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Accident Analysis and Prevention journal homepage: www.elsevier.com/locate/aap

Multilevel analysis in road safety research Emmanuelle Dupont a , Eleonora Papadimitriou b,∗ , Heike Martensen a , George Yannis b a b

IBSR, Belgian Road Safety Institute, Belgium National Technical University of Athens, Greece

a r t i c l e

i n f o

Article history: Received 7 July 2011 Received in revised form 1 March 2013 Accepted 29 April 2013 Keywords: Road safety Multilevel models Hierarchical structures Geographical dependences Road accident process dependences

a b s t r a c t Hierarchical structures in road safety data are receiving increasing attention in the literature and multilevel (ML) models are proposed for appropriately handling the resulting dependences among the observations. However, so far no empirical synthesis exists of the actual added value of ML modelling techniques as compared to other modelling approaches. This paper summarizes the statistical and conceptual background and motivations for multilevel analyses in road safety research. It then provides a review of several ML analyses applied to aggregate and disaggregate (accident) data. In each case, the relevance of ML modelling techniques is assessed by examining whether ML model formulations (i) allow improving the ﬁt of the model to the data, (ii) allow identifying and explaining random variation at speciﬁc levels of the hierarchy considered, and (iii) yield different (more correct) conclusions than single-level model formulations with respect to the signiﬁcance of the parameter estimates. The evidence reviewed offers different conclusions depending on whether the analysis concerns aggregate data or disaggregate data. In the ﬁrst case, the application of ML analysis techniques appears straightforward and relevant. The studies based on disaggregate accident data, on the other hand, offer mixed ﬁndings: computational problems can be encountered, and ML applications are not systematically necessary. The general recommendation concerning disaggregate accident data is to proceed to a preliminary investigation of the necessity of ML analyses and of the additional information to be expected from their application. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Most of the data of interest for road safety research happen to be hierarchically organized, i.e., to belong to structures with several hierarchically ordered levels. This implies that the observations can be unambiguously attributed to one and only one unit at higher level(s).1 For a part, these hierarchical structures result from the spatial (and temporal) spread of the data: Observations belong to larger geographical areas or units (road sites, segments, or intersections, counties, regions, etc.), or are made on a recurrent basis over a given time period. For another part, this hierarchical organization of observations results from the very nature of accidents, as each road-user, driver, or vehicle observation “pertains” to one and only one accident. One of the main problems associated with hierarchical data organization is the dependence that it generates among the observations (Hox, 2002). Observations that are sampled from the same

∗ Corresponding author. Tel.: +30 2107721380. E-mail address: [email protected] (E. Papadimitriou). 1 There are also cases where observations can simultaneously be attributed to different higher-level units. These cases are discussed later on in this article (Section 5.1). 0001-4575/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.aap.2013.04.035

geographical units have in common a series of unobserved characteristics that are proper to these larger geographical areas (Langford et al., 1999). One can think of risk studies that are based on crashfrequency data aggregated over a sample of road intersections or segments, which may themselves exhibit different road geometrics, trafﬁc, or other unobserved environmental characteristics that are all likely to affect accident frequency. In a similar vein, observations that are made at time points that are close from each other will also tend to be more similar than observations that are made at two remote time points. One can doubt of the possibility to exhaustively account for these heterogeneities by measuring and including them as covariates in a model. One can also doubt that all of these heterogeneities will be measurable at all (Huang and Abdel-Aty, 2010). Similarly, observations made on individuals occupying the same vehicles and involved in the same accident are likely to resemble each other more than observations made on individuals involved in different vehicles or accidents. This is so because these observations will be commonly inﬂuenced by vehicle and accident characteristics that are often left unobserved in a given analysis. The estimations obtained from most standard analysis techniques rest on the assumption that the observations are sampled from a single homogeneous population, and that the residuals are independent. However, the hierarchical organization of data fundamentally challenges these assumptions. Hence, applying traditional

E. Dupont et al. / Accident Analysis and Prevention 60 (2013) 402–411

statistical techniques (linear or generalized linear models) to hierarchically organized data is likely to result in underestimated standard errors and exaggeratedly narrow conﬁdence intervals (Kreft and De Leeuw, 1999). The risk is consequently that incorrect conclusions be derived about the signiﬁcance of the parameters whose effects are investigated. Statistical models have been developed that allow accounting for hierarchical data structures, and taking into account the dependence they introduce among the data. Because the hierarchical structure is speciﬁed in the model, predictors that characterize the different levels considered can also be correctly deﬁned (no need for aggregation or disaggregation). These models are labelled multilevel models, hierarchical models, mixed-effect models, random coefﬁcients or random parameter models. In the remainder of this article the terms multilevel (ML) or hierarchical (HL) models will be used indifferently. Although there are good statistical and conceptual arguments for the application of ML models in road safety research, so far no review based on road safety analyses has been conducted to assess the actual added value they can offer compared to “traditional” modelling techniques in this ﬁeld of research. This article starts with a description of the hierarchical structures most commonly encountered in road safety studies. HL models are then deﬁned and their statistical and conceptual interest is discussed. The second part of this article provides a review of several ML analyses conducted on the basis of three types of road safety data: (1) aggregated accident data, (2) disaggregated accident data, and (3) behavioural indicators. In each case, the review focuses on the questions of knowing whether ML model formulations (i) allow identifying signiﬁcant random variation of the observations at the various levels of the hierarchy considered, (ii) allow improving the ﬁt of the model to the data, and (iii) yield different conclusions than single-level model formulations with respect to the signiﬁcance of the estimates of the effects of explanatory variables. The necessity and feasibility of applying ML models is ﬁnally discussed distinguishing the three types of data.

