Modelling car accidents’ victims using Zero-Inflated Negative Binomial regression
Cezary Kuźmowicz (473169) & Wojciech Hrycenko (473173)
2025-06-04
library(DBI)
library(RSQLite)
library(dplyr)
library(tidyr)
library(knitr)
library(lubridate)
library(ggplot2)
library(pscl)
library(hms)
library(pscl)
library(lmtest)
library(modelsummary)
library(broom)
library(kableExtra)1 Abstract
Understanding the dynamics of road traffic accidents requires models that can address both the frequency and severity of such events. Traditional count models often fail to capture the excess zeros present in accident datasets, where many crashes result in no injuries. To address this limitation, we apply a Zero-Inflated Negative Binomial (ZINB) regression to model the number of victims in road accidents. This approach distinguishes between two processes: the likelihood of a crash being non-injurious and the count of victims in injurious crashes. Using real-world traffic accident data from California, we test multiple hypotheses concerning driver characteristics (age, gender, alcohol use, race), vehicle attributes (age, insurance status), and environmental conditions (weather). Our findings reveal nuanced relationships: for example, older vehicles and poor weather increase the likelihood of injury crashes, while younger drivers are more frequently involved in non-injurious incidents. Interestingly, male drivers are more likely to be involved in crashes without injuries, while crashes involving female drivers tend to result in a higher number of victims. These insights highlight the value of dual-structure models in traffic safety research and support more targeted and evidence-based policymaking in road safety and driver education.
2 Introduction
Despite major advances in automotive safety and infrastructure development, road accidents remain a leading cause of injury and death globally (WHO, 2023). With millions of collisions occurring every year, it is not only of scientific interest to understand the factors that contribute to accident frequency and severity, but also a public health and economic priority. Identifying the characteristics of drivers, vehicles and environments that lead to more severe outcomes could lead to better road safety policies, insurance models and educational campaigns.
Traditional studies of road accidents have often focused on binary outcomes (whether an accident occurred or not) or have relied on simple counting models to assess the severity of injuries (Berhanu et. al., 2023). However, traffic accident data typically contains a high proportion of zeros - many crashes result in no injuries at all. This can lead to biased estimates and misleading conclusions. Moreover, some variables influence the likelihood of injury, while others affect the number of victims, conditional on an injury occurring. In such cases, standard modelling techniques struggle to capture this complexity
To address this issue, we apply a Zero-Inflated Negative Binomial (ZINB) regression model. This allows us to analyse the probability of injury and the severity of outcomes within a single framework. The zero-inflated structure handles the large number of non-injury crashes, while the negative binomial (count model) component adjusts for overdispersion in the number of victims (Yau et. al., 2003). This dual perspective enables us to form a deeper understanding of the impact of various factors, such as driver age and gender, vehicle age, alcohol use or insurance status on road accident outcomes.
Our aim is to contribute to a more comprehensive understanding of road safety risks by modelling both injury probability and injury count in a unified framework. The insights derived from this analysis can support more data-driven safety measures and better-informed policy decisions.
3 Literature Review
The history of the first car can be traced back to 1885, when German engineer Karl Benz created the first “modern” automobile. This machine was self-propelled, gasoline-powered, had four wheels and could be steered and stopped by the driver (Ratiu, 2003). Since that moment the car industry has revolutionised transportation all over the globe. With increasing popularity of this new method of travel, the frequency of car accidents also started to grow.
Drivers’ age is believed to be one of the major factors affecting the probability of a car crash. Lawrence (2002) conducted a study focused on this aspect. He analysed nearly 64 000 accidents in New South Wales, Australia, between 1996 and 2000 and also considered various distractions such as phone use. Results showed that injury-crash rates were highest for teenagers and young adults (16–24 years). Moreover, the pattern was “U-shaped” – crash rates were lowest for middle-age groups and again higher for the oldest drivers. When it came to phone use, a significant difference appeared only for the 25–29 group, whose crash risk while phoning was 2.4 times higher than without a phone. These results may be outdated – nowadays phones are much more popular across every age group. Not only driver’s age matters – the age of the vehicle also appears to have an impact. According to a study conducted by Australian researchers in 2003, cars built before 1984 (> 15 years) had a significantly greater chance of being involved in an injury-producing crash than those manufactured after 1994 (< 5 years). They calculated that older vehicles carried roughly triple the injury risk of younger ones (Blows et al., 2003). Furthermore, risk rose by 5 % for every additional year of a vehicle’s age.
Another highly important factor affecting accident risk is alcohol. Another Australian paper highlighted the extreme results of drinking even small amount of this substance. Drivers with 0.3 and 0.5‰ alcohol in blood had 3 times higher crash risk compared to fully sober driver (Connor et. al., 2004). That amount corresponds to only one or two drinks. Consuming more than two drinks within six hours raised the crash risk eight times. The greatest change was visible after breaking the law. If the legal limit of 0.5‰ was exceeded, the risk was 23 times higher compared to sober driver (Connor et. al., 2004). That study underlined the extreme effect of consuming alcohol before driving and its dangerous consequences not only for driver.
Sex differences have also been documented. In a 2021 Australian cohort study of young drivers, men had higher overall crash rates, whereas women showed greater odds of crash-related hospitalisation (Cullen et al., 2021). Because the cohort was followed for 13 years, authors were able to show that risk decreased with each additional year of driving experience. Influence of race, especially in the United States, is another topic often raised when comparing statistics. In a 2001 article, Braver (2001) computed passenger-vehicle deaths per 10 million trips among persons aged 25–64. Black men had a 48 % higher risk of dying in a crash than white men. Similar patterns were observed for women, though exact differences were not presented. Hispanic men also faced elevated risk (+26 %) but no significant difference was found between Hispanic and white women. Education modified these effects: men without a high-school diploma had 3.5 times higher risk than graduates, and women 2.8 times, with the highest risk recorded among whites without a diploma – indicating that low education may override racial patterns (Braver, 2001).
Car crash often indicates overwhelming economical costs for drivers. Insurance coverage can mitigate such costs, yet not all motorists purchase it. According to research from 2003, the uninsured drivers were far more likely to be involved in a crash that hospitalized or killed someone. The risk was 4.8 times higher when compared to insured drivers (Blows et. al., 2003). All previous factor related to individuals’ characteristics or their decisions. Weather is independent and contribute to higher risk of car accidents. Bad conditions, especially heavy rain, ice or poor visibility, significantly raise the number of crashes (Brijs et. al., 2008).
3.1 Hypotheses
After reviewing existing literature, we propose to test eight hypotheses:
- Young drivers (under 25) have higher risk of car crash with victims
- Vehicles older than 15 years are at higher risk of car crash with injuries
- Drivers who consumed alcohol have higher risk of car crash with victims
- Men have higher overall risk of car crash than women
- Women have higher risk of car crash with victims than men
- Non-white drivers have higher risk of car crash
- Uninsured drivers have higher risk of car crash with victims
- Bad weather conditions indicate higher risk of car crash
4 Data
We decided to use data from the California Highway Patrol which covers collisions from January 1st, 2001 until mid-December, 2020. The Statewide Integrated Traffic Records System (SWITRS) contains all crashes that were reported to CHP by local and governmental agencies.
The data have a hierarchical structure.
- The CRASH table contains information on each crash, one line per crash.
- The PARTY table contains information from all parties involved in the crash, one line per party. Parties are the major players in a traffic crash - drivers, pedestrians, bicyclists, and parked vehicles. The information includes personal descriptors and vehicle descriptors.
- The VICTIM table contains information about the victims - persons associated with each party. For example, a motorcyclist and his passenger are each a victim. Injury severity is included in the VICTIM table.
4.1 Loading Data from sqlite
In this section, we extract and prepare a sample dataset of traffic collisions from a local SQLite database (switrs.sqlite), focusing on drivers who were determined to be at fault. The steps include:
- Connecting to the SQLite database.
- Filtering the parties table to include only drivers at fault with complete demographic and vehicle-related information.
- Randomly sampling up to 100,000 such drivers, keeping only one driver per collision (case_id).
- Retrieving the corresponding collisions data for the selected drivers.
- Merging the driver and collision data into a single dataset.
- Displaying a preview of the resulting data in a formatted, scrollable table.
- Exporting the final merged dataset to a CSV file for further analysis.
# Connect to SQLite database
conn <- dbConnect(RSQLite::SQLite(), dbname = "switrs.sqlite")
# Query drivers who were at fault and have complete information
parties_query <- "
SELECT *
FROM parties
WHERE party_type = 'driver'
AND at_fault = 1
AND TRIM(party_age) != ''
AND TRIM(party_sex) != ''
AND TRIM(party_race) != ''
AND TRIM(cellphone_in_use) != ''
AND TRIM(financial_responsibility) != ''
AND TRIM(party_sobriety) != ''
AND TRIM(vehicle_make) != ''
AND TRIM(vehicle_year) != ''
"
parties_filtered <- dbGetQuery(conn, parties_query)
# Sample up to 100,000 drivers
set.seed(123)
parties_sample_raw <- parties_filtered %>%
sample_n(size = min(100000, nrow(.)))
