category: literaturenote citekey: hewsonepidemiologychildpedestriancasualty2005 title: "Epidemiology of child pedestrian casualty rates: Can we assume spatial independence?" authors: "Hewson, Paul J." year: 2005 date: 2005-00-00 2005 doi: 10.1016/j.aap.2005.03.007 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2005.03.007" zotero_key: QE2M5695 zotero_storage: WNJANTGF collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Hewson, Paul J."

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Accident Analysis and Prevention 37 (2005) 651–659

Epidemiology of child pedestrian casualty rates: Can we assume spatial independence?

Paul J. Hewson ∗

Environment Directorate, Devon County Council Topsham Road, Exeter EX2 4QW, UK Received 26 January 2005; accepted 13 March 2005

Abstract

Child pedestrian injuries are often investigated by means of ecological studies, yet are clearly part of a complex spatial phenomena. Spatial dependence within such ecological analyses have rarely been assessed, yet the validity of basic statistical techniques rely on a number of independence assumptions. Recent work from Canada has highlighted the potential for modelling spatial dependence within data that was aggregated in terms of the number of road casualties who were resident in a given geographical area. Other jurisdictions aggregate data in terms of the number of casualties in the geographical area in which the collision took place. This paper contrasts child pedestrian casualty data from Devon County UK, which has been aggregated by both methods. A simple ecological model, with minimally useful covaraties relating to measures of child deprivation, provides evidence that data aggregated in terms of the casualty's home location cannot be assumed to be spatially independent and that for analysis of these data to be valid there must be some accounting for spatial auto-correlation within the model structure. Conversely, data aggregated in terms of the collision location (as is usual in the UK) was found to be spatially independent. Whilst the spatial model is clearly more complex it provided a superior fit to that seen with either collision aggregated or non-spatial models. Of more importance, the ecological level association between deprivation and casualty rate is much lower once the spatial structure is accounted for, highlighting the importance using appropriately structured models. © 2005 Elsevier Ltd. All rights reserved.

Keywords: Bayesian; Spatial modelling; Child Pedestrian; Deprivation

1. Background

There is a plentiful literature modelling child pedestrian casualty rates, in particular these demonstrate a strong association with areal socio-economic status (Kendrick, 1993; Jolly et al., 1993; Braddock et al., 1991; Roberts et al., 1992; Chichester et al., 1998; Lyons et al., 2003) Recently, a study released in the UK by the Institute of Public Policy Research (IPPR) offered an econometric analysis of child pedestrian casualty rates (Grayling et al., 2002) and claimed there was little evidence for an association between a number of traffic related environmental factors and casualty rates. This contrasts with other studies on the subject such as Pitt et al.

E-mail address: paul.hewson@plymouth.ac.uk.

(1990) who identified traffic speed greater than 30 mph as a specific risk factor. The IPPR study aggregated data in terms of the number of casualties injured in collisions in any given ward and modelled this against an official ward level measure of deprivation commissioned by the Department for the Environment, Transport and the Regions (Noble et al., 2000). The child deprivation component of this index captured information solely on the income deprivation of households containing children. This deprivation indicator therefore provides a single summary measure which is related to the number of children in each ward living in households in receipt of a number of state benefits. The analysis therefore suggests that wards with large numbers of pedestrian casualties (who may or may not have been children resident in that ward) can be modelled in terms of a "deprivation score" relating to the children resident in that ward. More recently, Graham et al. (2005) have published a similar study which investigated the effect of traffic flow, physical nature, characteristics of

^∗ Present address: School of Mathematics and Statistics, University of Plymouth, Plymouth PL4 8AA, UK. Tel.: +44 1752 232778; fax: +44 1752 232780.

local infrastructure, and "other" specific factors, albeit many of these covariates were measured by a range of proxy variables, and reported an association between deprivation and casualty rate once these variables had been accounted for. However, this model did not consider spatial dependencies within the casualty rates and will be discussed later. In contrast to the collision location based analysis popular in the UK, MacNab (2003, 2004) has recently presented a spatial analysis from Canada which used hospital derived data aggregated by the area in which the casualty is resident.

