How Risky Are Unrestrained Vehicle Occupants?

Abstract

Seatbelt use is well established as a life-saving measure. Nevertheless, many drivers and passengers continue to neglect seatbelt use. This study examines the risks associated with unrestrained occupants involved in motor vehicle crashes. Using data from the Fatality Analysis Reporting System from 2000 to 2018, the relative risk of fatal traffic accidents for unrestrained vehicle occupants in the United States was estimated using the maximum likelihood estimation method. The findings indicate that unrestrained passengers make up about 12% of all passengers on the road and face a roughly 4.3 times greater likelihood of fatality in severe crashes. Additionally, unrestrained drivers, whose higher risk profiles are linked not only to their lack of restraint but also to broader patterns of hazardous driving behavior, account for over 8% of all drivers and exhibit a risk approximately 5.4 times higher in causing fatal crashes compared to restrained drivers. The findings of this study reveal the prevalence and consequences of unrestrained vehicle occupants and supports ongoing efforts to promote seatbelt utilization and bolster road safety protocols. By doing so, we can alleviate the burden of preventable injuries and fatalities on individuals, families, and society at large, thus fostering a safer and more secure transportation environment for all.

1. Introduction

The severity of injuries in traffic accidents depends on various factors, such as occupant’s age, gender, and physical attributes, vehicle type and speed, collision type, driver impairment due to alcohol or drugs, and whether the driver uses safety restraints [1,2,3,4]. In 2022, 42,514 people were killed in road crashes in the United States [5]. In the same year based on the known restraint use, 50 percent of the drivers killed in passenger vehicles were unrestrained [6]. According to the National Safety Council (NSC), since 1975, estimates show that seat belts have saved 374,276 lives [7]. However, the national seatbelt usage rate in 2022 was found to be around 91 percent [7]. This explains that unrestrained occupants and drivers were overrepresented in fatal crashes. Among risky behaviors such as driving under the influence and speeding, unrestrained driving has been pinpointed as a significant factor in causing traffic accidents.

Over the years, a vast amount of literature documented the effectiveness of seatbelts in saving lives and reducing injury severity [8,9]. States considered seatbelt laws to reduce fatalities, and these laws were found to be effective in improving safety [10]. Studies have even shown the differences among primary and secondary enforcement laws on seatbelt usage rates and injury severities [11,12,13,14]. Educational campaigns have been proven to be effective in increasing seatbelt usage rates [15]. In addition to the policy and educational campaigns, in-vehicle technologies such as seatbelt reminders have also been found to increase seatbelt usage rates [16,17].

Fernandes et al. [18] analyzed various demographic, personality, and attitude-related variables (including age, gender, etc.) to forecast behaviors such as speeding, driving under the influence of alcohol, driving while fatigued, and neglecting seat belt usage among a group of young student drivers. Gender was found to moderate the relationship between perceived relative risk and seat belt usage in road-related contexts. It has been found that male drivers have a lower perceived relative risk associated with road-related incidents and less frequent intended seat belt use, whereas, for female drivers, no such relationship was observed in their study. In a study conducted by Wilson [19], the absence of seat belt use was correlated with other indicators of problematic behavior, such as substance abuse, specific personality traits, irresponsible attitudes, and a heightened propensity for risky driving. On average, individuals who did not use seat belts tended to be younger, less educated, more likely to be male, and unmarried. Even after accounting for these demographic distinctions, non-users exhibited higher levels of sensation-seeking behavior, impulsiveness, alcohol and drug consumption, and accrued more traffic violations.

Kim et al. [20] found that engaging in driver behaviors like alcohol or drug use and not using seat belts significantly raise the likelihood of experiencing more severe crashes and injuries. Janssen [21] offers empirical support indicating that individuals who do not use seat belts tend to drive notably faster compared to seat belt users, even after adjusting for factors such as gender, age, annual mileage, and years of holding a driver’s license. Petridou and Moustaki [22] as well as Shinar [23] also reference this tendency for risk-taking behavior among individuals who do not wear seat belts. However, a few researchers treated the seatbelt use variable as endogenous, meaning inherently unsafe drivers opt not to use seat belts and are consequently more prone to being engaged in severe crashes with high injury rates due to their reckless driving tendencies [24].

Estimating the relative risk and prevalence of unrestrained vehicle occupants presents several challenges. Firstly, unrestrained drivers often exhibit a propensity for engaging in risky driving behaviors, thereby increasing accident likelihood and potentially biasing conclusions about the riskiness of being unrestrained. Secondly, it is impractical to observe every individual on the road, making it difficult to accurately ascertain the proportion of unrestrained occupants, which is crucial for risk estimation (e.g., predicting injuries or fatalities). In this study, we build upon a methodology pioneered by Levitt and Porter [25] and updated and repurposed by Karl et al. [26] to evaluate the risks associated with unrestrained vehicle occupants and their prevalence.

