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category: literaturenote citekey: grahamdecomposingimpactdeprivationchild2008 title: Decomposing the impact of deprivation on child pedestrian casualties in England authors: "Graham, Daniel J.; Stephens, David A." year: 2008 date: 2008-00-00 2008 doi: 10.1016/j.aap.2008.02.006 publication: "Accident Analysis & Prevention" url: "https://doi.org/10.1016/j.aap.2008.02.006" zotero_key: AQ7ZZMFU zotero_storage: SKHABIDP collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Graham, Daniel J."
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Accident Analysis and Prevention
journal homepage: www.elsevier.com/locate/aap
Decomposing the impact of deprivation on child pedestrian casualties in England
Daniel J. Grahama,∗, David A. Stephens b,1
- a Imperial College London, London, UK
- b McGill University, Montreal, Canada
article info
Article history: Received 18 July 2007 Received in revised form 18 February 2008 Accepted 20 February 2008
Keywords: Deprivation Child pedestrian casualty Negative binomial GLM Quasi-likelihood Zero-inflation
abstract
The incidence of child injury due to road traffic accidents appears to be positively associated with socioeconomic deprivation. However, the reasons behind this relationship have proven difficult to identify. In this paper, we present results from a cross-sectional analysis using generalized linear regression models of child pedestrian casualties for the wards of England. We find that there are distinct and substantial effects on casualty rates from characteristics associated with area deprivation across diverse environments. These associations exist over and above influences arising from local environmental characteristics. Distinct dimensions of deprivation appear to affect the incidence of pedestrian casualties to varying degrees and sometimes in different directions. The results identify a relationship between income deprivation and the incidence of child pedestrian casualties, but they also show that poverty is only one aspect of deprivation that matters. In particular, we find a consistent positive influence from crime related deprivation.
© 2008 Elsevier Ltd. All rights reserved.
1. Introduction
The existence of a statistical relationship between socioeconomic deprivation and health has frequently been proposed in the literature (e.g. Lancaster and Green, 2002; Cooper, 2002; Lorant et al., 2001; Gould and Duncan, 2000; Diamond et al., 1999; Carstairs and Morris, 1991; Townsend et al., 1988; Jarman, 1983). Specifically, higher levels of deprivation are found to be associated with poor health. Research has sought to test the presence and strength of such associations, but has also frequently addressed issues relating to how deprivation is actually measured and how deprivation indices are constructed.
Recent findings indicate that the incidence of injuries amongst children is particularly influenced by deprivation. Hippisley-Cox et al. (2002) find that children from the most deprived social classes have a mortality rate from injury five times that of children from the least deprived social class, while Roberts and Powers (1996) have shown that the social class gradient is prevalent over the most common mechanisms of injury.
1.1. Deprivation and child pedestrian road traffic accidents
Pedestrian road accidents are a major source of injuries to children. Research on the relationship between child pedestrian casualties and the socio-economic characteristics of victims supports the deprivation hypothesis: children of the lowest social class are approximately four times more likely to be killed on the road than those of the highest social class (see White et al., 2000; Christie, 1995 for reviews of this literature).
The factors that lie behind this observed relationship are multifaceted and often highly inter-correlated. Although the existing literature does not provide us with an exact or comprehensive understanding, individual level studies of child pedestrian casualties do provide some indication of factors that may be of importance. These include poor housing accommodation and lack of open space for playing at home (Roberts and Pless, 1995), low levels of access to car passenger transport for children (Erskine, 1996), increased exposure to hazardous traffic environments and increased proximity to busy streets (Bagley, 1992; Christie, 1995; Roberts et al., 1995), lower levels of child supervision within households (Christie, 1995), the scarcity of publicly available play spaces (Al-Balbissi et al., 1990), and the tendency of deprived children to play unsupervised close to home (Sharples et al., 1990).
Despite evidence such as this we still do not have a precise understanding of how socioeconomic status affects road accident rates. This is because many of the factors cited in the individ-
∗ Corresponding author at: Centre for Transport Studies, Imperial College London, London SW7 2AZ, UK. Tel.: +44 20 7594 6088; fax: +44 20 7594 6107.
E-mail address: d.j.graham@imperial.ac.uk (D.J. Graham).
1 Department of Mathematics and Statistics, McGill University, Montreal H3A 2K6, Canada. Tel.: +1 514 398 2005; fax: +1 514 398 3899. d.stephens@math.mcgill.ca.
ual studies could be prevalent to equal degrees in both deprived and non-deprived locations. Indeed, for several of the factors cited above we could argue that variation is based at least as much on urban density as it is on deprivation. Thus, we need to be clear about what we are trying to capture when we measure deprivation, and we need to try to make deprivation measures distinct from other prominent influences on the incidence of child casualties.
1.2. Plan of this paper
In this paper we focus on the measurement of deprivation and the construction of the deprivation index in an attempt to shed light on relationships between area deprivation and the incidence of child pedestrian casualties.We decompose the deprivation effect by considering six separate dimensions related respectively to income, employment, health, education, housing and services, and crime. Our principal research question is whether the different dimensions of deprivation affect the incidence of child pedestrian casualties in different ways. If they do, we wish to obtain statistical quantification of the association between each distinct aspect of deprivation and the incidence of child pedestrian casualties in England.
We present results from models derived to explain variation in child pedestrian casualties (CPCs) according to a range of locationspecific characteristics, a subset of which are the components of deprivation. It is important to emphasize that our analysis is area based. Unlike the individual studies quoted above, our observations are spatial units, not people or family groups. The deprivation scores characterize the socio-economic profile of 7925 small areas of England. As such we cannot directly determine an association between the incidence of casualty and deprivation at an individual or household level. From a policy perspective, however, a location based model is very useful because it allows us to identify the area-level factors that are associated with casualties. Furthermore, our objective is to disentangle the effect on child casualties of ward characteristics and the dimensions of ward deprivation.
The paper is structured as follows. Section 2 describes the deprivation indices. Section 3 develops an area-level model to examine the incidence of child pedestrian casualties. Section 4 discusses issues ofmodel fit and estimation. Results from cross-sectional negative binomial GLMs are presented in Section 5. Conclusions are given in Section 6.
2. Variation in socio-economic characteristics
2.1. The english indices of deprivation
The Indices of Deprivation, published by the Office for the Deputy Prime Minister (ODPM, 2004), provide a comprehensive source of data on spatial variance in socio-economic conditions for England. The principal conviction underpinning the production of these indices is that there are distinct dimensions of deprivation that can be identified and measured separately. The first stage in the construction of the indices is to gather data describing a diverse range of characteristics of small areas, called Super Output Areas (SOAs), of which there are 32,842 in England. These data are then used to construct a series of indicators that measure spatial variance in some particular attribute. Certain indicators are then combined arithmetically or by using factor analyses to produce seven individual domain indices that each describes a separate aspect of deprivation.
2.2. Seven deprivation domain indices
The seven domain indices are as follows.
- (i) Income (INCSCR): the income domain index measures the proportion of the population in receipt of some form of income 'benefit' from the state. These benefits include income support for the unemployed and for working households below a low income threshold.
- (ii) Employment (EMPSCR): employment deprivation measures the proportion of the population of working age that are involuntarily excluded from work, including the unemployed, incapacitated, physically disabled, and those unable to undertake work.
- (iii) Housing and services (HOUSCR): this domain measures 'barriers of access to housing and services'. Barriers to services are principally geographic in nature, for instance, measuring distance to doctors, supermarkets, primary schools, post offices and the like. Barriers to housing include overcrowding, but also the affordability of owner-occupation.
- (iv) Health (HEASCR): the health domain identifies areas with poor health characteristics including premature death, illness and disability, and emergency admissions to hospital.