# Keep one driver per case_id
parties_sample <- parties_sample_raw %>%
group_by(case_id) %>%
slice_sample(n = 1) %>%
ungroup()
# Get matching collisions data
case_ids <- paste0("'", parties_sample$case_id, "'", collapse = ", ")
collisions_query <- paste0("
SELECT *
FROM collisions
WHERE case_id IN (", case_ids, ")
")
collisions_subset <- dbGetQuery(conn, collisions_query)
# Merge datasets
df <- parties_sample %>%
left_join(collisions_subset, by = "case_id")
dbDisconnect(conn)| id | case_id | party_number | party_type | at_fault | party_sex | party_age | party_sobriety | party_drug_physical | direction_of_travel | party_safety_equipment_1 | party_safety_equipment_2 | financial_responsibility | hazardous_materials | cellphone_in_use | cellphone_use_type | school_bus_related | oaf_violation_code | oaf_violation_category | oaf_violation_section | oaf_violation_suffix | other_associate_factor_1 | other_associate_factor_2 | party_number_killed | party_number_injured | movement_preceding_collision | vehicle_year | vehicle_make | statewide_vehicle_type | chp_vehicle_type_towing | chp_vehicle_type_towed | party_race | jurisdiction | officer_id | reporting_district | chp_shift | population | county_city_location | county_location | special_condition | beat_type | chp_beat_type | city_division_lapd | chp_beat_class | beat_number | primary_road | secondary_road | distance | direction | intersection | weather_1 | weather_2 | state_highway_indicator | caltrans_county | caltrans_district | state_route | route_suffix | postmile_prefix | postmile | location_type | ramp_intersection | side_of_highway | tow_away | collision_severity | killed_victims | injured_victims | party_count | primary_collision_factor | pcf_violation_code | pcf_violation_category | pcf_violation | pcf_violation_subsection | hit_and_run | type_of_collision | motor_vehicle_involved_with | pedestrian_action | road_surface | road_condition_1 | road_condition_2 | lighting | control_device | chp_road_type | pedestrian_collision | bicycle_collision | motorcycle_collision | truck_collision | not_private_property | alcohol_involved | statewide_vehicle_type_at_fault | chp_vehicle_type_at_fault | severe_injury_count | other_visible_injury_count | complaint_of_pain_injury_count | pedestrian_killed_count | pedestrian_injured_count | bicyclist_killed_count | bicyclist_injured_count | motorcyclist_killed_count | motorcyclist_injured_count | primary_ramp | secondary_ramp | latitude | longitude | collision_date | collision_time | process_date |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 90 | 0000048 | 1 | driver | 1 | male | 41 | had not been drinking | NA | east | lap/shoulder harness used | NA | proof of insurance obtained | NA | 0 | cellphone not in use | NA | NA | NA | NA | NA | none apparent | NA | 0 | 0 | changing lanes | 1996 | ford | pickup or panel truck | pickups & panels | 00 | hispanic | 9575 | 9820 | NA | 1400 thru 2159 | 50000 to 100000 | 1902 | los angeles | 0 | chp state highway | interstate | NA | chp other | 065 | RT 210 | BALDWIN AV | 20 | east | 0 | clear | NA | 1 | los angeles | 7 | 210 | NA | R | 30.85 | highway | NA | eastbound | 0 | property damage only | 0 | 0 | 2 | vehicle code violation | NA | unsafe lane change | 21658 | A | not hit and run | sideswipe | other motor vehicle | no pedestrian involved | dry | normal | NA | daylight | none | 1 | 0 | 0 | 0 | 0 | 1 | NA | pickup or panel truck | pickups & panels | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | NA | NA | NA | NA | 2002-01-12 | 15:15:00 | 2002-05-28 |
| 427 | 0000227 | 1 | driver | 1 | male | 45 | had not been drinking | NA | south | lap/shoulder harness used | NA | proof of insurance obtained | NA | 0 | cellphone not in use | NA | NA | NA | NA | NA | none apparent | NA | 0 | 0 | proceeding straight | 1992 | chevrolet | passenger car | mini-vans | 00 | white | 9530 | 16779 | NA | 0600 thru 1359 | >250000 | 1941 | los angeles | 0 | chp state highway | interstate | NA | chp other | 072 | RT 710 | DEL AMO BL | 528 | north | 0 | clear | NA | 1 | los angeles | 7 | 710 | NA | NA | 10.72 | highway | NA | northbound | 0 | property damage only | 0 | 0 | 2 | vehicle code violation | NA | NA | NA | NA | not hit and run | rear end | other motor vehicle | no pedestrian involved | dry | normal | NA | daylight | none | 1 | 0 | 0 | 0 | 0 | 1 | NA | passenger car | mini-vans | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | NA | NA | NA | NA | 2002-02-05 | 08:20:00 | 2002-03-15 |
| 717 | 0000378 | 1 | driver | 1 | male | 18 | had not been drinking | NA | south | lap/shoulder harness used | NA | proof of insurance obtained | NA | 0 | cellphone not in use | NA | NA | NA | NA | NA | none apparent | NA | 0 | 0 | proceeding straight | 1998 | dodge | passenger car | passenger car, station | 00 | hispanic | 9530 | 12519 | NA | 0600 thru 1359 | >250000 | 1941 | los angeles | 0 | chp state highway | interstate | NA | chp other | 430 | RT 405 | BELLFLOWER BL | 136 | north | 0 | clear | NA | 1 | los angeles | 7 | 405 | NA | NA | 2.33 | highway | NA | southbound | 0 | property damage only | 0 | 0 | 2 | vehicle code violation | NA | speeding | 22350 | NA | not hit and run | rear end | other motor vehicle | no pedestrian involved | dry | normal | NA | daylight | none | 1 | 0 | 0 | 0 | 0 | 1 | NA | passenger car | passenger car, station | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | NA | NA | NA | NA | 2002-02-04 | 07:40:00 | 2002-06-19 |
| 1087 | 0000578 | 1 | driver | 1 | male | 36 | had not been drinking | NA | south | lap/shoulder harness used | NA | proof of insurance obtained | NA | 0 | cellphone not in use | NA | NA | NA | NA | NA | none apparent | NA | 0 | 0 | other | 1994 | ford | passenger car | passenger car, station | 00 | white | 9565 | 15286 | NA | 0600 thru 1359 | 25000 to 50000 | 1918 | los angeles | 0 | chp county roadline | county road line | NA | chp other | 021 | RT 405 | JEFFERSON BL | 50 | north | 0 | clear | NA | 1 | los angeles | 7 | 405 | NA | NA | 25.97 | highway | NA | southbound | 1 | property damage only | 0 | 0 | 1 | vehicle code violation | NA | improper turning | 22107 | NA | not hit and run | hit object | fixed object | no pedestrian involved | dry | normal | NA | daylight | none | 1 | 0 | 0 | 0 | 0 | 1 | NA | passenger car | passenger car, station | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | NA | NA | NA | NA | 2002-01-11 | 07:25:00 | 2002-06-19 |
| 1442 | 0000781 | 1 | driver | 1 | male | 24 | had not been drinking | NA | south | lap/shoulder harness used | NA | no proof of insurance obtained | NA | 0 | cellphone not in use | NA | NA | NA | NA | NA | none apparent | NA | 0 | 0 | proceeding straight | 1994 | nissan | passenger car | passenger car, station | 00 | hispanic | 9690 | 13096 | NA | 1400 thru 2159 | 25000 to 50000 | 3045 | orange | 0 | chp state highway | interstate | NA | chp other | 055 | RT 5 | EL TORO RD | 50 | south | 0 | clear | NA | 1 | orange | 12 | 5 | NA | NA | 18.68 | highway | NA | southbound | 0 | property damage only | 0 | 0 | 2 | vehicle code violation | NA | speeding | 22350 | NA | not hit and run | rear end | other motor vehicle | no pedestrian involved | dry | normal | NA | dark with street lights | none | 1 | 0 | 0 | 0 | 0 | 1 | NA | passenger car | passenger car, station | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | NA | NA | NA | NA | 2002-02-15 | 18:30:00 | 2002-05-28 |
| 1474 | 0000797 | 1 | driver | 1 | male | 51 | had not been drinking | NA | south | lap/shoulder harness used | NA | proof of insurance obtained | NA | 0 | cellphone not in use | NA | NA | NA | NA | NA | none apparent | NA | 0 | 0 | proceeding straight | 1991 | honda | passenger car | passenger car, station | 00 | white | 9530 | 13970 | NA | 0600 thru 1359 | >250000 | 1941 | los angeles | 0 | chp state highway | interstate | NA | chp other | 072 | RT 710 | PACIFIC AV | 200 | west | 0 | clear | NA | 1 | los angeles | 7 | 405 | NA | NA | 7.06 | ramp | ramp exit, last 50 feet | southbound | 1 | property damage only | 0 | 0 | 2 | vehicle code violation | NA | speeding | 22350 | NA | not hit and run | rear end | other motor vehicle | no pedestrian involved | dry | normal | NA | daylight | none | 1 | 0 | 0 | 0 | 0 | 1 | NA | passenger car | passenger car, station | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | TR | NA | NA | NA | 2002-02-05 | 09:10:00 | 2002-06-19 |
| 1710 | 0000923 | 1 | driver | 1 | male | 48 | had not been drinking | NA | south | lap/shoulder harness used | NA | proof of insurance obtained | NA | 0 | cellphone not in use | NA | NA | NA | NA | NA | none apparent | NA | 0 | 0 | entering traffic | 1992 | oldsmobile | passenger car | NA | NA | black | 101 | 44 | NA | not chp | 50000 to 100000 | 0101 | alameda | 0 | not chp | not chp | NA | not chp | NA | BROADWAY | SANTA CLARA AV | 200 | north | 0 | cloudy | NA | 0 | NA | NA | NA | NA | NA | NA | NA | NA | NA | 0 | property damage only | 0 | 0 | 2 | vehicle code violation | NA | unsafe starting or backing | 22106 | NA | not hit and run | sideswipe | other motor vehicle | no pedestrian involved | dry | normal | NA | daylight | none | 0 | 0 | 0 | 0 | 0 | 1 | NA | passenger car | NA | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | NA | NA | NA | NA | 2002-02-18 | 15:50:00 | 2002-03-16 |
| 2752 | 0001531 | 2 | driver | 1 | male | 43 | had not been drinking | NA | east | lap/shoulder harness used | NA | proof of insurance obtained | NA | 0 | cellphone not in use | NA | NA | NA | NA | NA | none apparent | NA | 0 | 0 | changing lanes | 1998 | volvo | truck or truck tractor with trailer | truck tractor | semi | hispanic | 9575 | 9307 | NA | 0600 thru 1359 | >250000 | 1942 | los angeles | 0 | chp state highway | state route | NA | chp other | 043 | RT 134 | SAN FERNANDO RD | 1000 | west | 0 | clear | NA | 1 | los angeles | 7 | 134 | NA | R | 5.72 | highway | NA | eastbound | 0 | property damage only | 0 | 0 | 3 | vehicle code violation | NA | unsafe lane change | 21658 | A | not hit and run | sideswipe | other motor vehicle | no pedestrian involved | dry | normal | NA | daylight | none | 1 | 0 | 0 | 0 | 1 | 1 | NA | truck or truck tractor with trailer | truck tractor | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | NA | NA | NA | NA | 2002-01-31 | 08:45:00 | 2002-05-24 |
| 2810 | 0001565 | 1 | driver | 1 | female | 55 | had not been drinking | NA | south | lap/shoulder harness used | NA | proof of insurance obtained | NA | 0 | cellphone not in use | NA | NA | NA | NA | NA | inattention | NA | 0 | 0 | proceeding straight | 1995 | dodge | other bus | NA | NA | white | 4003 | 5529 | 02 | not chp | 2500 to 10000 | 4003 | san luis obispo | 0 | not chp | not chp | NA | not chp | 001 | EMBARCADERO | FRONT ST | 60 | north | 0 | clear | NA | 0 | NA | NA | NA | NA | NA | NA | NA | NA | NA | 0 | property damage only | 0 | 0 | 2 | other improper driving | NA | other improper driving | NA | NA | not hit and run | sideswipe | parked motor vehicle | no pedestrian involved | dry | normal | NA | daylight | none | 0 | 0 | 0 | 0 | 0 | 1 | NA | other bus | NA | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | NA | NA | NA | NA | 2002-02-08 | 10:35:00 | 2002-03-28 |
| 2836 | 0001584 | 1 | driver | 1 | male | 45 | had not been drinking | NA | west | lap/shoulder harness used | NA | proof of insurance obtained | NA | 0 | cellphone not in use | NA | NA | NA | NA | NA | entering/leaving ramp | NA | 0 | 0 | changing lanes | 2001 | volvo | passenger car | passenger car, station | 00 | asian | 9340 | 013316 | NA | 1400 thru 2159 | 100000 to 250000 | 4314 | santa clara | 0 | chp county roadline | county road line | NA | chp other | 530 | MONTAGUE EXPWY | LAURELWOOD DR | 50 | west | 0 | clear | NA | 0 | NA | NA | NA | NA | NA | NA | NA | NA | NA | 0 | property damage only | 0 | 0 | 2 | vehicle code violation | NA | unsafe lane change | 21658 | A | not hit and run | sideswipe | other motor vehicle | no pedestrian involved | dry | normal | NA | daylight | none | 0 | 0 | 0 | 0 | 0 | 1 | NA | passenger car | passenger car, station | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | NA | NA | NA | NA | 2002-01-16 | 15:15:00 | 2002-03-25 |
4.2 Cleaning nulls
In this section, we clean the merged dataset by handling missing or incomplete values. The process involves:
- Loading the previously saved dataset of drivers at fault.
- Defining a helper function to compute the count and percentage of missing or empty values for each column.
- Generating a summary table of missing data across all variables and displaying the top 50 columns with the most missingness.
- Dropping columns with more than 40% missing or empty values to reduce noise and sparsity.
- Saving a version of the dataset with the retained columns.
- Recalculating and displaying the missing data statistics after column filtering.
- Removing all rows that still contain any missing or empty fields, ensuring the dataset is complete.
- Saving the final, fully cleaned dataset to a new CSV file.
- Printing the number of remaining observations after cleaning.
This prepares the dataset by ensuring a consistent and complete structure.