2. Prevailing hierarchies in road safety research: spatial distributions of data and the nature of the accident process One can distinguish two prevailing hierarchies in road safety data, namely: geographical and accident hierarchies. As illustrated in Fig. 1, road safety data are organized in geographical units that are nested into each other (for example: road-sites nested into counties that are themselves nested into regions and countries). Similarly, the observations made on individual road users involved in accidents are nested into vehicles, which are themselves nested into different accidents. The two hierarchies are actually complementary and have been incorporated into a single framework to represent prevailing data structures in road safety (Huang and Abdel-Aty, 2010). An adapted version of this general hierarchical framework is presented in Fig. 2. Because road sites can be considered to belong to both types of hierarchies, they constitute the link between geographical and accident hierarchies, the macro- and microscopic ML structures. Repeated measurements in particular can be included as a horizontal ‘time’ dimension in this framework (Huang and Abdel-Aty, 2010; Aguero-Valverde and Jovanis, 2006). The multilevel structure can also be a multiple membership structure, as indicated by the double arrow inside the pyramid, or a cross-classiﬁcation structure, as indicated by the crossed arrows inside the pyramid. These complex structures are detailed in Section 5.1.

403

Depending on the research question, driver characteristics can be associated to the “vehicle” (e.g., all information about driver behaviour or manoeuvres) or to the “road users” level (e.g., the characteristics that are likely to affect the severity of accident outcomes such as age or gender). The “measurements/responses” level has been included in Fig. 2 to specify the capacity of multilevel models to handle complex types of response variables as being nested within individuals (i.e., multivariate responses, e.g. Duncan et al. (1999) multinomial responses, or repeated measurements). Intuitively, geographical hierarchies call for macroscopic analysis, while accident hierarchies, with individual road users or drivers as unit of analysis are the ideal basis for microscopic analysis (e.g., “What are the accident, vehicle, or driver characteristics that help predicting the occurrence of accidents and/or their outcomes?”).

3. “Hierarchical/multilevel models” – deﬁnition and general model formulation ML/HL models are regressions (linear or generalized linear models) in which the parameters (intercept and/or estimates of covariates effects) are assigned a probability model. As a consequence, this “higher-level (probability) model has parameters of its own (mean, variance). These are termed the “hyperparameters” of the model–which are also estimated from the data” (Huang and Abdel-Aty, 2010: p. 1560). In this sense, hierarchical models are grounded in the Bayesian paradigm: The model parameters are assigned a probability distribution that summarizes the knowledge the researcher has about each parameter, prior to any data observation. These “prior distributions” may be either informative (when, for example, existing knowledge allows reasonable assumptions to be made about the mean value of the parameter and its variance), or vague. In the latter case, “typical” distributions with relatively large variances are assigned to the parameters, so as to account for the lack of knowledge prior to observation. In the Bayesian approach, inference about the parameters is based on the posterior distribution, which combines the prior information (deﬁned by the prior distribution) with information derived from the observations. Carriquiry and Pavlovich (2004), as well as Miaou and Lord (2003) provide a thorough discussion of hierarchical model formulation in relation to the distinction between Empirical and Full Bayes estimation. Following Lord and Mannering (2010), it is important to distinguish between models allowing random variation of the parameters and “truly” hierarchical models. In the ﬁrst case, the intercept and covariate parameters are allowed to vary across the observations, and are thus assigned a probability distribution. HL models, on the other hand, specify the observations units (the lowest level of observation, for example, crash counts aggregated at various road intersections) as being clustered into higher-level units (for example, the “corridors” to which the various road segments belong to). In the latter case, the higher-level units are themselves considered a sample from a larger population (a sample from the “corridor population”). In such cases, the hyperparameters of the model deﬁne the random variation of the model’s parameters across the units at the higher level(s) (the corridors). The total variation in the observations can consequently be partitioned, or structured, along the different levels included in the model. As we will see, although the ﬁrst type of model takes account of the unobserved extra variations, it does not account for the hierarchical structure in itself, and does not offer any information about the proportion of variation in the observation that is

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E. Dupont et al. / Accident Analysis and Prevention 60 (2013) 402–411

Fig. 1. Geographical (top panel) and accident (bottom panel) hierarchies of road safety outcomes.

situated at the different levels. This distinction – which has often been ignored in previous articles discussing the use of ML models (e.g., Huang and Abdel-Aty, 2010) – needs to be explicitly stressed. In the present article, only research involving the second type of models is reported. Results from random coefﬁcients or random parameters models (e.g., Milton et al., 2008; Li et al., 2008; Gkritza

and Mannering, 2008) – although interesting in their own rights – will not be considered. To further deﬁne the principles underlying ML models, a simpliﬁed two-level model will be used. The response variable corresponds to the probability for driver i involved in accident j to die as a result of this accident.