- (v) Education (EDUSCR): the education domain measures spatial variance in education, skills and training deprivation. It makes use of indicators on the performance of children at school, the proportion of students continuing education after 16, secondary school absence rates, the proportion of under-21s not entering higher education, and the proportion of working age adults with no or low qualifications.
- (vi) Crime (CRMSCR): the crime domain reflects the rate of recorded crime combining indicators on burglary, theft, criminal damage and violence.
- (vii) Environment (ENVSCR): the living environment domain measures the 'quality' of the internal and external environment. Indicators of the internal environment are the conditions of social and private housing and the existence of central heating. Characteristics of the external environment considered are air quality and the number of road traffic accidents involving injury to cyclists or pedestrians.
The seven individual domain indices are also weighted and combined into a single overall Index of Multiple Deprivation (IMDSCR). In order to combine the domain indices they are first standardized to a uniform metric and then transformed to a common distribution. They are then weighted according to available research evidence about the nature of multiple deprivation, which in fact results in income and employment deprivation being awarded the most importance and therefore being weighted most highly.
2.3. Current deprivation data
Deprivation data are published at the SOA level. This is a much finer level of spatial disaggregation than we can achieve in our model because other necessary sources of data are not available at such a fine scale. For this reason, we have constructed weighted average deprivation domain scores for larger geographical units defined by the Census Area Statistic (CAS) ward disaggregation of Britain. There are 7927 CAS wards in England. To construct the aggregated scores we use the ward share of population in each of the smaller areas as the weight. The SOA resident population data are taken from the UK Census of Population 2001.
3. Data analysis and modelling
3.1. Dependent variables
Our study is based on a cross-sectional analysis of child pedestrian casualties in the CAS wards of England. The casualty data are based on records completed by police officers each time an incident is reported to them. The individual police records are collated and processed by the UK Department for Transport (DfT) as "Road Accident Data - GB", generally known as the STATS 19 data base. The data record the age of the victim and we define children as persons under the age of 16. Using a Geographical Information System (GIS), we allocated each of the child pedestrian casualties that took place between 1998 and 2002 to a ward. The data allow us to identify those incidents in which the child was classified as killed or seriously injured (CKSIs). The dependent variables used in our model are ward counts of the absolute number of CPCs and CKSIs.
Our study is based on a cross-sectional observational rather than causal analysis, that is, the effects that we report do not necessarily have a causal interpretation. The predictors of child pedestrian casualty rates by ward are themselves correlated, and hence their effects are possibly confounded. Because of this confounding, we do not in this paper report the causal effect on incidence rate of any individual predictor from our analysis. However, this limitation is not necessarily serious; recall that the causal effect of predictor X1 in the presence of confounding predictor X2 is the expected change in incidence rate when X1 changes from x1(0) to x1(1), say, either with X2 held constant, or with the effect of X2 averaged out. However, the causal measure may be of little interest if the change between potential values of X1 cannot be practically achieved. For example, the causal effect of Health Deprivation may be of little interest if the Health Deprivation score cannot be changed by intervention without concomitant change in the other predictors. Despite this, a causal analysis is possible, and we will report on this elsewhere.
3.2. Independent variables
The first set of independent variables of interest relate to the deprivation of the localities in which the pedestrian casualties occur. For these we use the indices of deprivation produced by the ODPM which have been described in the previous section. Note however, that the environment deprivation score actually makes use of data on road pedestrian and cycling casualties, in addition to a variety of other attributes, to describe the quality of the internal and external environment. For this reason the environment score could be endogenous and we do not include this particular component in our models. We do, though, need to provide some representation of variance in the external built and traffic environment because this could be crucial in explaining the incidence of pedestrian casualties.
Accordingly, our model emphasizes that deprivation can represent only some of many factors that could potentially influence the number of child pedestrian casualties in any ward. In addition to deprivation, our model hypothesizes that spatial variation in the absolute number of CPCs and CKSIs is a function of five local factors:
- (i) The absolute number of children-ward population under 16. Children resident in any ward is likely to affect the incidence of CPCs and CKSIs.
- (ii) The volume of traffic flows. Traffic flows are not observable at the ward level. Instead, we use a basic gravity form for trip distribution which assumes that traffic flows between wards i and j are directly proportionate to the level of activity in each ward and inversely proportionate to the distance
between them1. The gravity model yields for each ward a direct traffic generation potential measures that is proportionate to the level of population and employment in that ward, and a measure of through traffic potential generated by employment and population in proximate wards. For ward i we have constructed the following proximate employment and population measure:
$$PEPi = \sum{j \neq i} \frac{(E_j + Pj)}{d{ij}} \tag{1}$$
where dij is the distance from ward i to ward j, Ej is employment in ward j, and Pj is population in ward j.
- (iii) The physical nature of the local environment. We seek to model effects arising from the extent of development in the ward, and from land use mix and density. We use the number of traffic network nodes, N, defined as the number of meeting points of two or more road, to control for the extent of built development, and measures of employment per node and population per node as indicative of the relative densities of economic and residential land uses. We also have a measure of the physical area of the ward.
- (iv) The characteristics of the local road infrastructure. To model the characteristics of the road network we have generated data on road capacity across England. For each ward, we measure the length of A-roads, B-roads, minor roads, and motorway.
- (v) Other local specific factors. These include an average measure of annual rainfall and sunshine from one of 31 weather stations and also the x (north–south) and y (east–west) coordinates of the geographical location of the centre of each ward. We hypothesize that differences in climate could affect pedestrian casualties; motor vehicles may have less control on wet roads, or it may be that the length of sunshine influences the exposure of children to the traffic environment. We also hypothesize that there are a number of unobservable effects on pedestrian casualties that are related to the position of any ward within the country as a whole. These could include differences in culture, social habits, and other unknown effects.
Clearly, the different dimensions of deprivation form only a subset of the many factors that will influence the number of child pedestrian casualties in any ward. Using the variables described above, we have tried to represent some other key influences. An important issue to emphasise in this regard is that we do not have a direct representation of child exposure to the traffic environment, and this would in fact be impossible to achieve with the data we have available in the UK context. Instead, we have derived the set of variables described above which together we hope will account, in addition to other effects, for variance in rates of exposure. We can, of course, make no direct statements about the impact of exposure on accidents using this approach, and the inferences we make about the components of deprivation are contingent upon our modelling strategy having controlled effectively for variance in exposure.
1 The gravity model has a long established used in transport modelling and is still the most commonly used approach to estimate trip distributions between aggregate areas. Indeed, the traffic data that the UK Government use in their National Transport Models for forecasting and prediction is synthetic data generated using a gravity specification. A recent review of traffic potential measures derived from Gravity models is provided by El-Geneidy and Levinson (2006).
| Income, Employment, Health, Education, Housing, Crime |
|---|
| INCSCR, EMPSCR, HEASCR, EDUSCR, HOUSCR, CRMSCR |
| Ward employment E, Proximate Employment PEP |
| Number of network nodes N, Employment density E/N |
| Ward area and transformations A, √ A, A2,log(A), |
| Network node density N/A |
| Ward child population CP, Ward population P, |
| Population density per node P/N, Population densities per |
| unit Area |
| PopDens = P/AREA, CPopDens = CP/AREA |
| Road lengths for Motorway, A- and B-roads, minor roads |
| RLmo, RLA, RLB, RLmi |
| Annual rainfall and sunshine AR, AS |
| x- and y-coordinates of the centroid of the ward x, y |
A summary of the variables included in the generalized linear additive models is included in the table below. The empirical distribution of some of these independent variables indicates that a modified model should be used. For example, the distribution of road lengths of motorways, A- and B- roads is a mixture of a high frequency at length zero (many wards contain no motorways, A- or B-roads) and a monotone decreasing density. This mixture density form is acknowledged in our analysis by including indicator variables, RLmo0, RLA0 and RLB 0, which act as factor predictors taking the value one if the corresponding road length is zero, and zero otherwise. Correspondingly, the variables RLmo+, RLA+ and RLB + record the road length in thousands of miles if the length is non-zero. This simple extension provides a flexible way to model dependence on these variables.
