--- category: literaturenote citekey: singhdevelopmentpedestrianvehiclerelated2021 title: Development of pedestrian- and vehicle-related safety performance functions using Bayesian bivariate hierarchical models with mode-specific covariates authors: "Singh, Mankirat; Cheng, Wen; Samuelson, Dean; Kwong, Jerry; Li, Bengang; Cao, Menglu; Li, Yihua" year: 2021 date: 2021-00-00 2021 doi: 10.1016/j.jsr.2021.05.008 publication: Journal of Safety Research url: "https://doi.org/10.1016/j.jsr.2021.05.008" zotero_key: MEHPK3AV zotero_storage: 6KTGKDUS collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Singh, Mankirat" status: converted --- ![](_page_0_Picture_1.jpeg) # Journal of Safety Research journal homepage: www.el [sevier.com/locate/jsr](http://www.elsevier.com/locate/jsr) ![](_page_0_Picture_5.jpeg) # Development of pedestrian- and vehicle-related safety performance functions using Bayesian bivariate hierarchical models with modespecific covariates ![](_page_0_Picture_7.jpeg) Mankirat Singh a , Wen Cheng a,⇑ , Dean Samuelson b , Jerry Kwong c , Bengang Li a , Menglu Cao a , Yihua Li d - aDepartment of Civil Engineering, California State Polytechnic University, Pomona, CA 91768, United States - b Traffic Safety Investigations Branch, Department of Transportation California, United States - cDivision of Research, Innovation and System Information, Department of Transportation California, United States - dDepartment of Logistics Engineering, Logistics and Traffic College, Central South University of Forestry and Technology, Hunan 410004, China ### article info #### Article history: Received 20 October 2020 Received in revised form 15 February 2021 Accepted 21 May 2021 Keywords: Pedestrian-Vehicles crashes Crash frequency Models Bivariate models Pedestrian count Safety Performance Function Available online 4 June 2021 # abstract Introduction: Pedestrian safety is a major concern as traffic crashes are the leading cause of fatalities and injuries for commuters. Traffic safety research in the past has developed various strategies to counteract traffic crashes, including the safety performance function (SPF). However, there is still a need for research dedicated to enhancing the SPF for pedestrians from perspectives of methodological framework and data input. To fill this gap, this study aims to add to the current SPF development practice literature by focusing on pedestrian-involved collisions, while considering the typical vehicle ones as well. Methods: First, bivariate models are used to account for the common unobserved heterogeneity shared by the pedestrian- and vehicle-related crashes at the same intersections. Second, variable importance ranking technique is used, along with correlation analysis, to determine mode-specific feature input. Third, the exposure information for both modes, annual pedestrian count, and annual daily vehicles traveled are used for model development. Fourth, a recent Bayesian inference approach (integrated nested Laplace approximation (INLA)) was adopted for bivariate setting. Finally, different evaluation criteria are used to facilitate comprehensive model assessment. Results: The results reveal different statistically significant factors contributing to each of the modes. The offset intersection provides better safety performance for both pedestrians and drivers as compared to other intersection designs. The model findings also corroborate the sensibility of using the bivariate models, rather than the separate univariate ones. Practical Applications: The study shows that pedestrians are more vulnerable to various intersection features such as left-turn channelization, intersection control, urban and rural population group, presence of signal mastarm on the cross-street, and mainline average daily traffic. Greater focus should be directed toward such intersection features to improve pedestrian safety. 2021 The Authors. Published by the National Safety Council and Elsevier Ltd. This is an open access article under the CC BY license [\(http://creativecommons.org/licenses/by/4.0/](http://creativecommons.org/licenses/by/4.0/)). #### 1. Introduction Walking is considered an important travel mode due to its immense benefits. However, the percentage of total trips undertaken by pedestrians compared to other modes is very low. In the United States, the National Household Travel Survey ([NHTS,](#page-7-0) [2018\)](#page-7-0) reported that trips made by walking accounted for only 0.6% of total person-miles travel (PMT). Past studies reveal that there are many reasons for this, but one main reason is that pedestrians are among the most vulnerable and unsafe road users from [et al., 2003](#page-7-0)). According to national statistics [\(Governor Highway](#page-7-0) [Safety Association, 2019\)](#page-7-0), a total of 6,590 pedestrian fatalities and around 70,000 injuries were estimated in pedestrian-vehicle crashes in 2019. These consequences create an urgent need to prevent pedestrian-involved crashes by implementing better policies and strategies to provide a safe traffic environment for pedestrians. the viewpoint of traffic crashes [\(De Hartog et al., 2010; Retting](#page-7-0) Given this context, many researchers have investigated various factors, such as roadway-built characteristics [\(Mansfield et al.,](#page-7-0) [2018; Miranda-Moreno et al., 2011\)](#page-7-0), pedestrian behavior [\(Xu](#page-8-0) [et al., 2018; Dommes et al., 2014\)](#page-8-0), driver behavior ([Baker et al.,](#page-7-0) [1974; Geruschat & Hassan, 2005; Schroeder & Rouphail, 2011\)](#page-7-0), traffic