Fig. 2. Framework for multilevel analysis in road safety research. Adapted from Huang and Abdel-Aty (2010).

E. Dupont et al. / Accident Analysis and Prevention 60 (2013) 402–411

The response yij is deﬁned as being a function of the expected value for accident j (ij ) and of driver-speciﬁc variation (Rij ). yij = ij + Rij

(1)

Logit (ij ) = 0 +

r

h xhij + uoj

(2)

h=1

The expected probability for driver i to die in accident j is now deﬁned as being a logit function of the linear combination of an average value holding for the accident population (0 ), of the r , and of effect of level-1 (and/or) level-2 predictors x h hij h=1 accident-related random variation u0j . We can deﬁne Logit(j ) = 0 + u0j

(3)

where 0 represents the average of Logit(i ) across accidents and u0j the accident-speciﬁc deviation from this population average value (the “accident-level” random variation). These deviations are assumed to be normally distributed, with mean 0 and variance u20 The model in (3) allows the intercept to vary across accidents, so that the expected probability of dying can be higher for some accidents than for others. This variation is meant to represent the inﬂuence exerted by the unobserved characteristics of the accidentlevel units on the observations made at the “individual driverslevel”, and thus to account for the dependence introduced at that level. When a random parameter model is ﬁtted as an unconditional model (i.e., without any explanatory variable speciﬁed), the proportion of variance in the outcome that is associated with each of the two levels can be estimated by means of the so-called intra-class correlation coefﬁcient (ICC). In the present case2 the ICC can be calculated assuming that the logistic distribution for the individual level residual implies a variance of 2 /3 =3.29. As a consequence, for a two-level logistic random intercept model with an intercept 2 , the ICC for between-accident residuals is: variance of 0 =

u20 u20

+ (2 /3)

An ICC that is close to 0 indicates that there is almost no variation in the observations associated with the second level of the hierarchy, a case in which the use of ML modelling is generally not warranted. In anext coefﬁcients for the covariates included in the r step, the x can also be deﬁned as varying randomly across model h=1 h hij the higher-level units. The accident-level random variation of the covariate (u1j x1ij ) is then added to the model which is now written as: Logit(ij ) = 0 +

r

h xhij + uoj + u1j x1ij

(4)

h=1

Signiﬁcant variation of coefﬁcients at the accident level would suggest an interaction between some unobserved accident characteristic with a driver-level characteristic to determine the probability of the driver to die or survive the accident. One can imagine, for example, that the effect of driver age is stronger for certain types of impact than for others. All residuals components deﬁned at higher level (u0j and u1j ) are assumed to be:

2 The reader is referred to Goldstein (2003) and Bryk and Raudenbush (1992) for methods to appropriately estimate the ICC for non-normally distributed responses as well as for methods of calculation of the ICC for models including more than 2 levels.

405

(1) normally distributed with mean 0, variance u20 and u21 , and (possibly) covariance u01 ; (2) independent across the j-units; (3) independent from the residuals at other levels of the model. In the case of a correctly speciﬁed ML model, the residuals at the lowest level can be said to be independent, conditional on other effects in the model. In other words, the error term at the lowest level is “cleaned” from the inﬂuences of the higher levels by the speciﬁcation of the corresponding random effects, and can thus be considered to display independence. 4. The conceptual importance of using multilevel models in road safety The use of hierarchical models improves the correctness of the estimation and inferences made from hierarchically structured data. It is thus warranted and recommended for purely statistical reasons, as we have seen earlier. Apart from that, it is important to recognize that, in comparison to traditional models, the speciﬁcation of ML models forces the researcher to reﬁne his/her view of the phenomenon investigated. ML models require that attention is paid to the level at which the observation units are deﬁned, as well as to the level(s) at which the predictors of interest are expected to exert their inﬂuence. Actually, many problems in road safety research cannot be understood correctly if only one level is considered, and interpreting results without taking the hierarchical framework into account can lead to erroneous conclusions. The error that is frequently made consists of considering that the relationships observed at given levels of the hierarchy also hold for the others. This is nicely illustrated by a simulation study conducted by Davis on the probability of car–pedestrian collisions (2002, quoted in Davis (2004)). Vehicle/pedestrian encounters were simulated on the basis of actual observations, and the relation between collision probability and variables such as car speed or the trafﬁc volume was assessed. The data were simulated at three levels of aggregation: the individual pedestrian/vehicle encounter, the road site level, and the “population level”, aggregating the information from all road sites. On the basis of the population-level aggregate data, it appeared that collision probability was related to trafﬁc volume, but not to speed. At ﬁrst sight, this suggests that speed is unimportant for pedestrian safety. At the road-site level, however, it appeared that no relation was observable between speed and collision probability for a series of road-sites, while a positive relation was identiﬁed between both variables at others. Actually, the relation could only be observed on those road sites where trafﬁc volume was not too important (and thus, did not constrained speed). Working at the road-site level allowed holding that characteristic constant, and the relation between the 2 predictors at the lower level to show up. The term “ecological fallacy” is generally used to refer to instances where inferences made at one level of analysis are erroneously and straightforwardly applied to other levels (Diez-Roux, 2002). This said, it is also necessary to stress the importance of hierarchies when it comes to comparing and synthesizing results published around a given road safety topic – be they derived from multilevel or from “single-level” analyses. Indeed, in different studies treating a given topic, the same “generic” dependent variable – for example “accident severity”-often appears to be operationalized at different hierarchical levels. Accident severity can be deﬁned and measured at the accident level – when the response variable is the most severe injury recorded in each accident; at the vehicle level – when the response variable is deﬁned as the level of injury sustained by the driver; or at the level of the car occupants