# Load dataset
df <- read.csv("driver_at_fault_sample.csv", stringsAsFactors = FALSE)
# Helper function: count and percent of missing or empty values
null_stats <- function(x) {
total <- length(x)
nulls <- sum(is.na(x) | trimws(x) == "")
percent <- round(nulls / total * 100, 2)
c(Count = nulls, Percent = percent)
}
# Compute missing value summary for all columns
null_summary_all <- as.data.frame(t(sapply(df, null_stats)))
null_summary_all$Column <- rownames(null_summary_all)
null_summary_all <- null_summary_all %>%
select(Column, everything()) %>%
arrange(desc(Percent))| Column | Count | Percent | |
|---|---|---|---|
| pcf_violation_code | pcf_violation_code | 99996 | 100.00 |
| oaf_violation_code | oaf_violation_code | 99991 | 99.99 |
| hazardous_materials | hazardous_materials | 99979 | 99.98 |
| route_suffix | route_suffix | 99882 | 99.88 |
| school_bus_related | school_bus_related | 99870 | 99.87 |
| road_condition_2 | road_condition_2 | 99540 | 99.54 |
| secondary_ramp | secondary_ramp | 99128 | 99.13 |
| oaf_violation_suffix | oaf_violation_suffix | 97611 | 97.61 |
| primary_ramp | primary_ramp | 97478 | 97.48 |
| other_associate_factor_2 | other_associate_factor_2 | 96815 | 96.82 |
| weather_2 | weather_2 | 96791 | 96.79 |
| party_drug_physical | party_drug_physical | 95701 | 95.70 |
| city_division_lapd | city_division_lapd | 95675 | 95.67 |
| ramp_intersection | ramp_intersection | 94034 | 94.03 |
| alcohol_involved | alcohol_involved | 89462 | 89.46 |
| postmile_prefix | postmile_prefix | 89244 | 89.24 |
| oaf_violation_category | oaf_violation_category | 88599 | 88.60 |
| oaf_violation_section | oaf_violation_section | 88548 | 88.55 |
| reporting_district | reporting_district | 73066 | 73.07 |
| caltrans_county | caltrans_county | 66926 | 66.93 |
| caltrans_district | caltrans_district | 66926 | 66.93 |
| state_route | state_route | 66926 | 66.93 |
| postmile | postmile | 66926 | 66.93 |
| location_type | location_type | 66926 | 66.93 |
| side_of_highway | side_of_highway | 66927 | 66.93 |
| pcf_violation_subsection | pcf_violation_subsection | 64524 | 64.52 |
| latitude | latitude | 58132 | 58.13 |
| longitude | longitude | 58132 | 58.13 |
| chp_vehicle_type_towed | chp_vehicle_type_towed | 45782 | 45.78 |
| direction | direction | 19813 | 19.81 |
| chp_vehicle_type_towing | chp_vehicle_type_towing | 8633 | 8.63 |
| chp_vehicle_type_at_fault | chp_vehicle_type_at_fault | 8633 | 8.63 |
| party_safety_equipment_2 | party_safety_equipment_2 | 7845 | 7.85 |
| statewide_vehicle_type | statewide_vehicle_type | 6128 | 6.13 |
| statewide_vehicle_type_at_fault | statewide_vehicle_type_at_fault | 6128 | 6.13 |
| beat_number | beat_number | 5320 | 5.32 |
| party_safety_equipment_1 | party_safety_equipment_1 | 1687 | 1.69 |
| other_associate_factor_1 | other_associate_factor_1 | 1364 | 1.36 |
| pcf_violation | pcf_violation | 1011 | 1.01 |
| pcf_violation_category | pcf_violation_category | 631 | 0.63 |
| type_of_collision | type_of_collision | 467 | 0.47 |
| tow_away | tow_away | 456 | 0.46 |
| intersection | intersection | 451 | 0.45 |
| road_surface | road_surface | 448 | 0.45 |
| road_condition_1 | road_condition_1 | 361 | 0.36 |
| control_device | control_device | 298 | 0.30 |
| motor_vehicle_involved_with | motor_vehicle_involved_with | 274 | 0.27 |
| direction_of_travel | direction_of_travel | 261 | 0.26 |
| officer_id | officer_id | 263 | 0.26 |
| lighting | lighting | 261 | 0.26 |
# Drop columns with more than 40% missing data
columns_to_keep <- null_summary_all %>%
filter(Percent <= 40) %>%
pull(Column)
df_cleaned <- df[, columns_to_keep]
# Save cleaned dataset (columns only)
write.csv(df_cleaned, "driver_at_fault_sample_cleaned.csv", row.names = FALSE)
# Recalculate missing stats after dropping columns
null_summary_cleaned <- as.data.frame(t(sapply(df_cleaned, null_stats)))
null_summary_cleaned$Column <- rownames(null_summary_cleaned)
null_summary_cleaned <- null_summary_cleaned %>%
select(Column, everything()) %>%
arrange(desc(Percent))| Column | Count | Percent | |
|---|---|---|---|
| direction | direction | 19813 | 19.81 |
| chp_vehicle_type_towing | chp_vehicle_type_towing | 8633 | 8.63 |
| chp_vehicle_type_at_fault | chp_vehicle_type_at_fault | 8633 | 8.63 |
| party_safety_equipment_2 | party_safety_equipment_2 | 7845 | 7.85 |
| statewide_vehicle_type | statewide_vehicle_type | 6128 | 6.13 |
| statewide_vehicle_type_at_fault | statewide_vehicle_type_at_fault | 6128 | 6.13 |
| beat_number | beat_number | 5320 | 5.32 |
| party_safety_equipment_1 | party_safety_equipment_1 | 1687 | 1.69 |
| other_associate_factor_1 | other_associate_factor_1 | 1364 | 1.36 |
| pcf_violation | pcf_violation | 1011 | 1.01 |
| pcf_violation_category | pcf_violation_category | 631 | 0.63 |
| type_of_collision | type_of_collision | 467 | 0.47 |
| tow_away | tow_away | 456 | 0.46 |
| intersection | intersection | 451 | 0.45 |
| road_surface | road_surface | 448 | 0.45 |
| road_condition_1 | road_condition_1 | 361 | 0.36 |
| control_device | control_device | 298 | 0.30 |
| motor_vehicle_involved_with | motor_vehicle_involved_with | 274 | 0.27 |
| direction_of_travel | direction_of_travel | 261 | 0.26 |
| officer_id | officer_id | 263 | 0.26 |
| lighting | lighting | 261 | 0.26 |
| weather_1 | weather_1 | 180 | 0.18 |
| collision_time | collision_time | 163 | 0.16 |
| jurisdiction | jurisdiction | 135 | 0.14 |
| movement_preceding_collision | movement_preceding_collision | 130 | 0.13 |
| population | population | 16 | 0.02 |
| pedestrian_action | pedestrian_action | 24 | 0.02 |
| chp_beat_class | chp_beat_class | 6 | 0.01 |
| state_highway_indicator | state_highway_indicator | 12 | 0.01 |
| primary_collision_factor | primary_collision_factor | 8 | 0.01 |
| id | id | 0 | 0.00 |
| case_id | case_id | 0 | 0.00 |
| party_number | party_number | 0 | 0.00 |
| party_type | party_type | 0 | 0.00 |
| at_fault | at_fault | 0 | 0.00 |
| party_sex | party_sex | 0 | 0.00 |
| party_age | party_age | 0 | 0.00 |
| party_sobriety | party_sobriety | 0 | 0.00 |
| financial_responsibility | financial_responsibility | 0 | 0.00 |
| cellphone_in_use | cellphone_in_use | 0 | 0.00 |
| cellphone_use_type | cellphone_use_type | 0 | 0.00 |
| party_number_killed | party_number_killed | 0 | 0.00 |
| party_number_injured | party_number_injured | 0 | 0.00 |
| vehicle_year | vehicle_year | 0 | 0.00 |
| vehicle_make | vehicle_make | 0 | 0.00 |
| party_race | party_race | 0 | 0.00 |
| chp_shift | chp_shift | 0 | 0.00 |
| county_city_location | county_city_location | 0 | 0.00 |
| county_location | county_location | 0 | 0.00 |
| special_condition | special_condition | 0 | 0.00 |
# Drop rows with any missing or empty values
df_final <- df_cleaned %>%
filter(across(everything(), ~ !(is.na(.) | trimws(.) == "")))
# Save final dataset
write.csv(df_final, "driver_at_fault_sample_cleaned_no_na.csv", row.names = FALSE)Number of rows after full cleanup: 65552 ### Feature Pruning and Engineering for Modeling In this section, we finalize our feature selection and perform additional feature engineering to prepare the dataset for modeling. Specifically:
- We load the pre-cleaned dataset containing no missing values.
- A broad set of columns is removed, including identifiers, redundant location codes, officer/jurisdiction data, detailed injury/fatality breakdowns, and fields that are either irrelevant, redundant, or too granular for our modeling goals.
- The collision date is converted to a Date object and used to derive the vehicle’s age at the time of the crash.
- Injury and fatality counts are aggregated into a single total_victims variable.
- The season of the year in which each collision occurred is inferred based on the collision date.
- Vehicle make is used to infer the origin region of the vehicle (e.g., USA, Japan, Korea, Europe, etc.).
- The collision time is parsed to extract the hour and categorize it into a general time of day (e.g., Morning, Afternoon, Night).
- The processed dataset is saved for downstream modeling steps, and the final number of features (columns) is printed.
- These transformations help reduce dimensionality, engineer informative features, and ensure consistency across variables.
| chp_vehicle_type_at_fault | party_safety_equipment_1 | type_of_collision | road_surface | motor_vehicle_involved_with | direction_of_travel | lighting | weather_1 | movement_preceding_collision | population | party_sex | party_age | party_sobriety | financial_responsibility | cellphone_in_use | party_race | county_location | chp_beat_type | party_count | hit_and_run | car_age | total_victims | season | year | region | time_of_day |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| passenger car, station | lap/shoulder harness used | broadside | dry | other motor vehicle | south | daylight | clear | proceeding straight | 25000 to 50000 | female | 22 | had not been drinking | no proof of insurance obtained | 0 | hispanic | san mateo | us highway | 3 | not hit and run | 4 | 0 | Winter | 2002 | Japan | Morning |
| passenger car, station | lap/shoulder harness used | broadside | wet | other motor vehicle | south | daylight | raining | proceeding straight | unincorporated | male | 26 | had not been drinking | proof of insurance obtained | 0 | other | santa cruz | state route | 2 | not hit and run | 8 | 0 | Winter | 2002 | USA | Afternoon |
| passenger car, station | lap/shoulder harness used | hit object | dry | fixed object | west | dark with street lights | clear | proceeding straight | >250000 | male | 17 | had not been drinking | no proof of insurance obtained | 0 | asian | los angeles | state route | 1 | not hit and run | 7 | 0 | Spring | 2002 | Japan | Night |
| passenger car, station | air bag deployed | sideswipe | dry | other motor vehicle | north | daylight | clear | stopped | unincorporated | female | 36 | had not been drinking | proof of insurance obtained | 0 | hispanic | los angeles | county road line | 2 | not hit and run | 13 | 0 | Fall | 2002 | USA | Afternoon |
| passenger car, station | lap/shoulder harness used | rear end | dry | other motor vehicle | north | daylight | clear | proceeding straight | unincorporated | male | 21 | had not been drinking | proof of insurance obtained | 0 | white | tulare | county road area | 2 | not hit and run | 1 | 0 | Winter | 2003 | Japan | Morning |
| passenger car, station | lap/shoulder harness used | rear end | dry | parked motor vehicle | west | dark with no street lights | clear | backing | unincorporated | male | 22 | had not been drinking | proof of insurance obtained | 0 | white | kern | state route | 2 | not hit and run | 1 | 0 | Spring | 2003 | USA | Evening |
| passenger car, station | lap belt used | sideswipe | dry | other motor vehicle | south | daylight | clear | changing lanes | 50000 to 100000 | male | 46 | had been drinking, under influence | no proof of insurance obtained | 0 | white | san diego | interstate | 2 | not hit and run | 37 | 1 | Summer | 2003 | USA | Afternoon |
| passenger car, station | not required | hit object | dry | fixed object | north | daylight | clear | ran off road | unincorporated | male | 19 | had been drinking, under influence | proof of insurance obtained | 0 | white | el dorado | county road line | 1 | not hit and run | 15 | 0 | Summer | 2003 | USA | Evening |
| passenger car, station | air bag not deployed | rear end | dry | other motor vehicle | south | daylight | clear | changing lanes | >250000 | female | 31 | had not been drinking | proof of insurance obtained | 0 | black | los angeles | interstate | 2 | not hit and run | 12 | 0 | Summer | 2003 | Japan | Afternoon |
| passenger car, station | air bag not deployed | sideswipe | dry | other motor vehicle | south | daylight | clear | passing other vehicle | unincorporated | male | 38 | had been drinking, not under influence | proof of insurance obtained | 0 | asian | los angeles | interstate | 2 | not hit and run | 1 | 0 | Summer | 2003 | Japan | Afternoon |
Number of columns after pruning: 26
5 Method/Model
This section focuses on modeling the number of victims in traffic accidents using count regression techniques. It begins by preparing the dataset and examining the distribution of the target variable, total_victims, including an assessment of zero inflation. A custom score test is implemented to test for excess zeros beyond the Poisson assumption.