2. Data and Methods

This study utilized data from the Fatality Analysis Reporting System (FARS) spanning from 2000 to 2018 to assess both the relative risk and the prevalence of unrestrained vehicle occupants. The FARS database comprises over 140 coded variables, meticulously detailing each fatal crash occurrence in the United States, the District of Columbia, and Puerto Rico. Stringent data validation and quality assurance measures are performed by dedicated FARS data analysts. It is worth noting that the National Highway Traffic Safety Administration (NHTSA) has recognized FARS as “the most referenced motor vehicle crash data system in the world.” Furthermore, the methodologies and standards employed are consistently applied across all states, ensuring the provision of dependable and standardized data on this matter.

Driving without a seatbelt not only increases the risk of severe injury and death during a crash but can also indicate the risk preference of the driver, hence the probability of causing a crash. Therefore, unlike drinking and driving or distracted driving, the relative risk of an unrestrained driver estimated from Karl et al.’s [26] methodology contains information on both effects. In this study, these effects will be disentangled in the later stage of the analysis.

Two sets of analyses were performed: one on drivers and the other on passengers. The analysis on drivers utilized fatal crash data from 2000 to 2018. For analyzing passengers, data from the same period was used, however, with an additional condition: vehicles should have only one driver and one passenger. This is done because an unrestrained passenger has a negligible effect, if any, on the driver’s driving behavior. Examining the effects of both the drivers and the passengers allows to explore both the risk of vehicle occupants being unrestrained, and the risk resulting from the implied driving behavior of an unrestrained driver.

Table 1. Summary statistics with unrestrained driving details for fatal crashes (2000–2018, no constraints).

Total number of fatal crashes	555,650
Total number of fatal one-car crashes	332,350
Total number of fatal two-car crashes	196,445
Total number of fatal three-car crashes	26,855
Percentage of unrestrained drivers in all fatal crashes	28.08
Percentage of fatal one-car crashes with:
One unrestrained driver	39.90
One restrained driver	60.10
Percentage of fatal two-car crashes with:
Two unrestrained drivers	5.36
One unrestrained driver and one restrained driver	31.27
Two restrained drivers	63.38
Percentage of fatal three-car crashes with:
Three unrestrained drivers	0.80
Two unrestrained drivers and one restrained driver	5.15
One unrestrained driver and two restrained drivers	28.78
Three restrained drivers	65.27

and

Table 2. Summary statistics with unrestrained passengers details for fatal crashes (2000–2018, only vehicles with 1 driver and 1 passenger).

Total number of fatal crashes	84,483
Total number of fatal one-car crashes	73,292
Total number of fatal two-car crashes	10,784
Total number of fatal three-car crashes	407
Percentage of unrestrained drivers in all fatal crashes	37.11
Percentage of fatal one-car crashes with:
One unrestrained passenger	41.69
One restrained passenger	58.31
Percentage of fatal two-car crashes with:
Two unrestrained passengers	7.44
One unrestrained passenger and one restrained passenger	30.61
Two restrained passengers	61.95
Percentage of fatal three-car crashes with:
Three unrestrained passengers	1.72
Two unrestrained passengers and one restrained passenger	8.11
One unrestrained passenger and two restrained passengers	26.04
Three restrained passengers	64.13

provide summaries of the numbers of fatal crashes and offer statistics on the composition of each type of fatal crash within the sample data.

A total of over half a million fatal crashes were considered for analysis in this study. Table 1 illustrates an interesting trend: a substantial proportion of drivers involved in fatal crashes, more than 28%, were unrestrained. Meanwhile, Table 2 indicates that the number of crashes involving vehicles with precisely one driver and one passenger is notably smaller compared to the full sample. Interestingly, within this subset, the prevalence of unrestrained passengers in fatal crashes is even higher, with over 37% of the passengers being unrestrained.

This study employs Karl et al.’s [26] methodology to evaluate both the level of risk and the incidence of unrestrained vehicle occupants. The methodology, originally developed by Levitt and Porter [25] to estimate the riskiness of drinking and driving, was updated by Karl et al. [26] to examine the risks associated with distracted driving. Zhuang et al. [27] applied this methodology to identify the characteristics and predictors of drivers engaging in various dangerous driving behaviors. In order to undertake this analysis, it is essential to establish a foundational set of axiomatic assumptions, which include:

• There are two types of drivers/passengers, unrestrained (U) and restrained (R).
• There is equal mixing of unrestrained and restrained drivers/passengers on the road, meaning:
  • The number of interactions that a driver/passenger has with other cars is independent of the driver/passenger’s type.
  • A driver/passenger’s type does not affect the composition of the driver/passenger types with which he or she interacts.
• A fatal car crash results from a single driver’s error.
• The composition of driver/passenger type(s) in one fatal crash is independent of the composition of driver/passenger type(s) in other fatal crashes.
• The relative likelihood of an unrestrained driver, or the driver of an unrestrained passenger, causing a one-car fatal crash is equal to the relative likelihoods of such a driver causing a two- or three-car fatal crash.
• A passenger’s decision about whether to use a seatbelt or not has no impact on the driver’s driving behavior.

The first five assumptions are similar to those of Karl et al.’s [26]. The first assumption rules out the presence of any “undetermined” vehicle occupants. In essence, a vehicle occupant cannot be both restrained and unrestrained at the same time.