We use the node density per unit area (N/A) variable as it is a proxy for the area of any ward taken up by built development, on the assumption that built development generally only exists where there are roads, while the population (P/N) and employment (E/N) density variables give some measure of the density of that built development.
Finally, in our analyses, for numerical stability in the maximization routines, the covariate values were rescaled so that they lie approximately on the range 0–10. For E, this involved rescaling by 106; P and CP were rescaled by 104; PEP, E/N and P/N and allmileage variables were rescaled by 103; x and y coordinates (Easting and Northing) were rescaled by 105.
3.3. Limitations of analyses based on ward-level data
It is important to stress that our deprivation data describe the socio-economic conditions of the geographical unit (ward) and not necessarily of the victim of the accident. For reasons of confidentiality, data are not available on the location of residence of the victim. In effect we are using the deprivation data to test hypotheses both about the area and the travel patterns of children living in those areas. We do know from previous research that most child accidents take place very close to home. Sharples et al. (1990) recorded the distance of child accidents to the home in 235 cases and found that for child pedestrian casualties 34% of these incidents took place within 0.4 km of residence and 80% within 1.6 km. This compares to an average area of a British ward of 14.0 km2. In any case, all independent variables included in our model describe characteristics of location, not the casualties.
4. Model fit and estimation
Plots of the incidence data indicate distinctly different relationships to predictor variables when wards are grouped in relation to the level of urbanization. Accordingly, we want to stratify the data in a way that broadly reflects the degree of urbanization of the environment and estimate the model separately for these groupings.
Table 1 Description of area subsets and types
| Subset | Area code | Description |
|---|---|---|
| A: Conurbation | 1 | Central London |
| 2 | Inner London | |
| 3 | Outer London | |
| 4 | Inner Conurbation | |
| 5 | Outer Conurbation | |
| B: Urban | 6 | Urban area > 250, 000population |
| 7 | Urban area 100,000–250,000 population | |
| 8 | Urban area 25,000–150,000 population | |
| 9 | Urban area 10,000–25,000 population | |
| C: Rural | 10 | Rural |
The DfT provide an area type classification for the wards of England based on the level of urbanization as measured by population density. The area types are defined in Table 1. Using these codes we define three data subgroups; A: conurbation (1–5), B: urban (6–9) and C: rural (10). This stratification is based on the reasoning that population and population density in a region are likely to be influential in determining traffic density, and is motivated from exploratory inspection of graphical summaries for the outcome data, and their relationship to potentially influential predictors, when the data are grouped by these urbanization categories.
4.1. Exploratory analysis
Fig. 1 depicts the counts and rates of CKSIs and CPCs in wards categorized by Area Code as in Table 1. It indicates that incidences and rates of child pedestrian accidents are lower in more rural areas, even when the lower density of population of rural areas is accounted for (see bottom panel) (Fig. 2). In these figures, the impact of deprivation on incidence is hidden. Figs. 3 and 4 depict the counts and rates of the incidences of CKSIs and CPCs in wards categorized by Area Subset as given in Table 1, plotted using quintiles of the Crime and Housing components of deprivation. The increasing incidence of casualties with increasing crime score is evident, but this relationship is not apparent for all deprivation scores - there is no apparent relationship with increasing housing deprivation. Also, the deprivation scores are correlated, the pattern of correlation differs between data subsets. Fig. 5 displays the relationship between several scores and how the relationship changes across the conurbation, urban and rural subsets.
After stratifying on the level of urbanization, we distinguish a broad three way categorization of the wards that seems appropriate for the analysis of child casualties, and that the different dimensions of deprivations describe quite different phenomena and consequently are likely to have different implications for the incidence of child pedestrian casualties.
4.2. Statistical modelling with overdispersion and zero-inflation
We adopt a generalized linear model (GLM) for the response data, the numbers of CPCs or CKSIs observed per ward2. For a full discussion of GLMs and related models, see the classic text of McCullagh and Nelder (1989). Essentially, for independent data, a GLM models the expected response via a transformation of a linear combination of covariate terms x (the linear predictor) with
2 Poisson and negative binomial approaches to the modelling of accidents have been used before in the literature (e.g. Jovanis and Chang, 1986; Miaou, 1994; Shankar et al., 1995, 1997; Noland and Quddus, 2004).
Fig. 1. Boxplots of counts (top row) and rates per 10,000 population by ward categorized by area code (1, Inner London; 10, Rural).
vector of coefficients ˇ, that is,
$$E[Y] = \mu = g(\boldsymbol{\beta}^T \boldsymbol{x})$$
say. In most parametric models, this assumption also defines the variance of the response, and, typically
$$Var[Y] = V(\mu)$$
for some variance function V(·). For parametric GLMs, estimation is most commonly based on maximum likelihood procedures.
A standard model is the Poisson GLM with log-link and rate i, where, for datum i,
$$\log \lambda_i = \boldsymbol{\beta}^T \mathbf{x}_i \tag{2}$$
in which i is the expected number of events per period, xi is a vector of regressors. In the linear predictor, the coefficient ˇi quantifies the proportional change in the expected number of events per period from a unit change in an independent variable, xi. However, it can be quickly established that this model is not sufficient to capture the variability in the data subsets, for one of the two reasons listed below
• Overdispersion: The mean–variance relationship for the Poisson model (mean/variance = 1) is not valid for some of our observed data; typically, the sample variance is larger than the sample mean. A standard approach to modelling overdispersion is to assume a negative binomial rather than Poisson model. The negative binomial model parameterization we use has pdf
$$f_{Y}(y; \mu, \theta) = \frac{\Gamma(\theta + y)}{\Gamma(\theta)\Gamma(y + 1)} \frac{\mu^{y} \theta^{\theta}}{(\theta + \mu)^{\theta + y}} \qquad y = 0, 1, 2, \dots$$
where $$E[Y] = \mu$$ $Var[Y] = \mu + \mu^2/\theta$ .
Parameter > 0 allows for the modelling of data where the variance exceeds the mean. For the regression component, we
Fig. 2. Boxplots of populations by ward categorized by area code (1, Inner London; 10, Rural).
adopt a similar log-linear structure to the Poisson case, with log i = ˇT xi.
• Zero-inflation: In rural wards, there is an excess (relative to a Poisson distribution) of observed zeros, leading to a reduction in distribution variance. To account for the zero-inflation, we use the mixture model for the response Y which has
$$Y \stackrel{\mathcal{L}}{=} \begin{cases} 0 & \text{with prob. } \pi \ Y_{\lambda} & \text{with prob. } 1 - \pi \end{cases}$$ (3)
where Y ∼ Poisson(), for some parameter where 0 ≤ ≤ 1. In this model
$$E[Y] = (1 - \pi)\lambda$$ $Var[Y] = (1 - \pi)^2 \lambda$
A coupled binary regression GLM for the zero incidences can also be used.
Recently, Lord et al. (2005) and Lord et al. (2007) have outlined reasons why zero-inflated models may not be suitable
Fig. 3. Boxplots of counts of incidents by quintile of Crime Deprivation Score in area subsets A: Conurbation, B: Urban and C: Rural.