Four models are fitted to the data:
- Poisson Regression
- Negative Binomial Regression
- Zero-Inflated Poisson (ZIP)
- Zero-Inflated Negative Binomial (ZINB)
Likelihood ratio tests are used to compare nested models, and model fit is evaluated using AIC and log-likelihood criteria. The results guide the choice of the most appropriate model for handling both overdispersion and zero inflation in the count outcome.
library(MASS)
# Load the dataset
df <- read.csv("driver_at_fault_sample_final.csv", stringsAsFactors = FALSE)
df <- df[1:10000, ]
# Ensure the 'total_victims' column is numeric
df$total_victims <- as.numeric(df$total_victims)
# Calculate the count and percentage of zeros in 'total_victims'
zero_count <- sum(df$total_victims == 0, na.rm = TRUE)
total_count <- sum(!is.na(df$total_victims))
zero_percent <- round(100 * zero_count / total_count, 2)Number of zeros: 6 438 out of 10 000 observations (64.38%)
lr_test <- function(model1, model2) {
lr_stat <- 2 * (logLik(model2)[1] - logLik(model1)[1])
df_diff <- attr(logLik(model2), "df") - attr(logLik(model1), "df")
p_val <- pchisq(lr_stat, df = df_diff, lower.tail = FALSE)
cat("LR test comparing models:\n")
cat(" Statistic =", round(lr_stat, 4), "\n")
cat(" Degrees of freedom =", df_diff, "\n")
cat(" p-value =", p_val, "\n\n")
return(list(statistic = lr_stat, df = df_diff, p.value = p_val))
}
# Perform zero-inflation test on 'total_victims'
zero_test_result <- zero.test(df$total_victims)
knitr::kable(zero_test_result[, 1:4], caption = "Zero Inflation Test Result", align = c("l", "r", "r", "r"))| Test | Statistic | DF | P-value |
|---|---|---|---|
| Score test for zero inflation | 577.4755 | 1 | 1.33e-127 |
| Test | Statistic | DF | P-value |
|---|---|---|---|
| Score test for zero inflation | 577.4755 | 1 | 1.33e-127 |
— Interpretation of zero.test result — Reject H0: Evidence of zero inflation beyond the Poisson model. Consider zero-inflated models (ZIP or ZINB).
5.1 Fitting Models
# Creating 4 general models for comparison
# Poisson
poisson_model <- glm(total_victims ~ road_surface + direction_of_travel + lighting + weather_1 + party_sex + party_age + party_sobriety + financial_responsibility +
cellphone_in_use + party_race + + party_count + hit_and_run + car_age + season + year + region + time_of_day + chp_beat_type + chp_vehicle_type_at_fault + party_safety_equipment_1
+ type_of_collision + county_location + population + motor_vehicle_involved_with + movement_preceding_collision, family = "poisson", data = df)
# Negative Binomial
nb_model <- glm.nb(total_victims ~ road_surface + direction_of_travel + lighting + weather_1 + party_sex + party_age + party_sobriety + financial_responsibility +
cellphone_in_use + party_race + + party_count + hit_and_run + car_age + season + year + region + time_of_day + chp_beat_type + chp_vehicle_type_at_fault + party_safety_equipment_1
+ type_of_collision + county_location + population + motor_vehicle_involved_with + movement_preceding_collision, data = df)
# ZIP
zip_model <- zeroinfl(total_victims ~ road_surface + direction_of_travel + lighting + weather_1 + party_sex + party_age + party_sobriety + financial_responsibility +
cellphone_in_use + party_race + + party_count + hit_and_run + car_age + season + year + region + time_of_day + chp_beat_type + chp_vehicle_type_at_fault + party_safety_equipment_1
+ type_of_collision + county_location + population + motor_vehicle_involved_with + movement_preceding_collision, data = df, dist = "poisson")
# ZINB
zinb_model <- zeroinfl(total_victims ~ road_surface + direction_of_travel + lighting + weather_1 + party_sex + party_age + party_sobriety + financial_responsibility
+ cellphone_in_use + party_race + party_count + hit_and_run + car_age + season + year + region + time_of_day + chp_beat_type + chp_vehicle_type_at_fault + party_safety_equipment_1
+ type_of_collision + county_location + population + motor_vehicle_involved_with + movement_preceding_collision, data = df, dist = "negbin")5.2 Models Comparision
| Comparison | Statistic | DF | p.value |
|---|---|---|---|
| Poisson vs Negative Binomial | 185.761 | 1 | 2.676615e-42 |
| Poisson vs Zero-Inflated Poisson | 663.473 | 200 | 4.681003e-51 |
| Negative Binomial vs ZINB | 486.163 | 200 | 7.509073e-26 |
| ZIP vs ZINB | 8.451 | 1 | 3.647446e-03 |
Interpretation:
- Poisson vs Negative Binomial:
- The Likelihood Ratio (LR) statistic is 185.76 with 1 degree of freedom.
- p < 0.0001 → This is a highly significant result.
- Interpretation: The Negative Binomial model provides a significantly better fit than the Poisson model. This suggests overdispersion is present in the data (variance > mean), which Poisson cannot handle properly.
- Poisson vs Zero-Inflated Poisson (ZIP):
- LR statistic = 663.47, df = 200, p < 0.0001.
- Interpretation: The ZIP model is significantly better than the standard Poisson. This indicates that there is an excess of zero counts in the data that ZIP accounts for effectively.
- Negative Binomial vs Zero-Inflated Negative Binomial (ZINB):
- LR statistic = 486.16, df = 200, p < 0.0001.
- Interpretation: The ZINB model outperforms the standard Negative Binomial model. So, even with overdispersion addressed, there’s still zero inflation that needs to be modeled.
- ZIP vs ZINB:
- LR statistic = 8.45, df = 1, p ≈ 0.0036.
- Interpretation: The ZINB model is also significantly better than the ZIP model. That means both overdispersion and zero inflation are present in the data, and ZINB is the most appropriate model among those tested.
Conclusion:
- Based on all comparisons, the Zero-Inflated Negative Binomial (ZINB) model provides the best fit to your data, as it handles both excess zeros and overdispersion, which are not adequately addressed by simpler models like Poisson, NB, or ZIP.
| Model | AIC | Log-Likelihood | |
|---|---|---|---|
| ZINB | ZINB | 18425.64 | -8811.82 |
| ZIP | ZIP | 18432.09 | -8816.04 |
| NegBin | NegBin | 18511.80 | -9054.90 |
| Poisson | Poisson | 18695.56 | -9147.78 |
Interpretation:
- ZINB (Zero-Inflated Negative Binomial):
- Lowest AIC (18425.64) and highest log-likelihood (-8811.82) among all models.
- Interpretation: This is the best-fitting model according to both AIC and log-likelihood. It captures both overdispersion and excess zeros effectively.
- ZIP (Zero-Inflated Poisson):
- Second-best in terms of AIC and log-likelihood.
- Interpretation: While it accounts for zero inflation, it does not handle overdispersion as well as ZINB. Still, it performs better than the standard Poisson and Negative Binomial models.
- Negative Binomial (NegBin):
- Has a lower AIC than Poisson, indicating improvement when accounting for overdispersion, but still worse than ZIP and ZINB.
- Interpretation: Handles overdispersion, but fails to account for excess zeros.
- Poisson:
- Worst model with the highest AIC (18695.56) and lowest log-likelihood (-9147.78).
- Interpretation: It fails to account for both overdispersion and zero inflation, making it unsuitable for this dataset.
Conclusion:
- Based on both AIC and log-likelihood, the ZINB model is clearly the best choice. It provides the most accurate representation of the data, outperforming all other models in terms of goodness-of-fit.
| Name | df |
| Number of rows | 65000 |
| Number of columns | 26 |
| _______________________ | |
| Column type frequency: | |
| character | 20 |
| numeric | 6 |
| ________________________ | |
| Group variables | None |
Variable type: character
| skim_variable | n_missing | complete_rate | min | max | empty | n_unique | whitespace |
|---|---|---|---|---|---|---|---|
| chp_vehicle_type_at_fault | 0 | 1 | 8 | 91 | 0 | 58 | 0 |
| party_safety_equipment_1 | 0 | 1 | 5 | 40 | 0 | 19 | 0 |
| type_of_collision | 0 | 1 | 5 | 10 | 0 | 8 | 0 |
| road_surface | 0 | 1 | 3 | 8 | 0 | 4 | 0 |
| motor_vehicle_involved_with | 0 | 1 | 1 | 30 | 0 | 12 | 0 |
| direction_of_travel | 0 | 1 | 4 | 5 | 0 | 4 | 0 |
| lighting | 0 | 1 | 8 | 39 | 0 | 5 | 0 |
| weather_1 | 0 | 1 | 3 | 7 | 0 | 7 | 0 |
| movement_preceding_collision | 0 | 1 | 5 | 26 | 0 | 18 | 0 |
| population | 0 | 1 | 5 | 16 | 0 | 8 | 0 |
| party_sex | 0 | 1 | 4 | 6 | 0 | 2 | 0 |
| party_sobriety | 0 | 1 | 14 | 38 | 0 | 6 | 0 |
| financial_responsibility | 0 | 1 | 14 | 35 | 0 | 4 | 0 |
| party_race | 0 | 1 | 5 | 8 | 0 | 5 | 0 |
| county_location | 0 | 1 | 4 | 15 | 0 | 58 | 0 |
| chp_beat_type | 0 | 1 | 7 | 23 | 0 | 8 | 0 |
| hit_and_run | 0 | 1 | 6 | 15 | 0 | 3 | 0 |
| season | 0 | 1 | 4 | 6 | 0 | 4 | 0 |
| region | 0 | 1 | 3 | 6 | 0 | 6 | 0 |
| time_of_day | 0 | 1 | 5 | 9 | 0 | 4 | 0 |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| party_age | 0 | 1 | 36.53 | 16.20 | 0 | 23 | 32 | 47 | 121 | ▅▇▃▁▁ |
| cellphone_in_use | 0 | 1 | 0.02 | 0.15 | 0 | 0 | 0 | 0 | 1 | ▇▁▁▁▁ |
| party_count | 0 | 1 | 1.98 | 0.77 | 1 | 2 | 2 | 2 | 10 | ▇▂▁▁▁ |
| car_age | 0 | 1 | 9.44 | 6.84 | 0 | 4 | 9 | 14 | 89 | ▇▁▁▁▁ |
| total_victims | 0 | 1 | 0.55 | 0.89 | 0 | 0 | 0 | 1 | 20 | ▇▁▁▁▁ |
| year | 0 | 1 | 2012.39 | 4.91 | 2002 | 2008 | 2013 | 2017 | 2021 | ▃▇▆▇▆ |
5.3 Sample values from columns
| Sample_1 | Sample_2 | Sample_3 | Sample_4 | Sample_5 | |
|---|---|---|---|---|---|
| chp_vehicle_type_at_fault | passenger car, station | mini-vans | motorcycle | pickups & panels | two axle truck |
| party_safety_equipment_1 | lap/shoulder harness used | air bag deployed | lap belt used | not required | air bag not deployed |
| type_of_collision | broadside | hit object | sideswipe | rear end | overturned |
| road_surface | dry | wet | slippery | snowy | NA |
| motor_vehicle_involved_with | other motor vehicle | fixed object | parked motor vehicle | non-collision | train |
| direction_of_travel | south | west | north | east | NA |
| lighting | daylight | dark with street lights | dark with no street lights | dusk or dawn | dark with street lights not functioning |
| weather_1 | clear | raining | cloudy | other | fog |
| movement_preceding_collision | proceeding straight | stopped | backing | changing lanes | ran off road |
| population | 25000 to 50000 | unincorporated | >250000 | 50000 to 100000 | 100000 to 250000 |