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Table 1 Overview of the unit of observations and operationalization of the dependent variable for various accident severity analyses. Reference

Data selection

Response variable

Level of observation

Evans (1983) Chang and Mannering (1999)

Drivers only All occupants – Truck involved – Non-truck involved Drivers only Fatalities and uninjured only Vehicle occupants only Single-vehicle accidents only Multiple-vehicle accidents (no pedestrians) – All crashes – 2-vehicle crashes – Single crashes – 2-vehicle crashes – single-vehicle crashes – No crashes with only occupants killed or injured

Driver injury Most severe injury (in vehicle)

Accident Vehicle

Severity of driver injury Severity of occupant injury Severity of occupant injury Most severe injury (in crash) Most severe injury (in crash)

Driver-vehicle Individual road user Individual road user Accident Accident

Driver injury

Driver-vehicle

Driver injury

Driver-vehicle and accident (2 analyses)

Khorashadi et al. (2005) Shibata and Fukuda (1994) O’Donnell and Connor (1996) Yau (2004) Yau et al. (2006) Kockelman and Kweon (2002)

Martin and Lenguerrand (2008)

– when the response variable is the probability for each occupant to die, or to sustain injuries of a given severity level. As an example, Table 1 provides an overview of various severity studies (none of them analysed by means of ML models). The way the “severity” dependent variable was operationalized clearly indicates that different levels of analysis are at play in the different studies, with potentially important implications for the comparability of the results observed, as the conclusions (for example, in terms of factors signiﬁcantly affecting accident severity) observed on the response variable deﬁned at one level of the hierarchical accident structure may not necessarily apply to cases where the “severity” outcome is deﬁned at other hierarchical levels.

5. Review of ML applications in road safety research We have seen that hierarchical organization is a prevailing characteristic of road safety data, and that ML formulations allow accounting for hierarchical data organization but also provide the researcher with valuable additional information compared to traditional approaches (partitioning of the variation in observations over the various levels, information about the ﬁxed vs. random nature of the effect of explanatory variables, etc. (for a detailed discussion, see also Dupont and Martensen, 2007). ML models have been used in the ﬁeld of road safety for some years by now. Yet, no review has been conducted so far to examine their actual relevance and usefulness in the ﬁeld. In the following, several applications of these models are reviewed that are based on three main types of analyses: the analysis of aggregate accident data, of disaggregate accident data, and ﬁnally the analysis of behavioural indicators. In each case, the review focuses on the questions of knowing whether ML model formulations (i) allow identifying signiﬁcant random variation of the observations at the various levels of the hierarchy considered, (ii) allow improving the ﬁt of the model to the data, and (iii) yield different conclusions than single-level model formulations with respect to the signiﬁcance of the estimates of covariate effects. For the sake of clarity, we will use the term “aggregate data” to refer to cases in which the response variable – indicators of accident frequency in most cases – is aggregated over some geographically deﬁned units (road sites or intersections, for example). The terms “disaggregate data” will be used whenever the response variable is deﬁned at the accident level (for example: the most severe injury sustained by road users involved in each accident), at the vehicle level (the severity of the drivers’ injuries), or at the road-user level (the severity of the injuries sustained by each and every road-user involved) (see Table 2).

5.1. Multilevel modelling of aggregate accident data The spatial distribution is an important source of dependence among observations. One way to account for spatial dependence is to specify the hierarchical organization of the observations in the statistical model. The type of geographical units deﬁned and the number of geographical levels speciﬁed in the model is potentially unlimited and depends primarily on the focus and scope of each research question. The hierarchy can be much more detailed than the broad ‘accidents into road sites into regions’ structure described in Fig. 2. This variety is illustrated by the studies reviewed here, which include models of annual accident counts aggregated over 50 counties and clustered within 12 regions of Greece (Yannis et al., 2007, 2008); or aggregated over a sample of road segments/road sites and clustered into larger corridors (El-Basyouny and Sayed, 2009: 392 segments clustered in 58 corridors; Guo et al., 2010: 174 intersections clustered in 25 corridors). All three analyses indicate that a substantial proportion of the variation in the observations can be attributed to the higher levels included in the model, and that the effects of the predictors investigated can be considered to vary randomly across the geographical units deﬁned. All studies yield the conclusion that ﬁtting a ML model allowed improving the model’s ﬁt to the data, compared to that of a “traditional”, single-level model. Importantly enough, all studies showed that the effects of some of the predictors investigated – while being signiﬁcant when calculated on the basis of a standard regression model – were not signiﬁcant any more once the hierarchical organization was accounted for. Sometimes, the spatial structure one whishes to account for is more complex. One may need to classify observations into two distinct but not strictly hierarchical dimensions at the same time, for example (cross-classiﬁcations). This would be the case of road users nested both within geographical units and within transport modes. In other cases, observations belong to several geographical units at the same time (multiple memberships) – for instance, when participants change place of domicile in the course of a longitudinal study. Models of these types are usually referred to as “spatial models”, and can include complex spatio-temporal analyses (AgueroValverde and Jovanis, 2006; Eksler, 2008). When applied to road safety, the basic idea behind spatial modelling techniques is the decomposition of the random variation of the observations into two distinct components: a “structured” component, which is assumed to represent the spatial structure of the road safety outcomes, and an “unstructured” one, which is assumed to be random. Therefore, the road safety outcome Yi of county (i) is considered to be the