Fig. 4. Boxplots of counts of incidents by quintile of log Housing Deprivation Score in area subsets A: Conurbation, B: Urban and C: Rural.
for modelling highway safety data, citing different design and analysis issues. In particular they doubt the validity in context of a "dual-state" justification for such models, preferring explanations relating to low exposure rates and other data gathering issues or model inadequacy. We acknowledge these points, but note that factors such as exposure rates can be accommodated explicitly into the model for the zero component probability .
Fig. 5. Scatterplots of deprivation score components.
In our analysis, Poisson and negative binomial GLMs are fitted separately for each of the three data subsets for CPCs and CKSIs. In addition to these models, a quasi-likelihood-based Poisson-like model which relaxes the mean/variance relationship of the Poisson and gives a robust analysis is also fitted: an estimating equation of the form
$$\sum_{i=1}^{n} \frac{(Y_i - \mu_i(\boldsymbol{\beta}))}{\sqrt{V(\mu_i(\boldsymbol{\beta}))}} = 0$$
is specified, where E[Yi] = i and Var[Yi] = V(i) are presumed to take specific functional forms that do not correspond to the expectation and variance for a parametric distribution (see McCullagh and Nelder, 1989, chapter 9, for full description).
The full set of independent variables described in the previous section were included in the models along with a set of dummy variables corresponding to the area type classification of the ward (dn, n = 1, 2,... 10). We have no a priori information regarding functional forms for the relationships we have hypothesized, so an additive model (in the linear predictor) without interaction was fitted. Computation was carried out in R.
4.3. Model selection and goodness of fit
Model selection was carried out using the Bayesian Information Criterion (BIC), a criterion that rewards fidelity of the model fit to data, but penalizes complex models; for model Mˇ with *p*dimensional vector parameter ˇ and log-likelihood function , the BIC is computed as
$$BIC(M_{\beta}) = -2\ell(\hat{\beta}) + p\log n$$
where n is the sample size. The BIC is a consistent model selection procedure, in that it selects the true model from a set of possible models with probability one as n → ∞ if the true model is present, and has reasonable small sample properties; see Kass and Raftery (1995) for full details. Model selection criteria purely based on loglikelihood ratio statistics that do not penalize for complexity are sometimes prone to selecting overly complex models. However, the Vuong statistic is a likelihood ratio statistic (Vuong, 1989) has proved popular in testing non-nested regression models (see, for example, Lord et al., 2007 for a recent discussion).
There is no universally accepted optimal goodness-of-fit measure for GLMs. In what follows, we report the maximum likelihood pseudo R2 quantity (see for example Long, 1997 pp. 104–106) available from the pR2 function in R that takes the form
$$R^2 = 1 - \exp{-2(\ell(\hat{\beta}) - \ell_0)/n}$$
where 0 is the maximized log-likelihood from the null model, and n is the sample size. An alternative measure is the scaled deviance, or log-likelihood ratio compared with the saturated model, D, defined by
$$\mathcal{D}(\boldsymbol{y}, \hat{\boldsymbol{\beta}}) = -\frac{2}{\phi} (\ell(\hat{\boldsymbol{\beta}}) - \ell_{S})$$
where is the (estimated) dispersion, and S is the saturated model log-likelihood. D is also termed the residual deviance, and n − p is the residual degrees of freedom (rdf). Classical statistical theory indicates that D has an asymptotic chi-squared distribution with n − p degrees of freedom, hence, as a rule of thumb, if n is large and D ≈ n so that D/n ≈ 1, then the model is an adequate simplification of the saturated model, whereas significantly larger values of D indicate a model deficiency. Note that whereas BIC penalizes for model complexity, the R2 and deviance quantities do not. We also refer to residual plots to support the adequacy of the selected models.
5. Results
As mentioned previously the deprivation data include an overall index of multiple deprivation score (IMDSCR) as well as six component indices. To get a general feel for the effect of deprivation, we estimated our models using the composite IMD score for all child pedestrian casualties (CPC) and for child pedestrian casualties involving a serious or fatal injury (CKSI). Then the six individual components of deprivation were used in place of the composite score. As described above, the models were estimated separately for the conurbation, urban, and rural subsets.
Some data transformations were also considered. Whilst the health and crime scores are typically symmetrically distributed around zero, the composite score and remaining component scores are strictly positive and left skewed. This feature suggests that a log transformation may be useful, and in fact a log transform appears approximately to symmetrize the data, although the resulting transformed variables are not zero mean. We therefore inspected the results for models where log deprivation scores were fitted, in addition to models using variables on the original scale.
We thus have eight models to consider: (a) composite score, (b) component scores, (c) log composite score and (d) log component scores, each for Poisson and Negative Binomialmodels.We compare these models using BIC, but also report a pseudo- R2 statistic that gives an indication of global model adequacy.
5.1. Conurbation analysis
5.1.1. CKSI: initial analysis
The conurbation subset includes wards from Greater London, Greater Birmingham, Greater Manchester, Merseyside, West Yorkshire and Tyne & Wear, totalling 1448 wards. We found the eight best-fitting models for the CKSI conurbation data subset, in the eight classes identified above. The models were compared using BIC, and in general, models which utilize the log composite score fit better than the component or original scale scores. It is also evident that the negative binomial models fit better than the Poisson models.
The best-fitting model (BIC = 6520.291, pseudo- R2 = 0.475), selected using a stepwise method and validated manually, is the negative binomial model which uses the log composite deprivation score. The model shows a positive association between CKSIs and deprivation, with a coefficient (standard error) of 0.631 (0.037), which confirms prior opinion. The estimate indicates that a 10% increase in the deprivation score across the sample is associated with a 6.3% increase in the incidence of CKSIs. The final model fit is presented in Table 2: the selected model includes total
Table 2 Conurbation CKSI data: best fitting Negative Binomial model
| Term | Est. | S.E. | t | p |
|---|---|---|---|---|
| (Intercept) | 0.385 | 0.220 | 1.753 | 7.97e −02 |
| Employment | 0.534 | 0.140 | 3.806 | 1.41e −04 |
| log Employment | 0.076 | 0.023 | 3.282 | 1.03e −03 |
| Population | −0.755 | 0.170 | −4.433 | 9.31e −06 |
| log Population | 1.519 | 0.212 | 7.180 | 6.96e −13 |
| Child population | 3.146 | 0.461 | 6.822 | 8.96e −12 |
| log(IMDSCR) | 0.631 | 0.037 | 17.017 | 6.09e −65 |
| y-Coordinate | −0.078 | 0.016 | −4.938 | 7.88e −07 |
| Child population density | −2.739 | 0.607 | −4.512 | 6.42e −06 |
| log Area | −0.176 | 0.036 | −4.918 | 8.73e −07 |
| ˆ | 17.20 (2.85) | |||
| BIC | 6520.691 | |||
| Pseudo R2 | 0.475 |
t is Wald Test Statistic (n = 1448).
population and employment on the original and log scales, and indicates a non-linear but increasing risk with increasing population and employment over the observed ranges of these rescaled variables. Total child population and child population density are also included in the final model. The y-coordinate predictor has a moderately large negative coefficient, indicating that there is a gradient of increasing risk in England from north to south. Each of these variables can be thought of as representing exposure to road traffic risk within the ward, which allow us to estimate the effect of deprivation more accurately.
The model fitted with log scale components of deprivation instead of the composite score yields an inferior BIC (6539.279), and a slightly lower pseudo- R2 (0.465). In this model, log income score (INCSCR) and log education score (EDUSCR) were associated with significantly increased risk, with coefficients (standard errors) 0.328 (0.048) and 0.230 (0.037) respectively. In addition to those covariates mentioned in the previous paragraph, proximate employment density was also strongly associated with increasing risk, with coefficient 0.271 (0.059). For both the resulting models, the residual deviances indicate good fits of the negative binomial model (1551 on 1438 rdf and 1549.1 on 1439 rdf), and residual plots reveal no pattern, indicating that the models fit well overall.