The second assumption requires an “equal mixing” of vehicle occupant types on the road, meaning that vehicle occupants are evenly distributed across space and time, and both types interact at rates proportional to their population shares. As the analysis shifts to smaller time–space units, this assumption is gradually relaxed.

The third assumption rules out the possibility that multiple drivers jointly contribute to a crash. The probability that multiple drivers make errors simultaneously is sufficiently small to be ignored for the sake of parsimony. For example, the likelihood that two drivers make a mistake is on the order of 10⁻¹⁸ [25], while the probability that three drivers err simultaneously is on the order of 10⁻²⁷ [26].

The fourth allows the use of multinomial distribution to model the joint distribution of vehicle occupant types involved in fatal crashes.

Finally, the fifth assumption permits the inclusion of additional data in the estimation. It does not require the probabilities of causing a one-, two-, or three-car fatal crash by a certain type of driver to be equal, only that the relative risk levels remain stable. The sixth assumption enables to estimate the direct risk of vehicle occupants being unrestrained.

Given the assumptions outlined above, we first derive the probabilities that a driver or passenger is of a particular type and interacts with zero, one, or two other vehicles, as well as the types of individuals involved in those interactions. Let $N_U$ and $N_R$ denote the average number of unrestrained and restrained drivers/passengers, respectively, present on the road within a specified geographic area and time period. At any given moment, a proportion $p$ of all drivers/passengers are involved in interactions with one other vehicle ($I_2=1$), where a two-vehicle crash may occur. Similarly, a proportion $q$ are engaged in interactions with two other vehicles ($I_3=1$), where a three-vehicle fatal crash is possible. Consequently, the remaining share of the population, $1-p-q$, does not interact with any other vehicles, thereby limiting the potential crash scenario to a single-vehicle fatality. Accordingly, the probability that a driver or passenger is of type $i$ and does not interact with any other individuals, conditional on being present on the road ($Dr=1$), is given by:

$$\begin{array}{c}Pr\left(i,I_2=0,I_3=0|Dr=1\right)={\displaystyle \frac{N_i}{N_U+N_R}}\left(1-p-q\right).\end{array}$$

(1)

The corresponding joint distributions for a driver/passenger interacting with one or two other vehicles, along with the types of drivers/passengers in those vehicles, conditional on being present on the road, are given by:

$$Pr\left(ij,I_2=1|Dr=1\right)={\displaystyle \frac{N_i{N}_j}{{\left(N_U+N_R\right)}^2}}p,$$

(2)

and

$$Pr\left(ijk,I_3=1|Dr=1\right)={\displaystyle \frac{N_i{N}_j{N}_k}{{\left(N_U+N_R\right)}^3}}q.$$

(3)

We now derive the joint probabilities that the randomly selected driver(s) or passenger(s), as described in Equation (1) through (3), are involved in a fatal crash. Let ${\mathsf{\theta }}_i$ denote the probability that a driver of type $i$ (or the driver of a vehicle carrying a type $i$ passenger) causes a fatal single-vehicle crash. Under Assumption 5, the probabilities that a type $i$ driver causes a fatal crash involving two or three vehicles are proportional to the probability of a fatal single-vehicle crash. Let $\mathsf{\delta }$ and $\mathsf{\lambda }$ denote the respective scaling parameters, such that a type $i$ driver has probabilities $\mathsf{\delta }{\mathsf{\theta }}_i$ and $\mathsf{\lambda }{\mathsf{\theta }}_i$ of causing a fatal two-vehicle or three-vehicle crash, respectively. In accordance with Assumption 3, the resulting joint probabilities over crash type (i.e., number of vehicles), driver type(s), and the occurrence of a fatal crash ($A=1$), conditional on being on the road, are expressed as follows:

$$Pr\left(i,I_2=0,I_3=0,A=1|Dr=1\right)={\displaystyle \frac{N_i{\mathsf{\theta }}_i}{N_U+N_R}}\left(1-p-q\right),$$

(4)

$$Pr\left(ij,I_2=1,A=1|Dr=1\right)\approx {\displaystyle \frac{N_i{N}_j\mathsf{\delta }\left({\mathsf{\theta }}_i+{\mathsf{\theta }}_j\right)}{{\left(N_U+N_R\right)}^2}}p,$$

(5)

and

$$Pr\left(ijk,I_3=1,A=1|Dr=1\right)\approx {\displaystyle \frac{N_i{N}_j{N}_k\mathsf{\lambda }\left({\mathsf{\theta }}_i+{\mathsf{\theta }}_j+{\mathsf{\theta }}_k\right)}{{\left(N_U+N_R\right)}^3}}q.$$

(6)

In Equations (5) and (6), the approximately equal symbols are used to reflect the assumption that the probability of multiple drivers simultaneously committing an error is vanishingly small and can be reasonably disregarded for analytical tractability (The likelihood of simultaneous error by two drivers is estimated to be on the order of ${10}^{-18}$, and for three drivers, on the order of ${10}^{-27}$ [25]). For notational simplicity, we will henceforth use equality symbols in place of approximations.