characteristics [\(Shi et al., 2007; Barton & Morrongiello,](#page-7-0) [2011\)](#page-7-0), drug/alcohol use [\(Li et al., 2019; Plurad et al., 2006\)](#page-7-0), social E-mail address: [wcheng@cpp.edu](mailto:wcheng@cpp.edu) (W. Cheng). ⇑ Corresponding author. and demographic attributes [\(LaScala et al., 2000; Tabibi et al.,](#page-7-0) [2012\)](#page-7-0) to highlight the crucial insights related to pedestrianinvolved crashes. Among the distinct strategies, the development of safety performance functions (i.e., crash frequency models) is one of the most popular strategies to address traffic safety that not only aids in screening out the significant influential factors, but also predicts the crash counts for various purposes [\(Ukkusuri](#page-8-0) [et al., 2011; Wu et al., 2018; Harwood et al., 2008](#page-8-0)). Poisson regression models were initially widely adopted due to the popularity of Poisson distribution for discrete outcomes [\(Miranda-Moreno,](#page-7-0) [2006](#page-7-0)). However, Poisson regression models are not able to provide reliable and unbiased results in the case of over-dispersion (i.e., variance greater than the associated mean). In response, researchers employed alternate model formulations such as Poisson gamma or negative binomial ([Hauer, 2001\)](#page-7-0), Poisson lognormal ([Park & Lord, 2007](#page-7-0)), zero-inflated models [\(Aguero-Valverde,](#page-7-0) [2013\)](#page-7-0), which can better address over-dispersion issues and, hence, provide more valid inferences. Among the above-mentioned models, the univariate model framework has seen widespread applications in traffic safety studies due to its ease of implementation with only one dependent variable being involved ([Anarkooli et al., 2019](#page-7-0)). However, the univariate model is not capable of addressing the unobserved heterogeneities shared by various crash types or severities occurring in the same locations or situations ([Mannering & Bhat, 2014](#page-7-0)). To overcome this issue, multivariate models have been proposed due to their enhanced capabilities to tackle the common heterogeneity among different crash types via the explicit consideration of correlated random effects ([Lee et al., 2015; Park & Lord, 2007\)](#page-7-0). As a special case of multivariate setting, the bivariate model is dedicated to the crashes of two categories and also enjoys frequent applications. For example, [Russo et al. \(2014\)](#page-7-0) used bivariate framework to examine the factors pertaining to crash injury severities involved in angled collisions. The results demonstrate that bivariate models provide more insightful findings related to the factors influencing the propensity of crashes. [Zheng and Sayed \(2019\)](#page-8-0) developed bivariate models to integrate the traffic conflict indicators for crash estimation. The finding showed that the bivariate model improved the crash estimation precision and accuracy. Another dimension of model classification resides in the transportation modes. For instance, as previously stated, the SPF can be divided into motor vehicle-oriented, non-motorist-centered, and so on. Overall, the vehicle-related SPF dominates the SPF development given the largest proportion of such mode in the current transportation system. However, with the constant promotion of active transportation provided by various levels of government agencies in the past decades, ever-increasing interests were directed toward the SPF development for pedestrians ([Wier et al., 2009;](#page-8-0) [Rasciute & Downward, 2010](#page-8-0)). For example, [Thomas and DeRobertis](#page-7-0) [\(2013\)](#page-7-0) investigated road crashes involving pedestrians and bicyclists based on a very short period of volume count data available for these two modes. The results showed that the risk for an individual pedestrian/bicyclist to be involved in a crash decreases with an increase in the number of pedestrians/bicyclists. Subsequently, [McArthur et al. \(2014\)](#page-7-0) conducted a study to develop SPF to estimate the pedestrian crashes using five-year data, including socioeconomic and demographic characteristics. Another study done by [Tulu et al. \(2015\)](#page-7-0) developed SPF to investigate the pedestrian crashes on two-way two-lane rural roads by incorporating the short period counts of daily pedestrian crossing volumes in Ethiopia. [Gates et al. \(2016\)](#page-7-0) developed SPFs for pedestrian and bicyclist crashes at road segments and intersections. The results demonstrated that pedestrian and bicycle crashes tend to increase when vehicle traffic volumes increase investigated two pedestrian crash types (total pedestrian crashes at intersections and a subset of intersection crashes involving pedestrian-motorists collision), by developing SPFs and incorporating many factors such as roadway, built environment, census, and activity measures. A common theme among these papers is the lack of or very limited exposure information directly related to active transportation modes such as pedestrian counts. The potential reasons for such data scarcity are due to lack of definite paths or routes, followed by pedestrian and bicyclists, limited use of emerging technologies (e.g., crowdsourcing), expensive data collection devices, and so on. To address these issues, different strategies have been used in the past. Some studies employed daily vehicle miles traveled as the proxy for active transportation modes based on the assumption that most non-motorist-pertinent collisions are related to vehicles [\(Cheng](#page-7-0) [et al., 