5.1.2. CKSI: analysis including london factor
The presence of the y-coordinate in the selectedmodel, implying a risk gradient increasing toward the south, raises the possibility that the Greater London conurbation is a region of elevated risk. Therefore, to check the impact of geographical location within the UK for the conurbation subset, we extended the model to fit separate models subsets in Greater London (codes 1–3), which accounts for 620 of the 1448 wards in the conurbation data subset, and for the Other Conurbations (codes 4–5).
The BIC values for the refitted models for the Greater London subset indicate that the negative binomial models with deprivation scores on the log scale fit better than the other models. For the models with log composite score, the BIC value is 2795.616 (pseudo-R2 equal to 0.367), and for the model with log component scores, the BIC value is 2805.240 (pseudo- R2 equal to 0.357). The residual deviances are 1549.1 (rdf 1439) and 1551.3 (rdf 1438) indicating an adequate fit overall for the negative binomial model. The final model fits with the composite and component scores is presented in Table 3. The deprivation score terms have large positive coefficients, indicating a positive association with increased risk.
For the Other Conurbations subset, the composite score models again provide better fits than the component score models. The results are presented in Table 4. A notable feature of the fits for thesemodels is that the Poisson and Negative Binomialmodels have similar BIC and pseudo- R2 values (the best fitting Poisson model has BIC 3718.461, compared to 3711.632 for the best-fitting negative binomial model), indicating that the overdispersion relative to the Poisson model is minimal.
Inspection of the combined log-likelihoods and BICs indicates that it is statistically advantageous to analyze Greater London/Other Conurbation subsets separately, and that, as before, increasing deprivation – whether in composite or component form – is associated with an increase in risk, with similar coefficients to those in the previous analyses. The residual quantile −-quantile plot and plot against response depicted in Fig. 6 indicates that the model fit is good.
5.1.3. CPC: analysis including london factor
We now fit the models to the conurbation CPC data. For these data, on the basis of exploratory analysis, we again fit separate models to the London conurbation data (region codes 1–3) and the remaining conurbation data (codes 4–5). We find that for both
Table 3 Conurbation CKSI data: Greater London Conurbation (codes 1–3, n = 620)
| Term | Est. | S.E. | t | p | |
|---|---|---|---|---|---|
| Best fitting NB models with log composite scores | |||||
| (Intercept) | −0.414 | 0.195 | −2.121 | 3.39e −02 | |
| Employment | 0.794 | 0.188 | 4.233 | 2.31e −05 | |
| log Employment | 0.105 | 0.036 | 2.951 | 3.17e −03 | |
| log Population | 0.699 | 0.185 | 3.785 | 1.54e −04 | |
| Child population | 2.591 | 0.714 | 3.629 | 2.84e −04 | |
| log(IMDSCR) | 0.534 | 0.047 | 11.461 | 2.07e −30 | |
| ˆ |
12.52 (2.72) | ||||
| BIC | 2795.616 | ||||
| Pseudo R2 | 0.367 | ||||
| Term | Est. | S.E. | t | p | |
| Best fitting NegBin models with log component scores | |||||
| (Intercept) | 1.898 | 0.251 | 7.562 | 3.97e −14 | |
| Employment | 0.835 | 0.191 | 4.376 | 1.21e −05 | |
| log Employment | 0.106 | 0.035 | 3.058 | 2.23e −03 | |
| log Population | 1.068 | 0.138 | 7.734 | 1.04e −14 | |
| log(INCSCR) | 0.370 | 0.060 | 6.130 | 8.81e −10 | |
| log(EDUSCR) | 0.157 | 0.051 | 3.055 | 2.25e −03 | |
| ˆ | 12.07 (2.56) | ||||
| BIC | 2805.240 | ||||
| Pseudo R2 | 0.357 | ||||
subsets, the negative binomial model with log scale composite deprivation scores provides the best fitting model under BIC (values 4204.950 and 5667.009 for codes 1–3 and 4–5 respectively), although the component score data are only slightly inferior. For the component score models, the BIC values are 4202.355 and 5672.199 (pseudo- R2 0.646 and 0.760) for codes 1–3 and 4–5 respectively, whereas for the composite score, the BIC values are 4194.857 and 5669.025 (pseudo- R2 0.640 and 0.756). Results for the component and composite scores are presented in Tables 5.1.3. Residual deviance and residual plots indicate that the fit of the model is adequate in all cases.
In terms of component deprivation indices we find significant positive effects from the education and crime scores in both
Table 4 Conurbation CKSI data: other Conurbation (codes 4–5, n = 828)
| Term | Est. | S.E. | t | p |
|---|---|---|---|---|
| Best fitting NB models with log composite scores | ||||
| (Intercept) | 0.696 | 0.399 | 1.744 | 8.11e −02 |
| log Employment | 0.111 | 0.022 | 4.964 | 6.90e −07 |
| log Population | 0.589 | 0.146 | 4.030 | 5.57e −05 |
| log Child population | 0.547 | 0.121 | 4.517 | 6.27e −06 |
| log(IMDSCR) | 0.650 | 0.048 | 13.431 | 4.00e −41 |
| log Area | −0.124 | 0.028 | −4.482 | 7.41e −06 |
| ˆ | 28.65 (9.05) | |||
| BIC | 3711.632 | |||
| Pseudo R2 | 0.547 | |||
| Term | Est. | S.E. | t | p |
| Best fitting NB model with log component scores | ||||
| (Intercept) | 1.646 | 0.326 | 5.056 | 4.28e −07 |
| log Employment | 0.111 | 0.022 | 5.019 | 5.19e −07 |
| Population | −0.476 | 0.184 | −2.591 | 9.57e −03 |
| log Population | 1.511 | 0.235 | 6.437 | 1.22e −10 |
| Child population | 1.653 | 0.513 | 3.223 | 1.27e −03 |
| log(EMPSCR) | 0.248 | 0.075 | 3.290 | 1.00e −03 |
| log(EDUSCR) | 0.259 | 0.046 | 5.663 | 1.49e −08 |
| CRMSCR | 0.152 | 0.046 | 3.315 | 9.17e −04 |
| log Area | −0.144 | 0.028 | −5.162 | 2.45e −07 |
| ˆ | 29.37 (9.46) | |||
| BIC | ||||
| Pseudo R2 | 3727.829 0.550 |
|||
Table 5 Conurbation CPC data: best fitting Negative Binomial models with log component score. (n = 1448)
| Term | Est. | S.E. | t | p |
|---|---|---|---|---|
| Greater London Conurbation (codes 1–3) | ||||
| (Intercept) | 2.563 | 0.591 | 4.337 | 1.45e −05 |
| log Employment | 0.084 | 0.024 | 3.495 | 4.74e −04 |
| Population | 0.897 | 0.129 | 6.964 | 3.3012 |
| Child population | 2.796 | 0.958 | 2.920 | 3.50e −03 |
| log Child population | −0.688 | 0.163 | −4.224 | 2.40e −05 |
| Area | 0.277 | 0.063 | 4.416 | 1.01e −05 |
| log(EMPSCR) | 0.262 | 0.062 | 4.250 | 2.14e −05 |
| log(EDUSCR) | 0.161 | 0.032 | 5.082 | 3.73e −07 |
| CRMSCR | 0.250 | 0.042 | 5.887 | 3.92e −09 |
| Network nodes | 2.626 | 0.617 | 4.259 | 2.05e −05 |
| Child population density | 3.220 | 0.927 | 3.473 | 5.16e −04 |
| Employment density | 0.874 | 0.319 | 2.739 | 6.16e −03 |
| log Area | 1.770 | 0.307 | 5.769 | 7.97e −09 |
| √area | −2.977 | 0.542 | −5.496 | 3.89e −08 |
| ˆ | 15.78 (1.60) | |||
| BIC | 4202.355 | |||
| Pseudo R2 | 0.646 | |||
| Term | Est. | S.E. | t | p |
| Remaining Conurbation (codes 4–5) | ||||
| (Intercept) | 4.476 | 0.230 | 19.483 | 1.54e −84 |
| log Employment | 0.098 | 0.015 | 6.643 | 3.08e −11 |
| log Population | 0.401 | 0.091 | 4.402 | 1.07e −05 |
| log Child population | 0.604 | 0.082 | 7.397 | 1.40e −13 |
| HEASCR | 0.185 | 0.027 | 6.876 | 6.15e −12 |
| log(EDUSCR) | 0.122 | 0.023 | 5.395 | 6.86e −08 |
| CRMSCR | 0.222 | 0.028 | 7.834 | 4.72e −15 |
| Population/network nodes | −3.624 | 0.568 | −6.383 | 1.74e −10 |
| y-Coordinate | −0.088 | 0.016 | −5.527 | 3.27e −08 |
| Child population density | 2.140 | 0.589 | 3.636 | 2.77e −04 |
| √area | −0.047 | 0.012 | −3.755 | 1.73e −04 |
| ˆ | 24.56 (2.41) | |||
| BIC | 5672.199 | |||
| Pseudo R2 | 0.760 |
Greater London and Other Conurbation subsets. Each of these components has a significant positive coefficient. In Greater London, employment is also a significantly associated with increased risk, whereas in Other Conurbation wards, health is included in the best model.