The next step is to apply Equation (4) through (6) to derive the joint probabilities of the number of vehicles and driver/passenger types involved in a crash, conditional on the occurrence of a fatal crash. These results are presented in Equation (7) through (9).

$$\begin{array}{c}Pr\left(i,I_2=0,I_3=0|A=1\right)={\displaystyle \frac{Pr\left(i,I_2=0,I_3=0,A=1|Dr=1\right)}{Pr\left(A=1|Dr=1\right)}}=\\ {\displaystyle \frac{Pr\left(i,I_2=0,I_3=0,A=1|Dr=1\right)}{Pr\left(i,I_2=0,I_3=0,A=1|Dr=1\right)+Pr\left(ij,I_2=1,A=1|Dr=1\right)+Pr\left(ijk,I_3=1,A=1|Dr=1\right)}}=\\ {\displaystyle \frac{{\left(N_U+N_R\right)}^2{N}_i{\mathsf{\theta }}_i\left(1-p-q\right)}{{\left(N_U+N_R\right)}^2\left(N_U{\mathsf{\theta }}_U+N_R{\mathsf{\theta }}_R\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} ,\end{array}$$

(7)

$$Pr\left(ij,I_2=1|A=1\right)={\displaystyle \frac{\left(N_U+N_R\right)N_i{N}_j\mathsf{\delta }\left({\mathsf{\theta }}_i+{\mathsf{\theta }}_j\right)p}{{\left(N_U+N_R\right)}^2\left(N_U{\mathsf{\theta }}_U+N_R{\mathsf{\theta }}_R\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} ,$$

(8)

$$Pr\left(ijk,I_3=1|A=1\right)={\displaystyle \frac{N_i{N}_j{N}_k\mathsf{\lambda }\left({\mathsf{\theta }}_i+{\mathsf{\theta }}_j+{\mathsf{\theta }}_k\right)q}{{\left(N_U+N_R\right)}^2\left(N_U{\mathsf{\theta }}_U+N_R{\mathsf{\theta }}_R\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} .$$

(9)

We now turn to estimating the unknown variables in Equation (7) through (9). Because the number of unknowns exceeds the number of equations, a direct solution is not possible. However, we can still determine the relative terms. Let $P_i$, $P_{ij}$, and $P_{ijk}$ denote the probabilities that the composition of driver/passenger type(s) is (are) of type $i$, types $i$ and $j$, and types $i$, $j$, and $k$, respectively, given that a fatal crash occurs. Define $\mathsf{\theta }={\mathsf{\theta }}_U/{\mathsf{\theta }}_R$ and $N=N_U/N_R$, where $\mathsf{\theta }$ captures the relative likelihood that an unrestrained driver (or a driver with an unrestrained passenger, this interpretation applies throughout) causes a fatal crash compared to a restrained counterpart. Similarly, $N$ represents the average on-road ratio of unrestrained to restrained drivers in a given geographic area and time period. Using these definitions, we can now express the probabilities associated with each accident and driver type explicitly in terms of $\mathsf{\theta }$ and $N$.

$$P_U={\displaystyle \frac{{\left(N+1\right)}^{2}N\mathsf{\theta }\left(1-p-q\right)}{{\left(N+1\right)}^2\left(N\mathsf{\theta }+1\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} ,$$

(10)

$$P_R={\displaystyle \frac{{\left(N+1\right)}^2\left(1-p-q\right)}{{\left(N+1\right)}^2\left(N\mathsf{\theta }+1\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} .$$

(11)

Equation (10) specifies the probability of a fatal crash involving a single vehicle operated by an unrestrained driver, while Equation (11) gives the corresponding probability for a vehicle driven by a restrained driver.

$$P_{UU}={\displaystyle \frac{2\left(N+1\right)N^2\mathsf{\theta }\mathsf{\delta }p}{{\left(N+1\right)}^2\left(N\mathsf{\theta }+1\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} ,$$

(12)

$$\begin{array}{c}{P}_{UR}=Pr\left(i=1,j=2,I_2=1|A=1\right)+Pr\left(i=2,j=1,I_2=1|A=1\right)\\ ={\displaystyle \frac{2\left(N+1\right)N\left(\mathsf{\theta }+1\right)\mathsf{\delta }p}{{\left(N+1\right)}^2\left(N\mathsf{\theta }+1\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} ,\end{array}$$

(13)

$$P_{RR}={\displaystyle \frac{2\left(N+1\right)\mathsf{\delta }p}{{\left(N+1\right)}^2\left(N\mathsf{\theta }+1\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} .$$

(14)

Equation (12) through (14) represent the probabilities of a fatal crash involving two vehicles, accounting for the three possible pairings of driver types: both drivers being unrestrained, both being restrained, or one driver of each type.

$$P_{UUU}={\displaystyle \frac{3N^3\mathsf{\theta }\mathsf{\lambda }q}{{\left(N+1\right)}^2\left(N\mathsf{\theta }+1\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} ,$$

(15)

$$\begin{array}{c}{P}_{UUR}=Pr\left(i=1,j=1,k=2,I_3=1|A=1\right)+Pr\left(i=1,j=2,k=1,I_3=1|A=1\right)+Pr\left(i=2,j=1,k=1,I_3=1|A=1\right)\\ ={\displaystyle \frac{3N^2\left(2\mathsf{\theta }+1\right)\mathsf{\lambda }q}{{\left(N+1\right)}^2\left(N\mathsf{\theta }+1\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} ,\end{array}$$

