Bivariate ordered probit modelling of motorcycle riders and pillion passengers’ injury severities relationship and associated risk factors

Abstract

This study simultaneously modelled the injury severity of motorcycle riders and their pillion passengers and determine the associated risk factors. The analysis is based on motorcycle crashes data in Ashanti region of Ghana spanning from 2017 to 2019. The study implemented bivariate ordered probit model to identify the possible risk factors under the premise that the injury severity of pillion passenger is endogenously related to that of the rider in the event of crash. The model provides more efficient estimates by considered the common unobserved factors shared between rider and pillion passenger. The result shows a significant positive relationship between the two injury severities with a correlation coefficient of 0.63. Thus, the unobservable factors that increase the probability of the rider to sustain more severe injury in the event of crash also increase that of their corresponding pillion passenger. The rider and their pillion passenger injury severities have different propensity to some of the risk factors including passengers’ gender, day of week, road width and light condition. In addition, the study found that time of day, weather condition, collision type, and number of vehicles involved in the crash jointly influence the injury severity of both rider and pillion passenger significantly.

Keywords:

• Bivariate ordered probit model
• injury severity
• motorcycle riders
• pillion passenger
• risk factors

1. Introduction

Road traffic crashes have been one of the major concerns to public health officials and policymakers. Studies have indicated that over 1.3 million individuals are killed each year due to road crashes and as many as 50 million people suffer non-fatal injuries worldwide (WHO, 2018). It is also characterised with social distress, emotional, and psychological burden on non-fatally injured crash victims, families of victims, and the community at large (Marzoug et al., 2022; WHO, 2018). Globally, road traffic crashes has been estimated to cost the world’s economy approximately US$1.8 trillion within 2015 and 2030 (Chen et al., 2019b).

Various studies have shown that among road traffic crashes, motorcycle injuries are more deadly than crashes resulting from four-wheel vehicles (Ijaz et al., 2022; Venkatraman et al., 2021). For instance, a study by Se et al. (2021) revealed that over 70% of all death caused by road accidents in Thailand were related to motorcycle crashes. Similarly, Adeleye et al. (2019) established that motorcycle injuries account 57% of all road traffic injuries in developing countries in Africa with 73.4% of these crashes involve riders and pillion passengers. This makes motorcycle related crashes an inevitably dangerous in transportation and injury prevention. According to Baptistella et al. (2023), a motorcycle crash victim on an average will incur over US$3000 and is likely to be hospitalised for about 1 week to receive treatment for injuries related to motorcycle crashes. Most motorcycle injuries occur as a result insufficient protection such as seatbelts, airbags, and collision avoidance technology in most four-wheel vehicles which are unavailable in motorcycles (Rifaat et al., 2012). Crashes resulting from motorcycles are considered to be a significant concern in transportation safety particularly in the low-and middle-income countries (Ijaz et al., 2021; Wang et al., 2021). According to Wang et al. (2021), the incidence of death cases associated with motorcycle crashes in low and middle-income countries are more than two times of death cases in developed countries.

There have been several studies that have explored the injury severity among motorcycle users. For instance, Zhao et al. (2011) compared injuries severities sustained by riders and pillion passengers in fatal head-on motorcycle crashes in Chongqing, China using Chi-square test. The study found a significant difference in the distribution of injuries between riders and pillion passengers. The study further explained that riders were more likely to suffer more severe chest and abdominal injuries than passengers. In a prospective cohort pilot study conducted in Douala, Cameroon, Chichom-Mefire et al. (2015) found that riders were more likely to die or sustain head and neck injury compared to pillion passengers in the event of road traffic crash. However, they both have equal risk of lower limb injuries. In a similar study involving matched hospitalised rider and pillion passenger in Shiraz, Iran, Yadollahi and Jamali (2019) concluded that the injury severity score for rider was significantly higher than pillion passengers. Furthermore, the study found that riders were more likely to suffer from injuries in the abdomen, extremities, spine, and pelvis than their pillion passengers. Other recent studies (Aidoo & Amoh-Gyimah, 2020; Tamakloe et al., 2022; Vajari et al., 2020; Wang et al., 2021) have also been conducted to understand the contributing factors influencing the injury severities associated with motorcycle riders and their pillion passengers. In most previous studies, the level of injury severity differ for motorcycle riders without pillion passengers compared to riders with pillion passengers when involved in crashes (Aidoo & Amoh-Gyimah, 2020; Kashani et al., 2014).