2018](#page-7-0)). Others relied on the formulation of the pedestrian volume models using predictors such as land use, transportation system attributes, and neighborhood socioeconomic characteristics. One recent example is the pedestrian count model developed by UC Berkeley researchers ([Griswold et al., 2019\)](#page-7-0), which can be used in the pedestrian SPF as a major estimation of pedestrian exposure, rather than other proxy information. Building upon previous research, this paper aims to add to the current literature with microscopic level research pertaining to pedestrian-involved collisions, with some enhancements to current SPF development practice. First, bivariate models are used to account for the common unobserved heterogeneity shared by the pedestrian and vehicle-related crashes at the same intersections. Second, random forest (RF) method and Pearson's correlation test are employed to determine the variables to be included in the models, which are different for each of the joint models. Such practice leads to the proper covariates not only striking a balance between multi-collinearity and omitted variable bias issues, but also enhancing model flexibility with different inputs to specific transportation mode. Third, the exposure information for both modes, annual pedestrian count, and annual daily vehicles traveled is used for the model development. The former counts were generated based on research by [Griswold et al. \(2019\)](#page-7-0). Including exposure of both pedestrians and vehicles are anticipated to enhance model estimate precision. Fourth, in comparison with the typical Bayesian hierarchical models based on the Markov chain Monte Carlo (MCMC) algorithm, the integrated nested Laplace approximation (INLA) approach is selected due to faster calculation and more robust results ([Taylor & Diggle, 2014](#page-7-0)). Finally, for a comprehensive comparison of the predictive accuracy of the models, distinct goodness-of-fit measurements that include DIC (deviance information criterion), D (posterior mean deviance), PD (effective number of parameters) and LPML (log pseudo marginal likelihoods) are employed. The results of this study are expected to provide additional insights into SPF development, especially for the pedestrian mode. ## 2. Materials and methods The paper features Bayesian joint hierarchical models with different inputs of covariates, variable selection using both random forest (RF) for predictor importance ranking and Pearson's correlation test for variable correlation analysis, and a set of performance evaluation criteria. The following subsections cover the corresponding methodological details in order. # 2.1. Bayesian joint hierarchical model specification The paper employed Poisson lognormal model, which assumes the crash count to be Poisson distributed with the logarithm of Poisson rate following normal distribution. The model formulation is shown as follows ([Cheng et al., 2018](#page-7-0)). $$\mathbf{y} \ Poisson(\lambda)$$ (1) $$\ln(\lambda) = \beta_0 + \beta X + \varepsilon \tag{2}$$ where y is a matrix consisting of crash counts of different modes and intersections, k is a matrix consisting of the corresponding Poisson rates of different modes and intersections, b0 represents a global intercept vector for the two modes, b is a coefficient vector, X is the covariate matrix, and e represents the white noise matrix. To better describe the joint models with different predictor input, let kp and kv denote the pedestrian- and vehicle-involved Poisson rate vector, respectively. The model framework for each of the transportation modes can be expressed using the following equation. $$\ln(\lambda_{\mathbf{p}}) = \beta_{0\mathbf{p}} + \beta_{\mathbf{c}\mathbf{p}}\mathbf{X}_{\mathbf{c}} + \beta_{\mathbf{d}\mathbf{p}}\mathbf{X}_{\mathbf{d}\mathbf{p}} + \boldsymbol{\varepsilon}_{\mathbf{p}}$$ (3) $$\ln(\lambda_{\mathbf{v}}) = \beta_{0\mathbf{v}} + \beta_{\mathbf{c}\mathbf{v}} \mathbf{X}_{\mathbf{c}} + \beta_{\mathbf{d}\mathbf{v}} \mathbf{X}_{\mathbf{d}\mathbf{v}} + \boldsymbol{\varepsilon}_{\boldsymbol{v}}$$ (4) where the subscripts v and p represent the vehicle and pedestrian modes, b0 is the global intercept, bc is the vector of coefficients for the independent variables common to both modes, Xc is the matrix of covariates common to both modes, bd is the vector of coefficients for the independent variables that are different between the two modes, Xd is the corresponding covariate matrix, and e is the vector of error term. The two models are developed simultaneously with the two error vectors, ep and ev , following the bivariate normal distribution: $$\varepsilon MN(\mu, \sum)$$ (5) where $$: \mathbf{\varepsilon} = \begin{pmatrix} \mathbf{\varepsilon}_p \\ \mathbf{\varepsilon}_v \end{pmatrix}, \boldsymbol{\mu} = \begin{pmatrix} \mu_p \\ \mu_v \end{pmatrix}, \sum_{n} = \begin{pmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \end{pmatrix}$$ (6) In the above equations, MN represents multivariate (or, bivariate in the present study) normal distribution, e is the random effect matrix that captures the extra-Poisson heterogeneity among intersections,l is the vector of the mean values for the bivariate normal distribution, and P is the variance–covariance matrix where the diagonal elements (i.e., r11 and r22) in the matrix represent the variances of the random effects, while the off-diagonal element represents the covariance. The inverse of the variance–covariance matrix represents the precision matrix, which can be formulated using the Wishart distribution: $$\sum^{-1} Wishart(I,J)$$ (7) where I is the J identity matrix [\(Congdon, 2006\)](#page-7-0), and J is the degree of freedom, J = 2 herein representing two transportation modes. The non-informative specifications ([Heydari et al., 2017](#page-7-0)) for various coefficients were specified with a normally distributed vague priors N (0,100). Such diffused normal distribution with zero mean and a large variance is commonly employed as a vague prior to posterior estimates due to the absence of sufficient knowledge of priori distribution ([Cheng et al., 2018](#page-7-0)). #### 2.2. Random forest (RF), and various evaluation criteria Decision tree is one of the predictive models that come up with an item's target value (leaves) via the observations about the item (branches). Based on the nature of the target value, the decision tree model can be used for both regression and classification purposes. Compared with the other typical regression techniques, the decision tree gained its popularity as it closely mirrors the human decision-making process. As implied by the name, random forest (RF) ([Fatholahzade et al., 2018\)](#page-7-0) consists of a collection of individual decision trees that operate as an ensemble. The method combines bagging and the random selection of features to construct different decision trees with a controlled variation. Using ensembles of predictors have proven to give more accurate results than using a single predictor. This technique has an advantage over the traditional decision trees in obtaining unbiased error estimates without separating the cross-validation test dataset. When a particular tree in the RF is grown from a bootstrap sample, usually one-third of the training cases are left out (also called out-of-bag, OOB, data) from the tree-growing. The OOB data are then used later for the determination of the optimum number of predictors for each tree and the optimum number of trees in the RF, which results in the minimum OOB error rate. As a robust data mining technique with the implicit wisdom: ''a large number of uncorrelated individuals operating as a committee will make a better decision than do these individuals," RF has seen wide applications in various fields including traffic safety [\(Harb](#page-7-0) [et al., 2009\)](#page-7-0). For the classical purposes of regression and classification, RF has been frequently used for determining the importance of various response variables, based on the mean decrease of either prediction accuracy or node purity with a specific variable being excluded from the model [\(Jiang et al., 2016](#page-7-0)). The multiple steps are involved when the former metric is implemented. First, the prediction accuracy of the OOB sample is estimated. Second, the values of the variable in the OOB sample are randomly shuffled, with all other variables remaining the same. Third, the decreased prediction accuracy on the shuffled data is calculated. Finally, the average drop of accuracy across all trees is reported for the variable. The more decreased prediction accuracy in the OOB data, the more predictive power the variable tends to have. The second method follows a similar process as does the first one, except that the prediction accuracy is replaced with the node purity (or, Gini), which has the largest value when only one single class or value is involved in the node. Compared with the predictive accuracyoriented metric, the node purity one has the advantage of faster computation and is, therefore, chosen in the study [\(Nicodemus,](#page-7-0) [2011\)](#page-7-0). In addition to the variable importance ranking, this paper adopted various evaluation criteria that include Deviance Information Criterion (DIC), D (posterior mean deviance), PD (effective number of parameters) and Log pseudo marginal likelihoods (LPML) for the assessment of predictive accuracy and goodnessof-fit. As a hierarchical modeling generalization of the Akaike Information Criterion (AIC) which uses maximum likelihood estimates ([Hurvich & Tsai, 1998](#page-7-0)), DIC has been used extensively to assess the complexity and goodness of fit of the Bayesian models based on the posterior mean. The calculation of DIC can be done via the following expression [\(Spiegelhalter et al., 2003](#page-7-0)): $$DIC = D + P_D (8)$$ where, Dis the posterior mean deviance that measures the closeness of the fitted data to the original observations, and PD denotes the effective number of parameters in a model representing the model complexity. In general, models with more parameters tend to overfit the data, resulting in smaller deviance. Therefore, the PD term can be considered as compensation for this effect by favoring models with a smaller number of parameters. Different than DIC and D, which are based on within-sample predictive errors, other alternatives are based on the test data using cross-validation techniques. Nonetheless, the typical approaches of cross-validation are prone to selection bias related with data-splitting into subsets. To circumvent such bias, a robust conditional predictive ordinate (CPO) based on CV-1 (leave-oneout) was employed in this study ([Pettit, 1990\)](#page-7-0). Within the INLA framework, the estimate of CPO for each observation i can be calculated as ([Gelfand, 1996; Liu & Sharma, 2017\)](#page-7-0): $$CPO_i = \left(\frac{1}{T} \sum_{t=1}^{T} \frac{1}{f(Y_i | \beta^{(t)})}\right)^{-1} \tag{9}$$ where Yi is the ith observation (i = 1, 2, 3, n) for all intersections and b represents the estimated model parameters. Based on the CPO, the Log pseudo marginal likelihoods (LPML) can be calculated and have been employed in recent safety literature [\(Heydari et al., 2017; Cheng et al., 2018a](#page-7-0)). The computation for LPML can be performed using the following equation: $$LPML = \sum_{i=1}^{n} log(CPO_i)$$ (10) where i, n, and CPO are denoted in Equation (9). The large value of LPML signifies a better predictive capability related to the candidate model. ## 2.3. Data description The analysis in this study was based on data derived from California Traffic Accident Surveillance and Analysis System (TASAS). TASAS is a traffic records system that includes a crash database and infrastructure database consisting of highway segments, intersections, ramps, and other data. The study focused on crashes occurring at the intersections that have 73 variables available in the raw file in TASAS. Nonetheless, some of these variables were not associated with pedestrian or vehicle collisions like intersection location information (district, county, route, and milepost), date of intersection update (begin date of intersection update, the end date of intersection update), and so on. After data cleaning, 20 covariate variables were selected from a total of 6,198 intersections in the state routes, where the estimated annual pedestrian volume at each intersection was available through the pedestrian count model developed by [Griswold et al. \(2019\)](#page-7-0). Overall, a total Table 1 Descriptive Statistics of Collected Data. Numerical Variables | X-ADT
IRG
APV
Veh counts
Ped counts | Cross street - average daily traffic
Intersection rate group
Estimated annual pedestrian volume (2016)
Vehicle related accidents counts
Pedestrian related accidents count | 0
1
520
0
0 | 77,000
29
9,400,000
137
6 | 1,911
17.91
116,636
6.88
0.09 | 4272.05
7.61
481942.50
11.79
0.39 | |-------------------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|----------------------------------------------------------|-------------------------------------------|-----------------------------------------------| | Categorical Variables | | | | | | | Variables | Description | | Details of categories (frequency, percentage) | | | | Highway Group | Highway group of mainline in the intersection | 1-Divided Highway (3,294, 53.15%); 2-Undivided
Highway (2,881, 46.48%); 3-Right or Left Independent
Alignment (23, 0.37%) | | | | | Population Group | Population code of the intersection | -Urban (1,539, 24.83%); 2-Rural (1,278, 20.62%); 3-
Urbanized (3,381, 54.55%) | | | | | Intersection Design | Intersection design | 1-Four legged (2,328, 37.56%); 2- >Four legs (67, 1.08%);
3-Offset (349, 5.63%); 4- Tee (3,182, 51.34%); 5 - Wye
(206, 3.32%); 6-Other (66, 1.06%) | | | | | Light Condition | Presence of light condition at Intersection | | 1-No Lighting (1,561, 25.19%); 2-Lighted (4,637, 74.81%) | | | | Mastarm | Presence of signal mastarm on the mainline of the intersection | 1-No Mastarm (4,901, 79,07%); 2-Yes, Mastarm (1,297,
20.93%) | | | | | Left Turn | Left turn channelization on mainline at the intersection | 1-Curbed Median Left Turn Channelization (808, 13.04%);
2-No Left Turn Channelization (3,005, 48.48%); 3 -
Painted Left Turn Channelization (2,355, 37.00%); 4 -
Others (30, 0.48%) | | | | | Right Turn | Right turn channelization on mainline at the intersection | 1-No Right Turn Channelization (5,579, 90.01%); 2-Others
(617, 9.99%) | | | | | Traffic Flow | Traffic flow on the mainline of the intersection | 1-Two-Way Traffic, No Left Turns Permitted (297, 4.81%);
2-Two-Way Traffic, Left Turn Permitted (5,839, 94.21%); 3
- Others (61, 0.98%) | | | | | X-Mastarm | Presence of signal mastarm on the cross-street of the intersection | 1-No Mastarm (5,341, 86.17 %); 2-Yes, Mastarm (857,
13.83%) | | | | | X-Left Turn | Left turn channelization on the cross-street | 1-Curbed Median Left Turn Channelization (131, 2.11%);
2-No Left Turn Channelization (5,421, 87.47%); 3-Painted
Left Turn Channelization (622, 10.04%); 4-Others (24,
0.39%) | | | | | X-Right Turn | Right turn channelization on the cross-street. | 1-No Right Turn Channelization (5,631, 90.85%); 2-Others
(567, 9.15%) | | | | | X-Traffic Flow | Traffic flow on the cross-street of the intersection | 1–Two Way Traffic, No Left Turns Permitted (269, 4.34%);
2–Two-Way Traffic, Left Turn Permitted (5,846, 94.32%);
3-Others (83, 01.34%) | | | | | Intersection Control Condition | Intersection control condition | 1-No Control (210, 3,039%); 2-Stop signs on Cross Street
Only (4,514, 72.83%); 3-Signals Pretimed (2 Phase) (152,
2.45%); 4- Signals Semi-Traffic Actuated, Two-phase (125,
2.02%); 5 - Signals Full Traffic Actuated, Multi-Phase (993,
16.02%); 6-Others (204, 3.29%) | | | | Variables Description Minimum Maximum Mean S.D. MNL Mainline - number of lanes 2 8 3.33 1.39 MOL Mainline - override length (buffer) 15 350 187.90 61.94 X-NL Cross street - number of lanes 0 6 2.13 0.55 X-OL Cross street - override length 0 250 2.25 22.81 MADT Mainline - average daily traffic 180 125,000 20,198 15254.28 of 43,705 pedestrian and vehicle collisions spanning over six years (2012 to 2017) were aggregated for research purposes. The detailed information for all data including variable names, description, and other descriptive statistics are illustrated in [Table 1.](#page-3-0) ## 3. Results To develop the bivariate SPFs, the distinct covariates were selected for pedestrians and vehicles by using RF metric and correlation analysis. Under the INLA framework, models were developed with the posterior mean serving as the estimate for the model parameters. Different evaluation criteria