written:

$$N = L\exp(a+b\ln(\text{AADT}))$$

Here $N$ is the predicted annual frequency for the defined segment outcome, $L$ is length in the units required by the model, and $a$ and $b$ are estimated coefficients. This is an illustration of structure, not an official SPF with supplied design coefficients. Other models can use different variables or forms.

A volume exponent describes the model's relationship within its calibration range. It may be below, equal to, or above one depending on facility and outcome. Do not infer that congestion universally reduces all crash rates or that every model has the same exponent. Predictions outside the observed range can be unreliable.

Time units matter. An annual prediction must be combined appropriately over a multi-year observation period. If volumes change, calculate the relevant annual predictions and sum them rather than multiplying one convenient year by the period without justification. Length, volume, and outcome units must remain compatible throughout.

Why count models are useful

Crashes are nonnegative integer counts. A simple unconstrained linear model can produce negative predictions and may poorly represent variance. Poisson models assume variance equals the mean under the specified model. Crash datasets often show overdispersion due to differences among sites not fully explained by the predictors.

A negative-binomial model can represent variance exceeding the mean. In a common parameterization:

$$\operatorname{Var}(Y)=\mu+k\mu^2$$

The dispersion parameter $k$ and mean $\mu$ depend on the model and units. As $k$ approaches zero, this variance approaches the Poisson case. Overdispersion is a modeling issue, not proof that every individual road has one specific hidden defect. Models require appropriate fitting, diagnostics, and validation.

Select a suitable SPF

First match facility: rural or urban, segment or intersection, number of lanes, control, and other defined characteristics. Then match outcome: total, injury, specific type, or severity. Check the traffic and geometry range, observation period, region, definitions, and data quality. An intersection model cannot be substituted into a segment analysis merely because both use AADT.

Review the model's sample and predictive performance. A model developed from a narrow population may transfer poorly. An attractive equation with weak documentation is not preferable to a well-supported model that clearly explains limitations. Examine whether needed inputs are actually available at adequate quality.

Local calibration can adjust the level of predicted frequency to local conditions under the relevant method. In a simplified concept, a calibration factor compares observed totals with model-predicted totals for a suitable representative sample. It does not automatically repair an incorrect functional shape, wrong crash definition, or inappropriate facility model. Document the sample and procedure.

Connect SPFs and CMFs without confusing them

An SPF estimates baseline frequency; a CMF represents a relative change associated with a treatment or feature. A predictive procedure may multiply base predictions by applicable CMFs and calibration, but avoid double counting an attribute already included in the model. Review whether the SPF assumes base geometry or already represents the actual feature.

For an illustrative baseline of 10 targeted crashes per year and an applicable CMF of 0.8, the modeled treated frequency is 8 under the assumptions. The SPF supplied the baseline level; the CMF supplied the relative effect. Neither establishes that exactly two crashes will be prevented at a particular site next year.

Work a model-choice example

An analyst needs expected injury frequency at an urban signalized intersection. Candidate A concerns rural stop-controlled intersections and all crashes; Candidate B concerns urban signalized intersections and injury crashes within comparable volumes. Candidate B is the more relevant starting point, assuming adequate evidence and inputs. Candidate A's larger sample does not overcome the outcome and facility mismatch.

Next check local coding, entering volumes, geometry, and calibration requirements. If inputs are incomplete, report the resulting uncertainty or choose a defensible alternative analysis. Do not silently replace missing data with arbitrary values and then present the prediction as precise.

Model suitability checklist

• Match facility type and crash outcome.
• Confirm input units, periods, and calibration range.
• Review data quality and local calibration needs.
• Check which features the model already represents.

Critique the output and use it appropriately

Compare predictions with observed data across relevant groups and inspect systematic differences. A model that underpredicts one facility class can bias screening and investment toward that class. Check changes in traffic, reporting, and model applicability before calling an observed difference a safety problem.

Use the SPF as one component of evidence-based analysis. Diagnosis, field review, applicable treatment evidence, and implementation constraints remain necessary. A prediction helps identify expected performance under defined conditions; it does not certify a road as safe, determine legal fault, or select a project without further reasoning.