In the case of riders with pillion passengers injury severity modelling, most of the existing studies have developed separate univariate models to depict the injury severities of the rider and pillion passengers with an assumption that the two variables (injury severities) are independent (Aidoo & Amoh-Gyimah, 2020; Khan et al., 2015a; Wang & Kim, 2019). However, for crashes involving both riders and pillion passengers, the injury severities of both are intertwined due to some common characteristics they share together including violations/risky behaviour, lack of protective gear, road environment, collision type and other unobservable characteristics (Chiou et al., 2020). If the interrelationship between the two injury severities (injury severities of rider and pillion passenger) is significant and being influenced by common characteristics, then univariate models for such data may be inefficient, and the parameters will be biased (Chiou et al., 2020). This is because the common risk factors that simultaneously affect the injury severity of both the rider and pillion passenger will be ignored. Hence, accurately depicting the risk factors in both injury severities will require a simultaneous modelling approach that accounts for such intercorrelation. Existing literature shows that there are limited studies that models these two injury severities jointly and account for such intercorrelation.

This study contributes to the ongoing discussion by developing a joint model for the injury severity of motorcycle riders and pillion passengers using a bivariate ordered probit modelling approach. The use of the bivariate ordered probit model as opposed to separate univariate ordered probit model for each injury severity data is based on the premise that the injury severity of a motorcycle rider is endogenously related to that of their pillion passenger and other commonly shared characteristics. Thus, the use of such model makes it possible to estimate and test the intercorrelation between the injury severity of riders and their pillion passengers’ if any. The bivariate ordered probit modelling approach and other variants have successfully been implemented to model similar intercorrelated dependent variables in the transportation studies (Tsavdari et al., 2022). For instance, Sahebi et al. (2023) examined factors affecting driving risk perception of truck and passenger car drivers in Iran using bivariate ordered probit model. Bangladesh. Ibrahim et al. (2022) utilised simultaneous bivariate ordered probit approach to assess traveller characteristics and trip information influencing the boarding and alighting experience at bus stops in developing urban city. Rahman et al. (2021) developed bivariate ordered probit model to examine the effect of socio-economic and demographic factors on motorcycle and car ride-sourcing in Dhaka city. Hasnine et al. (2020) investigated the link between distance travelled and health condition of e-bike users and the associated factors in Toronto using bivariate ordered probit model. The effectiveness of enforcement levels of speed limit and drink driving laws and associated factors were explored by Wali et al. (2017) using bivariate ordered probit model. Russo et al. (2014) developed random parameters bivariate ordered probit model to account for crash correlation and unobserved heterogeneity in factors affecting the degree of injury sustained by at-fault and not-at-fault drivers involved in angle collisions. Anastasopoulos et al. (2012) investigated factors that affect household automobile and motorcycle ownership using random parameters bivariate ordered probit model to account for unobserved heterogeneity in the data and commonly shared characteristics. The injury severity of drivers and passengers in collisions with fixed object was examined by Yamamoto and Shankar (2004) using bivariate ordered probit model to control for unobservable factors influencing both injury severities.

Although, few studies have been conducted in Ghana to improve our understanding on motorcycle use injury severity and associated risk factors. These studies have either narrowly focused on head injury of crash survivors (Appiah et al., 2022) and particular road segments (Agyemang et al., 2021; Tamakloe et al., 2022) or too general across the country (Aidoo & Amoh-Gyimah, 2020) limiting its usage for regional safety purposes. In addition, the models used in the previous study assumed no possible relationship between the injury severity of the rider and that of their pillion passengers. To address these drawbacks, this study explores the use of bivariate ordered probit model to characterise the injury severity of motorcycle users in the Ashanti region of Ghana. To the best knowledge of the authors’, this is the first attempt of modelling motorcycle users’ injury severities in Ghana using a bivariate ordered probit model accounting for dependency in the injury serveries which is likely to improve model estimation. The findings of this study will provide potential risk factors for safety management and also provide basis for road safety data analysis particularly the choice of model and their impact on the results.