Fig. 6. Residuals from fit to CKSI Conurbation Data: normal quantile-quantile plots of residuals from the fit with London factor included (see Section 5.1.2).
Table 6 Conurbation CPC data: best fitting Negative Binomial models with log composite score. (n = 1448)
| Term | Est. | S.E. | t | p |
|---|---|---|---|---|
| Greater London Conurbation (codes 1–3) | ||||
| (Intercept) | 0.068 | 0.835 | 0.082 | 9.35e −01 |
| log Employment | 0.087 | 0.024 | 3.682 | 2.32e −04 |
| Population | 0.860 | 0.089 | 9.624 | 6.35e −22 |
| Proximate employment | −0.315 | 0.098 | −3.202 | 1.37e −03 |
| Area | 0.198 | 0.060 | 3.288 | 1.01e −03 |
| log(IMDSCR) | 0.647 | 0.034 | 18.859 | 2.48e −79 |
| Network nodes | 3.352 | 0.609 | 5.508 | 3.62e −08 |
| x-Coordinate | 0.401 | 0.140 | 2.864 | 4.19e −03 |
| Employment density | 0.778 | 0.280 | 2.780 | 5.43e −03 |
| log Area | 0.838 | 0.239 | 3.511 | 4.46e −04 |
| √area | −1.913 | 0.497 | −3.852 | 1.17e −04 |
| ˆ | 15.25 (1.52) | |||
| BIC | 4194.857 | |||
| Pseudo R2 | 0.640 | |||
| Term | Est. | S.E. | t | p |
| Remaining Conurbation (codes 4–5) | ||||
| (Intercept) | 2.679 | 0.246 | 10.895 | 1.22e −27 |
| Employment | −0.661 | 0.239 | −2.760 | 5.79e −03 |
| log Employment | 0.078 | 0.019 | 4.069 | 4.73e −05 |
| log Population | 0.397 | 0.088 | 4.498 | 6.86e −06 |
| log Child population | 0.696 | 0.076 | 9.153 | 5.52e −20 |
| Proximate employment | 0.586 | 0.084 | 6.977 | 3.01e −12 |
| log(IMDSCR) | 0.547 | 0.027 | 20.417 | 1.18e −92 |
| Population/network node | −3.893 | 0.508 | −7.656 | 1.91e −14 |
| Employment density | 3.855 | 1.470 | 2.622 | 8.75e −03 |
| √area | −0.056 | 0.011 | −5.201 | 1.99e −07 |
| ˆ | 24.22 (2.36) | |||
| BIC | 5669.025 | |||
5.2. Urban analysis
We first fit the models to the urban CKSI data, encompassing 3801 wards with area codes 6,7,8 and 9. For these data, as for the previous subset, the negative binomial model with log scale component or composite deprivation scores provides the best fitting model under BIC. The best fitting models with component and composite scores have very similar BIC values (11512.74 and 11509.83 respectively) and pseudo- R2 values (0.380 and 0.382). The fit in each case is quite parsimonious; the coefficients for the component score model are displayed in Table 7. Again the log crime score (0.205 (0.032)) and the health (0.407 (0.030)) component are selected as influential predictors. The x-coordinate appears as an influential variable, with a positive risk gradient going west to east. This is in part possibly related to the high level of urbanization in the south east surrounding the London conurbation. We also find posi-
Table 7 Urban CKSI data: best fitting Negative Binomial model (n = 3801)
| Est. | S.E. | t | p | |
|---|---|---|---|---|
| (Intercept) | 2.343 | 0.155 | 15.066 | 2.70e −51 |
| log Employment | 0.140 | 0.017 | 8.251 | 1.57e −16 |
| log Child population | 0.790 | 0.038 | 20.641 | 1.17e −94 |
| Proximate employment | 0.458 | 0.097 | 4.714 | 2.43e −06 |
| HEASCR | 0.407 | 0.030 | 13.399 | 6.16e −41 |
| CRMSCR | 0.205 | 0.032 | 6.457 | 1.07e −10 |
| Population / network node | −3.110 | 0.624 | −4.982 | 6.30e −07 |
| x-Coordinate | 0.109 | 0.019 | 5.690 | 1.27e −08 |
| Child population density | 2.148 | 0.666 | 3.226 | 1.25e −03 |
| ˆ | 5.61 (0.60) | |||
| BIC | 11512.74 | |||
| Pseudo R2 | 0.380 |
Table 8 Urban CPC data: best fitting Negative Binomial model (n = 3801)
| Est. | S.E. | t | p | |
|---|---|---|---|---|
| (Intercept) | 4.645 | 0.115 | 40.342 | 0.00e+00 |
| log Employment | 0.124 | 0.014 | 9.076 | 1.13e −19 |
| log Population | 0.917 | 0.029 | 31.531 | 3.31e −218 |
| Proximate employment | 0.281 | 0.058 | 4.881 | 1.06e −06 |
| log(INCSCR) | 0.182 | 0.041 | 4.441 | 8.96e −06 |
| log(EMPSCR) | 0.320 | 0.050 | 6.417 | 1.39e −10 |
| log(HOUSCR) | −0.158 | 0.023 | −6.957 | 3.49e −12 |
| CRMSCR | 0.184 | 0.019 | 9.591 | 8.71e −22 |
| Population/network node | −3.001 | 0.375 | −8.011 | 1.14e −15 |
| x-Coordinate | 0.050 | 0.011 | 4.762 | 1.92e −06 |
| y-Coordinate | −0.047 | 0.009 | −5.114 | 3.16e −07 |
| Population density | −0.918 | 0.152 | −6.030 | 1.64e −09 |
| Child population density | 7.778 | 0.954 | 8.153 | 3.55e −16 |
| Employment density | 3.282 | 0.701 | 4.678 | 2.89e −06 |
| √area | −0.046 | 0.013 | −3.682 | 2.32e −04 |
| ˆ | 7.57 (0.36) | |||
| BIC | 20243.43 | |||
| Pseudo R2 | 0.662 |
tive and significant effects from population, employment and child population on the log scale, child population density, proximate population, and employment density. Note that the ward area type (code 5–9) is not selected as an influential variable. The residual deviance for this best-fitting model is 4131.1 on 3791 rdf, indicating that there may be mild overdispersion. We re − examine these data using a quasi-likelihood approach in Section 5.4.