| party_sex | female | male | NA | NA | NA |
| party_age | 22 | 26 | 17 | 36 | 21 |
| party_sobriety | had not been drinking | had been drinking, under influence | had been drinking, not under influence | impairment unknown | had been drinking, impairment unknown |
| financial_responsibility | no proof of insurance obtained | proof of insurance obtained | not applicable | officer called away before obtained | NA |
| cellphone_in_use | 0 | 1 | NA | NA | NA |
| party_race | hispanic | other | asian | white | black |
| county_location | san mateo | santa cruz | los angeles | tulare | kern |
| chp_beat_type | us highway | state route | county road line | county road area | interstate |
| party_count | 3 | 2 | 1 | 4 | 9 |
| hit_and_run | not hit and run | misdemeanor | felony | NA | NA |
| car_age | 4 | 8 | 7 | 13 | 1 |
| total_victims | 0 | 1 | 4 | 2 | 3 |
| season | Winter | Spring | Fall | Summer | NA |
| year | 2002 | 2003 | 2004 | 2005 | 2006 |
| region | Japan | USA | Korea | Europe | Other |
| time_of_day | Morning | Afternoon | Night | Evening | NA |
| Vehicle.Types.at.Fault |
|---|
| all terrain vehicle |
| ambulance |
| dune buggy |
| farm labor transporter |
| fire truck |
| fork lift |
| general public paratransit |
| go-ped, zip electric scooter, motoboard |
| hwy. construction equip. |
| implement of husbandry |
| low speed vehicle |
| mini-vans |
| misc. motor vehicle (snowmobile, golf cart) |
| mobile equipment |
| motor driven |
| motor home |
| motor home > 40 feet |
| motorcycle |
| motorized bicycle |
| non-commercial bus |
| other commercial |
| paratransit |
| passenger car, station |
| passenger car, station wagon, jeep: hazardous material |
| passenger car, station wagon, jeep: hazardous waste or hazardous waste/material combination |
| pickup and camper: hazardous material |
| pickup w/camper |
| pickups & panels |
| pickups and panels: hazardous material |
| pickups and panels: hazardous waste or hazardous waste/material combination |
| police car |
| police motorcycle |
| public transit authority |
| school bus contractual type i |
| school bus contractual type ii |
| school bus private type i |
| school bus private type ii |
| school bus public type i |
| school bus public type ii |
| school pupil activity bus type i |
| school pupil activity bus type ii |
| sport utility vehicle |
| three-axle tank truck: hazardous material |
| three-axle tow truck |
| three axle tank truck |
| three or more axle truck |
| three or more axle truck: hazardous material |
| three or more axle truck: hazardous waste or hazardous waste/material combination |
| tour bus |
| truck tractor |
| truck tractor: hazardous material |
| two-axle tank truck: hazardous waste or hazardous waste/material combination |
| two-axle tow truck |
| two-axle truck: hazardous material |
| two-axle truck: hazardous waste or hazardous waste/material combination |
| two axle tank truck |
| two axle truck |
| youth bus |
- proceeding straight
- stopped
- backing
- changing lanes
- ran off road
- passing other vehicle
- entering traffic
- other
- crossed into opposing lane
- making right turn
- making left turn
- other unsafe turning
- slowing/stopping
- merging
- making u-turn
- traveling wrong way
- parking maneuver
- parked
# aggregating movement into more general groups
df$move_general <- with(df, case_when(
movement_preceding_collision %in% c("proceeding straight", "entering traffic", "merging") ~ "Straight Travel",
movement_preceding_collision %in% c("making right turn", "making left turn", "making u-turn", "other unsafe turning", "other") ~ "Turning",
movement_preceding_collision %in% c("changing lanes", "passing other vehicle", "crossed into opposing lane") ~ "Lane Change/Passing",
movement_preceding_collision %in% c("slowing/stopping", "stopped", "backing", "parking maneuver", "parked") ~ "Stopping/Slowing",
movement_preceding_collision %in% c("ran off road", "traveling wrong way") ~ "Loss of Control/Irregular",
TRUE ~ "Other"
))
df$move_general <- factor(df$move_general)- lap/shoulder harness used
- air bag deployed
- lap belt used
- not required
- air bag not deployed
- other
- none in vehicle
- passenger, motorcycle helmet used
- unknown
- driver, motorcycle helmet used
- passive restraint not used
- shoulder harness not used
- no child restraint in vehicle
- lap/shoulder harness not used
- shoulder harness used
- passive restraint used
- child restraint in vehicle, use unknown
- child restraint in vehicle, improper use
- lap belt not used
# aggregating safety equipment into air_bag variable
df$air_bag <- factor(
ifelse(df$party_safety_equipment_1 == "air bag deployed", "Yes",
ifelse(df$party_safety_equipment_1 == "air bag not deployed", "No", "Other safety")),
levels = c("Yes", "No", "Other safety")
)| Air Bag | Count |
|---|---|
| Yes | 14 301 |
| No | 44 601 |
| Other safety | 6 098 |
- clear
- raining
- cloudy
- other
- fog
- snowing
- wind
# aggregating weather into more general groups
df$weather <- factor(
ifelse(df$weather_1 == "clear", "clear_sky",
ifelse(df$weather_1 %in% c("raining", "snowing", "fog"), "harder_conditions", "other"))
)| Weather | Count |
|---|---|
| clear_sky | 52 511 |
| harder_conditions | 2 713 |
| other | 9 776 |
- had not been drinking
- had been drinking, under influence
- had been drinking, not under influence
- impairment unknown
- had been drinking, impairment unknown
- not applicable
# aggregating sobriety into more general groups
df$sobriety <- factor(
ifelse(df$party_sobriety == "had not been drinking", "not drinking",
ifelse(df$party_sobriety %in% c(
"had been drinking, under influence",
"had been drinking, not under influence",
"had been drinking, impairment unknown"
), "drinking", "other"))
)| Sobriety | Count |
|---|---|
| drinking | 7 004 |
| not drinking | 55 988 |
| other | 2 008 |
- dry
- wet
- slippery
- snowy
# aggregating road surface into more general groups
df$surface_dry <- ifelse(df$road_surface == "dry", "dry", "non-dry")
df$surface_dry <- factor(df$surface_dry)# converting age into 2 intervals: under and over 25 years old
df$age_group <- cut(
df$party_age,
breaks = c(-Inf, 24, Inf),
labels = c("under 25", "over 25"),
right = TRUE
)| Column | Missing_Values |
|---|---|
| chp_vehicle_type_at_fault | 0 |
| party_safety_equipment_1 | 0 |
| type_of_collision | 0 |
| road_surface | 0 |
| motor_vehicle_involved_with | 0 |
| direction_of_travel | 0 |
| lighting | 0 |
| weather_1 | 0 |
| movement_preceding_collision | 0 |
| population | 0 |
| party_sex | 0 |
| party_age | 0 |
| party_sobriety | 0 |
| financial_responsibility | 0 |
| cellphone_in_use | 0 |
| party_race | 0 |
| county_location | 0 |
| chp_beat_type | 0 |
| party_count | 0 |
| hit_and_run | 0 |
| car_age | 0 |
| total_victims | 0 |
| season | 0 |
| year | 0 |
| region | 0 |
| time_of_day | 0 |
| vehicle_general | 0 |
| move_general | 0 |
| air_bag | 0 |
| weather | 0 |
| sobriety | 0 |
| surface_dry | 0 |
| age_group | 0 |
# aggregating weather into 2 seasons
df$LA_season <- ifelse(df$season %in% c("Winter", "Spring"), "Wet", "Dry")
df$LA_season <- factor(df$LA_season, levels = c("Dry", "Wet"))| Season | Count |
|---|---|
| Fall | 17 018 |
| Spring | 15 745 |
| Summer | 16 064 |
| Winter | 16 173 |
- hispanic
- other
- asian
- white
- black
# aggregating race into white and non-white
df$race_group <- ifelse(df$party_race == "white", "white", "non-white")
df$race_group <- factor(df$race_group, levels = c("white", "non-white"))- no proof of insurance obtained
- proof of insurance obtained
- not applicable
- officer called away before obtained
# aggregating insurance into 2 groups
df$financial_responsibility_bi <- ifelse(df$financial_responsibility == "proof of insurance obtained",
"proof", "no or unknown proof")
df$financial_responsibility_bi <- factor(df$financial_responsibility_bi, levels = c("proof", "no or unknown proof"))# aggregating car age into 2 intervals
df$car_age_group <- ifelse(df$car_age <= 15, "under_15", "over_15")
df$car_age_group <- factor(df$car_age_group, levels = c("under_15", "over_15"))# aggregating lighting into 2 groups
df$lighting_simple <- ifelse(df$lighting == "daylight", "daylight", "dark")
df$lighting_simple <- factor(df$lighting_simple, levels = c("daylight", "dark"))5.4 Interactions
# sex x race
df$sex_race <- interaction(df$party_sex, df$race_group, drop = TRUE)| Sex x Race | Count |
|---|---|
| female.white | 10 264 |
| male.white | 16 389 |
| female.non-white | 12 537 |
| male.non-white | 25 810 |
# time of the day x weather
df$timeofday_weather <- interaction(df$time_of_day, df$weather, drop = TRUE)| Time of Day x Weather | Count |
|---|---|
| Afternoon.clear_sky | 22 383 |
| Evening.clear_sky | 11 170 |
| Morning.clear_sky | 13 770 |
| Night.clear_sky | 5 188 |
| Afternoon.harder_conditions | 865 |
| Evening.harder_conditions | 616 |
| Morning.harder_conditions | 785 |
| Night.harder_conditions | 447 |
| Afternoon.other | 3 419 |
| Evening.other | 1 668 |
| Morning.other | 3 655 |
| Night.other | 1 034 |
# age group x car age group
df$age_car <- interaction(df$age_group, df$car_age_group, drop = TRUE)| Age Group x Car Age Group | Count |
|---|---|
| under 25.under_15 | 15 809 |
| over 25.under_15 | 37 359 |
| under 25.over_15 | 3 840 |
| over 25.over_15 | 7 992 |
# lighting x surface dry
df$lighting_surface <- interaction(df$lighting_simple, df$surface_dry, drop = TRUE)| Lighting x Surface Dry | Count |
|---|---|
| daylight.dry | 41 021 |
| dark.dry | 17 244 |
| daylight.non-dry | 3 920 |
| dark.non-dry | 2 815 |
5.5 Modelling
# Previous analysis showed that Zero-Inflated Negative Binomial is the most appropriate choice
# Starting with the most general model
general <- zeroinfl(total_victims ~ weather + party_sex + age_group + sobriety
+ financial_responsibility_bi + cellphone_in_use + race_group
+ car_age_group + LA_season + time_of_day
+ air_bag + lighting_surface, data = df, dist = "negbin")| Estimate | Std. Error | z value | Pr(>|z|) | |
|---|---|---|---|---|
| (Intercept) | 0.074 | 0.033 | 2.234 | 2.55e-02 |
| weatherharder_conditions | 0.059 | 0.048 | 1.233 | 2.17e-01 |