2. Materials and methods

2.1. Data description

The data used in this study were police-reported road traffic crashes involving motorcycle riders and pillion passengers in the Ashanti Region of Ghana. The data were extracted from the National Road Traffic Accident Databased managed by the Building and Road Research Institute (BRRI) of the Council for Scientific and Industrial Research, Ghana. The records in the database were compiled from road traffic crashes files from the Motor Traffic and Transport Unit (MTTU) of the Ghana Police Service. The data covers a 3-year span from 2017 to 2019. The total number of observations in the study was 294 motorcycle crashes, with each crash involving at least one pillion passenger and a rider. The injury severities of the rider and pillion passenger were categorised into three levels: fatal, serious and minor. Fatal injury describes a situation where the casualty died of injuries sustained within 30 days of occurrence of the crash. Serious injury describes a situation where the casualty is hospitalised for more than 24 h. Minor injury describes a situation where the casualty only sustained minor injury which requires at most first-aid attention (NRSA, 2020). Several independent variables that describe the characteristics of the rider/pillion passenger, roadway geometry, and environmental condition were considered.

The description of the data and the associated independent variables are presented in Table 1. It is observed that most of the injury severity relating to riders and pillion passengers were serious, and constituted 46% and 58%, respectively, of the total crashes. All the riders involved in crashes were found to be males whilst about 74% of the pillion passengers were males. Although, the rider’s gender variable will not be useful in further analysis since there is no variation. It has been included in the table to describe pattern of motorcycle crashes in relation to gender in the study area. The majority of the riders and pillion passengers were between 20 and 40 years. Most of the crash cases (71.43%) occurred during weekdays, usually daytime during off-peak periods. and involved multiple vehicles (83%). Furthermore, crashes occurring during the daytime are slightly higher than those at night and accounted for 63% of the total crashes. More so, it can be seen from Table 1 that the distribution of injury severity for crashes during daylight conditions is similar to crashes on straight and flat roads among riders.

Table 1. Distribution of riders and pillion passengers’ injury severities classified by other observed variables in the study.

Variables	Number of crashes (%)	Injury severity for rider (%)			Injury severity for pillion passenger (%)
		Fatal	Serious	Minor	Fatal	Serious	Minor
Total	294 (100.00)	17.35	46.26	36.39	17.35	57.48	25.17
Rider’s gender
Male	294 (100.00)	100.00	100.00	100.00	100.00	100.00	100.00
Female	0 (0.00)	0.00	0.00	0.00	0.00	0.00	0.00
Passenger’s gender
Male	218 (74.15)	94.12	70.59	69.16	86.27	69.82	75.68
Female	76 (25.85)	5.88	29.41	30.84	13.73	30.18	24.32
Rider’s age
Less than 20	31 (10.54)	1.96	13.97	10.28	11.76	11.24	8.11
20–40	217 (73.81)	88.2	69.12	72.90	78.43	73.37	71.62
40+	46 (15.65)	9.80	16.91	16.82	9.80	15.38	20.27
Passenger’s age
Less than 20	54 (18.37)	7.84	22.06	18.37	9.80	21.30	17.57
20–40	184 (62.59)	62.75	60.29	65.42	62.75	60.36	67.57
40+	56 (19.04)	29.41	17.65	15.89	27.45	18.34	14.86
Rider action
Going ahead	270 (91.84)	92.16	95.59	86.92	92.16	93.49	87.84
Other	24 (8.16)	7.84	4.41	13.08	7.84	6.51	12.16
Motor defects
Yes	5 (1.70)	1.96	2.21	0.93	13.73	13.02	12.16
No	289 (98.30)	98.04	97.79	99.07	86.27	86.98	87.84
Number of vehicles
Single	49 (16.67)	15.69	18.38	14.95	15.69	21.30	6.76
Multiple	245 (83.33)	84.31	81.38	85.05	84.31	78.70	93.24
Day of week
Weekday	210 (71.43)	74.51	75.00	65.42	76.47	71.60	67.57
Weekend	84 (28.57)	25.49	25.00	34.58	23.53	28.40	32.43
Time of day
AM Peak (6:00–10:00am)	48 (16.33)	9.80	11.76	25.23	7.84	13.61	28.38
PM Peak (3:00–7:00pm)	99 (33.67)	37.25	34.56	30.84	27.45	37.28	29.73
Off Peak (All other times)	147 (50.00)	52.94	53.68	43.93	64.71	49.11	41.89
Weather conditions
Clear	261 (88.78)	78.43	88.97	93.46	84.31	88.76	91.89
Others	33 (11.22)	21.57	11.03	6.54	15.69	11.24	8.11
Light conditions
Day	185 (62.93)	52.94	65.44	64.49	70.59	62.72	58.11
Night	109 (37.07)	47.06	34.56	35.51	29.41	37.28	41.89
Road descriptions
Straight and flat	251 (85.37)	80.39	86.03	86.92	82.35	84.62	89.19
Other	43 (14.63)	19.61	13.97	13.08	17.65	15.38	10.81
Road surface type
Paved	209 (71.09)	82.35	63.97	74.77	72.55	65.09	83.78
Unpaved	85 (28.91)	17.65	36.03	25.23	27.45	43.91	16.22
Shoulder condition
Good	177 (60.20)	72.55	55.88	59.81	54.90	57.99	68.92
Not good	65 (22.11)	11.76	27.94	19.63	29.41	22.49	16.22
No shoulder	52 (17.69)	15.69	16.18	20.56	15.69	19.53	14.86
Location type
Not at junction	250 (85.03)	90.20	86.76	80.03	86.27	86.98	79.73
At junction	44 (14.97)	9.80	13.24	19.63	13.73	13.02	20.27
Traffic control
Non-signalized	267 (90.82)	92.16	92.65	87.85	88.24	92.31	89.19
Signalized	27 (9.18)	7.84	7.35	12.15	11.76	7.69	10.81
Collision type
Head-on	88 (29.93)	54.90	27.94	20.56	43.14	29.59	21.62
Rear-end	84 (28.57)	23.53	25.74	34.58	31.37	23.67	37.84
Others	122 (41.50)	21.57	46.32	44.86	25.49	46.75	40.54
Road separation
Median	29 (9.86)	7.84	11.76	8.41	7.84	10.65	9.46
No median	265 (90.14)	92.16	88.24	91.59	92.16	89.35	90.54
Road width
Less than 6	145 (49.32)	29.41	49.26	58.88	43.14	49.11	54.05
Between 6 and 12	149 (50.68)	70.59	50.74	41.12	56.86	50.89	45.95