(16)

$$\begin{array}{c}{P}_{URR}=Pr\left(i=1,j=2,k=2,I_3=1|A=1\right)+Pr\left(i=2,j=1,k=2,I_3=1|A=1\right)+Pr\left(i=2,j=2,k=1,I_3=1|A=1\right)\\ ={\displaystyle \frac{3N\left(\mathsf{\theta }+2\right)\mathsf{\lambda }q}{{\left(N+1\right)}^2\left(N\mathsf{\theta }+1\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} ,\end{array}$$

(17)

$$P_{RRR}={\displaystyle \frac{3\mathsf{\lambda }q}{{\left(N+1\right)}^2\left(N\mathsf{\theta }+1\right)\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}} .$$

(18)

Equation (15) through (18) represent the probabilities of a fatal crash involving three vehicles, covering the four possible combinations of driver types: all three drivers are unrestrained; two unrestrained drivers and one restrained; one unrestrained driver and two restrained; or all three drivers are restrained.

Let $A_i$, $A_{ij}$, and $A_{ijk}$ be the numbers of fatal crashes involving driver(s)/passenger(s) of type $i$, types $i$ and $j$, and types $i$, $j$, and $k$, respectively, and denote $A_{total}$ as the total number of fatal crashes. Using the ratios of the various accident types, we can derive the following additional expressions for our model parameters:

$$\begin{array}{c}{\displaystyle \frac{N_U{\mathsf{\theta }}_U\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}{N_R{\mathsf{\theta }}_R\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]}}=N\\ ={\displaystyle \frac{A_{UUR}\left(\frac{2\mathsf{\theta }}{2\mathsf{\theta }+1}\right)+A_{URR}\left(\frac{\mathsf{\theta }}{\mathsf{\theta }+2}\right)+A_{UR}\left(\frac{\mathsf{\theta }}{\mathsf{\theta }+1}\right)+A_{UUU}+A_{UU}+A_U}{A_{UUR}\left(\frac{1}{2\mathsf{\theta }+1}\right)+A_{URR}\left(\frac{2}{\mathsf{\theta }+2}\right)+A_{UR}\left(\frac{1}{\mathsf{\theta }+1}\right)+A_{RRR}+A_{RR}+A_R}},\end{array}$$

(19)

$$\begin{array}{c}{\displaystyle \frac{\left(1-p-q\right)\left(N_U{\mathsf{\theta }}_U+N_R{\mathsf{\theta }}_R\right)}{\left[\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }\right]\left(N_U{\mathsf{\theta }}_U+N_R{\mathsf{\theta }}_R\right)}}\\ ={\displaystyle \frac{\left(1-p-q\right)}{\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }}}={\displaystyle \frac{A_U+A_R}{A_{total}}},\end{array}$$

(20)

$${\displaystyle \frac{2p\mathsf{\delta }}{\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }}}={\displaystyle \frac{A_{UU}+A_{UR}+A_{RR}}{A_{total}}},$$

(21)

$${\displaystyle \frac{3q\mathsf{\lambda }}{\left(1-p-q\right)+2p\mathsf{\delta }+3q\mathsf{\lambda }}}={\displaystyle \frac{A_{UUU}+A_{UUR}+A_{URR}+A_{RRR}}{A_{total}}}.$$

(22)

Equation (19) expresses the ratio of fatal crashes involving unrestrained drivers to those involving restrained drivers in terms of $\mathsf{\theta }$ and the observed fatal crash counts. Similarly, Equations (20)–(22) represent the proportions of single-car, two-car, and three-car fatal crashes relative to the total, also as functions of $\mathsf{\theta }$.

Given Assumption 4, which ensures the independence of driver type composition in fatal crashes, the joint distribution of involved driver types follows a multinomial distribution. Accordingly, the likelihood function is derived as follows:

$$\begin{array}{c}\mathit{Pr}\left(A_U, A_R, A_{UU}, A_{UR}, A_{RR}, A_{UUU}, A_{UUR}, A_{URR}, A_{RRR}|A_{total}\right)\\ ={\displaystyle \frac{{(A}_U+A_R+A_{UU}+A_{UR}+A_{RR}+A_{UUU}+A_{UUR}+A_{URR}+A_{RRR})!}{A_U!A_R!A_{UU}!A_{UR}!A_{RR}!A_{UUU}!A_{UUR}!A_{URR}!A_{RRR}!}} \times \\ {{(P}_U)}^{A_U}{{(P}_R)}^{A_R}{{(P}_{UU})}^{A_{UU}}{{(P}_{UR})}^{A_{UR}}{{(P}_{RR})}^{A_{RR}}{{(P}_{UUU})}^{A_{UUU}}{{(P}_{UUR})}^{A_{UUR}}{{(P}_{URR})}^{A_{URR}}{{(P}_{RRR})}^{A_{RRR}},\end{array}$$

(23)

The probabilities in terms of $\mathsf{\theta }$ and $N$ for each crash type and driver/passenger type are explicitly expressed. By doing so, we can estimate $\mathsf{\theta }$ by directly maximizing the log likelihood function (the natural logarithm of Equation (1)) solely in terms of $\mathsf{\theta }$. This optimization is achieved using a nonlinear programming (NLP) method.