Table 2 Review of several ML analyses in road safety research. Reference

Case study

Response variable

Level of observation

Statistical model

Levels included in model

Random effects

Tested Vehicle

Disaggregate analyses Jones and Injury severity Jørgensen (2003) Lenguerrand et al. Injury severity (2006) Yannis et al. (2010) Injury severity in fatal accidents Dupont et al. Injury severity in (2010) car-car fatal accidents Helai et al. (2008) Injury severity at intersections Kim et al. (2007) Accident risk at intersections Behavioural data Vanlaar (2005a) Vanlaar (2005b)

a

Seat-belt survey Drink-driving survey

Road site

Corridor Region

Signiﬁcant

Country InterceptSlope

InterceptSlope

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

Yes

•

•

•

Yes

•

•

•

Yes

•

•

–

•

•

–

•

•

•

Yes

•

•

•

•

–

Binomial

•

•

Driver-vehicle

Binomial

•

•

Accident

Binomial

•

Driver-vehicle

Binomial

Driver-vehicle

Binomial

Accidents and fatality counts Accident counts

County County

Multivariate Poisson CAR, MMMC

Accidents counts

Road segment

Poisson log-normal

Accidents counts

Road site

Poisson, CAR

Accidents counts

County

Neg. binomial, CAR

Casualty counts

Ward

Poisson, CAR

Fatality rates

Municipality (BE), District (DE), Department (FR), NUTS 2 (EU)

Poisson, CAR

Probability of injury severity Probability of injury severity Probability of injury severity Probability of injury severity

Casualties

Binomial

Vehicle occupants

Binomial

•

Road-user

Multinomial

Vehicle occupant

Probability of injury severity Probability of accident occurrence Probability of seatbelt use Probability of exceeding legal BAC

•

•

•

•

•

•

•

•

•

•

•

Yes Yes

•

–

–

•

•

Yes

•

•

•

No

•

•

•

•

Yes

•

•

•

•

–

E. Dupont et al. / Accident Analysis and Prevention 60 (2013) 402–411

Aggregate analyses Yannis et al. (2008) Road accidents and enforcement Papadimitriou et al. Spatial road safety (2012) analysis El-Basyouny and Accident prediction Sayed (2009) with corridor effects Guo et al. (2010) Spatial analysis of intersection accidents Aguero-Valverde Fatal and injury and Jovanis (2006) accidents Hewson (2005) Child pedestrian casualties Eksler (2008) Fatality rates in Europe

Accident

Improvement over base modela

Base model is deﬁned as the ordinary – non ML – version of the model.

407

408

E. Dupont et al. / Accident Analysis and Prevention 60 (2013) 402–411

result of systematic variables effects and of random variation εi , which is further decomposed into “structured” random variation (ui ) and unstructured random variation (vi ). A detailed presentation of the formulation and statistical properties of spatial models is beyond the scope of this article, the reader is referred to Eksler and Lassarre (2008) for a complete presentation. Most often, the models used to account for these complex structures are the MMMC (Multiple Membership Multiple Classiﬁcation) and the CAR (Conditional AutoRegressive) model. The MMMC model is a multilevel model which combines a cross-nested structure and a multiple membership structure. The cross-nested structure is used to describe the fact that the variation in the observations comes for a part from the geographical unit they belong to (i.e., overdispersion in the accident counts), and for another part from the neighbourhood structure (i.e., spatial effects). The MMMC model assumes that geographical units are separate entities. The CAR model uses a slightly different approach, in which geographical units are no longer considered as separate entities (Browne et al., 2001) and a conditional auto-regressive distribution is initially assumed for the structured variation. Consequently, in the CAR model, one (global) random neighbourhood effect is estimated for each observation. Moreover, no exchangeable random effect (overdispersion) is initially assumed in the CAR model. Incorporating overdispersion gives the CAR convolution model. In general, the CAR method results are considered to be more reliable, for both theoretical and practical reasons. For details over the differences of MMMC and CAR models the reader is referred to Papadimitriou et al. (2012). Papadimitriou et al. (2012) applied a neighbourhood matrix to the analyses of the Greek-counties accident counts described earlier. The results suggested that the spatial structure examined accounted for an important part of the variation in road accidents in the Greek counties, revealing a general pattern of risk increase from northern to southern Greece. Importantly, the results indicated that the effect of one explanatory variable (enforcement) would have been quite overestimated had spatial effects not been taken into account. Hewson (2005) examined spatial dependence in child pedestrian casualties aggregated by casualty’s home location, by means of both a simple model, and a model accounting for spatial auto-correlation. It appears that the simple ecological model was not sufﬁcient to account for spatial correlation. On the contrary, when casualties were aggregated on the basis of the accident location they appeared to be spatially independent. The added value of using spatial models, either within or across European countries was demonstrated in recent research (Eksler, 2008), where a number of interesting examples highlight the importance of investigating the spatial structure of road safety outcomes: • At European level, a series of neighbouring (spatially dependent) regions was identiﬁed, stretching from north-west Spain to north-east Poland, which presents increased mortality risk. • A clear east–west spatial pattern can be identiﬁed in Germany, accounting for part of the random variation in mortality risk. In particular, the eastern regions of Germany present increased mortality risk compared to the western ones, and this pattern can only be identiﬁed once the spatial dependence among regions is examined separately. • A distinct south–north pattern is observed in Belgium, where the spatial analysis revealed increased mortality risk in the southern region of Wallonia.