Also, about 89% of minor injuries occurred among pillion passengers on straight and flat roads. Even though few (15%) crashes occurred at junctions, 71% of crashes happened on paved road surfaces, with 82% of riders and 73% of pillion passengers experiencing fatal crashes on paved roads. However, 91% of crashes occurred at non-signalised intersections, and 51% occurred on road widths between 6 and 12 meters. From the study, 60% of crashes occurred on roads with good shoulder conditions.

2.2. Bivariate ordered probit model

The study adopts a bivariate ordered probit model to determine the significant risk factors that influence the severity of injuries traffic crash event, and account for the intercorrelation between the rider and pillion passenger. The bivariate ordered probit model is an extended form of the univariate ordered probit model, which accounts for the variation in an ordered category by estimating the response variable as a function of one or more explanatory variables (Sajaia, 2008). The latent utility function for the injury severity of the rider is represented by ${\mathrm{y}}_{\text{ri}}^{\text{*}}$ while the corresponding injury severity of the pillion passenger is represented by ${\mathrm{y}}_{\text{pi}}^{\text{*}}$. The joint utility equation becomes: (1) $$y_{\text{ri}}^{\text{*}}=X_i^{\prime }{\beta }_r+{\epsilon }_{\text{ri}}$$(1) (2) $$y_{\text{pi}}^{\text{*}}=X_i^{\prime }{\beta }_p+{\epsilon }_{\text{pi}}$$(2) where $\mathbf{X}$ represents a vector of independent variables that is assumed to influence the dependent variables ${\mathrm{y}}_{\text{ri}}^{\text{*}}$ and ${\mathrm{y}}_{\text{pi}}^{\text{*}}$, ${\beta }_r$ and ${\beta }_p$ are vectors of estimable model parameters, and ${\epsilon }_{\text{ri}}$ and ${\epsilon }_{\text{pi}}$ denote the random components that capture all latent factors connected with the two injury severities and follow a bivariate normal distribution such that: (3) $$\left(\begin{array}{c}{\epsilon }_{\text{ri}}\\ {\epsilon }_{\text{pi}}\end{array}\right)\sim N\left[\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right)\right]$$(3) where ρ represents the correlation parameter between the two random components. For bivariate ordered probit model assumption to hold for this data, the estimated value for $\rho$ must be significantly different from zero. If $\rho =0,$ it indicates that the two dependent variables are not correlated and thus, two independent univariate ordered probit models should be fitted (Ibrahim et al., 2022). Since ${\mathrm{y}}_{\text{ri}}^{\text{*}}$ and ${\mathrm{y}}_{\text{pi}}^{\text{*}}$ are latent unobserved variables, we assumed ${\mathrm{y}}_{\text{ri}}$ and ${\mathrm{y}}_{\text{pi}}$ as the observed variable which captures the injury severity of the rider and pillion passenger respectively, for each motorcycle crash. These two observed ordered variables relate to the unobserved latent variables through the following relation: (4) $$y_{\mathit{\text{ri}}}=\{\begin{array}{cccc}1\hfill & \text{if}\hfill & {\mathrm{y}}_{\text{ri}}^{\text{*}}\le {\mu }_{r1}\hfill & \mathrm{(}\text{minor}\mathrm{)}\hfill \\ 2\hfill & \text{if}\hfill & {\mu }_{r1}<{\mathrm{y}}_{\text{ri}}^{\text{*}}\le {\mu }_{r2}\hfill & \mathrm{(}\text{serious}\mathrm{)}\hfill \\ 3\hfill & \text{if}\hfill & {\mu }_{r2}<{\mathrm{y}}_{\text{ri}}^{\text{*}}\hfill & \mathrm{(}\text{fatal}\mathrm{)}\hfill \end{array}\text{and}{y}_{\mathit{\text{pi}}}=\{\begin{array}{cccc}1\hfill & \text{if}\hfill & {\mathrm{y}}_{\text{pi}}^{\text{*}}\le {\mu }_{p1}\hfill & \mathrm{(}\text{minor}\mathrm{)}\hfill \\ 2\hfill & \text{if}\hfill & {\mu }_{p1}<{\mathrm{y}}_{\text{pi}}^{\text{*}}\le {\mu }_{p2}\hfill & \mathrm{(}\text{serious}\mathrm{)}\hfill \\ 3\hfill & \text{if}\hfill & {\mu }_{p2}<{\mathrm{y}}_{\text{pi}}^{\text{*}}\hfill & \mathrm{(}\text{fatal}\mathrm{)}\hfill \end{array}$$(4) where ${\mu }_r$ and ${\mu }_p$ represent threshold parameters used to identify the observed injury severity levels of the rider and pillion passenger, respectively relative to their injury propensity in a crash. The threshold parameters satisfy the condition that ${\mu }_{r1}<{\mu }_{r2}$ and ${\mu }_{p1}<{\mu }_{p2}$. Given that the two random components ${\epsilon }_{\text{ri}}$ and ${\epsilon }_{\text{pi}}$ are bivariate standard normally distributed with correlation parameter $\rho$ then the joint probability that explains the injury severity of a crash between a rider and pillion passenger can be expressed as (Chen et al., 2019a): (5) $$\begin{array}{ccc}\mathrm{\text{Pr}}(y_{\mathit{\text{ri}}}=j,y_{\mathit{\text{pi}}}=k)\hfill & =\hfill & \mathrm{\text{Pr}}({\mu }_{\mathit{\text{rj}}-1}<{\mathrm{y}}_{\text{ri}}^{\text{*}}\le {\mu }_{\mathit{\text{rj}}},{\mu }_{\mathit{\text{pk}}-1}<{\mathrm{y}}_{\text{pi}}^{\text{*}}\le {\mu }_{\mathit{\text{pk}}})\hfill \\ \hfill & =\hfill & \mathrm{\text{Pr}}({\mu }_{\mathit{\text{rj}}-1}<{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_r+{\mathrm{\epsilon }}_{\text{ri}}\le {\mu }_{\mathit{\text{rj}}},{\mu }_{\mathit{\text{pk}}-1}<{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_p+{\mathrm{\epsilon }}_{\text{pi}}\le {\mu }_{\mathit{\text{pk}}})\hfill \\ \hfill & =\hfill & \mathrm{\text{Pr}}({\mu }_{\mathit{\text{rj}}-1}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_r<{\mathrm{\epsilon }}_{\text{ri}}\le {\mu }_{\mathit{\text{rj}}}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_r,{\mu }_{\mathit{\text{pk}}-1}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_p<{\mathrm{\epsilon }}_{\text{pi}}\le {\mu }_{\mathit{\text{pk}}}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_p)\hfill \end{array}$$(5) where $j(j=1,2,3)$ and $k(k=1,2,3)$ represent a specific injury severity level associated with the rider and pillion passenger, respectively. The joint likelihood function for the model can be expressed as: (6) $$\begin{array}{c}\mathrm{\text{Pr}}(y_{\mathit{\text{ri}}}=j,y_{\mathit{\text{pi}}}=k)={\Phi }_2({\mu }_{\mathit{\text{rj}}}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_r,{\mu }_{\mathit{\text{pk}}}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_p,\rho )-{\Phi }_2({\mu }_{\mathit{\text{rj}}-1}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_r,{\mu }_{\mathit{\text{pk}}}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_p,\rho )\hfill \\ -{\Phi }_2({\mu }_{\mathit{\text{rj}}}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_r,{\mu }_{\mathit{\text{pk}}-1}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_p,\rho )+{\Phi }_2({\mu }_{\mathit{\text{rj}}-1}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_r,{\mu }_{\mathit{\text{pk}}-1}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_p,\rho )\hfill \end{array}$$(6)