Most of the analyses highlighted above reported the number of children who have been injured in any given area expressed as the proportion of the number of children resident in that area. The adequacy of data collection in terms of child casualties is well studied and will be considered in the context of this study later. However, the denominator, which is often assumed to be measured accurately is in fact mis-defined. Resident population is not a true measure of the "population at risk" in a given area. For example, a town center or the environs of a school may have a much higher albeit transient "population at risk". It is also not clear how the child deprivation score relates to the "population at risk". It can be argued that there are environmental features of "deprived" wards that may relate to the deprivation score of the children resident in an area, but it is not clear that the deprivation score of children resident in any given area will necessary relate well to the children injured in any given area. An extreme example with these data are that the wards with the most injuries tend to be town centers with higher deprivation scores. But many children, from a range of backgrounds, will be present (and exposed to pedestrian injury risk) in town centers for reasons that are not captured by the deprivation score of the ward where the collision took place. However, the the alternative approach, of aggregating data by the area in which the casualty was resident is also problematical. The first problem is that details on area of residence have to be captured in collision reports, as will be seen later this seems difficult to arrange in the UK. Additionally, it is not always possible to obtain information from outside any given region as to children for that region who may have been injured elsewhere. The best way of incorporating a spatial element into a modelling exercise with these data may then differ according to precisely how the data have been aggregated.

The significance of these analyses are that they inform choice of possible remedy. Either it is intended to alter the behavior of the child to reduce the risk of an injury or attempts are made to alter the environment in which the child is active so making it more forgiving of an error. There is evidence from systematic review that the former is effective in terms of altering behavior but it has not been possible to demonstrate any convincing reduction in casualty involvement (Duperrex et al., 2003). Conversely, systematic review has been able to demonstrate that environmental modification can indeed reduce casualty involvement generally (Bunn et al., 2003). A "micro"-spatial analysis of collision locations has been reported highlighting important environmental features (LaScala et al., 2004). Nevertheless, for a variety of reasons, including confidentiality of collision participants, population based analyses are likely to be a mainstay of injury prevention research for the foreseeable future. It follows, given the consequences of the interpretation of such an analysis, that effort is justified in ensuring that the model assumptions are valid for these rather complex real-world data.

2. Methods

For the purposes of this study a child is defined as being in the age range 0–15 years.

2.1. Geographical context

Devon County is located in the south west peninsula of England. The County Council came into being following the Local Government Act of 1888, the most recent structure reflects local government reorganization which took place in 1998 and removed the two largest urban areas from Devon's governance. The population of Devon is currently estimated as 706,000 with a child population of 120,000. Administratively Devon is subdivided into 226 wards with an average of 530 children and 2600 adults per ward, a frequency table of these data is given in Table 1. The 1998 ward boundaries have been used for this study (Office of the Deputy Prime Minister, Office of the Deputy Prime Minister). There is one major urban settlement within Devon, Exeter (population 110,000), and a number of smaller urban settlements such as Exmouth (population 32,400), Newton Abbot (23,600) and Barnstaple (20,800). Devon is however typical of shire counties in England, being 10th out of 38 shire counties in terms of its rural nature, and having 22% of the population resident in rural areas. Excluding the seven major metropolitan areas of England the average number of children in each county reported killed or seriously injured in 2002 was 16 with a further 65 reported slight casualties. The corresponding figures for Devon were 18 and 95.

Table 2 also shows the ward sizes throughout Devon County, the largest ward is the Forest of Dartmoor which is mainly unpopulated and un-trafficked moorland.

Table 1 Frequency table showing the range of population sizes for wards in Devon County

| Population in ward | Number of wards | | | |--------------------|-----------------|--|--| | 0–500 | 139 | | | | 501–1000 | 59 | | | | 1001–1500 | 22 | | | | 1501–2000 | 4 | | | | 2001–2500 | 1 | | | | ≥2501 | 1 | | |

Table 2 Frequency table showing the rang of ward sizes (acres) in Devon County

| Area (acres) | Number of wards | | |--------------|-----------------|--| | 0 | 83 | | | 5000 | 34 | | | 10,000 | 31 | | | 15,000 | 15 | | | 20,000 | 5 | | | 25,000 | 0 | | | 30.000 | 1 | | | >35,000 | 1 | |

2.2. Collision data

Administrative data on road casualties in the UK comes via the police from a recording system referred to as "Stats 19" after the form on which the data are initially collected. There are known problems with these data (Bull and Roberts, 1973). In particular, collisions involving vulnerable road users (in this context these may be defined as pedestrians, cyclists and motorcyclists) and young road users are most likely to be under-reported (James, 1991). However, hospital based studies also suffer from disadvantages; firstly only casualties with injuries requiring certain medical attention will be included and secondly, much of the specific locational information of the collision will not be recorded. The "Stats 19" data remains the definitive source in terms of policy definition of the scale and scope of the road casualty problem in the UK. There are specified national arrangements for collection of road collision data in the UK, but additional information is collected locally and it is some of this information that is used here, to determine the ward of residence of the casualty. Abdalla et al. (1997) have described a similar attempts using the postcode to extract demographic information about casualties.