Once the estimates of $\mathsf{\theta }$ for various geographic and temporal observation units were obtained, these $\mathsf{\theta }$ values are substituted back into the equations to express the log likelihood function solely in terms of $N$. This allows us to estimate the implied relative exposure of unrestrained drivers/passenger for each geographic and temporal unit by maximizing the log likelihood function. In summary, this study first estimates $\mathsf{\theta }$ by maximizing the log likelihood function with respect to $\mathsf{\theta }$, and then, by using these estimated $\mathsf{\theta }$ values, the relative exposure of unrestrained drivers/passengers ($N$) was estimated by maximizing the log likelihood function with respect to $N$ for different geographic and temporal observation units.

Note that in the second set of analyses, the relative riskiness and prevalence specifically for drivers of unrestrained passengers were estimated, not for unrestrained passengers themselves. It is important to understand that the prevalence of drivers with unrestrained passengers is equivalent to the prevalence of unrestrained passengers in these vehicles. This equivalence arises because, in the analysis, each vehicle accommodates only one passenger, who can either be restrained or unrestrained. Assumption 6a further implies that the percentage of unrestrained passengers in one-passenger vehicles is the same as the percentage of unrestrained passengers overall. Consequently, the estimated fraction of drivers with unrestrained passengers is equal to the fraction of unrestrained passengers overall.

Next, the relative risk of vehicle occupants being unrestrained were calculated. Denote the fraction of unrestrained passengers as ${F(p}_u)$ and the fraction of restrained passengers as ${F(p}_r)$, the ratio of deaths from unrestrained passengers to deaths from all passengers ${R(p}_u)$, can be written as:

$${R(p}_u)={\displaystyle \frac{{F(p}_u) \times {P(p}_u){D(p}_u)}{{F(p}_u) \times {P(p}_u) \times { D(p}_u)+{F(p}_r) \times {P(p}_r) \times {D(p}_r)}},$$

(24)

where ${P(p}_u)$ and ${P(p}_r)$ are the probabilities of getting into a crash that is severe enough to cause fatalities, of an unrestrained passenger and a restrained passenger, respectively. ${D(p}_u)$ and ${D(p}_r)$ are the probabilities of being killed from such crash, of an unrestrained passenger and a restrained passenger, respectively.

Because passengers are not driving the vehicle, from assumption 6b, ${P(p}_u)= {P(p}_r)$. Furthermore, it can be observed ${R(p}_u)$ from the data, therefore, the relative risk of being killed in a crash that is severe enough to cause fatalities of an unrestrained passenger can be calculated as:

$$D={\displaystyle \frac{{D(p}_u)}{{D(p}_r)}}={\displaystyle \frac{{F(p}_r) \times {R(p}_r)}{{F(p}_u) \times [1-{R(p}_r\left)\right]}},$$

(25)

where ${\displaystyle \frac{{F(p}_r)}{{F(p}_u)}}={\displaystyle \frac{1}{N}}$, and $N$ is the estimate of fraction that is obtained in the second set of the analyses.

3. Results

Table 3. Estimates for relative riskiness and fraction of unrestrained drivers (2000–2018, no constraints).

	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
Unit of Observation For “Equal Mixing” Assumption	All Data	Hour	Hour × Year	Hour × Year × Weekend	Hour × Region × Year	Hour × Region × Year × Weekend	Hour × State × Year	Hour × State × Year × Weekend
Relative riskiness of unrestrained drivers	5.30	4.51	4.76	4.64	5.16	5.07	5.41	5.37
	(0.00)	(0.19)	(0.08)	(0.08)	(0.07)	(0.08)	(0.10)	(0.10)
Implied fraction of unrestrained drivers	0.0972	0.1066	0.1063	0.1049	0.0984	0.0966	0.0829	0.0805
	(0.0000)	(0.0042)	(0.0018)	(0.0015)	(0.0001)	(0.0009)	(0.0009)	(0.0008)
Degrees of freedom	3	26	458	914	4106	8128	19,741	32,140

presents the maximum likelihood estimates of the relative fatal crash risks and fractions of unrestrained drivers (

Table 4. Estimates for relative riskiness and fraction of drivers of unrestrained passengers and fractions of unrestrained passengers (2000–2018, only vehicles with 1 driver and 1 passenger).

	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
Unit of Observation for “Equal Mixing” Assumption	All Data	Hour	Hour × Year	Hour × Year × Weekend	Hour × Region × Year	Hour × Region × Year × Weekend	Hour × State × Year	Hour × State × Year × Weekend
Relative riskiness of drivers of unrestrained passengers	4.72	3.84	4.07	4.13	4.39	4.56	4.48	4.79
	(0.00)	(0.27)	(0.18)	(0.21)	(0.22)	(0.24)	(0.30)	(0.36)
Implied fraction of unrestrained passengers	0.1260	0.1464	0.1471	0.1415	0. 1384	0.1319	0.1241	0.1176
	(0.0000)	(0.0076)	(0.0028)	(0.0022)	(0.0017)	(0.0015)	(0.0018)	(0.0016)
Degrees of freedom	3	26	458	914	3923	6521	11,107	12,693

for fractions of unrestrained passengers). The degrees of freedom reported in Table 3 and Table 4 correspond to the number of observational units that contain at least one crash (plus two). These estimates are provided for progressively finer subsets of the data. With each step from column (1) to column (8), the “equal mixing” assumption was relaxed, which in turn mitigates potential biases in the relative risk estimates, ultimately leading to more accurate results.