were used to assess the predictive accuracy of the models. #### 3.1. Feature importance ranking by random forest (RF) The importance of variables was reported and ranked using the R package ''randomforest" ([Cutler et al., 2012](#page-7-0)). When estimating the RF model, m = 4 variables were randomly sampled as a candidate at each split, with the OOB error rate reaching a minimum value of 0.132 and 59.24% of data variability being explained by the model. The variable importance plots for both pedestrian and vehicles are shown in Fig. 1 with the decreasing order of ''IncNodePurity," which represents the mean decrease of node purity in predictions on OOB samples with a given variable being excluded from the model. #### 3.2. Correlation analysis Variable importance ranking was used along with the correlation of numerical variables for the determination of the covariate inputs to the model development. The correlation tests were conducted using the Harrell Miscellaneous package in R software, which allowed the calculation of Pearson's correlation coefficient and the accompanying p-values. The variables were observed to be correlated by using the popular cut line of 0.6 for the correlation coefficient and with a significance level of 0.05 were eliminated in multiple steps using engineering judgment to choose the minimum subset of variables, while maintaining the maximum data variability. In other words, the selection procedure strived to strike the balance between omitted variable bias and multi-collinearity issues. As shown in Table 2, the upper portion values are Pearson's correlation coefficient magnitudes and lower shaded cells represent the associated p-values. Based on the results of the correlation test, out of eight numerical variables, six of them that include MOL, X-OL, MADT, X-ADT, IRG, and APV were retained. Combining both results from RF and correlation analysis, the final list of predictors to be included into subsequent model development can be found in [Table 3.](#page-5-0) It is important to note that the variables ''Right Turn" and ''X-OL" were retained only for pedestrians, while ''X-Mastarm" and ''X-Right Turn" were included only for vehicles. They were not considered for both modes at the same time since they had little influence on one of the modes according to the variable importance ranking results via RF. ![](_page_4_Figure_10.jpeg) Fig. 1. A Variable Importance Plot for (a) Vehicle Crash Counts and (b) Pedestrian Crash Counts. Table 2 Correlation Coefficients and P-Value for the Numerical Variables. | | MNL | MOL | X-NL | X-OL | MADT | X-ADT | IRG | APV | |-------|-------|-----------|-------|-----------|-----------|-------|-----------|-----------| | MNL | 1.000 | 0.116
| 0.230 | 0.001 | 0.722 | 0.273 | 0.118 | 0.254 | | MOL | 0.000 | 1.000 | 0.046 | 0.001
| 0.088
| 0.032 | 0.139
| 0.152
| | X-NL | 0.000 | 0.000 | 1.000 | 0.118 | 0.227 | 0.600 | 0.087
| 0.112 | | X-OL | 0.925 | 0.917 | 0.000 | 1.000 | 0.005 | 0.187 | 0.034
| 0.194 | | MADT | 0.000 | 0.000 | 0.000 | 0.683 | 1.000 | 0.269 | 0.154 | 0.257 | | X-ADT | 0.000 | 0.014 | 0.000 | 0.000 | 0.000 | 1.000 | 0.118
| 0.184 | | IRG | 0.000 | 0.000 | 0.000 | 0.008 | 0.000 | 0.000 | 1.000 | 0.046
| | APV | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 1.000 | #### Notes. 1. The upper triangle of the matrix shows the correlation coefficients of the variables, and the gray grids in the lower triangle of the matrix shows the p-values. 2. Highly correlated estimate with correlation coefficient greater than 0.6 are marked as bold font. 3. Refer to [Table 1](#page-3-0) for details of variable definition. Table 3 Description of Model Parameter Estimates. | b1 (Pedestrian) | b2 (Vehicle) | | | | |---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|--| | Mean | SD | Mean | SD | | | | | | | | | | | | 0.735 | | | | | | | | | | | | 0.141 | | | | | | 0.831 | | | | | | | | | | | | 0.321 | | | | | | 0.141 | | | | | | | | | | | | 0.448 | | | | | | 0.21 | | | | | | 0.485 | | | | | | 1.107 | | | | | | 0.779 | | | | | | | | | | | | 0.213 | | | | | | | | | | | | 0.271 | | | | | | | | | | | | 0.183 | | | | | | 0.14 | | | | | | 0.559 | | | | | | | | | | | | 0.153 | | | | | | | | | | | | 0.336 | | | | | | 0.455 | | | | | | | | | | | | N/A | | | | | | | | | | | | N/A | | | | | | | | | | | | 0.264 | | | | | | 0.258 | | | | | | 1.125 | | | | | | | | | | | | 0.345 | | | | | | 0.473 | | | | | | | | | | | | 0.503 | | | | | | 0.554 | | | | | | 0.59 | | | | | | 0.566 | | | | | | 0.578 | | | | | | 0.001 | | | | | | 0.001 | | | | | | 0.497 | | | | | | 0.718 | | | | | | 0.031 | | | | | | 0.775 | | | | | | | | | | | | 0.462 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Highway Group 1 (Base)
Highway Group 2
Highway Group 3
Population Group 1 (Base)
Population Group 2
Population Group 3
Intersection Design 1 (Base)
Intersection Design 2
Intersection Design 3
Intersection Design 4
Intersection Design 5
Intersection Design 6
Light Condition (1) (Base)
Light Condition (2)
Mastarm 1 (Base)
Mastarm 2
Left Turn 1 (Base)
Left Turn 2
Left Turn 3
Left Turn 4
Right Turn 1
Right Turn 2
Traffic Flow 1 (Base)
Traffic Flow 2
Traffic Flow 3
X-Mastarm 1 (Base)
X-Mastarm 2
X-Right Turn 1 (Base)
X-Right Turn 2
X-Left Turn 1 (Base)
X-Left Turn 2
X-Left Turn 3
X-Left Turn 4
X-Traffic Flow 1 (Base)
X-Traffic Flow 2
X-Traffic Flow 3
Intersection Control Condition (1) (Base)
Intersection Control Condition (2)
Intersection Control Condition (3)
Intersection Control Condition (4)
Intersection Control Condition (5)
Intersection Control Condition 6 | Fixed Effects
5.222