5.2. Multilevel modelling of disaggregate accident data The analysis of disaggregated accident data, focusing on individual road safety casualties, requires the accident hierarchy to be taken into account (cfr. Fig. 2). Of course, road users or accidents can be further nested into road sites, areas, regions, and so on, expanding the hierarchy towards geographical elements. A fundamental characteristic of disaggregate accident data needs to be taken into account concerning the ML analyses: While the number of accidents can be rather large, that of car per accident and of individual per car is typically very low. This is a major difference with the studies reviewed in the previous section, where the number of observations per higher-level (geographical) unit was generally rather high. In the case of disaggregate accident data, the low numbers of observations within the accident and vehicle levels can create computational problems and compromise the estimation of the variance of the random effects for these two levels (Dupont and Martensen, 2008). Lenguerrand et al. (2006) modelled the probability of vehicle occupants to die as a result of accidents on the basis of observed and simulated data. The model speciﬁed the nesting of occupants into vehicles and of vehicles into crashes. They observed that the variance of the vehicle random effect is falsely estimated as being zero in 36% of the cases. These incorrect estimations are clearly related to the small number of observations available per vehicle and per accident.3 Fitting a model where the vehicle level was ignored allowed overcoming that difﬁculty. A series of studies focusing on indicators of accident severity disaggregated at the individual level provided mixed evidence with respect to the proportion of variation in these observation that can be attributed to the vehicle or accident level. Jones and Jørgensen (2003) observed that 16% of the random variation in the probability of individual casualties to be fatal is associated with the accident level, while Helai et al. (2008) estimated that 28% of the random variation in the probability of driver-vehicle damages to be severe was situated at the accident level. One study (Dupont et al., 2010) could not identify signiﬁcant random variation of individual indicators of severity, be it at the vehicle, accident, or country level. Another one, modelling the probability for road-users to sustain injuries of different severity levels yielded mixed results: no random variation was observed either at vehicle or accident level for the probability to sustain a fatal injury. The probability of serious and slight injury, however, appeared to vary signiﬁcantly across vehicles (Yannis et al., 2008). One should note that the last 2 studies were characterized by relatively small datasets (Dupont et al., 2010: 1296 accidents involving 3444 road-users, Yannis et al., 2008: 1300 accidents and 3500 road users) as compared to Lenguerrand et al. (12,030 accidents for a total of 26,918 occupants) or Jones and Jorgensen (12,943 accidents with 16,332 casualties). The dataset used by Helai et al. was of moderate size (4095 accidents), but characterized by a relatively high rate of driver-vehicle units per accident (70,840, thus an average of 1.91 units per accident). Finally, Kim et al. (2007) examined the issue of crash frequency on the basis of disaggregate data by modelling the probability of occurrence of 5 different crash types by means of binomial multilevel models (crashes were clustered into intersections). The random variation of the intercept across intersections was significant for all except one crash type. The results also showed that ML models overall provide results similar to traditional estimation techniques, except for one crash type, for which several predictors

3 The correctness of the estimates of the variances of the random parameters increases when the number of observations per car or accident units is increased in the simulated data.