For the urban CPC data, the negative binomial model with log scale component deprivation scores provides the best fitting model under BIC (value 20243.43) with a reasonably high pseudo- R2 value (0.662). The residual deviance is 4190.4 on 3786 degrees of freedom, again suggesting mild overdispersion. The model uses 16 terms; parameters estimates are displayed in Table 8. The log income, employment, and crime scores again appear in the best fitting model with positive coefficients, but here housing also appears but with a negative coefficient respectively. All terms appear as main effects only, and no other deprivation score components appear in the selected model. Area Type is not selected as an influential variable.
5.3. Rural analysis
We fitted the models to the rural CKSI data, encompassing 2668 wards with area codes 10. For these data, the Poisson model with original scale composite deprivation score provides the best fitting model under BIC (value 4489.886, with pseudo- R2 0.218), although all the Poisson fits in the different classes have similar optimal BICs; the Poisson model with log component scores has BIC 4490.645. The coefficients for this model are displayed in Table 9. The deprivation scores selected as influential predictors
Table 9 Rural CKSI data: best fitting Poisson model (n = 2668)
| Est. | S.E. | t | p | |
|---|---|---|---|---|
| (Intercept) | 1.825 | 0.331 | 5.521 | 3.37e −08 |
| log Employment | 0.207 | 0.048 | 4.311 | 1.63e −05 |
| log Population | 0.892 | 0.077 | 11.559 | 6.66e −31 |
| Proximate employment | 0.679 | 0.202 | 3.368 | 7.56e −04 |
| log(INCSCR) | 0.368 | 0.095 | 3.875 | 1.07e −04 |
| HEASCR | 0.247 | 0.070 | 3.510 | 4.48e −04 |
| x-Coordinate | 0.090 | 0.032 | 2.860 | 4.24e −03 |
| RLmo0 | −0.271 | 0.077 | −3.528 | 4.18e −04 |
| RLB+ | −0.132 | 0.048 | −2.734 | 6.26e −03 |
| ˆ | – | |||
| BIC | 4490.645 | |||
| Pseudo R2 | 0.218 |
Table 10 Rural CPC data: best fitting Negative Binomial model (n = 2668)
| Est. | S.E. | t | p | |
|---|---|---|---|---|
| (Intercept) | 4.393 | 0.141 | 31.245 | 2.60e −214 |
| log Employment | 0.177 | 0.027 | 6.680 | 2.38e −11 |
| log Population | 1.080 | 0.044 | 24.670 | 2.23e −134 |
| log(INCSCR) | 0.452 | 0.035 | 12.912 | 3.87e −38 |
| CRMSCR | 0.131 | 0.032 | 4.080 | 4.51e −05 |
| √area | −0.060 | 0.007 | −9.020 | 1.88e −19 |
| ˆ | 7.639 (0.967) | |||
| BIC | 8988.322 | |||
| Pseudo R2 | 0.451 |
in this model are the log of the income score and the health score. Log transformed employment, population, and proximate population and employment density also appears as influential variables with positive coefficients. In this analysis, the binary variable RL0 mo, recording whether there are zero miles of motorway in the ward, is also selected as an influential variable with a negative coefficient, indicating that the absence of motorways reduces risk. However, the fit of this model is rather poor, again due to the very low level of counts; of the 2668 rural wards, 1768 had count equal to zero. This is reflected in the residual deviance, which is 2414.1 on 2659 degrees of freedom, indicating underdispersion, although this may be unreliable for Poisson count data with low rate parameters. We examine zero-inflation models for these data in Section 5.4.
For the rural CPC data, the negative binomial model with log scale component and composite deprivation scores again provide the best fitting models under BIC (values 8988.322 and 8987.467 respectively, pseudo- R2 equal to 0.440 and 0.451), although the non-logged component model also fits well (BIC 8988.284). See Table 10 for the log component score estimated coefficients; this model uses 7 terms. Again the log income score appears in the best fitting model, as does crime both with positive coefficients, and so these serve to increase the rate of CPCs. The model again does not fit completely adequately due to the excess of zero counts, although the situation is better here, and the residual deviance of 3005.4 on 2662 degrees of freedom indicates mild overdispersion.
5.4. Modelling extensions
5.4.1. Overdispersion modelling using quasi-likelihood
Fitting a quasi-likelihood model with an identity link using the estimating equation given in Section 4.2 is straightforward in R. For the Urban CKSI data, the results of the quasi-likelihood analysis are presented in Table 11. The results show that both health and education component scores are influential, having significant positive coefficients even under this distribution free analysis. For the Rural CPC data that also exhibited overdispersion, a quasilikelihood fit led to coefficients (standard errors) of 1.139 (0.083) and 0.245 (0.073) for log(INCSCR) and CRMSCR respectively.
Table 11 Urban CKSI data: best fitting model estimated with quasi-likelihood
| p |
|---|
| 1.11e −71 |
| 6.32e −15 |
| 3.84e −74 |
| 4.73e −05 |
| 3.62e −38 |
| 1.15e −12 |
| 1.10e −07 |
| 1.08e −06 |
| 8.20e −07 |
5.4.2. Zero inflation modelling
We fitted a zero-inflated Poisson models using the zeroinflpackage in R, to the Rural CKSI data which exhibited underdispersion. The conclusions from the analysis were broadly similar to those obtained using the Poisson model above, but the best fitting zero-inflated model did not yield a higher BIC than the standard Poisson.
Another model that has exhibits zero-inflation relative to the Poisson distribution is the logarithmic distribution with mass function
$$f_Y(y;\theta) = -\frac{1}{\log(1-\theta)} \frac{\theta^{y+1}}{y+1}$$ $y = 0, 1, 2, ...$
where
$$E[Y] = \mu = -\frac{1}{\log(1-\theta)} \frac{\theta}{1-\theta} - 1$$
The logarithmic series distribution is commonly used for modelling species-incidence data, and can form the basis of a GLM analysis with a logistic link function. We fitted this model using the VGAM library in R, and found an improvement in BIC over Poisson, negative binomial and zero-inflated models; the final BIC for a model including log employment, log population, proximate employment and log(INCSCR) was 4764.772. The coefficient (S.E.) ofincome was again positive (0.692 (0.107), indicating that increasing income deprivation was associated with increasing risk.
5.4.3. Spatial models
For count data which arise in spatially varying locations, it is common to fit regression models that take such spatial variation into account. This is often achieved using spatially-correlated random effects, that is, a collection of (usually Gaussian distributed) correlated random variables where the amount of correlation depends on the distance between the observation points. The most common models include conditional autoregressive models (see, for example Besag et al., 1991), where correlation is induced by consideration of local neighbourhood structure, specifically, in our analysis, the random effect contribution to the observed variation in response is correlated with the random contribution in neighbouring wards. For generalized linear models, the spatial random effect effects are introduced into the linear predictor in an additive fashion. Such models can be fit in a reasonably straightforward fashion using R.
In our analysis, the need for modelling spatial random effects appears to be minimal. Fig. 7 depicts (a) the correlogram for the deviance residuals from a fit of the negative binomial GLM to the conurbation CKSI data , and (b) a plot of the deviance residuals at nearest-neighbouring wards. Both plots indicate that there is little spatial correlation amongst neighbouring wards, and this general conclusion held for the analyses carried out for other data subsets. This issue needs further investigation, but our analysis seems to be relatively satisfactory.