| weatherother | -0.011 | 0.024 | -0.467 | 6.40e-01 |
| party_sexmale | -0.080 | 0.016 | -5.071 | < 1e-03 |
| age_groupover 25 | -0.014 | 0.016 | -0.908 | 3.64e-01 |
| sobrietynot drinking | -0.033 | 0.024 | -1.374 | 1.69e-01 |
| sobrietyother | -0.162 | 0.052 | -3.095 | 1.97e-03 |
| financial_responsibility_bino or unknown proof | 0.176 | 0.022 | 8.128 | < 1e-03 |
| cellphone_in_use1 | -0.098 | 0.049 | -2.006 | 4.48e-02 |
| race_groupnon-white | 0.014 | 0.015 | 0.897 | 3.70e-01 |
| car_age_groupover_15 | -0.005 | 0.019 | -0.241 | 8.10e-01 |
| LA_seasonWet | -0.019 | 0.015 | -1.293 | 1.96e-01 |
| time_of_dayEvening | 0.018 | 0.026 | 0.698 | 4.85e-01 |
| time_of_dayMorning | -0.113 | 0.019 | -6.000 | < 1e-03 |
| time_of_dayNight | -0.204 | 0.034 | -6.043 | < 1e-03 |
| air_bagNo | -0.543 | 0.020 | -26.515 | < 1e-03 |
| air_bagOther safety | -0.420 | 0.022 | -19.413 | < 1e-03 |
| lighting_surfacedark.dry | -0.050 | 0.026 | -1.939 | 5.25e-02 |
| lighting_surfacedaylight.non-dry | -0.156 | 0.040 | -3.886 | < 1e-03 |
| lighting_surfacedark.non-dry | -0.133 | 0.049 | -2.730 | 6.34e-03 |
| Log(theta) | 1.088 | 0.044 | 24.451 | < 1e-03 |
| Estimate | Std. Error | z value | Pr(>|z|) | |
|---|---|---|---|---|
| (Intercept) | -14.422 | 54.178 | -0.266 | 7.90e-01 |
| weatherharder_conditions | 0.629 | 0.339 | 1.855 | 6.36e-02 |
| weatherother | 0.081 | 0.122 | 0.659 | 5.10e-01 |
| party_sexmale | 0.379 | 0.089 | 4.234 | < 1e-03 |
| age_groupover 25 | -0.378 | 0.080 | -4.734 | < 1e-03 |
| sobrietynot drinking | 0.014 | 0.142 | 0.095 | 9.24e-01 |
| sobrietyother | 0.469 | 0.229 | 2.044 | 4.09e-02 |
| financial_responsibility_bino or unknown proof | 0.441 | 0.103 | 4.287 | < 1e-03 |
| cellphone_in_use1 | 0.029 | 0.261 | 0.113 | 9.10e-01 |
| race_groupnon-white | 0.413 | 0.089 | 4.621 | < 1e-03 |
| car_age_groupover_15 | -0.249 | 0.105 | -2.364 | 1.81e-02 |
| LA_seasonWet | 0.064 | 0.078 | 0.828 | 4.07e-01 |
| time_of_dayEvening | -0.005 | 0.128 | -0.039 | 9.69e-01 |
| time_of_dayMorning | -0.084 | 0.098 | -0.859 | 3.90e-01 |
| time_of_dayNight | -0.490 | 0.221 | -2.220 | 2.64e-02 |
| air_bagNo | 12.975 | 54.178 | 0.239 | 8.11e-01 |
| air_bagOther safety | 1.833 | 62.848 | 0.029 | 9.77e-01 |
| lighting_surfacedark.dry | -0.082 | 0.129 | -0.633 | 5.27e-01 |
| lighting_surfacedaylight.non-dry | -1.414 | 0.444 | -3.184 | 1.45e-03 |
| lighting_surfacedark.non-dry | -0.435 | 0.327 | -1.331 | 1.83e-01 |
# To analyze it properly we'll use the Likelihood Ratio Test. Hypotheses:
# H0: the removed variable is jointly significant
# H1: the removed variable is not jointly significant
# Starting with dropping the variable that is insignificant in both parts – LA_season
step1 <- zeroinfl(total_victims ~ weather + party_sex + age_group + sobriety
+ financial_responsibility_bi + cellphone_in_use + race_group + car_age_group
+ time_of_day + air_bag + lighting_surface, data = df, dist = "negbin")| Estimate | Std. Error | z value | Pr(>|z|) | |
|---|---|---|---|---|
| (Intercept) | 0.065 | 0.032 | 2.005 | 4.49e-02 |
| weatherharder_conditions | 0.058 | 0.048 | 1.210 | 2.26e-01 |
| weatherother | -0.014 | 0.024 | -0.564 | 5.73e-01 |
| party_sexmale | -0.079 | 0.016 | -5.059 | < 1e-03 |
| age_groupover 25 | -0.014 | 0.016 | -0.887 | 3.75e-01 |
| sobrietynot drinking | -0.033 | 0.024 | -1.362 | 1.73e-01 |
| sobrietyother | -0.162 | 0.052 | -3.092 | 1.99e-03 |
| financial_responsibility_bino or unknown proof | 0.177 | 0.022 | 8.140 | < 1e-03 |
| cellphone_in_use1 | -0.098 | 0.049 | -1.993 | 4.63e-02 |
| race_groupnon-white | 0.013 | 0.015 | 0.893 | 3.72e-01 |
| car_age_groupover_15 | -0.005 | 0.019 | -0.282 | 7.78e-01 |
| time_of_dayEvening | 0.019 | 0.026 | 0.706 | 4.80e-01 |
| time_of_dayMorning | -0.114 | 0.019 | -6.061 | < 1e-03 |
| time_of_dayNight | -0.203 | 0.034 | -6.035 | < 1e-03 |
| air_bagNo | -0.544 | 0.021 | -26.518 | < 1e-03 |
| air_bagOther safety | -0.420 | 0.022 | -19.412 | < 1e-03 |
| lighting_surfacedark.dry | -0.050 | 0.026 | -1.944 | 5.19e-02 |
| lighting_surfacedaylight.non-dry | -0.159 | 0.040 | -3.971 | < 1e-03 |
| lighting_surfacedark.non-dry | -0.136 | 0.049 | -2.792 | 5.23e-03 |
| Log(theta) | 1.087 | 0.044 | 24.440 | < 1e-03 |
| Estimate | Std. Error | z value | Pr(>|z|) | |
|---|---|---|---|---|
| (Intercept) | -12.660 | 22.781 | -0.556 | 5.78e-01 |
| weatherharder_conditions | 0.654 | 0.341 | 1.920 | 5.49e-02 |
| weatherother | 0.093 | 0.122 | 0.760 | 4.47e-01 |
| party_sexmale | 0.382 | 0.090 | 4.268 | < 1e-03 |
| age_groupover 25 | -0.376 | 0.080 | -4.704 | < 1e-03 |
| sobrietynot drinking | 0.013 | 0.142 | 0.091 | 9.28e-01 |
| sobrietyother | 0.469 | 0.229 | 2.044 | 4.09e-02 |
| financial_responsibility_bino or unknown proof | 0.438 | 0.103 | 4.266 | < 1e-03 |
| cellphone_in_use1 | 0.035 | 0.260 | 0.134 | 8.94e-01 |
| race_groupnon-white | 0.413 | 0.089 | 4.629 | < 1e-03 |
| car_age_groupover_15 | -0.249 | 0.106 | -2.360 | 1.83e-02 |
| time_of_dayEvening | -0.005 | 0.127 | -0.036 | 9.71e-01 |
| time_of_dayMorning | -0.093 | 0.097 | -0.959 | 3.38e-01 |
| time_of_dayNight | -0.488 | 0.220 | -2.218 | 2.65e-02 |
| air_bagNo | 11.241 | 22.780 | 0.493 | 6.22e-01 |
| air_bagOther safety | 0.594 | 33.774 | 0.018 | 9.86e-01 |
| lighting_surfacedark.dry | -0.085 | 0.129 | -0.663 | 5.07e-01 |
| lighting_surfacedaylight.non-dry | -1.433 | 0.450 | -3.184 | 1.45e-03 |
| lighting_surfacedark.non-dry | -0.457 | 0.330 | -1.384 | 1.66e-01 |
| #Df | LogLik | Df | Chisq | Pr(>Chisq) | |
|---|---|---|---|---|---|
| step1 | 39 | -63170.65 | NA | NA | NA |
| general | 41 | -63168.03 | 2 | 5.257 | 0.072 |
- p-value = 0.072 -> we fail to reject H0. No evidence that “LA_season” is jointly significant – removing it.
# In step 2 we are dropping the "weather" variable
step2 <- zeroinfl(total_victims ~ party_sex + age_group + sobriety + financial_responsibility_bi
+ cellphone_in_use + race_group + car_age_group + time_of_day
+ air_bag + lighting_surface, data = df, dist = "negbin")| Estimate | Std. Error | z value | Pr(>|z|) | |
|---|---|---|---|---|
| (Intercept) | 0.065 | 0.032 | 2.011 | 4.44e-02 |
| party_sexmale | -0.079 | 0.016 | -5.025 | < 1e-03 |
| age_groupover 25 | -0.014 | 0.016 | -0.877 | 3.80e-01 |
| sobrietynot drinking | -0.034 | 0.024 | -1.437 | 1.51e-01 |
| sobrietyother | -0.165 | 0.052 | -3.149 | 1.64e-03 |
| financial_responsibility_bino or unknown proof | 0.175 | 0.022 | 8.053 | < 1e-03 |
| cellphone_in_use1 | -0.096 | 0.049 | -1.958 | 5.03e-02 |
| race_groupnon-white | 0.014 | 0.015 | 0.945 | 3.44e-01 |
| car_age_groupover_15 | -0.006 | 0.019 | -0.295 | 7.68e-01 |
| time_of_dayEvening | 0.018 | 0.026 | 0.671 | 5.02e-01 |
| time_of_dayMorning | -0.115 | 0.019 | -6.133 | < 1e-03 |
| time_of_dayNight | -0.204 | 0.034 | -6.066 | < 1e-03 |
| air_bagNo | -0.542 | 0.020 | -26.548 | < 1e-03 |
| air_bagOther safety | -0.420 | 0.022 | -19.416 | < 1e-03 |
| lighting_surfacedark.dry | -0.050 | 0.026 | -1.927 | 5.40e-02 |
| lighting_surfacedaylight.non-dry | -0.142 | 0.033 | -4.370 | < 1e-03 |
| lighting_surfacedark.non-dry | -0.113 | 0.040 | -2.809 | 4.97e-03 |
| Log(theta) | 1.089 | 0.044 | 24.497 | < 1e-03 |
| Estimate | Std. Error | z value | Pr(>|z|) | |
|---|---|---|---|---|
| (Intercept) | -11.373 | 12.180 | -0.934 | 3.50e-01 |
| party_sexmale | 0.382 | 0.089 | 4.276 | < 1e-03 |
| age_groupover 25 | -0.372 | 0.080 | -4.679 | < 1e-03 |
| sobrietynot drinking | -0.005 | 0.139 | -0.038 | 9.70e-01 |
| sobrietyother | 0.438 | 0.228 | 1.924 | 5.44e-02 |
| financial_responsibility_bino or unknown proof | 0.424 | 0.102 | 4.139 | < 1e-03 |
| cellphone_in_use1 | 0.052 | 0.255 | 0.202 | 8.40e-01 |
| race_groupnon-white | 0.417 | 0.089 | 4.679 | < 1e-03 |
| car_age_groupover_15 | -0.249 | 0.105 | -2.364 | 1.81e-02 |
| time_of_dayEvening | -0.016 | 0.127 | -0.125 | 9.01e-01 |
| time_of_dayMorning | -0.091 | 0.097 | -0.939 | 3.47e-01 |
| time_of_dayNight | -0.491 | 0.219 | -2.242 | 2.50e-02 |
| air_bagNo | 9.983 | 12.180 | 0.820 | 4.12e-01 |
| air_bagOther safety | -0.715 | 28.474 | -0.025 | 9.80e-01 |
| lighting_surfacedark.dry | -0.076 | 0.128 | -0.590 | 5.55e-01 |
| lighting_surfacedaylight.non-dry | -1.069 | 0.376 | -2.843 | 4.47e-03 |
| lighting_surfacedark.non-dry | -0.076 | 0.226 | -0.338 | 7.35e-01 |
| #Df | LogLik | Df | Chisq | Pr(>Chisq) | |
|---|---|---|---|---|---|
| step2 | 35 | -63173.43 | NA | NA | NA |
| step1 | 39 | -63170.65 | 4 | 5.549 | 2.35e-01 |
- p-value = 0.23 -> we fail to reject H0. No evidence that “weather” is jointly significant – removing it.
# In step 3 we are dropping the "time_of_day" variable
step3 <- zeroinfl(total_victims ~ party_sex + age_group + sobriety + financial_responsibility_bi
+ cellphone_in_use + race_group + car_age_group + air_bag + lighting_surface,
data = df, dist = "negbin")| #Df | LogLik | Df | Chisq | Pr(>Chisq) | |
|---|---|---|---|---|---|
| step3 | 29 | -63234.01 | NA | NA | NA |
| step2 | 35 | -63173.43 | 6 | 121.173 | <1e-03 |
- p-value = <1e-03 -> we reject H0. “time_of_day” is jointly significant.
- We are going back to model from step 2
# In step 4 we bring back "time_of_day" and remove "sobriety"
step4 <- zeroinfl(total_victims ~ party_sex + age_group + sobriety + financial_responsibility_bi
+ cellphone_in_use + race_group + car_age_group + time_of_day
+ air_bag + lighting_surface, data = df, dist = "negbin")| #Df | LogLik | Df | Chisq | Pr(>Chisq) | |
|---|---|---|---|---|---|
| step4 | 35 | -63173.43 | NA | NA | NA |
| step2 | 35 | -63173.43 | 0 | 0 | 1e+00 |
- p-value = 7.551e-07 -> we reject H0. “sobriety” is jointly significant.
- We are going back to model from step 2
# In step 5 we bring back "sobriety" and remove "lighting_surfacedark"
step5 <- zeroinfl(total_victims ~ party_sex + age_group + sobriety + financial_responsibility_bi
+ cellphone_in_use + race_group + car_age_group + time_of_day + air_bag,
data = df, dist = "negbin")| #Df | LogLik | Df | Chisq | Pr(>Chisq) | |
|---|---|---|---|---|---|
| step5 | 29 | -63190.43 | NA | NA | NA |
| step2 | 35 | -63173.43 | 6 | 34.004 | <1e-03 |
- p-value = <1e-03 -> we reject H0. “lighting_surfacedark” is jointly significant.