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were evaluated as non-signiﬁcant on the basis of the single-level model, and as signiﬁcant on the basis of the multilevel one. None of the studies reviewed has investigated whether the effects of explanatory variables varied randomly across the units of the higher levels of the accident hierarchy. Of course, the same difﬁculties apply in this case than when estimating the random variation in the observations that is associated with each level. ML formulations seem to improve the ﬁt of the models, as Lenguerrand et al. report that ML models provide the least biased estimates for covariate effects – even when the higher level random variation is incorrectly estimated.4 The biases observed on the basis of the single-level model formulation were small, however (slight under-estimation of the conﬁdence intervals of the parameters), and did not affect the conclusions as to the signiﬁcance level of the covariates tested. Dupont et al. (2010) also concluded that the single- and multi-level model formulations provide similar values and signiﬁcance levels for the coefﬁcients estimated. According to Lenguerrand et al., compared to the hierarchical organization of data that are traditionally processed by ML models, the hierarchical structure of disaggregate road accident data seems closer to that of independent observations. 5.3. Multilevel modelling of behavioural and attitudinal data The importance of monitoring the behaviour and attitudes of road-users (speed, drink-driving, seatbelt wearing etc.) is now increasingly acknowledged as being integral part of an efﬁcient monitoring of the performances of the road safety system (Koornstra et al., 2002). Such attitudinal and behavioural data are most often collected on the basis of a sampling scheme that is likely to introduce dependence in the observations. Multi-stage sampling, for example, consists of randomly selecting a set of geographical units, within which the observations themselves are randomly selected. (see, for example Kelley-Baker et al., 2013, or Hakkert and Gitelman, 2007 for a review of roadside survey methods applied in different European member states).5 This situation is close to the one of accident count data collected on several road sites or road segments which are themselves nested within larger geographical areas (counties, regions etc.). The geographical hierarchies described in Fig. 2 are thus also prevailing in the case of attitudinal or behavioural data. Yet, despite the potential dependence introduced by the sampling scheme used, examples of multilevel applications using these types of data are strikingly rare in the road safety literature. Vanlaar (2005a) analysed data from a Belgian roadside survey on seatbelt use in a single-level and a multilevel framework. These data were collected at randomly selected road sites. The probability of seatbelt use was modelled by means of a binary logistic regression model (yes/no). The results showed signiﬁcant variation between road sites in the average probability of wearing a seatbelt (intercept), suggesting that the ML approach was appropriate. Some of the predictors included in the model, while considered signiﬁcant on the basis of the single-level model formulation, proved non-signiﬁcant on the basis of the ML model. Vanlaar (2005b) also compared the results of single and multilevel models for another roadside survey conducted in Belgium, this time focusing on drink-driving. The results indicated that the average probability for a driver to be above the limit varied signiﬁcantly across road-sites.

4 According to these authors, this is because the covariate estimates are adjusted for other confounding factors related to the occupant–vehicle–accident structure. 5 The example given here is actually a so-called “2-stage” sampling scheme. Depending on the research, the sampling scheme can be more complex and involve 3, 4 or more stages.

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6. Discussion The results described above conﬁrm that hierarchical dependences are characteristic of a large array of road safety data, and that a multilevel representation may offer a solid conceptual and computational basis for road safety research. The empirical examples reviewed in this article also clearly illustrate the fact that ML models can be implemented for a wide variety of response distributions, including those that are most appropriate for accident count data (i.e., Poisson, negative binomial). A major argument for the use of HL modelling techniques is their ability to handle dependence in data, which is revealed by signiﬁcant random variation of the intercept across higher level(s) units. A related argument is the fact that biased parameter estimates can be obtained if the dependence in the observations is not properly handled, with the consequence that the effects of some parameters may appear to be signiﬁcant while in fact they are not. The literature review presented in this article suggests, however, that these arguments do not apply to the same extent depending on whether aggregate data (aggregate accident data, data on behavioural indicators), or disaggregate accident data are analysed. The application of ML techniques to aggregate accident data is rather straightforward: The number of observations generally allows the estimation of the random variation located at higher (geographical) levels of interest, or within more complex spatial structures. Besides, the additional precision and information gained does not require much additional efforts in terms of data collection. The literature reviewed showed that higher level effects were found signiﬁcant in basically all the studies, regardless of the type of geographical units investigated. Cases where conclusions about the signiﬁcance of parameters had to be revised on the basis of ML models were also frequently observed. Hierarchical organization is an inherent, natural characteristic of disaggregate accident data. Using only aggregate accident data, one runs the risk of erroneously generalizing observed relations between accident characteristics and aggregate accident outcomes, to accident outcomes for individual road users (ecological fallacy). ML formulations can help avoiding this pitfall and properly examine outcomes at different levels of the accident hierarchy. One will note, however, that ML models allow avoiding such fallacious conclusions to be derived from the data only to the extent that data are effectively collected at the different levels considered (Gelman et al., 2001). To put it otherwise: Their added value in this case comes at the costs of increased efforts on the ground of data collection. However, with the expansion of in-depth accident research, the question of the necessity and feasibility of ML applications to disaggregate accident data will become increasingly acute. The review of studies based on disaggregate data however illustrated that applying ML models is in this case not as straightforward as in that of aggregate accident data. Indeed, the average numbers of observations at the vehicle and at the accident level being generally low, they do not always enable the efﬁcient estimation of the random effects associated with these levels. As a consequence, in several of the studies reviewed (e.g., Dupont et al., 2010) no signiﬁcant variation could be observed at higher levels of the hierarchy. In other cases, however, the conclusion was that a substantial part of the variation in the outcome variable was to be attributed to higher level(s) of the hierarchy. One recommendation to be made when dealing with disaggregate accident data is to “ignore” the vehicle level, and to directly model the nesting of individuals within accidents instead, especially when the size of the accident sample is reduced (Lenguerrand et al., 2006; Jones and Jørgensen, 2003). In any case, one needs to bear in mind that, when using disaggregate accident data, ML models are best applied to large accident datasets, accident data characterized by large numbers

