# Predicting Pedestrian Crashes in Texas' Intersections and Midblock Segments
## 4 Crash Count Modeling  
An NB count model was used for pedestrian crash counts. The expected number of pedestrian crash counts \(\text{E}(Y_{i})\) along the \(i\)th intersection or midblock segment is expressed as follows:  
\[\text{E}(Y_{i})=\exp\left(\beta_{0}\ +\sum_{k}x_{ik}\beta_{k}+\varepsilon_{i}\right) \tag{1}\]  
where:  
* \(Y_{i}\) represents the total pedestrian count at intersection \(i\) with: \(\circ\) mean \(\text{E}(Y_{i})=\mu_{i}\)
* variance \(\text{Var}(Y_{i})=\mu_{i}+\rho\mu_{i}{}^{2}\), where \(\rho\) is the dispersion parameter. As the dispersion parameter becomes smaller and smaller, the variance converges to the same value as the mean, and the negative binomial turns into a Poisson distribution. Therefore, \(\rho=0\) for a Poisson model. * \(\beta_{k}\) is the \(k\)th covariate (e.g., speed limit, number of lanes, lane width, among others). * \(\varepsilon_{i}\) is a random error term which follows a one-parameter Gamma distribution \(\varepsilon_{i}\)-Gamma(\(\gamma,\ \gamma\)) with \(mean=1\), where \(\gamma\) is the scale parameter. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Variables**} & \multicolumn{4}{c}{**Intersections**} & \multicolumn{4}{c}{**Roadway Segments (0.1 mi Uniform)**} \\ \cline{2-9}  & **Mean** & **S.D.** & **Min.** & **Max.** & **Mean** & **S.D.** & **Min.** & **Max.** \\ \hline \#Pedestrian crashes per year & 0.074 & 0.489 & 0 & 13 & 0.042 & 0.274 & 0 & 9 \\ \hline Signalized intersection indicator & 0.042 & 0.201 & 0 & 1 & & & & \\ \hline Number of approaches & 3.043 & 0.659 & 0 & 5 & & & & \\ \hline Intersections crossed & & & & & 0.994 & 2.113 & 0 & 5 \\ \hline Walk-miles traveled per pop. dens. & 756 & 808 & 7 & 8180 & 647 & 736 & 7 & 8180 \\ \hline Daily vehicle-miles traveled & 2627 & 7182 & 0 & 133,254 & 4133 & 11,904 & 1 & 133,254 \\ \hline Speed limit (miles/h) & 56.737 & 5.698 & 10 & 65 & 57.359 & 5.001 & 10 & 70 \\ \hline Number of lanes & 2.259 & 0.731 & 1 & 6 & 2.266 & 0.736 & 1 & 6 \\ \hline Design: one-way road indicator & 0.013 & 0.114 & 0 & 1 & 0.049 & 0.216 & 0 & 1 \\ \hline Annual average daily traffic lane & 1499 & 2753 & 58 & 97,049 & 2123 & 4645 & 2 & 97,049 \\ \hline Percentage of trucks & 3.294 & 0.708 & 0 & 20 & 3.398 & 1.024 & 0 & 20 \\ \hline Functional class: local & 0.718 & 0.450 & 0 & 1 & 0.708 & 0.455 & 0 & 1 \\ \hline Functional class: collector & 0.165 & 0.371 & 0 & 1 & 0.161 & 0.368 & 0 & 1 \\ \hline Functional class: arterial & 0.117 & 0.321 & 0 & 1 & 0.131 & 0.337 & 0 & 1 \\ \hline On system roadway indicator & 0.039 & 0.194 & 0 & 1 & 0.123 & 0.328 & 0 & 1 \\ \hline Distance to nearest hospital (miles) & 2.092 & 1.279 & 0 & 10 & 2.258 & 1.417 & 0 & 10 \\ \hline Transit stops within 0.25-mi buffer & 0.220 & 0.414 & 0 & 1 & 0.211 & 0.408 & 0 & 1 \\ \hline Number of stops (0.25-mi buffer) & 1.022 & 2.391 & 0 & 18 & 1.038 & 2.562 & 0 & 21 \\ \hline Population density (per sq. mile) & 4443 & 3398 & 0 & 64,812 & 3775 & 3282 & 0 & 64,812 \\ \hline Employment density (per sq. mile) & 2344 & 9526 & 0 & 419,403 & 2196 & 8469 & 0 & 419,403 \\ \hline Median household income in the traffic & 7.193 & 3.891 & 0 & 25 & 7.228 & 4.109 & 0 & 25 \\ \hline Central business district indicator & 0.011 & 0.105 & 0 & 1 & 0.008 & 0.089 & 0 & 1 \\ \hline \hline \end{tabular} Note: S.D. = standard deviation, Min. = minimum, Max. = maximum, and the Speed limit, Number of lanes, Lane width, Median width, traffic count, and roadway class variables apply only to the major approach segment at each intersection.  
\end{table} 
Table 3: Summary statistics of variables, city of Austin. Additionally, a sensitivity analysis was applied to the NB estimates to understand the covariates' effects. Specifically, for each covariate, one standard deviation or binary change is applied. The modified variables are passed to the model to calculate the prediction. Then, the difference between the mean of original prediction and permuted prediction is calculated to represent the contribution of that covariate.