A noticeable pattern emerges as one moves from the left to the right in the table. The bias observed in this method is quite mild, and the estimates remain remarkably consistent across the various columns. This consistency indicates that the approach is robust in estimating both the relative riskiness and the relative exposure. It is important to clarify that the estimated relative risks disclosed in Table 4 pertain to drivers of unrestrained passengers rather than unrestrained passengers themselves. This distinction arises because the direct determination of relative risks for unrestrained passengers being killed in a severe crash is not feasible. However, these relative risks for unrestrained passengers can still be calculated using Equation (3).

The degrees of freedom presented in Table 4 imply that when breaking down the data from column (7) to column (8), a considerable amount of information is lost. This is attributed to numerous time–space units lacking observations of fatal crashes involving at least one unrestrained passenger or any crashes altogether. The implication is that the levels of disaggregation become overly refined (going beyond column (7)), primarily due to the limited number of fatal crashes, especially in the context of multi-car fatal crashes where all vehicles have precisely one passenger. Consequently, the results derived from column (7) are likely the most accurate estimates for both relative riskiness and prevalence. We will employ these estimates in the subsequent steps of the analysis.

The findings presented in Table 3 and Table 4 reveal that the fatal crash risk for unrestrained drivers is approximately 4.51 to 5.41 times higher than that for restrained drivers. Similarly, the risk for drivers of unrestrained passengers is about 3.84 to 4.79 times greater than that for drivers of restrained passengers. Furthermore, these results also highlight that at any given moment, between 8.05 to 10.66 percent of drivers and 11.76 to 14.71 percent of passengers on the road are not using restraints.

With the fractions of unrestrained passengers now estimated, the risk of unrestrained passengers being killed in severe crashes relative to restrained passengers using Equation (3) can be calculated. The FARS data reveals that ${R(p}_r)=0.6052974$, indicating that over 60.5% of all passengers killed were unrestrained. Utilizing the estimated fraction of unrestrained passengers from column (7) of Table 4, the calculation yields a result suggesting that unrestrained passengers are approximately 4.27 times more likely to be killed than their restrained counterparts.

The estimated relative risk of unrestrained passengers is statistically indistinguishable from that of drivers with unrestrained passengers (4.48 with a standard error of 0.30). This confirms that the decision of passengers not to wear seatbelts has no discernible impact on the driving behavior of the driver. However, this estimate is significantly lower than the estimated risk for unrestrained drivers (5.37 with a standard error of 0.10). Assuming the risk of being unrestrained is the same for both drivers and passengers, this result suggests that not wearing a seatbelt not only increases the risk of the driver being killed but also indicates that the driver exhibits riskier driving behavior, thereby elevating the risk of causing a fatal crash.

Moving forward, the number of potential avoidable deaths if all vehicle occupants were restrained were calculated. Assuming the risk of unrestrained vehicle occupants being killed is consistent for both drivers and passengers, and using the estimated relative risk ($\mathsf{\theta }$) of 4.27, then $(\mathsf{\theta }-1)/\mathsf{\theta }$, or 76.58% of all deaths from unrestrained vehicle occupants could be avoided.

Table 5. Estimates of avoidable deaths.

Year	Total Number of Deaths from Unrestrained Vehicle Occupants	Potential Avoidable Deaths
2000	19,523	14,951
2001	19,445	14,891
2002	19,748	15,123
2003	18,944	14,507
2004	18,725	14,340
2005	18,769	14,373
2006	18,112	13,870
2007	17,102	13,097
2008	15,603	11,949
2009	13,884	10,632
2010	11,000	8424
2011	10,522	8058
2012	10,640	8148
2013	9894	7577
2014	9662	7399
2015	10,224	7830
2016	10,845	8305
2017	10,468	8016
2018	10,222	7828
Average	14,386	11,017
Total	273,332	209,320

provides such estimates by year.

As shown in Table 5, the total number of deaths among unrestrained vehicle occupants was approximately 18,000–19,000 per year between 2000 and 2006, with an estimated 14,000–15,000 of these classified as potentially avoidable deaths. Both figures began to gradually decline thereafter: by 2010, total deaths had decreased to about 11,000 per year, while potentially avoidable deaths fell to roughly 8000. From 2010 to 2018, both measures remained relatively stable. Improvements in vehicle safety features in newer cars may have contributed to the observed decline in these estimates.

The U.S. Department of Transportation annually reports estimates of the value of a statistical life. In the most recent estimate from 2023, they state this value as $13.2 million. Multiplying this value by the estimated 209,320 avoidable deaths occurring during our sample period suggests that the value of lives lost due to unrestrained vehicle occupants totaled approximately $2763 billion between 2000 and 2018, exceeding $2.76 trillion.