For this study, information has been gathered on collisions in which 645 children in Devon were injured between January 1999 and December 2003. Of these 645 children it was possible to extract valid postcode information for 323 casualties, 315 of which were resident within Devon. Postcode information is not available in many cases, either because the child couldn't provide the information, were not asked for it or there were transcription errors and omissions during the data input stages. Patterns of missingness were investigated to check for bias in as much as examining possible outliers from a binomial model applied to the presence or absence of postcode information by the ward in which the collision occurred. There was no striking evidence of systematic bias in the missing data in terms of the ward where the collision occurred. It should be acknowledged that this cannot provide a check on missing-ness by ward of casualty residence. Table 3 presents summary information regarding the distribution of casualties around wards within Devon with respect to either the ward where the collision occurred or the ward where the casualty was resident.

Table 3 Frequency distribution comparing the two methods of aggregation, the total for the first row, (n = 323), for the second (n = 315)

Casualties per ward	Number where collision reported	Number where casualty resident
0	114	96
1	38	49
2	25	31
3	11	20
4	9	12
5	10	5
6	6	4
7	5	4
8	2	1
9	2
10
11	1	1

Based on all collisions reported in Devon between 1999 and 2003.

2.3. Modelling approach

Firstly, the role of the generalized linear mixed model (GLMM) in modelling casualty rates will be considered. Then, extensions to this model which assume a spatial structure for the random effects will be considered as these models offer one way of accounting for the spatial dependence seen in the data. Finally, some information will be provided on the Bayesian approach to such models will be given. It should be noted that this framework, realised using a computationally intensive process known as Markov Chain Monte Carlo (MCMC), provides a feasible way of modelling real world complexities, such as spatial dependence, in a given dataset.

Child pedestrian casualty rates, where the casualty count is denoted by Yi and the denominator denoted by oi can be conveniently modelled assuming the data to be Poisson distributed:

$$Y_i \sim \text{Poisson}(o_i \lambda_i)$$

Where λⁱ is the Poisson rate parameter, often referred to in this context as the relative risk. It is also possible to derive the standardized morbidity ratio as Yi/*oi*λi. In the more usual generalised linear model the λⁱ for each ward contains covariates and parameters which indicate the association between the covariates and the casualty rate). This model can be extended to form a generalised linear mixed model (GLMM) which also incorporates a "random effect" within λi. In this case we have:

$$\log\left(\lambda{i}\right) = \beta{0} + \beta{1}x{i} + \nu_{i}$$

where β⁰ is a population level intercept, β¹ is a parameter which measures the strength of association between the deprivation measure and the casualty rate which denotes the rate of increase in the casualty rate as the deprivation score increases. The νⁱ are random effects.

As stated, aggregated casualty data are often relatively easy to obtain and therefore popular for analysis. However, there is a wide range of exposures experienced within any given grouping and these studies are susceptible to multiple sources of bias and confounding. The hope of spatial modelling is that some of these unrecorded covariates may vary in a spatial manner (Richardson and Monfort, 2000). If such a spatial model correctly incorporates these effects, the relationships between principal covariates and the response rates may therefore be more accurately modelled. In this case, vi ∼ Normal(vˆi, τ/mi) where vˆⁱ is spatially weighted with reference to adjacent wards. It may be noted that this is a very well established approach in small-area spatial epidemiology and the idea of "smoothing" the risk surface across adjacent areas is a well studied approach.