0.008

0.494
0.834

0.247
0.215

0.341

0.391
0.485
0.681
0.307
0.207

0.264

0.048
0.572
N/A
0.871
0.338
0.251
0.038
0.118
0.159
0.079
0.143
0.292
1.134
1.975
2.218
1.899
1.611
0.002
N/A
2.997
3.581
0.076

0.146

Random Effects
0.643
29113.63
24753.94
43596.89 | 0.772
0.053
0.37
0.092
0.051
0.176
0.085
0.157
0.189
0.246
0.054
0.131
0.078
0.063
0.284
N/A
0.148
0.256
0.077
0.068
0.14
0.136
0.353
0.144
0.224
0.137
0.201
0.219
0.192
0.186
0.001
N/A
0.215
0.388
0.01
0.451
0.018 | 3.788
0.082

0.941

0.38

0.088
0.34

0.401

1.761

3.634

1.994

0.747
0.087

0.041

0.152

1.682
0.246

0.025

0.784
N/A
N/A
0.182
0.232
1.311

0.169
0.234
0.708
0.922
1.446
1.168
0.662
0.001

0.002
1.845
2.598
0.048
0.429
2.297 | | Notes. # 3.3. Model estimates The posterior model estimates of model parameters across pedestrian and vehicle crash counts are shown in Table 3. The estimated coefficients for 10 influential variables including 'Intersection Design 2' (>four legs), 'Intersection Design 3' (offset), 'Intersection Design 4' (tee), 'Intersection Design 5' (wye), 'Intersection Design 6' (others), 'Light Condition [\(2\)](#page-2-0)' (Lighted), 'Left turn 4' (No left turn channelization), 'Intersection Control Condition [\(4\)'](#page-2-0) (signals semi-traffic actuated, two phase), 'Intersection Control Condition [\(5\)'](#page-2-0) (signals full traffic actuated, multi-phase), 'MADT' (mainline-average daily traffic), and 'X-ADT' (crossline-average daily traffic), appeared to be statistically significant across both pedestrian and vehicle crash counts. Interestingly, among these 1. S.D. represents standard deviation; DIC represents deviance information criterion; D represents posterior mean deviance; PD represents an effective number of parameters; LPML represents log pseudo marginal likelihood; NA means Not Applicable. 2. Refer to [Table 1](#page-3-0) for details of variable definition. 3. The bold fonts represent the variables with a statistically significant impact. significant covariates, five variables were found to have a negative impact, in which one variable ('Intersection Design 3') was common for both modes and other four variables were observed for vehicles crash count only (or, 'Intersection Design 4,' Intersection Design 5,' Intersection Design 6,' 'Right Turn 2'). It follows that, compared with the base condition of the four-legged intersection, offset intersection seems to be more advantageous in terms of traffic safety for both pedestrians and vehicle drivers. This finding is aligned with the previous studies [\(Bared & Kaisar, 2001; Jonsson](#page-7-0) [et al., 2007; Cunningham et al., 2020](#page-7-0)), which illustrates that the implementation of offset intersection might help to decrease pedestrian-vehicle collisions. For vehicle drivers only, the tee and wye intersections and those without right-turn channels tend to provide more safety benefits compared with the base conditions of four-leg and intersection with right turn channels, respectively. The better safety performance associated with those without right turn channels is somewhat counterintuitive, which warrants further verifications from other studies. At the individual mode level, 10 covariates that contain 'Population Group 2' (rural), 'Population Group 3' (urbanized), 'Left Turn 2' (no left turn channelization), 'Traffic Flow 2' (two-way traffic, left turn permitted), 'X-Mastarm 2' (presence of signal mastarm), 'Intersection Control Condition [\(2\)](#page-2-0)' (stop signs on cross street only), 'Intersection Control Condition [\(3\)](#page-2-0)' (signals pretimed), 'Intersection Control Condition 6' (others), 'MOL' (mainline - override length), and 'IRG' (intersection rate group) observed to be statistically significant for pedestrians. Similarly, for vehicles, there are two statistically significant variables that include 'Right Turn 2' (right turn channelization) and 'X-OL' (cross street - override length). Such phenomenon indicates that, relative to drivers, pedestrians are not only subject to more injury severities, but also sensitive to more intersection features such as left turn channelization, intersection control, and so on. This finding is consistent with previous studies ([Pulugurtha & Sambhara, 2011; Zegeer & Bushell,](#page-7-0) [2012; Yue et al., 2020](#page-7-0)). This study also employed various types of evaluation criteria including (D) (measure of training errors), DIC (indirect measure of test errors), and LPML (measure of test errors) to assess the models from different perspectives. Under the close review of the evaluation results, it is obvious that DIC is the sum of D and PD, where PD serves as the correction term to the in-sample error so that DIC can approximate the out-of-sample error. Different from DIC, the LPML provide a direct cross validation-oriented error. Both values have relatively large magnitude (or, 29113.63 and -32668.16) due to the large sample size of the intersections (6,198) used in the study. To better explore the suitability of using the bivariate setting, the random effects of the two transportations modes were also collected. Their correlation and covariance are shown in Table 4. The statistically significant correlation coefficient signifies the strong positive correlation between the two types of crashes, corroborat- Table 4 Correlation and Covariance Matrix Between the Random Effects of Pedestrian and Vehicle Counts. | Observation. ID | b1 (Pedestrian) | b2 (Vehicle) | |-----------------|-----------------|--------------| | b1 (Pedestrian) | 1.000 | 0.899 | | b2 (Vehicle) | 0.344 | 1.000 | # Notes. - 1. The lower triangle of the matrix shows the covariance, while the upper triangle of the matrix shows the associated correlation coefficient. - 2. The bold font indicates the statistics are statistically significant at the significance level of 0.05. - 3. Correlation coefficients