where ${\Phi }_2(.)$ represents a cumulative distribution function of a bivariate standard normal distribution. For N observations with independent assumption, the unknown parameters of the model are obtained by maximising the joint log-likelihood function defined as (Yamamoto & Shankar, 2004): (7) $$\begin{array}{c}\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}L=\sum _{i=1}^N\sum _{j=1}^J\sum _{k=1}^K\hspace{0.17em}\mathrm{\text{ln}}\hspace{0.17em}[{\Phi }_2({\mu }_{\mathit{\text{rj}}}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_r,{\mu }_{\mathit{\text{pk}}}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_p,\rho )-{\Phi }_2({\mu }_{\mathit{\text{rj}}-1}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_r,{\mu }_{\mathit{\text{pk}}}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_p,\rho )\hfill \\ -{\Phi }_2({\mu }_{\mathit{\text{rj}}}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_r,{\mu }_{\mathit{\text{pk}}-1}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_p,\rho )+{\Phi }_2({\mu }_{\mathit{\text{rj}}-1}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_r,{\mu }_{\mathit{\text{pk}}-1}-{\mathbf{X}}_{\mathrm{i}}^{\prime }{\beta }_p,\rho )]\hfill \end{array}$$(7)

3. Results and discussion

3.1. Model specification

The risk factors for the motor riders and their corresponding pillion passengers were jointly explored using a bivariate ordered probit model defined in the previous section. The parameters and the associated p- values of the simultaneously estimated model are presented in Table 2. The table also summarizes the goodness-of-fit statistics of the fitted model. The selection of the variables retained in the model was based on 10% significance level rule. Since all the variables are categorical the rule was such that the variables are retained if at least one of the categories is statistically significant at 10% alpha level. The significance of the fitted model was evaluated by comparing it to the null model using likelihood ratio test. The p value of the test as reported in the goodness-of-fit section of the table suggests that the fitted model is highly significant. In addition, the correlation coefficient between the two dependent variables was found to be 0.63 and statistically significant at 5%. This suggests a positive relationship between the injury severities of rider and their corresponding pillion passenger. Thus, the higher the injury severity of the rider, the higher the injury severity of the passenger. This result also indicates that the unobservable factors that influence the likelihood of more severe injury of riders also increase that of their pillion passengers. This result would not have been seen if the injury severities of riders and their passengers were modelled independently. Thus, the use of bivariate ordered probit model for analyzing the data is better compared to two independent univariate ordered probit models. Among the twenty variables considered in the study, seven and five variables were found to influence riders and pillion passengers, respectively. Only four variables jointly influence both riders and pillion passengers. These include time of crash, weather condition, collision type and number of vehicles involved in the crash.

Table 2. Parameters of the bivariate ordered probit model jointly estimated for riders and pillion passengers.