Much could be said about the Bayesian approach to statistics, it is sufficient to note at this stage that it is favored here because it makes the spatial modelling attainable. In a Bayesian modelling exercise, β⁰ and β¹ are believed to be random variables. If νⁱ are simply assumed to be zero mean normal variables, that is νⁱ ∼ N(0,σ) the model takes on the fairly standard GLMM form and only σ has to be estimated during the model fitting procedure. However, here νⁱ can be considered as a latent variable (Banerjee et al., 2004). This latent variable captures all the uncertainty regarding the differential casualty rate in each ward, such as that arising due to measurement error, missing covariates, over-dispersion or even genuine underlying differences in casualty rate. The particular attraction of Bayesian models in this context is that the "random effect" can be given a spatial prior structure, which has the effect of "shrinking" the estimate of casualty rate across adjacent wards (Clayton and Kaldor, 1987) This is a powerful approach to dealing with small area data. This class of models is referred to as a conditionally autoregressive model (CAR), the random effects distribution for each row of the correlated random effects can be expressed as a conditional distribution.

$$v|\sigma \propto \frac{1}{\sigma^n} \exp \left{ \frac{1}{\sigma^2} \sum{i=1}^n \sum{j < i} w_{ij} (v_i - v_j)^2 \right}$$

These "conditional auto regressive" models (CAR) have been extensively used for over a decade (Besag et al., 1991).

The statistic used to assess the goodness of fit of these models is the deviance information criteria (DIC) (Spiegelhalter et al., 2002). The deviance should be familiar as the generally used diagnostic for generalized linear models. However, the DIC also includes a penalty term which penalizes over-complex models (Spiegelhalter et al., 2002) and is based on assessing D¯ , the posterior mean of the Deviance, as well as Dˆ , the deviance at the posterior mean of the various model parameters. The difference between these two is the penalty term pD, which estimates the effective number of parameters and is used to penalize unduly complex models.

Markov Chain Monte Carlo is a generic name given to a whole set of model fitting procedures that has revolutionized the use of this set of statistical procedures by providing a very convenient method for fitting models which cannot be solved analytically. A tractable introduction to the approach is available in a road safety context (Tunaru, 1999). Briefly, MCMC enables an estimate of the posterior distribution, something that cannot be estimated analytically for these models. However, unlike many numerical algorithms, this is a simulation approach which is not guaranteed to converge around the correct solution, hence in addition to model fitting diagnostics it is important to carry out convergence diagnostics with these models (Cowles and Carlin, 1996). These models have been fit with the WinBUGs software (Win referring to Windows and BUGs referring to Bayesian Updating by Gibbs Sampling -one of a number of MCMC methods) as this software is particularly capable of fitting these spatial models (Spiegelhalter et al., 1998).

For the models fit here, there were two sets of casualty counts, Yi, the number of child pedestrians injured in a road collision in a given ward i and Zi the number resident in ward i who were injured elsewhere in Devon. In fact, the denominator for the casualty rate is problematical. In essence it is derived from the child population resident in any given ward. However, in order to correct for misdefinition of the "population at risk" when considering the data aggregated by the ward in which the casualty was injured, it is also possible to use a geographically weighted child population for any given ward where the population of each ward can be modified in accordance with the population of adjacent wards. Regardless of how the ward population at risk is estimated, for computational convenience, the denominator is rescaled as an expected casualty count (referred to as oi, for all the n = 226 wards in Devon where the offset oi = ((Population/ ⁿ ⁱ=1Populationi) ^× ⁿ ⁱ=¹Yi) (or by analogy for Zi). Given the expected casualty count oi, both Yi and Zi are generally believed in principle to be capable of being modelled by a Poisson distribution having mean λⁱ (albeit λ will be different for each model).

The Bayesian methodology requires prior assumptions about the model to be made. β⁰ has been assumed to follow a uniform (−∞,∞) distribution, β¹ a normal (0,100,000) distribution. Each νⁱ is assumed to be conditional on its immediate neighbors only with a precision (the inverse of the variance) σ−¹ assumed a Gamma (0.5, 0.0005) distribution.

3. Results

3.1. Diagnostic tests for spatial dependence

There are conceptual reasons for believing that there may be lack of independence in terms of the casualty home location rather than the collision location (MacNab, 2004). For example, if there are indeed casualty related covariates influencing risk, these would tend to be taken with the casualty to other locations. One initial investigative measure for spatial independence is the Geary C statistic (Cliff and Ord, 1981) Results are given in Table 4 which suggest a value of 1.0272 for the collision location aggregated data and 0.8984 for the

Table 4 Geary's C test under randomization

	Geary C statistic	Expectation	S.D.	p-Value
Collision location aggregated (n = 323)	1.0272	1.0000	0.0608	0.6726
Home location aggregated (n = 315)	0.8984	1.00000	0.0592	0.0435

casualty home location aggregated data. A value of 1 would suggest that these data were indeed spatially independent and significance tests suggest that the home location aggregated data may indeed have spatial dependencies.