- We are going back to model from step 2
# In step 6 we bring back "lighting_surfacedark" and remove "cellphone_in_use"
step6 <- zeroinfl(total_victims ~ party_sex + age_group + sobriety + financial_responsibility_bi
+ race_group + car_age_group + time_of_day + air_bag
+ lighting_surface, data = df, dist = "negbin")| #Df | LogLik | Df | Chisq | Pr(>Chisq) | |
|---|---|---|---|---|---|
| step6 | 33 | -63176.62 | NA | NA | NA |
| step2 | 35 | -63173.43 | 2 | 6.389 | 4.1e-02 |
- p-value = 4.1e-02 -> we reject H0. “cellphone_in_use” is jointly significant
- We are going back to model from step 2
# In step 7 we bring back "cellphone_in_use" and remove "car_age_group"
step7 <- zeroinfl(total_victims ~ party_sex + age_group + sobriety + financial_responsibility_bi
+ cellphone_in_use + race_group + time_of_day + air_bag + lighting_surface, data = df, dist = "negbin")| #Df | LogLik | Df | Chisq | Pr(>Chisq) | |
|---|---|---|---|---|---|
| step7 | 33 | -63177.36 | NA | NA | NA |
| step2 | 35 | -63173.43 | 2 | 7.866 | 1.96e-02 |
- p-value = 1.96e-02 -> we reject H0. “car_age_group” is jointly significant
- We are going back to model from step 2
# In step 8 we bring back "car_age_group" and remove "race_group"
step8 <- zeroinfl(total_victims ~ party_sex + age_group + sobriety + financial_responsibility_bi
+ cellphone_in_use + car_age_group + time_of_day + air_bag
+ lighting_surface, data = df, dist = "negbin")| #Df | LogLik | Df | Chisq | Pr(>Chisq) | |
|---|---|---|---|---|---|
| step8 | 33 | -63189.12 | NA | NA | NA |
| step2 | 35 | -63173.43 | 2 | 31.393 | <1e-03 |
- p-value = <1e-03 -> we reject H0. “race_group” is jointly significant
- We are going back to model from step 2
# In step 9 we bring back "race_group" and remove "air_bag"
step9 <- zeroinfl(total_victims ~ party_sex + age_group + sobriety + financial_responsibility_bi
+ cellphone_in_use + race_group + car_age_group + time_of_day
+ lighting_surface, data = df, dist = "negbin")| #Df | LogLik | Df | Chisq | Pr(>Chisq) | |
|---|---|---|---|---|---|
| step9 | 31 | -64944.40 | NA | NA | NA |
| step2 | 35 | -63173.43 | 4 | 3541.952 | <1e-03 |
- p-value = <1e-03 -> we reject H0. “air_bag” is jointly significant
- We are going back to model from step 2
# In step 10 we bring back "air_bag" and remove "age_group"
step10 <- zeroinfl(total_victims ~ party_sex + sobriety + financial_responsibility_bi
+ cellphone_in_use + race_group + car_age_group + time_of_day
+ air_bag + lighting_surface, data = df, dist = "negbin")| #Df | LogLik | Df | Chisq | Pr(>Chisq) | |
|---|---|---|---|---|---|
| step10 | 31 | -64944.40 | NA | NA | NA |
| step2 | 35 | -63173.43 | 4 | 3541.952 | <1e-03 |
- p-value = <1e-03 -> we reject H0. “age_group” is jointly significant
- We are going back to model from step 2
# We checked all variables and kept all jointly significant ones
final_model <- zeroinfl(total_victims ~ party_sex + age_group + sobriety + financial_responsibility_bi
+ cellphone_in_use + race_group + car_age_group + time_of_day
+ air_bag + lighting_surface, data = df, dist = "negbin")| Estimate | Std. Error | z value | Pr(>|z|) | |
|---|---|---|---|---|
| (Intercept) | 0.065 | 0.032 | 2.011 | 4.44e-02 |
| party_sexmale | -0.079 | 0.016 | -5.025 | < 1e-03 |
| age_groupover 25 | -0.014 | 0.016 | -0.877 | 3.80e-01 |
| sobrietynot drinking | -0.034 | 0.024 | -1.437 | 1.51e-01 |
| sobrietyother | -0.165 | 0.052 | -3.149 | 1.64e-03 |
| financial_responsibility_bino or unknown proof | 0.175 | 0.022 | 8.053 | < 1e-03 |
| cellphone_in_use1 | -0.096 | 0.049 | -1.958 | 5.03e-02 |
| race_groupnon-white | 0.014 | 0.015 | 0.945 | 3.44e-01 |
| car_age_groupover_15 | -0.006 | 0.019 | -0.295 | 7.68e-01 |
| time_of_dayEvening | 0.018 | 0.026 | 0.671 | 5.02e-01 |
| time_of_dayMorning | -0.115 | 0.019 | -6.133 | < 1e-03 |
| time_of_dayNight | -0.204 | 0.034 | -6.066 | < 1e-03 |
| air_bagNo | -0.542 | 0.020 | -26.548 | < 1e-03 |
| air_bagOther safety | -0.420 | 0.022 | -19.416 | < 1e-03 |
| lighting_surfacedark.dry | -0.050 | 0.026 | -1.927 | 5.40e-02 |
| lighting_surfacedaylight.non-dry | -0.142 | 0.033 | -4.370 | < 1e-03 |
| lighting_surfacedark.non-dry | -0.113 | 0.040 | -2.809 | 4.97e-03 |
| Log(theta) | 1.089 | 0.044 | 24.497 | < 1e-03 |
| Estimate | Std. Error | z value | Pr(>|z|) | |
|---|---|---|---|---|
| (Intercept) | -11.373 | 12.180 | -0.934 | 3.50e-01 |
| party_sexmale | 0.382 | 0.089 | 4.276 | < 1e-03 |
| age_groupover 25 | -0.372 | 0.080 | -4.679 | < 1e-03 |
| sobrietynot drinking | -0.005 | 0.139 | -0.038 | 9.70e-01 |
| sobrietyother | 0.438 | 0.228 | 1.924 | 5.44e-02 |
| financial_responsibility_bino or unknown proof | 0.424 | 0.102 | 4.139 | < 1e-03 |
| cellphone_in_use1 | 0.052 | 0.255 | 0.202 | 8.40e-01 |
| race_groupnon-white | 0.417 | 0.089 | 4.679 | < 1e-03 |
| car_age_groupover_15 | -0.249 | 0.105 | -2.364 | 1.81e-02 |
| time_of_dayEvening | -0.016 | 0.127 | -0.125 | 9.01e-01 |
| time_of_dayMorning | -0.091 | 0.097 | -0.939 | 3.47e-01 |
| time_of_dayNight | -0.491 | 0.219 | -2.242 | 2.50e-02 |
| air_bagNo | 9.983 | 12.180 | 0.820 | 4.12e-01 |
| air_bagOther safety | -0.715 | 28.474 | -0.025 | 9.80e-01 |
| lighting_surfacedark.dry | -0.076 | 0.128 | -0.590 | 5.55e-01 |
| lighting_surfacedaylight.non-dry | -1.069 | 0.376 | -2.843 | 4.47e-03 |
| lighting_surfacedark.non-dry | -0.076 | 0.226 | -0.338 | 7.35e-01 |
# Preparing baseline models with all initial variables
poisson_model <- glm(total_victims ~ weather + party_sex + age_group + sobriety
+ financial_responsibility_bi + cellphone_in_use + race_group
+ car_age_group + LA_season + time_of_day + air_bag + lighting_surface,
family = "poisson", data = df)
# Negative Binomial model
nb_model <- glm.nb(total_victims ~ weather + party_sex + age_group + sobriety
+ financial_responsibility_bi + cellphone_in_use + race_group
+ car_age_group + LA_season + time_of_day + air_bag + lighting_surface,
data = df)
# Zero-inflated Poisson model
zip_model <- zeroinfl(total_victims ~ weather + party_sex + age_group + sobriety
+ financial_responsibility_bi + cellphone_in_use + race_group
+ car_age_group + LA_season + time_of_day + air_bag + lighting_surface,
data = df, dist = "poisson")| Poisson | Negative Binomial | ZIP | Full model (ZINB) | Final model (ZINB) | |
|---|---|---|---|---|---|
| (Intercept) | 0.090*** | 0.091** | |||
| (0.025) | (0.029) | ||||
| weatherharder_conditions | 0.006 | 0.005 | |||
| (0.036) | (0.041) | ||||
| weatherother | -0.021 | -0.023 | |||
| (0.018) | (0.020) | ||||
| party_sexmale | -0.123*** | -0.126*** | |||
| (0.011) | (0.013) | ||||
| age_groupover 25 | 0.032** | 0.032* | |||
| (0.012) | (0.013) | ||||
| sobrietynot drinking | -0.037* | -0.036+ | |||
| (0.018) | (0.021) | ||||
| sobrietyother | -0.232*** | -0.236*** | |||
| (0.038) | (0.042) | ||||
| financial_responsibility_bino or unknown proof | 0.123*** | 0.121*** | |||
| (0.016) | (0.019) | ||||
| cellphone_in_use1 | -0.105** | -0.105* | |||
| (0.036) | (0.041) | ||||
| race_groupnon-white | -0.033** | -0.035** | |||
| (0.011) | (0.012) | ||||
| car_age_groupover_15 | 0.020 | 0.023 | |||
| (0.014) | (0.016) | ||||
| LA_seasonWet | -0.026* | -0.027* | |||
| (0.011) | (0.012) | ||||
| time_of_dayEvening | 0.019 | 0.021 | |||
| (0.019) | (0.022) | ||||
| time_of_dayMorning | -0.100*** | -0.102*** | |||
| (0.013) | (0.015) | ||||
| time_of_dayNight | -0.158*** | -0.156*** | |||
| (0.025) | (0.029) | ||||
| air_bagNo | -0.797*** | -0.796*** | |||
| (0.012) | (0.014) | ||||
| air_bagOther safety | -0.413*** | -0.412*** | |||
| (0.019) | (0.022) | ||||
| lighting_surfacedark.dry | -0.040* | -0.037+ | |||
| (0.019) | (0.022) | ||||
| lighting_surfacedaylight.non-dry | -0.039 | -0.031 | |||
| (0.029) | (0.033) | ||||
| lighting_surfacedark.non-dry | -0.095** | -0.090* | |||
| (0.036) | (0.041) | ||||
| count_(Intercept) | 0.125*** | 0.074* | 0.065* | ||
| (0.035) | (0.033) | (0.032) | |||
| count_weatherharder_conditions | 0.045 | 0.059 | |||
| (0.052) | (0.048) | ||||
| count_weatherother | -0.018 | -0.011 | |||
| (0.026) | (0.024) | ||||
| count_party_sexmale | -0.040* | -0.080*** | -0.079*** | ||
| (0.017) | (0.016) | (0.016) | |||
| count_age_groupover 25 | -0.039* | -0.014 | -0.014 | ||
| (0.017) | (0.016) | (0.016) | |||
| count_sobrietynot drinking | -0.004 | -0.033 | -0.034 | ||
| (0.025) | (0.024) | (0.024) | |||
| count_sobrietyother | -0.082 | -0.162** | -0.165** | ||
| (0.057) | (0.052) | (0.052) | |||
| count_financial_responsibility_bino or unknown proof | 0.169*** | 0.176*** | 0.175*** | ||
| (0.023) | (0.022) | (0.022) | |||
| count_cellphone_in_use1 | -0.102+ | -0.098* | -0.096+ | ||
| (0.053) | (0.049) | (0.049) | |||
| count_race_groupnon-white | 0.073*** | 0.014 | 0.014 | ||
| (0.017) | (0.015) | (0.015) | |||
| count_car_age_groupover_15 | -0.002 | -0.005 | -0.006 | ||
| (0.021) | (0.019) | (0.019) | |||
| count_LA_seasonWet | -0.013 | -0.019 | |||
| (0.016) | (0.015) | ||||
| count_time_of_dayEvening | 0.024 | 0.018 | 0.018 | ||
| (0.028) | (0.026) | (0.026) | |||
| count_time_of_dayMorning | -0.115*** | -0.113*** | -0.115*** | ||
| (0.020) | (0.019) | (0.019) | |||
| count_time_of_dayNight | -0.256*** | -0.204*** | -0.204*** | ||
| (0.037) | (0.034) | (0.034) | |||
| count_air_bagNo | -0.435*** | -0.543*** | -0.542*** | ||
| (0.019) | (0.020) | (0.020) | |||
| count_air_bagOther safety | -0.429*** | -0.420*** | -0.420*** | ||
| (0.033) | (0.022) | (0.022) | |||
| count_lighting_surfacedark.dry | -0.034 | -0.050+ | -0.050+ | ||
| (0.027) | (0.026) | (0.026) | |||
| count_lighting_surfacedaylight.non-dry | -0.172*** | -0.156*** | -0.142*** | ||
| (0.043) | (0.040) | (0.033) | |||
| count_lighting_surfacedark.non-dry | -0.121* | -0.133** | -0.113** | ||
| (0.052) | (0.049) | (0.040) | |||
| zero_(Intercept) | -2.478*** | -14.422 | -11.373 | ||
| (0.136) | (54.178) | (12.180) | |||
| zero_weatherharder_conditions | 0.189 | 0.629+ | |||
| (0.187) | (0.339) | ||||
| zero_weatherother | 0.012 | 0.081 | |||
| (0.079) | (0.122) | ||||
| zero_party_sexmale | 0.333*** | 0.379*** | 0.382*** | ||
| (0.055) | (0.089) | (0.089) | |||
| zero_age_groupover 25 | -0.275*** | -0.378*** | -0.372*** | ||
| (0.051) | (0.080) | (0.080) | |||
| zero_sobrietynot drinking | 0.138 | 0.014 | -0.005 | ||
| (0.091) | (0.142) | (0.139) | |||
| zero_sobrietyother | 0.535*** | 0.469* | 0.438+ | ||
| (0.156) | (0.229) | (0.228) | |||
| zero_financial_responsibility_bino or unknown proof | 0.182** | 0.441*** | 0.424*** | ||
| (0.069) | (0.103) | (0.102) | |||
| zero_cellphone_in_use1 | 0.002 | 0.029 | 0.052 | ||
| (0.171) | (0.261) | (0.255) | |||
| zero_race_groupnon-white | 0.433*** | 0.413*** | 0.417*** | ||
| (0.059) | (0.089) | (0.089) | |||
| zero_car_age_groupover_15 | -0.104 | -0.249* | -0.249* | ||
| (0.067) | (0.105) | (0.105) | |||
| zero_LA_seasonWet | 0.052 | 0.064 | |||
| (0.049) | (0.078) | ||||
| zero_time_of_dayEvening | 0.020 | -0.005 | -0.016 | ||
| (0.082) | (0.128) | (0.127) | |||
| zero_time_of_dayMorning | -0.049 | -0.084 | -0.091 | ||