Variables	Rider				Pillion Passenger
	Estimates	Standard error	Z	p value	Estimate	Standard Error	Z	p value
Constant	0.913	0.278	3.284	0.001	1.757	0.300	5.867	0.000
Passenger’s gender (Ref: Male)
Female	−0.293	0.140	−2.090	0.037
Time of day (Ref: Off Peak)
AM peak	−0.501	0.204	−2.460	0.014	−0.828	0.205	−4.040	0.000
PM peak	0.003	0.151	0.020	0.984	−0.204	0.151	−1.350	0.177
Day of week (Ref: Weekday)
Weekend	−0.250	0.133	−1.880	0.060
Weather conditions (Ref: Others)
Clear	−0.448	0.215	−2.080	0.037	−0.424	0.229	−1.850	0.065
Light conditions (Ref: Day)
Night					−0.473	0.140	−3.370	0.001
Collision type (Ref: Others)
Head-on	0.805	0.190	4.230	0.000	0.631	0.186	3.390	0.001
Rear-end	0.170	0.189	0.900	0.370	0.184	0.186	0.990	0.322
Number of vehicles (Ref: Single)
Multiple	−0.470	0.213	−2.210	0.027	−0.628	0.214	−2.930	0.003
Road width (Ref: Less than 6)
Between 6 and 12	0.389	0.122	3.170	0.001
${\mu }_1$	1.442	0.106	13.560	0.000	1.736	0.111	15.70	0.000
Ρ	0.631	0.052	12.135	0.000
Goodness-of-fit statistics
Log-likelihood at convergence	−501
Log-likelihood at intercept only	−526
Likelihood ratio test statistic	50
Likelihood ratio test p values	0.0001

3.2. Pillion passenger characteristics on injury severity

Among the variables considered in the model, pillion passengers’ gender significantly influenced crash severity among motorcycle users. This variable only affected the rider but not the pillion passenger (Table 2). The likelihood of a rider sustaining more severe injury decreases when the corresponding pillion passenger is a female. This result is in line with the findings in existing literature (Lin & Kraus, 2009; Wang et al., 2021). These findings may be explained by the fact that riders are less likely to engage in risky behaviour when their pillion passengers are females (Lin & Kraus, 2009). In addition, female pillion passengers usually caution riders about risky behaviour such as excessive speeding, unnecessary manoeuvring and overtaking thereby reducing their likelihood of severe injury in the event of crash. Contrary to this result, Khan et al. (2015b) found an adverse result attributing to loose clothes worn by female pillion passengers as the cause of severe injuries experienced by riders.

3.3. Temporal characteristics on injury severity

Regarding time of the day, it was found that the injury severities of both the riders and their pillion passengers were significantly influenced by the AM peak period. That is, the injury severities of both rider and their pillion passenger were less likely to be fatal when involved in a crash during AM peak compared to off peak period. This result is consistent with the findings in the existing studies (Vajari et al., 2020). This result may be explained by the fact that AM peak periods, also referred to as “morning rush hour” are associated with high traffic volume and congestion in both developed and developing countries (Luo et al., 2019). In Ghana, the Motor Traffic and Transport Department of the Ghana Police Service usually assists vehicular movement during the AM period due to traffic congestion. The appearance of a Police Officer during these periods keeps riders vigilant and cautious since they will be charged should they infringe on road laws. This leads to less impact and severe injury in the event of crash during AM peak periods.

Day of the week influenced the injury severity of only the rider when involved in a crash. The risk of more severe injury decreases for riders when involved in crashes during the weekend compared to weekdays. This result is in agreement with a study conducted in Pakistan (Waseem et al., 2019). The direction of the result may be influenced by the fact that riders are less likely to take risky behaviour such as unnecessary manoeuvring and overtaking during weekend compared to weekdays because of less traffic on weekend. However, other studies in Ghana (Aidoo & Amoh-Gyimah, 2020; Tamakloe et al., 2022), and Australia (Allen et al., 2017; Vajari et al., 2020) reported a contradicting results. These studies argued that riders might drive recklessly during weekends, since there may be no police officers to punish offenders. However, there is a high chance that a rider will moderate the rate of speeding during weekends unlike weekdays due to less traffic. The differences in the results may also be influenced by the differences in the methodological approaches used. Thus, further investigation regarding the effect of the modelling approach and its effect on the sign of the estimates may be worth studying.

3.4. Environmental characteristics on injury severity

Clear weather condition was less likely to influence more severe injuries sustained by both riders and pillion passengers in the event of traffic crash compared to unclear weather conditions such as rainy and foggy situations, etc. This result is supported by the findings in the existing literature (Asgarzadeh et al., 2018; Jalayer et al., 2018). These results may be explained by the fact that good weather conditions provide motorcycle riders with a clearer vision to journey on roads. This supports riders to escape crashes as well as take good initiatives when crashes are about to occur, hence decreasing the risk of engaging in high impact and severe injuries should crashes occur.