Table 5 confirms this impression. Monte Carlo tests (these are replicate tests not to confused with the Markov Chain Monte Carlo methodology used to fit the spatial models) have been carried out with 1000 replicates which placed the observed data for the home location aggregated data 46th least likely to have occurred by chance and suggesting a Monte Carlo based p-value of 0.0460 for testing a hypothesis of spatial independence versus spatial dependence. Whilst this may be indicative of a lack of spatial independence it has to be acknowledged that the Geary C statistic can be difficult to calculate correctly when there is a wide range of ward sizes.

3.2. Fitting spatial models

The most obvious concern with the data aggregated in terms of the collision location is that these models underestimate the "population at risk". One simple remedy would be to "weight" the population of each ward essentially by increasing it in accordance with the population in neighboring wards. Alternatively, if it could be argued that population movement would tend to be from rural wards to urban wards, perhaps the rural population needing to make use of leisure, educational and retail facilities in the urban centers, only the urban wards would need to have their population weighted. This was attempted and the basic models fitted, in every case this was found to significantly deteriorate model fit.

Therefore, Bayesian methods have been used to fit a spatial dependence model to these data. The validity of the model fitting exercise is very dependent on the algorithm fitting the model. In particular, inference is only valid if it can be shown that the output from the model fitting algorithm has reached a viable and stable state, referred to as convergence. Fig. 1 appears to show a MCMC chain which is fluctuating around a stationary value. This visual impression can be confirmed by both the Heidelberger and Welch stationarity and inter-

Table 5 Monte-Carlo simulation of Geary's C

	Statistic	Observed rank	p-Value
Collision location aggregated (n = 323)	1.0272	700	0.6993
Home location aggregated (n = 315)	0.8984	46	0.0460

Fig. 1. Trace from MCMC fitting algorithm for parameter β¹ for 50,000 samples.

Table 6 Diagnostic checks on model fitting

	D¯	Dˆ	pD	DIC
Collision location aggregated (CAR model, n = 323)	718.878	627.059	91.819	810.698
Home address aggregated (CAR model, n = 315)	591.997	587.795	4.202	596.199

A smaller value of DIC indicates a better fitting model. D¯ is the posterior mean of the deviance, Dˆ is the deviance at the posterior mean of the various model parameter, pD is the effective number of parameters.

val halfwidth tests and a Geweke test which all suggested that the chain had converged (Heidelberger and Welch, 1983; Geweke, 1992). It should be noted that in most cases, the first 5000 samples of output were discarded (referred to as the "burn in" part of the MCMC fitting procedure) and 50,000 samples were kept for assessment. In the case of the spatially correlated models, the first 25,000 samples were discarded, and every fifth sample from a subsequent chain of 250,000 repeats was kept. Another source of error derives from the nature of the principles of Monte Carlo integration: the Monte Carlo (MC) error must be small for inference to be valid.

As highlighted above, the "deviance information criteria" is a model diagnostic related to the deviance and also whereby the smaller the DIC the better the model fit is believed to be. It can be seen from Table 6 that the DIC resulting from fitting a spatial model to data aggregated by home location aggregated is much lower than one fitted to the data aggregated by collision location.

Table 7 gives details for the mean and standard deviation of the posterior distribution of the parameters estimated by the model. As previously mentioned, MacNab (2004) has provided results when the data are aggregated by casualty home location. Here it is possible to compare the two possible aggregation methods and it can be seen that there are large

Table 7 Comparison of posterior distribution of parameter values from the two spatial models

Parameter	Accident location aggregated		Casualty home aggregated
	Posterior mean (S.D.)	MC error	Posterior mean (S.D.)	MC error
β0	−1.3820 (0.1664)	0.0025	−0.7103 (0.1585)	0.0014
β1	0.0364 (0.0051)	0.0000	0.0226 (0.0045)	0.0000
σ	1.0650 (0.1161)	0.0000	0.0750 (0.0728)	0.0035