are listed in the diagonal of the matrix at the same time. - 4. Refer to Equation 6 for the definition of covariance of the two random effects. ing the importance of developing the joint models where the correlation between the two response variables was explicitly considered. ### 4. Conclusions Compared with the vehicle modes, much less research has been dedicated to the development of SPF for active transportation modes such as pedestrians. There are multiple reasons behind this, which include the dominant use of vehicle modes and the difficulty in obtaining exposure information of pedestrians. For enhancing current SPF development practice from perspectives of methodology and data usage, the paper aims to develop a joint SPF involving both pedestrian- and vehicle-related crashes. The following conclusions were drawn based on the research results: - 1. Compared with the base condition of four-legged intersection, offset intersection demonstrates better safety performance for both pedestrians and drivers. This finding suggests that implementation of offset intersection may help to reduce pedestrian-vehicle collisions significantly. - 2. For drivers only, the tee and wye intersections and those without right-turn channels tend to provide more safety benefits compared with the base conditions of intersections with fourleg and right turn channels, respectively. The better safety performance associated with the intersections without right turn channels is relatively contradictory, which warrants further investigations from other studies. - 3. There are much more statistically significant variables associated with pedestrians, suggesting that pedestrians are more sensitive to various intersection features than the vehicle drivers. - 4. The correlation and covariance matrix between the random effects of both pedestrian and vehicle counts demonstrate the existence of strong correlation, indicating the sensibility of using the bivariate models, which explicitly consider the correlation between the two modes. The aforementioned findings from this study reflect an improvement to current SPF development with mode-specific inputs of predictors and count model-estimated pedestrian exposure being used. Such improvement can help safety practitioners in reducing the pedestrian-related crashes by centralizing the resources toward potential improvements of intersection features. However, it is important to mention that the current findings are based on the empirical results obtained from the intersectionrelated crash data in California. Some of the model findings may not hold true when employing data at a different spatial level. The present paper employed Poisson-lognormal model for modeling the pedestrian-related crash data. Future studies may explore the other formulations such as zero-inflated models (both for the Poisson and negative binomial models), Gamma model, and Conway-Maxwell-Poisson model, which might lead to different findings from the present study. Moreover, only crashes of two modes are investigated. More modes involved might lead to different results, given more complex interrelationships are introduced among all crash outcomes. Finally, this study considered timely aggregated crashes only. The consideration of serial correlation among various years of crashes is also worth further investigation. #### Author contributions The authors confirm contribution to the paper as follows: study conception and design: Wen Cheng, Mankirat Singh, Jerry Kwong, Dean Samuelson; data collection: Menglu Cao, Dean Samuelson, Jerry Kwong; analysis and interpretation of results: Bengang Li, Wen Cheng; draft manuscript preparation: Mankirat Singh, Wen Cheng. 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Pedestrian crash trends and potential](http://refhub.elsevier.com/S0022-4375(21)00069-4/h0320) [countermeasures from around the world.](http://refhub.elsevier.com/S0022-4375(21)00069-4/h0320) Accident Analysis & Prevention, 44 [\(1\), 3–11](http://refhub.elsevier.com/S0022-4375(21)00069-4/h0320). - [Zheng, L., & Sayed, T. \(2019\). A full Bayes approach for traffic conflict-based before–](http://refhub.elsevier.com/S0022-4375(21)00069-4/h0325) [after safety evaluation using extreme value theory.](http://refhub.elsevier.com/S0022-4375(21)00069-4/h0325) Accident Analysis & [Prevention, 131](http://refhub.elsevier.com/S0022-4375(21)00069-4/h0325), 308–315. Mankirat Singh is a research associate in the Department of Civil Engineering at California State Polytechnic University, Pomona. His research focuses on highway safety and statistical modeling. Wen Cheng Ph.D., P.E., PTOE, is a professor in the Department of Civil Engineering at California State Polytechnic University, Pomona. He received a M.S. from University of Arizona, and a Ph.D. from Arizona State University. His research focuses on highway safety, statistical modeling, traffic signal design and traffic simulation. Dean Samuelson is the Chief of Traffic Safety Investigations Branch at Caltrans. He has extensive experience in traffic safety data analysis. Jerry Kwong is one project manager of Division of Research, Innovation and System Information in Caltrans. He has extensive experience in traffic safety and infrastructure design. Bengang Li M.S.C.E., is a research assistant in the Department of Civil Engineering at California State Polytechnic University, Pomona. His research focuses on highway safety and deep learning. Menglu Cao M.S.C.E., is a research assistant in the Department of Civil Engineering at California State Polytechnic University, Pomona. Her research focuses on highway safety and data mining. Yihua Li Ph.D., is an associate professor in Department of Logistics Engineering, Logistics and Traffic College, at Central South University of Forestry and Technology, Hunan, China . His research focuses on traffic safety, logistics, and big data.