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Fig. 3. Needs and feasibility of multilevel modelling in road safety research.

of occupants per vehicle, or when the results obtained from traditional models seem questionable, because they are in obvious contradiction with previous ﬁndings, for example, or on the basis of a preliminary comparison of results obtained on the basis of single and multi-level models (Lenguerrand et al., 2006). A parallel modelling of the response variable by means of single level and multilevel techniques thus seems to be generally warranted in cases where disaggregate accident data is worked with. Finally, we have also seen that the application of ML models is both straightforward (number of observations) and relevant for research focusing on road user attitudes and behaviour. In this case, the need for ML modelling is determined mainly by the technique used to sample the observations (multi-stage sampling). In such cases indeed, the characteristics of the higher-level sampling units (for example, the speed limit at the different road sites) are often part of the predictors of interest. However, given that the sample of road-users is most of the times a lot larger than that of road sites, testing the effects of these “geographical predictors” in a singlelevel model generally implies that the value corresponding to each road site is merely replicated across all the road-users who have been sampled from it. In this case, the risk of making incorrect inferences is also considerably higher. This was actually illustrated by one of the two researches reviewed. The two studies examined also unambiguously revealed signiﬁcant variation of the observations at higher levels of the hierarchy. Considering all this, the scarcity of ML applications in this type of research is striking and deserves to be stressed. Fig. 3 summarizes the theoretical and practical conditions for using multilevel analysis in road safety research. As a general rule, it can be said that multilevel modelling appears to be both more meaningful and easier to apply in the context of aggregate data analyses. On the other hand, the application of multilevel models in the context of disaggregate data appears to be less straightforward, and also less critical for the outcomes of the analysis – although the research reviewed provides mixed evidence in this respect. Additional insight may be indeed provided by a multilevel model speciﬁcation, but the relevance of ML analyses deserves preliminary investigation on a case-by-case basis. Despite the fact that they often appear to be recommended for the analysis of road safety data, there is one major drawback of multilevel models that needs to be mentioned: the fact that they can grow very complex very quickly. The number of levels that can be speciﬁed is theoretically unlimited, random slopes can be deﬁned at any level, for any of the predictors tested, etc. The number of assumptions that need veriﬁcation also increases with the complexity of the model (e.g., normal distribution of the random effects and independence from the covariates included in the model), and as Wakeﬁeld (2009) notes, “outside of a linear mixed-effect model, little theory exists to support the reliability of the estimation when

violations of assumptions occur. Ideally, normality assumptions should be checked before the model has been ﬁtted, as “estimates of random effects reﬂect both the data and the assumed random effects distribution” (Wakeﬁeld, 2009, p. 331). One general recommendation is thus to check beforehand that the use of ML model is indeed warranted (variance partition coefﬁcients, signiﬁcance of the variances of the random effects etc.), and to ﬁt parsimonious models, introducing random slopes for particular predictors very sparsely, and preferably on the basis of theoretical reasons. Indeed, the recommendation to apply ML techniques does not only rest on statistical considerations. The hierarchies characterizing the data investigated in this ﬁeld of research – especially accident data – should be considered as an integral part of the theoretical framework we are working with. Research comparison and synthesis thus also requires that the hierarchies behind data be taken into account. 7. Conclusion Although multilevel models are commonly applied in many scientiﬁc areas, they are relatively newer to the ﬁeld of road safety. This article started on a discussion of the common hierarchical framework characterizing road safety data, namely geographical and accident hierarchies. The general model formulation and basic principles underlying ML techniques were then discussed, and two possible consequences of ignoring a hierarchical structure in the data were described: statistical and conceptual. The ﬁrst consequence is the statistical inaccuracy resulting from the underestimation of standard errors due to the dependence of nested observations. The second consequence is a conceptually impoverished representation of the research topic investigated. Research applications of multilevel models for three types of data were ﬁnally reviewed: studies using aggregate accident data, studies focusing on disaggregate accident data, and ﬁnally studies aiming at collecting behavioural data and at making inferences on the determinants of the behaviours in question. The efﬁciency and usefulness of ML model formulations in the case of aggregate accident data was uniformly supported. Disaggregate accident data are, however, those for which the application of ML models was the least straightforward, although hierarchical organization is an obvious characteristic of the accident phenomenon when accidents are examined in a “disaggregate way”. Several recommendations were made on the basis of the research reviewed, and it was also stressed that the (computational) difﬁculties encountered on the basis of disaggregate data should not lead to downplay the theoretical importance of the hierarchical framework when deﬁning research questions and hypotheses, or when comparing results from different researches: the relationships observed at a given level of the hierarchy do not necessarily apply at others. Finally, ML analysis seems as straightforward and as warranted for studies aiming at collecting behavioural and attitudinal data on the basis of “hierarchical” sampling schemes (e.g., multi-stage sampling). However, examples of ML applications to this type of data is strikingly scarce in the road safety research literature. Acknowledgements The authors would like to address special thanks to Ward Vanlaar for coordinating and contributing to the earlier stages of this work. The authors would also like to thank all partners involved in the SafetyNet group on “Data analysis and synthesis”, namely Constantinos Antoniou, Christian Brandstaetter, Ruth Bergel, Frits

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