Fig. 2. Overlaid caterpillar plots (ordered according to median values from casualty home aggregated model) from both models, dots denoting the posterior median for νi, the extremities of each band the 95th percentiles.

differences in the estimate of σ which is much smaller for the casualty home address aggregated data. The effect of this on the estimation of individual random effects can be seen in Fig. 2 which overlays "caterpillar plots", similar to a the plots of mean ± 2S.D. in common usage but denoting the median and 95th percentiles as a way of summarizing the posterior distribution. This indicates that the random effects posterior distributions are much narrower (i.e. have been estimated with much less uncertainty) when the casualty home data aggregated model has been used.

Having considered the evidence that accounting for spatial dependence improves the fit of the casualty home aggregated data, attention turns to differences in the estimate of β1, the parameter indicating the strength of association between deprivation and casualty rate also varies considerably between the two models. The mean of the posterior estimate for β¹ is 0.0364 for the collision location aggregated data, but is only 0.0226 for the casualty home location aggregated data. In attempting a Bayesian analysis, it is necessary to appreciate that the parameter is considered a random variable, and so a summary of the posterior distribution is given in Fig. 3. It can be seen that there is some overlap in terms of β¹ obtained from the two models, but generally speaking the evidence for an association between deprivation and casualty rate is much lower when considering the home location aggregated data (the better fitting model).

One minor point deserves mention in respect of the posterior distribution. In terms of policy and resource allocation greater consideration needs to be given to the uncertainty implicit in any measure of casualty rate or relative risk. Fig. 4 presents "caterpillar" plots of the modelled relative risk for each ward (sorted in ascending order of relative risk) derived from the better fitting model, based on the casualty home lo-

Fig. 3. Child pedestrian casualties injured in Devon 1999–2003 and reported in "Stats 19" system: Density plot summarising the posterior distribution of β1, the association between deprivation and casualty rate according to the two models.

cation aggregated data. Each row of the plot features an arrow indicating the 95th percentiles for the posterior distribution of the relative risks, with a point denoting the median. It is abundantly clear that there is considerable overlap in terms of the estimated relative risk. It is therefore not possible to identify wards that are substantially different in terms of relative risk, and hence it would be difficult to support an argument from these data for prioritizing resources in one area rather than another.

Fig. 4. Child pedestrian casualties injured in Devon 1999–2003 and reported in "Stats 19" system: "Caterpillar" plot summarizing the posterior distribution (median and 95th percentiles) of λi, the relative risk for each ward. Wards have been ordered in size of the posterior median along the y axis.

4. Discussion

There have been attempts recently using multilevel modelling in order to attempt to disentangle areal effects from individual effects (Reading et al., 1999; Haynes et al., 2003; Jones and Jorgensen, 2003) which would go a long way toward addressing ecological concerns surrounding this kind of analysis. However, the situation remaining is clearly one where there is considerable confounding and interaction between individual and environmental factors. It would be therefore most interesting to integrate both spatial models in a multilevel formulation (or at least include spatially correlated casualty covariates in a multilevel model of collision locations), where both areal and individual factors could be modelled simultaneously. This requires much closer attention to data collection than is practiced at present in the UK. For example, it would be useful to know whether the child had been transported to the collision location (for example to attend school, visit retail outlets), or was roaming around the home location. It is however known that much of the difference in pedestrian risk is due to widely varying exposure (Stevenson et al., 1996). There may be differences in the amount rather than the way in which individuals from different socio-economic groups are exposed (Roberts et al., 1994). In particular, groups of low socio-economic status are less likely to own a car and therefore will be more exposed as pedestrians (Rao et al., 1997). This could imply that the association between deprivation and casualty rate is due at an individual level to the amount of exposure, and at an areal level to the type of environment.