Light condition as a factor was found to influence only pillion passenger’s injury severity. There was no significant relationship between light condition and rider’s injury severity. The likelihood of pillion passengers sustaining more severe injury in the event of crash significantly decreases during night-time compared to day-time. The direction of the results may be explained by the fact that riders maybe more careful during the night, particularly when carrying pillion passenger due to low visibility thereby leading to less impact and low severe injury in an event of crash. Although most studies (Haque et al., 2009; Manan et al., 2018; Manan & Várhelyi, 2012) have found that night time crashes involving motorcycle users are associated with more severe injuries, Abdel-Aty et al. (2011) attributed more severe night time crashes to roads without street lights or poor lighting systems on roads.

3.5. Crash characteristics on injury severity

The likelihood of sustaining severe injuries significantly decreases for riders and their pillion passengers when involved in multi-vehicle crashes compared to single-vehicle crashes. These results was supported by Sivasankaran et al. (2021) and Zhou and Chin (2019) who concluded that single-vehicle motorcycle crashes were more likely to be fatal due to the fact that such crashes usually include collision with fixed object, run of road, etc., leading to higher impact.

Furthermore, it was observed that motorcycle collision type substantially influenced injury severity for both riders and pillion passengers. Among the injuries experienced by motorcycle users, riders and pillion passengers were more likely to sustain more severe injuries for head-on collision compared to other forms of collision. Although, both riders and pillion passengers were more likely to experience severe injuries in rear-end collision, this relationship was not statistically significant. The result of head-on collision confirmed the findings in existing studies (Champahom et al., 2020; Gårder, 2006; Hosseinpour et al., 2014; Qi et al., 2013). Head-on collision that leads to severe injuries may be associated with high red-lights violation which is common among motorcycle riders in Ghana (Nkrumah et al., 2022).

3.6. Road characteristics on injury severity

Road width was found to be a significant predictor of rider’s injury severity but not that of their pillion passengers. The results revealed that injuries sustained by riders were more likely to be severe on wide roads (roads with a width between 6 and 12 m) compared to narrow roads (roads with a width of less than 6 m). This result is in agreement with the findings in existing literature (Haque et al., 2010; Kvasnes et al., 2021; Russo & Savolainen, 2018; Se et al., 2021). The direction of the results may be explained by the fact that riders are more likely to engaged in risky riding such as unnecessary overtaking and manoeuvring in between cars on a wider road compared to narrow roads where such activities may be limited. These actions, however, are more likely to result in severe crash injuries when done improperly (Vieira & Larocca, 2017).

4. Conclusion

Studies that account for riders and pillion passengers’ injury severity relationship in the modelling approach play a pivotal role in improving model performance and efficiency. This study used the bivariate probit ordered logit model to determine the risk factors for riders and pillion passengers crash injury severities. In the bivariate approach, the two injury severities variables were jointly models to accounts for the existence of possible interdependencies between them. From the results, a positive highly intercorrelation was established between the injury severities of riders and that of their corresponding pillion passengers with a correlation coefficient of 0.63 was established. The direction of this intercorrelation between the two injury severities suggests that the unobservable factors that increases the likelihood of more severe injury of riders also increase the pillion passenger’s injury severity propensity. This part of the results would not have been observed if the injury severities of the rider and pillion passengers were modelled together as two independent variables. The study found that pillion passenger’s gender, time of day, day of week, weather condition, light condition, collision type, number of vehicles involved, and road width significantly influences the injury severity of the rider or pillion passenger or both.

The implementation of the bivariate ordered probit model in this research provides an exclusive methodology for future research on injury severity estimation. Findings from this research can be used for safety educational programs and policy formulation to gradually alleviate the severity of injuries caused by motorcycle riding and usage for commercial purposes. For instance, safety education for motorcycle users regarding nighttime travelling and adverse weather condition should be intensified as visibility is usually low during such periods. In addition, there is the need to develop their awareness on the risk of travelling on one-lane road to address head-on collision problems. In addition, it will also serve as resource information for future study. Future research will consider how rider and pillion passenger injury severity differ in relations to different collision types. It is worth mentioning that database containing the data is subject it two shortfalls: non-reporting and under-recording.

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The author(s) reported there is no funding associated with the work featured in this article.