4. Discussion and Conclusions

The findings of this study underscore the critical importance of seatbelt usage in mitigating the severity of injuries and fatalities in traffic accidents. Despite extensive efforts to promote seatbelt use through legislation, educational campaigns, and technological interventions, a disconcerting reality persists: a substantial proportion of drivers and passengers continue to disregard this fundamental safety measure.

In this paper, we disentangle the risk associated with vehicle occupants being unrestrained from the risk stemming from the driving behavior attributed to an unrestrained driver. The analysis uncovers a troubling pattern: unrestrained passengers are exposed to greater crash risks, underscoring the interdependence of seatbelt usage and the safety of vehicle occupants. Additionally, unrestrained drivers are substantially more likely to be involved in fatal crashes compared to their restrained counterparts, attributable to both their lack of restraint and their propensity for engaging in risky driving behaviors. These estimates rely on the assumption of equal mixing of restrained and unrestrained drivers/passengers on the road and the abstraction that fatal crashes can be attributed to a single driver’s error.

Contextually, restraint use is best situated within the during-crash phase of the Haddon matrix [28], because its primary function is to reduce occupant injury severity once a crash occurs. Given that, the passenger-related findings of this study are consistent with that framework, as they correspond to the relative likelihood of fatality in crashes severe enough to result in death. For drivers, although restraint use is also during crash protective factor, the estimated relative risk for unrestrained drivers may additionally reflect broader behavioral characteristics associated with restraint non-use. Therefore, driver related estimates should be interpreted as representing a combined effect of restrained non- use and correlated risky driving behavior. Also, the interpretation of passenger-related estimates assumes that passenger restraint decisions do not influence driver behavior, which enables a closer approximation of the direct fatality risk associated with restraint non-use.

More specifically, our investigation delves into both the relative risk of fatal crashes involvement for unrestrained drivers and the relative risk of being killed for unrestrained passengers, providing useful insights into the potential consequences of flouting seatbelt regulations. The results indicate that unrestrained passengers, constituting approximately 12% of all passengers on the road, encounter roughly 4.3 times greater likelihood of fatality in severe crashes, emphasizing the pressing necessity for enhanced seatbelt compliance among all vehicle occupants. It should be noted that these estimates are derived from vehicles with exactly one driver and one passenger, and their broader generalization depends on the extent to which this configuration reflects overall passenger behavior.

Equally pertinent are the implications of our findings for unrestrained drivers, whose elevated risk profiles extend beyond mere personal choice to reflect broader patterns of hazardous driving behavior. Our analysis demonstrates that unrestrained drivers comprise over 8% of all drivers on the road and present a risk approximately 5.4 times higher in causing fatal crashes compared to their restrained counterparts. This is based on the assumption that the relative risk associated with restraint status remains stable across one-, two-, and three-vehicle fatal crashes. Moreover, the revelation necessitates a comprehensive re-evaluation of existing strategies aimed at promoting seatbelt compliance, with a particular focus on addressing the underlying factors contributing to non-adherence, including risk perceptions, safety attitudes, and socio-demographic determinants.

Furthermore, the analysis offers compelling evidence of the substantial human and economic toll associated with unrestrained vehicle occupants. By extrapolating the estimated relative risk to calculate the number of avoidable deaths if all vehicle occupants were restrained, the study underscores the magnitude of lives lost due to non-compliance with seatbelt usage. The staggering economic cost of these preventable fatalities underscores the urgent need for continued efforts to promote seatbelt use and enhance road safety measures. These estimates assume that the relative risks remain stable across crash types and the aggregated dataset, and should therefore be interpreted as approximate, model-dependent measures.

However, achieving meaningful progress in this endeavor demands more than just exhortations to buckle up; it necessitates a multifaceted approach encompassing targeted interventions tailored to address the diverse array of factors influencing seatbelt non-compliance. These interventions must be informed by rigorous research and data-driven insights to ensure their efficacy and sustainability over time, as well as to monitor progress towards improving seatbelt usage rates. The effectiveness of such interventions, however, may depend on behavioral, demographic, and contextual factors not explicitly captured within the scope of this study.

The study limitations stem from the set of assumptions required to enable tractable estimation of relative risk and prevalence. The assumption that a fatal crash results from a single driver’s error represents an abstraction of complex real-world crash dynamics, where multiple drivers, environmental conditions, and interactions may contribute. Violations of this assumption may influence the estimated risk parameters by attributing shared responsibility to a single driver type. Additionally, a formal sensitivity analysis was not conducted; therefore, the robustness of the estimates under alternative assumptions cannot be fully assessed. Accordingly, the results should be interpreted as conditional on the modeling framework and its underlying assumptions. It is therefore recommended that policies and crash countermeasures rely on further robust data analysis, one that includes additional variables to fully capture the wide range of factors directly influencing seatbelt use and, by extension, the severity of crashes.

Despite the limitations of this study, the findings serve as a clarion call for concerted action to promote seatbelt utilization and bolster road safety protocols. By doing so, we can alleviate the burden of preventable injuries and fatalities on individuals, families, and society at large, thus fostering a safer and more secure transportation environment for all.