A conventional analysis of very similar data been reported before suggesting a better model fit with the collision location aggregated data (Hewson, 2004). However, the results presented here indicate that when compensation is made for spatial dependence not only can model fit be improved considerably, but the preference for data aggregation method is reversed. The spatial dependence in the casualty home aggregated data can be understood as implying that casualties tend to have collisions near their home, but not necessarily within their home ward. However, in preferring the spatial model, it should be noted that belief in the strength of the relationship between deprivation and casualty rate is reduced considerably. This implies that part of the strength of the association is derived from a lack of independence in the way data are aggregated; children from both deprived and un-deprived wards are still at greater injury risk in more deprived wards explaining the earlier finding. But given estimates of the distance between home and collision location of around 600 m it is clear that injury will tend to occur within either the home or a neighbouring ward. Therefore, when spatial dependence is incorporated within a model, a better model fit can be obtained by virtue of the risk surface being shared between wards with higher resident casualties, less smoothing is possible between the collision aggregated hotspots. This may make sense in the light of individual level studies which are strongly suggestive of factors acting at an individual level but which are realized within the environmental context (Rao et al., 1997).

One major problem with these data are the under-reporting problems alluded to earlier. Literature suggests that perhaps 60% of slight casualties are recorded, a figure that gets lower for vulnerable road users and for young road user (James, 1991). Having lost information on many casualties, these data have postcode information for just under half of the reported casualties. In mail delivered survey terms this might be considered quite a high response rate, but it clearly means that analysis is based on a minority of casualties. Whilst the obvious bias checks suggested no major problem, neither is there any reason to suppose that we have data missing completely at random. For example, it is entirely possible that the spatial effect described by these models represents a "propensity to report" which varies in a spatial manner.

Despite the complexity of the spatially structured random effects, it can also be seen that the model reported here is relatively simple when compared, for example with Graham et al. (2005). However, their "special factors" are not required here as this is a small area study based in Devon. Of more interest are their proxy variables; traffic flow was estimated by means of a proxy variable based on employment and population; the physical nature of the road environment was again a proxy variable based on the number of traffic nodes and the density of population and employment. Notwithstanding the risk of collinearity with the deprivation indicators, these proxy measures clearly have potential for spatial dependence. For example, population and employment in one ward will clearly contribute to traffic in adjacent wards and there are strong reasons for assuming that these proxy measures will not be spatially independent.

There are many concerns about the ecological fallacy; phenomena observed or not observed at an areal level are not necessarily similar at the individual level. Indeed, Davis (2004) very strongly highlights the particular risks of aggregate level analysis in road safety. Nevertheless, we cannot avoid the finding in this study, regardless of the exact nature of the spatial phenomena, that corrections are required for the lack of spatial independence before the casualty home aggregated data can be used, the very data where one would assume the association between deprivation measures and casualty rates to be most meaningful. However, in using these data, our belief in the strength of association between the casualty rate and a known and important covariate (deprivation) is reduced. In attempting to conduct a valid population level analysis which attempts to infer the role of postulated casualty related covariates it is important that this is taken into account.

5. Conclusions

Ecological analyses are popular largely because the data are widely available, there are both cost and confidentiality reasons which limit the amount of research conducted at an individual level and are likely to continue to provide part of the body of evidence used to guide interventions. Firstly it needs to be noted that the underlying data are based entirely on an administrative abstraction of an observed phenomenon. Secondly, it is important that methodological developments underpinning these types of studies are reflected in areas such as child pedestrian injury research. This paper confirms the importance of formulating ecological models to take account of spatial dependence in child pedestrian casualty data. It demonstrates that without such an accounting, there is potential to overstate the relationship between covariates, such as the deprivation index, and the casualty rate.

Profitable avenues for further work would clearly involve specifically considering models for missing data and measurement error. Individual level studies clearly remain central to any serious attempt to understand and develop interventions for child pedestrian casualties. However, in this paper one aspect of casualty rate modelling has been made very clear, the ability to fit models which make allowance for spatial dependence may in turn depend on the method of aggregation used and at the very least provides a warning of the potential for ecological fallacies when interpreting these data. The Geary C statistic provides evidence that this is a necessary step when using data that has been aggregated by the ward (or other areal concept) of residence of the casualty. It is impossible to escape the result that our belief in the strength of association between casualty rate and a measure of deprivation are lower when using a spatial model fitted to the data aggregated by home location of the casualty. By UK standards, Devon is a relatively sparsely populated county with relatively larger wards. It seems reasonable to suggest that child pedestrian roaming will be more pronounced in metropolitan areas and perhaps spatial dependencies stronger. Nevertheless, this paper does illustrate concerns raised by Davis (2004) in respect of ecological analysis. The same data, aggregated in two different ways behave very differently with respect to spatial dependence, and the better fitting ecological model suggests a weaker relationship between deprivation and child casualty rate.

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