| (0.062) | (0.098) | (0.097) | |||
| zero_time_of_dayNight | -0.446*** | -0.490* | -0.491* | ||
| (0.134) | (0.221) | (0.219) | |||
| zero_air_bagNo | 1.607*** | 12.975 | 9.983 | ||
| (0.094) | (54.178) | (12.180) | |||
| zero_air_bagOther safety | -0.069 | 1.833 | -0.715 | ||
| (0.246) | (62.848) | (28.474) | |||
| zero_lighting_surfacedark.dry | 0.029 | -0.082 | -0.076 | ||
| (0.082) | (0.129) | (0.128) | |||
| zero_lighting_surfacedaylight.non-dry | -0.612*** | -1.414** | -1.069** | ||
| (0.174) | (0.444) | (0.376) | |||
| zero_lighting_surfacedark.non-dry | -0.118 | -0.435 | -0.076 | ||
| (0.180) | (0.327) | (0.226) | |||
| Num.Obs. | 65000 | 65000 | 65000 | 65000 | 65000 |
| R2 | 0.040 | 0.043 | 0.043 | ||
| R2 Adj. | 0.040 | 0.043 | 0.043 | ||
| AIC | 128844.3 | 126737.4 | 127156.2 | 126418.0 | 126416.9 |
| BIC | 129025.9 | 126928.1 | 127519.5 | 126790.4 | 126734.7 |
| Log.Lik. | -64402.136 | -63347.708 | |||
| F | 276.214 | 204.683 | |||
| RMSE | 0.86 | 0.86 | 0.86 | 0.86 | 0.86 |
| + p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001 |
6 Results
We tested seven hypotheses related to the risk and severity of car crashes using a two-part model: a count model for the number of victims and a zero model for the likelihood of zero-victim crashes. Below are the results and interpretations for each hypothesis.
1. Young drivers (under 25) have higher risk of car crash with victims
- The count model shows no significant difference in the number of victims (p = 0.380), but the zero model indicates that older drivers are significantly less likely to be in zero-victim crashes (β = –0.372, p < 0.001).
- Hypothesis supported: Younger drivers are more often involved in harmless crashes, while older drivers are more likely to cause injury.
2. Vehicles older than 15 years are at higher risk of car crash with injuries
- The count model shows no significant difference (p = 0.768), but the zero model shows that crashes involving older vehicles are less likely to be zero-victim (β = –0.249, p = 0.018).
- Hypothesis supported: Older vehicles are associated with a higher risk of harmful crashes, even if not more frequent overall.
3. Drivers who consumed alcohol have higher risk of car crash with victims - Comparing to the baseline of drinking drivers:
“Not drinking” status was not significant in either model.
“Other” sobriety states (e.g., drugs, fatigue) were significant in the count model only (β = –0.165, p = 0.002), associated with fewer victims.
Hypothesis not supported: Alcohol consumption did not significantly increase crash severity or likelihood in this model. This may be due to underreporting or other contextual factors.
4. Men have higher overall risk of car crash than women
- Men are associated with fewer victims in the count model (β = –0.079, p < 0.001), and more likely to be in zero-victim crashes (β = +0.382, p < 0.001).
- Hypothesis rejected: Male drivers appear to be involved in less harmful crashes, contrary to expectations.
5. Women have higher risk of car crash with victims than men
- Contrary to expectations, women are more likely to be involved in crashes with victims than men.
- Model shows a positive β coefficient of +0.382 (p < 0.001) for women, indicating higher likelihood of victim-involved crashes.This finding rejects the hypothesis that women have a lower risk of crashes with victims.
6. Non-white drivers have higher risk of car crash
- The count model showed no significant difference, but the zero model shows that non-white drivers are more likely to be in zero-victim crashes (β = +0.417, p < 0.001).
- Hypothesis rejected: Non-white drivers are involved in less harmful crashes overall.
7. Uninsured drivers have higher risk of car crash with victims
- Uninsured/unknown drivers are associated with more victims (β = +0.175, p < 0.001) and more likely to be in zero-victim crashes (β = +0.424, p < 0.001).
- Hypothesis supported: While results suggest a paradox (more victims and more harmless crashes), this may reflect underreporting or bimodal crash types (very minor vs. serious).
8. Bad weather conditions indicate higher risk of car crash
- Daylight, wet and dark, wet conditions are associated with fewer victims, likely due to cautious driving. However, daylight, wet crashes are much less likely to be harmless (β = –1.069, p < 0.01).
- Hypothesis partially rejected: Bad conditions don’t increase crash frequency, but may increase severity when they occur.
7 Findings
In this study, we employed the Zero-Inflated Negative Binomial (ZINB) model to examine the determinants of traffic accidents and their severity. This approach enabled us to analyze each variable’s effect in two distinct dimensions:
- Zero-inflation (logit) component: assessed the likelihood of an accident being non-injurious (zero-victim crash)
- Count (negative binomial) component: estimated the expected number of victims, conditional on the crash involving injuries.
1. Gender
Driver gender emerged as an influential factor. Male drivers were more likely to be involved in zero-victim accidents, while female drivers were associated with injury crashes resulting in approximately 7.6% more victims.
2. Age of driver
Despite the traditional expectations, younger drivers (under 25) were more likely to be involved in non-injury crashes. It suggests that older drivers may contribute more frequently to crashes with physical harm.
3. Age of vehicle
Vehicle age also played a notable role. Cars older than 15 years were less likely to be involved in zero-victim accidents, indicating a higher likelihood of causing injury-related crashes (potentially due to outdated safety features or poorer maintenance).
4. Insurance status Lack of verified insurance was significantly associated with both crash dimensions. Uninsured drivers were more likely to be involved in zero-victim crashes, possibly reflecting minor or underreported incidents. However, if such drivers caused injury crashes, they were linked to ~19% more victims.
5. Driver’s race Driver race also influenced outcomes. Non-white drivers were more likely to be involved in crashes with no injuries, implying that white drivers may be more frequently responsible for injury-related accidents.
6. Time of day The time of day significantly shaped both injury risk and crash severity. Nighttime driving increased the likelihood of injury crashes, but such incidents involved ~18.5% fewer victims (possibly due to fewer passengers). Morning accidents also resulted in ~11% fewer victims than afternoon ones.
7. Airbag and safety equipment Surprisingly, accidents in which airbags did not deploy were associated with ~42% fewer victims. This may reflect lower-impact collisions that do not activate airbags. Similarly, crashes involving other safety features (e.g. seatbelts) led to ~34% fewer injuries, aligning with expectations regarding passive safety measures.
8. Weather and road surface Environmental conditions had a measurable effect. During daytime on wet roads, crashes were more likely to involve injuries but included ~13% fewer victims. It was likely due to more cautious driving. A similar pattern emerged under dark and wet conditions, where victim counts were ~11% lower (once again possibly due to fewer passengers).
9. Sobriety Interestingly, no significant difference was observed between drinking and non-drinking drivers. However, drivers under the influence of other substances (e.g. drugs) were associated with ~15% fewer victims in injury crashes.
10. Phone use In our model phone usage by driver was also included. Despite being jointly significant, it hasn’t provided insightful comparison results.
7.1 Next Steps and Alternative Approaches
While the ZINB model allows for an enchanted understanding of both occurrence and severity of crashes, it comes with some limitations. The model’s dual nature can complicate interpretation for policymakers or practitioners seeking straightforward insights.
A potential alternative would be to separate the analysis into two simpler models:
- Logistic regression predicting whether any injuries occurred (binary outcome),
- Truncated count model (e.g. negative binomial) estimating the number of victims in injury-only crashes.
This modular approach may improve interpretability while still addressing overdispersion and zero inflation where relevant. Future research could also explore spatial effects (e.g. region or county-level risk), driver behaviour history or usage of machine learning classifiers to capture nonlinear interactions across variables.
8 Bibliography
Berhanu Y., Alemayehu E., Schröder D. (2023). Examining Car Accident Prediction Techniques and Road Traffic Congestion: A Comparative Analysis of Road Safety and Prevention of World Challenges in Low-Income and High-Income Countries. Journal of Advanced Transportation, https://doi.org/10.1155/2023/6643412
Blows S., Ivers R., Connor J., Ameratunga S., Norton R. (2003). Car insurance and the risk of car crash injury. Accident Analysis & Prevention, 35(6), 987–990. https://doi.org/10.1016/S0001-4575(02)00106-9
Blows, S., Ivers, R. Q., Woodward, M., Connor, J., & Norton, R. (2003). Vehicle year and the risk of car crash injury. Injury Prevention, 9(4), 353–356. https://doi.org/10.1136/ip.9.4.353
Braver, E. R. (2003). Race, Hispanic origin, and socioeconomic status in relation to motor vehicle occupant death rates and risk factors among adults. Accident Analysis & Prevention, 35(3), 295–309. https://doi.org/10.1016/S0001-4575(01)00106-3
Brijs, T., Karlis, D., & Wets, G. (2008). Studying the effect of weather conditions on daily crash counts using a discrete time-series model. Accident Analysis & Prevention, 40(3), 1180–1190. https://doi.org/10.1016/j.aap.2008.01.001
Connor J., Norton R., Ameratunga S. & Jackson, R (2004). The contribution of alcohol to serious car crash injuries. Epidemiology, 15(3), 337-344. 10.1097/01.ede.0000120045.58295.86
Cullen P., Möller H., Woodward M., Senserrick t., Boufous S, Rogers K., Brown J., Ivers R. (2021). Are there sex differences in crash and crash-related injury between men and women? A 13-year cohort study of young drivers in Australia. SSM - Population Health, 14, 100816. https://doi.org/10.1016/j.ssmph.2021.100816
Lam L. T. (2002). Distractions and the risk of car crash injury: The effect of drivers’ age. Journal of Safety Research, 33(3), 411–419. https://doi.org/10.1016/S0022-4375(02)00034-8
Ratiu S. (2003). The history of the internal combustion engine. Annals of the Faculty of Engineering Hunedoara, Tome I, 3. https://annals.fih.upt.ro/pdf-full/2003/ANNALS-2003-3-21.pdf
World Health Organization. (2023). Road traffic injuries. https://www.who.int/news-room/fact-sheets/detail/road-traffic-injuries
Yau, K. K. W., Wang, K., & Lee, A. H. (2003). Zero-inflated negative binomial mixed regression modeling of over-dispersed count data with extra zeros. Biometrical Journal, 45(4), 437–452. https://doi.org/10.1002/bimj.200390024