BrTS Count Model

The HSM best practice is to use the NB crash count modeling method to estimate the number of BrTSs. The form of the bridge SPFs is as follows:

$Crashe{s}_i=n\times StructureLength\times AAD{T}^{{\alpha }_2}\times e^{{\beta }_0+{\displaystyle {\sum }_{j=1}^q{\beta }_j{x}_{ij}}}$

Where:

- Crashes_i denotes the estimated bridge strikes at bridge i,

- Structure Length:1) length of the bridge structure for on-bridge strikes (e.g., truss, parapet, railing), or 2) length of the deck for both under-bridge strikes with overhead structure (e.g., girder) and under-bridge strikes with pier or support (e.g., bent, support, pier, abutment),

- n equals the number of years considered for the study (i.e., 5 years),

- AADT is the annual average daily traffic (based on the type of the estimated crash),

- x_ij is the quantitative measure of each explanatory variable j associated with bridge i,

- α₁, and β_j are estimated model coefficients for the explanatory variable j, and

- β₀ represents the estimated constant.

All variables used in the model development are from the NBI. Three sets of models have been developed with selected structure variables to correspond to each bridge-related crash type. Tables D2 to D4 show the model results of variables that contribute to the number of each bridge-related crash type, including coefficients (estimates), standard errors, and associated p-values (or significance level) of the significant variables.

Severity Distribution Function (SDF)

The MNL model was chosen as the personal injury severity model. The MNL model can be shown in following equation:

$P_{ij}=\frac{exp\left({\alpha }_j-X_{ij}^{\text{'}}{\beta }_j\right)}{{\displaystyle {\sum }_{j\in J}exp\left({\alpha }_j-X_{ij}^{\text{'}}{\beta }_j\right)}}$

Where:

- P_ij is the probability of ith vehicle-bridge strike with jth personal injury severity level outcome;

- X_ij denotes the vector of observable factors (variables) for ith vehicle-bridge strike with jth injury severity level;

- α_j is the estimated constant;

- β_j represents the vector of estimated coefficients.

An important feature of the MNL is that it estimates k-1 models, where k is the number of levels of the dependent variable. In this case, the dependent variable is injury severity level (i.e., k = 5, KABCO scale). In this instance, “No Injury/PDO (O)” is the referent group. Since the parameter estimates are relative to the referent group, the standard interpretation of the MNL model is as follows: for a unit change in the predictor variable, the logit of outcome m relative to the referent group is expected to change by its respective parameter estimate, given the variables in the model are held constant.

Selected variables from both crash data and the NBI are utilized for model development. Table 25 shows brief results including the list of remaining variables that contribute to the injury severity of each crash type.