5.5. Discussion
For the conurbation, urban and rural subsets the best models specified included some combination of ward employment and population, proximate employment and population, population and employment densities, resident child population, the deprivation scores, and x and/or y coordinates. Thus, we find positive associations with the variables that were included to control for the volumes of activity and traffic flows in each area, supporting the intuitively reasonable proposition that it is busier wards, and those with higher potential for direct and through traffic generation that tend to have the highest casualty rates.
Fig. 7. Conurbation CKSI Data: residuals from Negative Binomial model. Numbers in (a) are counts of pairs of wards used to estimate the correlation. Plots indicate that there is little residual spatial correlation.
The variables of most interest in this analysis are the deprivation indices. We determine a statistically significant positive effect from the composite or component deprivation scores on the incidence of CPCs and CKSIs. These effects are identified in each of the three subsets of the data corresponding to conurbation, urban and rural wards. Furthermore, they exist over and above the variables included in the model to control for levels of activity, traffic generation potential, and other area specific effects.
From models based on the log composite deprivation score we actually estimate fairly similar effects from deprivation in each of our data sub samples. The estimates and standard errors are presented in the following table:
| Subset | CPC | CKSI |
|---|---|---|
| London | 0.647 (0.034) | 0.534 (0.047) |
| Other Conurbation | 0.650 (0.048) | 0.547 (0.027) |
| Urban | 0.621 (0.030) | 0.557 (0.017) |
| Rural | 0.582 (0.059) | 0.544 (0.036) |
These results indicate a substantial effect on the incidence of CPCs and CKSIs from characteristics associated with area deprivation.
We also find that the decomposition of the deprivation indices into six component scores provides a good fit to the data. The results show that the deprivation components have different effects across the three broad environmental subsets. The elasticities associated with each component vary in magnitude and even sign, and not all components feature in the models we estimate.
The income deprivation score does not show a positive association with the incidence of CPCs for the conurbation wards. It does, however, feature as a positive predictor for the incidence of CPCs in the urban and rural models. This is a very interesting result. It indicates that for the most highly urbanized wards it may not principally be relative poverty, as measured by the level of income benefits received by the population, that is associated with higher rates of CPCs. Furthermore, even for the urban and rural subsets, it is not poverty alone that explains the association between deprivation and pedestrian casualties.
The employment deprivation index has a significant positive effect on the incidence of CPCs in the conurbation and urban wards. The Housing and access to services index has a negative effect on CPCs in urban wards. This component does not feature in any other model. We find that education deprivation is significant and positive only in the conurbation CKSI and CPC models. Thus, the evidence for these effect is rather patchy: exclusion from work, access to housing and services, and the level of qualification of the population do not show consistent influences on the incidence of CPCs and CKSIs.
Interestingly, of the deprivation components, it is the crime score that produces the most consistent and compelling evidence of a positive association with the incidence of CPCs and CKSIs. With the exception of the rural CKSI model, which suffers from underdispersion due to the high number of zero values of the dependent variable, the results show large positive effects from crime deprivation on CPCs and CKSIs. Furthermore, the effect of this particular component of deprivation is identified in all three environments.
The data do not reveal why should we find this relationship between child casualties and levels of burglary, theft, criminal damage and violence. We could propose some hypothesis about why criminal activity, on the part of adults or children, could gives rise to higher accident rates. But these could only be satisfactorily put to test through detailed survey work and case study approaches rather than aggregate data analysis. One particularly interesting theme for future research would be to test whether variance in criminal activity is associated with different pedestrian and driver behavior. For instance, differences could include levels of consumption of alcoholic drink or drugs by pedestrian and motorists, the prevalence of reckless driving behavior, or some change in travel and activity patterns of people that increase their exposure to risk.
The results we present in this paper demonstrate that there are disparate effects from the different dimensions of deprivation on the incidence of CPCs and CKSIs. We have attempted to test for the individual deprivation effects while controlling for some other fundamental influences on the incidence of child pedestrian casualties relating to demographics, traffic volumes, the physical nature of the environment, the characteristics of the road network, and other observed and unobserved local specific factors. Our models show that there are distinct effects from different components of deprivation and that these exist over and above the other effects included in the models. The data do not reveal why these particular deprivation effects on child pedestrian casualty rates exist and for this reason it is not appropriate to speculate about precise mechanisms. In addition, it is entirely possible that the indices reflect not only what they are intended to measure, but also act as proxies for some other inter-correlated aspects of deprivation. Only so much can be achieved from the statistical analysis of aggregate data, but we hope that the results presented here may be of use in generating hypothesis and guiding future research on the relationship between deprivation, in particular crime deprivation, and child pedestrian casualties.
6. Conclusions
In this paper we have developed an area-based model to analyze the incidence of child pedestrian casualties. Our results are consistent with the general conclusion of previous research in this field which has shown that there is an association between the incidence of child pedestrian casualties and socio-economic deprivation. Furthermore, we find that the positive association of deprivation exits in diverse environments.
Our models have sought to disentangling the effects of area type from deprivation. By doing so we are able to confirm that the reasons why children from lower socio-economic groupings have a higher probability of being a pedestrian casualty, is due to some characteristics of deprived areas over and above influences arising from local environmental characteristics.
We have also been able to show that distinct dimensions of deprivation affect the incidence of pedestrian casualties to varying degrees and sometimes in different directions. The data do not show a consistent relationship between employment, education and housing deprivation and the incidence of child pedestrian casualties across area types. The results identify a relationship between income deprivation and the incidence of child pedestrian casualties, but they also show that the poverty aspect of deprivation cannot provide a complete explanation. In particular, we find a consistent positive influence associated with the crime deprivation score. Areas with high levels of burglary, theft, criminal damage and violent crimes also tend to have high incidences of CPCs and CKSIs. The effect of crime deprivation is robust across conurbation, urban and rural area types. The data do not reveal the actual mechanisms of causality that run from crime to increased child casualties. It may be useful in future research to explore the ways in which a crime-affected environment can influence activity patterns, travel behaviour, and child exposure to the motorized environment.
Individual level studies, concerned with analyzing the exposure rates of children to traffic and the behaviour and travel patterns of children, have dominated the child casualty literature. Research along these lines has proven extremely fruitful. Our analysis differs. It is not possible from our results to make inference at an individual level, but we have identified some important locational influences on the incidence of child casualties. This is crucially important for policy because travel demand has clearly defined spatial manifestations that can be address through area-based initiatives.
The authorities that are charged with the task of reducing child pedestrian casualties typically have jurisdiction over relatively small areas. In Britain, these are local authorities which divide the country into approximately 500 areas. At this disaggregate spatial scale it is extremely difficult to observe the influence of local characteristics on the pedestrian casualty rate due to the small number of incidents taking place. The model we develop can be used to give an indication of where child accidents may take place given a range of local characteristics. In other words it gives pointers about the locational factors that increase the probability of incidents, even if these factors have not been observed by authorities due to the scarcity of casualties.
This is particularly important regarding deprivation and the components of deprivation, because the effect of these variables may be very difficult to observe locally. Ourmodel shows that deprivation does count, but it is only one of several local characteristics that do and perhaps not even the most important. Furthermore, we need to look closely at the different dimensions of deprivation to understand the mechanisms that give rise to this effect. On the other hand, our findings on deprivation are relevant in policy formulation for two reason. First, because they confirm that road safety strategies can be compatible with other local regeneration or redistributive policy measures aimed at alleviating deprivation. Second, because they indicate that authorities should examine whether road safety measures are currently unevenly distributed with respect to their socio-economic environments.
Acknowledgments
We are grateful to two anonymous referees for helpful comments and suggestions. The second author's work was partially supported by the National Science and Engineering Council of Canada (NSERC). For John Graham.
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